# Porosity of the Frontier Between Classical and Quantum Mechanics

Patrice Melinon

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## To cite this version:

Patrice Melinon. Porosity of the Frontier Between Classical and Quantum Mechanics. International journal of quantum foundations, 2025. hal-05030530

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Submitted on 11 Apr 2025

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Original Paper

# Porosity of the Frontier Between Classical and Quantum Mechanics

Patrice Mélinon

Université de Lyon, F-69000, France; Univ. Claude Bernard Lyon 1, Institut Lumière Matière; CNRS, UMR 5306; F69622 Villeurbanne Cedex, France.

E-mail: patrice.melinon@univ-lyon1.fr

Received: 7 December 2024 / Accepted: 14 March 2025 / Published: 21 March 2025

**Abstract:** Despite its status as a core part of contemporary physics, there is no consensus among physicists on the question of what, if anything, the success of quantum theory is telling us about the physical world. This gives rise to the collection of debates and discussions known as "the interpretation of quantum mechanics". Describing quantum mechanics is complicated by there being so many interpretations of it. In text books, classical mechanics is an approximation of the quantum mechanics when $\hbar \to 0$. In a first part, we recall the fundamental differences between the classical mechanics in different formalisms: Newtonian, Lagrangian and Hamiltonian (Jacobi). A second part is dedicated to the deep similarities and differences between classical and quantum mechanics. The third part deals with the misunderstanding between scientific communities about the meaning $\hbar \to 0$. The last part deals with the porous frontier between classical and quantum physics to the light of the algebraic topology illustrated by two major theorems: Borsuk Ulam and Non squeezing Gromov’s theorem.

**Keywords:** Topology, quantum mechanics
# 1. Introduction: Classical physics; Mechanics

## 1.1. Outlook

Newtonian mechanics is a classical theory of physics that describes the motion of particles under the influence of forces $F$. It is based on Isaac Newton's three independent laws of motion. The forces depend on positions, velocities and time $F(\dot{x}, x, t) = m\ddot{x}$ that is to say forces "live" in $\mathbb{R}^3 \times \mathbb{R}^3 \times \mathbb{R}$ space. Lagrangian mechanics is a new formulation of classical mechanics that uses the principle of least action to derive the equations of motion. The Noether principle [1] asserts that there is a one-to-one correspondence between symmetries and integrals of motion. The role of the symmetry in physics is natural in Lagrangian formalism. Lagrangian $\mathcal{L}(x_i, \dot{x}_i, t)$ here is regarded as a function of three variables, the position $x_i$, the velocity $\dot{x}_i$, and time $t$. The evolution of the system writes

$$ \frac{d}{dt} \left( \frac{\partial \mathcal{L}}{\partial \dot{x}_i} \right) = \frac{\partial \mathcal{L}}{\partial x_i} \qquad (1) $$

Note that Lagrangian $\mathcal{L}(x_i, \dot{x}_i, t)$ is kinetic energy $E_c$ minus potential energy $U(\mathcal{L} = E_c - U)$, $E_c$ and $U$ being two real (scalar) valued functions.

The position coordinates and velocity components are all independent variables but velocity is formally the first derivate of position.

Hamiltonian mechanics is another reformulation of classical mechanics. It introduces the concept of canonical coordinates $x_i$ and momenta $p_i$. One obtains a set of two equations instead one in Lagrangian formalism but the two variables are really independent.

$$ \frac{\partial x_i}{\partial t} = \frac{\partial \mathcal{H}}{\partial p_i}(x_i, p_i) \qquad \frac{\partial p_i}{\partial t} = -\frac{\partial \mathcal{H}}{\partial x_i}(x_i, p_i) \qquad (2) $$

Hamiltonian $H$ is the sum ($\mathcal{H} = E_c + U$). At a first glance all the formulations are identical assuming kinematic equivalence stating that the kinematics of the system is enough to reconstruct the system and vice-versa. For a given time $t$, the Legendre transformation of $\mathcal{L}$ is defined as the map $(x_i, \dot{x}_i) \to (x_i, p_i)$ which is assumed to have a smooth inverse $(x_i, p_i) \to (x_i, \dot{x}_i)$. They are some counterexamples where matrices are not invertible (see annex .6). Even though there are some links between, not all Hamiltonian systems are Newtonian and not all Newtonian systems are Hamiltonian [2]. All Lagrangian systems are both Newtonian and Hamiltonian [3,4]. In summary, Newtonian formalism cannot be generalize to relativistic or quantum mechanics. Lagrangian is ideally suited for relativity and permits the extension to quantum field theory as mentioned by Feynman [5]. Relation between Hamiltonian and relativity is not obvious but the evolution of the system in Hamiltonian mechanics traduce the symplectic flow equations, symplectic structure being the natural way for introducing quantum mechanics as discussed now.
**Figure 1.** Illustration of the difference between Riemannian and symplectic geometries. (A) An ellipsoid (the symplectic egg introduced by de Gosson) [6] is cut through its center by different planes with $\theta$ angle. (B) the cross-sectional areas have different values in Riemannian geometry and a constant value in symplectic geometry that is hard to imagine!

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## 1.2. Hamiltonian system

### 1.2.1. Symplectic geometry versus Riemannian geometry

Before to discuss the properties of the hamiltonian, let us introduce the geometry where hamiltonian live that is to say the symplectic geometry and everything you need to understand the rest. A symplectic space first introduced by Weyl [7] can be built as follows. At each point $x_i, i = 1, 2, 3$ in the three dimensional space (our universe) corresponding to the coordinates that specify a particle's position, we can assign three coordinates $p_i, i = 1, 2, 3$ associated to the momentum. The combination forms a vector with six components in $\mathbb{R}^6$ that is a six-dimensional phase space or symplectic space.

A symplectic [7] form is a 2-form $(M, \omega)$ where $M$ is a differentiable (orientable) manifold without boundary, $\omega$ the symplectic structure $\omega = \sum_{i=1}^n dx_i \wedge dp_i$, satisfying an algebraic condition the non degeneracy $(\omega(x, p) = 0)$ for all $p$ then $x = 0$) and an analytical condition, the closedness $(d(\omega(x, p)) = 0)$. If Riemannian may exist in any dimension, symplectic exists only in even dimension. There are no local invariants (apart from dimension) in symplectic geometry while curvature is a local invariant in Riemannian geometry. Riemannian metric is a non degenerate form, while symplectic is as well. Note $S^{2n}$ is not symplectic, for any $n > 1$: $S^2$ is symplectic ($S^n$ is the n-sphere). Another difference is the vector field. In Riemann geometry for a smooth function $f: M \to \mathbb{R}$ its gradient vector $\nabla f$ is normal to the level surface, in symplectic geometry the Hamiltonian
vector field $X_f$ is tangent to the level surface. So the vector field in symplectic geometry looks like a "skew-gradient" (skew symplectic means $\omega(x, p) = -\omega(p, x)$). Formally in algebraic topology, a symplectic manifold does not allow non-trivial vector/tangent sub bundles [8] of the tangent bundle, while we can define subbundles of the tangent bundle [9] in a Riemannian manifold.

So what is behind these abstract concepts for an average physicist. A celebrated theorem of Darboux [10] asserts that any symplectic manifold is locally equivalent to an Euclidean space with its standard symplectic structure. Therefore, all symplectic manifolds are (locally) symplectomorphic (a closed orbit can also be thought of as a "fixed point" of a special kind of function called a symplectomorphism). In particular we can take the standard symplectic space $M \equiv \mathbb{R}^{2n}$ with coordinates $\{x_1, x_2, ..., x_n, p_1, p_2, ..., p_n\}$ and symplectic structure $\omega = \sum_{i=1}^{n} dx_i \wedge dp_i$ that corresponds to Hamiltonian's formalism ($x_i$ the coordinates, $y_i \equiv p_i$ the momentum). We can also define the symplectic volume. Defining a volume measure over a submanifold is equivalent to the integration over that submanifold $V$. In Riemann geometry, the procedure is easy by integrating over an element $dx^n$ with the limit $dx^n \to 0^+$ that is to say $Vol = \int_V dx^n$. Such procedure needs the notion of distance "x". Since for a symplectic manifold the notion of distance has no sense, we cannot do this integration. We can define the symplectic volume $Vol_S$

$$Vol_S = k! \sum_{1 \le i_1 \le i_2 \dots \le i_k \le 1} [dx_1 \wedge dp_1 \wedge dx_2 \wedge dp_2 \dots \wedge dx_k \wedge dp_k] \quad (3)$$

For example consider $(M, \omega) \in \mathbb{R}^4$, the symplectic volume writes $2(dx_1 \wedge dp_1 \wedge dx_2 \wedge dp_2)$. Liouville's theorem states that this phase space volume is conserved over time with respect to the dynamical system generated by Hamilton's equations (ie symplectic geometry). This theorem also states that the motion of an ensemble behaves like the flow of an incompressible fluid (ie volume preservation) in phase space. This is a sort of invariant property.

It can be shown that Hamiltonian flows possess an even deeper property than volume preservation (Poincaré-Birkhoff fixed point theorem [11]). For a symplectic shape, the cross-sectional areas will be the same no matter which plane cuts the shape as illustrated in figure 1. In other words, if we consider motion of a differential element in the symplectic phase space, even if the shape is distorted during the motion the area is always preserved by the symmetry. This is an amazing feature that we cannot imagine in our standard representation (Riemann geometry) but this property will be the key issue for our discussion (the proof of the symplectic egg property is given in reference [6]).

The area preservation needs two properties:

i) *The symplectic geometry*

ii) *The use of fundamental conjugate coordinates ($x_i, p_i$)*

Once again, the fundamental conjugate coordinates are the coordinates underlying the
Hamiltonian that is symplectic. This is in contrast to $(x_i, \dot{x}_i)$ which are the coordinates that underlie the Lagrangian.

