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# A new quantum mechanical formalism based on the probability representation of quantum states

jaykov foukzon, E. Menkova, S.A. Podosenov

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jaykov foukzon, E. Menkova, S.A. Podosenov. A new quantum mechanical formalism based on the probability representation of quantum states. 2023. hal-04097022

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Distributed under a Creative Commons Attribution 4.0 International License
# A new quantum mechanical formalism based on the probability representation of quantum states.

J.Foukzon¹, A.A.Potapov², E. Menkova³, S.A. Podosenov³

1. Israel Institute of Technology.

2. IRE RAS,

3. All-Russian Scientific-Research Institute

**Abstract:** A new quantum mechanical formalism based on the probability representation of quantum states is proposed. This paper in particular deals with the special case of the measurement problem, known as Schrödinger's cat paradox. We pointed out that Schrödinger's cat demands to reconcile Born's rule. Using new quantum mechanical formalism we find the collapsed state of the Schrödinger's cat always shows definite and predictable outcomes even if cat also consists of a superposition

$$|\text{cat}\rangle = c_1|\text{live cat}\rangle + c_2|\text{death cat}\rangle \\ |c_1|^2 + |c_2|^2 = 1.$$

Using new quantum mechanical formalism the EPRB-paradox is considered successfully. We find that the EPRB-paradox can be resolved by nonprincipal and convenient relaxing of the Einstein's locality principle.

## Content

### Part I. Schrödinger's cat paradox resolution using GRW collapse model.
Von

#### Neumann measurement postulate revisited.

##### I. Introduction

###### I.1. The canonical interpretations of the Schrödinger experiment.

##### I.1.1. The Copenhagen interpretation.

###### I.1.2. The Objective collapse theories.

##### I.1.3. The Ensemble interpretation.

###### I.2. The canonical collapse models. Quantum Mechanics with Spontaneous Localizations

(QMSL).

##### I.2.1. The classical GRW model.

###### I.2.2. Tails of Schrödinger's cat. Schrödinger's cat demands to reconcile canonical
collapse models postulates.

I.2.3. The stochastic nonlinear Schrödinger equation.

I.2.4. The nonclassical collapse models with spontaneous localizations based on generalized measurement postulates.

I.2.5. The generalization of nonclassical collapse models.

I.3. The nonlocal Schrödinger equations and nonlocal nature of the wave function collapse.

I.4. The nonlocal evolution equation for the statistical operator.

I.5. The nonlocal stochastic nonlinear Schrödinger equation.

I.6. Schrödinger's cat through Stern-Gerlach experiment. Schrödinger's cat demands to reconcile Born's rule.

I.7. A new quantum mechanical formalism based on the probability representation of quantum states.

I.7.1. Generalized Postulates for Continuous Valued Observables.

I.7.2. The Position Representation. Position observable of a particle in one dimension.

I.8. The EPR paradox.

I.8.1. Einstein's 1927 gedanken experiment.

I.8.2. The continuous variable EPR paradox.

I.8.3. The EPR-Bohm paradox. Reid's criteria for EPR-Bohm paradox.

II. Generalized Gamow theory of the alpha decay via tunneling using GRW collapse model. Nonlocal Schrödinger equation corresponding to alpha decay.

II.1. Generalized Gamow theory of the alpha decay via tunneling using GRW collapse model.

II.2. Nonlocal Schrödinger equation corresponding to alpha decay.

II.3. The nonlocal stochastic nonlinear Schrödinger equation corresponding to alpha decay.

II.4. Einstein's 1927 gedanken experiment revisited.

III. Schrödinger's Cat paradox resolution

III.1. Resolution of the Schrödinger's cat paradox using canonical von Neumann postulate.

III.2. Resolution of the Schrödinger's cat paradox using generalized von Neumann postulate.

III.3. Schrödinger's cat does not a live cat and dead cat samultaniously.

III.4. Schrödinger's cat paradox and Stern-Gerlach experiment revisited.

# Part II. EPR Paradox Resolution. Generalized EPR argument and Postulate of
## Nonlocality.

IV.EPR Paradox Resolution.

IV.1.1.Generalized EPR argument and Postulate of Nonlocality.

IV.1.2.The EPR-Reid criterion.

IV.2.Nonlocal Schrödinger equation implies the Postulate of Nonlocality.

IV.3。「Position-momentum entangled photon pairs in non-linear wave-guide.

IV.4。「Position-momentum entangled photon pairs and the experimental verification of the postulate of nonlocality.

IV.5.EPR-B experiment.

Appendix A.

Appendix B.

Appendix C. Generalized Postulates for Continuous Valued Observables.

# Part I.

Schrödinger's cat paradox resolution using GRW collapse model. Von Neumann measurement postulate revisited.

## I. Introduction

In his famous thought experiment, Schrödinger (1935) imagined a cat that measures the value of an quantum mechanical observable with its life. Since Schrödinger's time, no any interpretations or modifications of quantum mechanics have been proposed which gives clear unambiguous answers to the questions posed by Schrödinger's cat of how long superpositions last and when (or whether) they collapse? In this paper appropriate modification of quantum mechanics are proposed. We claim that canonical interpretation of the wave function $\psi = c_1\psi_1 + c_2\psi_2$ is correct only when the supports the wave functions $\psi_1$ and $\psi_2$ essentially overlap. When the wave functions $\psi_1$ and $\psi_2$ have separated supports (as in the case of the experiment that we are considering in this paper) we claim that canonical interpretation of the wave function $\psi = c_1\psi_1 + c_2\psi_2$ is no longer valid for a such cat state. Possible solution of the Schrödinger's cat paradox are considered. We pointed out that the collapsed state of the cat always shows definite and predictable outcomes even if cat also consists of a superposition [16]-[17] :

$$|\text{cat}\rangle = c_1|\text{live cat}\rangle + c_2|\text{death cat}\rangle.$$

### I.1. The canonical interpretations of the Schrödinger experiment.

As Weinberg recently reminded us [1], the measurement problem remains a
fundamental conundrum. During measurement the state vector of the microscopic system collapses in a probabilistic way to one of a number of classical states, in a way that is unexplained, and cannot be described by the time-dependent Schrödinger equation [1]. To review the essentials, it is sufficient to consider two-state systems. Suppose a nucleus **n**, whose Hilbert space is spanned by orthonormal states $|s_i(t)\rangle$, $i = 1,2$, where $|s_1(t)\rangle = |\text{undecayed nucleus at instant } t\rangle$ and $|s_2(t)\rangle = |\text{decayed nucleus at instant } t\rangle$ is in the superposition state,

$$|\Psi_t\rangle_n = c_1|s_1(t)\rangle + c_2|s_2(t)\rangle, |c_1|^2 + |c_2|^2 = 1. \qquad (1.1.1)$$

A measurement apparatus **A**, which may be microscopic or macroscopic, is designed to distinguish between states $|s_i(t)\rangle$ by transitioning at each instant $t$ into state $|a_i(t)\rangle$ if it finds **n** is in $|s_i(t)\rangle$, $i = 1,2$. Assume the detector is reliable, implying the $|a_1(t)\rangle$ and $|a_2(t)\rangle$ are orthonormal at each instant $t$, i.e., $\langle a_1(t)|a_2(t)\rangle = 0$ and that the measurement interaction does not disturb states $|s_i\rangle$ -i.e., the measurement is “ideal”. When **A** measures $|\Psi_t\rangle_n$, the Schrödinger equation’s unitary time evolution then leads to the “measurement state” $|\Psi_t\rangle_n$:

$$|\Psi_t\rangle_{nA} = c_1|a_1(t)\rangle + c_2|a_2(t)\rangle, |c_1|^2 + |c_2|^2 = 1. \qquad (1.1.2)$$

of the composite system **nA** following the measurement.

Standard formalism of continuous quantum measurements [2],[3],[4],[5-10] leads to a definite but unpredictable measurement outcome, either $|a_1(t)\rangle$ or $|a_2(t)\rangle$ and that $|\Psi_t\rangle_n$ suddenly “collapses” at instant $t'$ into the corresponding state $|s_i(t')\rangle$. But unfortunately the equation (1.1.2) does not appear to resemble such a collapsed state at instant $t'$?

The measurement problem is as follows:

(I) How do we reconcile canonical collapse models postulate’s

(II) How do we reconcile the measurement postulate’s definite outcomes with the

“measurement state” $|\Psi_t\rangle_n$ at each instant $t$ and

(III) how does the outcome become irreversibly recorded in light of the Schrödinger

equation’s unitary and, hence, reversible evolution?

The Part I of this paper in particular deals with the special case of the measurement problem, known as Schrödinger’s cat paradox. For a good and complete explanation of this paradox see Albert [7], Leggett [11], Hobson [12] and Schrödinger[13], see also [14]-[16].
<figure><img src="image_4.png" /><figcaption>Pic.1.1.1.Schrödinger's generic cat.</figcaption></figure>

In his famous thought experiment [11], Schrödinger(1935) imagined a cat that measures the value of an quantum mechanical observable with its life. Adapted to the measurement of position of an alpha particle, the experiment is this. A cat, a flask of poison, and a radioactive source are placed in a sealed box. If an internal monitor detects radioactivity (i.e. a single atom decaying), the flask is shattered, releasing the poison that kills the cat. The Copenhagen interpretation of quantum mechanics implies that after a while, the cat is simultaneously alive and dead. Yet, when one looks in the box, one sees the cat either alive or dead, not both alive and dead.

*This poses the question of when exactly quantum superposition ends and reality collapses into one possibility or the other?*

Since Schrödinger's time, no any interpretations or extensions of quantum mechanics have been proposed which gives clear unambiguous answers to the questions posed by Schrödinger's cat of how long superpositions last and when (or whether) they collapse.

# The canonical interpretations of the Schrödinger experiment.

## I.1.1. The Copenhagen interpretation

The most commonly held interpretation of quantum mechanics is the Copenhagen interpretation.[14-15] In the Copenhagen interpretation, a system stops being a superposition of states and becomes either one or the other when an observation takes place. This thought experiment makes apparent the fact that the nature of measurement, or observation, is not well-defined in this interpretation. The experiment can be interpreted to mean that while the box is closed, the system simultaneously exists in a superposition of the states "decayed nucleus/dead cat" and "undecayed nucleus/living cat", and that only when the box is opened and an
observation performed does the wave function collapse into one of the two states.

However, one of the main scientists associated with the Copenhagen interpretation, Niels Bohr, never had in mind the observer-induced collapse of the wave function, so that Schrödinger's cat did not pose any riddle to him. The cat would be either dead or alive long before the box is opened by a conscious observer [14-15]. Analysis of an actual experiment found that measurement alone (for example by a Geiger counter) is sufficient to collapse a quantum wave function before there is any conscious observation of the measurement.[15] The view that the "observation" is taken when a particle from the nucleus hits the detector can be developed into objective collapse theories. The thought experiment requires an "unconscious observation" by the detector in order for magnification to occur.

### I.1.2. The Objective collapse theories

According to objective collapse theories, superpositions are destroyed spontaneously (irrespective of external observation) when some objective physical threshold (of time, mass, temperature, irreversibility, etc.) is reached. Thus, the cat would be expected to have settled into a definite state long before the box is opened. This could loosely be phrased as "the cat observes itself", or "the environment observes the cat".

Objective collapse theories require a modification of standard quantum mechanics to allow superpositions to be destroyed by the process of time evolution. This process, known as "decoherence", is among the fastest processes currently known to physics [5], [].

### I.1.3. The Ensemble interpretation

The ensemble interpretation states that superpositions are nothing but subensembles of a larger statistical ensemble. The state vector would not apply to individual cat experiments, but only to the statistics of many similarly prepared cat experiments. Proponents of this interpretation state that this makes the Schrödinger's cat paradox a trivial matter, or a non-issue. This interpretation serves to discard the idea that a single physical system in quantum mechanics has a mathematical description that corresponds to it in any way.

**Remark 1.1.1.** Ensemble interpretation in a good agreement with a canonical interpretation of the wave function (ψ-function) in canonical QM-measurement theory. However under rigorous consideration of the dynamics of the Schrödinger's cat this interpretation gives obviously unphysical result, see section III, Proposition 3.1.2.(ii).

## I.2. The canonical collapse models. Quantum Mechanics with Spontaneous Localizations (QMSL)

We remind that Quantum Mechanics with Spontaneous Localizations [2],[3] is based on the following assumptions:

(1) Each particle of a system of $n$ distinguishable particles experiences, with a mean rate $\lambda_i$, a sudden spontaneous localization process.
(2) In the time interval between two successive spontaneous processes the system evolves according to the usual Schrödinger equation.

(3) The sudden spontaneous process is a localization given by:

$$|\psi\rangle \xrightarrow{\text{localization}} \frac{|\psi_x^i\rangle}{\|\psi_x^i\|}, \quad \mathbf{x} \in \mathbb{R}^3, \qquad (1.2.1)$$

where

$$|\psi_x^i\rangle = L_x^i |\psi\rangle. \qquad (1.2.2)$$

Here $L_x^i$ is a norm-reducing, positive, self-adjoint, linear operator in the n-particle projective Hilbert space $\mathbf{H}$, representing the localization of particle i around the point $\mathbf{x}$.

(4) The probability density for the occurrence of a localization at point $\mathbf{x}$ is assumed to be

$$P_i(\mathbf{x}) = \| |\psi_x^i\rangle \|^2. \qquad (1.2.3)$$

Eq.(1.2.3) requires that $\int d^3x [L_x^i]^2 = 1$.

(5) The localization operators $L_x^i$ have been chosen to have the form:

$$L_x^i = \left(\frac{1}{\delta\pi}\right)^{3/4} \exp\left[-\frac{1}{2\delta}(\hat{q}_i - \mathbf{x})^2\right], \qquad (1.2.4)$$

$\hat{q}_i$ being the position operator for particle i.

## 1.2.1. The classical GRW model

In order to appreciate how canonical collapse models work, and what they are able to achieve, we briefly review classical GRW model [2]. Let us consider a system of n particles which, only for the sake of simplicity, we take to be scalar and spinless; the GRW model is defined by the following postulates: (1) The state of the system is represented by a wave function $\psi_t(\mathbf{x}_1, \mathbf{x}_2, \dots, \mathbf{x}_n)$ belonging to the Hilbert space $\mathcal{L}_2(\mathbb{R}^{3n})$. (2) At random times, the wave function experiences a sudden jump of the form:

$$\psi_t(\mathbf{x}_1, \mathbf{x}_2, \dots, \mathbf{x}_n) \rightarrow \psi_t(\mathbf{x}_1, \mathbf{x}_2, \dots, \mathbf{x}_n; \tilde{\mathbf{x}}_m) = \frac{\Re_m(\tilde{\mathbf{x}}_m)\psi_t(\mathbf{x}_1, \mathbf{x}_2, \dots, \mathbf{x}_n)}{\|\Re_m(\tilde{\mathbf{x}}_m)\psi_t(\mathbf{x}_1, \mathbf{x}_2, \dots, \mathbf{x}_n)\|_2}, \qquad (1.2.5)$$

where $\psi_t(\mathbf{x}_1, \mathbf{x}_2, \dots, \mathbf{x}_n)$ is the state vector of the whole system at time $t$, immediately prior to the jump process and $\Re_m(\tilde{\mathbf{x}}_m)$ is a linear operator which is conventionally chosen equal to:

$$\Re_m(\tilde{\mathbf{x}}_m) = (\pi r_c^2)^{-3/4} \exp\left[-\frac{(\hat{\mathbf{x}}_m - \tilde{\mathbf{x}}_m)^2}{2r_c^2}\right], \qquad (1.2.6)$$

where $r_c$ is a new parameter of the model which sets the width of the localization process, and $\hat{\mathbf{x}}_m$ is the position operator associated to the m-th particle of the
system and the random variable $\tilde{x}_m$ corresponds to the place where the jump occurs. (3) It is assumed that the jumps are distributed in time like a Poissonian process with frequency $\lambda = \lambda_{GRW}$ this is the second new parameter of the model. (4) Between two consecutive jumps, the state vector evolves according to the standard Schrödinger equation.

Let us consider a single particle. Suppose it suffers a hitting process: its wave function $|\psi\rangle$ changes it into the new wavefunction $|\psi_x\rangle$. We do not know where the hitting occurs, but only the probability for it to occur around position $x$. Accordingly, the pure state is transformed into the following statistical mixture:

$$|\psi\rangle\langle\psi| = \int d^3x P(\mathbf{x}) \frac{|\psi_x\rangle\langle\psi_x|}{||\psi_x\rangle\rangle^2} = \int d^3x L_x |\psi\rangle\langle\psi| L_x = T[|\psi\rangle\langle\psi|] \quad (1.2.7)$$

Of course, if the initial state of the particle is not pure but a statistical mixture given by the operator $\rho$, the effect of a hitting process is the same as the one described above: $\rho$ changes into $T[\rho]$. We derive now the evolution equation for $\rho(t)$. In a time interval $dt$, the statistical operator evolves in the following way: since the localization mechanism is Poissonian, there is a probability $\lambda dt$ for a hitting to occur during that time interval, in which case $\rho$ changes to $T[\rho]$, and a probability $1 - \lambda dt$ for no hittings to occur so that the statistical operator evolves according to the usual Schrödinger equation:

$$\rho(t + dt) = (1 - \lambda dt) \left[ \rho(t) - \frac{i}{\hbar} \left[ \hat{H}, \rho(t) \right] dt \right] + \lambda dt T[\rho(t)] \quad (1.2.8)$$

Thus 1-particle master equation of the GRW model takes the form [2]-[5]

$$\frac{d}{dt} \rho(t) = -\frac{i}{\hbar} \left[ \hat{H}, \rho(t) \right] - \lambda(\rho(t) - T[\rho(t)]). \quad (1.2.9)$$

Here $\hat{H}$ is the standard quantum Hamiltonian of the particle, and $T[\cdot]$ represents the effect of the spontaneous collapses on the particle's wave function. In the position representation, this operator becomes:

$$\langle \mathbf{q}' | T[\rho(t)] | \mathbf{q}'' \rangle = \exp \left[ - \frac{(\mathbf{q}' - \mathbf{q}'')^2}{4r_c^2} \right] \langle \mathbf{q}' | \rho(t) | \mathbf{q}'' \rangle. \quad (1.2.10)$$

Since, owing to Eq.(1.2.10) $\langle \mathbf{q} | T[\rho(t)] | \mathbf{q} \rangle = \langle \mathbf{q} | \rho(t) | \mathbf{q} \rangle$, equation (1.2.9) is obviously trace preserving. Moreover, using equation (1.2.9), it can be proved that

$$\frac{d}{dt} \mathrm{Tr}[\rho^2(t)] < 0. \quad (1.2.11)$$

This implies that the dynamical evolution transforms pure states into statistical mixtures.

## 1.2.2. Tails of Schrödinger's cat.Schrödinger's cat demands to reconcile canonical collapse models
postulates

GRW collapse of the Schrödinger's cat state considered in [7]. The orthodox account attempts to solve the measurement problem by claiming that in measurement interactions the quantum state of the measured system plus measuring system does not evolve in accord with the Schrodinger equation but instead collapses into one of the states that is an eigenstate of the measurement observable (i.e. the observable that records the result of the measurement) with a probability proportional to the square of the coefficient of that state. For example, if Schrodinger's cat measures the x-spin, then the post-measurement state $|CAT\rangle$ collapses either into $|\uparrow\rangle|Alive\rangle$ or $|\downarrow\rangle|Dead\rangle$ each with a probability proportional to the square of the coefficient associated with each state. So on the orthodox account, there are two fundamental laws that govern the evolution of quantum states. In non-measurement situations, Schrodinger's deterministic law holds away. But in measurement situations, the collapse dynamics takes over.

<figure><img src="image_5.png" /><figcaption>Pic.1.2.1.(i) The GRW collapse of the Schrödinger's cat state $|CAT\rangle$:</figcaption></figure>

$$|CAT\rangle \xrightarrow{GRW \ collapse} |\uparrow\rangle|Alive\rangle$$

<figure><img src="image_6.png" /><figcaption>Pic.1.2.1.(ii) The GRW collapse of the Schrödinger's cat state $|CAT\rangle$:</figcaption></figure>

$$|CAT\rangle \xrightarrow{GRW \ collapse} |\downarrow\rangle|Dead\rangle \quad \text{Adopted from [7].}$$

There are difficulties, however. Look more closely, for example, at a post GRW-collapse state like GRWCAT(Alive), illustrated in Pic.1.2.1(i)-(ii). Note that while most of GRWCAT(Alive)'s amplitude is indeed (as we mentioned above) concentrated in the "Alive" region of the state space, it also has non-zero tails which extend into the "Dead" region.

And so it follows from the eigenstate-eigenvalue rule that the cat is, as a matter of fact, not determinately alive (or dead), when GRWCAT(Alive) obtains, after all.

And so the GRW theory, as we have stated it above, patently fails to solve Schrödinger's paradox.

**Remark 1.2.1.** In order to avoid the problem of the tails of Schrödinger's cat we replace now postulate (5) by the next postulate:

(5') The localization operators $L_x$ have been chosen to have the form:
$$L_x^i = L_x^i(\delta, \epsilon) = \begin{cases} \left(\frac{1}{\delta\pi_\delta}\right)^{3/4} \exp\left[-\frac{1}{2\delta}(\hat{q}_i - \mathbf{x})^2\right] & \text{iff } \|q_i - \mathbf{x}\| \le \epsilon \ll 1, \\ 0 & \text{iff } \|q_i - \mathbf{x}\| > \epsilon. \end{cases} \quad (1.2.12)$$

Here $\delta \in (0, 1]$ and $\lim_{\delta \to 0} \pi_\delta = \pi$.

### I.2.3. The stochastic nonlinear Schrödinger equation

Another modern approach to stochastic reduction is to describe it using a stochastic nonlinear Schrödinger equation [2]-[10],[46] an elegant simplified example of which is the following one particle case known as Quantum Mechanics with Universal Position Localization [QMUPL]:

$$d|\psi_t(x)\rangle = \left[-\frac{i}{\hbar}\hat{\mathbf{H}} - \frac{k}{2}(\hat{q} - \langle q_t \rangle)^2 dt\right] |\psi_t(x)\rangle dt + \sqrt{k}(\hat{q} - \langle q_t \rangle) dW_t |\psi_t(x)\rangle. \quad (1.2.13)$$

Here $\hat{q}$ is the position operator, $\langle q_t \rangle = \langle \psi_t | \hat{q} | \psi_t \rangle$ it is its expectation value, and $k$ is a constant, characteristic of the model, which sets the strength of the collapse mechanics, and it is chosen proportional to the mass $m$ of the particle according to the formula: $k = (m/m_0)\lambda_0$, where $m_0$ is the nucleon's mass and $\lambda_0$ measures the collapse strength. It is easy to see that Eqn.(1.2.5) contains both non-linear and stochastic terms, which are necessary to induce the collapse of the wave function. For an example let us consider a free particle ($\hat{\mathbf{H}} = p^2/2m$), and a Gaussian state:

$$\psi_t(x) = \exp\{-a_t(x - \bar{x}_t)^2 + i\bar{k}_t x\}. \quad (1.2.14)$$

It is easy to see that $\psi_t(x)$ given by Eq.(1.2.14) is solution of Eq.(1.2.13), where

$$\frac{da_t}{dt} = k - \frac{2ih}{m}a_t^2, \quad \frac{d\bar{x}_t}{dt} = \frac{\hbar}{m}\bar{k}_t + \frac{\sqrt{k}}{2\operatorname{Re}(a_t)}\dot{W}_t, \quad \frac{d\bar{k}_t}{dt} = -\sqrt{k}\frac{\operatorname{Im}(a_t)}{\operatorname{Re}(a_t)}\dot{W}_t. \quad (1.2.15)$$

The many-particle equation [2]:

$$d|\psi_t(x)\rangle = \left[-\frac{i}{\hbar}\hat{\mathbf{H}} - \sum_{i=1}^{N} \frac{k_i}{2}(\hat{q}_i - \langle q_{i,t} \rangle)^2 dt\right] |\psi_t(x)\rangle dt + \sum_{i=1}^{N} \sqrt{k_i}(\hat{q}_i - \langle q_{i,t} \rangle) dW_{i,t} |\psi_t(x)\rangle, \quad (1.2.16)$$

where $\hat{\mathbf{H}}$ is the quantum Hamiltonian of the composite system, the operators $\hat{q}_i$ ($i = 1, \dots, N$) are the position operators of the particles of the system, and $W_{i,t}$ ($i = 1, \dots, N$) are $N$ independent standard Wiener processes.
The CSL model is defined by the following stochastic differential equation in the Fock space [2]:

$$d|\psi_t(\mathbf{x})\rangle = \left[-\frac{i}{\hbar}\hat{\mathbf{H}} - \frac{k}{2}(\hat{M}(\mathbf{x}) - \langle M_t(\mathbf{x}) \rangle)^2 dt\right] |\psi_t(\mathbf{x})\rangle dt + \sqrt{k} (\hat{M}(\mathbf{x}) - \langle M_t(\mathbf{x}) \rangle) dW_t(\mathbf{x}) |\psi_t(\mathbf{x})\rangle. \quad (1.2.17)$$

The parameter k is a positive coupling constant which sets the strength of the collapse process, while $\hat{M}(\mathbf{x})$ is a smeared mass density operator:

$$\hat{M}(\mathbf{x}) = \sum_j m_j N_j(\mathbf{x}), \quad (1.2.18)$$

$$N_j(\mathbf{x}) = \int dy g(y - \mathbf{x}) \psi_j^\dagger(\mathbf{x}) \psi_j(\mathbf{x}),$$

$\psi_j^\dagger(\mathbf{x}), \psi_j(\mathbf{x})$ being, respectively, the creation and annihilation operators of a particle of type j in the space point y. The smearing function $g(\mathbf{x})$ is taken equal to

$$g(\mathbf{x}) = \left(\frac{1}{2\pi\delta}\right)^{3/4} \exp\left[-\frac{\mathbf{x}^2}{2\delta}\right] \quad (1.2.19)$$

where $\delta$ is the second phenomenological constant of the model. $W_t(\mathbf{x})$ is an ensemble of independent Wiener processes, one for each point in space.

### 1.2.4. The nonclassical collapse models with spontaneous localizations based on generalized measurement postulates

The nonclassical collapse models attempt to overcome the difficulties that standard quantum mechanics meets in accounting for the measurement (or macro-objectification) problem, an attempt based on the consideration of nonlinear and nonlocal stochastic modifications of the Schroedinger equation. The proposed new nonlocal dynamics is characterized by the feature of not contradicting any known fact about microsystems and of accounting, on the basis of a unique, universal dynamical principle, for wavepacket reduction and for the classical behavior of macroscopic systems.

**Quantum Mechanics with Nonclassical Spontaneous Localizations is based on the following assumptions:**

(1) Each particle of a system of n distinguishable particles experiences, with a mean rate $\lambda i$, a sudden spontaneous localization process.

(2) In the time interval between two successive spontaneous processes the system evolves according to the usual Schrödinger equation.

(3) Let $|\psi\rangle_{cl}$ be the classical pure state correspond to an vector $|\psi\rangle_{cl} \in S^\infty \subsetneq \mathbf{H}$
in a non projective Hilbert space $\mathbf{H}$, see Subsection I.7.1, Def I.7.1-I.7.2. Then the sudden spontaneous process is a localization given by:

$$ |\psi\rangle_{cl} \xrightarrow{\delta, \varepsilon \text{-localization}} \frac{|\psi^i_{\delta, \varepsilon, \mathbf{x}}\rangle_{cl}}{\| |\psi^i_{\delta, \varepsilon, \mathbf{x}}\rangle_{cl} \|}, \mathbf{x} \in \mathbb{R}^3, \qquad (1.2.20) \\ \delta \in (0, 1], \varepsilon \ll 1, $$

where

$$ |\psi^i_{\delta, \varepsilon, \mathbf{x}}\rangle_{cl} = \hat{L}_{\mathbf{x}}^i(\delta, \varepsilon) |\psi\rangle_{cl}. \qquad (1.2.21) $$

Here $\hat{L}_{\mathbf{x}}^i(\delta, \varepsilon)$ is a norm-reducing, positive, self-adjoint, linear operator with a symbol $L_{\mathbf{x}}^i(\delta, \varepsilon)$ in the $n$-particle non projective Hilbert space $\mathbf{H}$, representing the localization of particle $i$ around the point $\mathbf{x}$.

**Definition 1.2.1.** Such localization is called $\delta, \varepsilon$-localization or $\delta, \varepsilon$-collapse of the state

$$ |\psi\rangle_{cl}. $$

(4) The probability density $p_i(\mathbf{x}, \delta, \varepsilon)$ for the occurrence of a localization at point $\mathbf{x}$ is assumed to be

$$ p_i(\mathbf{x}, \delta, \varepsilon) = \frac{\| |\psi^i_{\delta, \varepsilon, \mathbf{x}}\rangle_{cl} \|^2}{\iiint_{\mathbb{R}^3} \| |\psi^i_{\delta, \varepsilon, \mathbf{x}}\rangle_{cl} \|^2 d^3 \mathbf{x}}. \qquad (1.2.22) $$

(5) Let $|\psi\rangle_{n.cl}$ be the nonclassical pure state correspond to an vector $|\psi^\zeta\rangle = \zeta|\psi\rangle \in \mathbf{H} \setminus \mathbf{S}^\infty$,

where $|\psi\rangle \in \mathbf{S}^\infty$, $|\zeta| \neq 1$, see Appendix C, Def.C.3. Then the sudden spontaneous process is a localization given by:

$$ |\psi\rangle_{n.cl} \xrightarrow{\delta, \varepsilon \text{-localization}} \frac{\zeta |\psi^i_{\delta, \varepsilon, \mathbf{x}}\rangle_{n.cl}}{\| |\psi^i_{\delta, \varepsilon, \mathbf{x}}\rangle_{n.cl} \|}, \mathbf{x} \in \mathbb{R}^3, \qquad (1.2.23) $$

where

$$ |\psi^i_{\delta, \varepsilon, \mathbf{x}}\rangle_{n.cl} = \hat{L}_{\mathbf{x}}^i(\delta, \varepsilon) |\psi\rangle_{n.cl}. \qquad (1.2.24) $$

**Definition 1.2.2.** Such localization is called $\delta, \varepsilon$-localization or $\delta, \varepsilon$-collapse of the state

$$ |\psi\rangle_{n.cl}. $$

(6) The probability density $p_i(\mathbf{x}, \zeta, \delta, \varepsilon,)$ for the occurrence of a localization at point $\mathbf{x} \in \mathbb{R}^3$ in

acordance to postulate Q.IV.3 (see Subsection I.7.1, Eq.(1.7.8)) is assumed to be

$$ p_i(\mathbf{x}, \zeta, \delta, \varepsilon, ) = \frac{|\zeta|^{-6} \| |\psi^i_{\delta, \varepsilon, |\zeta|^{-2}\mathbf{x}}\rangle_{n.cl} \|^2}{\iiint_{\mathbb{R}^3} \| |\psi^i_{\delta, \varepsilon, \mathbf{x}}\rangle_{cl} \|^2 d^3 \mathbf{x}}. \qquad (1.2.25) $$

(7) The localization operators $\hat{L}_{\mathbf{x}}^i(\delta, \varepsilon)$ have been chosen to have the form:
$$ \hat{L}_{\mathbf{x}}^{i}(\delta, \varepsilon) = \begin{cases} \left(\frac{1}{\delta\pi_{\delta}}\right)^{3/4} \exp\left[-\frac{1}{2\delta}(\hat{\mathbf{q}}_{i} - \mathbf{x})^{2}\right] & \text{iff } \|\mathbf{q}_{i} - \mathbf{x}\| \le \varepsilon \ll 1, \\ 0 & \text{iff } \|\mathbf{q}_{i} - \mathbf{x}\| > \varepsilon. \end{cases} \quad (1.2.26) $$

Here $\delta \in (0, 1] \int d^3x [L_x^i(\delta, \varepsilon)]^2 = 1$ and $\lim_{\delta \to 0} \pi_\delta = \pi$.

