# On the Velocity of Moving Relativistic Unstable Quantum Systems

Authors: K. Urbanowski
Journal: Advances in High Energy Physics
 (2015)

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## Abstract

We study properties of moving relativistic quantum unstable systems. We show that in contrast to the properties of classical particles and quantum stable objects the velocity of freely moving relativistic quantum unstable systems cannot be constant in time. We show that this new quantum effect results from the fundamental principles of the quantum theory and physics: it is a consequence of the principle of conservation of energy and of the fact that the mass of the quantum unstable system is not defined. This effect can affect the form of the decay law of moving relativistic quantum unstable systems.


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## Body

## 1. Introduction

Physicists studying the decay processes of unstable quantum systems moving with the velocityv→ relative to the rest reference frame of an observer and trying to derive theoretically the decay law of such systems are confronted with the following problem: Which of the two possible assumptions, v→=const or perhaps p→=const (where p→ is the momentum of the moving unstable system), will get decay law correctly describing the real properties of such system. When one considers classical physics decay processes, the mentioned assumptions both lead to the decay law of the same form. Namely, from the standard, text book considerations, one finds that if the decay law of the unstable particle in rest has the exponential form P0(t)=exp[-Γ0t], then the decay law of the moving particle with momentum p→≠0 is Pp(t)=P0(t/γ)≡exp[-Γ0t/γ]≡Pv(t), where t denotes time, Γ0 is the decay rate (time t and Γ0 are measured in the rest reference frame of the particle), γ is the relativistic Lorentz factor, γ≡1/1-v2, v=|v→|, and Pv(t) is the decay law of the particle moving with the constant velocity v→ (we use ħ=c=1 units, and thus v<1). It is almost common belief that this equality is valid also for any t in the case of quantum decay processes and does not depend on the model of the unstable system considered. The cases p→=const and v→=const both were studied in the literature. The assumption p→=const was used in [1, 2] to derive the survival probability Pp(t). From these studies, it follows that in the case of moving quantum unstable systems the relation Pp(t)≃P0(t/γ) is valid to a sufficient accuracy only for not more than a few lifetimes and that for times much longer than a few lifetimes there is Pp(t)>P0(t/γ) (see [2, 3]). The assumption v→=const was used, for example, in [4], to derive the decay law of moving quantum unstable systems. Unfortunately, the result obtained in [4] is similar to the case p→=const: Pv(t)≃P0(t/γ) only for no more than a few lifetimes. What is more, it appears that the assumption v→=const may lead to the relation Pv(t)=P0(γt), that is, to the result never observed in experiments [5, 6].Unfortunately, the experiments did not give any decisive answer for the problem which is the correct assumption:p→=const or v→=const? It is because all known tests of the relation Pv(p)(t)≃P0(t/γ) were performed for times t~τ0 (where τ0 is the lifetime) (see, e.g., [7, 8]). Note that the same relation obtained in [1, 2, 4] is approximately valid for the same times t (see also discussion in [9–13]). The problem seems to be extremely important in accelerator physics where the correct interpretation of the obtained results depends on knowledge of the properly calculated decay law of the moving unstable particles created in the collisions observed. Similarly, the proper interpretation of results of observations of astrophysical processes in which a huge numbers of elementary particles (including unstable one) are produced is impossible without knowing the correct form of the decay law of unstable particles created in these processes. So the further theoretical studies of the above-described problem are necessary and seem to be important.In this paper, we analyze general properties of unstable quantum system from the point of view of fundamental principles of physics and quantum theory. Here, we show that the principle of the conservation of the energy does not allow any moving quantum unstable system to move with the velocityv→ constant in time.