### 1.2.2. Hamiltonian mechanics

We now define the phase space $M$ as the space of all positions and momenta of the particle. Hamiltonian systems tend to have a lot of periodic orbits (ie closed orbits [12]). It is reasonable to expect that a given Hamiltonian system with a proper smooth Hamiltonian has a closed orbit on every regular energy level. This is a consequence of the Poincaré recurrence theorem [13] stating that certain dynamical systems will, after a sufficiently long but finite time, return to a state arbitrarily close to (for continuous state systems), or exactly the same as (for discrete state systems), their initial state. Note that examples of Hamiltonian flows without periodic orbits on a compact energy level are very scarce, they are few counterexamples [14]. Bound orbits are valid when central forces (radial) scalar potentials coincide with inverse-square central force (gravitation, Coulomb) or radial harmonic oscillator (Bertrand's theorem [15]). Let us consider closed orbits, Hamilton's equations of motion have the antisymmetric form

$$ \frac{\partial x_i}{\partial t} = \frac{\partial H}{\partial p_i}(x, p) \quad \frac{\partial p_i}{\partial t} = -\frac{\partial H}{\partial x_i}(x, p) \qquad (4) $$

The utility to change to Hamiltonian mechanics is the existence of a Poisson structure which uses the symplectic structure. More generally, if a transformation $q(x, p) = (x_0, p_0)$ of phase space $\mathbb{R}^{2n}$ is canonical, then its Jacobian matrix calculated at any phase space point $(x, p)$ were $q$ is defined, is symplectic. To check it, define the symplectic matrix, with the standard symplectic Weyl matrix $\bar{J}$

$$ \bar{J} = \begin{pmatrix} 0 & I \\ -I & 0 \end{pmatrix} \qquad (5) $$

where $I$ is the $n \times n$ identity matrix. A real $2n \times 2n$ matrix $\bar{S}$ is symplectic if it satisfies the two conditions

$$ \bar{S}^T \cdot \bar{J} \cdot \bar{S} = \bar{J} \quad \bar{S} \cdot \bar{J} \cdot \bar{S}^T = \bar{J} \qquad (6) $$

Equations 4 are of this form $(q = (x, p))$

$$ \frac{\partial q}{\partial t} = \bar{J} \nabla_q H(q) \qquad (7) $$

This equation represents the flow of the Hamiltonian vector field.
## 2. Classical versus quantum mechanics: old and modern picture

Classical mechanics as the limit of quantum mechanics was already reported in the literature [16]. Writing the time dependent Schrödinger equation for one particle of mass *m* with $\Psi$ wavefunction in a *U* scalar potential

$$i\hbar \frac{\partial \Psi}{\partial t} = -\frac{\hbar^2}{2m} \nabla_i \nabla_i \Psi + U \Psi \quad (8)$$

The wavefunction is of the form $\Psi = A \exp iS(t, x_i)/\hbar$ where *S* is termed the Hamilton principal function (ie the action). The previous equation reads

$$\frac{\partial S(t, x_i)}{\partial t} + \frac{|\nabla_i S(t, x_i)|^2}{2m} + U = 0 \quad (9)$$

This is the standard form for the Hamilton Jacobi function [17]

$$\frac{\partial S(t, x_i)}{\partial t} + H = 0 \quad (10)$$

with

$$H = \frac{|\nabla_i S(t, x_i)|^2}{2m} + U \quad (11)$$

Note that the classical Hamiltonian is for a particle with mass *m*, the analogy with quantum particle (Schrödinger equation) needs the duality wave-particle which is one the cornerstones of old quantum mechanics. For a wave plane $\Psi_i = A \exp(ik_i x)$ then $H = \frac{|p_i|^2}{2m} + U$ with $p_i = \hbar k_i$. Note that the particle restores the non duality (in classical picture the particle is a mass no a wave, the "mass" is just finite frequency at zero wave number) taking $\hbar \to 0$ that justify in text books: classical mechanics is an approximation of the quantum mechanics when $\hbar \to 0$.

Now, suppose that $f(p_i, x_i, t)$ is a function on the solution's trajectory. Hamilton's equations of motion (equations 4) have an equivalent expression in terms of the Poisson bracket

$$\frac{d}{dt} = \{H, f\} + \frac{\partial f}{\partial t} \quad (12)$$

with $\{\cdot, \cdot\}$ the Poisson brackets

$$\{A, B\} = \frac{\partial A}{\partial x_i} \frac{\partial B}{\partial p_i} - \frac{\partial B}{\partial x_i} \frac{\partial A}{\partial p_i} \quad (13)$$

with the anticommutativity property $\{AB\} = -\{BA\}$

The Poisson brackets of the canonical coordinates are

$$\{x_i, x_j\} = 0, \{p_i, p_j\} = 0 \text{ and } \{x_i, p_j\} = \delta_{ij} \quad (14)$$
In a "modern" picture, we can replace the Poisson brackets in classical mechanics by the corresponding commutators

$$ \frac{-i}{\hbar} [\hat{A}, \hat{B}] = \frac{-i}{\hbar} [\hat{A}\hat{B} - \hat{B}\hat{A}] \qquad (15) $$

In summary, a classical observable A with no explicit time dependence in Hamiltonian mechanics evolves according to $\frac{\partial A}{\partial t} = \{A, H\}$ and a quantum observable evolves as $\frac{\partial \hat{A}}{\partial t} = \frac{-i}{\hbar} [\hat{A}, \hat{B}]$ that is the equivalence of Heisenberg's equation to the Schrödinger equation where the operator has no explicit time dependence.

### 3. Puzzling of the previous picture

#### 3.1. Gromov's theorem

Prior introducing Gromov's theorem let us define an open Ball (does not contain the boundary ie. the surface $\delta B^{2n}(r)$) with r radius

$$ B^{2n}(r) = \{(x, p) \in \mathbb{R}^{2n} : \sum_{i=1}^{n} |x_i^2| + |p_i^2| < r^2\} \qquad (16) $$

For a physicist, it's the description of a full sphere without its surface. For the sake of simplicity, we discuss one dimensional system $n = 1$

$$ B^2(r) = \{(x, p) \in \mathbb{R}^2 : |x|^2 + |p|^2 < r^2\} \qquad (17) $$

The symplectic Ball is defined as follows: if $|z|^2 = |x|^2 + |p|^2$ then we can construct the image $SB^2(r)$ of $B^2(r)$ by a symplectic matrix $S$. It consists of all points $z'$ such as $z = S^{-1}z'$. The symplectic Ball comes

$$ B^2(r) = \{(x, p) \in \mathbb{R}^2 : |S^{-1}z'|^2 < r^2\} \qquad (18) $$

Gromov's theorem [18] which is often called the "symplectic non-squeezing theorem" is summarized as follows:

Let define an open symplectic Ball of radius $r$

$$ B^{2n}(r) = \{(x, p) \in \mathbb{R}^{2n} : \sum_{i=1}^{n} |x_i^2| + |p_i^2| < r^2\} \qquad (19) $$

Let define an open symplectic cylinder of radius $R$

$$ Z^{2n}(r) = \{(x, p) \in \mathbb{R}^{2n} : x_1^2 + p_1^2 < R^2\} \qquad (20) $$

Cylinder and Ball endowed with the symplectic form $\omega = \sum_{i=1}^{n} dx_i \wedge dp_i$
**Figure 2. Restoring point zero energy in classical mechanics.** Classical harmonic motion of a particle governed by the Hamiltonian $H = \frac{\omega}{2}(\bar{p}^2 + \bar{x}^2)$, where the scaled variables are defined as $\bar{p} = p/\sqrt{m\omega}$ and $\bar{x} = \sqrt{m\omega x}$. **Left panel:** motion of the particle in the potential labeled from 1 to 5 (increasing time), **right panel (top)** phase space trajectory and **right panel (down)** the time series $\bar{x}(t)$ and $\bar{p}(t)$. Prior the amazing Gromov's theorem, people thank that the impulsion and the position could be merged on a single value zero. This is not true and the minimum area enclosed in the minimal orbit trajectory is shown in yellow. This corresponds in the potential curve to a minimal value $1/2(\hbar\omega)$. Gromov's theorem makes a bridge between classical and quantum mechanics.

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*Theorem:* No canonical transformation can squeeze a ball $B^{2n}(r)$ through a circular hole in a plane $\Sigma_j$ of conjugate coordinates $x_i, p_i$ with radius $r < R$.

## 3.2. Heisenberg, Maslov and Gromov

If $r < R$, we cannot embed $B^{2n}(R)$ into $Z^{2n}(r)$ by any canonical transformation that is to say that if a collection of particles initially spread out all over the unit ball $B^{2n}(R)$, then one cannot squeeze the collection into a statistical state in which the momentum and position in the $(q_1, p_1)$-direction spreads out less than initially [22] (area preservation, see Poincaré-Birkoff's theorem)). Gromov's theorem brings a new insight in the Heisenberg formula. This can be briefly summarized as follows

We assume that the phase-space evolution of a system is governed by Hamilton-Jacobi's equations [23,24]
**Figure 3.** Relationship between Maslov indice, Heisenberg and Gromov's **non squeezing theorem** (A) Intersection of the $\gamma$ loop corresponding for example with a 1D harmonic oscillator with the Maslov cycle (see text). In this case the Maslov term $\mu(\gamma) = 2$. The line labeled $\Lambda(1)$ is the 1-dimensional Lagrangian Grassmannian $\mathcal{G}(1)$. The Grassmannian $Gr(n, k)$ is the set of $k$-dimensional subspaces in an $n$-dimensional vector space. (B) Constant energy curves (energy contours) in phase space for the double well potential $V(x) = -ax^2+bx^4$ ($a > 0, b > 0$) (label (C)). For convenience, we take the values proposed by Morita et al $a = 1, b = 0.2$ [19]. (D) critical value corresponding to the zero energy in the classical mechanics approach. Taking into account the winding number (red arrow for counter clockwise (positive Maslov index), blue arrow for clockwise (negative Maslov index)) [20] the Maslov indice is zero that corresponds to $\hbar \to 0$ [21]. (E) Applying Gromov's theorem, this is forbidden. Non squeezing theorem needs a gap opening at the critical point, the gap (the diameter of the symplectic cylinder) corresponds to $2\sqrt{\hbar}$ the associated energy being $1/2\hbar\omega$ (zero point energy). This curve has a non zero Maslov index restoring a quantum behavior with the zero point energy $1/2\hbar$. Note that the curve in (D) has a transverse crossing. For the two values $x \approx \pm 1.581$ corresponding to the ground state in each valley, we restore the pure harmonic oscillator displayed in figure 2. Since Maslov index is a topological invariant, the crossing induces a topological phase transition ($\mu(\gamma) : 2 \to 0$) that is forbidden. Keeping the invariance of the Maslov index induces the gap opening as predicted by Gromov's non squeezing theorem. Note that the gap opening corresponds to the well known degeneracy lift by Jahn's Teller effect.