**Remark 1.2.2.** In one dimension case it follows that

$$ \hat{L}_{\mathbf{x}}^{i}(\delta, \varepsilon) = \begin{cases} \left(\frac{1}{\delta\pi_{\delta}}\right)^{1/4} \exp\left[-\frac{1}{2\delta}(\hat{\mathbf{q}}_{i} - \mathbf{x})^{2}\right] & \text{iff } |q_{i} - \mathbf{x}| \le \varepsilon \ll 1, \\ 0 & \text{iff } |q_{i} - \mathbf{x}| > \varepsilon. \end{cases} \quad (1.2.27) $$

**Remark 1.2.3.** Note that from Eq.(1.2.22) and Eq.(1.2.26) follows that a probability density $p_i(\mathbf{x}, \zeta, \delta, \varepsilon, \cdot)$ for the occurrence of a localization inside sphere $S(\mathbf{x}, \varepsilon) = \{\mathbf{q}_i \in \mathbb{R}^3 \mid \|\mathbf{q}_i - \mathbf{x}\| \le \varepsilon\}$ is given by

$$ \begin{align} p_i(\mathbf{x}, \delta, \varepsilon) &= \frac{\|\psi_{\delta, \varepsilon, \mathbf{x}}^i\|_{cl}^2}{\Omega(\delta, \varepsilon)}, \quad \Omega(\delta, \varepsilon) = \iiint_{\mathbb{R}^3} \|\psi_{\delta, \varepsilon, \mathbf{x}}^i\|_{cl}^2 d^3\mathbf{x}, \notag \\ \|\psi_{\delta, \varepsilon, \mathbf{x}}^i\|_{cl}^2 &= \left(\frac{1}{\delta\pi_{\delta}}\right)^{3/2} \int_{\|\mathbf{q}_i - \mathbf{x}\| \le \varepsilon} d^3\mathbf{q}_i \psi^i(\mathbf{q}_i) \exp\left[-\frac{1}{\delta}(\mathbf{q}_i - \mathbf{x})^2\right], \tag{1.2.28} \\ \psi^i(\mathbf{q}_i) &= \langle \mathbf{q}_i \|\psi^i\|_{cl} \rangle, \notag \end{align} $$

and therefore

$$ \begin{align} p_i(\mathbf{x}, \varepsilon) &= \lim_{\delta \to 0} p_i(\mathbf{x}, \delta, \varepsilon) = \nonumber \\ &\lim_{\delta \to 0} \Omega^{-1}(\delta, \varepsilon) \left(\frac{1}{\delta\pi_{\delta}}\right)^{3/2} \int_{\|\mathbf{q}_i - \mathbf{x}\| \le \varepsilon} d^3\mathbf{q}_i \psi^i(\mathbf{q}_i) \exp\left[-\frac{1}{\delta}(\mathbf{q}_i - \mathbf{x})^2\right] = \psi^i(\mathbf{x}). \tag{1.2.29} \end{align} $$

**Remark 1.2.4.** In one dimension case it follows that a probability density $p_i(x, \delta, \varepsilon, \cdot)$ for the occurrence of a localization inside interval $[x - \varepsilon, x + \varepsilon]$ is given by

$$ \begin{align} p_i(x, \delta, \varepsilon) &= \|\psi_{\delta, \varepsilon, x}^i\|_{cl}^2 = \left(\frac{1}{\delta\pi_{\delta}}\right)^{1/2} \int_{|q_i - x| \le \varepsilon} d^3 q_i \psi^i(q_i) \exp\left[-\frac{1}{\delta}(q_i - x)^2\right], \tag{1.2.30} \\ \psi^i(q_i) &= \langle q_i \|\psi^i\|_{cl} \rangle, \notag \end{align} $$

and therefore

$$ \begin{align} p_i(x, \varepsilon) &= \lim_{\delta \to 0} p_i(x, \delta, \varepsilon) = \nonumber \\ &= \lim_{\delta \to 0} \Omega^{-1}(\delta, \varepsilon) \left(\frac{1}{\delta\pi_{\delta}}\right)^{1/2} \int_{|q_i - x| \le \varepsilon} dq_i \psi^i(q_i) \exp\left[-\frac{1}{\delta}(q_i - x)^2\right] = \psi^i(x). \tag{1.2.31} \end{align} $$

### I.2.5. The generalization of nonclassical collapse models
(8) Let $|\psi_t\rangle_{cl}, t \in [0, T]$ be the classical pure states correspond to an vector-function $|\psi_t\rangle_{cl} : [0, T] \times \mathbb{S}^\infty \to \mathbb{S}^\infty$ such that $|\psi_t\rangle_{cl} \in \mathbb{S}^\infty \subset \mathbf{H}$, $t \in [0, T]$, where is a non projective Hilbert space $\mathbf{H}$, see Subsection I.7.1, Def I.7.1-I.7.2. Then the sudden spontaneous process is a localization along classical trajectory $\mathbf{x}_t : [0, T] \times \mathbb{R}^3 \to \mathbb{R}^3$ given by:

$$ |\psi_t\rangle_{cl} \xrightarrow{\delta, \varepsilon, \mathbf{x}_t\text{-localization}} \frac{|\psi_{t, \delta, \varepsilon, \mathbf{x}_t}^i\rangle_{cl}}{\| |\psi_{t, \delta, \varepsilon, \mathbf{x}_t}^i\rangle_{cl}\|}, \qquad (1.2.32) \\ \delta \in (0, 1], \varepsilon \ll 1, \mathbf{x}_t \in \mathbb{R}^3, t \in [0, T]. $$

where

$$ |\psi_{t, \delta, \varepsilon, \mathbf{x}_t}^i\rangle_{cl} = \hat{L}_{\mathbf{x}_t}^i(\delta, \varepsilon) |\psi_t\rangle_{cl}. \qquad (1.2.33) $$

Here $\hat{L}_{\mathbf{x}_t}^i(\delta, \varepsilon)$ is a norm-reducing, positive, self-adjoint, linear operator with a symbol $L_{\mathbf{x}_t}^i(\delta, \varepsilon)$ in the $n$-particle non projective Hilbert space $\mathbf{H}$, representing the localization of particle $i$ at each instant $t \in [0, T]$ around the point $\mathbf{x}_t$.

**Definition 1.2.3.** Such localization as mentioned above is called $\delta, \varepsilon, \mathbf{x}_t$-localization or $\delta, \varepsilon, \mathbf{x}_t$-collapse of the state $|\psi_t\rangle_{cl}$.

(9) The probability density $p_i(t, \mathbf{x}_t, \delta, \varepsilon)$ for the occurrence of a localization at point $\mathbf{x}_t$ at instant $t$ is assumed to be

$$ \begin{gathered} p_i(t, \mathbf{x}_t, \delta, \varepsilon) = \frac{\| |\psi_{t, \delta, \varepsilon, \mathbf{x}_t}^i\rangle_{cl} \|^2}{\Omega(t, \delta, \varepsilon)}, \qquad (1.2.34) \\ \Omega(t, \delta, \varepsilon) = \iiint_{\mathbb{R}^3} \| |\psi_{\delta, \varepsilon, \mathbf{x}_t}^i\rangle_{cl} \|^2 d^3x. \end{gathered} $$

(10) Let $|\psi_t\rangle_{n.cl}$ be the nonclassical pure state correspond to an vector-function $|\psi_t^\zeta\rangle = \zeta|\psi_t\rangle \in \mathbf{H} \setminus \mathbb{S}^\infty$, where $|\psi_t\rangle \in \mathbb{S}^\infty$, $|\zeta| \neq 1, t \in [0, T]$ see Subsection I.7.1, Def.I.7.3.

Then the sudden spontaneous process is a localization along classical trajectory $\mathbf{x}_t : [0, T] \times \mathbb{R}^3 \to \mathbb{R}^3$ given by:

$$ |\psi_t\rangle_{n.cl} \xrightarrow{\delta, \varepsilon, \mathbf{x}_t\text{-localization}} \frac{\zeta |\psi_{t, \delta, \varepsilon, \mathbf{x}_t}^i\rangle_{n.cl}}{\| |\psi_{t, \delta, \varepsilon, \mathbf{x}_t}^i\rangle_{n.cl}\|}, \qquad (1.2.35) \\ \mathbf{x}_t \in \mathbb{R}^3, t \in [0, T] $$

where

$$ |\psi_{t, \delta, \varepsilon, \mathbf{x}_t}^i\rangle_{n.cl} = \hat{L}_{\mathbf{x}_t}^i(\delta, \varepsilon) |\psi_t\rangle_{n.cl}. \qquad (1.2.36) $$

**Definition 1.2.4.** Such localization is called $\delta, \varepsilon, \mathbf{x}_t$-localization or $\delta, \varepsilon, \mathbf{x}_t$-collapse of the state $|\psi\rangle_{n.cl}$.
(11) The probability density $p_i(t, \mathbf{x}_t, \zeta, \delta, \varepsilon, \cdot)$ for the occurrence of a localization at point

$\mathbf{x}_t \in \mathbb{R}^3$ at instant $t \in [0, T]$ in accordance to postulate Q.IV.3 (see Subsection I.7.1,

Eq.(1.7.8)) is assumed to be

$$
\begin{align}
p_i(t, \mathbf{x}_t, \zeta, \delta, \varepsilon, \cdot) &= \frac{|\zeta|^{-6} \left\| \psi_{t, \delta, \varepsilon, |\zeta|^{-2} \mathbf{x}_t}^i \right\|_{n,cl}^2}{\Omega(t, \delta, \varepsilon)}, \nonumber \\
\Omega(t, \delta, \varepsilon) &= \iiint_{\mathbb{R}^3} \left\| \psi_{\delta, \varepsilon, \mathbf{x}_t}^i \right\|_{cl}^2 d^3 x. \tag{1.2.37}
\end{align}
$$

(12) The localization operators $\hat{L}_{x_t}^i(\delta, \varepsilon)$ have been chosen to have the form:

$$
\hat{L}_{x_t}^i(\delta, \varepsilon) = \begin{cases} \left(\frac{1}{\delta\pi_\delta}\right)^{3/4} \exp\left[-\frac{1}{2\delta}(\hat{\mathbf{q}}_i - \mathbf{x}_t)^2\right] & \text{iff } \|\mathbf{q}_i - \mathbf{x}_t\| \le \varepsilon \ll 1, \\ 0 & \text{iff } \|\mathbf{q}_i - \mathbf{x}_t\| > \varepsilon. \end{cases} \quad (1.2.38)
$$

Here $\delta \in (0, 1]$ and $\lim_{\delta \to 0} \pi_\delta = \pi$.

Remark 1.2.5. In one dimension case it follows that

$$
\hat{L}_{x_t}^i(\delta, \varepsilon) = \begin{cases} \left(\frac{1}{\delta\pi_\delta}\right)^{1/4} \exp\left[-\frac{1}{2\delta}(\hat{\mathbf{q}}_i - \mathbf{x}_t)^2\right] & \text{iff } |\mathbf{q}_i - \mathbf{x}_t| \le \varepsilon \ll 1, \\ 0 & \text{iff } |\mathbf{q}_i - \mathbf{x}_t| > \varepsilon. \end{cases} \quad (1.2.39)
$$

**Remark 1.2.6.** Note that from Eq.(1.2.34) and Eq.(1.2.38) follows that a probability density

$p_i(t, \mathbf{x}_t, \zeta, \delta, \varepsilon, \cdot)$ for the occurrence of a localization at instant $t$ inside sphere
$S(\mathbf{x}_t, \varepsilon) = \{\mathbf{q}_i \in \mathbb{R}^3 | \|\mathbf{q}_i - \mathbf{x}_t\| \le \varepsilon\}$ is given by

$$
\begin{align}
p_i(t, \mathbf{x}_t, \delta, \varepsilon) &= \frac{\left\| \psi_{t, \delta, \varepsilon, \mathbf{x}_t}^i \right\|_{cl}^2}{\Omega(t, \delta, \varepsilon)} \nonumber \\
&\quad \left\| \psi_{t, \delta, \varepsilon, \mathbf{x}_t}^i \right\|_{cl}^2 = \left(\frac{1}{\delta\pi\delta}\right)^{3/2} \int_{\|\mathbf{q}_i - \mathbf{x}_t\| \le \varepsilon} d^3\mathbf{q}_i \psi_t^i(\mathbf{q}_i) \exp\left[-\frac{1}{\delta}(\mathbf{q}_i - \mathbf{x}_t)^2\right], \tag{1.2.40} \\
\psi_t^i(\mathbf{q}_i) &= \langle \mathbf{q}_i \|\psi_t^i\|_{cl} \rangle, \nonumber
\end{align}
$$

and therefore

$$
\begin{align}
p_i(t,x,\varepsilon) &= \lim_{\delta \to 0} p_i(t,x,\delta,\varepsilon) = \nonumber \\
&= \lim_{\delta \to 0} \Omega^{-1}(t,\delta,\varepsilon) \left(\frac{1}{\delta\pi\delta}\right)^{1/2} \int_{|q_i - x| \le \varepsilon} dq_i \psi^i(q_i) \exp\left[-\frac{1}{\delta}(q_i - x_i)^2\right] = \psi^i(x_i). \tag{1.2.41}
\end{align}
$$
### I.3. The nonlocal Schrödinger equations and nonlocal nature of the wave function collapse

In this subsection we introduce new additional QM-postulates namely nonlocal Schrödinger equations. This nonlocal equations in a good consent with nonlocal nature of the wave function collapse. We obtain main GRW postulates given by Eq.(1.2.20) and Eq.(1.2.32) using nonlocal Schrödinger equations (1.3.3) and (1.3.6).

**Assumption 1.3.1.** We assume now that a wave function $\Psi(x, t) = \langle x | \psi_t \rangle$ collapses at instant: $t = 0$.

**Definition 1.3.1.** Let us consider the time-dependent canonical Schrödinger equation:

$$i\hbar \frac{\partial \Psi(x, t)}{\partial t} = \hat{\mathbf{H}}(t) \Psi(x, t), \qquad (1.3.1) \\ t \in [0, T], \mathbf{x} = (\mathbf{x}_1, \mathbf{x}_2, \dots, \mathbf{x}_n) \in \mathbb{R}^{3n}.$$

Let $\Psi(x, t)$ be a classical solution of the equation (1.3.1). The time-dependent Schrödinger equation (1.3.1) is a weakly well preserved by corresponding to $\Psi(x, t)$ collapsed along classical trajectory

$$\tilde{\mathbf{X}}_{m_1, \dots, m_k}(t, t') = [\tilde{\mathbf{x}}_{m_1}(t, t'), \dots, \tilde{\mathbf{x}}_{m_i}(t, t'), \dots, \tilde{\mathbf{x}}_{m_k}(t, t')], 1 \le i \le k \\ \tilde{\mathbf{x}}_{m_i}(t, t') = \begin{cases} \mathbf{x}_{m_i}(t) & \text{iff } t' < t \le T \\ 0 & \text{iff } 0 \le t \le t' \end{cases}$$

wave function $\Psi^{\#}(x, t, t')$ :

$$\Psi^{\#}(\mathbf{x}_1, \mathbf{x}_2, \dots, \mathbf{x}_n, t, t') =$$

$$\Psi^{\#}(\mathbf{x}_1, \mathbf{x}_2, \dots, \mathbf{x}_n, t, t'; \tilde{\mathbf{x}}_{m_1}(t, t'), \dots, \tilde{\mathbf{x}}_{m_k}(t, t')) =$$

$$\frac{\Re_{m_1, \dots, m_k}(\tilde{\mathbf{x}}_{m_1}(t, t'), \dots, \tilde{\mathbf{x}}_{m_k}(t, t')) \Psi(\mathbf{x}_1, \mathbf{x}_2, \dots, \mathbf{x}_n, t)}{\| \Re_{m_1, \dots, m_k}(\tilde{\mathbf{x}}_{m_1}(t, t'), \dots, \tilde{\mathbf{x}}_{m_k}(t, t')) \Psi(\mathbf{x}_1, \mathbf{x}_2, \dots, \mathbf{x}_n, t) \|_2}, \\ \Re_{m_1, \dots, m_k}(\tilde{\mathbf{x}}_{m_1}(t, t'), \dots, \tilde{\mathbf{x}}_{m_k}(t, t')) = \prod_{i=1}^{k} \Re_{m_i}(\tilde{\mathbf{x}}_{m_i}(t, t')), \qquad (1.3.2) \\ \Re_{m_i}(\tilde{\mathbf{x}}_{m_i}(t, t')) =$$

$$\left\{ \begin{array}{ll} (\pi\delta)^{-3/4} \exp\left[-\frac{(\hat{\mathbf{x}}_{m_i} - \tilde{\mathbf{x}}_{m_i}(t, t'))^2}{2\delta}\right] & \text{iff } |\mathbf{x}_{m_i} - \tilde{\mathbf{x}}_{m_i}(t, t')| \le \varepsilon, \\ 0 & \text{iff } \|\mathbf{x}_m - \tilde{\mathbf{x}}_m(t, t')\| > \varepsilon. \end{array} \right.$$
in region $\Gamma \subseteq \mathbb{R}^{3d}$ if the estimate

$$ \int_0^T \int_\Gamma \left\{ i\hbar \frac{\partial \Psi^\#(\mathbf{x}, t, t')}{\partial t} - \hat{\mathbf{H}}(t) \Psi^\#(\mathbf{x}, t, t') \right\} d^{3n}x = O(\hbar^\alpha), 1/4 < \alpha < 1/2, \quad (1.3.3) $$

$$ t \in [0, T], \mathbf{x} \in \mathbb{R}^{3n}, $$

is satisfied.

**Definition 1.3.2.** (i) The time-dependent integral equation (1.3.3) is colled the time- dependent nonlocal Schrödinger equation of the order $\hbar^\alpha$.

(ii) Such collapsed wave function $\Psi^\#(\mathbf{x}, t, t')$ as mentioned in Definition 1.3.1 is colled the

$\hbar^\alpha$- solution of the nonlocal Schrödinger equation (1.3.3) of the order $\alpha$.

**Definition 1.3.3.** Let us consider the time-independent canonical Schrödinger equation:

$$ \begin{gathered} \hat{\mathbf{H}}\Psi(\mathbf{x}) = 0, \tag{1.3.4} \\ t \in [0, T], \mathbf{x} = (\mathbf{x}_1, \mathbf{x}_2, \dots, \mathbf{x}_n) \in \mathbb{R}^{3n}. \end{gathered} $$

Let $\Psi(\mathbf{x})$ be a classical solution of the equation (1.3.4). The time-independent Schrödinger equation (1.3.4) is a weakly well preserved by corresponding to $\Psi(\mathbf{x})$ collapsed wave function $\Psi^\#(\mathbf{x})$ :

$$ \Psi^\#(\mathbf{x}_1, \mathbf{x}_2, \dots, \mathbf{x}_n) = $$

$$ \Psi(\mathbf{x}_1, \mathbf{x}_2, \dots, \mathbf{x}_n; \tilde{\mathbf{x}}_{m_1}, \dots, \tilde{\mathbf{x}}_{m_k}) = $$

$$ = \frac{\Re_{m_1, \dots, m_k}(\tilde{\mathbf{x}}_{m_1}, \dots, \tilde{\mathbf{x}}_{m_k}) \Psi(\mathbf{x}_1, \mathbf{x}_2, \dots, \mathbf{x}_n)}{\| \Re_{m_1, \dots, m_k}(\tilde{\mathbf{x}}_{m_1}, \dots, \tilde{\mathbf{x}}_{m_k}) \Psi(\mathbf{x}_1, \mathbf{x}_2, \dots, \mathbf{x}_n) \|_2}, \quad (1.3.5) $$

$$ \Re_{m_1, \dots, m_k}(\tilde{\mathbf{x}}_{m_1}, \dots, \tilde{\mathbf{x}}_{m_k}) = \prod_{i=1}^{k} \Re_{m_i}(\tilde{\mathbf{x}}_{m_i}), $$

$$ \Re_m(\tilde{\mathbf{x}}_m) = \begin{cases} (\pi\delta)^{-3/4} \exp\left[-\frac{(\tilde{\mathbf{x}}_m - \tilde{\mathbf{x}}_m)^2}{2\delta}\right] & \text{iff } \|\tilde{\mathbf{x}}_m - \tilde{\mathbf{x}}_m\| \le \varepsilon, \\ 0 & \text{iff } \|\tilde{\mathbf{x}}_m - \tilde{\mathbf{x}}_m\| > \varepsilon. \end{cases} $$

in region $\Gamma \subseteq \mathbb{R}^{3d}$ if the estimate

$$ \int_{\Gamma} \hat{\mathbf{H}} \Psi^{\#}(\mathbf{x}) d^{3d}x = O(\hbar^{\alpha}), 1/4 < \alpha < 1/2, \quad (1.3.6) $$

$$ \mathbf{x} \in \mathbb{R}^{3n}, $$
is satisfied.

**Definition 1.3.4.** (i) The stationary integral equation (1.3.6) is called nonlocal stationary Schrödinger equation of the order $\hbar^\alpha$.

(ii) Such collapsed wave function $\Psi^#(x)$ as mentioned in Definition 1.3.3 is colled the

$\hbar^\alpha$- solution of the time-independent nonlocal Schrödinger equation (1.3.6) of the

order $\alpha$.

**Definition 1.3.5.** The time-dependent integral equation

$$ \int_0^T \int_\Gamma \left\{ i\hbar \frac{\partial \Psi^#(\mathbf{x}, t)}{\partial t} - \hat{\mathbf{H}}(t) \Psi^#(\mathbf{x}, t) \right\} d^{3n}x = 0, \quad (1.3.7) \\ t \in [0, T], \mathbf{x} \in \mathbb{R}^{3n}, $$

is colled the time-dependent nonlocal Schrödinger equation.

**Definition 1.3.6.** The stationary integral equation

$$ \int_{\Gamma} \hat{\mathbf{H}} \Psi^{\#}(\mathbf{x}) d^{3n}x = 0, \mathbf{x} \in \mathbb{R}^{3n}, \quad (1.3.8) $$

is called nonlocal stationary Schrödinger equation

**Remark 1.3.1.** We have introduced in consideration Eq.(1.3.3) and Eq.(1.3.6) in order to obtain good approximation of the solutions of the Eq.(1.3.7) and Eq.(1.3.8) correspondingly.

**Lemma 1.3.1.** [24]. Let $\Phi(\lambda)$ be a function

$$ \Phi(\lambda) = \int_{0}^{a} x^{\beta-1} \exp(-\lambda x^{\alpha}) f(x) dx, \quad (1.3.9) $$

where $\lambda \gg 1, 0 < a < \infty, 0 < \beta, 0 < \alpha$. Assume that $f(x)$ is continuous on $[0, a]$. Then

$$ \Phi(\lambda) = \alpha^{-1} \Gamma\left(\frac{\beta}{\alpha}\right) [f(0) + o(1)] \lambda^{-\beta/\alpha} \quad (1.3.10) $$

**Theorem 1.3.1.** We assume now for a short that $m = n = 1$. Let $\Psi(x), x \in \mathbb{R}$ be a classical

solution of the equation (1.3.4), where

$$ \hat{\mathbf{H}} = \frac{\partial^2}{\partial x^2} + V(x). \quad (1.3.11) $$

Assume that

$$ \begin{aligned} |\Psi(x)| &= O(\hbar^{-1/4}), \\ |\partial\Psi(x)/\partial x| &= O(\hbar^{-5/4}). \end{aligned} \quad (1.3.12) $$

Then any collapsed wave function $\Psi^#(x)$ given by Eq.(1.3.5) with
$\sqrt{\hbar/\delta} = \hbar^\alpha, 1/4 < \alpha < 1/2$ that is $\hbar^\alpha$-solution of the time-independent nonlocal Schrödinger equation (1.3.6) of the order $\alpha$.

**Proof.** Note that

$$ 
\begin{aligned}
\Psi^{\#}(x) &= \phi_{\delta}(x)\Psi(x), \\
\phi_{\delta}(x) &= (\pi_{\delta}\delta)^{-1/4} \exp\left[-\frac{(x-\tilde{x})^2}{2\delta}\right] \quad \text{iff } |x-\tilde{x}| \le \varepsilon, \\
\phi_{\delta}(x) &= 0 \quad \text{iff } |x-\tilde{x}| > \varepsilon.
\end{aligned} 
\tag{1.3.13}
$$

From Eq.(1.3.13) one obtains

$$ 
\begin{aligned}
\frac{\partial \phi_{\delta}(x)}{\partial x} &= -(\pi_{\delta}\delta)^{-1/4} \delta^{-1}(x - \tilde{x}) \exp\left[-\frac{(x - \tilde{x})^2}{2\delta}\right] + \\
&\quad + ([\phi_{\delta}]_{\tilde{x}-\varepsilon})\delta(x - \tilde{x} + \varepsilon) + ([\phi_{\delta}]_{\tilde{x}+\varepsilon})\delta(x - \tilde{x} - \varepsilon) \quad \text{iff } |x - \tilde{x}| \le \varepsilon, \\
\frac{\partial \phi_{\delta}(x)}{\partial x} &= 0 \quad \text{iff } |x - \tilde{x}| > \varepsilon.
\end{aligned}
\tag{1.3.14}
$$
$$
\begin{aligned}
\frac{\partial^2 \phi_{\delta}(x)}{\partial x^2} &= -(\pi_{\delta}\delta)^{-1/4} \delta^{-1} \exp\left[-\frac{(x - \tilde{x})^2}{2\delta}\right] + \\
&\quad + (\pi_{\delta}\delta)^{-1/4} \delta^{-2} (x - \tilde{x})^2 \exp\left[-\frac{(x - \tilde{x})^2}{2\delta}\right] + \\
&\quad + ([\phi_{\delta}]_{\tilde{x}-\varepsilon})\delta'(x - \tilde{x} + \varepsilon) + ([\phi_{\delta}]_{\tilde{x}+\varepsilon})\delta'(x - \tilde{x} - \varepsilon) \quad \text{iff } |x - \tilde{x}| \le \varepsilon, \\
\frac{\partial^2 \phi_{\delta}(x)}{\partial x^2} &= 0 \quad \text{iff } |x - \tilde{x}| > \varepsilon.
\end{aligned}
$$

and

$$ 
\begin{aligned}
\frac{\partial^2 \Psi^{\#}(x)}{\partial x^2} &= \frac{\partial^2 [\phi_{\delta}(x) \Psi(x)]}{\partial x^2} = \frac{\partial}{\partial x} \left[ \Psi(x) \frac{\partial \phi_{\delta}(x)}{\partial x} + \phi_{\delta}(x) \frac{\partial \Psi(x)}{\partial x} \right] = \\
&= 2 \frac{\partial \Psi(x)}{\partial x} \frac{\partial \phi_{\delta}(x)}{\partial x} + \Psi(x) \frac{\partial^2 \phi_{\delta}(x)}{\partial x^2} + \phi_{\delta}(x) \frac{\partial^2 \Psi(x)}{\partial x^2}.
\end{aligned}
\tag{1.3.15}
$$

Substitution Eq.(1.3.11) and Eq.(1.3.15) into LHS of the Eq.(1.3.8) gives
$$
\begin{align}
\int_{\Gamma} \hat{\mathbf{H}} \Psi^{\#}(x) dx &= \int_{\Gamma} \left[ \hbar^2 \frac{\partial^2}{\partial x^2} \Psi^{\#}(x) + V(x) \Psi^{\#}(x) \right] dx = \nonumber \\
&= \int_{\Gamma} \phi_{\delta}(x) \left[ \hbar^2 \frac{\partial^2 \Psi(x)}{\partial x^2} + V(x) \Psi(x) \right] dx + 2\hbar^2 \int_{\Gamma} \left[ \frac{\partial \Psi(x)}{\partial x} \frac{\partial \phi_{\delta}(x)}{\partial x} \right] dx + \nonumber \\
&\qquad + \hbar^2 \int_{\Gamma} \left[ \Psi(x) \frac{\partial^2 \phi_{\delta}(x)}{\partial x^2} \right] dx = \nonumber \\
&= 2\hbar^2 \int_{\Gamma} \left[ \frac{\partial \Psi(x)}{\partial x} \frac{\partial \phi_{\delta}(x)}{\partial x} \right] dx + \hbar^2 \int_{\Gamma} \left[ \Psi(x) \frac{\partial^2 \phi_{\delta}(x)}{\partial x^2} \right] dx. \tag{1.3.16}
\end{align}
$$

From Eq.(1.3.16) and Eq.(1.3.12) one obtains

$$
\begin{align}
\left| \int_{\Gamma} \hat{\mathbf{H}} \Psi^{\#}(x) dx \right| &\le 2\hbar^2 \int_{\Gamma} \left[ \left| \frac{\partial \Psi(x)}{\partial x} \right| \left| \frac{\partial \phi_{\delta}(x)}{\partial x} \right| \right] dx + \hbar^2 \int_{\Gamma} \left[ \left| \Psi(x) \right| \left| \frac{\partial^2 \phi_{\delta}(x)}{\partial x^2} \right| \right] dx \le \nonumber \\
&\quad 2\hbar^2 O(\hbar^{-5/4}) \int_{\Gamma} \left| \frac{\partial \phi_{\delta}(x)}{\partial x} \right| dx + \hbar^2 O(\hbar^{-1/4}) \int_{\Gamma} \left| \frac{\partial^2 \phi_{\delta}(x)}{\partial x^2} \right| dx \le \nonumber \\
&\quad O(\hbar^{3/4}) (\pi_{\delta}\delta)^{-1/4} \delta^{-1} \int_{\Gamma} |x - \tilde{x}| \exp \left[ -\frac{(x - \tilde{x})^2}{2\delta} \right] dx + \tag{1.3.17} \\
&\quad + O(\hbar^{3/2}) (\pi_{\delta}\delta)^{-1/4} \delta^{-1} \int_{\Gamma} \exp \left[ -\frac{(x - \tilde{x})^2}{2\delta} \right] dx + \nonumber \\
&\quad + O(\hbar^{7/4}) (\pi_{\delta}\delta)^{-1/4} \delta^{-2} \int_{\Gamma} (x - \tilde{x})^2 \exp \left[ -\frac{(x - \tilde{x})^2}{2\delta} \right] dx. \nonumber
\end{align}
$$

Having applied Lemma 1.3.1 to RHS of the (1.3.17) we have finalized the proof.