## 2. Quantum Unstable Systems

The main information about properties of quantum unstable systems is contained in their decay law, that is, in their survival probability. Let the reference frameO be the common inertial rest frame for the observer and for the unstable system. Then, if one knows that the system in the rest frame is in the initial unstable state ϕ, which was prepared at the initial instant t0=0, one can calculate its survival probability, P0(t), which equals P0(t)=|a(t)|2, where a(t) is the survival amplitude, a(t)=ϕ∣ϕ;t, |ϕ;t〉=e-itH|ϕ〉, H is the total self-adjoint Hamiltonian of the system under considerations, |ϕ〉,|ϕ;t〉∈H, and H is the Hilbert space of states of the considered system. So in order to calculate the amplitude a(t), one should know the state |ϕ〉. Within the standard approach, the unstable state |ϕ〉 is modeled as the following wave packet [14–18]:(1)ϕ=∫μ0∞cmmdm,where μ0 is the lower bound of the continuous part σc(H) of the spectrum of H and vectors |m〉 solve the following:(2)Hm=mm,m∈σcH.Eigenvectors |m〉 are normalized as follows:(3)m∣m′=δm-m′. We require the state |ϕ〉 to be normalized; so it has to be ∫μ0∞|c(m)|2dm=1. Thus, (4)ϕ;t=e-itHϕ≡∫μ0∞cme-itmmdm,which allows one to represent the amplitude a(t) as the Fourier transform of the mass (energy) distribution function, ω(m)≡|c(m)|2:(5)at=∫ωme-itmdm,where ω(m)≥0 and ω(m)=0, for m<μ0 [14–23] (see also [1–6]). From the last relation and from the Riemann-Lebesque lemma, it follows that |a(t)|→0 as t→∞. It is because, from the normalization condition ∫μ0∞|c(m)|2dm≡∫μ0∞ω(m)dm=1, it follows that ω(m) is an absolutely integrable function (note that this approach is also applicable in Quantum Field Theory models [24, 25]). The typical form of the survival probability P0(t) is presented in Figure 1, where the calculations were performed for ω(m) having the Breit-Wigner form:(6)ωBWm≡N2πΘm-μ0Γϕ0m-Mϕ02+Γϕ0/22,assuming for simplicity that s0=(Mϕ0-μ0)/Γϕ0=50. Here, Θ(m) is a step function: consider Θ(m)=0 for m≤0 and Θ(m)=1 for m>0.Figure 1
Decay curve obtained forω(m) given by (6). Axes: y=P0(t), the logarithmic scale; x=t/τ0 (time is measured in lifetimes).Note that(7)Hϕ≡∫μ0∞mcmmdm,which means that the vector |ϕ〉 corresponding to an unstable state is not the eigenvector for the Hamiltonian H. In other words, in the rest frame considered, there does not exist any number; let us denote it by mϕ0, such that it would be H|ϕ〉=mϕ0|ϕ〉. This means that the mass (i.e., the rest mass mϕ0) of the unstable quantum system described by the vector |ϕ〉 is not defined. What is more, in such a case, the mass of this system cannot be constant in time in the state considered. Simply, the mass of the unstable system cannot take the exact constant value in the state |ϕ〉; otherwise, it would not be any decay; that is, it would be P0(t)≡|ϕexp-itH|ϕ〉|2=1, for all t. In general, such quantum systems are characterized by the time independent mass (energy) distribution density ω(m), that is, by the modulus of the expansion coefficient c(m), but not by the exact value of the mass. In this case, instead of the mass, the average mass, 〈mϕ〉, of the unstable system can be determined knowing ω(m) or the instantaneous mass of this system [23, 26–28]. The average mass is defined by means of the standard formula: 〈mϕ〉=∫μ0∞mω(m)dm. The instantaneous mass mϕ0(t) (energy) can be found using the exact effective Hamiltonian hϕ(t) governing the time evolution in the subspace of states spanned by the vector |ϕ〉:(8)hϕt=iat∂at∂t,(9)≡ϕHϕ;tϕ∣ϕ;t,which results from the Schrödinger equation when one looks for the exact evolution equation for the mentioned subspace of states (for details, see [23, 26–29]). Within the assumed system of units, the instantaneous mass (energy) of the unstable quantum system in the rest reference frame is the real part of hϕ(t):(10)mϕ0t=Rhϕt,and Γϕ(t)=-2I[hϕ(t)] is the instantaneous decay rate.Using relation (9), one can find some general properties of hϕ(t) and mϕ0(t). Indeed, if to rewrite the numerator of the right-hand side of (9) as follows,(11)ϕHϕ;t≡ϕHϕat+ϕHϕ;t⊥,where |ϕ;t〉⊥=Q|ϕ;t〉, Q=I-P is the projector onto the subspace od decay products, P=|ϕ〉〈ϕ|, and ϕ∣ϕ;t⊥=0, then one can see that there is a permanent contribution of decay products described by |ϕ;t〉⊥ to the instantaneous mass (energy) of the unstable state considered. The intensity of this contribution depends on time t. Using (9) and (11), one finds that(12)hϕt=ϕHϕ+ϕHϕ;t⊥at.From this relation, one can see that hϕ(0)=ϕHϕ if the matrix elements ϕHϕ exist. It is because |ϕ(t=0)〉⊥=0 and a(t=0)=1. Now, let us assume that ϕHϕ exists and i∂a(t)/∂t≡ϕHϕ;t is a continuous function of time t for 0≤t<∞. If these assumptions are satisfied, then hϕ(t) is a continuous function of time t for 0≤t<∞ and hϕ(0)=〈ϕ|H|ϕ〉 exists. Now, if to assume that for 0≤t1≠t2 there is R[hϕ(0)]=R[hϕ(t1)]=R[hϕ(t2)]=const, then from the continuity of hϕ(t) it immediately follows that there should be R[hϕ(t)]=hϕ(0)≡〈ϕ|H|ϕ〉=const for any t≥0. Unfortunately, such an observation contradicts implications of (12): from this relation, it follows that R[〈ϕ|H|ϕ;t〉⊥/a(t)]≠0 for t>0 and thus R[hϕ(t>0)]≠〈ϕ|H|ϕ〉≡R[hϕ(0)] which shows that mϕ0(t)≡R[hϕ(t)] cannot be constant in time. Results of numerical calculations presented in Figure 2 (or those one can find in [28]) confirm this conclusion.Figure 2
