Nonclassical behavior of moving relativistic
We study the survival probability of moving relativistic unstable particles with definite momentum . The amplitude of the survival probability of these particles is calculated using its integral representation. We found decay curves of such particles for the quantum mechanical models considered. These model studies show that late time deviations of the survival probability of these particles from the exponential form of the decay law, that is the transition times region between exponential and non-exponential form of the survival probability, should occur much earlier than it follows from the classical standard approach resolving itself into replacing time by (where is the relativistic Lorentz factor) in the formula for the survival probability and that the survival probabilities should tend to zero as much slower than one would expect using classical time dilation relation. Here we show also that for some physically admissible models of unstable states the computed decay curves of the moving particles have fluctuating form at relatively short times including times of order of the lifetime.
PACS numbers: 03.65.-w, 11.10.St, 03.30.+p
Key words: Non–exponential decay, relativistic unstable particles, Einstein time dilation.
Physicists studying the decay processes are often confronted with the problem of how to predict the form of the decay law of the particle moving in respect to the rest reference frame of the observer knowing the decay law of this particle decaying in its rest frame. From the standard, text book considerations one finds that if the decay law of the unstable particle in rest has the exponential form then the decay law of the moving particle with momentum is , where denotes time, is the decay rate (time and are measured in the rest reference frame of the particle) and is the relativistic Lorentz factor, , is the velocity of the particle. This equality is the classical physics relation. It is almost common belief that this equality is valid also for any in the case of quantum decay processes and does not depend on the model of the unstable particles considered. For the proper interpretation of many accelerator experiments with high energy unstable particles as well as of results of observations of astrophysical processes in which a huge numbers of elementary particles (including unstable one) are produced we should be sure that this belief is supported by theoretical analysis of quantum models of decay processes. The problem seems to be extremely important because from some theoretical studies it follows that in the case of quantum decay processes this relation is valid to a sufficient accuracy only for not more than a few lifetimes [1, 2, 3, 4]. What is more it appears that this relation may not apply in the case of the famous result of the GSI experiment, where an oscillating decay rate of the ionized isotopes Pr and Pm moving with relativistic velocity () was observed [5, 6]. So we can see that the problem requires a deeper analysis. In this paper the basis of such an analysis will be the formalism developed in [1, 2] where within the quantum field theory the formula for the survival amplitude of moving particles was derived. We will follow the method used in  and we will analyze numerically properties of the survival probability for a model of the unstable particle based on the Breit–Wigner mass distribution considered therein and as well as the other different one. Here we show that the relativistic treatment of the problem within the Stefanovich–Shirokov theory [1, 2] yields decay curves tending to zero as much slower than one would expect using classical time dilation relation which confirms and generalizes some conclusions drawn in . We show also that for some physically admissible models of unstable states decay curves of the moving particles computed using the above mentioned approach have analogous fluctuating form as the decay curve measured in the GSI experiment and that in the model considered these fluctuations begin from times much shorter than the lifetime. Our results shows that conclusions relating to the quantum decay processes of moving particles based on the use of the classical physics time dilation relation need not be universally valid.
One of the aims of this paper is to analyze numerically properties of the survival probability in a wide range of times from very short trough until of moving unstable particles derived in [1, 2] and to present results of calculations of decay curves of such particles for the model considered in  but for the more realistic parameters of this model and to confront them with results obtained for another more realistic model. Another intention is to demonstrate that when considering the relativistic quantum unstable system the only rational assumption seems to be the assumption that the momentum of such a system is constant. The paper is organized as follows: Sec. 2 contains preliminaries and the main steps of the derivation of all relations necessary for the numerical studies, which results are presented in Sec. 3. Consequences of the assumption that the momentum of the moving freely quantum unstable system is constant are analyzed in Sec. 4. Sec. 5 contains a discussion and conclusions.