**Theorem 1.3.2.** We assume now for a short that $m = n = 1$. Let $\Psi(x,t), x \in \mathbb{R}$ be a classical solution of the equation (1.3.1), where

$$
\hat{\mathbf{H}} = \hbar^2 \frac{\partial^2}{\partial x^2} + V(x,t). \tag{1.3.18}
$$

Assume that

$$
\begin{align}
& |\Psi(x,t)| = O(\hbar^{-1/4}), \nonumber \\
& |\partial\Psi(x,t)/\partial x| = O(\hbar^{-5/4}), \tag{1.3.19} \\
& |\partial\Psi(x,t)/\partial t| = O(\hbar^{-5/4}). \nonumber
\end{align}
$$

Then any collapsed wave function $\Psi^{\#}(x,t) = \Psi^{\#}(x,t, t' = 0)$ given by Eq.(1.3.3) with $t' = 0$ and $\sqrt{\hbar/\delta} = \hbar^{\alpha}, 1/4 < \alpha < 1/2$ that is $\hbar^{\alpha}$-solution of the time-dependent nonlocal Schrödinger equation (1.3.7) of the order $\alpha$.
**Proof.** Note that

$$ 
\begin{aligned}
\Psi^{\#}(x,t) &= \phi_{\delta}(x,t)\Psi(x,t), \\
\phi_{\delta}(x,t) &= \begin{cases}
(\pi_{\delta}\delta)^{-1/4}\eta(t)\exp\left[-\frac{(x-\tilde{x}_t)^2}{2\delta}\right] & \text{iff } |x-\tilde{x}_t| \le \varepsilon, \\
0 & \text{iff } |x-\tilde{x}_t| > \varepsilon,
\end{cases} \\
\eta(t) &= (\|\Re(\tilde{x}_t)\Psi(x,t)\|_2)^{-1}.
\end{aligned}
\quad (1.3.20)
$$

From Eq.(1.3.20) one obtains

$$ 
\begin{aligned}
\frac{\partial \phi_{\delta}(x,t)}{\partial x} &= -(\pi_{\delta}\delta)^{-1/4}\delta^{-1}(x-\tilde{x}_t)\eta(t) \exp\left[-\frac{(x-\tilde{x}_t)^2}{2\delta}\right] + \\
&\quad + ([\phi_{\delta}]_{\tilde{x}_t-\varepsilon})\delta(x-\tilde{x}_t + \varepsilon) + ([\phi_{\delta}]_{\tilde{x}_t+\varepsilon})\delta(x-\tilde{x}_t - \varepsilon) \quad \text{iff } |x-\tilde{x}_t| \le \varepsilon, \\
\frac{\partial \phi_{\delta}(x,t)}{\partial x} &= 0 \quad \text{iff } |x-\tilde{x}_t| > \varepsilon. \\
\frac{\partial^2 \phi_{\delta}(x,t)}{\partial x^2} &= -(\pi_{\delta}\delta)^{-1/4}\delta^{-1}\eta(t) \exp\left[-\frac{(x-\tilde{x}_t)^2}{2\delta}\right] + \\
&\quad + (\pi_{\delta}\delta)^{-1/4}\delta^{-2}(x-\tilde{x}_t)^2\eta(t) \exp\left[-\frac{(x-\tilde{x}_t)^2}{2\delta}\right] + \\
&\quad + \left(\left[\frac{\partial \phi_{\delta}}{\partial x}\right]_{\tilde{x}_t-\varepsilon}\right)\delta(x-\tilde{x}_t + \varepsilon) + \left(\left[\frac{\partial \phi_{\delta}}{\partial x}\right]_{\tilde{x}_t+\varepsilon}\right)\delta(x-\tilde{x}_t - \varepsilon) \\
&\quad + ([\phi_{\delta}]_{\tilde{x}_t-\varepsilon})\delta'(x-\tilde{x}_t + \varepsilon) + ([\phi_{\delta}]_{\tilde{x}_t+\varepsilon})\delta'(x-\tilde{x}_t - \varepsilon) \quad \text{iff } |x-\tilde{x}_t| \le \varepsilon, \\
\frac{\partial^2 \phi_{\delta}(x,t)}{\partial x^2} &= 0 \quad \text{iff } |x-\tilde{x}_t| > \varepsilon. \\
\frac{\partial \phi_{\delta}(x,t)}{\partial t} &= (\pi_{\delta}\delta)^{-1/4}\eta'(t) \exp\left[-\frac{(x-\tilde{x}_t)^2}{2\delta}\right] - \\
&\quad - (\pi_{\delta}\delta)^{-1/4}\eta(t)\tilde{x}'_t(x-\tilde{x}_t) \exp\left[-\frac{(x-\tilde{x}_t)^2}{2\delta}\right].
\end{aligned}
\quad (1.3.21)
$$

Note that

$$ i\hbar \frac{\partial \Psi^{\#}(x,t)}{\partial t} - \hat{H}(t) \Psi^{\#}(x,t) = i\hbar \frac{\partial \Psi^{\#}(x,t)}{\partial t} - \left[ h^2 \frac{\partial^2}{\partial x^2} \Psi^{\#}(x,t) + V(x,t) \Psi^{\#}(x,t) \right]. \quad (1.3.22) $$

Substitution Eq.(1.3.20) into LHS of the Eq.(1.3.22) gives
$$i\hbar \frac{\partial \Psi^{\#}(x,t)}{\partial t} - \hat{H}(t) \Psi^{\#}(x,t) = i\hbar \frac{\partial \phi_{\delta}(x,t)}{\partial t} \Psi(x,t) - \hbar^2 \left[ 2 \frac{\partial \Psi(x,t)}{\partial x} \frac{\partial \phi_{\delta}(x,t)}{\partial x} + \Psi(x,t) \frac{\partial^2 \phi_{\delta}(x)}{\partial x^2} \right] - \tag{1.3.23}$$

$$\phi_{\delta}(x,t) \left[ i\hbar \frac{\partial \Psi(x,t)}{\partial t} - \hbar^2 \frac{\partial^2 \Psi(x,t)}{\partial x^2} - V(x,t)\Psi(x,t) \right] = i\hbar \frac{\partial \phi_{\delta}(x,t)}{\partial t} \Psi(x,t) - \hbar^2 \left[ 2 \frac{\partial \Psi(x,t)}{\partial x} \frac{\partial \phi_{\delta}(x,t)}{\partial x} + \Psi(x,t) \frac{\partial^2 \phi_{\delta}(x)}{\partial x^2} \right]$$

Substitution Eq.(1.3.23) into LHS of the Eq.(1.3.7) gives

$$\int_{\Gamma} \left\{ i\hbar \frac{\partial \Psi^{\#}(x,t)}{\partial t} - \hat{H}(t) \Psi^{\#}(x,t) \right\} dx = \tag{1.3.24} \\ \int_{\Gamma} \left\{ i\hbar \frac{\partial \phi_{\delta}(x,t)}{\partial t} \Psi(x,t) - \hbar^2 \left[ 2 \frac{\partial \Psi(x,t)}{\partial x} \frac{\partial \phi_{\delta}(x,t)}{\partial x} + \Psi(x,t) \frac{\partial^2 \phi_{\delta}(x)}{\partial x^2} \right] \right\} dx.$$

From Eq.(1.3.24) we obtain

$$\left| \int_{\Gamma} \left\{ i\hbar \frac{\partial \Psi^{\#}(x,t)}{\partial t} - \hat{H}(t) \Psi^{\#}(x,t) \right\} dx \right| \le \tag{1.3.25} \\ \int_{\Gamma} \left\{ \hbar \left| \frac{\partial \phi_{\delta}(x,t)}{\partial t} \Psi(x,t) \right| + \hbar^2 \left[ 2 \left| \frac{\partial \Psi(x,t)}{\partial x} \frac{\partial \phi_{\delta}(x,t)}{\partial x} \right| + \left| \Psi(x,t) \frac{\partial^2 \phi_{\delta}(x)}{\partial x^2} \right| \right] \right\} dx.$$

Substitution Eq.(1.3.19),Eq.(1.3.19) into RHS of the (1.3.25) gives

$$\left| \int_{\Gamma} \left\{ i\hbar \frac{\partial \Psi^{\#}(x,t)}{\partial t} - \hat{H}(t) \Psi^{\#}(x,t) \right\} dx \right| \le \hbar O(\hbar^{-1/4}) \int_{\Gamma} \left| \frac{\partial \phi_{\delta}(x,t)}{\partial t} \right| dx + 2\hbar^2 O(\hbar^{-5/4}) \int_{\Gamma} \left| \frac{\partial \phi_{\delta}(x,t)}{\partial x} \right| dx + \hbar^2 O(\hbar^{-3/4}) \int_{\Gamma} \left| \frac{\partial \phi_{\delta}(x,t)}{\partial x} \right| dx = O(\hbar^{3/4}) \int_{\Gamma} \left| \frac{\partial \phi_{\delta}(x,t)}{\partial t} \right| dx + O(\hbar^{3/4}) \int_{\Gamma} \left| \frac{\partial \phi_{\delta}(x,t)}{\partial x} \right| dx + O(\hbar^{3/4}) \int_{\Gamma} \left| \frac{\partial^2 \phi_{\delta}(x,t)}{\partial x^2} \right| dx$$

Substitution Eq.(1.3.20),Eq.(1.3.21) into RHS of the (1.3.26) gives
$$ \left| \int_{\Gamma} \left\{ i\hbar \frac{\partial \Psi^{\#}(x,t)}{\partial t} - \hat{H}(t) \Psi^{\#}(x,t) \right\} dx \right| \le $$

$$ \begin{aligned} & O(\hbar^{3/4}) \int_{\Gamma} \left| \frac{\partial \phi_{\delta}(x,t)}{\partial t} \right| dx + O(\hbar^{3/4}) \int_{\Gamma} \left| \frac{\partial \phi_{\delta}(x,t)}{\partial x} \right| dx + O(\hbar^{3/4}) \int_{\Gamma} \left| \frac{\partial^2 \phi_{\delta}(x,t)}{\partial x^2} \right| dx \le \\ & \qquad O(\hbar^{3/4}) (\pi_{\delta}\delta)^{-1/4} |\eta'(t)| \int_{\Gamma} dx \exp \left[ -\frac{(x - \tilde{x}_t)^2}{2\delta} \right] + \\ & \qquad + O(\hbar^{3/4}) (\pi_{\delta}\delta)^{-1/4} |\eta(t)| |\tilde{x}'_t| \int_{\Gamma} dx |x - \tilde{x}_t| \exp \left[ -\frac{(x - \tilde{x}_t)^2}{2\delta} \right] + \\ & \qquad O(\hbar^{1/2}) (\pi_{\delta}\delta)^{-1/4} \delta^{-1} |\eta(t)| \int_{\Gamma} dx |x - \tilde{x}_t| \exp \left[ -\frac{(x - \tilde{x}_t)^2}{2\delta} \right] + \\ & \qquad O(\hbar^{3/4}) (\pi_{\delta}\delta)^{-1/4} \delta^{-2} |\eta(t)| \int_{\Gamma} dx (x - \tilde{x}_t)^2 \exp \left[ -\frac{(x - \tilde{x}_t)^2}{2\delta} \right]. \end{aligned} \quad (1.3.27) $$

Having applied Lemma 1.3.1 to RHS of the (1.3.27) we have finalized the proof.

**Assumption 1.3.2.** We assume now that a wave function $\Psi(x,t) = \langle x | \psi_t \rangle$ collapses at instant: $t' > 0$.

**Definition 1.3.7.** Let $\Psi(x,t)$ be a classical solution of the equation (1.3.1). The time-dependent Schrödinger equation (1.3.1) is a weakly well preserved by corresponding to classical solution $\Psi(x,t)$ collapsed along classical trajectory $\tilde{\mathbf{X}}_{m_1, \dots, m_k}(t, t') = [\tilde{\mathbf{X}}_{m_1}(t, t'), \dots, \tilde{\mathbf{X}}_{m_i}(t, t'), \dots, \tilde{\mathbf{X}}_{m_k}(t, t')], 1 \le i \le k$ a wave function $\Psi^{\#}(x, t, t')$ given by Eq.(1.3.2) with $t' > 0$ in region $[0, T] \times \Gamma \subseteq [0, T] \times \mathbb{R}^{3n}$, if the estimate

$$ \begin{gathered} \int_{\Gamma} d^{3n}x \int_{0}^{T} dt \left\{ i\hbar \frac{\partial \Psi^{\#}(x, t, t')}{\partial t} - \hat{H}(t) \Psi^{\#}(x, t, t') \right\} = O(\hbar^a), 0 < a \le 1, \quad (1.3.28) \\ t \in [0, T], x \in \mathbb{R}^{3n}, \end{gathered} $$

is satisfied.

**Definition 1.3.8.** (i) The time-dependent integral equation (1.3.28) is colled the time-dependent nonlocal Schrödinger equation of the order $\hbar^a$.

(ii) Such collapsed wave function $\Psi^{\#}(x, t, t')$ as mentioned in Definition 1.3.7 is colled the

$\hbar^a$- solution of the nonlocal Schrödinger equation (1.3.28) of the order $\alpha$.

**Definition 1.3.9.** The time-dependent integral equation
$$ \int_{\Gamma} d^{3n}x \int_{0}^{T} dt \left\{ i\hbar \frac{\partial \Psi^{\#}(\mathbf{x}, t, t')}{\partial t} - \hat{\mathbf{H}}(t) \Psi^{\#}(\mathbf{x}, t, t') \right\} = 0, \qquad (1.3.29) \\ t \in [0, T], \mathbf{x} \in \mathbb{R}^{3n}, $$

is colled the time-dependent nonlocal Schrödinger equation.

**Remark 1.3.2.** We have introduced in consideration Eq.(1.3.28) in order to obtain good approximation of the solutions of the time-dependent nonlocal Schrödinger equation (1.3.29).

**Theorem 1.3.4.** Then any collapsed wave function $\Psi^{\#}(\mathbf{x}, t, t')$ given by Eq.(1.3.) that is $\hbar^a$- solution of the time-independent nonlocal Schrödinger equation (1.3.29) of the order $\hbar^a$.

**Proof.** The proof similarly as the proof of the Theorem 1.3.2.

## 1.4. The nonlocal evolution equation for the statistical operator

Let us consider the evolution equation for the statistical operator [25]:

$$ \frac{d}{dt} \rho(t) = -\frac{i}{\hbar} \left[ \hat{\mathbf{H}}, \rho(t) \right]. \qquad (1.4.1) $$

Here $\hat{\mathbf{H}}$ is the standard quantum Hamiltonian. In the coordinate representation one has, according to Eq.(1.4.1)

$$ \begin{gathered} \frac{d}{dt} \rho(t, \mathbf{q}', \mathbf{q}'') = -\frac{i}{\hbar} \left( \hat{\mathbf{H}}_{\mathbf{q}'} - \left( \hat{\mathbf{H}}_{\mathbf{q}''} \right)^* \right) \rho(t, \mathbf{q}', \mathbf{q}''), \\ \rho(0, \mathbf{q}', \mathbf{q}'') = \rho(\mathbf{q}', \mathbf{q}''), \\ \mathbf{q}', \mathbf{q}'' \in \mathbb{R}^{3d}. \end{gathered} \qquad (1.4.2) $$

Here

$$ \rho(t, \mathbf{q}', \mathbf{q}'') = \Psi(t, \mathbf{q}') \Psi^*(t, \mathbf{q}'') \qquad (1.4.3) $$

and $\Psi(t, \mathbf{q}')$ is the classical solution of the Schrödinger equation (1.3.1), $H_{\mathbf{q}'}$ is the Hamiltonian of the system acting on a functions of variable $\mathbf{q}'$ and $\hat{H}_{\mathbf{q}''}$ is the same Hamiltonian of the system acting on a functions of variable $\mathbf{q}''$.

**Definition 1.4.1.** (i) Let $\rho(t, \mathbf{q}', \mathbf{q}'')$ be an classical solution of the equation (1.4.2). Collapsed statistical operator $\rho^{\#}(t, \mathbf{q}', \mathbf{q}'')$ corresponding to statistical operator $\rho(t, \mathbf{q}', \mathbf{q}'')$ given by

$$ \rho^{\#}(t, \mathbf{q}', \mathbf{q}'') = \Psi^{\#}(t, \mathbf{q}') \Psi^{\#*}(t, \mathbf{q}'') \qquad (1.4.4) $$

and where $\Psi^{\#}(t, \mathbf{q}')$ given by Eq.(1.3.2) with $t' = 0$.

(ii) Let $\rho(t, \mathbf{q}', \mathbf{q}'')$ be classical solution of the equation (1.4.2). The time-dependent equation (1.4.5) is a weakly well preserved by corresponding to $\rho(t, \mathbf{q}', \mathbf{q}'')$ collapsed statistical operator $\rho^{\#}(t, \mathbf{q}', \mathbf{q}'')$ in region $\Gamma \subseteq \mathbb{R}^{3d}$ if the estimate
$$ \frac{d}{dt} \int_{\Gamma} \rho^{\#}(t, \mathbf{q}', \mathbf{q}'') d^{3d} \mathbf{q}' d^{3d} \mathbf{q}'' = -\frac{i}{\hbar} \int_{\Gamma} (H_{\mathbf{q}'} - (\hat{H}_{\mathbf{q}''})^{*}) \rho^{\#}(t, \mathbf{q}', \mathbf{q}'') d^{3d} \mathbf{q}' d^{3d} \mathbf{q}'' + O(\hbar^{\alpha}), \quad (1.4.5) $$
$$ 1/4 < \alpha \le 1/2, $$

$$ \rho^{\#}(0, \mathbf{q}', \mathbf{q}'') = \rho^{\#}(\mathbf{q}', \mathbf{q}''), $$
$$ \mathbf{q}', \mathbf{q}'' \in \mathbb{R}^{3d} $$

is satisfied. Here

$$ \rho^{\#}(\mathbf{q}', \mathbf{q}'') = \Psi^{\#}(0, \mathbf{q}') \Psi^{**}(0, \mathbf{q}'') \quad (1.4.6) $$

and a function $\Psi^{\#}(0, \mathbf{q}')$ is the solution of the nonlocal stationary Schrödinger equation (1.3.34).

**Definition 1.4.2.** (i) The integral equation (1.4.5) is called nonlocal equation of the order $\hbar^\alpha$

for the statistical operator $\rho^{\#}(t, \mathbf{q}', \mathbf{q}'')$.

(ii) Such collapsed statistical operator $\rho^{\#}(t, \mathbf{q}', \mathbf{q}'')$ as mentioned in Definition 1.4.1 is called

the $\hbar^\alpha$- solution of the time-dependent nonlocal equation (1.4.5) of the order $\alpha$.

**Definition 1.4.3.** The time-dependent integral equation

$$ \frac{d}{dt} \int_{\Gamma} \rho^{\#}(t, \mathbf{q}', \mathbf{q}'') d^{3d} \mathbf{q}' d^{3d} \mathbf{q}'' = -\frac{i}{\hbar} \int_{\Gamma} (H_{\mathbf{q}'} - (\hat{H}_{\mathbf{q}''})^{*}) \rho^{\#}(t, \mathbf{q}', \mathbf{q}'') d^{3d} \mathbf{q}' d^{3d} \mathbf{q}'' \quad (1.4.7) $$

$$ \rho^{\#}(0, \mathbf{q}', \mathbf{q}'') = \rho^{\#}(\mathbf{q}', \mathbf{q}''), $$
$$ \mathbf{q}', \mathbf{q}'' \in \mathbb{R}^{3d} $$

is called nonlocal equation for the statistical operator $\rho^{\#}(t, \mathbf{q}', \mathbf{q}'')$.

**Remark 1.4.1.** We have introduced in consideration Eq.(1.4.5) in order to obtain good approximation of the solutions of the Eq.(1.4.7)

**Definition 1.4.4.** Let $\rho(t, \mathbf{q}'_1, \mathbf{q}'_2, \dots, \mathbf{q}'_n, \mathbf{q}''_1, \mathbf{q}''_2, \dots, \mathbf{q}''_n)$ be an statistical operator. We define statistical operator $\rho^{\#}(t, t', \mathbf{q}'_1, \mathbf{q}'_2, \dots, \mathbf{q}'_n, \mathbf{q}''_1, \mathbf{q}''_2, \dots, \mathbf{q}''_n)$ corresponding to statistical operator $\rho(t, \mathbf{q}'_1, \mathbf{q}'_2, \dots, \mathbf{q}'_n, \mathbf{q}''_1, \mathbf{q}''_2, \dots, \mathbf{q}''_n)$ and collapsed along classical trajectories
$$
\begin{align}
\tilde{\mathbf{X}}'_{m_1, \dots, m_k}(t, t') &= [\tilde{\mathbf{x}}'_{m_1}(t, t'), \dots, \tilde{\mathbf{x}}'_{m_i}(t, t'), \dots, \tilde{\mathbf{x}}'_{m_k}(t, t')], 1 \le i \le k, \nonumber \\
\tilde{\mathbf{x}}'_{m_i}(t, t') &= \begin{cases} \mathbf{x}'_{m_i}(t) & \text{iff } t' < t \le T \\ 0 & \text{iff } 0 \le t \le t' \end{cases} \nonumber \\
\tilde{\mathbf{X}}''_{m_1, \dots, m_k}(t, t') &= [\tilde{\mathbf{x}}''_{m_1}(t, t'), \dots, \tilde{\mathbf{x}}''_{m_i}(t, t'), \dots, \tilde{\mathbf{x}}''_{m_k}(t, t')], 1 \le i \le k, \tag{1.4.8} \\
\tilde{\mathbf{x}}''_{m_i}(t, t') &= \begin{cases} \mathbf{x}''_{m_i}(t) & \text{iff } t' < t \le T \\ 0 & \text{iff } 0 \le t \le t' \end{cases} \nonumber
\end{align}
$$

by formulae

$$
\begin{align*}
\rho^{\#}(t, t', \mathbf{q}'_1, \mathbf{q}'_2, \dots, \mathbf{q}'_n, \mathbf{q}''_1, \mathbf{q}''_2, \dots, \mathbf{q}''_n) &= \\
\rho^{\#}(\mathbf{x}_1, \mathbf{x}_2, \dots, \mathbf{x}_n, t, t'; \tilde{\mathbf{x}}_{m_1}(t, t'), \dots, \tilde{\mathbf{x}}_{m_k}(t, t')) &= \\
\mathfrak{R}_{m_1, \dots, m_k}(\tilde{\mathbf{x}}'_{m_1}(t, t'), \dots, \tilde{\mathbf{x}}'_{m_i}(t, t'), \dots, \tilde{\mathbf{x}}'_{m_k}(t, t')) \times \\
\mathfrak{R}_{m_1, \dots, m_k}(\tilde{\mathbf{x}}''_{m_1}(t, t'), \dots, \tilde{\mathbf{x}}''_{m_i}(t, t'), \dots, \tilde{\mathbf{x}}''_{m_k}(t, t')) \rho(\mathbf{q}'_1, \mathbf{q}'_2, \dots, \mathbf{q}'_n, \mathbf{q}''_1, \mathbf{q}''_2, \dots, \mathbf{q}''_n), \\
\mathfrak{R}_{m_1, \dots, m_k}(\tilde{\mathbf{x}}'_{m_1}(t, t'), \dots, \tilde{\mathbf{x}}'_{m_i}(t, t'), \dots, \tilde{\mathbf{x}}'_{m_k}(t, t')) &= \prod_{i=1}^{k} \mathfrak{R}_{m_i}(\tilde{\mathbf{x}}'_{m_i}(t, t')), \\
\mathfrak{R}_{m_1, \dots, m_k}(\tilde{\mathbf{x}}''_{m_1}(t, t'), \dots, \tilde{\mathbf{x}}''_{m_i}(t, t'), \dots, \tilde{\mathbf{x}}''_{m_k}(t, t')) &= \prod_{i=1}^{k} \mathfrak{R}_{m_i}(\tilde{\mathbf{x}}''_{m_i}(t, t')), \\
\mathfrak{R}_{m_i}(\tilde{\mathbf{x}}'_{m_i}(t, t')) &= \tag{1.4.9} \\
\left\{
\begin{array}{@{}l@{}}
(\pi \delta)^{-3/4} \exp \left[ - \frac{(\hat{\mathbf{q}}'_{m_i} - \tilde{\mathbf{x}}'_{m_i}(t, t'))^2}{2\delta} \right] \quad \text{iff } \|\mathbf{q}'_{m_i} - \tilde{\mathbf{x}}'_{m_i}(t, t')\| \le \varepsilon, \\
0 \quad \text{iff } \|\mathbf{q}'_{m_i} - \tilde{\mathbf{x}}'_{m_i}(t, t')\| > \varepsilon.
\end{array}
\right. \\
\mathfrak{R}_{m_i}(\tilde{\mathbf{x}}''_{m_i}(t, t')) &= \\
\left\{
\begin{array}{@{}l@{}}
(\pi \delta)^{-3/4} \exp \left[ - \frac{(\hat{\mathbf{q}}''_{m_i} - \tilde{\mathbf{x}}''_{m_i}(t, t'))^2}{2\delta} \right] \quad \text{iff } \|\mathbf{q}''_{m_i} - \tilde{\mathbf{x}}''_{m_i}(t, t')\| \le \varepsilon, \\
0 \quad \text{iff } \|\mathbf{q}''_{m_i} - \tilde{\mathbf{x}}''_{m_i}(t, t')\| > \varepsilon.
\end{array}
\right.
\end{align*}
$$

Let us consider now the 1-particle master equation of the classical GRW model
$$ \frac{d}{dt} \rho(t) = -\frac{i}{\hbar} \left[ \hat{H}, \rho(t) \right] - \lambda(\rho(t) - T[\rho(t)]). \qquad (1.4.10) $$

Here $\hat{H}$ is the standard quantum Hamiltonian of the particle, and $T[\cdot]$ represents the effect of the spontaneous collapses on the particle's wave function. In the position representation, this equation becomes:

$$ \frac{d}{dt} \rho(t, \mathbf{q}', \mathbf{q}'') = -\frac{i}{\hbar} \left( \hat{H}_{\mathbf{q}'} - (\hat{H}_{\mathbf{q}''})^* \right) \rho(t, \mathbf{q}', \mathbf{q}'') - \lambda \left\{ 1 - \exp \left[ -\frac{(\mathbf{q}' - \mathbf{q}'')^2}{4r_c^2} \right] \right\} \rho(t, \mathbf{q}', \mathbf{q}''), \qquad (1.4.11) $$

$$ \rho(0, \mathbf{q}', \mathbf{q}'') = \rho(\mathbf{q}', \mathbf{q}''), \\ \mathbf{q}', \mathbf{q}'' \in \mathbb{R}^{3d}. $$

Here $H_{\mathbf{q}'}$ is the Hamiltonian of the system acting on a functions of variable $\mathbf{q}'$ and $\hat{H}_{\mathbf{q}''}$ is the same Hamiltonian of the system acting on a functions of variable $\mathbf{q}''$.

**Definition 1.4.5.** Let $\rho(t, \mathbf{q}', \mathbf{q}'')$ be classical solution of the equation (1.4.11). The time-dependent equation (1.4.11) is a weakly well preserved by corresponding to $\rho(t, \mathbf{q}', \mathbf{q}'')$ collapsed statistical operator $\rho^{\#}(t, \mathbf{q}', \mathbf{q}'')$ given by (1.4.9) with $t' = 0$ in region $\Gamma \subseteq \mathbb{R}^{3d}$ if the estimate

$$ \frac{d}{dt} \int_{\Gamma} \rho^{\#}(t, \mathbf{q}', \mathbf{q}'') d^{3d} \mathbf{q}' d^{3d} \mathbf{q}'' = -\frac{i}{\hbar} \int_{\Gamma} \left( H_{\mathbf{q}'} - (\hat{H}_{\mathbf{q}''})^* \right) \rho^{\#}(t, \mathbf{q}', \mathbf{q}'') d^{3d} \mathbf{q}' d^{3d} \mathbf{q}'' - \lambda \int_{\Gamma} \left\{ 1 - \exp \left[ -\frac{(\mathbf{q}' - \mathbf{q}'')^2}{4r_c^2} \right] \right\} \rho^{\#}(t, \mathbf{q}', \mathbf{q}'') d^{3d} \mathbf{q}' d^{3d} \mathbf{q}'' + O(\hbar^a), \qquad (1.4.12) $$

$$ 1/4 < \alpha \le 1/2, $$

$$ \rho^{\#}(0, \mathbf{q}', \mathbf{q}'') = \rho^{\#}(\mathbf{q}', \mathbf{q}''), \\ \mathbf{q}', \mathbf{q}'' \in \mathbb{R}^{3d} $$

is satisfied.

**Definition 1.4.6.** The time-dependent integral equation
$$
\begin{align}
& \frac{d}{dt} \int_{\Gamma} \rho^{\#}(t, \mathbf{q}', \mathbf{q}'') d^{3d} \mathbf{q}' d^{3d} \mathbf{q}'' = \nonumber \\
& -\frac{i}{\hbar} \int_{\Gamma} (H_{\mathbf{q}'} - (\hat{H}_{\mathbf{q}''})^{*}) \rho^{\#}(t, \mathbf{q}', \mathbf{q}'') d^{3d} \mathbf{q}' d^{3d} \mathbf{q}'' - \nonumber \\
& -\lambda \int_{\Gamma} \left\{ 1 - \exp\left[-\frac{(\mathbf{q}' - \mathbf{q}'')^2}{4r_c^2}\right] \right\} \rho^{\#}(t, \mathbf{q}', \mathbf{q}'') d^{3d} \mathbf{q}' d^{3d} \mathbf{q}'' \tag{1.4.13} \\
& \qquad \qquad \qquad \qquad \qquad \qquad 1/4 < \alpha \le 1/2, \nonumber
\end{align}
$$

$$
\begin{gathered}
\rho^{\#}(0, \mathbf{q}', \mathbf{q}'') = \rho^{\#}(\mathbf{q}', \mathbf{q}''), \\
\mathbf{q}', \mathbf{q}'' \in \mathbb{R}^{3d}
\end{gathered}
$$

is called nonlocal master equation for the statistical operator $\rho^{\#}(t, \mathbf{q}', \mathbf{q}'')$.

**Remark 1.4.2.** We have introduced in consideration Eq.(1.4.12) in order to obtain good approximation of the solutions of the Eq.(1.4.13)

**Theorem 1.4.1.** We assume now for a short that $m = n = 1$. Let $\Psi(x,t), x \in \mathbb{R}$ be a classical solution of the equation (1.3.1), where

$$
\hat{\mathbf{H}} = \hbar^2 \frac{\partial^2}{\partial x^2} + V(x,t). \qquad (1.4.14)
$$

Assume that

$$
\begin{align}
& |\Psi(x,t)| = O(\hbar^{-1/4}), \nonumber \\
& |\partial\Psi(x,t)/\partial x| = O(\hbar^{-5/4}), \tag{1.4.15} \\
& |\partial\Psi(x,t)/\partial t| = O(\hbar^{-5/4}). \nonumber
\end{align}
$$

Then for any collapsed wave function $\Psi^{\#}(x,t) = \Psi^{\#}(x,t, t' = 0)$ given by Eq.(1.3.2) with $t' = 0$ and $\sqrt{\hbar/\delta} = \hbar^\alpha, 1/4 < \alpha < 1/2$ corresponding collapsed statistical operator $\rho^{\#}(t, \mathbf{q}', \mathbf{q}'')$ given by Eq.(1.4.4) that is $\hbar^\alpha$-solution of the time-dependent nonlocal equation (1.4.5) of the order $\alpha$.