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$$ \frac{\partial x_i}{\partial t} = \frac{\partial H}{\partial p_i}(x, p) \quad \frac{\partial p_i}{\partial t} = -\frac{\partial H}{\partial x_i}(x, p) \qquad (21) $$

In the case of one degree of freedom (that is to say the phase space is $\mathbb{R}^n \times \mathbb{R}^n = \mathbb{R}^{2n} = \mathbb{R}^2$, $n = 1$) one considers a cloud of points lying in the phase plane, and consisting of a great number points that corresponds to a couple of position/momentum measurement of a physical system with one degree of freedom $(x_1, p_1), (x_2, p_2), (x_3, p_3), \dots, (x_k, p_k), \dots$. Suppose that this cloud is, at time $t$ spherical so it is represented by a phase space closed to a metric ball $B(r)$ with center $(0,0)$ and radius $r$ [24–26] in Euclidean space:

$$ B^2(r) : |x|^2 + |p|^2 \le r^2 \qquad (22) $$

This inequality is moreover conserved in time under a Hamiltonian evolution: it is true for all times, past, initial time ($t = 0$) and future (this is the same remark for the entanglement).

The orthogonal projection of that ball on any plane of coordinates will always be a circle with area $\pi r^2$ (see figure 1). According to Liouville’s theorem [27], it will distort and may take after a while a very different shape, while keeping constant volume. However, Gromov’s [18] "symplectic non-squeezing theorem" states that the projections of that deformed ball on any plane of conjugate coordinates $(x_i, p_i)$ will never decrease below its original value $\pi R^2$ (contrary to non conjugate coordinates $(x_i, p_j)_{i \neq j}$). In other words, Gromov showed that it is impossible for a Hamiltonian phase to squeeze a ball into a cylinder of a smaller radius. As a consequence, the symplectic area of a ball $B^{2n}(R)$ that correspond to the symplectic cylinder, is independent of $n$ ($\mathbb{R}^{2n}$) as it is constant ($A[B^{2n}(R)] = \pi R^2$). Conversely, the volume of the Ball $B^{2n}(R)$ is a function of $n$ (the Ball is defined with the standard Euclidean distance)

$$ V[B^{2n}(R)] = \pi R^{2n}/n! \qquad (23) $$

Note that subsets of phase space with very different shapes and volumes can have the same symplectic area. Now let us discuss the connection between symplectic form and physics. As mentioned in section 1.2.2 we consider close orbits (Poincaré recurrence theorem [13]). Then, for close orbits (we take one dimensional case $n = 1$), the usual quantization [28] writes in the old quantum picture (with no boundary)

$$ \frac{1}{2\pi\hbar} \oint_{\gamma} pdx = m \quad \text{with} \ m \in \mathbb{N} \qquad (24) $$

The first term is the classical action, $\gamma$ is the loop (periodic orbit) on the invariant torus, $m$ an integer (named main quantum number). This procedure makes a bridge between quantum and classical physics (hence the name semi classical theory). This equation was
modified (Einstein-Brillouin-Keller (EBK) equation) to include the Heisenberg principle through the zero point energy. Keller-Maslov quantization gets the condition [29]

$$ \frac{1}{2\pi\hbar} \oint_{\gamma} pdx = \frac{1}{4}\mu(\gamma) + m \text{ with } m \in \mathbb{N} \qquad (25) $$

$\mu(\gamma)$ is the Maslov index [30]. The particular case $\mu(\gamma) = 0$ restores the semi classical point of view given by equation 24. They are different ways to determine Maslov index. This index can be computed by an integral. For highest dimensions ($n > 1$), its calculation is out of scope of this paper (see for further details [20,31–35]). The Maslov index is a topological invariant. It is an integer for close loops [20]. It is defined as the degree of a closed curve in the Lagrangian Grassmanian manifold (for low dimensions ($n = 1$ in our case), Grassmanian is a projective space). An alternative definition involves counting intersections with the Maslov cycle (figure 3 A). Note that Maslov index carries all information about Heisenberg principle as shown in figure 3 (figure 3D-E). Taking a 1D inverse harmonic oscillator (IHO) which mimics the double well potential [19] (figure 3B-C), the Maslov index is non zero if we introduce the Gromov’s cylinder with a radius $r = \sqrt{\hbar}$. If Maslov index $\mu(\gamma) = 0 \mod 4$, the action is quantized in integer units of $\hbar$, whereas if $\mu(\gamma) = 0 \mod 2$ in half-integer units [32]. For simple cases such as 1D pendulum or 1D harmonic oscillator Maslov index is two (see figure 3 A). Then for a simple curve such as a circle, for the fundamental state ($m = 0$)

$$ \frac{1}{2\pi\hbar} \oint_{\gamma} pdx = 1/2 \qquad (26) $$

The additional 1/2 can be explained geometrically in terms of the non-projectability of the classical energy level curves (a linear transformation $\mathbf{P}$ does not meet the condition $\mathbf{P}^2 = \mathbf{P}$). Using Stokes’s theorem with $n = 1$ the symplectic area writes

$$ \mathcal{A} = \oint_{\gamma} pdx = \pi r^2 \qquad (27) $$

Applying Gromov’s theorem, one defines the minimal loop $\gamma_{min}$ [23]

$$ \mathcal{A} = \oint_{\gamma_{min}} pdx = \pi R^2 \qquad (28) $$

From equations 26 and 28 a minimal periodic orbit satisfies the relationship

$$ \oint_{\gamma_{min}} pdx = \pi\hbar = \pi R^2 \qquad (29) $$

For a particular value $R = \sqrt{\hbar}$ we restore Heisenberg inequality. This is a symplectic topological property. We do not need to replace the Poisson brackets in classical mechanics by the corresponding commutators in quantum mechanics. The $\hbar$ constant can be considered as a symplectic (topological) property. Figure 2 displays the Gromov’s theorem
in the simple case of a classical 1D harmonic oscillator (see supplementary information) where the zero point energy $1/2(\hbar\omega)$ is restoring in a classical picture [6,24,25,36]. Even though the area is constant, the volume collapses with a cloud of particles, equation 23 writes

$$VB^{2n}(R) = \pi\hbar^n/n! \qquad (30)$$

For $N$ particles in three dimensional space ($n = 3$ for one particle $N = 1$) $n$ grows as $3N$ hence $VB^{6N}(R) = \pi\hbar^{6N}/N!$. The phase space collapses rapidly with $N$ that makes sense in classical mechanics. However, textbooks report the minimal volume $\sim h^3$ (for one particle) because of Heisenberg principle. As a matter of fact, the minimal volume is not the best parameter, this is the minimal area that makes a "deep" sense.

It summary, within the topology, we can repostulate the Heisenberg principle as well [36]:

*No periodic orbits exist, in quantum mechanics, on subsets of phase space with symplectic capacity smaller than $1/2h$.*

We can also heuristically define the existence of zero-point energy, i.e. the Heisenberg principle, pointing out that the presence of $\hbar$ is necessary to avoid breaking the topological invariance of the Maslov constant, which is equivalent in this simple case to Gromov's theorem (see figure 3 E).

# 4. Entanglement

## 4.0.1. Borsuk Ulam theorem

Prior discussing the entanglement let us introduce a major theorem of algebraical topology: the Borsuk Ulam theorem (BUT) [37–39] displayed in figure 4.

*Theorem:* Every continuous maps $f: S^n \to \mathbb{R}^n$ has a point $x \in S^n$ such that $f(x) = f(-x)$. It is true also when the domain of $f$ is the boundary of any open bounded symmetric subset of $\mathbb{R}^n$ containing the origin.

As a result, there is no continuous antipode preserving map $f: S^n \to S^{n-1}$.