A typical form of the instantaneous massmϕ(t) as a function of time obtained for ωBW(m). Axes: y=κ(t), where κ(t) is defined by (16); x=t/τϕ: time is measured in lifetimes. The horizontal dashed line represents the value of mϕ0(t)=Mϕ0.In the general case, the mass (energy) distribution functionω(E) has properties similar to the scattering amplitude; that is, it can be decomposed into a threshold factor, a pole-function P(m) with a simple pole at m=Mϕ0-i/2Γϕ0 (often modeled by a Breit-Wigner), and a smooth form factor F(m). So there is (see, e.g., [18])(13)ωm=Θm-μ0m-μ0λ+lPmFm,where l is the angular momentum; 1>λ≥0. In such a case,(14)hϕ0t≃Mϕ0-i2Γϕ0,t~τϕ,at canonical decay times, that is, when the survival probability has the exponential form (here, τϕ is the lifetime), and(15)mϕ0t=Rhϕt≃Mϕ0=mϕ+Δmϕ0,t~τϕ,at these times to a good accuracy (see [26, 27, 29]). The parameters Mϕ0 and Γϕ0 are the quantities that are measured in decay and scattering experiments. If the state vector |ϕ〉=|ϕα〉 is an eigenvector for H corresponding to the eigenvalue mα, then there is hϕα(t)≡mα. Beyond the canonical decay times, mϕ0(t) differs from Mϕ0 significantly (for details, see [26–28]). At canonical decay times, values of mϕ0(t) fluctuate (faster or slower) around Mϕ0. One can see a typical behavior of mϕ0(t) in Figure 2, where the function,(16)κt=mϕ0t-μ0Mϕ0-μ0,is presented. These results were obtained numerically for the Breit-Wigner mass (energy) distribution function ω(m)=ωBW(m) and for s0=50. From Figure 2, one can see that fluctuations of mϕ0(t) take place at all stages of the time evolution of the quantum unstable system. At times of order of the lifetime, t~τϕ, and at shorter times, the amplitude of these fluctuations is so small that their impact on results of the mass (energy) measurements can be neglected (see (15)). With increasing time, their amplitude grows up to the maximal values, which take place at the transition times, that is, when the late time nonexponential deviations of the survival probability, P0(t), begin to dominate. Thus, with the increasing time, for t>τϕ, the impact of these fluctuations on behavior of the quantum unstable systems increases.Now, let us consider the case when the unstable quantum system is moving with a velocityv→ relative to reference frame O. It is obvious that an unstable quantum system moving with the relativistic velocity does not turn into a classical system but still subjects to the laws of quantum physics. So when one searches for properties of such systems, the implications following from rules of the quantum theory are decisive. Let us assume that this quantum object is moving freely with the constant velocity v→:(17)v→=const,and let us admit that v→ is so large that the relativistic effects can take place. The energy Eϕ of the quantum unstable system described in the rest frame by vector |ϕ〉 and moving with the constant velocity v→ can be expressed within the system of units used as follows:(18)Eϕ=mϕ0γ,where mϕ0 is the mass parameter (i.e., the rest mass) and γ=const. Thus, knowing the energy Eϕ and the velocity v→, that is, the Lorentz factor γ, one can determine the mass parameter mϕ0.From the fundamental principles, it follows that the total energy of the freely moving objects, both quantum and classical, stable and unstable, must be conserved. This means that if an experiment indicates the energy,E, of such an object to be equal to E(t1)=Eϕ at an instant t1, then at any instant t2>t1 there must be E(t2)=E(t1)≡Eϕ. Now, if the energy, E, of the moving quantum unstable system is conserved, E=Eϕ=const, then from the assumption it trivially results that there must be(19)mϕ0=Eϕγ≡const.This observation concerns also the instantaneous mass mϕ0(t): if it was Eϕ=Eϕ(t), it would be Eϕ(t)/γ=mϕ0(t). The conservation of the energy means that at any instant of time the energy has the same value, so there must be Eϕ(t)≡Eϕ=const. Therefore, if the energy is conserved and assumption (17) holds, then there must be mϕ0(t)=Eϕ(t)/γ≡Eϕ/γ=const.On the basis of this analysis, one can conclude that the rest massmϕ0 as well as the instantaneous mass mϕ0(t) of the moving quantum unstable system are constant at all instants of time t. But, unfortunately, such a conclusion is in sharp contrast to the conclusion following from the relation (7) and its consequences. This means that one should consider the following possible situations: either (a) conclusions following from the quantum theoretical treatment of the problem are wrong (i.e., the quantum theory is wrong) or (b) the energy of moving quantum unstable systems is not conserved (i.e., the principle of the conservation of the energy does not apply to moving quantum unstable systems), or simply (c) the assumption (17) cannot be realized in the case of moving quantum unstable systems. The probability that situations (a) or (b) occur is rather negligible small. So the only reasonable conclusion is that case (c) takes place.This situation has a simple explanation. Namely, despite the conclusions resulting from relation (7) in experiments with unstable particles, one observes them as massive objects. This is not in contradiction to the implications of relation (7). The conclusion that the mass of the unstable quantum system (i.e., the rest mass of such a system) cannot be defined and constant in time means that in the case of such system only the instantaneous mass varying in time can be considered: it can only be mϕ0≡mϕ0(t).So in the case of the moving quantum unstable system, the principle of the conservation of the energy takes the following form:(20)Eϕ=mϕ0tγv→t=const.Thus, the principle of conservation of energy forces the compensation of changes in the instantaneous mass mϕ0(t) through appropriate changes in the velocity v→≡v→(t) so that the product mϕ0(t)γ(v→(t)) was fixed and constant in time: for any times t1>t2 (in general, t1≠t2), there must be mϕ0(t1)γ(v→(t1))=mϕ0(t2)γ(v→(t2))=Eϕ (it is a pirouette like effect). In other words, the principle of the conservation of the energy does not allow any moving quantum unstable system to move with the velocity v→ constant in time.