**Proof.** The proof similarly as the proof of the Theorem 1.3.2.

Let us consider now the equation (1.4.11) in the case in which $\hat{\mathbf{H}}$ is the Hamiltonian for a free particle; for simplicity we work in one dimension. In the coordinate representation we get

$$
\frac{d}{dt} \rho(t, \mathbf{q}', \mathbf{q}'') = \frac{i\hbar}{2m} \left( \frac{\partial^2}{\partial q'^2} - \frac{\partial^2}{\partial q''^2} \right) \rho(t, \mathbf{q}', \mathbf{q}'') - \lambda \left\{ 1 - \exp\left[ - \frac{(\mathbf{q}' - \mathbf{q}'')^2}{4r_c^2} \right] \right\} \rho(t, \mathbf{q}', \mathbf{q}''). \quad (1.4.16)
$$

One can express the solution of the above equation satisfying given initial conditions in terms of the solution $\rho_{\text{Sch}}(t, q', q'')$ of the usual Schrödinger equation ($\lambda = 0$) satisfying the same initial conditions, according to [3]:
$$
\begin{equation} \tag{1.4.17}
\begin{aligned}
\rho(t, q', q'') &= \frac{1}{2\pi\hbar} \int_{-\infty}^{\infty} dk \int_{-\infty}^{\infty} dy F(k, q' - q'', t) \rho_{Sch}(t, q' + y, q'' + y) \exp\left(\frac{iky}{\hbar}\right), \\
F(k, q, t) &= \exp\left[-\lambda t + \lambda \int_0^t d\tau \exp\left(-\frac{1}{4r_c^2}\left[q - \frac{k\tau}{m}\right]^2\right)\right].
\end{aligned}
\end{equation}
$$

By Theorem 1.4.1 corresponding to $\rho(t, q', q'')$ given by

$$
\begin{equation}
\begin{aligned}
\rho^{\#}(t, q', q'') &= \Re(\tilde{q}'(t))\Re(\tilde{q}''(t))\rho(t, q', q''), \\
\Re(\tilde{q}'(t)) &= \begin{cases}
(\pi\delta)^{-1/4} \exp\left[-\frac{(\hat{q}' - \tilde{q}'(t))^2}{2\delta}\right] & \text{iff } |q' - \tilde{q}'(t)| \le \varepsilon, \\
0 & \text{iff } |q' - \tilde{q}'(t)| > \varepsilon.
\end{cases} \tag{1.4.18} \\
\Re(\tilde{q}''(t)) &= \begin{cases}
(\pi\delta)^{-1/4} \exp\left[-\frac{(\hat{q}'' - \tilde{q}''(t))^2}{2\delta}\right] & \text{iff } |q'' - \tilde{q}''(t)| \le \varepsilon, \\
0 & \text{iff } |q'' - \tilde{q}''(t)| > \varepsilon.
\end{cases}
\end{aligned}
\end{equation}
$$

**Assumption 1.4.1.** We assume now that a wawe function collapses at

instant $t' > 0$.

**Definition 1.4.7.** Let $\rho(t, q', q'')$ be a classical solution of the equation (1.4.2). The time-dependent equation (1.4.2) is a weakly well preserved by corresponding to $\rho(t, q', q'')$ collapsed statistical operator $\rho^{\#}(t, t', q', q'')$ in region $[0, T] \times \Gamma, \Gamma \subseteq \mathbb{R}^{3d}$ iff the estimate

$$
\begin{equation}
\begin{aligned}
& \int_0^T dt \int_\Gamma \frac{d}{dt} \rho^{\#}(t, t', \mathbf{q}', \mathbf{q}'') d^{3d} \mathbf{q}' d^{3d} \mathbf{q}'' = \\
& \quad -\frac{i}{\hbar} \int_0^T dt \int_\Gamma (H_{\mathbf{q}'} - (\hat{H}_{\mathbf{q}''})^*) \rho^{\#}(t, t', \mathbf{q}', \mathbf{q}'') d^{3d} \mathbf{q}' d^{3d} \mathbf{q}'' + O_T(\hbar^\alpha), \quad (1.4.19) \\
& \frac{1}{4} < \alpha \le \frac{1}{2}, \\
& \rho^{\#}(0, \mathbf{q}', \mathbf{q}'') = \rho^{\#}(\mathbf{q}', \mathbf{q}''), \\
& \mathbf{q}', \mathbf{q}'' \in \mathbb{R}^{3d}
\end{aligned}
\end{equation}
$$

is satisfied.

**Definition 1.4.8.** The time-dependent integral equation
$$ \int_0^T dt \int_\Gamma \frac{d}{dt} \rho^{\#}(t, \mathbf{q}', \mathbf{q}'') d^3\mathbf{q}' d^3\mathbf{q}'' = -\frac{i}{\hbar} \int_0^T dt \int_\Gamma (H_{\mathbf{q}'} - (\hat{H}_{\mathbf{q}''})^*) \rho^{\#}(t, \mathbf{q}', \mathbf{q}'') d^3\mathbf{q}' d^3\mathbf{q}'' \quad (1.4.20) $$

$$ \rho^{\#}(0, \mathbf{q}', \mathbf{q}'') = \rho^{\#}(\mathbf{q}', \mathbf{q}''), \\ \mathbf{q}', \mathbf{q}'' \in \mathbb{R}^{3d} $$

is called nonlocal master equation for the statistical operator $\rho^{\#}(t, \mathbf{q}', \mathbf{q}'')$.

**Definition 1.4.9.** Let $\rho(t, \mathbf{q}', \mathbf{q}'')$ be a classical solution of the equation (1.4.13). The time-dependent equation (1.4.13) is a weakly well preserved by corresponding to $\rho(t, \mathbf{q}', \mathbf{q}'')$ collapsed statistical operator $\rho^{\#}(t, t', \mathbf{q}', \mathbf{q}'')$ given by (1.4.9) in region $[0, T] \times \Gamma, \Gamma \subseteq \mathbb{R}^{3d}$ if the estimate

$$ \int_0^T dt \int_\Gamma \frac{d}{dt} \rho^{\#}(t, \mathbf{q}', \mathbf{q}'') d^3\mathbf{q}' d^3\mathbf{q}'' = -\frac{i}{\hbar} \int_0^T dt \int_\Gamma (H_{\mathbf{q}'} - (\hat{H}_{\mathbf{q}''})^*) \rho^{\#}(t, \mathbf{q}', \mathbf{q}'') d^3\mathbf{q}' d^3\mathbf{q}'' - \lambda \int_0^T dt \int_\Gamma \left\{ 1 - \exp\left[ -\frac{(\mathbf{q}' - \mathbf{q}'')^2}{4r_c^2} \right] \right\} \rho^{\#}(t, \mathbf{q}', \mathbf{q}'') d^3\mathbf{q}' d^3\mathbf{q}'' + O_T(\hbar^a), \quad (1.4.21) $$
$$ 1/4 < \alpha \le 1/2, $$

$$ \rho^{\#}(0, \mathbf{q}', \mathbf{q}'') = \rho^{\#}(\mathbf{q}', \mathbf{q}''), \\ \mathbf{q}', \mathbf{q}'' \in \mathbb{R}^{3d} $$

is satisfied.

**Definition 1.4.10.** The time-dependent integral equation

$$ \int_0^T dt \int_\Gamma \frac{d}{dt} \rho^{\#}(t, \mathbf{q}', \mathbf{q}'') d^3\mathbf{q}' d^3\mathbf{q}'' = -\frac{i}{\hbar} \int_0^T dt \int_\Gamma (H_{\mathbf{q}'} - (\hat{H}_{\mathbf{q}''})^*) \rho^{\#}(t, \mathbf{q}', \mathbf{q}'') d^3\mathbf{q}' d^3\mathbf{q}'' - \lambda \int_0^T dt \int_\Gamma \left\{ 1 - \exp\left[ -\frac{(\mathbf{q}' - \mathbf{q}'')^2}{4r_c^2} \right] \right\} \rho^{\#}(t, \mathbf{q}', \mathbf{q}'') d^3\mathbf{q}' d^3\mathbf{q}'' \quad (1.4.22) $$
$$ 1/4 < \alpha \le 1/2, $$

$$ \rho^{\#}(0, \mathbf{q}', \mathbf{q}'') = \rho^{\#}(\mathbf{q}', \mathbf{q}''), \\ \mathbf{q}', \mathbf{q}'' \in \mathbb{R}^{3d} $$
is called nonlocal master equation for the statistical operator $\rho^{\#}(t, q', q'')$.

## 1.5. The nonlocal stochastic nonlinear Schrödinger equation

Let us consider the stochastic nonlinear Schrödinger equation [9],[10],[46]:

$$d|\psi_t(\mathbf{x})\rangle = \left[-\frac{i}{\hbar}\hat{\mathbf{H}} - \frac{k}{2}(\hat{q} - \langle q_t \rangle)^2 dt\right] |\psi_t(\mathbf{x})\rangle dt + \sqrt{k}(\hat{q} - \langle q_t \rangle) dW_t(\omega) |\psi_t(\mathbf{x})\rangle. \quad (1.5.1)$$

Here $\mathbf{x} \in \mathbb{R}^r$, $\hat{q}$ is the position operator, $\langle q_t \rangle = \langle \psi_t | \hat{q} | \psi_t \rangle$ it is its expectation value, and $k$ is a constant.

**Definition 1.5.1.** Let $(w_{\epsilon,t}(\omega))_{\epsilon \in (0,1]} \in \mathcal{H}$ be smoothed with respect to $\mathbb{R}^r$ white noise, (see [20],[21]) and let $(W_{\epsilon,t}(\omega))_{\epsilon \in (0,1]} \in \mathcal{H}$ be Kolmbeau-Wiener process $(\dot{W}_{\epsilon,t}(\omega))_{\epsilon \in (0,1]} \in \mathcal{H}$.

We rewrite now Eq. (1.5.1) in Kolmbeau-Ito form:

$$(d|\psi_{\epsilon,t}(\mathbf{x})\rangle)_{\epsilon} = \left( \left[-\frac{i}{\hbar}\hat{\mathbf{H}}_{\epsilon} - \frac{k}{2}(\hat{q} - \langle q_{\epsilon,t} \rangle)^2 dt\right] |\psi_{\epsilon,t}(\mathbf{x})\rangle dt + \sqrt{k}(\hat{q} - \langle q_{\epsilon,t} \rangle) dW_{\epsilon,t}(\omega) |\psi_{\epsilon,t}(\mathbf{x})\rangle \right)_{\epsilon}. \quad (1.5.2)$$

Here $\mathbf{x} \in \mathbb{R}^r$, $\hat{q}$ is the position operator, $(\langle q_{\epsilon,t} \rangle)_{\epsilon} = (\langle \psi_{\epsilon,t} | \hat{q} | \psi_{\epsilon,t} \rangle)_{\epsilon}$ it is its expectation value, and $k$ is a constant.

**Definition 1.5.2.** Let $(\psi_{\epsilon,t}(\mathbf{x}))_{\epsilon} = (\psi_{\epsilon}(\mathbf{x},t))_{\epsilon}$, $\mathbf{x}=(\mathbf{x}_1, \mathbf{x}_2, \dots, \mathbf{x}_n)$ be an Kolmbeau solution of the equation (1.5.2) and let

$$(\tilde{\mathbf{X}}_{m_1, \dots, m_i, \dots, m_k}(t, t', \omega, \epsilon))_{\epsilon} = [(\tilde{\mathbf{x}}_{m_1}(t, t', \omega, \epsilon))_{\epsilon}, \dots, (\tilde{\mathbf{x}}_{m_i}(t, t', \omega, \epsilon))_{\epsilon}, \dots, (\tilde{\mathbf{x}}_{m_k}(t, t', \omega, \epsilon))_{\epsilon}], 1 \le i \le k \quad (1.5.3)$$

be Kolmbeau stochastic trajectory. Then we define collapsed Kolmbeau solution $(\psi^{\#}_{\epsilon,t,t'}(\mathbf{x}))_{\epsilon}$ corresponding to Kolmbeau solution $(\psi_{\epsilon,t}(\mathbf{x}))_{\epsilon}$ by formula
nonlocal equation (1.5.5) of the order $\alpha$.

**Theorem 1.5.1.** Let $(\psi_{\epsilon,t,t'}(\mathbf{x}))_{\epsilon}, \mathbf{x} \in \mathbb{R}^{3n}$ be a Colombeau solution of the equation (1.5.2), where

$$ (\hat{\mathbf{H}}_{\epsilon})_{\epsilon} = \hbar^2 \sum_{i=1}^{3n} \frac{\partial^2}{\partial x_i^2} + (V_{\epsilon}(\mathbf{x},t))_{\epsilon}. \qquad (1.5.6) $$

Assume that

$$ \begin{gathered} (|\psi_{\epsilon,t,t'}(\mathbf{x})|)_{\epsilon} = (O_{\epsilon}(\hbar^{-3n/4}))_{\epsilon}, \\ (|\partial\psi_{\epsilon,t,t'}(\mathbf{x})/\partial x_i|)_{\epsilon} = (O_{\epsilon}(\hbar^{-3n/4-1}))_{\epsilon}, \quad i = 1, \dots, 3n, \\ (|\partial\psi_{\epsilon,t,t'}(\mathbf{x})/\partial t|)_{\epsilon} = (O_{\epsilon}(\hbar^{-3n/4-1}))_{\epsilon}. \end{gathered} \qquad (1.5.7) $$

Then any collapsed Colombeau wave function $(|\psi_{\epsilon,t,t'}(\mathbf{x})|)_{\epsilon}$ given by Eq.(1.5.4) with $\sqrt{\hbar/\delta} = \hbar^{\alpha}, 1/4 < \alpha < 1/2$ that is $\hbar^{\alpha}$-solution of the time-dependent nonlocal Schrödinger equation (1.5.5) of the order $\alpha$.

**Proof.** The proof similarly as the proof of the Theorem 1.3.2.

## 1.6. The reconciled Bohr rule. Schrödinger's cat through Stern-Gerlach experiment. Schrödinger's cat demands to reconcile Bohr rule.

Another known in literature special sort of the Schrödinger cat paradox can be simply illustrated with the famous Stern-Gerlach experiment (Fig.1.6.1). Silver atoms boiled off from a furnace are sent through a non-uniform magnetic field, and impinge on a photographic plate. Instead of a continuous distribution of spots, one sees two spots, corresponding to spin up and spin down relative to the magnetic field axis. Each atom goes up OR down, but one cannot predict which in any given run – the results of the experiment are probabilistic. There is a 50% chance of an atom going up, and a 50% chance that it will go down.

<figure><img src="image_7.png" /><figcaption>Fig.1.6.1.Stern-Gerlach experiment.
Adapted from [27].</figcaption></figure>

**Remark 1.6.1.** We remind that from the point of view of the Schrödinger equation of quantum theory, this result has no any rigorous explanation.
In quantum theory, the state of the particle is described by its wave function, and the Schrödinger equation says that at a post-measurement final time $t_f$, the wave function is related to that at a pre-measurement initial time $t_i$, by known deterministic relation

$$ \Psi(t_f) = U(t_f, t_i)\Psi(t_i), \\ U(t_f, t_i) = \exp[i\hat{H}(t_f - t_i)] \qquad (1.6.1) $$

with the transition unitary operator $U$ completely specified by the Hamiltonian $\hat{H}$. To explain what is observed, the Schrödinger equation must be supplemented by the reduction postulate and the Born rule. These state that the wave function only gives a description of probabilities when a measurement is made, with the probabilities for an “up” outcome and a “down” outcome given by the squares of the coefficients of the corresponding components in the initial wave function $\Psi(t_i)$,

<figure><img src="image_8.png" /><figcaption>Fig.1.6.2. The Stern-Gerlach apparatus with a Schrödinger cat as the outcome registration. Adapted from [27].</figcaption></figure>

Born's Rule for Probabilities

$$ \Psi(t_i) = c_{\text{up}}\psi_{\text{up}} + c_{\text{down}}\psi_{\text{down}}, \qquad (1.6.2) \\ \text{prob}(\psi_{\text{up}}) = |c_{\text{up}}|^2, \text{prob}(\psi_{\text{down}}) = |c_{\text{down}}|^2, \\ |c_{\text{up}}|^2 + |c_{\text{down}}|^2 = 1. $$

with the sum of the up and down probabilities equal to one. The reduction postulate and Born rule are an add-on to the Schrödinger equation. According to the Copenhagen interpretation of quantum mechanics, the Schrödinger equation applies when a microscopic system, the silver atom, is time-evolving in isolation. But when the atom interacts with a macroscopic measuring apparatus, as in the Stern-Gerlach setup, you have to use the reduction postulate and Born rule.

**Remark 1.6.2.** This situation leads to puzzles that have been debated for over eighty years. If quantum mechanics describes the whole universe, then why can't
one use the Schrödinger equation to describe the system consisting of the silver atom plus the measuring apparatus? But we never see a superposition state of the atom plus apparatus. This is Schrödinger’s famous cat paradox. Arrange the experiment so that an “up” outcome triggers a mechanism that kills the cat, while a “down” outcome keeps the cat alive. Of course we don’t do this, but if we were to do it, we would always see a live cat OR a dead one, never a superposition of the two (Fig. 1.6.2). So we have the problem of definite outcomes: where does the “either”-“or” dichotomy arise?

We remind now some fundamental notions from probability theory.

**Definition 1.6.1.** In probability theory, the **sample space** (observation space) of an experiment or random trial is the set of all possible outcomes or results of that experiment. A sample space is usually denoted using set notation, and the possible outcomes are listed as elements in the set. It is common to refer to a sample space by the label Ω.

**Remark 1.6.3.** A well-defined sample space (observation space) is one of three basic elements in a probabilistic model (a probability space $\Theta = \{\Omega, \Sigma, \mathbf{P}\}$); the other two are a well-defined set of possible events (a sigma-algebra $\Sigma$) and a probability assigned to each event (a probability measure function $\mathbf{P}$).

**Remark 1.6.4.** An simple example of a sample phase space and corresponding probability space closer to our Stern-Gerlach experiment, is a coin toss. Consider 1000 coin tosses. If the coin is tossed without bias, you will find close to 500 heads and 500 tails, corresponding to $prob_{heads} = 0.5$ and $prob_{tails} = 0.5$. Here the sample space consists of the 1000 detailed trajectories of the toss, which your eye cannot follow,

<figure><img src="image_9.png" /><figcaption>Fig.1.6.3. An sample space.Trajectories in a coin toss.Adapted from [27].</figcaption></figure>

but which if analyzed by a very fast computer could predict which toss would give a head and which a tail (Fig.1.6. 3). Again, the probabilities are just reflections of our ignorance of the details, but the details are there. So we have the questions – are there hidden details underlying the probabilities in quantum mechanics? Is there a hidden sample space and corresponding probability space?

At a phenomenological level, there are very interesting models for the emergence of probabilities within the usual wave function formulation of nonrelativistic quantum theory, pioneered by Ghirardi, Rimini, and Weber [2]-[4]. These models postulate that space is filled with a very low level noise with a
coupling to matter proportional to the imaginary unit $i$, rather than with a
real-valued coupling (more technically, they couple through an anti-Hermitian
Hamiltonian term). For example, there could be a small, rapidly fluctuating
contribution to the gravitational potential or $g_{00}$ metric component proportional to
the imaginary unit $i$. If such a theory obeys two general properties, (1) the total
probability of a particle being present remains one for all times (that is, the wave
function normalization is preserved), and (2) there is no faster than light signaling,
then the extra terms in the Schrödinger equation equation must have a special
structure. This special structure allows one to prove definite outcomes obeying the
Born rule!

<figure><img src="image_10.png" /><figcaption>Fig.1.6.4. Different noise histories, in objective reduction models, can explain “up” and “down” registrations in the Stern-Gerlach experiment. Adapted from [27].</figcaption></figure>

In these models, for each repetition of the Stern–Gerlach experiment, the noise variable takes different values. For a large apparatus, these have a measurable effect, whereas for an atom not interacting with an apparatus, the effect is not measurable. The noise leads to different outcomes for different runs, with probabilities given by the Born rule. The different noises for different runs of the experiment are analogous, in the coin toss example I gave earlier, to different details of the tumbling coin trajectories for the different coin tosses (Fig. 1.6.4).

In the de Broglie-Bohm interpretation: a particle has an initial position and follows a path whose velocity at each instant is given by an classical equation. On the basis of this assumption we conduct a simulation experiment by drawing random initial positions of the electrons in the initial wave acket ("quantum equilibrium hypothesis").
<figure><img src="image_11.png" /><figcaption>Fig.1.6.5. An sample space in Bohmian QM.
100 electron trajectories for the
Jönsson experiment. Adapted from [55].</figcaption></figure>

Figure 1.6.5 shows, after its initial starting position, 100 possible quantum trajectories of an electron passing through one of the two slits: We have not represented the paths of the electron when it is stopped by the first screen. Figure 1.6.6 shows a close-up of these trajectories just after they leave their slits.

<figure><img src="image_12.png" /><figcaption>Fig.1.6.6. Close-up on the 100 trajectories of
the electrons just after the slits.
Adapted from [55].</figcaption></figure>

**Remark 1.6.5.** The different trajectories explain both the impact of electrons on the detection screen and the interference fringes. This is the simplest and most natural interpretation to explain the impact positions: "The position of an impact is simply the position of the particle at the time of impact." This was the view defended by Einstein at the Solvay Congress of 1927. **The position is the only measured variable of the experiment.**
<figure><img src="image_13.png" /><figcaption>Fig.1.6.6. Ten silver atom trajectories within initial spin orientation $\theta_0 = \pi/3$ and initial position $z_0$; arrows represent the spin orientation $\theta(z, t)$ along the trajectories. Adapted from [55].</figcaption></figure>

Figure 1.6.6. presents, for a silver atom with the initial spinor orientation ($\theta_0 = \pi/3, \phi_0 = 0$), a plot in the ($Oyz$) plane of a set of 10 trajectories whose initial position $z_0$ has been randomly chosen from a Gaussian distribution with standard deviation $\sigma_0$. The spin orientations $\theta(z,t)$ are represented by arrows

<figure><img src="image_14.png" /><figcaption>Fig.1.6.7. Ten silver atom trajectories where the initial orientation ($\theta_0, \phi_0$) has been randomly chosen; arrows represent the spin orientation $\theta(z,t)$ along the trajectories. Adapted from [55].</figcaption></figure>

Now let us consider a mixture of pure states where the initial orientation ($\theta_0, \phi_0$) from the spinor has been randomly chosen. These are the conditions of the initial Stern and Gerlach experiment. Figure 1.6.7 represents a simulation of 10 quantum trajectories of silver atoms from which the initial positions $z_0$ are also randomly chosen.

**Definition 1.6.2.** A probability space consists of three parts:
1. A sample space (observation space) $\Omega$, which is the set of all possible single outcomes

$$ \omega \in \Omega. $$

2. A set of events $\Sigma$, where each event is a set containing $\emptyset$ or more outcomes.

3. The assignment of probabilities to the events; that is, a function $P$ from events to probabilities.

An outcome is the result of a single execution of the model. Since individual outcomes might be of little practical use, more complex events are used to characterize groups of outcomes. The collection of all such events is a $\sigma$-algebra $\Sigma$. Finally, there is a need to specify each event's likelihood of happening. This is done using the probability measure function, $P : \Sigma \to [0, 1]$.

**Remark 1.6.6.** Note that:

(i) In conventional quantum mechanics we dealing with probabilities without any

$$ \text{probability space } \Theta = \{\Omega, \Sigma, P\}. $$

(ii) However a wave function $\psi$ in quantum mechanics is a description of the quantum state

$|\psi\rangle$ of a quantum system $\Xi$. The wave function is a complex-valued probability amplitude,

and the probabilities for the possible results of measurements of an observable

$Q = Q_\Xi$ (represented by oerator $\hat{Q}$) made on the system $\Xi$ in state $|\psi\rangle$ can be derived from

a wave function $\psi$.

(iii) From (ii) follows that there exist an probability space $\Theta_\Xi = \{\Omega_\Xi, \Sigma_\Xi, P_\Xi\}$ and random

variable $Q_{\hat{Q}|\psi\rangle} : \Omega_\Xi \to E_\Xi$, i.e. $X_{\hat{Q}|\psi\rangle}$ is a measurable function from the set of possible

outcomes $\Omega_\Xi$ to some set $E_\Xi$.

**Example 1.6.1.** We now, consider as an example, the simple case of a non-relativistic single particle, without spin, in one spatial dimension.

Note that:

(i) The state of such a particle is completely described by its position-space wave function, $\psi(x)$ where $x$ is position of a particle. This is a complex-valued function of real variable $x$. For one spinless particle in 1D, if the wave function is interpreted as a probability amplitude, the square modulus of the wave function, the positive real number

$$ |\psi(x)|^2 = \psi^*(x)\psi(x) = \rho(x) $$

is interpreted as the probability density that the particle is at $x$.

(ii) If the particle's position is measured, its location cannot be determined from the wave function, but is described by a probability distribution. The probability that its position $x$ will be in the interval $a \le x \le b$ is the integral of the density over this
interval:

$$P(a \le x \le b) = \int_{a}^{b} |\psi(x)|^2 dx.$$

This leads to the normalization condition

$$\int_{-\infty}^{\infty} |\psi(x)|^2 dx = 1$$

because if the particle is measured, there is 100% probability that it will be somewhere.

(iii) Assume that particle in state $|\psi\rangle$. From a statement (ii) follows that the coordinate $x$ of

the particle wave function, $\psi(x)$ under measurement by an measuring device is a random

variable $x_{\psi}(\omega) \triangleq X_{\hat{x}|\psi}(\omega), X_{\hat{x}|\psi}\Omega_{|\psi\rangle} \rightarrow E_{|\psi\rangle}$ which well defined on an probability

space

$$\Theta_{|\psi\rangle} = \{\Omega_{|\psi\rangle}, \Sigma_{|\psi\rangle}, \mathbf{P}\}.$$

(iv) However in conventional quantum mechanics as mentioned above [see Remark

1.6.6(i)] such probability space $\Theta_{|\psi\rangle} = \{\Omega_{|\psi\rangle}, \Sigma_{|\psi\rangle}, \mathbf{P}\}$ is missing.

**Remark 1.6.7.** For a given system, the set of all possible normalizable wave functions (at any given time) forms an abstract mathematical vector space, meaning that it is possible to add together different wave functions, and multiply wave functions by complex numbers.

Note that:

(i) Technically, because of the normalization condition, wave functions form a **projective space** $H_p$ rather than an ordinary infinite-dimensional vector space **H**. Also **H** is a Hilbert space, because the inner product of two wave functions $\psi_1$ and $\psi_2$ can be defined as the complex number

$$(\psi_1, \psi_2) = \int_{-\infty}^{\infty} \psi_1^*(x) \psi_2(x) dx.$$

(ii) $H_p = S^\infty \subsetneq H$.

(iii) The all values of the wave function $\psi(x)$ are components of an vector $|\psi\rangle$. There are uncountably infinitely many of them and integration is used in place of summation. In Bra-ket notation, this vector is written

$$|\psi\rangle = \int_{-\infty}^{\infty} dx \psi(x) |x\rangle,$$

where $\langle x'|x\rangle = \delta(x' - x)$.

Let us consider QM system which consists of one particle with a wave function $\psi(x)$, $x \in [a, b]$, such that $\text{supp}(\psi(x)) \subseteq [a, b]$ and $\int_{\mathbb{R}} |\psi(x)|^2 dx = 1$. We go to construct now corresponding probability space $\Theta_{|\psi\rangle} = \{\Omega_{|\psi\rangle}, \Sigma_{|\psi\rangle}, \mathbf{P}\}$. In one dimension, the position $x$ of a such particle can range over the values $a \le x \le b$.
Consider now measurement of coordinate of such QM particle. Obviously a sample space for such coordinate measurement is $\Omega_{|\psi\rangle} = \Omega_{a,b} = [a,b]$. Note that in practice observable $x$ is measured to an accuracy $\delta x$ determined by the measuring device. Thus $\forall x \forall \delta x_1 \forall \delta x_2 [(x - \delta x_1, x + \delta x_2) \subseteq [a,b] \rightarrow (x - \delta x_1, x + \delta x_2) \in \Sigma_{|\psi\rangle}]$ and therefore $\sigma$-algebra $\Sigma_{a,b} = B([a,b])$ is the Borel algebra on the set $[a,b]$. The probability measure function, $\mathbf{P}: \Sigma_{a,b} \rightarrow [0,1]$ we choose of the form

$$ \mathbf{P}(A) = \int_A \sigma(x) d\mu(x), \qquad (1.6.3) $$

where $A \in \Sigma_{a,b}$ and $d\mu(x)$ is the Lebesgue measure.

**Definition 1.6.3.** The probability measure $\mathbf{P}_{|\psi\rangle}: \Sigma_{a,b} \rightarrow [0,1]$ corresponding to a wave function $\psi(x), \|\psi(x)\|_2^2 = 1$, we choose in the following form:

$$ \mathbf{P}_{|\psi\rangle}(A) = \int_A |\psi(x)|^2 d\mu(x), \qquad (1.6.4) $$

where $A \in \Sigma_{a,b}$ and $d\mu(x)$ is the Lebesgue measure.

**Definition 1.6.4.** A random variable $X_{|\psi\rangle}: \Omega_{|\psi\rangle} \rightarrow E_{|\psi\rangle}$ is a measurable function from the set of possible outcomes $\Omega$ to some set $E_{|\psi\rangle}$. The technical axiomatic definition requires $\Omega_{|\psi\rangle}$ to be a probability space and $E_{|\psi\rangle}$ to be a measurable space. Note that although $X_{|\psi\rangle}$ is usually a real-valued function $X_{|\psi\rangle}: \Omega_{|\psi\rangle} \rightarrow [a,b]$, it does not return a probability. The probabilities of different outcomes or sets of outcomes (events) in our case are already given by the probability measure $\mathbf{P}_{|\psi\rangle}$ with which $\Omega_{|\psi\rangle}$ is equipped above.

**Definition 1.6.5.** (Real-valued random variables) In a case mentioned above the observation space is a set $[a,b]$. Recall, $\{\Omega_{a,b}, \Sigma_{a,b}, \mathbf{P}\}$ is the probability space. For real observation space, the function $X_{|\psi\rangle}: \Omega_{a,b} \rightarrow [a,b]$ is a real-valued random variable, i.e. $\forall r[\{\omega: X_{|\psi\rangle}(\omega) \le r\} \in \Sigma_{a,b}]$.