This theorem means that if a sphere is mapped continuously into a plane set, there is at least one pair of antipodal points having the same image; that is, they are mapped in the same point of the plane [40,41] (see figure 4). In other words, a point embedded in a $\mathbb{R}^n$ manifold is projected to two opposite points on a $S^{n+1}$-sphere, and vice versa [42] (this concept was first introduced by Peters and Tozzi [43,44]). Borsuk Ulam theorem (BUT see supplementary information) states that entanglement can be seen as just 3D space surface points that have a corresponding pair of $S^3$ antipodal points which are pure states. The
**Figure 4. BUT theorem in S¹ and S².** part A: Illustration of the mapping $f : S^1 \to \mathbb{R}$ with the two antipodal points "merging" in one point in $\mathbb{R}$. The mapping $f$ is defined by a rule that tells us how to find $f(x)$. The mapping function $f(x)$ is in $\mathbb{R}^1$. part B: same mapping with a distortion of the circle showing that the theorem is independent of the mapping path (without crossing). part C: antipodal points and mapping $f : S^2 \to \mathbb{R}^2$. We recall that $\mathbb{R}^2$ is topologically equivalent to an open disk, which can be embedded into $S^2$. part D: BUT in $S^1$ from the intermediate value theorem. Since $g(x) = f(x) - f(-x)$ is an odd function and can be positive, zero or negative, intermediate value theorem states that its value is zero (P point) then $g(x) = 0$ ie $f(x) = f(-x)$. For example, in classical mechanics, $g(x)$ can be the difference between the temperature and/or the pressure (assuming that these functions are continuous). BUT theorem states that the difference for the antipodal points is zero. This is to say a macroscopic entanglement and BUT theorem violates the locality principle. part E: Humorous illustration of a classical entanglement deduced from Borsuk Ulam theorem. Note that in $S^2$, in this example, the $f(x)$ function is a two variable function (temperature and pressure) then it is a 2D vector.

<figure><img src="image_8.png" /></figure>
**Figure 5. BUT theorem for a $\Phi^+$ Bell state.** The two antipodal states x,-x (ie $|00\rangle$ and $|11\rangle$) represented in the q sphere $S^2 \to \mathbb{R}^2$ has a point $x \in S^2$ such that $f(x) = f(-x)$. Note that $S^2$ is embedded ($\leftrightarrow$) in $\mathbb{R}^3$. Likewise, $S^3 \to \mathbb{R}^3$ has a point $x \in S^3$ such that $f(x) = f(-x)$ (for the $S^3$ representation: direct projection of 3-sphere into 3D space and covered with surface grid, showing structure as stack of 3D spheres (2-spheres): Credit Eugène Antipov). Of course $\mathbb{R}^2$ is a subspace of $\mathbb{R}^3$, the element $A(x_1, y_1)$ of $\mathbb{R}^2$ is identified with $(x_1, y_1, 0) \in \mathbb{R}^3$. Shortly, the two states are entangled at $S^2$ level and disentangled (pure states) at $S^3$ level. The q sphere is provided by the IBM's open-source software Qskit [45]. Each node's radius is proportional to the probability of its basis state, whereas the node color indicates the quantum phase $\Phi$. In this example $\Phi = 0$

<figure><img src="image_9.png" /></figure>

system is entangled at the $S^n$ level ($n=2$ in the real world), but un-entangled at the $S^{n+1}$ hypersphere level.

## 4.1. Quantum entanglement

For a sake of simplicity, we consider a Qu-bit

$$ | \Psi \rangle = | \Psi_A \rangle \otimes | \Psi_B \rangle \quad (31) $$

with $\Psi_{A,B} >= \alpha_{A,B} | 0 \rangle + \beta_{A,B} | 1 \rangle$, $|\alpha_{A,B}|^2 + |\beta_{A,B}|^2 = 1$ ($\alpha, \beta \in \mathbb{C}^2$). In general, any state, pure or mixed, that is neither a product state $| \Psi \rangle = | \Psi_A \Psi_B \rangle$ nor a mixture of product states, is called an entangled state. We consider Bell states that are quantum states of two Qu-bits with the maximum of entanglement
$$ \Phi^{\pm}(A, B) = \frac{1}{\sqrt{2}}(|0\rangle_A \otimes |0\rangle_B \pm |1\rangle_A \otimes |1\rangle_B) \quad (32) $$

$$ \Psi^{\pm}(A, B) = \frac{1}{\sqrt{2}}(|0\rangle_A \otimes |1\rangle_B \pm |1\rangle_A \otimes |0\rangle_B) \quad (33) $$

shortly

$$ \Phi^{\pm} = \frac{1}{\sqrt{2}}(|00\rangle \pm |11\rangle) \quad (34) $$

$$ \Psi^{\pm} = \frac{1}{\sqrt{2}}(|01\rangle \pm |10\rangle) \quad (35) $$

If the composite system is in one of this state, it is impossible to attribute to either system A or system B a definite pure state. The entanglement can be routinely estimated from the trace of the square of the reduced density matrix $tr(\rho_A^2)$ where the density matrix $\rho = |\Psi\rangle\langle\Psi|$ and $\rho_A$ the reduced density matrix for the first Qu-bit A given by the trace over the subspaces of the Hilbert space for the second Qu-bit $\rho_A = Tr_B(\rho)$. As an example for $\Phi^+$, a straightforward calculation gives $Tr(\rho_A^2) = 1/2$, $Tr(\rho_A^2) = 1$ when entanglement vanishes.

The Bloch sphere gives a geometrical representation of the pure state space of a 2-level quantum system, a two Qu-bits pure state can be explicitly parameterized by three unit 2-spheres (quasi Bloch spheres) and a phase factor [46,47]. Formally, Borsuk Ulam theorem is replaced by Borsuk Ulam type theorems on product spaces [48]. Any generalization to multi Qu-bit states tends to be less intuitive, less mathematically convenient and inappropriate for quantum computing. An alternative is given by que q sphere developed by IBM (IBM's open-source software Qskit) [45]. Contrary to Bloch sphere, the q-sphere provides a visual representation of the quantum state, and thus this global viewpoint. The q-sphere represents the state of a system of Qu-bits by associating each computational basis state with a point on the surface of a sphere. A node is visible at each point. Each node's radius is proportional to the probability of its basis state, whereas the node color indicates the quantum phase (see figure 5).

The nodes are laid out on the q-sphere which means that we consider the properties of $S^2$ manifold. The basis state with all zeros is located at north pole, and the basis state with all ones is located at south pole. Basis states with the same number of zeros (or ones) lie on a shared latitude of the q-sphere. We can also define a metric with the Hamming distance $d_{Hamm}$: consider the set of all strings of a length $n \in \mathbb{N}$. The Hamming distance between two strings is the number of places in which the two strings differ. Considering the two-Qubits (Bell states) in equations 32 and 33 the Hamming distance is maximum ($d_{Hamm} = 2$) both states $|0\rangle$ and $|1\rangle$ are antipodal in the q-sphere and span the most degree of symmetry in the sphere $S^2$.
Now, let us examine the three Qu-bits to the light of BUT (figure 6 A). It is possible for three "particles" to be mutually entangled. An example is the simplest GHZ (Greenberger-Horne-Zeilinger) state (figure 6 B).

$$|GHZ> = \frac{1}{\sqrt{2}}(|000> + |111>)(36)$$

The Hamming distance is maximal ($d_H = 3$) and the states are antipodal. According to BUT the states are maximally entangled in agreement with the definition of Absolutely Maximally Entangled states [49]. However, the $|111>^>$ state is fully separable after loss of one Qu-bit. This is easy to check that for example for the three states 1,2,3 the reduced matrix $\rho_{23}$ writes as a convex combination of separable states

$$\rho_{23}(GHZ) = Tr_1(\rho) = Tr_1(|GHZ><GHZ|) = \frac{1}{2}(|0><0|\otimes|0><0| + |1><1|\otimes|1><1|) \quad (37)$$

The second example is the W state (figure 6 C)

$$|W> = \frac{1}{\sqrt{3}}(|100> + |010> + |001>)(38)$$

The Hamming distance $d_H = 2$ is not maximal. Neither couple of points are antipodal and the fully entanglement cannot be achieved in the sense of Borsuk Ulam theorem. However the states are highly symmetric (equilateral triangle in the q sphere). If we trace out the third Qu-bit we obtain

$$\rho_{23}(W) = Tr_1(\rho) = Tr_1(W><W|) = \frac{1}{3}(|0><0|\otimes|0><0| + 2|\Phi^+><\Phi^+|) \quad (39)$$

Contrary to GHZ state, if one of the three Qu-bits is lost, the state of the remaining two Qu-bits system is still partially entangled. This is in agreement with Coffman-Kundu-Wooters inequality [50] which basically states that if two systems are maximally entangled then entangling a third system with them requires a lack of entanglement from the previous one.

## 4.2. "Classical" entanglement

### 4.2.1. BUT theorem versus quantum mechanics

We can understand the EPR paradox [51] to the light of BUT. Mathematically, entangled particles are described as parts of the same equation, not as if they are two different particles interacting. Experimental tests [52–56] of Bell's inequalities remove the last doubts that we should renounce local realism (the cause of a physical change must be local) [57]. An
<figure><img src="image_10.png" /><figcaption>**Figure 6.** **Q-sphere for three states.** Part A: representation of few states in the q-sphere. Part B: GHZ states. Part C: W state. Part D: illustration of the antipodal points (stars with the same color) where BUT theorem applies.</figcaption></figure>

entangled system constitutes one indivisible, spatiotemporally disconnected, event with no spatiotemporal parts. Then the "instantaneous nature" where the light speed does not take place is considered as an intrinsic property of the quantum mechanics. The origin is the subject of much philosophical debate such as the "ontological holism conflicts" [58–61]. Since BUT is time independent (topology is true nowadays, in the past and in the future), the "instantaneous nature" is well understood as long the entangled states come from a pure state in a $S^3$ manifold [43]. Since we live in $S^2$ we observe the consequence not the origin. Moreover, applying BUT for two states (ie two singletons for example two spins $|↑⟩$ and $|↓⟩$) in $S^0$, BUT states that every continuous maps $f: S^n \to \mathbb{R}^n$ has a point $x \in S^n$ such that $f(x) = f(-x)$. Taking $n=0$ the two opposite states are antisymmetric (there is no continuous function) and then for a spin the single state in $\mathbb{R}^0$ is formed by a pair of opposite spins $|↑↓⟩$ that is the Pauli principle. This is not a principle but a theorem ... and intrinsically a topological property.