2 Decay law of moving relativistic particles
Let us begin our considerations from the following assumptions: Suppose that in a laboratory a large number of unstable particles was created at the instant of time and then their decay process is observed there. Suppose also that all these unstable particles do not move or are moving very slowly in relation to the rest frame of the reference of the observer and that the observer counts at instants , (where ), how many particles survived up to these instants of time. All collected results of these observations can be approximated by a function of time forming a decay curve. If is large then can be considered as the survival probability of the unstable particle. The standard procedure is to confront results of the experiment with theoretical predictions. Within the quantum theory, when one intends to analyze the survival probability of the unstable state or particle, say , in the rest system, one starts from the calculation of the probability amplitude . This amplitude defines the survival probability we are looking for. There is and , where is the total, self–adjoint Hamiltonian of the system considered. Studying the properties of the amplitude it is convenient to use the integral representation of as the Fourier transform of the energy or, equivalently mass distribution function, , (see, eg. [7, 8, 9, 10, 11, 12]), with and for , ( is the lower bound of the spectrum of ). It appears that the general form of the decay law of the particle in its rest reference frame practically does not depend on the form of the all physically acceptable (see, eg. [13, 9, 10, 11, 14, 15, 16]): There is, , starting from times slightly longer than the extremely short times [14, 15, 16]. Here , ( is the energy of the system in the unstable state measured at the canonical decay times when has the exponential form, is the normalization constant). The component exhibits inverse power–law behavior at the late time region. The late time region denotes times , where is the cross–over time and it can be found by solving the following equation, . There is for and for .
We came to the place where a flux of moving relativistic unstable particles investigated by an observer in his laboratory should be considered. According to the fundamental principles of the classical physics and quantum theory (including relativistic quantum field theory) the energy and momentum of the moving particle have to be conserved. There is no an analogous conservation law for the velocity . These conservation laws are one of the basic and model independent tools of the study of reactions between the colliding or decaying particles. So it seems to be reasonable to assume, as it was done in [1, 2, 17], that momentum of the moving unstable particles measured in the rest frame of the observer is constant (see also a discussion in ). The question is what is the picture seen by the observer in such a case and what is the relation between this picture and the picture seen by this observer in the case of non moving unstable particles? In other words we should compare the decay law with the decay law of the moving relativistic unstable particle with the definite, constant momentum const.. It is important to remember that the decay law does not describe the quantum decay process of the moving particle in its rest frame but describes the decay process of this particle seen by the observer in his rest laboratory frame. Such a picture one meets in numerous experiments in the field of high energy physics or when detecting cosmic rays: Detectors of a finite volume are stationary in the frame of reference of the observer and stable or unstable particles together with their decay products passing through the detector are recorded. According to the broadly shared opinion reproduced in many textbooks one expects that it should be,
in the considered case. This relation is a simple extension of the standard time dilation formula to quantum decay processes. The question is how does the time dilation formula being the classical physics formula work in the case of quantum decay processes? From the results reported in [1, 2, 4] and obtained there for the model defined by Breit–Wigner mass (energy) distribution function it follows that the relation (2). works in this model only within a limited range of times: For no more than a few lifetimes What is more, it has been shown in  that for times longer than few lifetimes the difference between the correctly obtained survival probability and is significant.
Now let us follow [1, 2] and calculate survival probabilities and . Hamiltonian and the momentum operator have common eigenvectors . Momentum is the eigenvalue of the momentum operator . There is in units:
In the coordinate system of the unstable quantum state at rest, when , we have ,
where and is the continuous part of the spectrum of the Hamiltonian . Operators and act in the state space . Eigenvectors are normalized as follows
Now we can model the moving unstable particle with constant momentum, , as the following wave–packet ,
where expansion coefficients are functions of the mass parameter , that is of the rest mass , which is Lorentz invariant and therefore the scalar functions of are also Lorentz invariant. (Here is the lower bound of the spectrum of ). We require the state to be normalized: So it has to be .