**Assumption 1.6.1.** We assume now that $\mathbf{P}_{|\psi\rangle} \ll \mathbf{P}$, i.e. $\mathbf{P}_{|\psi\rangle}$ is absolutely continuous with respect to $\mathbf{P}$. Then by Radon-Nicodym theorem we obtain for any $A \in \Sigma_{a,b}$:

$$ \begin{align} \mathbf{P}_{|\psi\rangle}(A) &= \int_{A^+} X_{|\psi\rangle}^+(\omega) d\mathbf{P} + \int_{A^-} X_{|\psi\rangle}^-(\omega) d\mathbf{P} = \mathbf{P}_{|\psi\rangle}^+(A) + \mathbf{P}_{|\psi\rangle}^-(A), \notag \\ \mathbf{P}_{|\psi\rangle}^+(A) &= \int_{A^+} X_{|\psi\rangle}^+(\omega) d\mathbf{P}, \mathbf{P}_{|\psi\rangle}^-(A) = \int_{A^-} X_{|\psi\rangle}^-(\omega) d\mathbf{P}, \tag{1.6.5.a} \end{align} $$

$$ A^+ = A \cap [0, +\infty], A^- = A \cap [-\infty, 0), $$

$$ X_{|\psi\rangle}^{+}(\omega) = \frac{d\mathbf{P}_{|\psi\rangle}^{+}}{d\mathbf{P}}, X_{|\psi\rangle}^{-}(\omega) = \frac{d\mathbf{P}_{|\psi\rangle}^{-}}{d\mathbf{P}}. $$

Using (1.6.5.a) we define random variable $X_{|\psi\rangle}: \Omega_{|\psi\rangle} \rightarrow [a,b]$ by formula

$$ X_{|\psi\rangle} = X_{|\psi\rangle}^{+}(\omega) - X_{|\psi\rangle}^{-}(\omega). \qquad (1.6.5.b) $$

**Definition 1.6.6.** The cumulative distribution function of a real-valued random variable
$X_{|\psi\rangle}(\omega)$ is the function given by

$$F_{X_{|\psi\rangle}}(x) = \mathbf{P}(\omega \in \Omega_{a,b} | X_{|\psi\rangle}(\omega) \le x), \qquad (1.6.6)$$

where the right-hand side represents the probability that the random variable $X_{|\psi\rangle}(\omega)$ takes on a value less than or equal to $x$. The probability that $X_{|\psi\rangle}(\omega)$ lies in the semi-closed interval $(a_1, b_1] \subset [a, b]$, where $a_1 < b_1$, is therefore
$$\mathbf{P}_{|\psi\rangle}(a_1 < X_{|\psi\rangle} \le b_1) = F_{X_{|\psi\rangle}}(b_1) - F_{X_{|\psi\rangle}}(a_1).$$

We remind that:

(i) The CDF of any continuous random variable $X_{|\psi\rangle} = X_{|\psi\rangle}(\omega)$ can be expressed as the

integral of its probability density function $p_{X_{|\psi\rangle}}(x)$ as follows:

$$F_{X_{|\psi\rangle}}(x) = \int_{-\infty}^{x} p_{X_{|\psi\rangle}}(t)dt = \int_{-\infty}^{x} |\psi(t)|^2 dt. \qquad (1.6.7.a)$$

(ii) In the case of any random variable $X_{|\psi\rangle}$ which has distribution having a discrete component at a value $b$,

$$\mathbf{P}(\{X_{|\psi\rangle} = b\}) = F_{X_{|\psi\rangle}}(b) - \lim_{x \to b^{-}} F_{X_{|\psi\rangle}}(x). \qquad (1.6.7.b)$$

(iii) Every cumulative distribution function $F_{X_{|\psi\rangle}}(x)$ is non-decreasing and right-continuous,

which makes it a càdlàg function.

(iv)

$$\lim_{x \to -\infty} F_{X_{|\psi\rangle}}(x) = 0, \lim_{x \to +\infty} F_{X_{|\psi\rangle}}(x) = 1. \qquad (1.6.7.c)$$

The next result from probability theory it well known.

**Theorem 1.6.1.** (i) Every function $F_{X_{|\psi\rangle}}(x)$ with these properties (i)-(iv) is a CDF, i.e., for every such function, a random variable can be defined such that the function is the cumulative distribution function of that random variable.

(ii) If $X_{|\psi\rangle}$ is a purely discrete random variable, then it attains values $x_1, x_2, \dots, x_i, \dots$ with probability $p_i = \mathbf{P}(\{X_{|\psi\rangle} = x_i\})$, and the CDF of $X^B$ will be discontinuous at the points $x_i$ and constant in between

$$F_{X_{|\psi\rangle}}(x) = \mathbf{P}(X_{|\psi\rangle} \le x) = \sum_{x_i \le x} \mathbf{P}(X_{|\psi\rangle} = x_i) = \sum_{x_i \le x} p_{X_{|\psi\rangle}}(x_i) \qquad (1.6.7.d)$$

(iii) If the CDF $F_{X_{|\psi\rangle}}(x)$ of a real valued random variable $X_{|\psi\rangle}$ is continuous, then $X_{|\psi\rangle}$ is a continuous random variable.

(iv) If furthermore $F_{X_{|\psi\rangle}}(x)$ is absolutely continuous, then there exists a Lebesgue-integrable function $f(x)$ such that

$$F_{X_{|\psi\rangle}}(x) = \int_{-\infty}^{x} f(t)dt. \qquad (1.6.7.e)$$

In that caseы when $\neg(\mathbf{P}_{|\psi\rangle} \ll \mathbf{P})$ we define random variable $X_{|\psi\rangle} = X_{|\psi\rangle}(\omega)$ by using Theorem 1.6.1. Thus from Eq.(1.6.7.a) we obtain
$$ E_{\Omega_{a,b}}[X_{|\psi\rangle}(\omega)] = \int_{\Omega_{a,b}} X_{|\psi\rangle}(\omega) d\mathbf{P} = \int_{-\infty}^{+\infty} xp_{X_{|\psi\rangle}}(x)dx. \quad (1.6.8) $$

Using canonical QM-abbreviations (see subsection I.7.2)

$$ |\psi\rangle = \int_{-\infty}^{+\infty} |x\rangle\langle x|\psi\rangle dx, \quad (1.6.9) $$

where $\langle x|\psi\rangle = \psi(x), |\psi\rangle \in S^{\infty} \not\subseteq \mathbf{H}$, from Eq.(1.6.8)-Eq.(1.6.9) we obtain

$$ \langle \psi | \hat{x} | \psi \rangle = \int_{\Omega_{a,b}} X_{|\psi\rangle}(\omega) d\mathbf{P} = \int_{-\infty}^{+\infty} xp_{X_{|\psi\rangle}}(x) dx. \quad (1.6.10) $$

**Assumption 1.6.2.** We assume now that:

(i) for any $\psi(x) \in \mathbf{H}$: (a) $\int_{\Omega_{a,b}} |X_{|\psi\rangle}|(\omega)d\mathbf{P} < \infty$, (b) $\int_{\Omega_{a,b}} X_{|\psi\rangle}^2(\omega)d\mathbf{P} < \infty$,

(ii) for any $\psi(x) \in \mathbf{H}: X_{|\psi\rangle} \in \mathcal{L}_{1,2}(d\mathbf{P}) = \mathcal{L}_1(d\mathbf{P}) \cap \mathcal{L}_2(d\mathbf{P})$.

**Definition 1.6.7.** We will write the Eq.(1.6.10) in the following form

$$ \langle \psi | \hat{x} | \psi \rangle = \int_{\Omega_{a,b}} X_{\hat{x}|\psi\rangle}(\omega) d\mathbf{P} = \int_{-\infty}^{+\infty} xp_{X_{\hat{x}|\psi\rangle}}(x) dx, \quad (1.6.11) $$

where $\hat{x}|\psi\rangle = x|\psi\rangle$. This form remind that continuous random variable $X_{|\psi\rangle} = X_{|\psi\rangle}(\omega)$ corresponds to measurement of the coordinate of an particle with a state vector $|\psi\rangle$.

**Remark 1.6.8.** If $X_{|\psi\rangle} \in \mathcal{L}_{1,2}(d\mathbf{P})$ then for any $\eta \in \mathbb{R}_+ : \eta X_{|\psi\rangle} \in \mathcal{L}_{1,2}(d\mathbf{P})$. From RHS of the Eq.(1.6.10) by change of random variable: $Y(\omega) = \eta X_{|\psi\rangle}(\omega)$ we obtain

$$ \int_{\Omega_{a,b}} \eta X_{|\psi\rangle}(\omega) d\mathbf{P} = \int_{\Omega_{a,b}} Y(\omega)(\omega) d\mathbf{P} = \int_{-\infty}^{+\infty} y \eta^{-1} p_{X_{|\psi\rangle}}(\eta^{-1}y) dy. \quad (1.6.12) $$

From LHS of the Eq.(1.6.10) and Eq.(1.6.12) for any $\xi \in \mathbb{C}\setminus\{0\}$, we obtain

$$ \langle \xi \psi | \hat{x} | \xi \psi \rangle = \int_{\Omega_{a,b}} |\xi|^2 X_{|\psi\rangle}(\omega) d\mathbf{P} = \int_{\Omega_{a,b}} \eta X_{|\psi\rangle}(\omega) d\mathbf{P} = \int_{-\infty}^{+\infty} y \eta^{-1} p_{X_{|\psi\rangle}}(\eta^{-1}y) dy, \quad (1.6.13) $$

where $\eta = |\xi|^2$.

**Remark 1.6.9.** Formula (1.6.13) allow us to change conventional projective Hilbert space $\mathbf{H}_p = S^{\infty} \not\subseteq \mathbf{H}$ (see Remark 1.6.7) by full nonprojective Hilbert space $\mathbf{H}$.

**Definition 1.6.8.** The probability measure $P_{\xi|\psi\rangle}: \Sigma_{a,b} \to [0,1]$ corresponding to a wave function $\xi\psi(x)$, where $|\xi|^2 \neq 1$, $\|\psi(x)\|_2^2 = 1$, we choose in the following form:

$$ P_{\xi|\psi\rangle}(A) = \int_A \eta^{-1} |\psi(\eta^{-1}y)|^2 d\mu(y), \quad (1.6.14) $$

where $\eta = |\xi|^2, A \in \Sigma_{a,b}$ and $d\mu(x)$ is the Lebesgue measure.

**Definition 1.6.9.** (Real-valued random variables) In this case the observation space is a set $[a, b]$. Recall, $\{\Omega_{a,b}, \Sigma_{a,b}, \mathbf{P}\}$ is the probability space. For real observation space, the function $X_{\xi|\psi\rangle}: \Omega_{a,b} \to [a', b']$ is a real-valued random
variable if $\forall r\{\omega : X_{\xi|\psi}(\omega) \le r\} \in \Sigma_{a,b}$.

By Radon-Nicodym theorem we obtain for $A^{\pm} \in \Sigma_{a,b}$:

$$ 
\begin{aligned}
\mathbf{P}_{\xi|\psi}^{\pm}(A) &= \int_{A^{\pm}} X_{\xi|\psi}^{\pm}(\omega) d\mathbf{P}, \\
X_{\xi|\psi}^{\pm}(\omega) &= \frac{d\mathbf{P}_{\xi|\psi}^{\pm}}{d\mathbf{P}}.
\end{aligned}
\qquad (1.6.15) $$

The CDF of a continuous random variable $X_{\xi|\psi} = X_{\xi|\psi}(\omega)$ can be expressed as the integral of its probability density function $p_{X_{\xi|\psi}}(y)$ as follows:

$$ F_{X_{\xi|\psi}}(y) = \int_{-\infty}^{y} p_{X_{\xi|\psi}}(t)dt. \qquad (1.6.16) $$

From Eq.(1.6.14)-Eq.(1.6.16) we obtain

$$ \langle \xi\psi | \hat{x} | \xi\psi \rangle = \mathbf{E}_{\Omega_{a,b}}[X_{\xi|\psi}(\omega)] = \int_{\Omega_{a,b}} X_{\xi|\psi}(\omega) d\mathbf{P} = \int_{-\infty}^{+\infty} y p_{X_{\xi|\psi}}(y) dy = \qquad (1.6.17) $$

$$ \int_{-\infty}^{+\infty} \eta^{-1}|\psi(\eta^{-1}y)|^2 y d\mu(y) = \int_{-\infty}^{+\infty} y \eta^{-1} p_{X_{\psi}(\eta^{-1}y)} dy = \int_{\Omega_{a,b}} \eta^{-1} X_{\psi}(\omega) d\mathbf{P} = \int_{\Omega_{a,b}} |\xi|^2 X_{\psi}(\omega) d\mathbf{P}. $$

**Definition 1.6.10.** We will write the Eq.(1.6.17) in the following symbolical form

$$ \langle \xi \psi | \hat{x} | \xi \psi \rangle = \int_{\Omega_{a,b}} X_{\hat{x}|\xi\psi}(\omega) d\mathbf{P} = \int_{-\infty}^{+\infty} x p_{X_{\hat{x}|\xi\psi}}(x) dx = \qquad (1.6.18) $$

$$ \int_{\Omega_{a,b}} X_{\hat{x}|\psi^{\xi}}(\omega) d\mathbf{P} = \int_{-\infty}^{+\infty} x p_{X_{\hat{x}|\psi^{\xi}}}(x) dx. $$

This symbolical form remind that continuous random variable

$X_{\xi\psi} = X_{\xi\psi}(\omega) \triangleq X_{\psi\xi}(\omega)$ corresponds to measurement of the coordinate of an particle with a wave function $\psi(x)$.

## The reconciled Bohr rule.

We choose now an constants $\xi_1, \xi_2, |\xi_1| \ne 1, |\xi_2| \ne 1$, and an wave functions $\psi_1(x)$ and $\psi_2(x)$ such that:

(i) $\|\psi_1(x)\|_2^2 = 1, \|\psi_2(x)\|_2^2 = 1,$

(ii) $\mathrm{supp}(\psi_1(x)) \subset [a_1, b_1], \mathrm{supp}(\psi_2(x)) \subset [a_2, b_2],$

(iii) $[a_1, b_1] \cap [a_2, b_2] = \emptyset$

(iv) we assum that: if $\psi_{1,2}(x_1, x_2)$ is a two particles wave function, then

$$ \psi_{1,2}(x_1, x_2) = \psi_1(x_1)\psi_2(x_2). $$

Therefore we obtain

$$ \langle \xi_1 \psi_1 + \xi_2 \psi_2 | \hat{x} | \xi_1 \psi_1 + \xi_2 \psi_2 \rangle = \langle \xi_1 \psi_1 | \hat{x} | \xi_1 \psi_1 \rangle + \langle \xi_2 \psi_2 | \hat{x} | \xi_2 \psi_2 \rangle. \qquad (1.6.17) $$
Substituting Eq.(1.6.16) into Eq.(1.6.17) gives

$$
\begin{align}
\langle \xi_1 \psi_1 + \xi_2 \psi_2 | \hat{x} | \xi_1 \psi_1 + \xi_2 \psi_2 \rangle &= \langle \xi_1 \psi_1 | \hat{x} | \xi_1 \psi_1 \rangle + \langle \xi_2 \psi_2 | \hat{x} | \xi_2 \psi_2 \rangle = \nonumber \\
&\quad \int_{\Omega_{a_1,b_1}} X_{\xi_1|\psi_1}(\omega) d\mathbf{P} + \int_{\Omega_{a_2,b_2}} X_{\xi_2|\psi_2}(\omega) d\mathbf{P} = \nonumber \\
&\quad \int_{\Omega_{a_1,b_1} \bigcup \Omega_{a_2,b_2}} [X_{\xi_1|\psi_1}(\omega) + X_{\xi_2|\psi_2}(\omega)] d\mathbf{P} = \tag{1.6.18} \\
&\quad \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} x [p_{X_{\hat{x}|\xi_1\psi_1}}(x) + p_{X_{\hat{x}|\xi_2\psi_1}}(x)] dx = \int_{-\infty}^{+\infty} x [p_{X_{\hat{x}|\xi_1\psi_1}}(x) * p_{X_{\hat{x}|\xi_2\psi_1}}(x)] dx = \nonumber \\
&\quad \int_{-\infty}^{+\infty} y \eta_1^{-1} \eta_2^{-1} [|\psi_1(\eta_1^{-1}y)|^2 * |\psi_2(\eta_2^{-1}y)|^2] dy, \nonumber
\end{align}
$$

where $\eta_1 = |\xi_1|^2$, $\eta_2 = |\xi_2|^2$. From Eq.(1.6.18) follows that if particle in a state $|\xi_1\psi_1 + \xi_2\psi_2\rangle$ then the probability density $p(x)$ that the particle is at $x$ is given by formula

$$p(x) = \eta_1^{-1} \eta_2^{-1} (|\psi_1(\eta_1^{-1}x)|^2 * |\psi_2(\eta_2^{-1}x)|^2) \quad (1.6.19)$$

but not by conventional formula

$$p(x) = |\xi_1|^2 |\psi_1|^2 + |\xi_2|^2 |\psi_2|^2 \quad (1.6.20)$$

even if $|\xi_1|^2 + |\xi_2|^2 = 1$.

**Remark 1.6.10.** Obviously formula (1.6.19) forced us to changed not only conventional

Born's rule (1.6.2) but even conventional interpretation of the wave function of the QM

particle must be changed if particle consist of superposition such that mentioned above.

**Remark 1.6.11.** An canonical quantum mechanical formalism based on the probability representation of states was also proposed S.Mancini and V.I.Man'ko in [28]-[37]. Note that the distribution $w(X, \mu, v)$ of the observable $\hat{X} = \mu\hat{q} + v\hat{p}$ one can expressed by the relation [32]:

$$w(X, \mu, v) = (2\pi)^{-2} \int dk dq dp W(q,p) \exp[-ik(X - \mu q - v p)], \quad (1.6.21)$$

where $W(q,p)$ that is a Wigner function, operators $\hat{q}$ and $\hat{p}$ that is operator of coordinate and momentum operator correspondingly. Note that [32]:

$$W(q,p) = (2\pi)^{-2} \int d\mu dv dX w(X, \mu, v) \exp[-ik(\mu q + v p - X)]. \quad (1.6.22)$$

Because the Wigner function completely determains the quantum system and the distribution $w(X, \mu, v)$ completely determains the Wigner function $W(q,p)$ the classical disribution of componets completely determains the quantum system. In fact we can say that the quantum system states is known if the distribution
$w(X, \mu, \nu)$ is known.

**Remark 1.6.12.** An general quantum mechanical formalism based on the probability representation of states was also proposed by A.Yu.Khrennikov [37].In [37] the main structures of quantum theory (interference of probabilities, Born's rule, complex probabilistic amplitudes, Hilbert state space, representation of observables by operators) are present in a latent form in the Kolmogorov model. In particular, “interference of probabilities” is obtained without appealing to the Hilbert space formalism.

However in this paper we dealing by using axiomatical approach, see section I.7 below.We have to combined canonical QM approach based on a Hilbert state space, representation of observables by operators, etc. and Kolmogorov probabilistic model which completed canonical QM postulates.

# I.7. A new quantum mechanical formalism based on the probability representation of quantum states.

## I.7.1. Generalized Postulates for Continuous Valued Observables.

Suppose we have an $n$-dimensional physical quantum system.

I. Then we claim the following:

**Q.I.1.** Any given $n$-dimensional quantum system is identified by a set $Q$ :

$$Q \triangleq \langle \mathbf{H}, \mathfrak{J}, \mathfrak{R}, \mathcal{L}_{2,1}, \mathfrak{T}^*(\mathbf{H}), \mathbf{G}, |\psi_t\rangle \rangle$$

where:

(i) $\mathbf{H}$ that is some infinite-dimensional complex Hilbert space,

(ii) $\mathfrak{J} = (\Omega, \mathcal{F}, \mathbf{P})$ that is complete probability space,

(iii) $\mathfrak{R} = (\mathbb{R}^n, \Sigma)$ that is measurable space,

(iv) $\mathcal{L}_{2,1}(\Omega)$ that is complete space of random variables $X : \Omega \to \mathbb{R}^n$ such that

$$\int_{\Omega} \|X(\omega)\| d\mathbf{P} < \infty, \quad \int_{\Omega} \|X(\omega)\|^2 d\mathbf{P} < \infty \qquad (1.7.1)$$

(v) $\mathbf{G} : \mathbf{C}^*(\mathbf{H}) \times \mathbf{H} \to \mathcal{L}_{2,1}(\Omega)$ that is one to one correspondence such that

$$\begin{gathered} |\langle\psi|\hat{Q}|\psi\rangle| = \int_{\Omega} (\mathbf{G}[\hat{Q}, |\psi\rangle](\omega)) d\mathbf{P} = \mathbf{E}_{\Omega}(\mathbf{G}[\hat{Q}|\psi\rangle](\omega)), \qquad (1.7.2) \\ \mathbf{G}[\hat{\mathbf{1}}, |\psi\rangle](\omega) = 1 \end{gathered}$$

for any $|\psi\rangle \in \mathbf{H}$ and for any Hermitian adjoint operator

$\hat{Q} : \mathbf{H} \to \mathbf{H}, \hat{Q} \in \mathbf{C}^*(\mathbf{H})$, where $\mathbf{C}^*(\mathbf{H})$ is $\mathbf{C}^*$-algebra of the Hermitian adjoint operators in $\mathbf{H}$ and $\mathfrak{T}^*(\mathbf{H})$ an commutative subalgebra of $\mathbf{C}^*(\mathbf{H})$.

(vi) $|\psi_t\rangle$ is a continuous vector function $|\psi_t\rangle : \mathbb{R}_+ \to \mathbf{H}$ which represented the evolution of the quantum system $Q$.

**Q.I.2.** For any $|\psi_1\rangle, |\psi_2\rangle \in \mathbf{H}$ and for any Hermitian operator $\hat{Q} : \mathbf{H} \to \mathbf{H}$ such that
$$ \langle \psi_1 | \hat{Q} | \psi_2 \rangle = \langle \psi_2 | \hat{Q} | \psi_1 \rangle = 0 \qquad (1.7.3) $$

valid the equality

$$ \mathbf{G}[\hat{Q}(|\psi_1\rangle + |\psi_2\rangle)](\omega) = \mathbf{G}[\hat{Q}|\psi_1\rangle](\omega) + \mathbf{G}[\hat{Q}|\psi_2\rangle](\omega). \qquad (1.7.4) $$

**Definition 1.7.1.** A random variable $X: \Omega \to E$ is a measurable function from the set of

possible outcomes $\Omega$ to some set $E$.

**Definition 1.7.2.** Given a probability space $\mathfrak{F} = (\Omega, \mathcal{F}, \mathbf{P})$ and a measurable space $\Re = (\mathbb{R}^n, \Sigma)$, any $\mathbb{R}^n$-valued stochastic process that is a collection of $\mathbb{R}^n$-valued random variables on $\Omega$, indexed by a totally ordered set $T$ ("time"). That is, a stochastic process $X_t(\omega)$ is a collection $\{X_t(\omega)|t \in T\}$, where each $X_t(\omega)$ is an $\mathbb{R}^n$-valued random variable on $\Omega$. The space $\mathbb{R}^n$ is then called the state space of the process.

**Q.I.3.** Suppose that the evolution of the quantum system is represented by continuous vector function $|\psi_t\rangle: \mathbb{R}_+ \to \mathbf{H}$. Then any process of continuous measurements on measuring observable $\hat{Q}$ for the system in state $|\psi_t\rangle$ one can to describe by an continuous $\mathbb{R}^n$-valued stochastic process

$$ X_t(\omega) = X_t\left(\omega; |\hat{Q}\psi_t\rangle\right) \triangleq X_{|\hat{Q}\psi_t\rangle}(\omega) $$

given on probability space $(\Omega, \mathcal{F}, \mathbf{P})$ and a measurable space $(\mathbb{R}^n, \Sigma)$.

**Remark 1.7.1.** We assume now for short but without loss of generality that $n = 1$.

**Remark 1.7.2.** Let $X(\omega)$ be random variable $X(\omega) \in \mathcal{L}_{2,1}(\Omega)$ such that $X(\omega) = \mathbf{G}[|\psi\rangle](\omega)$, then we denote such random variable by $X_{|\psi\rangle}(\omega)$. The probability density of random variable $X_{|\psi\rangle}(\omega)$ we denote by $p_{|\psi\rangle}(q), q \in \mathbb{R}$.

**Definition 1.7.3.** The classical pure states correspond to vectors $v \in \mathbf{H}$ of norm $\|v\|=1$. Thus the set of all classical pure states corresponds to the unit sphere $\mathbb{S}^\infty \subset \mathbf{H}$ in a Hilbert space $\mathbf{H}$.

**Definition 1.7.4.** The projective Hilbert space $P(\mathbf{H})$ of a complex Hilbert space $\mathbf{H}$ is the set of equivalence classes $[v]$ of vectors $v$ in $\mathbf{H}$, with $v \neq 0$, for the equivalence relation given by $v \sim_P w \Leftrightarrow v = \lambda w$ for some non-zero complex number $\lambda \in \mathbb{C}$. The equivalence classes for the relation $\sim_P$ are also called rays or projective rays.

**Remark 1.7.3.** The physical significance of the projective Hilbert space $P(\mathbf{H})$ is that in canonical quantum theory, the states $|\psi\rangle$ and $\lambda|\psi\rangle$ represent the same physical state of the quantum system, for any $\lambda \neq 0$. It is conventional to choose a state $|\psi\rangle$ from the ray $[|\psi\rangle]$ so that it has unit norm $\sqrt{\langle\psi|\psi\rangle} = 1$.

**Remark 1.7.4.** In contrast with canonical quantum theory we have used instead contrary to $\sim_P$ equivalence relation $\sim_Q$, a Hilbert space $\mathbf{H}$, see Definition 1.7.7.

**Definition 1.7.5.** The non-classical pure states correspond to the vectors $v \in \mathbf{H}$ of a norm $\|v\|\neq 1$. Thus the set of all non-classical pure states corresponds to the set $\mathbf{H}\setminus\mathbb{S}^\infty \subset \mathbf{H}$ in the Hilbert space $\mathbf{H}$.
Suppose we have an observable $\mathcal{Q}$ of a quantum system that is found through an exhaustive series of measurements, to have a set $\mathfrak{T}$ of values $q \in \mathfrak{T}$ such that $\mathfrak{T} = \bigcup_{i=1}^{m} (\theta_1^i, \theta_2^i), m \ge 2$, $(\theta_1^i, \theta_2^i) \cap (\theta_1^j, \theta_2^j) = \emptyset, i \ne j$. Note that in practice any observable $\mathcal{Q}$ is measured to an accuracy $\delta q$ determined by the measuring device. We represent now by $|q\rangle$ the idealized state of the system in the limit $\delta q \to 0$, for which the observable definitely has the value $q$.

**II. Then we claim the following:**

**Q.II.1.** The states $\{|q\rangle : q \in \mathfrak{T}\}$ form a complete set of $\delta$-function normalized basis states

for the state space $\mathbf{H}_{\mathfrak{T}}$ of the system. That the states $\{|q\rangle : q \in \mathfrak{T}\}$ form a complete set of

basis states means that any state $|\psi[\mathfrak{T}]\rangle \in \mathbf{H}_{\mathfrak{T}}$ of the system can be expressed as:

$$|\psi[\mathfrak{T}]\rangle = \int_{\mathfrak{T}} c_{\psi[\mathfrak{T}]}(q)dq, \qquad (1.7.5)$$

where $\text{supp}(c_{\psi[\mathfrak{T}]}) \subseteq \mathfrak{T}$ and while $\delta$-function normalized means that $\langle q|q'\rangle = \delta(q - q')$

from which follows $c_{\psi[\mathfrak{T}]}(q) = \langle q|\psi[\mathfrak{T}]\rangle$ so that

$$|\psi[\mathfrak{T}]\rangle = \int_{\mathfrak{T}} |q\rangle\langle q|\psi[\mathfrak{T}]\rangle dq. \qquad (1.7.6)$$

The completeness condition can then be written as

$$\int_{\mathfrak{T}} |q\rangle\langle q| dq = \hat{1}_{\mathbf{H}_{\mathfrak{T}}}. \qquad (1.7.7)$$

**Q.II.2.** For the system in state $|\psi[\mathfrak{T}]\rangle$ the probability $P(q, q + dq; |\psi[\mathfrak{T}]\rangle)$ of obtaining the result

$q \in \mathfrak{T}$ lying in the range $(q, q + dq) \subset \mathfrak{T}$ on measuring observable $\mathcal{Q}$ is given by

$$P(q, q + dq; |\psi[\mathfrak{T}]\rangle) = p_{|\psi[\mathfrak{T}]\rangle}(q) dq \qquad (1.7.8)$$

for any $|\psi[\mathfrak{T}]\rangle \in \mathbf{H}_{\mathfrak{T}}$.

**Remark 1.7.5.** Note that in general case $p_{|\psi[\mathfrak{T}]\rangle}(q) \ne |c_{\psi[\mathfrak{T}]}(q)|^2$.

**Q.II.3.** The observable $\mathcal{Q}_{\mathfrak{T}}$ is represented by a Hermitian operator $\hat{\mathcal{Q}}_{\mathfrak{T}} : \mathbf{H}_{\mathfrak{T}} \to \mathbf{H}_{\mathfrak{T}}$ whose eigenvalues are the possible results $\{q : q \in \mathfrak{T}\}$, of a measurement of $\mathcal{Q}_{\mathfrak{T}}$, and the associated eigenstates are the states $\{|q\rangle : q \in \mathfrak{T}\}$, i.e. $\hat{\mathcal{Q}}_{\mathfrak{T}}|q\rangle = q|q\rangle, q \in \mathfrak{T}$.

**Remark 1.7.6.** Note that the spectral decomposition of the operator $\hat{\mathcal{Q}}_{\mathfrak{T}}$ is then

$$\hat{\mathcal{Q}}_{\mathfrak{T}} = \int_{\mathfrak{T}} q|q\rangle\langle q| dq. \qquad (1.7.9)$$

**Definition 1.7.6.** A connected set in $\mathbb{R}$ is a set $X \subset \mathbb{R}$ that cannot be partitioned into two nonempty subsets which are open in the relative topology induced on the set. Equivalently, it is a set which cannot be partitioned into two nonempty subsets such that each subset has no points in common with the set closure of the other.

**Definition 1.7.7.** The well localized pure states $|\psi[\Theta]\rangle$ with a support $\Theta = (\theta_1, \theta_2)$ correspond to vectors of the norm 1 and such that: $\text{supp}(c_{\psi[\Theta]}(q)) = \Theta$ is a connected set in $\mathbb{R}$. Thus the set of all well localized pure states corresponds to the unit sphere $S_{\Theta}^{\infty} \subsetneq S^{\infty} \subset \mathbf{H}$ in the Hilbert space $\mathbf{H}_{\Theta} \subsetneq \mathbf{H}$.
Suppose we have an observable $Q_\Theta$ of a system that is found through an exhaustive series of measurements, to have a continuous range of values $q$ :
$\theta_1 < q < \theta_2$.

**III. Then we claim the following:**

**Q.III.1.** For the system in well localized pure state $|\psi[\Theta]\rangle$ such that:

(i) $|\psi[\Theta]\rangle \in S_\Theta^\infty$ and

(ii) $\text{supp}(c_{\psi[\Theta]}(q)) \triangleq \{q | c_{\psi[\Theta]}(q) \neq 0\}$ is a connected set in $\mathbb{R}$, then the probability

$P(q, q + dq; |\psi[\Theta]\rangle)$ of obtaining the result $q$ lying in the range $(q, q + dq)$ on measuring

observable $Q_\Theta$ is given by

$$P(q, q + dq; |\psi[\Theta]\rangle) = |\langle q | \psi[\Theta] \rangle|^2 dq = |c_{\psi[\Theta]}(q)|^2 dq. \quad (1.7.10)$$

**Q.III.2.** $p_{|\psi[\Theta]\rangle}(q)dq = |\langle q | \psi[\Theta] \rangle|^2 dq = |c_{\psi[\Theta]}(q)|^2 dq$.

**Q.III.3.** Let $|\psi[\Theta_1]\rangle$ and $|\psi[\Theta_2]\rangle$ be well localized pure states with $\Theta_1 = (\theta_1^1, \theta_2^1)$ and

$\Theta_2 = (\theta_1^2, \theta_2^2)$ correspondingly. Let $X_1(\omega) = X_{|\psi[\Theta_1]\rangle}(\omega)$ and $X_2(\omega) = X_{|\psi[\Theta_2]\rangle}(\omega)$

correspondingly. Assume that $\bar{\Theta}_1 \cap \bar{\Theta}_2 = \emptyset$ (here the closure of $\Theta_i, i = 1,2$ is denoted by

$\bar{\Theta}_i, i = 1,2)$ then random variables $X_1(\omega)$ and $X_2(\omega)$ are independent.