In summary the entanglement can be seen as a consequence of the duality between intrinsic and extrinsic geometries. Extrinsic geometry classically study objet X sitting in ambient $\mathbb{R}^3$ which is our "true physical world" (far from the Planck wall). Intrinsic geometry study X itself without reference to ambient space in other words our universe $\mathbb{R}^3$ that is a sub object of $\mathbb{R}^n$ with $n \ge 4$.

Coming back to BUT, a tentative to define the Bell's entangled states should be:

*The maximum of correlation involving the highest degree of symmetry in the system* [62].

Clearly, two states present the highest degree of symmetry in a sphere in antipodal
positions. Note that BUT needs the maximal Hamming distance. For example, figure 6 D reports the antipodal states (labeled by a star) in the q-sphere corresponding to the pairs ($|100angle$, $|011angle$), ($|000angle$, $|111angle$), ($|001angle$, $|1101angle$) and ($|010angle$, $|101angle$).

### 4.2.2. BUT theorem in classical mechanics

Entanglement is a natural occurrence in quantum mechanics. The biggest mistake would be to think that entanglement is a purely quantum phenomenon. BUT for the $S^2$ sphere states that: for any two continuous real functions f,g on the sphere, there exists a pair of antipodal points $(x, -x)$ where $f(x) = f(-x)$ and $g(x) = g(-x)$. In this case mathematicians cite as examples temperature and pressure data as two continuous functions (temperature and pressure on the 2-dimensional manifold that corresponds to the ground/water level). Even though weather is chaotic [63], there must always be at least two places on opposite ends of the earth (antipodal points) where a couple of weather data and no more: temperature and pressure, temperature and humidity, wind and temperature ... would be the same! [64] (see figure 5) (this issue is disputed by Jha et al [65] arguing that temperature or pressure are may be not a continuous function). Many authors think this is not really possible to check experimentally, work is in progress to fix the remaining problems. However, BUT theorem assumes the possible non locality for all the physical systems. Entanglement cannot be reduced to a quantum property. The origin of non-locality is no longer a hidden physical term but a topological property, in this case n+1 dimensionality. It is clear that $f(x)$ function ($x$: temperature, pressure ...) avoids any state of coherence (see next section) between the two antipodal points, the entanglement is just a symmetry property (antipodal points) with a pure mathematical origin.

### 4.3. Schrödinger cat in classical mechanics

Schrödinger's cat is a concept that has puzzled and fascinated scientists alike for decades but the concept firstly introduced in quantum mechanics as a consequence of Heisenberg's principle. The paradox is a clear consequence of quantum theory allowing the existence of entangled states. According to quantum mechanics, if a cat placed in a box with an ingenious mechanism containing different things (hammer, flask of poison, radioactive source and Geiger counter), until we open the box and observe, the cat is in a superposition of being both alive and dead. In a classical mechanics: anybody may be standing or seated, but nobody may simultaneously be standing and seated [61]. This paradox can be illustrated without quantum mechanics if we introduce the non orientability as reported in Dirac belt trick (see supplementary information). There is no physical justification for this demonstration, only the illustration that the topology of non-orientable surfaces allows us to understand the paradox within a classical mechanics framework.
**Figure 7.** Grandfather paradox: gedankenexperiment. **Left:** base space corresponding to a rigid classical space. The sequence is: *I am born* → *I am travel back in time* → *I kill my grandfather* → *My mother is not born* → *I am not born* → .... This is not consistent. **Right:** Introducing the non orientability (Moebius strip) the sequence gives a superposition of states (born/not born, killed/not killed like Schrödinger cat analogous to a pair or spins up and down). The quantum aspect is related to the non orientability.

<figure><img src="image_11.png" /></figure>

### 4.3.1. Grandfather paradox: gedankenexperiment

Taking the example of the gedankenexperiment named "infamous grand father paradox" (we do not discuss Novikov self consistency principle that refutes the concept). The paradox is cited in the novel "Le voyageur imprudent" published by René Barjavel in 1943 [66,67]. If you go back in time to kill your grandfather, a chronological sequence of events lead to several inconsistencies (see figure 7) and then an impossible scenario. At a first glance, this is the heuristic proof that we cannot go back in time (for a discussion [68,69]). Briefly, there are two ways to explain the paradox. If you go back in time to kill your grandfather, then a new timeline branch is created and the chronological sequence is not consistent [70] (the issue of multiverse is open). The second way was suggested by S.Hawkins in the quantum mechanics framework [71,72]. In the past, experimental simulation of closed timelike curves have been studied [73]. A possible scenario is based from the twisted Deutsch-Politzer (TDP) space that is non orientable. The DP space [69,74] is based from a cylinder, while the TDP space is based from a Moebius strip. For a sake of simplicity, we take the famous "Dirac's belt trick" [75] as a teaching illustration. The Dirac's belt trick is a demonstration that the path (for a fermion with 1/2 spin) through the space of
rotations $SO(3) \equiv \mathbb{R}^{3}$ corresponding to a $2\pi$ twist cannot be contracted to the trivial path (perfectly straight belt) while the path corresponding to a $4\pi$ twist can do. In brief, we must lift path from $SO(3)$ to $SU(2) \equiv S^3$ to rotate an electron. Algebraic topology tell us that the base space $SO(3)$ is covered by two $SU(2)$ illustrated by the Moebius strip (figure 7). In other words, $SO(3)$ is the set of all $3 \times 3$ rotation matrices. We can specify a rotation by a direction in $\mathbb{R}^3$ along with a magnitude in [$\sim \pi, \pi$], the a rotation by $\pi$ being equivalent to a rotation by $-\pi$.

In this case, the chronological sequence leads to superposition states (killed/no killed) (analogous to the spin up and down for the electron). The paradox is lift by quantum mechanics but in this experiment the paradox is lift by the topology. The technical relevance of this model to time travel is tenuous but it well illustrates the deep connection between classical and quantum physics. Here, the bridge is the non orientability. Once again, the quantum part is explained taking a manifold ($n \ge 4$) embedding our manifold ($n = 3$) (remember that Moebius strip is embedded in $\mathbb{E}^3$).

Now, let us examine the connection between the Schrödinger cat illustrated in this gedankenexperiment and the BUT (ie the entanglement). The connection appears with a few algebraic topology:

The real projective plane $\mathbb{R}^3$ is a 3-dimensional closed non-orientable manifold, which is a quotient of $S^3 \subseteq \mathbb{R}^4$ by the equivalence relation $\{x \sim -x\}$ [76]. We write its points $[x_1, x_2, x_3, x_4] = [-x_1, -x_2, -x_3, -x_4]$, where $(x_1, x_2, x_3, x_4) \in S^3$. These antipodal points correspond to the maximum of entanglement depicted by BUT. This also makes it easy to see the double cover $2SU(2)$ (or Spin(3) covering): It is just the map $S^3 \to \mathbb{R}^3$ that identifies opposite points, $SO(3)$ being diffeomorphic (for smooth manifolds, the "same" in a differentiable sense) to $\mathbb{R}^3$.

### 4.3.2. Rotational decoherence

Previous gedankenexperiment can be thought of as an example of rotational coherence. Consider two states $|0\rangle$ and $|1\rangle$, with a state $|\Psi\rangle = a|0\rangle + b|1\rangle$, the density matrix $\rho = |\Psi\rangle\langle\Psi|$ writes for a coherent state

$$ \rho = \begin{pmatrix} |a|^2 & ab^* \\ a^*b & |b|^2 \end{pmatrix} \qquad (40) $$

If the off diagonal elements are 0, this indicates there is merely classical uncertainty about the state and no quantum superposition going on. In other words, there is a total decoherence. For a partial rotation decoherence, one writes $|0\rangle$ state is the reference), $|0\rangle \to |0\rangle$ and $|1\rangle \to \exp(i\phi)|1\rangle$, $\phi$ is the random phase defined in the rotation matrix $\bar{R}(\phi) = \begin{pmatrix} 1 & 0 \\ 0 & \exp(i\phi) \end{pmatrix}$. Assuming that the probability measure of the random phase is
given by a Gaussian law with a variance $\sigma$, the density matrix average over the random phase writes [77]

$$ \rho = \begin{pmatrix} |a|^2 & ab^* \exp(-\sigma(t)^2/2) \\ a^*b \exp(-\sigma(t)^2/2) & |b|^2 \end{pmatrix} \quad (41) $$

The spread $\sigma(t)$ due to external parameters such as collisions increases over time giving a pure probabilistic signal when $t \to \infty$ (off diagonal terms are zero). Quantum correlations coherence and decoherence states can be estimated from positive Wigner distribution function [78,79] in the phase space (Wigner distribution function $W(x, p)$ of a pure quantum state is positive if the state is coherent) [80]. Wigner distribution function $W(x, p)$ is a special representation of the density matrix.

$$ W(x, p) = \frac{1}{\pi \hbar} \int \Psi^*(x+y) \exp(2ipy/\hbar) \Psi(x-y) dy \quad (42) $$

The merit of the Wigner function is the fact that the function makes a bridge between the classical and quantum dynamics in phase space.

In the theory of decoherence [81–83] it is admitted that quantum coherence will disappear on a decoherence time scale $\tau_D$ [84]

$$ \tau_D \approx \tau_R \left( \frac{\hbar}{\Delta x \sqrt{2mk_B T}} \right)^2 \quad (43) $$

$\tau_R$ is the relaxation time, the denominator is the thermal De Broglie wavelength, $\delta x$ is the separation between two objects (macroscopic in classical mechanics, atoms for example in quantum mechanics). Obviously, numerical application for massive macroscopic objects indicates that quantum coherence would be destroyed in a very short time! that it to say that two macroscopic objects cannot be entangled with a coherent phase. Figure 8 gives the decoherence dynamics illustrated by the Wigner positive function with a Gaussian distribution [85]. Spatial coherence is illustrated by the maximum value of the off diagonal terms. After a long time decoherence is assumed with zero off diagonal terms corresponding to a deterministic value for the states $|0\rangle$ and $|1\rangle$ (classical point of view). The maximal coherence is similar to the Dirac belt trick with $SU(2)$ the (double) covering space of $SO(3)$ (see supplementary information) as mentioned in the previous gedankenexperiment, each pair having a total decoherence (that is to say a classical state).