By means of the relation (2) we can define the state vector describing an unstable state in rest as follows,
This expansion and (2) allow one to find the amplitude and to write
We need also the probability amplitude , which defines the survival probability . There is in units. We have the vector (see (2)) but we still need eigenvalues solving Eq. (2). Vectors are elements of the same state space connected with the coordinate rest system of the observer : We are looking for the decay law of the moving particle measured by the observer . If to assume for simplicity that and that then there is for the eigenvalues of the momentum operator . Let be the Lorentz transformation from the reference frame , where the momentum of the unstable particle considered is zero, , into the frame where the momentum of this particle is and , or, equivalently, where its velocity equals , (where is the rest mass and ). In this case the corresponding 4–vectors are: within the considered system of units, and . There is in Minkowski space, which is an effect of the Lorentz invariance. (Here the dot ”” denotes the scalar product in Minkowski space). Hence, in our case: because and thus .
Another way to find is to use the unitary representation, , of the transformation , which acts in the Hilbert space of states : One can show that the vector is the common eigenvector for operators and , that is that there is
(see, eg. ). Indeed, taking into account that operators and form a 4–vector , and , we have
where (see, e.g., , Chap. 4). From this general transformation rule it follows that
Based on this relation, one can show that that vectors are eigenvectors for the Hamiltonian . There is
The Lorentz factor corresponds to the rest mass being the eigenvalue of the vector . There are and for . From (2) it follows that for , which means that using (2) the relation ( ‣ 2) can be rewritten as follows
Taking into account the form of the forced by the condition const one concludes that in fact the eigenvalue found, , equals . This is exactly the same result as that at the conclusion following from the Lorentz invariance mentioned earlier: , which shows that the above considerations are self–consistent.
Similarly one can show that vectors are the eigenvectors of the momentum operator for the eigenvalue , that is that which was to show.
Thus finally we come to desired result:
which replaces Eq. (2).
The above derivation of the expression for is similar to that of . It is based on  and it is reproduced here for the convenience of readers. This is a shortened and slightly changed, simplified version of the considerations presented in  and mainly in  and more explanations and more details can be found therein and in [20, 21], where this formula was derived using the quantum field theory theory approach.
3 Results of numerical studies
According to the literature a reasonable simplified representation of the density of the mass distribution is to choose the Breit–Wigner form for , which under rather general condition approximates sufficiently well many real systems [1, 9, 13],
where is a normalization constant and is the unit step function, is the rest mass of the particle and is the decay rate of the particle in the rest. Inserting into (2) and into ( ‣ 2) one can find decay curves (survival probabilities) and . Results of numerical calculations are presented in Figs (1), (2) where calculations were performed for , and . Values of these parameters correspond to , which is very close to from the experiment performed by the GSI team [5, 6] and this is why such values of them were chosen in our considerations. Similar calculations were performed in  but for different and less realistic values of the ratio : For and and different . According to the literature for laboratory systems a typical value of the ratio is (see eg. ) therefore the choice seems to be reasonable and more realistic than those used in .
Results presented in Figs (1) and (2) show that in the case of having the Breit–Wigner form the survival probabilities and overlap for not too long times when has the canonical, that is the exponential form. This observation confirms conclusions drawn in [1, 2, 4]. On the other hand results presented in the panel of Fig (1) and Fig (2) show that in the case of moving relativistic unstable particles the transition times region, when the canonical form of the survival probability transforms into inverse power like form of , begins much earlier than in the case of this particle observed in its rest coordinate system and described by . This observation agrees with results obtained in .