**Q.III.4.** If the system is in well localized pure state $|\psi[\Theta]\rangle$ the state $|\psi[\Theta]\rangle$ described by a

wave function $\psi(q, \Theta) = \langle q | \psi[\Theta] \rangle$ and the value of observable $Q_\Theta$ is measured once each

on many identically prepared system, the average value of all the measurements will be

$$\langle Q_{\Theta} \rangle = \frac{\int_{\Theta} q |\psi(q, \Theta)|^2 dq}{\int_{\Theta} |\psi(q, \Theta)|^2 dq}. \quad (1.7.11)$$

The completeness condition can then be written as $\int_{\Theta} |q\rangle\langle q| dq = \hat{1}_{H_\Theta}$.

Completeness means that for any state $|\psi[\Theta]\rangle \in S_\Theta^\infty$ it must be the case that $\int_{\Theta} |\langle q | \psi[\Theta] \rangle|^2 dq \neq 0$, i.e. there must be a non-zero probability to get some result on measuring observable $Q_\Theta$.

**Q.III.5.** (von Neumann measurement postulate) Assume that

(i) $|\psi\rangle \in S_\Theta^\infty$ and (ii) $\text{supp}(c_\psi(q)) = \Theta$ is a connected set in $\mathbb{R}$. Then if on performing a measurement of $Q_\Theta$ with an accuracy $\delta q$, the result is obtained in the range $(q - \frac{1}{2}\delta q, q + \frac{1}{2}\delta q)$, then the system will end up in the state
$$ \frac{\hat{P}(q, \delta q)|\psi[\Theta]\rangle}{\sqrt{\langle\psi|\hat{P}(q, \delta q)|\psi[\Theta]\rangle}} = \frac{\int_{|q-q'| \le \delta q/2} |q'\rangle\langle q'|\psi[\Theta]\rangle dq'}{\sqrt{\int_{|q-q'| \le \delta q/2} |\langle q' || \psi[\Theta] \rangle|^2 dq'}}. \quad (1.7.12) $$

IV. We claim the following:

**Q.IV.1** For the system in state $|\psi^a[\Theta]\rangle = a|\psi[\Theta]\rangle \in \mathbf{H}_\Theta$, where: (i) $|\psi[\Theta]\rangle \in S_\Theta^\infty, |a| \neq 1,$

(ii) $\text{supp}(c_{\psi[\Theta]}(q))$ is a connected set in $\mathbb{R}$ and (iii) $|\psi[\Theta]\rangle = \int_{\Theta_1}^{\Theta_2} c_{\psi[\Theta]}(q)|q\rangle dq$

$$ \mathbf{G}[\hat{Q}_{\Theta}|\psi^{a}[\Theta]\rangle] = |a|^{2}\mathbf{G}[\hat{Q}_{\Theta}|\psi[\Theta]\rangle]. \quad (1.7.13) $$

**Q.IV.2.** Assume that the system in state $|\psi^a[\Theta]\rangle = a|\psi[\Theta]\rangle \in \mathbf{H}_\Theta$, where (i) $|\psi[\Theta]\rangle \in S_\Theta^\infty,$

$|a| \neq 1$, (ii) $\text{supp}(c_{\psi[\Theta]}(q))$ is a connected set in $\mathbb{R}$ and (iii)

$$ |\psi[\Theta]\rangle = \int_{\Theta_1}^{\Theta_2} c_{\psi[\Theta]}(q)|q\rangle dq. $$

Then if the system is in state $|\psi^a[\Theta]\rangle$ described by a wave function $\psi^a(q; \Theta) = \langle q || \psi^a[\Theta] \rangle$ and the value of observable $Q_\Theta$ is measured once each on many identically prepared system, the average value of all the measurements will be

$$ \langle Q_{\Theta} \rangle = \int_{\Theta} q |\psi^a(q; \Theta)|^2 dq. \quad (1.7.14) $$

**Q.IV.3.** The probability $P(q, q + dq; |\psi^a[\Theta]\rangle)dq$ of obtaining the result $q$ lying in the range

$(q, q + dq)$ on measuring $Q_\Theta$ is

$$ P(q, q + dq; |\psi^a[\Theta]\rangle)dq = |a|^{-2}|c_{\psi[\Theta]}(q|a|^{-2})|^2 dq. \quad (1.7.15) $$

**Remark 1.7.7.** Note that Q.IV.3 immediately follows from Q.IV.1 and Q.III.2.

**Q.IV.4. (Generalized von Neumann measurement postulate)** If on performing a measurement of observable $Q_\Theta$ with an accuracy $\delta q$, the result is obtained in the range $(q - \frac{1}{2}\delta q, q + \frac{1}{2}\delta q)$, then the system immediately after measurement will end up in the state
$$
\begin{equation} \tag{1.7.16}
\begin{aligned}
\frac{\hat{P}(q, \delta q) |\psi^a[\Theta]\rangle}{\sqrt{\langle\psi|\hat{P}(q, \delta q)|\psi[\Theta]\rangle}} &= \frac{\int_{|q-q'| \le \delta q/2} |q'\rangle\langle q'| \|\psi^a[\Theta]\rangle dq'}{\sqrt{\int_{|q-q'| \le \delta q/2} |\langle q'| \psi^a[\Theta]\rangle|^2 dq'}} = \\
&\qquad \frac{a \int_{|q-q'| \le \delta q/2} |q'\rangle\langle q'| \|\psi[\Theta]\rangle dq'}{\sqrt{\int_{|q-q'| \le \delta q/2} |\langle q'| \|\psi[\Theta]\rangle|^2 dq'}} \in \mathbf{H}_{\Theta}.
\end{aligned}
\end{equation}
$$

**Q.V.1.** Let $|\Psi^{a_1,a_2}[\Theta_1, \Theta_2]\rangle = |\psi_1^{a_1}[\Theta_1]\rangle + |\psi_2^{a_2}[\Theta_2]\rangle \in \mathbf{H}_{1,2} \triangleq \mathbf{H}_{\Theta_1} \oplus \mathbf{H}_{\Theta_2} \not\subseteq \mathbf{H}$,
where

$$
(i) |\psi_i^{a_i}[\Theta_i]\rangle = a_i |\psi_i[\Theta_i]\rangle \in \mathbf{H}_{\Theta_i}, |\psi_i\rangle = |\psi_i[\Theta_i]\rangle \in \mathbf{S}_{\Theta_i}^{\infty}, |a_i| \neq 1, i = 1,2;
$$

$$
(ii) \supp(c_{\psi_i[\Theta_i]}(q)), i = 1,2 \text{ is a connected sets in } \mathbb{R};
$$

$$
(iii) (\mathrm{supp}(c_{\psi_1[\Theta_1]}(q))) \cap (\mathrm{supp}(c_{\psi_2[\Theta_2]}(q))) = \emptyset \text{ and}
$$

$$
(iv) |\psi_i[\Theta_i]\rangle = \int_{\Theta_1}^{\Theta_2} c_{\psi_i[\Theta_i]}(q) |q\rangle dq, i = 1,2.
$$

Then if the system is in a state $|\Psi^{a_1,a_2}[\Theta_1, \Theta_2]\rangle$ described by a wave function
$\Psi^{a_1,a_2}(q; \Theta_1, \Theta_2) = \langle q|\|\Psi^{a_1,a_2}[\Theta_1, \Theta_2]\rangle, q \in \Theta_1 \cup \Theta_2$ and the value of observable
$Q_{\Theta_1, \Theta_2}$ is measured once each on many identically prepared system, the average
value of all the measurements will be

$$
\langle Q_{\Theta_1, \Theta_2} \rangle = \int_{\Theta_1 \cup \Theta_2} q |\Psi^{a_1, a_2}(q; \Theta_1, \Theta_2)|^2 dq. \quad (1.7.17)
$$

**Q.V. 2.** The probability of getting a result $q$ with an accuracy $\delta q$ such that
$(q - \frac{1}{2}\delta q, q + \frac{1}{2}\delta q) \in \text{supp}(c_{\psi_1}(q))$ or $(q - \frac{1}{2}\delta q, q + \frac{1}{2}\delta q) \in \text{supp}(c_{\psi_2}(q))$ given by

$$
\int_{|q-q'| \le \delta q/2} \left[ (|\langle q' || \psi_1^{a_1} [\Theta_1] \rangle|^2) * (|\langle q' || \psi_2^{a_2} [\Theta_2] \rangle|^2) \right] dq'. \quad (1.7.18)
$$

**Remark 1.7.8.** Note that Q.IV.3 immediately follows from Q.III.3.

**Q.V. 3.** Assume that the system is initially in the state $|\Psi^{a_1,a_2}[\Theta_1, \Theta_2]\rangle$. If on performing a measurement of $Q_{\Theta_1, \Theta_2}$ with an accuracy $\delta q$, the result is obtained in the range $(q - \frac{1}{2}\delta q, q + \frac{1}{2}\delta q)$, then the state of the system immediately after measurement given by
$$ \frac{\hat{P}(q_i, \delta q) |\Psi^{a_1, a_2}[\Theta_1, \Theta_2]\rangle}{\sqrt{\langle\psi|\hat{P}(q_i, \delta q)|\psi\rangle}} = $$

$$ \frac{\int_{|q_i - q'| \le \delta q/2} (|q'|\langle q'|\psi_1^{a_1}[\Theta_1]\rangle + |q'|\langle q'|\psi_2^{a_2}[\Theta_2]\rangle) dq'}{\sqrt{\int_{|q_i - q'| \le \delta q/2} [|\langle q'|\psi_1[\Theta_1]\rangle|^2 + |\langle q'|\psi_2[\Theta_2]\rangle|^2] dq'}} = \qquad (1.7.19) $$

$$ \frac{\int_{|q_i - q'| \le \delta q/2} (a_1 |q'|\langle q'|\psi_1[\Theta_1]\rangle + a_2 |q'|\langle q'|\psi_2[\Theta_2]\rangle) dq'}{\sqrt{\int_{|q_i - q'| \le \delta q/2} [|\langle q'|\psi_1[\Theta_1]\rangle|^2 + |\langle q'|\psi_2[\Theta_2]\rangle|^2] dq'}} \in \mathbf{H}_{\Theta_i}, $$

$$ q_i \in \Theta_i, i = 1, 2. $$

**Definition 1.7.8.** Let $H_{1,2}$ be $H_{1,2} \triangleq H_{\Theta_1} \oplus H_{\Theta_2}$.

**Definition 1.7.9.** Let $|\psi^a\rangle$ be a state $|\psi^a\rangle = a|\psi\rangle$, where $|\psi\rangle \in S^\infty, |a| \neq 1$ and $|\psi\rangle = \int_{\theta_1}^{\theta_2} c_\psi(q)|q\rangle dq$. Let $|\psi_a\rangle$ be an state such that $|\psi_a\rangle \in S^\infty$. States $|\psi^a\rangle$ and $|\psi_a\rangle$ is a $Q$-equivalent: $|\psi^a\rangle \sim_Q |\psi_a\rangle$ iff

$$ P(q, q + dq; |\psi^a\rangle) = |a|^{-2} |c_\psi(q|a|^{-2})|^2 dq = P(qq + dq; |\psi_a\rangle) dq \qquad (1.7.20) $$

**Q.V.** For any state $|\psi^a\rangle = a|\psi\rangle$, where $|\psi\rangle \in S^\infty, |a| \neq 1$ and $|\psi\rangle = \int_{\theta_1}^{\theta_2} c_\psi(q)|q\rangle dq$ there exist an state $|\psi_a\rangle \in S^\infty$ such that: $|\psi^a\rangle \sim_Q |\psi_a\rangle$.

**Definition 1.7.10.** Let $|\psi^a\rangle$ be a state $|\psi^a\rangle = a|\psi\rangle$, where $|\psi\rangle \in S^\infty, |a| \neq 1$ and $|\psi\rangle = \int_{\theta_1}^{\theta_2} c_\psi(q)|q\rangle dq$. Let $|\psi_a\rangle$ be an state such that $|\psi_a\rangle \in S^\infty$. States $|\psi^a\rangle$ and $|\psi_a\rangle$ is a $\hat{Q}$-equivalent ($|\psi^a\rangle \sim_{\hat{Q}} |\psi_a\rangle$) iff: $\langle\psi^a|\hat{Q}|\psi^a\rangle = \langle\psi_a|\hat{Q}|\psi_a\rangle$.

**Q.VI.** For any state $|\psi^a\rangle = a|\psi\rangle$, where $|\psi\rangle \in S^\infty, |a| \neq 1$ and $|\psi\rangle = \int_{\theta_1}^{\theta_2} c_\psi(q)|q\rangle dq$ there exist an state $|\psi_a\rangle \in S^\infty$ such that: $|\psi^a\rangle \sim_{\hat{Q}} |\psi_a\rangle$.

## I.7.2. The Position Representation. Position observable of a particle in one dimension.

The position representation is used in quantum mechanical problems where it is the position of the particle in space that is of primary interest. For this reason, the position representation, or the wave function, is the preferred choice of representation.

**P.1.** In one dimension, the position $x$ of a particle can range over the values $-\infty < x < +\infty$. Thus the Hermitean operator $\hat{x}$ corresponding to this observable will have eigenstates $|x\rangle$ and associated eigenvalues $x$ such that: $\hat{x}|x\rangle = x|x\rangle, -\infty < x < +\infty$.
**P.2.** As the eigenvalues cover a continuous range of values, the completeness relation will be expressed as an integral: $|\psi_t\rangle = \int_{-\infty}^{+\infty}|x\rangle\langle x|\psi_t\rangle dx$, where $\langle x|\psi_t\rangle = \psi(x,t)$ is the wave function associated with the particle at each instant $t$. Since there is a continuously infinite number of basis states $|x\rangle$, these states are $\delta$-function normalized: $\langle x|x'\rangle = \delta(x-x')$.

**P.3.** The operator $\hat{x}$ itself can be expressed as: $\hat{x} = \int_{-\infty}^{+\infty} x|x\rangle\langle x|dx$.

**Definition 1.7.11.** A connected set is a set $X \subset \mathbb{R}$ that cannot be partitioned into two nonempty subsets which are open in the relative topology induced on the set. Equivalently, it is a set which cannot be partitioned into two nonempty subsets such that each subset has no points in common with the set closure of the other.

**P.4.** The wave function is, of course, just the components of the state vector $|\psi_t\rangle \in S^\infty$ with respect to the position eigenstates as basis vectors. Hence, the wave function is often referred to as being the state of the system in the position representation. The probability amplitude $\langle x|\psi_t\rangle$ is just the wave function, written $\langle x|\psi_t\rangle \triangleq \psi(x,t)$ and is such that $|\psi(x,t)|^2 dx$ is the probability $P(x,t;|\psi_t\rangle)$ of the particle being observed to have a coordinate in the range $x$ to $x+dx$

**Definition 1.7.12.** Let $|\psi_t^a\rangle, t \in [0, +\infty)$ be a state $|\psi_t^a\rangle = a|\psi_t\rangle$, where $|\psi_t\rangle \in S^\infty$, $|a| \neq 1$ and

$|\psi_t\rangle = \int_{-\infty}^{+\infty} \psi(x,t)|x\rangle dx$. Let $|\psi_{t,a}\rangle, t \in [0, +\infty)$ be an state such that $|\psi_{t,a}\rangle \in S^\infty$, $t \in [0, +\infty)$.

States $|\psi_t^a\rangle$ and $|\psi_{t,a}\rangle$ is $x$-equivalent ($|\psi_t^a\rangle \sim_x |\psi_{t,a}\rangle$) iff

$$P(x,t; |\psi_t^a\rangle) dx = |a|^{-2} |\psi(x|a|^{-2}, t)|^2 dx = P(x,t; |\psi_{t,a}\rangle) dx \quad (1.7.21)$$

**P.5.** From postulate Q.V above (see subsection 1.7.1) follows: for any state $|\psi_t^a\rangle = a|\psi_t\rangle$, where $|\psi_t\rangle \in S^\infty$, $|a| \neq 1, t \in [0, +\infty)$ and $|\psi_t\rangle = \int_{-\infty}^{+\infty} \psi(x,t)|x\rangle dx$ there exist an state $|\psi_{t,a}\rangle \in S^\infty$, $t \in [0, +\infty)$ such that: $|\psi_t^a\rangle \sim_x |\psi_{t,a}\rangle$.

**Definition 1.7.13.** Let $|\psi_t^a\rangle, t \in [0, +\infty)$ be a state $|\psi_t^a\rangle = a|\psi_t\rangle$, where

$|\psi_t\rangle \in S^\infty$, $|a| \neq 1$ and $|\psi_t\rangle = \int_{-\infty}^{+\infty} \psi(x,t)|x\rangle dx$. Let $|\psi_{t,a}\rangle, t \in [0, +\infty)$ be an state such that $|\psi_{t,a}\rangle \in S^\infty$, $t \in [0, +\infty)$. States $|\psi_t^a\rangle$ and $|\psi_{t,a}\rangle$ is $\hat{x}$-equivalent ($|\psi_t^a\rangle \sim_{\hat{x}} |\psi_{t,a}\rangle$) iff: $\langle \psi_t^a | \hat{x} | \psi_t^a \rangle = \langle \psi_{t,a} | \hat{x} | \psi_{t,a} \rangle$.

**D.6.** From postulate C.7 (see Appendix C) follows: for any state $|\psi_t^a\rangle = a|\psi_t\rangle$, where $|\psi_t\rangle \in S^\infty$, $|a| \neq 1, t \in [0, +\infty)$ and $|\psi_t\rangle = \int_{-\infty}^{+\infty} \psi(x,t)|x\rangle dx$ there exist an state $|\psi_{t,a}\rangle \in S^\infty$, $t \in [0, +\infty)$ such that: $|\psi_t^a\rangle \sim_{\hat{x}} |\psi_{t,a}\rangle$.

**Definition 1.7.14.** The pure state $|\psi_t\rangle \in S^\infty, t \in [0, +\infty)$, $|\psi_t\rangle = \int_{-\infty}^{+\infty} \psi(x,t)|x\rangle dx$ is a weakly

Gaussian in the position representation iff

$$|\psi(x,t)|^2 dx = \frac{1}{\sigma_t \sqrt{2\pi}} \exp\left[-\frac{(x - \bar{x}_t)^2}{\sigma_t^2}\right] dx, \quad (1.7.22)$$
where $\bar{x}_t$ and $\sigma_t$ are given functions which depend only on variable $t$.

**P.7.** From statement P.5 follows: for any state $|\psi_t^a\rangle = a|\psi_t\rangle$, where $|\psi_t\rangle \in S^\infty$, $|a| \neq 1$, $t \in [0, +\infty)$ and $|\psi_t\rangle = \int_{-\infty}^{+\infty} \psi(x,t)|x\rangle dx$ is a weakly Gaussian state there exist an weakly Gaussian state $|\psi_{t,a}\rangle \in S^\infty$, $t \in [0, +\infty)$ such that:

$$ 
\begin{aligned}
P(x,t; |\psi_t^a\rangle)dx &= |a|^{-1} |\psi(x|a|^{-1}, t)|^2 dx = \\
&= \frac{1}{|a|\sigma_t\sqrt{2\pi}} \exp\left[-\frac{(x - |a|\bar{x}_t)^2}{|a|^2\sigma_t^2}\right] dx.
\end{aligned} 
\quad (1.7.23) 
$$

# I.8. The EPR paradox

In 1935, Einstein, Podolsky and Rosen (EPR) originated the famous “EPR paradox” [38]. This argument concerns two spatially separated particles which have both perfectly correlated positions and momenta, as is predicted possible by quantum mechanics. The EPR paper spurred investigations into the nonlocality of quantum mechanics, leading to a direct challenge of the philosophies taken for granted by most physicists. The EPR conclusion was based on the assumption of local realism, and thus the EPR argument pinpoints a contradiction between local realism and the completeness of quantum mechanics.

## I.8.1. Einstein's 1927 gedanken experiment

Einstein never accepted orthodox quantum mechanics because he did not believe that its nonlocal collapse of the wavefunction could be real. When he first made this argument in 1927 [39], he considered just a single particle. The particle's wavefunction was diffracted through a tiny hole so that it 'dispersed' over a large hemispherical area before encountering a screen of that shape covered in photographic film. Since the film only ever registers the particle at one point on the screen, orthodox quantum mechanics must postulate a 'peculiar mechanism of action at a distance, which prevents the wave... from producing an action in two places on the screen. That is, according to the theory, the detection at one point must instantaneously collapse the wavefunction to nothing at all other points.

**Remark 1.8.1.** It was only in 2010, nearly a century after Einstein's original proposal, that a scheme to rigorously test Einstein's 'spooky action at a distance' [39], [40] using a single particle (a photon), as in his original conception, was conceived [41]. In this scheme, Einstein's 1927 gedankenexperiment is simplified so that the single photon is split into just two wavepackets, one sent to a laboratory supervised by Alice and the other to a distant laboratory supervised by Bob. However, there is a key difference, which enables demonstration of the nonlocal collapse experimentally: rather than simply detecting the presence or absence of the photon, homodyne detection is used. This gives Alice the power to make different measurements, and enables Bob to test (using tomography) whether Alice's measurement choice affects the way his conditioned state collapses,
without having to trust anything outside his own laboratory.

<figure><img src="image_15.png" /><figcaption>Pic. Pic. 1.8.1. Simplified version of Einstein's original gedankenexperiment Adapted from [42].</figcaption></figure>

**Simplified version of Einstein's original gedankenexperiment** [42]. A single photon is incident on a beam splitter of reflectivity R and then subjected to homodyne measurements at two spatially separated locations. Alice is trying to convince Bob that she can steer his portion of the single photon to different types of local quantum states by performing various measurements on her side. She does this by using different values of her LO phase $\theta$, and extracting only the sign $s \in \{+, -\}$ of the quadrature she measures. Meanwhile, Bob scans his LO and performs full quantum-state tomography to reconstruct his local quantum state. He reconstructs unconditional and conditional local quantum states to test if his portion of the single photon has collapsed to different states according to Alice's LO setting $\theta$, and result $s$ see Pic. 1.8.1.

The key role of measurement choice by Alice in demonstrating 'spooky action at a distance' was introduced in the famous Einstein-Podolsky-Rosen (EPR) paper [38] of 1935. In its most general form, this phenomenon has been called EPR-steering, to acknowledge the contribution and terminology of Schrödinger [43], who talked of Alice 'steering' the state of Bob's quantum system. From a quantum information perspective, EPR-steering is equivalent to the task of entanglement verification when Bob (and his detectors) can be trusted but Alice (or her detectors) cannot. This is strictly harder than verifying entanglement with both parties trusted [44], but strictly easier than violating a Bell inequality [45], where neither party is trusted [44].

**Remark 1.8.2.** A recent experimental test of entanglement for a single photon via an entanglement witness has no efficiency loophole [46] however, it demonstrates a weaker form on nonlocality than EPR-steering. In [42], it was demonstrated experimentally that there exist Einstein's elusive 'spooky action at a distance' for a single particle without opening the efficiency loophole without claim to have closed the separation loophole. That is the one-sided device-independent verification of spatial-mode entanglement for a single photon.

## 1.8.2. The continuous variable EPR
paradox.EPR-Reid's criteria

We remind that EPR treated the case of a non-factorizable pure state $|\psi\rangle$ which describes the results for measurements performed on two spatially separated systems at A and B (Fig.1.8.2). “Non- factorizable” means “entangled”, that is, we cannot express $|\psi\rangle$ as a simple product $|\psi\rangle = |\psi\rangle_A \otimes |\psi\rangle_B$, where $|\psi\rangle_A$ and $|\psi\rangle_B$ are quantum states for the results of measurements at A and B, respectively.

<figure><img src="image_16.png" /><figcaption>Fig.1.8.2. The original EPR gedanken experiment. Two particles move from the source into spatially separated regions A and B, and yet continue to have maximally correlated positions: $x_A + x_0 = x_B$ and anti-correlated momenta: $p_A = -p_B$. Adapted from [47].</figcaption></figure>

In the first part of their paper, EPR point out in a general way the problematic aspects of such entangled states. The key issue is that one can expand $|\psi\rangle$ in terms of more than one basis, that correspond to different experimental settings, which we parametrize by $\phi$. Let us consider the state

$$|\psi\rangle = \int dx |\psi_x\rangle_{\phi_A} \otimes |u_x\rangle_{\phi_B}, \qquad (1.8.1)$$

where the eigenvalue $x$ could be continuous or discrete. The parameter setting $\phi$ at the detector B is used to define a particular orthogonal measurement basis $|u_x\rangle_{\phi_B}$. On measurement at B, this projects out a wave-function $|\psi_x\rangle_{\phi_A}$ at A, the process called “reduction of the wave packet”.

**Remark 1.8.3.** The locality assumption postulates no action-at-a-distance, so that measurements at a location B cannot immediately “disturb” the system at a spatially separated location A.

**Remark 1.8.4.** The problematic issue is that different choices of measurements $\phi$ at B will cause reduction of the wave packet at A in more than one possible way. EPR state that, “as a consequence of two different measurements” at B, the “second system may be left in states with two different wavefunctions”. Yet, “no real change can take place in the second system in consequence of anything that may be done to the first system”.

The problem was established by EPR by a specific example, shown in Fig.1.8.2. EPR considered two spatially separated subsystems, at A and B, each with two observables $\hat{x}$ and $\hat{p}$ where $\hat{x}$ and $\hat{p}$ are non-commuting quantum
operators, with commutator

$$[\hat{x}, \hat{p}] = \hat{x}\hat{p} - \hat{p}\hat{x} = 2C \neq 0. \qquad (1.8.2)$$

The results of the measurements $\hat{x}$ and $\hat{p}$ are denoted $x$ and $p$ respectively, and this convention we follow throughout the paper. We note that EPR assumed a continuous variable spectrum and considered wavefunction $\psi$ defined in a position representation by

$$\psi(x, x^B) = \int e^{(ip/h)(x-x^B-x_0)} dp, \qquad (1.8.3)$$

where $x_0$ is a constant implying space-like separation. Here the pairs $x$ and $p$ refer to the results for position and momentum measurements at $A$, while $x^B$ and $p^B$ denote the position and momentum measurements at $B$. We leave off the superscript for system $A$, to emphasize the inherent asymmetry that exists in the EPR argument, where one system $A$ is steered by the other, $B$.

**Remark 1.8.5.** According to canonical quantum mechanics, one can “predict with certainty” that a measurement $\hat{x}$ will give result $x^B + x_0$, if a measurement $\hat{x}^B$, with result $x^B$, was already performed at $B$. One may also “predict with certainty” the result of measurement $\hat{p}$, for a different choice of measurement at $B$. If the momentum at $B$ is measured to be $p$, then the result for $\hat{p}$ is $-p$. These predictions are made “without disturbing the second system” at $A$, based on the assumption, implicit in the original EPR paper, of “locality”.

**Remark 1.8.6.** The locality assumption can be strengthened if the measurement events at $A$ and $B$ are causally separated (such that no signal can travel from one event to the other, unless faster than the speed of light).

**Remark 1.8.7.** The remainder of the EPR argument may be summarized as follows. Assuming local realism, one deduces that both the measurement outcomes, for $x$ and $p$ at $A$, are predetermined. The perfect correlation of $x$ with $x^B + x_0$ implies the existence of an “element of reality” for the measurement $\hat{x}$. Similarly, the correlation of $p$ with $-p^B$ implies an “element of reality” for $\hat{p}$. Although not mentioned by EPR, it will prove useful to mathematically represent the “elements of reality” for $\hat{x}$ and $\hat{p}$ by the respective variables $\mu_x^A$ and $\mu_p^A$, whose “possible values are the predicted results of the measurement”

**Remark 1.8.8.** To continue the argument, local realism implies the existence of two elements of reality, $\mu_x^A$ and $\mu_p^A$, that simultaneously predetermine, with absolute definiteness, the results for measurement $x$ or $p$ at $A$. These “elements of reality” for the localized subsystem $A$ are not themselves consistent with quantum mechanics. Simultaneous determinacy for both the position and momentum is not possible for any quantum state. Hence, assuming the validity of local realism, one concludes quantum mechanics to be incomplete or even inconsistent!

**Remark 1.8.9.** We claim that any assumption of local realism is completely wrong.

Such claim meant as minimum the weak postulate of nonlocality.

**The weak postulate of nonlocality for continuous variables.**

The Heisenberg uncertainty relations
$$ \Delta x^A \Delta p^A \geq 1 \qquad (1.8.4) $$

cannot be violated in any cases:

(i) of course according to quantum mechanics, the Heisenberg uncertainty relations (1.8.4)

cannot be violated if the coordinate $x^A$ and momentum $p^A$ of the particle **A** are measured

directly by measurements performed on the particle **A**,

(ii) the Heisenberg uncertainty relations (1.8.4) cannot be violated even if the coordinate $x^A$

and momentum $p^A$ of the particle **A** are measured indirectly, i.e. by using measurement on

particle **B**, as required in EPR gedanken experiment,

(iii) in any cases true coordinate $x^A$ and momentum $p^A$ of the particle **A** cannot be predicted

simultaneously with a sufficiently small uncertainty $\Delta x^A$ and $\Delta p^A$ such that the Reid's inequality [50]:

$$ \Delta x^A \Delta p^A < 1 \qquad (1.8.5) $$

based on local realism would be satisfied, i.e., always

$$ \Delta x^A \Delta p^A < 1. \qquad (1.8.6) $$

We claim strictly stronger assumptions of nonlocality than mentioned above.

**The strong postulate of nonlocality for continuous variables.**

Let $|\psi_t^x\rangle_A$ and $|\psi_t^x\rangle_B$ be a state vector in $x$-representation at instant $t$ of the particle **A** and

particle **B** correspondingly.

Let $|\psi_t^p\rangle_A$ and $|\psi_t^p\rangle_B$ be a state vector in $p$-representation at instant $t$ of the particle **A** and

particle **B** correspondingly.

Let $\psi_t^A(x) = \langle x|\psi_t^x\rangle_A, \psi_t^B(x) = \langle x|\psi_t^x\rangle_B$ be a wave functions in $x$-representation of the

particle **A** and particle **B** correspondingly.

Let $\psi_t^A(p) = \langle p|\psi_t^p\rangle_A, \psi_t^B(p) = \langle p|\psi_t^p\rangle_B$ be a wave functions in $p$-representation of the

particle **A** and particle **B** correspondingly.

Let $\psi_t^{A/B}(x_A, x_B)$ be corresponding two-particle wave function in $x$-representation and let

$\psi_t^{A/B}(p_A, p_B)$ be corresponding two-particle wave function in $p$-representation.