## 3. Concluding Remarks

The above conclusions result from the basic principles of the quantum theory. Taking this into account, one should consider a possibility that, in the case of moving quantum unstable systems, assumption (17) may lead to the wrong conclusions. In general, the relation Pv(t)=P0(t/γ) can be considered only as the approximate one and it cannot pretend to be rigorous. So instead of using such a relation, one should rather look for an effective formula for the survival probability of the moving quantum unstable systems. Such effective formula could be obtained, for example, by replacing γ=const in P0(t/γ) by an effective Lorentz factor γeff(t)=γ(v→(t))≠const, which varies with the changes of velocity v→(t). Similar analysis shows that the assumption p→=const leads to the conclusion analogous to that resulting from the assumption Eϕ=const that the velocity v→ of the moving quantum unstable systems cannot be constant in time. This is the consequence of the relativistic formula for the momentum p→.From results presented in Figure2, it is seen that, with increasing time, t, the amplitude of fluctuations of mϕ(t) grows. So according to (20), in order to compensate these growing fluctuations, the fluctuations of the velocity v→(t) of the unstable system have to grow. This means that, with increasing time, t, (at t>τϕ), deviations of the decay law of moving unstable system from the classical relation P0(t/γ) should be more visible and should grow. This effect explains the results presented in [3] where with the increasing time the increasing difference between P0(t/γ) and Pp(t) was indicated and analyzed.One more remark is as follows. Let us denote byO′ the reference frame which moves together with the moving quantum unstable system considered and in which this system is in rest. This reference frame moves relative to O with the velocity v→. The property that the velocity v→ of the moving quantum unstable system cannot be constant in time has an effect that dv→/dt≠0. Therefore, the rest reference frame O′ of such a system cannot be the inertial one. This observation means that there does not exist a Lorentz transformation describing a transition from the inertial rest reference frame O of the observer into the noninertial rest reference frame O′ of the moving quantum unstable system.The last remark is as follows. It seems that the above-described effect can be relatively easily verified experimentally. It is because the conclusion that the velocityv→ of the moving quantum unstable system must vary in time means that dv→/dt≠0. Therefore, the moving freely charged unstable particles (or neutral unstable particles with nonzero magnetic moment) should emit electromagnetic radiation of the very broad spectrum: from very small up to extremely large frequencies (see [28]). Thus, this effect can be verified by using currently carried out experiments, which use a beam of charged unstable particles (e.g., π± mesons or muons) or ions of radioactive elements moving along a straight line. A section of the track of these particles, where they are moving freely, should be surrounded by sensitive antennae connected to the receivers being able to register a broad spectrum of the electromagnetic radiation. Then, every signal coming from the beam registered by these receivers will be the proof that the above-described effect takes place.



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