To be sure that the above conclusions are valid not only in the approximate case of the density of the mass distribution we should consider a more general form of . The most general condition for following from (2) is that . So, if to assume that and additionally that for , and for , that is that
(where ), and , for and , (), exist and they are continuous in , and limits exist, and
where is Euler’s gamma function. Hence, one finds that, e.g. for , the leading term of has the following form
From an analysis of general properties of the mass (energy) distribution functions of real unstable systems it follows that they have properties similar to the scattering amplitude, i.e., they can be decomposed into a threshold factor, a pole-function, with a simple pole (often modeled by ) and a smooth form factor [9, 13]. This means that in ( ‣ 3) should have the following form , where as . Guided by this observation we follow  and assume that
with . The asymptotic form of the survival amplitude for such a is given by the relation ( ‣ 3). Hence one finds that at late times there is in the case considered. Decay curves corresponding to defined by ( ‣ 3) were find numerically for the case of the particle decaying in the rest system (the survival probability ) as well as for the moving particle (the non–decay probability ). Results are presented in Figs (3) and (4). In order to compare them with the results obtained for , calculations were performed for the same ratios as in that case: , and . The ratio was chosen to be (Fig. (3)) and (Fig. (4)).
From Figs (3), (4) it is seen that in the case of , e.g. when has the form given by Eq. ( ‣ 3), the survival probabilities and have an analogous form as the corresponding probabilities obtained for both for relatively short times and for long times . On the other hand, in the case of the survival probabilities the difference between decay curves calculated for the density given by formula ( ‣ 3) and for is significant: The decay curves calculated for defined in ( ‣ 3) have an oscillating form at times and shorter while those obtained for do not have. This is rather unexpected result but it shows that in the case of moving relativistic particles quantum decay processes may have nonclassical form even at times shorter than the lifetime.
4 Analysis of masses and velocities of unstable states
It was assumed in Sec. 2 and 3 that the momentum of the relativistic unstable particle moving like a free particle is conserved. Using this assumption one usually concludes that in such a case the velocity of the particle have to be conserved and constant in time. Such a conclusion is true in the case of the classical particles: In the case o a classical object moving like a free particle the conservation of the momentum means that the velocity of this object is constant in time. The question is whether such a conclusion is true in the case of moving quantum unstable objects or not. In order to solve this problem we should analyze relativistic formula for the momentum , which within the assumed system of units has the following form: . In this relation is the rest mass of the moving quantum or classical objects and is the velocity of these objects. From the point of view of the quantum theory the problem is that the state vector of the form (2) corresponding to such a quantum object can not be an eigenvector of the Hamiltonian (including the case ), otherwise it would be that for all times . The fact that the vector describing the unstable quantum object is not the eigenvector for means that the mass (energy) of this object is not defined. Simply the mass can not take the exact constant value in this state . In such a case quantum objects are characterized by the mass (energy) distribution density and the average mass
Let us analyze the properties of the instantaneous mass. The instantaneous mass (energy) can be found using the exact effective Hamiltonian governing the time evolution in the subspace of states spanned by the vector ,
which results from the Schrödinger equation when one looks for the exact evolution equation for the mentioned subspace of states (for details see [14, 16, 23, 24, 25, 26]). where the system of units is used. It is assumed that the vector is not an eigenvector of : There does not exist any number such that .
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 :
and is the instantaneous decay rate.
where , is the projector onto the subspace od decay products, and , then one can see that there is a permanent contribution of decay products described by to the instantaneous mass (energy) of the unstable state considered. The intensity of this contribution depends on time . Using ( ‣ 4) and ( ‣ 4) one finds that
From this relation one can see that and if the matrix elements exists. It is because and .
Now let us assume that exists and is a continuous function of time for . If these assumptions are satisfied then is a continuous function of time for and exists. Now if to assume that for there is then from the continuity of immediately follows that there should be for any . Unfortunately such an observation contradicts implications of ( ‣ 4), ( ‣ 4): From the relations ( ‣ 4), ( ‣ 4) one concludes that it is possible if, and only if,
for every such that . There is for , therefore
for every and . The relation ( ‣ 4) can take place if, and only if,
This last condition leads to the conclusion that
This observation means that
if and only if there is no any decay of the state considered (if there is no any transitions between and ). So, in the case of unstable systems , which means that in the case of unstable systems the instantaneous mass (energy) and the instantaneous decay rate can not be constant in time: and . Results of numerical calculations presented in Figs (5) — (7) (or those one can find in [23, 24]) confirm this conclusion. In Figs (5) — (7) the function,
is presented, which illustrates a typical form of time–varying . (All calculations were performed for ).