We claim that:

(i) whenever a measurement of the coordinate $x$ of a particle **B** is performed at instant

$t$ with result $\bar{x}^B \in [x^B - \epsilon, x^B + \epsilon]$, $\epsilon \ll 1$, then:

(a) according to quantum mechanics a state vector $|\psi_t^x\rangle_B$ collapses at instant $t$ to
the state
vector

$$|\psi_{t,\delta,\varepsilon,x^B}^x\rangle_B \sim \hat{L}_{x^B}(\delta, \varepsilon) |\psi_t^x\rangle_B \quad (1.8.7)$$

given by law (1.2.20), where $\hat{L}_{x^B}(\delta, \varepsilon)$ is a norm-reducing, positive, self-adjoint, linear

operator in the 2-particle non projective Hilbert space $\mathbf{H}$, representing the
localization of

particle **B** around the point $x^B$, (see subsection I.2.4),

(b) according postulate of nonlocality a state vector $|\psi_t^x\rangle_A$ immediately collapses
at instant $t$

to the state vector

$$|\psi_{t,\delta,\varepsilon,x^A}^x\rangle_A \sim \hat{L}_{x^B+x_0}^A(\delta, \varepsilon) |\psi_t^x\rangle_A \quad (1.8.8)$$

given by law (1.2.20) and this is true independent of the distance in Minkovski spacetime

$M_4 = \mathbb{R}^{1,3}$ that separates the particles. Thus

$$|\psi_t^x\rangle_B \xrightarrow{\text{col lapse}} |\psi_{t,\delta,\varepsilon,x^B}^x\rangle_B \Rightarrow |\psi_t^x\rangle_A \xrightarrow{\text{col lapse}} |\psi_{t,\delta,\varepsilon,x^{B+x_0}}^x\rangle_A \quad (1.8.9)$$

(ii) under conditions given by Eq.(1.8.7)-Eq.(1.8.9) two-particle wave function

$\psi_t^{A/B}(x_A, x_B)$ collapses at instant $t$ by law

$$\psi_t^{A/B}(x_A, x_B) \xrightarrow{\text{col lapse}} \hat{L}_{x^B+x_0}^A \hat{L}_{x^B}^B(\delta, \varepsilon) \psi_t^{A/B}(x_A, x_B) \quad (1.8.10)$$

(iii) whenever a measurement of the momentum $p^B$ of a particle **B** is performed
at instant

$t$ with result $\bar{p}^B \in [p^B - \varepsilon, p^B + \varepsilon]$, $\varepsilon \ll 1$, then:

(a) according to quantum mechanics a state vector $|\psi_t^p\rangle_B$ collapses at instant $t$
to the state

vector

$$|\psi_{t,\delta,\varepsilon,p^B}^p\rangle_B \sim \hat{L}_{p^B}^B(\delta, \varepsilon) |\psi_t^p\rangle_B, \quad (1.8.11)$$

where $\hat{L}_{p^B}(\delta, \varepsilon)$ is a norm-reducing, positive, self-adjoint, linear operator in the
2-particle

non projective Hilbert space $\mathbf{H}$, representing the localization of momentum of
the particle **B**

around the value $p^B$. The localization operators $\hat{L}_{p^B}(\delta, \varepsilon)$ have been chosen to
have the

following form:

$$\hat{L}_{p^B}^B(\delta, \varepsilon) = \left(\frac{1}{\delta\pi_\delta}\right)^{3/4} \exp\left[-\frac{1}{2\delta}(\hat{p} - p^B)^2\right] \quad (1.8.12)$$

where $\delta \in (0, 1]$ and $\lim_{\delta \to 0} \pi_\delta = \pi$.

(b) according postulate of nonlocality a state vector $|\psi_t^p\rangle_A$ immediately collapses
at instant $t$
to the state vector

$$|\psi_{t,\delta,\varepsilon,x}^p\rangle_A \sim \hat{L}_{-p}^A(\delta, \varepsilon) |\psi_t^p\rangle_A \quad (1.8.13)$$

and this is true independent of the distance in Minkovski spacetime $M_4 = \mathbb{R}^{1,3}$ that

separates the particles. Thus

$$|\psi_t^p\rangle_B \xrightarrow{\text{col lapse}} |\psi_{t,\delta,\varepsilon,p}^p\rangle_B \Rightarrow |\psi_t^p\rangle_A \xrightarrow{\text{col lapse}} |\psi_{t,\delta,\varepsilon,-p}^p\rangle_A \quad (1.8.14)$$

(iv) under conditions given by Eq.(1.8.11)-Eq.(1.8.13) two-particle wave function $\psi_t^{A/B}(p_A, p_B)$ collapses at instant $t$ by law

$$\psi_t^{A/B}(p_A, p_B) \xrightarrow{\text{col lapse}} \hat{L}_{-p}^A \hat{L}_{-p}^B(\delta, \varepsilon) \psi_t^{A/B}(p_A, p_B). \quad (1.8.15)$$

**Remark 1.8.10.** Let $p_t^A$ and $p_t^B$ be the momentum at instant $t$ of the particle **A** and particle **B** correspondingly. Note that whenever a measurement of the coordinate $x$ of a particle **B** is

performed at instant $t$ with an accuracy $\varepsilon_{xB} \ll 1$ then:

(i) immediately after this measurement the momentum $p_t^B$ at instant $t$ changed

according

to quantum mechanics by the Heisenberg uncertainty relations (1.8.4);

(ii) immediately after this measurement the momentum $p_t^A$ at instant $t$ changed

according

to postulate of nonlocality by the Heisenberg uncertainty relations (1.8.4)

**Remark 1.8.11.** Let $x_t^A$ and $x_t^B$ be the coordinate at instant $t$ of the particle **A** and particle **B** correspondingly. Note that whenever a measurement of the momentum $p$ of a particle **B** is

performed at instant $t$ with an accuracy $\varepsilon_{pB} \ll 1$ then:

(i) immediately after this measurement the coordinate $x_t^B$ at instant $t$ changed

(ii) immediately after this measurement the coordinate $x_t^A$ at instant $t$ changed

according

to quantum mechanics by the Heisenberg uncertainty relations (1.8.4);

**Remark 1.8.12.** Schrödinger [43] pointed out that the EPR two-particle wavefunction in Eq.(1.8.3) was verschränkten - which he later translated as entangled - i.e., not of the separable form $\psi_A\psi_B$. Schrödinger considered as a possible resolution of the paradox that this “entanglement” degrades as the particles separate spatially, so that EPR correlations would not be physically realizable.

**Definition 1.8.1.** Quantum inseparability (entanglement) for a general mixed quantum state is defined as the failure of

$$\hat{\rho} = \int d\lambda P(\lambda) \hat{\rho}_{\lambda}^{A} \otimes \hat{\rho}_{\lambda}^{B}, \quad (1.8.16)$$

where $\int d\lambda P(\lambda) = 1$ and $\hat{\rho}$ is the density operator. Here $\lambda$ is a discrete or continuous
label for component states, and $\hat{\rho}_{\lambda}^{A}$ and $\hat{\rho}_{\lambda}^{B}$ correspond to density operators that are
restricted to the Hilbert spaces $\mathbf{A}$ and $\mathbf{B}$ respectively.

**Remark 1.8.13.** The definition of inseparability extends beyond that of the EPR situation, in that one considers a whole spectrum of measurement choices, parametrized by $\theta$ for those performed on system $\mathbf{A}$, and by $\phi$ for those performed on $\mathbf{B}$. We use canonical notation $\hat{x}_{\theta}^{A}$ and $\hat{x}_{\phi}^{B}$ to describe all measurements at $\mathbf{A}$ and $\mathbf{B}$. Denoting the eigenstates of $\hat{x}_{\theta}^{A}$ by $|x_{\theta}^{A}\rangle$, we define $P_{Q}(x_{\theta}^{A}|\theta, \lambda) = \langle x_{\theta}^{A}|\hat{\rho}_{\lambda}^{A}|x_{\theta}^{A}\rangle$ and $P_{Q}(x_{\phi}^{B}|\phi, \lambda) = \langle x_{\phi}^{B}|\hat{\rho}_{\lambda}^{B}|x_{\phi}^{B}\rangle$, which are the localized probabilities for observing results $x_{\theta}^{A}$ and $x_{\phi}^{B}$ respectively. The separability condition (1.8.9) then implies that joint probabilities $P(x_{\theta}^{A}, x_{\phi}^{B})$ are given as [50]:

$$P(x_{\theta}^{A}, x_{\phi}^{B}) = \int d\lambda P(\lambda) P_{Q}(x_{\theta}^{A}|\lambda) P_{Q}(x_{\phi}^{B}|\lambda). \qquad (1.8.17)$$

**Remark 1.8.14.** We note the canonical restriction

$$\Delta^2(x^A|\lambda)\Delta^2(p^A|\lambda) \geq 1 \qquad (1.8.18)$$

where $\Delta^2(x^A|\lambda)$ and $\Delta^2(p^A|\lambda)$ are the variances of $P_Q(x_{\theta}^A|\theta, \lambda)$ for the choices $\theta$
corresponding to position $x$ and momentum $p$, respectively. Thus

$$\Delta^2(x^A|\lambda)\Delta^2(p^A|\lambda) < 1 \qquad (1.8.19)$$

is an EPR criterion, meaning that this would imply an EPR "paradox".

**Remark 1.8.13.** Note that the original EPR state of Eq. (1.8.3) is not separable.

Suppose that, based on a result $x^B$ for the measurement at $\mathbf{B}$, an estimate $x_{\text{est}}(x_{\mathbf{B}})$ is made of the result $x$ at $\mathbf{A}$. We may define the average error $\Delta_{\text{inf}}x$ of this inference as the root mean square (RMS) of the deviation of the estimate from the actual value, so that [50]:

$$\Delta_{\text{inf}}^2 x = \int dx dx^B P_t(x, x^B) (x - x_{\text{est}}(x^B))^2 . \qquad (1.8.20)$$

An inference variance $\Delta_{\text{inf}}^2 p$ is defined similarly, i.e.

$$\Delta_{\text{inf}}^2 p = \int dp dp^B P_t(p, p^B) (p - p_{\text{est}}(p^B))^2 . \qquad (1.8.21)$$

**Remark 1.8.14.** Let $\psi_t^{A/B}(x_A, x_B)$ be corresponding two-particle wave function in $x$-representation and let $\psi_t^{A/B}(p_A, p_B)$ be corresponding two-particle wave function in $p$-representation. Note that:

(i) $P_t(x, x^B)$ is the joint probability of obtaining an outcome $x$ at $\mathbf{A}$ and $x^B$ at $\mathbf{B}$ at instant $t$ is
of the form

$$P_t(x, x^B) \sim |\psi_t^{A/B}(x_A, x_B)|^2, \qquad (1.8.22)$$

(ii) $P_t(p, p^B)$ is the joint probability of obtaining an outcome $p$ at $\mathbf{A}$ and $p^B$ at $\mathbf{B}$ at instant $t$ is
of the form

$$P_t(p, p^B) \sim |\psi_t^{A/B}(p_A, p_B)|^2. \qquad (1.8.23)$$

The best estimate, which minimizes $\Delta_{\text{inf}}x$, is given by choosing $x_{\text{est}}$ for each $x^B$
to be the mean $\langle x|x^B \rangle$ of the conditional distribution $P_t(x|x^B)$. This is seen upon noting that for each result $x^B$, we can define the RMS error in each estimate as

$$ \Delta_{\inf}^2(t, x|x^B) = \int dx P_t(x|x^B)(x - x_{est}(x^B))^2. \qquad (1.8.24) $$

The average error in each inference is minimized for $x_{est} = \langle x|x^B \rangle$, when each $\Delta_{\inf}^2(t, x|x^B)$ becomes the variance $\Delta^2(t, x|x^B)$ of $P_t(x|x^B)$. We thus define the minimum inference error $\Delta_{\inf}x$ for position, averaged over all possible values of $x^B$, as

$$ V_{A|B}^x = \min(\Delta_{\inf}^2 x) = \int dx^B P_t(x^B) \Delta^2(t, x|x^B), \qquad (1.8.25) $$

where $P(x^B)$ is the probability density for a result $x^B$ upon measurement of $\hat{x}^B$. This minimized inference variance is the average of the individual variances for each outcome at $B$. Similarly, we can define a minimum inference variance, $V_{A|B}^p$, for momentum, i.e.

$$ V_{A|B}^p = \min(\Delta_{\inf}^2 p) = \int dp^B P_t(p^B) \Delta^2(t, p|p^B). \qquad (1.8.26) $$

**Remark 1.8.14.** Let $\psi_t^{A/B}(x_A, x_B)$ be corresponding two-particle wave function in $x$-representation and let $\psi_t^{A/B}(p_A, p_B)$ be corresponding two-particle wave function in $p$-representation. Note that:

(i) according to local realism the conditional distributions densities $P_{\text{loc}}(x|x^B)$ and

$P_{\text{loc}}(p|p^B)$ are given by formulae

$$ P_{\text{loc}}(x|x^B) \sim \hat{L}_{x^B}^{A}(\delta, \varepsilon) \psi_t^{A/B}(x, x_B) \qquad (1.8.27) $$

and

$$ P_{\text{loc}}(p|p^B) \sim \hat{L}_{p^B}^{A}(\delta, \varepsilon) \psi_t^{A/B}(p_A, p_B). \qquad (1.8.28) $$

(ii) distributions densities $P_{\text{loc}}(t, x^B)$ and $P_{\text{loc}}(t, p^B)$ are given by formulae

$$ P_{\text{loc}}(t, x^B) = \int dx P_{\text{loc}}(t, x|x^B) \qquad (1.8.29) $$

and

$$ P_{\text{loc}}(t, p^B) = \int dp P_{\text{loc}}(t, p|p^B). \qquad (1.8.30) $$

**Remark 1.8.15.** Let $\psi_t^{A/B}(x_A, x_B)$ be corresponding two-particle wave function in $x$-representation and let $\psi_t^{A/B}(p_A, p_B)$ be corresponding two-particle wave function in $p$-representation. Note that:

(i) according to postulates of nonlocality the conditional distributions densities

$P_{\text{n.loc}}(t, x|x^B)$ and $P_{\text{n.loc}}(t, p|p^B)$ are given by formulae

$$ P_{\text{n.loc}}(t, x|x^B) = \hat{L}_{x^B+x_0}^{A} \hat{L}_{x^B}^{B}(\delta, \varepsilon) \psi_t^{A/B}(x, x_B) \qquad (1.8.31) $$
and

$$P_{\text{n.loc}}(t, p|x^B) \sim \hat{L}_{-p^B}^A \hat{L}_{p^B}^B (\delta, \epsilon) \psi_i^{A/B}(p, p_B), \quad (1.8.32)$$

see Eq.(1.8.10) and Eq.(1.8.15) respectively.

(ii) distributions $P_{\text{n.loc}}(t, x^B)$ and $P_{\text{n.loc}}(t, p^B)$ are given by formulae

$$P_{\text{n.loc}}(t, x^B) = \int dx P_{\text{n.loc}}(t, x|x^B) \quad (1.8.33)$$

and

$$P_{\text{n.loc}}(t, p^B) = \int dp P_{\text{n.loc}}(t, p|p^B) \quad (1.8.34)$$

Thus we can define corresponding RMS errors as

$$\Delta_{\text{loc.inf}}^2(t, x|x^B) = \int dx P_{\text{loc}}(t, x|x^B) (x - x_{\text{est}}(x^B))^2 \quad (1.8.35)$$

$$\Delta_{\text{loc.inf}}^2(t, p|p^B) = \int dx P_{\text{loc}}(t, p|p^B) (p - x_{\text{est}}(p^B))^2$$

and

$$\Delta_{\text{n.loc.inf}}^2(t, x|x^B) = \int dx P_{\text{loc}}(t, x|x^B) (x - x_{\text{est}}(x^B))^2, \quad (1.8.36)$$
$$\Delta_{\text{n.loc.inf}}^2(t, p|p^B) = \int dx P_{\text{loc}}(t, p|p^B) (p - x_{\text{est}}(p^B))^2$$

respectively. We thus define the minimum inference error $\Delta_{\text{inf}}x$ for position, averaged over all possible values of $x^B$ and $p^B$ as

$$\min(\Delta_{\text{loc.inf}}^2 x) = \int dx^B P_{\text{loc}}(t, x^B) \Delta_{\text{loc.}}^2 (t, x|x^B), \quad (1.8.37)$$
$$\min(\Delta_{\text{loc.inf}}^2 p) = \int dp^B P_{\text{loc}}(t, p^B) \Delta_{\text{loc.}}^2 (t, p|p^B)$$

and

$$\min(\Delta_{\text{n.loc.inf}}^2 x) = \int dx^B P_{\text{n.loc}}(t, x^B) \Delta_{\text{n.loc.}}^2 (t, x|x^B), \quad (1.8.38)$$
$$\min(\Delta_{\text{n.loc.inf}}^2 p) = \int dp^B P_{\text{n.loc}}(t, p^B) \Delta_{\text{n.loc.}}^2 (t, p|p^B).$$

respectively. From Eq.(1.8.37) and Eq.(1.8.38) we obtain the EPR-nonlocality criteria

$$\begin{aligned} & \left| \min \Delta_{\text{loc.inf.}}^2 x - \min \Delta_{\text{n.loc.inf.}}^2 x \right| = \\ & \left| \int dx^B [P_{\text{loc}}(t, x^B) \Delta_{\text{loc.}}^2 (t, x|x^B) - P_{\text{n.loc}}(t, x^B) \Delta_{\text{n.loc.}}^2 (t, x|x^B)] \right| > 0, \quad (1.8.39) \\ & \left| \min \Delta_{\text{loc.inf.}}^2 p - \min \Delta_{\text{n.loc.inf.}}^2 p \right| = \\ & \left| \int dp^B [P_{\text{loc}}(t, p^B) \Delta_{\text{loc.}}^2 (t, p|p^B) - P_{\text{n.loc}}(t, p^B) \Delta_{\text{n.loc.}}^2 (t, p|p^B)] \right| > 0 \end{aligned}$$

and

$$\left| (\min \Delta_{\text{loc.inf.}}^2 x) (\min \Delta_{\text{loc.inf.}}^2 p) - (\min \Delta_{\text{n.loc.inf.}}^2 x) ((\Delta_{\text{n.loc.inf.}}^2 p)) \right| > 0. \quad (1.8.40)$$

## 1.9. The EPR-Bohm paradox. Reid's criteria for EPR-Bohm paradox.
Bohm [49] considered two spatially-separated spin-1/2 particles at A and B produced in an entangled singlet state (often referred to as the “EPR-Bohm state” or the “Bell-state”):

$$|\psi\rangle = \frac{1}{\sqrt{2}} \left( \left| \frac{1}{2} \right\rangle_A \left| -\frac{1}{2} \right\rangle_B - \left| -\frac{1}{2} \right\rangle_A \left| \frac{1}{2} \right\rangle_B \right) \qquad (1.9.1) \quad \text{Here } |\pm\frac{1}{2}\rangle_A$$

are eigenstates of the spin operator $\hat{J}_z^A$, and we use $\hat{J}_z^A$, $\hat{J}_x^A$, $\hat{J}_y^A$ to define the spin-components measured at location A. The spin-eigenstates and measurements at B are defined similarly. By considering different quantization axes, one obtains different but equivalent expansions of $|\psi\rangle$ in Eq. (1.8.1), just as EPR suggested.

<figure><img src="image_17.png" /><figcaption>Fig.1.9.1. The Bohm gedanken EPR experiment. Two spin-1/2 particles prepared in a singlet state from the source into spatially separated regions A and B, and give anti-correlated outcomes for $J_\theta^A$ and $J^B$, where $\theta$ is x, y or z. Adapted from [50].</figcaption></figure>

Bohm's paradox is based on the existence, for Eq. (1.9.1), of a maximum anti-correlation between not only $\hat{J}_z^A$ and $\hat{J}_z^B$, but $\hat{J}_y^A$ and $\hat{J}_y^B$, and also $\hat{J}_x^A$ and $\hat{J}_x^B$. An assumption of local realism would lead to the conclusion that the three spin components of particle A were simultaneously predetermined, with absolute definiteness. Since no such quantum description exists, this is the situation of an EPR paradox.

**Remark 1.9.1.** Bohm's paradox is based on the existence, for Eq. (1.9.1), of a maximum

anti-correlation between not only $\hat{J}_z^A$ and $\hat{J}_z^B$, but $\hat{J}_y^A$ and $\hat{J}_y^B$, and also $\hat{J}_x^A$ and $\hat{J}_x^B$.

**Remark 1.9.2.** Note that an assumption of local realism would lead to the conclusion that

the three spin components of particle A were simultaneously predetermined, with absolute

definiteness. Since no such quantum description exists, this is the situation of an EPR

paradox.

**Remark 1.9.3.** Criteria sufficient to demonstrate Bohm's EPR paradox can be derived using
Reid's canonical inferred uncertainty approach [41]. Using the Heisenberg spin uncertainty relation

$$ \Delta J_x^A \Delta J_y^A \geq |\langle J_z^A \rangle|/2, \qquad (1.9.2) \text{ one obtains the} $$

following canonical spin-EPR criterion that is useful for the Bell state given by Eq. (1.9.1)

$$ \Delta_{\inf} J_x^A \Delta_{\inf} J_y^A < \frac{1}{2} \sum_{J_z^B} P(J_z^B) |\langle J_z^A \rangle_{J_z^B}|. \qquad (1.9.3) $$

Here $\langle J_z^A \rangle_{J_z^B}$ is the mean of the conditional distribution $P(J_z^A|J_z^B)$. Calculations for Eq. (1.9.1) including the effect of detection efficiency $\eta$ reveals this EPR criterion to be satisfied for $\eta > 0.62$. The concept of spin-EPR has been experimentally tested in the continuum limit with purely optical systems for states with $\langle J_z^A \rangle \neq 0$. In this case the EPR criterion linked closely to definition of spin squeezing

$$ \Delta_{\inf} J_x^A \Delta_{\inf} J_y^A < |\langle J_z^A \rangle|. \qquad (1.9.4) $$

**Remark 1.9.4.** We claim that any assumption of local realism is completely wrong. The

three spin components of particle A were simultaneously predetermined, does not with

absolute definiteness but only with uncertainties which required by Heisenberg spin

uncertainty relations (1.9.5). Such claim meant as minimum the weak postulate of nonlocality.

**The weak postulate of nonlocality.**

The Heisenberg spin uncertainty relations

$$ \begin{aligned} \Delta J_x^A \Delta J_y^A &\geq |\langle J_z^A \rangle|/2, \\ \Delta J_x^A \Delta J_z^A &\geq |\langle J_y^A \rangle|/2, \\ \Delta J_z^A \Delta J_y^A &\geq |\langle J_x^A \rangle|/2 \end{aligned} \qquad (1.9.5) $$

does not violated in any cases:

(i) if the three spin components of the particle A are measured directly by measurements

performed on the particle A

(ii) and even if some spin components of the particle A are measured indirectly as required

in Bohm gedanken EPR experiment.

Think of the following situation: a particle with zero spin decays into two particles (A and B), each with 1/2-spin. Due to the fact that spin angular momentum must be conserved during the decay, if initially the total spin angular momentum
was zero, then after the decaying process it must still be zero. Therefore, particles **A** and **B** have opposite spin. Take as an example the dissociation of an exited hydrogen molecule into two hydrogen atoms. If the decaying mechanism does not change total angular momentum, then the spins on the hydrogen atoms will be anti-correlated.

**Remark 1.9.5.** Whenever a measurement of the spin of **A** is found to be positive with respect of the z-axis (we shall note this state as $|↑⟩_z$, then, under local realism, we could infer that the spin of the **B** particle must be negative $|↓⟩_z$, and this is true independent of the distance that separates the particles. The spin of these particles are then entangled.

**Remark 1.9.6.** We claim again that any assumption of local realism is completely wrong.

**The strong postulate of nonlocality.**

Let $|ψ_t⟩_A$ and $|ψ_t⟩_B$ a state at instant $t$ of the particle **A** and particle **B** correspondingly.

Let $|↑⟩_{z,A/B}$ be eigenstates of the spin operator $S^z_{A/B}$:

$$S^z_{A/B} = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} \qquad (1.9.6)$$

We claim that:

(i) whenever a measurement of the spin of a particle **A** is performed at instant $t_1 \ge t$ and

particle **A** is found in the state $|↑⟩_z$, i.e., a state $|ψ_{t_1}⟩_A$ collapses at instant $t_1$ to the state

$|↑⟩_{z,A}$ with respect of the Heisenberg spin uncertainty relations (1.9.5), then a state $|ψ_{t_1}⟩_B$

immediately collapses at instant $t_1$ to the state $|↓⟩_{z,B}$ with respect of the Heisenberg spin uncertainty relations (1.9.5), and this is true independent of the distance in Minkovski spacetime that separates the particles:

$$|ψ_{t_1}⟩_A \xrightarrow{\text{col lapse}} |↑⟩_{z,A} \Rightarrow |ψ_{t_1}⟩_B \xrightarrow{\text{col lapse}} |↓⟩_{z,B} \qquad (1.9.7)$$

(ii) whenever a measurement of the spin of a particle **A** is performed at instant $t_1 \ge t$ and

particle **A** is found in the state $|↓⟩_z$, i.e., a state $|ψ_{t_1}⟩_A$ collapses at instant $t_1$ to the state

$|↓⟩_{z,A}$ with respect of the Heisenberg spin uncertainty relations (1.9.5), then a state $|ψ_{t_1}⟩_B$

immediately $|ψ_t⟩_B$ collapses at instant $t_1$ to the state $|↑⟩_{z,B}$ with respect of the Heisenberg spin uncertainty relations (1.9.5), and this is true independent of the distance in Minkovski spacetime that separates the particles:
spacetime that separates the particles:

$$|\psi_{t_1}\rangle_A \xrightarrow{\text{col lapse}} |\downarrow\rangle_{z,A} \Rightarrow |\psi_{t_1}\rangle_B \xrightarrow{\text{col lapse}} |\uparrow\rangle_{z,B} \qquad (1.9.8)$$

Note that we can not predict which spin will be positive (or negative) with respect of the

z-axis, so the state that describes the spins of the particles could be for instance the spin

singlet state

$$|\psi\rangle = \frac{1}{\sqrt{2}}(|\downarrow\uparrow\rangle - |\uparrow\downarrow\rangle) \qquad (1.9.9)$$

We have a probability of 50% for the spin of particle **A** to be positive (and the spin of **B** negative) and a probability of 50% of it being the other way around.

**Remark 1.9.7.** So far we have assumed that we are performing a measurement along the z-axis, but measurements are not restricted to this particular election, we could measure for instance the spin of particle **A** along the **a**-axis and the spin of **B** along the **b**-axis. Let's see what happens if we decide to measure the spin along the x-axis: **a** = **b** = **x**. As it known for 1/2-spins, the spin operator $S_{A/B}^x$ can be represented by the 2 × 2 hermitian matrix

$$S_{A/B}^x = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix} \qquad (1.9.10)$$

By performing a change of basis we can rewrite the state $|\psi\rangle$ in terms of the eigenstates of the spin operator $S_{A/B}^x$:

$$|u\rangle = \frac{1}{\sqrt{2}}(|\downarrow\rangle + |\uparrow\rangle), \quad |v\rangle = \frac{1}{\sqrt{2}}(|\downarrow\rangle - |\uparrow\rangle), \qquad (1.9.11)$$

and using Eq. (1.9.10), we can rewrite the state $|\psi\rangle$ as

$$|\psi\rangle = \frac{1}{\sqrt{2}}(|vu\rangle - |uv\rangle). \qquad (1.9.12)$$

The strong postulate of nonlocality in this case takes the form similarly mentioned above. Just like before, by choosing to measure the spin of **A** along the x-axis we can determine its value and infer the value of the spin of particle $\tilde{B}$ ($\tilde{B} \neq B$) in the state $|\psi\rangle_{x,\tilde{B}} = |u\rangle_{x,B} \neq |\psi\rangle_{x,B}$ without the need to measure it (and vice versa).

Furthermore, it turns out that this is the case independent of the election of the axis we choose to measure! (Provided that $\mathbf{a} = \mathbf{b} = \mathbf{v}$).

This is exactly the same situation such that a simple choice of the axis along which to measure the spin **A** allow us to establish the value of the spin of **B** along this same axis without the need to measure it. And this is also the case (as we already saw) for physical properties described by non-commuting operators ($S^x$ and $S^z$ do not commute).
## II. Generalized Gamow theory of the alpha decay via tunneling using GRW collapse model. Nonlocal Schrödinger equation corresponding to alpha decay.

### II.1. Generalized Gamow theory of the alpha decay via tunneling using GRW collapse model.

By 1928, George Gamow had solved the theory of the alpha decay via tunneling [42]. The alpha particle is trapped in a potential well by the nucleus. Classically, it is forbidden to escape, but according to the (then) newly discovered principles of quantum mechanics, it has a tiny (but non-zero) probability of "tunneling" through the barrier and appearing on the other side to escape the nucleus. Gamow solved a model potential for the nucleus and derived, from first principles, a relationship between the half-life of the decay, and the energy of the emission.

The α-particle has total energy *E* and is incident on the barrier from the right to left.

<figure><img src="image_18.png" /><figcaption>Pic. 2.1.1. The particle has total energy *E* and is incident on the barrier *V(x)* from right to left. Adapted from [42].</figcaption></figure>

The Schrödinger equation in each of regions I = {x|x < 0}, II = {x|0 ≤ x ≤ l} and III = {x|x > l} takes the following form

$$ \frac{\partial^2 \Psi(x)}{\partial x^2} + \frac{2m}{\hbar^2} [E - U(x)] \Psi(x) = 0, \qquad (2.1.1) $$

where
$$ U(x) = \begin{cases} 0 & \text{for } x < 0 \\ U_0 & \text{for } 0 \le x \le l \\ 0 & \text{for } x > l \end{cases} \qquad (2.1.2) $$

The solutions reads [8]:

$$ \begin{aligned} \Psi_{\text{III}}(x) &= C_+ \exp(ikx) + C_- \exp(-ikx), \\ \Psi_{\text{II}}(x) &= B_+ \exp(k'x) + B_- \exp(-k'x), \\ \Psi_{\text{I}}(x) &= A \cos(kx) = \frac{A}{2} [\exp(ikx) + \exp(-ikx)], \end{aligned} \qquad (2.1.3) $$

where

$$ \begin{aligned} k &= \frac{2\pi}{h} \sqrt{2mE}, \\ k' &= \frac{2\pi}{h} \sqrt{2m(U_0 - E)}. \end{aligned} \qquad (2.1.4) $$

At the boundary $x = 0$ we have the following boundary conditions:

$$ \Psi_{\text{I}}(0)|_{x=0} = \Psi_{\text{II}}(0)|_{x=0}, \quad \left. \frac{\partial \Psi_{\text{I}}(x)}{\partial x} \right|_{x=0} = \left. \frac{\partial \Psi_{\text{II}}(x)}{\partial x} \right|_{x=0}. \qquad (2.1.5) $$

At the boundary $x = l$ we have the following boundary conditions

$$ \Psi_{\text{II}}(l)|_{x=l} = \Psi_{\text{III}}(l)|_{x=l}, \quad \left. \frac{\partial \Psi_{\text{II}}(x)}{\partial x} \right|_{x=l} = \left. \frac{\partial \Psi_{\text{III}}(x)}{\partial x} \right|_{x=l}. \qquad (2.1.6) $$

From the boundary conditions (2.1.5)-(2.1.6) one obtains [42]:

$$ \begin{aligned} B_+ &= \frac{A}{2} \left(1 + i \frac{k}{k'}\right), B_- = \frac{A}{2} \left(1 - i \frac{k}{k'}\right), \\ C_+ &= A[\mathbf{ch}(k'l) + iD\mathbf{sh}(k'l)], C_- = i(AS\mathbf{sh}(k'l) \exp(ikl)), \\ D &= \frac{1}{2} \left(\frac{k}{k'} - \frac{k'}{k}\right), S = \frac{1}{2} \left(\frac{k}{k'} + \frac{k'}{k}\right). \end{aligned} \qquad (2.1.7) $$

From Eqs.(2.1.7) one obtains the conservation law

$$ |A|^2 = |C_+|^2 - |C_-|^2. $$

Let us introduce now a function $E_{\text{II}}(x, l) = \theta_2(x, l)E_{\text{II}}(x, l)$ where
$$
\begin{align}
E_2(x, l) &= \begin{cases}
(\pi r_c^2)^{-1/4} \exp\left(-\frac{x^2}{2r_c^2}\right) & \text{for } -\infty < x < \frac{l}{2} \\
(\pi r_c^2)^{-1/4} \exp\left(-\frac{(x-l)^2}{2r_c^2}\right) & \text{for } \frac{l}{2} \le x < \infty
\end{cases} \tag{2.1.8} \\
\theta_2(x, l) &= \begin{cases}
1 & \text{for } x \in [0, l] \\
0 & \text{for } x \notin [0, l]
\end{cases}
\end{align}
$$

**Assumption 2.1.1.** We assume now that:

(i) at instant $t=0$ the wave function $\Psi_I(x)$ experiences a sudden jump of the form

$$
\Psi_I(x) \rightarrow \Psi_I^{\#}(x) = \frac{\Re_I(\hat{x})\Psi_I(x)}{\|\Re_I(\hat{x})\Psi_I(x)\|_2}, \qquad (2.1.9)
$$

where $\Re_I(\hat{x})$ is a linear operator which is chosen equal to:

$$
\Re_I(\hat{x}) = (\pi r_c^2)^{-1/4} \theta_1(\hat{x}, l) \exp\left[-\frac{\hat{x}^2}{2r_c^2}\right]; \qquad (2.1.10)
$$

where

$$
\theta_1(x, l) = \begin{cases} 1 & \text{for } x \in [-l, 0], \\ 0 & \text{for } x \notin [-l, 0]. \end{cases}
$$

**Remark 2.1.1.** Note that: $\text{supp}(\Psi_I^{\#}(x)) \subseteq [-l, 0]$

(ii) at instant $t=0$ the wave function $\Psi_{II}(x)$ experiences a sudden jump of the form

$$
\Psi_{II}(x) \rightarrow \Psi_{II}^{\#}(x) = \frac{\Re_{II}(\hat{x})\Psi_{II}(x)}{\|\Re_{II}(\hat{x})\Psi_{II}(x)\|_2}, \qquad (2.1.11)
$$

where $\Re_{II}(\hat{x})$ is a linear operator which is chosen equal to:

$$
\Re_{II}(\hat{x}) = E_{II}(\hat{x}, l); \tag{2.1.12}
$$

**Remark 2.1.2.** Note that: $\text{supp}(\Psi_{II}^{\#}(x)) \subseteq [0, l]$.

(iii) at instant $t=0$ the wave function $\Psi_{III}(x)$ experiences a sudden jump of the form
$$ \Psi_{\text{III}}(x) \rightarrow \Psi_{\text{III}}^{\#}(x) = \frac{\Re_{\text{III}}(\hat{x})\Psi_{\text{III}}(x)}{\|\Re_{\text{III}}(\hat{x})\Psi_{\text{III}}(x)\|_2}, \qquad (2.1.13) $$

where $\Re_{\text{III}}(\hat{x})$ is a linear operator which is chosen equal to:

$$ \Re_{\text{III}}(\hat{x}) = (\pi r_c^2)^{-1/4} \exp\left[-\frac{(\hat{x} - l)^2}{2r_c^2}\right]. \qquad (2.1.14) $$

**Remark 2.1.3.** Note that. We have choose operators (2.1.10),(2.1.12) and (2.1.14) such that the boundary conditions (2.1.5),(2.1.6) is satisfied.