# 5. Comparison with other models

## 5.1. Outlook

We can briefly sum up the core ideas in different quantum mechanics models developed
**Figure 8.** Top: Visualization of the decoherence dynamics in one dimension, showing the reduced density matrix representing a superposition of two Gaussian wavepackets separated in position space illustrated by the Wigner (positive) function with a Gaussian distribution [85]. (A) the initial density matrix before the onset of rotational decoherence, exhibiting largest off-diagonal terms that represent the maximal spatial coherence. (B) Partial cecoherence arising from entanglement with the environment (temperature, collisions ...) diminishes the weight of the off-diagonal terms over time. (C) total decoherence where we are in a classical probability distribution with an equal probability of finding the particle in either of the locations corresponding to the Gaussian wave packets (classical case). Bottom: double covering state $2SU(2) \rightarrow SO(3)$ showing the construction of coherent states from a topological point of view. The coherence (F) is seen as the superposition of two classical states (D) and (E) with a dephasing factor due to the Moebius "inversion".

<figure><img src="image_12.png" /></figure>
to date

Copenhagen Interpretation (Bohr, Heisenberg, Born): In this traditional view, quantum systems exist in superpositions until they are measured, at which point the wave function collapses into a definite state. This theory requires the wave function to collapse into a single outcome upon measurement. This, of course, is the "standard" in quantum mechanics interpretation in a probabilistic way. It takes advantage ok working in an $\mathbb{E}^3$ space in an non relativistic model. Within which some paradoxes (or not) emerge (*EPR*, the time when it was thought that a photon was either a particle or a wave in two slit experiments discussed in the Feynman Lectures on Physics, the role of measurement in quantum mechanics, the collapse of the wavefunction ...)

De Broglie Bohm interpretation [86]: the de Broglie-Bohm interpretation, also known as Bohmian mechanics or pilot-wave theory, is a deterministic (avoiding the inherent randomness of the Copenhagen interpretation.) and non-local interpretation of quantum mechanics. Given the initial positions of all particles and the wave function, the future evolution of the system is completely determined. However, because we cannot know the exact initial positions of all particles (they are "hidden variables"), the theory still predicts the probabilistic outcomes observed in quantum experiments. The lack of theory is the nature of the hidden variable introduced in Bohmian mechanics [87]. This theory was rejected (hidden variables are rejected) but recent experiments in two slits experiments have brought it up to date [88] within the concept of (strong/weak) measurement.

The Everett many-worlds interpretation (MWI) [89–93] [94–100]. This model of quantum mechanics is one of the most fascinating and controversial ideas in modern physics. The many-worlds interpretation suggests that all possible outcomes of quantum measurements actually occur, each in a separate, branching universe. MWI claims that the wave function never collapses (unlike the Copenhagen interpretation). Instead, every possible outcome is realised in its own "world" or "branch" of the universe. The implications are the existence of parallel universes, the determinism and no special role for the observer.

The "many interacting worlds" (MIW) [101–104]. The MIW hypothesis suggests that there are multiple classical worlds (or universes) that exist simultaneously and interact with each other. Unlike the many-worlds interpretation, where worlds branch and do not interact, MIW proposes that these worlds are always present and influence one another through a specific interaction mechanism. MIW rejects the need for wave function collapse. In the well-known "Many-Worlds Interpretation", each universe branches into a bunch of new universes every time a quantum measurement is made.
The main criticisms [105–107] of deterministic models question the ontological proliferation, the testability and the probability problem.

## 5.2. Multi world and topology

### 5.2.1. Topological multiverse

Topology points out the main challenge to the Everett interpretation and questions the preferred basis problem (in other words, the superpositions described above are supposed to represent different worlds, againsts which basis are these superpositions defined?). In Everett's model the Schrödinger's cat equation says that the two parts of the wave function are perpendicular to each other. This can be interpreted as two separate worlds. Here, the Schrödinger's cat is described in the Moebius band. The Moebius band displays the full duality of its structure at the classical level (for details see reference [108]). The superposed states shown in the section corresponds to Everett interpretation where the world is not orientable (section 4.3).

From a topological point of view, the multiverse (interacting or non interacting worlds) can be viewed naively in terms of the submanifolds sitting in $\mathbb{R}^n$ and manifolds (formally not necessarily embedded inside $\mathbb{R}^n$). In the non-relativistic world, we live in 3D Euclidean space a submanifold embedded in $\mathbb{E}^3$. Entanglement is observed in 3D space but the interpretation is rather simple in 4D space (see text) where the states are disentangled (a submanifold embedded in $\mathbb{E}^4$). Note that in topological manifold (ie multiworld), there is no time: all things can appear simultaneously (true in the past, now and in the future), contrary to the physical principle (Einstein proposed that the meaning of "simultaneous" is that two equidistant events are simultaneous if the light from both events reaches the observer at the same time). This is an attempt to explain the surprising experience of Megidish et al [109] who observed an entanglement swapping between photons that have never coexisted.

As shown in figure 2, the position in the trajectory is time-dependent, but the information is in the winding number. Time is only a tool for constructing the trajectory (iso-energy curves fig. 3), in other words the "measurement". In Copenhagen interpretation, a measurement is an interaction of the quantum system with a classical system which can be the observer (the origin of the collapse of the wave function). Topology does not provide any information about the measurement. The latter is obtained by time parameter that highlights the closed curve.

## 5.3. Condensed matter physics: the paradox of the nearly free electron model

Topology and dimensionality can be illustrated in a very simple model: the nearly free
electron model, the simplest way to explain the metal properties and a popular model for graduate students. For a sake of simplicity let us consider a 2D crystal in which an electron moves. Crystals have one fundamental property : the translation symmetry which is the base of the Bloch’s theorem (we do not discuss other symmetries such as rotation, inversion which are the base of the topological materials). The wave function takes into account the periodicity of the crystal through the Bloch functions. Assuming the Bloch hypothesis (Born Von Karman boundary conditions, acronym BVK), the wave function *a priori* probes an infinite periodicity. Breaking the periodicity introduces the so-called confinement effect, which leads to a slight shift in the electronic levels. This requires that the electron has a "knowledge" of how the crystal is made .... The model accounts for the effects of the periodic potential on electron motion in the framework of effective mass. This effective mass ( or inertial effective mass tensor) contains all the "physics" but sweeps it under the carpet! Furthermore, the BVK conditions states that "when an electron is at one side, it appears suddenly to the opposite side" (see figure 9). What does "suddenly" means? We are in the realm where everything happens as if ...

Topology has a different view. The classification theorem of closed surfaces states that every connected closed surface is homeomorphic to some member of one of the following three groups

The sphere $S^2$, with the standard topology of $\mathbb{R}^3$

The connected sum of $g$ tori for $g \ge 1$, $H_g = T^2\#T^2...\#T^2$ $g$ times

The connected sum of $k$ real projective planes for $k \ge 1$, $C_g = \mathbb{R}P^2\#\mathbb{R}P^2...\#\mathbb{R}P^2$ $g$ times. Real projective space, $\mathbb{R}P^2$ is the quotient of the sphere by the antipodal map: $\mathbb{R}P^2 = S^2/Z_2$ [110] ($Z_2$ (binary) symmetry involves only two possible values or states, $+1, -1$ for example).

This is illustrated in figure 9 for 2D surfaces with the gluing instructions.

This harmless operation has consequences for topology, geometry and dimension. Taking a 2D manifold, according to the classification theorem, the square with BVK conditions gives a flat torus. Such a torus has a hole $g=1$ (square has no hole $g=0$, $g$ the genus), the flat torus is embedded in $\mathbb{E}^3$ space, the square like the 2D crystal being embedded in $\mathbb{E}^2$ space. The global curvature of the torus is zero like the square (Gauss Bonnet $\int K dA = 0$ Euler's characteristic is zero) but local curvature is either, positive, zero or negative. Now let us imagine the complexity for a 3D crystal. In the same way a simple cubic lattice (3-cube) with translational symmetry gets a 3-torus ($S^1 \times S^1 \times S^1$) embedded in 4D Euclidean space. Although, the crystal is really described in a 3D space, fermions (electrons) have trajectories in the 4D space and their properties are related to this hyperspace. The simple fact for introducing the translational symmetry changes the topology and the manifold dimension. We call this "topological" multiverse. An open question that needs to be clarified is the classification of compact 3-manifolds similar to the 2-dimensional case. The reader understands that a simple boundary condition leads to
<figure><img src="image_13.png" /><figcaption>**Figure 9.** Four cases for the square gluing. The gluing instructions are given by arrows. A gluing diagram for a polygon is an assignment of a letter and an arrow to each edge of the polygon. The shaded figure (top left) gives the BVK conditions in standard crystallography, the square giving the torus corresponds to the first Brillouin zone (lower left side). The Z₂ gauge is of importance giving a correspondence between these surfaces (no two surfaces in this list are homeomorphic to each other)</figcaption></figure>

$$K^2 = \mathbb{RP}^2 \# \mathbb{RP}^2$$

$$K^2 = T^2/Z_2$$

$$\mathbb{RP}^2 = S^2/Z_2$$

a complex structure which is represented in a high-dimensional universe.