As it is seen from Figs (5), (6), (7) the amplitude of variations of needs not be large at relatively short times: It is almost negligible small but these variations always exist (see Figs (5) — (7) and results presented in ). When the time increases the amplitude of these variations grows and reaches maximal values for times . Now if this particle is a moving relativistic particle then within the assumed system of units its momentum equals , where is the rest mass of the particle , is the velocity. The total momentum (and energy) of the objects moving like a free particle both quantum and classical must be conserved. Thus it has to be , that is for any . It is possible only if changes of are compensated by suitable changes of , that is by corresponding changes in the velocity . (A similar mechanism was described in [23, 24], where its consequences were analyzed for times of the order of the cross–over time ). So the principle of conservation of the momentum forces compensation of changes in the instantaneous mass through appropriate changes in the velocity of the moving unstable system. (It is a pirouette like effect). This is why the assumption when considering moving quantum unstable objects leads to the result , i.e., to the result never observed in experiments . Thus the assumption mentioned seems to be the only acceptable choice in the case of moving quantum unstable systems (see also a discussion in ).
Let us analyze now implications of the observation that the velocity of the quantum unstable system moving like a free particle can not be constant in time and it has to vary in time . This property has an effect that . Now let us denote by 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 with the velocity measured in . The observation that means that the rest reference frame of the quantum unstable system moving like a free particle can not be the inertial one.
5 Discussion and Conclusions
Let us begin from a general remark: In any case we should remember that the relation (2) is the classical physics relation and that the quantum decay processes are analyzed in this paper. The relativistic time–dilation relation in its form known from classical physics does not need to manifest itself in quantum processes in the same way as in classical physics processes. It is also important to be aware that as it was shown in  the Quantum Field Theory models of the decay processes can be also described within the formalism used in Sec. 2.
All results presented in Figs (1) — (4) show decay curves seen by the observer in his rest reference frame (curves correspond to the situation when the classical dilation relation (2) is assumed to be true in the case of quantum decay processes). The time (the horizontal axes) in all these figures is the time measured by the observer in his rest system. These results show that Stefanovich–Shirokov theory [1, 2] predict a such form of the survival probability that the expected relation (2) holds to very good approximation only for times and only for . The visible difference between and takes place at times but this needs not mean that this theory is wrong: To this day there have been no published reports on experiments analyzing the form of the decay law of moving relativistic unstable particles at times or and .
Analyzing the results presented in Figs (1) — (4) we can conclude that properties of the survival probability of the moving unstable particle, , where is calculated using the Eq. ( ‣ 2) (i.e the formula derived in [1, 2]), are much more sensitive to the form of than properties of . It is a general observation. Another general conclusion following from these results is that starting from times from the transition time region, , the decay process of moving particles is much slower than one would expect assuming the standard dilation relation (2).
From Figs (3) and (4) it follows that in the case of moving relativistic unstable particles the standard relation (2) does not apply in the case of the density of the form ( ‣ 3) and leads to the wrong conclusions for such densities. Results presented in these Figures show also that a conclusion drawn in [1, 2, 4] on the basis of studies of the model defined by the Breit–Wigner density that the relation (2) is valid for not more than few lifetimes is true only for the density and need not be true for densities having a more general form. Similar limitations concern the result presented in , where it is stressed that the approximations used to derive the final result may work only for times no longer than a few lifetimes. What is more, a detailed analysis shows that the final result presented therein was obtained using the non–relativistic limit of