## II.2. The nonlocal Schrödinger equation corresponding to alpha decay.

Note that $\alpha$-particle moves through some medium and therefore the states of the atoms of the medium are changed as a result of the interaction with it. As a result, information about positions of the particle at different instants of time is recorded in the state of the medium. This means that the position is monitored. Simultaneously the motion of the particle is influenced by the medium. The resulting influence of the medium leads to the decoherence of the $\alpha$-particle.

<figure><img src="image_19.png" /><figcaption>Pic.2.2.1. Quantum diffusion: Decoherence by the structure of atoms. Adopted from [9].</figcaption></figure>

The following master equation was derived for the density matrix of the such $\alpha$-particle [9]:

$$ \begin{aligned} \dot{\rho} &= -\frac{i}{\hbar}[H, \rho] - \frac{k}{2}[\mathbf{r}, [\mathbf{r}, \rho]], \\ k &= \frac{2}{l^2\tau}, \end{aligned} \qquad (2.2.1) $$

where $l$ is the distance at which the particle excites an atom and $\tau$ a characteristic time of interaction between the atom and the $\alpha$-particle. In this case decoherence is caused by the internal structure of atoms. This illustrated by the Pic.2.2.1. The medium to be considered is a system of an infinite number of non-interacting
quantum oscillators located at the nodes of a cubic lattice. The oscillators are labelled by the index $k(k = 1, 2, \dots)$. The interaction of the k-th oscillator with the measured particle given by the following simple Hamiltonian [9]:

$$H_{\text{int}} = \sum_k H_k = \gamma \sum_k q_k \exp\left[-\frac{(\mathbf{r} - \mathbf{c}_k)^2}{l^2}\right], \qquad (2.2.2)$$

where $q_k$ is a canonical coordinate of the oscillator while $\mathbf{c}_k$ is its location, $\gamma$ is the interaction constant, $l$ characterizes its range, and $\mathbf{r}$ is the coordinate of the $\alpha$-particle. The complete Hamiltonian of the problem has the form

$$\hat{\mathbf{H}} = \frac{\mathbf{P}^2}{2M} + V(\mathbf{r}) + \sum_k \frac{p_k^2}{2m} + \sum_k \frac{m\omega^2 q_k^2}{2} + H_{\text{int}}, \qquad (2.2.3)$$

where $\mathbf{P}, \mathbf{M}$ are the linear momentum and the mass of the $\alpha$-particle.

**Remark 2.2.1.** (i) In classical handbooks it assumed that collapse of $\alpha$-particle is caused

only by the interection with atoms of the medium, see for example [9].

(ii) In contrast with a classical case we assume that $\alpha$-particle immediately collapses after

decay in accordance with nonclassical collapse models, see subsection I.2.4-I.2.5.

**Remark 2.2.2.** Schrödinger equation (2.1.1) can be solved by canonical Feynman propagator [9],[26].

**Definition 2.2.1.** Let $\Psi(x)$ be an solution of the Schrödinger equation (2.1.1). The stationary Schrödinger equation (2.1.1) is a weakly well preserved in region $\Gamma \subseteq \mathbb{R}$ by collapsed wave function $\Psi^{\#}(x)$ if the estimate

$$\int_{\Gamma} \left\{ \frac{\partial^2 \Psi^{\#}(x)}{\partial x^2} + \frac{2m}{\hbar^2} [E - U(x)] \Psi^{\#}(x) \right\} dx = O(\hbar^\alpha), \qquad (2.2.4)$$

where $1/4 < \alpha \le 1/2$, is satisfied.

**Definition 2.2.2.** The stationary integral equation (2.2.4) is called nonlocal Schrödinger equation of the order $\hbar^\alpha$ (corresponding to alpha decay).

**Proposition 2.2.1.** The Schrödinger equation (2.1.1) in each of regions I, II, III

is a weakly well preserved by collapsed wave function $\Psi_I^{\#}(x)$, $\Psi_{II}^{\#}(x)$ and $\Psi_{III}^{\#}(x)$ correspondingly.

**Proof.** See Appendix B.

**Definition 2.2.3.** Let us consider the time-dependent Schrödinger equation:

$$i\hbar \frac{\partial \Psi(x,t)}{\partial t} = \hat{\mathbf{H}}\Psi(x,t), \hat{\mathbf{H}} = \frac{\partial^2}{\partial x^2} + \frac{2m}{\hbar^2} [E - U(x)], \qquad (2.2.5) \\ t \in [0, T], x \in \mathbb{R}.$$

The time-dependent Schrödinger equation (2.2.5) is a weakly well preserved by corresponding to $\Psi(x, t)$ collapsed wave function $\Psi^{\#}(x, t)$
$$
\begin{gather*}
\Psi^{\#}(x,t) = \Psi(x,t; \tilde{x}) = \frac{\Re(\tilde{x})\Psi(x,t)}{\|\Re(\tilde{x})\Psi(x,t)\|_2}, \\
\Re(\tilde{x}) = (\pi r_c^2)^{-3/4} \exp\left[-\frac{(\hat{x} - \tilde{x})^2}{2r_c^2}\right],
\end{gather*}
$$

in region $\Gamma \subseteq \mathbb{R}$ if the estimate

$$
\begin{gather*}
\int_{\Gamma} \left\{ i\hbar \frac{\partial \Psi^{\#}(x,t)}{\partial t} - \hat{H} \Psi^{\#}(x,t) \right\} dx = O(\hbar^{\alpha}), \quad 1/4 < \alpha \le 1/2, \tag{2.2.6} \\
t \in [0, T], x \in \mathbb{R},
\end{gather*}
$$

is satisfied.

**Definition 2.2.4.** The time-dependent integral equation (2.2.6) is colled the time-dependent nonlocal Schrödinger equation of the order $\hbar^\alpha$.

**Definition 2.2.5.** Let $\Psi^{\#}(\mathbf{x},t)=\Psi^{\#}(\mathbf{x}_1,\mathbf{x}_2,\dots,\mathbf{x}_d,t)$ be a function
$\Psi(\mathbf{x}_1,\mathbf{x}_2,\dots,\mathbf{x}_d,t;\mathbf{\tilde{x}}_1,\dots,\mathbf{\tilde{x}}_d)$. Let us consider the Probability Current Law

$$
\begin{align}
& \frac{\partial}{\partial t} P(\Gamma, t) + \int_{\partial \Gamma} \mathbf{J}(\mathbf{x}_1, \mathbf{x}_2, \dots, \mathbf{x}_d, t) \cdot \mathbf{n} \, d^{2d}x = 0, \tag{2.2.7} \\
& \mathbf{J}(\mathbf{x}_1, \mathbf{x}_2, \dots, \mathbf{x}_d, t) = \Psi(\mathbf{x}, t) \overline{\nabla \Psi(\mathbf{x}, t)} - \overline{\Psi(\mathbf{x}, t)} \nabla \Psi(\mathbf{x}, t), \nonumber \\
& \qquad \qquad \qquad \qquad \qquad \qquad t \in [0, T], \mathbf{x} \in \mathbb{R}^{3d}, \nonumber
\end{align}
$$

corresponding to Schrödinger equation (2.2.16). Probability Current Law (2.2.7) is a

weakly well preserved by corresponding to $\Psi(\mathbf{x},t)$ collapsed wave function $\Psi^{\#}(\mathbf{x},t)$
in region $\Gamma \subseteq \mathbb{R}^{3d}$ if there exist an wave function $\Psi(\mathbf{x},t)$ such that the estimate

$$
\begin{align}
& \frac{\partial}{\partial t} P(\Gamma, t) + \int_{\partial \Gamma} \mathbf{J}^{\#}(\mathbf{x}_1, \mathbf{x}_2, \dots, \mathbf{x}_d, t) \cdot \mathbf{n} \, d^{2d}x = O(\hbar^\alpha), \tag{2.2.8} \\
& \mathbf{J}^{\#}(\mathbf{x}_1, \mathbf{x}_2, \dots, \mathbf{x}_d, t) = \Psi^{\#}(\mathbf{x}, t) \overline{\nabla \Psi^{\#}(\mathbf{x}, t)} - \overline{\Psi^{\#}(\mathbf{x}, t)} \nabla \Psi^{\#}(\mathbf{x}, t) \nonumber \\
& = O(\hbar^\alpha), \quad 1/4 < \alpha \le 1/2, \nonumber \\
& t \in [0, T], \mathbf{x} \in \mathbb{R}^{3d}, \nonumber
\end{align}
$$

is satisfied.

**Proposition 2.2.2.** Assume that there exist an wave function $\Psi(\mathbf{x}, t)$ such that
the estimate

(2.2.7) is satisfied. Then Probability Current Law (2.2.4) is a weakly well
preserved by corresponding to $\Psi(\mathbf{x}, t)$ collapsed wave function $\Psi^{\#}(\mathbf{x}, t)$ in region $\Gamma \subseteq \mathbb{R}^{3d}$, i.e. the estimate (2.2.8) is satisfied on the wave function $\Psi^{\#}(\mathbf{x}, t)$.

**Definition 2.2.6.** The time-dependent integral equation (2.2.9)

$$ \begin{array}{l} \displaystyle \int_{\Gamma \subseteq \mathbb{R}^{3d}} \left\{ i\hbar \frac{\partial \Psi(\mathbf{x}, t)}{\partial t} - \hat{\mathbf{H}} \Psi(\mathbf{x}, t) \right\} d^{3d}x = 0, \\ \displaystyle \hat{\mathbf{H}} = \frac{\partial^2}{\partial x^2} + \frac{2m}{\hbar^2} [E - U(x)] \\ \qquad t \in [0, T], x \in \mathbb{R}, \end{array} \qquad (2.2.9) $$

is called the time-dependent nonlocal Schrödinger equation corresponding to alpha decay.

### II.3. The nonlocal stochastic nonlinear Schrödinger equation corresponding to alpha decay

Let us consider the nonlocal stochastic nonlinear Colombeau-Schrödinger equation describing the (one-dimensional) evolution of a free quantum particle subject to spontaneous localizations in space. The stochastic dynamics is governed by a standard Colombeau-Wiener process $(W_{\epsilon,t})_{\epsilon}$, defined on the probability space $(\Omega, \mathcal{F}, \mathbb{P})$ with the natural filtration $\{\mathcal{F}_t : t \ge 0\}$ defined on i

$$ \left( \int_{\mathbb{R}} dx [\psi_{\epsilon,T}(x) - \psi_{\epsilon,0}(x)] \right)_{\epsilon} = \int_{\mathbb{R}} dx \times \qquad (2.3.1) $$

$$ \left( \left[ \int_{0}^{T} \left( -\frac{i}{\hbar} \frac{\hat{p}^2}{2m} dt + \sqrt{\lambda} (\hat{q} - \langle \hat{q} \rangle_{\epsilon,t}) \right) \psi_{\epsilon,t}(x) dW_{\epsilon,t} - \frac{\lambda}{2} \int_{0}^{T} (\hat{q} - \langle \hat{q} \rangle_{\epsilon,t})^2 \psi_{\epsilon,t}(x) dt \right]_{\epsilon} \right), $$

where $\hat{q}$ and $\hat{p}$ are the position and momentum operators, respectively, and $\langle \hat{q} \rangle_{\epsilon,t} = \langle \psi_{\epsilon,t} | \hat{q} | \psi_{\epsilon,t} \rangle$.

In order to obtain let us consider corresponding stochastic nonlinear Colombeau-Schrödinger equation

$$ \begin{aligned} (\psi_{\epsilon,T}(x) - \psi_{\epsilon,0}(x))_{\epsilon} = & \left( \int_{0}^{T} \left( -\frac{i}{\hbar} \frac{\hat{p}^2}{2m} \psi_{\epsilon,t}(x) - \frac{\lambda}{2} (q - \langle q \rangle_{\epsilon,t})^2 \psi_{\epsilon,t}(x) \right) dt \right)_{\epsilon} + \\ & + \left( \int_{0}^{T} \sqrt{\lambda} (\hat{q} - \langle \hat{q} \rangle_{\epsilon,t}) \psi_{\epsilon,t}(x) dW_{\epsilon,t} \right)_{\epsilon}. \end{aligned} \qquad (2.3.2) $$

In order to obtain an solution of the Eq.(2.3.2) Let us consider the following Colombeau linear stochastic differential equation
$$ (d\phi_{\epsilon,t}(x))_{\epsilon} = \left( \left[ -\frac{i}{\hbar} \frac{p^2}{2m} dt + \sqrt{\lambda} \hat{q} d\xi_{\epsilon,t} - \frac{\lambda}{2} \hat{q}^2 dt \right] \phi_{\epsilon,t}(x) \right)_{\epsilon}, \quad (2.3.3) $$

where $\xi_{\epsilon,t}$ is a standard Colombeau-Wiener process defined on the probability space $(\Omega, \mathcal{F}, \mathbb{Q})$, where $\mathbb{Q}$ is a new probability measure, whose connection with $\mathbb{P}$ will be clear in what follows. Contrary to Eq.(2.3.2), the above equation does not preserve the norm of statevectors, so let us define the normalized vectors:

$$ (\psi_{\epsilon,t})_{\epsilon} = \begin{cases} (\phi_{\epsilon,t} / \|\phi_{\epsilon,t}\|)_{\epsilon} & \text{if } (\|\phi_{\epsilon,t}\|)_{\epsilon} \neq 0 \\ \text{a fixed unit vector} & \text{if } (\|\phi_{\epsilon,t}\|)_{\epsilon} = 0 \end{cases} \quad (2.3.4) $$

By using Colombeau-Itô calculus, it is not difficult to show that $(\psi_{\epsilon,t})_{\epsilon}$ defined by (2.3.4) is a solution of Eq. (2.3.2), whenever $(\phi_{\epsilon,t})_{\epsilon}$ solves Eq. (2.3.3). We now briefly explain the relations between the two probability measures $\mathbb{Q}$ and $\mathbb{P}$, and between the two Colombeau-Wiener processes $(\xi_{\epsilon,t})_{\epsilon}$ and $(W_{\epsilon,t})_{\epsilon}$.

The key property of Eq.(2.3.3) is that $(p_{\epsilon,t})_{\epsilon} = (\|\phi_{\epsilon,t}\|^2)_{\epsilon}$ is a martingale satisfying the following Colombeau stochastic differential equation

$$ (dp_{\epsilon,t}) = 2\sqrt{\lambda} (\langle q \rangle_{\epsilon,t} p_{\epsilon,t} d\xi_{\epsilon,t})_{\epsilon}, \quad (2.3.5) $$

with $(\langle q \rangle_{\epsilon,t})_{\epsilon} = (\langle \psi_{\epsilon,t} | q | \psi_{\epsilon,t} \rangle)_{\epsilon}$. As a consequence of the martingale property (and assuming, as we shall always do, that $\|\phi_{\epsilon,0}\| = 1$) $(p_{\epsilon,t})_{\epsilon}$ can be used to generate a new probability measure $\tilde{\mathbb{P}}$ on $(\Omega, \mathcal{F})$ We choose now $\mathbb{Q}$ in such a way that the new measure $\tilde{\mathbb{P}}$ coincides with $\mathbb{P}$. Given Girsanov's theorem provides a simple relation between the Colombeau-Wiener process $(\xi_{\epsilon,t})_{\epsilon}$ defined on $(\Omega, \mathcal{F}, \mathbb{Q})$, and the Colombeau-Wiener process $(W_{\epsilon,t})_{\epsilon}$ defined on $(\Omega, \mathcal{F}, \mathbb{P})$

$$ (W_{\epsilon,t})_{\epsilon} = (\xi_{\epsilon,t})_{\epsilon} - 2 \left( \sqrt{\lambda} \int_{0}^{t} \langle q \rangle_{\epsilon,s} ds \right)_{\epsilon}. \quad (2.3.6) $$

The above results imply that one can find the solution $(\psi_{\epsilon,t})_{\epsilon}$ of Eq.(2.3.2), given the initial condition $(\psi_{\epsilon,0})_{\epsilon}$, by using the following procedure:(i) find the solution $(\phi_{\epsilon,t})_{\epsilon}$ of Eq. (2.3.3), with the initial condition $(\phi_{\epsilon,0})_{\epsilon} = (\psi_{\epsilon,0})_{\epsilon}$, (ii) normalize the solution: $(\phi_{\epsilon,t})_{\epsilon} \rightarrow (\psi_{\epsilon,t})_{\epsilon} = (\phi_{\epsilon,t})_{\epsilon} / (\|\phi_{\epsilon,t}\|)_{\epsilon}$, (iii) make the substitution: $(d\xi_{\epsilon,t})_{\epsilon} \rightarrow (dW_{\epsilon,t})_{\epsilon} + 2\sqrt{\lambda} (\langle q \rangle_{\epsilon,t})_{\epsilon}$. We choose now, as a solution, a single-Gaussian wavefunction:

$$ (\phi_{\epsilon,t}(x))_{\epsilon} = \left( [-a_t(x - \bar{x}_{\epsilon,t})^2 + i\bar{k}_{\epsilon,t}x + \gamma_{\epsilon,t}] \right)_{\epsilon}, \quad (2.3.7) $$

where $a_t$ and $(\gamma_{\epsilon,t})_{\epsilon}$ are supposed to be complex functions of time, while $(\bar{x}_{\epsilon,t})_{\epsilon}$ and $(\bar{k}_{\epsilon,t})_{\epsilon}$ are taken to be real. By inserting (2.3.7) into Eq.(2.3.3), one finds the following sets of Colombeau stochastic differential equations for the relevant parameters:
$$
\begin{align*}
& da_t = \left[ \lambda - \frac{2i\hbar}{m} (a_t)^2 \right] dt, \\
& (d\bar{x}_{\epsilon,t})_\epsilon = \frac{\hbar}{m} (\bar{k}_{\epsilon,t} dt)_\epsilon + \frac{\sqrt{\lambda}}{2 \operatorname{Re}(a_t)} \left[ (d\xi_{\epsilon,t})_\epsilon - 2\sqrt{\lambda} (\bar{x}_{\epsilon,t} dt)_\epsilon \right], \\
& (d\bar{k}_{\epsilon,t}) = -\sqrt{\lambda} \frac{\operatorname{Im}(a_t)}{\operatorname{Re}(a_t)} \left[ (d\xi_{\epsilon,t})_\epsilon - 2\sqrt{\lambda} (\bar{x}_{\epsilon,t} dt)_\epsilon \right], \\
& (d\operatorname{Re}(\gamma_{\epsilon,t}))_\epsilon = \left[ \lambda(\bar{x}_{\epsilon,t}^2)_\epsilon + \frac{\hbar}{m} \operatorname{Im}(a_t) \right] dt + \sqrt{\lambda} (\bar{x}_{\epsilon,t} [d\xi_{\epsilon,t} - 2\sqrt{\lambda} \bar{x}_{\epsilon,t} dt])_\epsilon, \\
& (d\operatorname{Im}(\gamma_{\epsilon,t}))_\epsilon = \left[ -\frac{\hbar}{m} \operatorname{Re}(a_t) - \frac{\hbar}{2m} (\bar{k}_{\epsilon,t}^2)_\epsilon \right] dt + \sqrt{\lambda} \frac{\operatorname{Im}(a_t)}{\operatorname{Re}(a_t)} (\bar{x}_{\epsilon,t} [d\xi_{\epsilon,t} - 2\sqrt{\lambda} \bar{x}_{\epsilon,t} dt])_\epsilon.
\end{align*}
$$

For a single–Gaussian wavefunction, the two equations for $(\gamma_{\epsilon,t})_{\epsilon}$ can be omitted since the real part of $(\gamma_{\epsilon,t})_{\epsilon}$ is absorbed into the normalization factor, while the imaginary part gives an irrelevant global phase. The normalization procedure is trivial, and also the Girsanov transformation (2.3.6) is easy since, for a Gaussian wavefunction like (2.3.7), one simply has $(\langle q \rangle_{\epsilon,t})_{\epsilon} = (\bar{x}_{\epsilon,t})_{\epsilon}$. We then have the following set of stochastic differential equations for the relevant parameters:

$$
\begin{align*}
& da_t = \left[ \lambda - \frac{2i\hbar}{m} (a_t)^2 \right] dt, a_t|_{t=0} = a_0, \\
& d\bar{x}_t = \frac{\hbar}{m} \bar{k}_t dt + \frac{\sqrt{\lambda}}{2 \operatorname{Re}(a_t)} dW_t, \bar{x}_t|_{t=0} = \bar{x}_0, \tag{2.3.9} \\
& d\bar{k}_t = -\sqrt{\lambda} \frac{\operatorname{Im}(a_t)}{\operatorname{Re} a_t} dW_t, \bar{k}_t|_{t=0} = \bar{k}_0.
\end{align*}
$$

Eq. (2.3.9) for $a_t$ can be easily solved

$$ a_t = c \tanh[b t + k], \qquad (2.3.10) $$

with:

$$ c = (1 - i) \frac{1}{2} \sqrt{\frac{m\lambda}{\hbar}}, \quad b = (1 + i) \sqrt{\frac{\hbar\lambda}{m}}, \quad k = \tanh^{-1}\left[\frac{a_0}{c}\right]. \qquad (2.3.11) $$

After some algebra, one obtains the following analytical expressions for the real and the imaginary parts of $a_t$:

$$
\begin{align}
\operatorname{Re}(a_t) &= \frac{\lambda}{\omega} \frac{\sinh(\omega t + \varphi_1) + \sin(\omega t + \varphi_2)}{\cosh(\omega t + \varphi_1) + \cos(\omega t + \varphi_2)}, \notag \\
\operatorname{Im}(a_t) &= -\frac{\lambda}{\omega} \frac{\sinh(\omega t + \varphi_1) - \sin(\omega t + \varphi_2)}{\cosh(\omega t + \varphi_1) + \cos(\omega t + \varphi_2)}. \tag{2.3.12}
\end{align}
$$

An important property of $\operatorname{Re}(a_t)$ is positivity: $a_0 > 0 \Rightarrow a_t > 0$.

The quantities $(\langle q \rangle_{\epsilon,t})_{\epsilon} = (\bar{x}_{\epsilon,t})_{\epsilon}$ and $(\langle p \rangle_{\epsilon,t})_{\epsilon} = \hbar(\bar{k}_{\epsilon,t})_{\epsilon}$, corresponding to the peak of the Gaussian wavefunction in the position and momentum spaces, respectively, satisfy the following stochastic differential equations:
$$ (d\langle q \rangle_{\epsilon,t})_{\epsilon} = \frac{1}{m} (\langle p \rangle_{\epsilon,t} dt)_{\epsilon} + \sqrt{\lambda} \frac{1}{2 \operatorname{Re}(a_t)} (dW_{\epsilon,t})_{\epsilon}, \\ (d\langle p \rangle_{\epsilon,t})_{\epsilon} = -\sqrt{\lambda} h \frac{\operatorname{Im}(a_t)}{\operatorname{Re}(a_t)} (dW_{\epsilon,t})_{\epsilon}. \qquad (2.3.13) $$

We now study the time evolution of the superposition of two Gaussian wavefunctions; such an analysis is interesting since it allows to understand in a quite simple and clear way how the reduction mechanism works, i.e. how the superposition of two different position states is reduced into one of them. To this purpose, let us consider the following wavefunction:

$$ \begin{aligned} \phi_t^D(x) &= \phi_{1t}(x) + \phi_{2t}(x) = \\ &= [-a_{1t}(x - \bar{x}_{1t})^2 + i\bar{k}_{1t}x + \gamma_{1t}] \\ &\quad + [-a_{2t}(x - \bar{x}_{2t})^2 + i\bar{k}_{2t}x + \gamma_{2t}]; \end{aligned} \qquad (2.3.14) $$

we follow the strategy outlined in Sec. II, by first finding the solution of the linear equation.

Because of linearity, $\phi_t^D(x)$ is automatically a solution of Eq. (ref: le), provided that its

## II.4. Einstein's 1927 gedanken experiment revisited.

During the famous 5th Solvay conference in 1927, Einstein [39] considered a single particle which, after diffraction in a pin-hole encounters a “detection plate” (e.g. in the case of photons, a photographic plate), see Fig 2.2.1. We simplify this thought experiment, though keeping the essence, by replacing the “detection plate” by two detectors. Einstein noted that there is no question that only one of them can detect the particle, otherwise energy would not be conserved. However, he was deeply concerned about the situation in which the two detectors are space-like separated, as this prevents - according to relativity - any possible coordination among the detectors: “It seems to me,” Einstein continued, “that this difficulty cannot be overcome unless the description of the process in terms of the Schrödinger wave is supplemented by some detailed specification of the localization of the particle during its propagation. I think M. de Broglie is right in searching in this direction.”
<figure><img src="image_20.png" /><figcaption>Pic. 2.4.1. Einstein's 1927 gedanken experiment.
A and B are points on the photographic plate,
for which the events of detection can be space-
like separated from each other. Adapted from [39].</figcaption></figure>

But what happened to Einstein's original "Gedanken experiment"? This simple -
with today's technology - experiment had been done originally by T. Guerreiro, B.
Sanguinetti, H. Zbinden N. Gisin, and A. Suarez, see [53]. This experiment consists
in verifying that when a single photon is thrown at a beam splitter, it is detected in
only one arm, i.e. the probability P<sub>A/B</sub> of getting a coincidence between the two
detectors A and B is much smaller than the product of the probabilities of detection
on each side P<sub>A</sub> × P<sub>B</sub>, as would be expected in the case of uncorrelated events. The
experimental setup is shown in Fig. 2.2.2 and consists of a source of heralded
single photons which is coupled into a single mode fiber and injected into a fiber
beamsplitter (BS). Each of the two outputs of the beamsplitter goes to a single
photon detector (IDQ ID200), detector A being close to the source and detector B
being separated by a distance of approximately 10 meters.
<figure><img src="image_21.png" /><figcaption>Pic.2.2.2. Experimental setup: photon pairs a regenerated by Spontaneous Parametric Down Conversion at the wavelengths of 1550 nm and 810 nm. These pairs are split by a dichroic mirror (DM), and the 810 nm photon is sent to detector D,used to herald the presence of the1550 nm photon which follows to the beam splitter (BS). Arbitrary electronic delays were applied beforeTDC to ensure the coincidence peak swould remain on scale. Adapted from [53].</figcaption></figure>

If we ensure that the fiber lengths before each detector are equal by inserting a
10 m (50ns) fiber delay loop before detector A, the detections will happen
simultaneously in some reference frame, thus being space-like separated (a signal
would take 33 ns to travel between the two detectors at the speed of light;
simultaneity of detection is guaranteed to within 1ns by the matched length of fiber
both before and inside the detectors). It is also possible to make the detections
time-like separated by removing the 10m delay line from detector A and adding it to
detector B.

<figure><img src="image_22.png" /><figcaption>Pic.2.2.4.[53]. Spacetime diagrams for spacelike (i) and timelike (ii) configurations. A and B represent the locations of the detectors. Adapted from [53].</figcaption></figure>

First one measure the probabilities of detecting a photon at detector A or at
detector B given that a heralding photon has been detected at H. We denote R<sub>HA</sub>
the total number of coincident counts at detector H and detector A during the time
of measurement, and $R_{H(A)}$ the total number of counts at detector H alone during
the same measurement; $R_{HB}$ and $R_{H(B)}$ denote similar quantities for the
measurement with H and B. Next we measure the probability of detectors A and B
clicking at the same time, again given a heralding signal. $R_{HAB}$ denotes the number
of triple coincident counts at the detectors H, A and B, and $R_{H(AB)}$ the total number
of counts at detector H alone during the same measurement. All these quantities
are measured directly for both a space-like configuration and a time-like
configuration.

Next one measure the probability of detectors A and B clicking at the same
time, again given a heralding signal. $R_{HAB}$ denotes the number of triple coincident
counts at the detectors H, A and B, and $R_{H(AB)}$ the total number of counts at
detector H alone during the same measurement. All these quantities are measured
directly for both a space-like configuration and a time-like configuration.

<figure><img src="image_23.png" /><figcaption>Pic.2.2.5.Coincidences between the heralding detector and each of the detectors A (red) and B (blue) with spacelike separation,measured in a window of 1ns during a time period of 10 minutes. $R_{HA} = 9.49 \times 104/10$ min, $R_{HB} = 6.39 \times 104/10$ min.The noise is on average:$R_N = 50/10$ min. Adapted from [53].</figcaption></figure>