Now let us use the other possibilities (see figure 9). The second case corresponds to a sphere in 3D space. The third is the Klein bottle which is not orientable. The Klein bottle is a two-dimensional manifold on which you cannot define a normal vector at every point that varies continuously over the whole manifold. It is homeomorphic to the connected sum of two projective planes and present striking similarities with the Moebius strip used in section 4.3. However, whereas a Moebius strip is a surface with a boundary, a Klein bottle has no boundary. The configuration is observed in non-symmetric crystal [111–115]. By definition non-symmetric space groups cannot be represented as semi-direct product groups of a discrete translation group (Bravais lattice) and a corresponding point group. Such a symmetry can even change the fundamental domain from a Brillouin torus to a Brillouin Klein bottle.

Now consider the last case. This is a real projective plane. The real projective plane $\mathbb{RP}^2$ is defined as the quotient space $S^2/\sim$, where $\sim$ identifies the antipodal points of $S^2/\sim$. Similarly, the projective plane is embedded in $\mathbb{R}^4$ which requires a 4D Euclidean space. Finally, all the cases differ from the gluing construction (a simple phase factor for the electron wave function).
It seems harmless, but the consequences are dramatic. The experiments are performed in 2D space, but the first two formers require a 3D Euclidean space and the last two a 4D Euclidean space for a complete understanding. All the considerations in the space where the system lives are incomplete leading to some paradoxes (sudden, positive or negative mass for the electron ... ). In other words , the "phase factor" (gluing instruction) involves the "upgrading" of the sub-manifold to another sub-manifold (or manifold). In this sense, this could be compared with Everett’s interpretation which considers that the size of the universe is larger than ours .

All the usual Euclidean spaces Rn are Hilbert spaces under the usual scalar product. The classification theorem says that two Hilbert spaces H₁ and H₂ are isometrically isomorphic if and only if they have the same dimension [116]. We understand that the topological dimension (in the Lebesgue sense) is the hidden parameter not in the physical sense but in a mathematical (topological) sense.

5.4. *Heisenberg’s principle and the problem of the measurement; limitation of the topology interpretation*

In the example shown in figure 3, the Heisenberg principle is the consequence of the
winding number invariance. Breaking the invariance requires external energy, as in a
common phase transition. This is the key point of decoherence with the interaction between
particles and the environment. Breaking the winding number invariance allows the wave
function to collapse. Consider a three-body problem, there is no general closed-form
solution to the three-body problem. From astronomical observations, we have known
for a long time that the orbits of planets are not exactly closed (in the regime of small
perturbation). They precess with time. Let us take a simple harmonic oscillator, when we
make a measurement, we introduce a *priori* a slight perturbation and the radius in phase
space changes a little bit with time. The open question is the contribution of this fluctuation
with respect to the the topological part (Gromov’s theorem). Contrary to the Copenhagen
interpretation where the wave function collapses (strong measurement), we need a weak
measurement to assume the topological rules. This is the weakness of the model.

**6. What does porosity mean?**

In this review, we have illustrated the porosity that results in a blurred context between
classical and quantum mechanics. If topology plays a fundamental role, the question arises
as to why it comes into play in quantum mechanics and seems to play an insignificant
role in classical mechanics. One possible answer is the number of particles (whatever
their size). It is clear that from a topological point of view it is only possible to formally
entangle two particles. This raises the question of the entanglement of a large number of Qu-bits. This contradicts the definition of topology: topology is the branch of mathematics concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending. This is no longer the classical development of a physical system subjected to forces. In other words, topology is everywhere. One answer concerns the number of particles (which may themselves be made up of several entities). If the number is small, topology plays a key role, if the number is large (which is the case for a classical system), topology plays no role at all. All matter is quantum, regardless of the number of basic constituents it contains. Topology only comes into play for a finite number of basic constituents. This is illustrated by the equation 30, where the phase space rapidly collapses with the number of basic ingredients. Since topology introduces no notion of time or metric, the physicist must judge which domain is relevant for a given system. This is the real between mathematics and physics. However, we should note that the time parameter, in addition to measurement (highlighting the closed orbit), also appears in the evolution of the system under deformation, without reference to a time scale.

## 7. Conclusion

With two examples, the diffuse frontier between classical and quantum mechanics justifies the term "porosity". Both examples are based from two major theorems of algebraic topology : Gromov's Non squeezing theorem and Borsuk Ulam theorem. The relationship between quantum and classical mechanics comes from the symplectic geometry in the Hamiltonian representation. The basic point of view $\hbar \to 0$ is an improper concept insofar as $\hbar$ is conceptually a part of the classical mechanics through Gromov's theorem. Clearly, this term is small enough to be ignored in the macroscopic world. Moreover, the non locality concept inherent of the quantum entanglement is an universal concept of the topology for all the physical systems modelized on a n-sphere(BUT) theorem. The open question is why the topology dictates these properties. The answer is obvious if one considers the definition of the topology. The only difference is the time. Physical system evolve under time while topology consider a "deformation" without time reference. It's time for physicists to discover all the tools (theorems) of topology that can be so useful in understanding physical mechanisms and seeming paradox.

## 8. Acknowledgment

I'd like to thank Professor Michel Broyer for being the devil's advocate when it came to the fundamentals of quantum mechanics, and thus keeping me on track.
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## Supplementary Materials

### 1. Harmonic oscillator

In particular for the n-dimensional anisotropic harmonic oscillator with Hamiltonian

$$H = \sum_{j=1}^{n} \frac{1}{2} (p_j^2/m_j + m_j \omega_j x_j^2) \quad (44)$$

Harmonic oscillator derives from a central force scalar potential $\propto r^2$ and then closed orbits exist (Bertrand's theorem) [15] (see figure 2). The (periodic [12]) closed (repeatedly returning to exactly the same path) orbits $\gamma_j = \int p_j dx_j$ as a function of time are given by harmonic functions

$$x_j(t) = x_j(0) \cos \omega_j(t) + \frac{1}{m_j \omega_j} p_j(0) \sin \omega_j(t) \quad (45)$$

$$p_j(t) = -m_j \omega_j x_j(0) \sin \omega_j(t) + p_j(0) \cos \omega_j(t) \quad (46)$$

Applying a gauge transformation with a matrix $\bar{S} = \begin{pmatrix} (m\omega)^{-1/2} & 0 \\ 0 & (m\omega)^{1/2} \end{pmatrix}$, one obtains a new Hamiltonian $H'$ derived from equation 44

$$H = \sum_{j=1}^{n} \frac{\omega_j}{2} (p_j^2 + x_j^2) \quad (47)$$

The matrix $\bar{S}$ is symplectic (see equation 6) then the orbits remain unchanged. In particular equation 47 corresponds to a symplectic cylinder of radius $R_j$ where the orbits are the same (equation 22)

$$Z_j(R_j) = \{(x, p) : \pi(x_j^2 + p_j^2) = \pi R_j^2\} \quad (48)$$

The minimum area corresponding to the smallest orbit $\gamma_{min}$ is given by Gromov's theorem, then $R_j \ge \sqrt{\hbar}$. Inspecting equation 47 gives the minimum energy ($p_j = 0, x_j = R_j$)

$$E(\gamma_0) = \sum_{j=1}^{n} \frac{\omega_j}{2} (R_j^2) = \sum_{j=1}^{n} \frac{\hbar \omega_j}{2} \quad (49)$$
This term corresponds to the zero point energy. In summary, this term is restore by the
properties of symplectic geometry in classical Hamilton's equations thanks to the topology
(Gromov's theorem).

## 2. Bell states

Consider two arbitrary quantum systems A and B, with respective Hilbert spaces $H_A$ and $H_B$. The Hilbert space of the composite system is the tensor product $H_A \otimes H_B$. If the first system is in state $|\psi\rangle_A$ and the second in state $|\phi\rangle_B$, the state of the composite system is $|\phi\rangle_A \otimes |\phi\rangle_B$. Not all states in the composite system are separable states. For example, given two basis vectors $\{|0\rangle_A, |1\rangle_A\}$ of $H_A$ and two basis vectors $\{|0\rangle_B, |1\rangle_B\}$ of $H_B$. The quantum entanglement E(A,B) which denote the entanglement of vectors A, B on a normed linear space (entangled state) on an n-Sphere $S^n$ embedded in a normed linear space X writes (we use the original notation of Peters et al [43]):

$A, B \in S^1$, is a vector on the circumference of a 1-sphere.

$$E(A, B) = \frac{1}{\sqrt{2}} (|0\rangle_A \otimes |1\rangle_B - |1\rangle_A \otimes |0\rangle_B) \quad (50)$$

More generally, one defines the four maximally entangled two-qubit states or Bell states

$$\Phi^{\pm}(A, B) = \frac{1}{\sqrt{2}} (|0\rangle_A \otimes |0\rangle_B \pm |1\rangle_A \otimes |1\rangle_B) \quad (51)$$

$$\Psi^{\pm}(A, B) = \frac{1}{\sqrt{2}} (|0\rangle_A \otimes |1\rangle_B \pm |1\rangle_A \otimes |0\rangle_B) \quad (52)$$

If the composite system is in one of this state, it is impossible to attribute to either
system A or system B a definite pure state.

## 3. Quantum BUT theorem in $S^1$

The two antipodal points are located in the circle that defines Bell states in $S^1$. For the
antipodal points $B = -A$ equations 51 and 52

$$\Phi^{\pm}(A, -A) = \frac{1}{\sqrt{2}} (|0\rangle_A \otimes |0\rangle_{-A} \pm |1\rangle_A \otimes |1\rangle_{-A}) \quad (53)$$

$$\Psi^{\pm}(A, -A) = \frac{1}{\sqrt{2}} (|0\rangle_A \otimes |1\rangle_{-A} \pm |1\rangle_A \otimes |0\rangle_{-A}) \quad (54)$$

The entanglement supposes that there is a function $f(x)$ defined by $q(A, B)$ where $q(-A) = q(A)$ in $\mathbb{R}^1$. In order to precise what is the function, we can define $(A, q(A))$ a vector on $S^2$, in which $q(A) \in \mathbb{R}$ that is a signal value. One writes
$$ \Phi^{\pm}(q(A), q(B)) = \frac{1}{\sqrt{2}} (|q(A)\rangle_A \otimes |q(A)\rangle_B \pm |q(B)\rangle_A \otimes |q(B)\rangle_B) \quad (55) $$

$$ \Psi^{\pm}(q(A), q(B)) = \frac{1}{\sqrt{2}} (|q(A)\rangle_A \otimes |q(B)\rangle_B \pm |q(B)\rangle_A \otimes |q(A)\rangle_B) \quad (56) $$

Let us consider the entanglement of the two antipodal points $B = -A$

$$ \Phi^{\pm}(q(A), q(-A)) = \frac{1}{\sqrt{2}} (|q(A)\rangle_A \otimes |q(A)\rangle_{-A} \pm |q(-A)\rangle_A \otimes |q(-A)\rangle_{-A}) \quad (57) $$

$$ \Psi^{\pm}(q(A), q(-A)) = \frac{1}{\sqrt{2}} (|q(A)\rangle_A \otimes |q(-A)\rangle_{-A} \pm |q(-A)\rangle_A \otimes |q(A)\rangle_{-A}) \quad (58) $$

Borsuk Ulam theorem states that $|q(A)\rangle = |q(-A)\rangle$. In other words the information provided by $q(A)$ matches the information provided by $q(-A)$, then

$$ \Phi^{\pm}(q(A), q(-A)) = \frac{1}{\sqrt{2}} (|q(A)\rangle_A \otimes |q(A)\rangle_{-A} \pm |q(A)\rangle_A \otimes |q(A)\rangle_{-A}) \quad (59) $$

$$ \Psi^{\pm}(q(A), q(-A)) = \frac{1}{\sqrt{2}} (|q(A)\rangle_A \otimes |q(-A)\rangle_{-A} \pm |q(-A)\rangle_A \otimes |q(A)\rangle_{-A}) \quad (60) $$

The (Bell) states entanglement in $\mathbb{S}^2$ "vanishes". Then, a definite pure state can be attributed to $|q(A)\rangle$. The system is entangled at the $\mathbb{S}^1$ level but un-entangled at the $\mathbb{S}^2$ level.

## 4. Quantum BUT theorem in $\mathbb{S}^2$

The same procedure is applied in the Bloch-Poincaré sphere $\mathbb{S}^2$ that defines Bell states in $\mathbb{S}^2$. The entangled states are given by equations 57 and 58. The mapping $f(x) = f(-x)$ is in $\mathbb{R}^2$. Now, writes in $\mathbb{S}^3$ hyper sphere. Based from the Borsuk Ulam theorem, Peters and Tozzi [43] obtains the Hypersphere Quantum Information Theorem [43]. If $f(q(A))$ defines a feature vector $(A, q(A), f(q(A)))$ (in term of machine learning, a feature vector is an n-dimensional vector of numerical features that represent some object) on the surface of a hypersphere $\mathbb{S}^3$, one writes for the antipodal points $B = -A$

$$ \frac{1}{\sqrt{2}} (|f(q(A))\rangle_{q(A)} \otimes |f(q(A))\rangle_{q(-A)} \pm |f(q(-A))\rangle_{q(A)} \otimes |f(q(-A))\rangle_{q(-A)}) = \Phi^{\pm}(f(q(A))), f(q(-A))) \quad (61) $$
$$ \Psi^{\pm}(f(q(A)), f(q(B))) = \frac{1}{\sqrt{2}} (|f(q(A))\rangle_{q(A)} \otimes |f(q(-A))\rangle_{q(-A)} \pm |f(q(-A))\rangle_{q(A)} \otimes |f(q(A))\rangle_{q(-A)}) \quad (62) $$

Using Hyper sphere quantum information [43] theorem, Borsuk Ulam theorem states that $|f(q(A))\rangle = |f(q(-A))\rangle$ then

$$ \Phi^{\pm}(f(q(A)), f(q(-A))) = \frac{1}{\sqrt{2}} (|f(q(A))\rangle_{q(A)} \otimes |f(q(A))\rangle_{q(-A)} \pm |f(q(A))\rangle_{q(A)} \otimes |f(q(A))\rangle_{q(-A)}) \quad (63) $$

$$ \Psi^{\pm}(f(q(A)), f(q(-A))) = \frac{1}{\sqrt{2}} (|f(q(A))\rangle_{q(A)} \otimes |f(q(A))\rangle_{q(-A)} \pm |f(q(A))\rangle_{q(A)} \otimes |f(q(A))\rangle_{q(-A)}) \quad (64) $$

The (Bell) states entanglement in $\mathbb{S}^3$ "vanishes". Then, a definite pure state can be attributed to $|f(q(A))\rangle$. This means that entanglement (i.e., a system not formed by two product states) can be seen as just 3D space surface points that have a corresponding pair of $\mathbb{S}^3$ antipodal points (pure states!). The system is entangled at the $\mathbb{S}^n$ level (n=2 in the real world), but un-entangled at the $\mathbb{S}^{n+1}$ hypersphere level.

For zero dimension, Borsuk Ulam theorem is evidenced. The pair of points at the ends of a (one-dimensional) line segment is a 0-sphere $\mathbb{S}^0$. Then, the two antipodal points are antisymmetric (see Pauling exclusion principle). Of course, in this example, one distinguish $x$ and $-x$ by the projection of the spin 1/2 and -1/2 in fermionic systems. For photons, the spin is replaced by the information provided by the polarization.

## 5. Dirac's string trick

Let us introduce basic tools that appear in the famous problem of Dirac's string trick.

Dirac's string trick is a good illustration of the entanglement between groups and topological spaces. It is intimately correlated to the notion of contractile loop. A complete rotation where one returns to the initial state needs a contractile loop. Rotating an object with strings attached to it by 360 degrees does not return the system to its original state, while a second rotation of 360 degrees, giving a total rotation of 720 degrees, allowe it. The illustration is given by figure 10 with a loop having a 2 winding number (a topological quantum number in physics). This is the origin of the half integer value for the electron spin. The 3 × 3 matrix rotation $R$ (we live in 3D Euclidean space and consider the position vector) is orthogonal $R^T R = 1$, the group of all rotation matrices is $SO(3)$ ($R \in SO(3)$) ($S$ for special $det(R) = 1$, $O$ for orthogonal, $3$ for three components). The full description
<figure><img src="image_14.png" /><figcaption>**Figure 10.** Sketch of the 4π rotation when in a loop with a winding number 2. This winding number appears in SU(2) group which is the double covering space of the rotational group SO(3).</figcaption></figure>

for the fundamental group of SO(3) writes $\pi_1 SO(3) = \{id, p\} = \mathbb{Z}_2$ (one reads $\mathbb{Z}$ modulo 2) and then 720 degrees to return in its original state, (p is a point on the $S^2$). According to quantum mechanics, the quantum spin states are vectors with two components in $\mathbb{C}^2$ U ($|\alpha|^2 + |\beta|^2 = 1$) contrary to a position vector with three components in $\mathbb{R}^3$. Note that U is orthogonal like $R$. The probability of the spin $|↑〉$ along z axis is $|\alpha|^2$, the probability of the spin $|↓〉$ along z axis is $|\beta|^2$

If we make a rotation $\theta$ we just introduce a phase term $\exp(i\theta)$ corresponding to the same transformation $U = U \exp(i\theta)$ (we take $detU = 1$). More generally, in order to rotate a spin vector, we must act on it by $U \in \mathbb{C}^2$. The set of such matrices that satisfy these conditions writes $U \in SU(2)$ ($detU = 1, U^T U = I$). $\alpha$ and $\beta$ terms have two components (real and imaginary parts) then the general $SU(2)$ matrix is given by four real numbers defined in $S^3$ ($SU(2)$ is topologically a sphere $S^3$). Since it not possible to represent $S^3$ space we use homeomorphism. $S^2$ can be looked the gluing of two disks glued along their boundary. The main idea is to view the 2-sphere as consisting of two hemispheres and an equator, ie as the disjoint union of two open two-dimensional disks or balls, $B^2$, and a circle $S^1$. Then $S^3$ can be looked the gluing of two balls glued along their boundary (equator). $SU(2)$ is the covering space of $SO(3)$ (double cover). It is shown that 2π twist is not contractible in $SO(3)$. The rotation of an electron corresponds to lift a path in $SO(3)$ to $SU(2)$ (Lie algebra). If we lift the path corresponding to the 2π twist in $SO(3)$ to $SU(2)$ then the start point and the end point are different (for example spin up to spin down). We need 4π twist to come back at the same position. There is a deep
connection between the rotation group and the symplectic group $Sp(n)$. In Lie algebras, there are isomorphisms $Sp(1) = SO(3) = SU(2)$ [117].

## 6. Photon in Hamiltonian mechanics

Taking the example of the photon treated as a classical particle (point particle). Its Hamiltonian writes $H = c|p|$ since $H$ does not depend to position $x$. Then the two Hamiltonian’s equation write (equation 7)

$$ \frac{\partial p}{\partial t} = -\frac{\partial H}{\partial x} = 0 \qquad (65) $$

$$ \frac{\partial x}{\partial t} = \frac{\partial H}{\partial p} = c \frac{p}{|p|} \qquad (66) $$

The phase space diagram $(x, p)$ gives a vector flux positive for $p > 0$ and negative for $p < 0$ (ie a $c$ velocity for $p > 0$ and $-c$ for $p < 0$). It is clear that the system is not Newtonian since $H$ is not differentiable at $p = 0$. Then we cannot define a vector flux for $x = 0$ and $p = 0$ simultaneously. This is not so far to the Heisenberg principle where at $x = 0$ (well defined) the vector flux is defined in the interval $\Delta p > 0$.

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