XJ0400115 MY76 E2-2004-4 M. I. Shirokov* DECAY LAW OF MOVING UNSTABLE PARTICLE Submitted to International Journal of Theoretical Physics>> *E-mail: shirokovthsunljinr.ru 1. INTRODUCTION Experimenters showed that the lifetime of unstable particles moving with velocity v is equal to 7-0-y, where To is the lifetime of the particle at rest and - = 1 _ V2/C2)-1/2. Usual explanation of the fact is based on the special theory of relativity. For example, Moller 1972) sets forth it as follows: "In view of the fact that an arbitrary physical system can be used as a clock, we see that any physical system which is moving relative to a system of inertia must have a slower course of development than the same system at rest. Consider for instance a radioactive process. The mean life of the radioactive substance, when moving with a velocity v, will thus be larger than the mean life o when the substance is at rest. From 2.36) we obtain immediately = - v 2 IC2) - 12, ro". This argumentation may be complemented by the following possible defin- ition of the unit of time which radioactive substance provides: this is the time interval during which the amount of the substance decreases twice, e.g. However, the standard clocks of the relativity theory are used when obtaining Eq. 2.36) At = t2 - t = 7(t2 - t' V2/C2)-1/2 Ar which Moller mentions. He begins the derivation of this equation with the phrase: "Consider a standard clock C' which is placed at rest in S' at a point on the x-axis with the coordinate x = x"'.I However, such a quantum clock as an unstable particle cannot be at rest (i.e., cannot have zero velocity or zero momentum) and simultaneously be at a definite point (due to the quantum uncertainty relation). So, the standard derivation of the moving clock dilation is inapplicable for the quantum clock. The related quantum-mechanical derivation must contain some reservations and corrections. Here another way of deriving the time dilation is used: to find relativistic quantum decay law of a moving unstable particle (momentum p --4 0) and to compare it with the decay law of the particle at rest. Both the laws refer to one Lorentz frame (e.g., laboratory frame). Lorentz transformations of the space-time coordinates are not needed as well as the space coordinates themselves. I This approach to the derivation of the time dilation was used by Exner ( 983) and Stefanovich 1996). A simple derivation of the probability amplitude Ap(t) of the decay (more exactly nondecay) of the particle with momentum p is suggested in Sect. 2. Distinctions from Exner and Stefanovich derivations are discussed. Let us define the terminology used below. The equation p = T-Y is equiva- lent to the the equation jAp(t) 12 = Ao(t/-y) 12 if JA, 12 and Ao 12 depend upon t as IA,(t)l' - exp(-t/-rp), JAo(t) 12 _ XP(_t/70). Eq. (1) or the substitution t - t/-y used in Eq. (1) will be called Einstein dilation ED (in this respect see also Eq. 2.37) in [Moller, 1972]). Calculations of quantum probabilities JA, (t)12 and jAo (t)12 show that ED does not hold exactly. Numerical calculations were performed by Stefanovich (1996). Here analytical evaluation of Ap(t) is carried out, see Sect. 3 and Appen- dix. It allows us to determine the region of times t, where ED is approximately valid, and its accuracy. It also shows that ED fails under some conditions. In particular, I shall consider in Sect. 4 a variant of the quantum clock for which appreciable deviations from ED take place at all times. For conclusion see Sect.'5. The analytical evaluation of Ap(t) is presented in Appendix. 2. NONDECAY LAW OF MOVING UNSTABLE PARTICLE Let us consider a relativistic theory which describes unstable particles, prod- ucts of their decay, and the corresponding interactions. A field theory may be an example. Such a theory must contain operators of total energy and momentum 1' 0 (the generators of time and space translations), total angular momentum, and generators of Lorentz boosts. Suppose that at the initial moment t = there is one unstable system (particle) having definite momentum p'. As there are no other particles p is an eigenvalue of the total momentum P. The state vector of this system is the related eigenvector which may be represented in the momentum representation as (14Y = 6#"700- (2) Here V50 describes the unstable system in its rest frame. For example, *0 may describe an excited state of the hydrogen atom in the atom rest frame. In Eq. 2), 6#,fi, is the Kronecker symbol: I suppose that P has a discrete spectrum (the 2 system is in a large space volume and usual periodicity conditions are imposed or the volume opposite boundaries are identified). Thus, Tp, has the unit norm. Consider a common eigenvector p,,, of and 1. Similarly to Eq. 2) one has (PI(DPI') (3) Here 0, is the common I and P eigenvector corresponding to the zero eigen- value of P. The related ft eigenvalue may be called the mass p. Let us expand IPP, in vectors fi,, TP = ACUL),Dfi", (4) Here S, denotes sum and/or integral over (it will be refined below). I suppose that c(p) in Eq. 4) does not depend on p'. It follows from Eqs. 2)-(4) that 0 = S'-cG')00"'- (5) One may consider Eq. (5) as a possible concrete definition of the vector . Let us determine the eigenvalue Ep, of ft which corresponds to the common eigenvector Dfi,, of ft and P, p 54 0. (Ep, assumes the value if p 0) In relativistic theory E'P - P" is the Lorentz invariant. From 2 /2 2 2 EP - P - E = 11 (6) the known relativistic formula Ep, follows. So we have f14,P11 /p2 p2.p PtA (7) (the prime over p is omitted). It follows from Eqs. 4) and 7) that lpp(t = e- p = c(p)1Dp,. exp( -itV'p2 +,u2). (8) So the probability amplitude of the nondecay (survival amplitude) is Ap(t) (,Pp, e-' HtXPP = SAIC(tt)12 exp(-itN/p-2+p (9) The state qPP is called unstable if Ap(t - as t - oo. This property holds only if S, in Eq. 9) is integral over continual i values. Further, the spectrum of tile Hamiltonian H must be bounded from below. Let us choose the energy origin so that the lower bound in S, would be zero. Thus S, in Eq. 9) must denote the integral over from zero to, e.g., infinity. When p = Eq. (8) turns into the known equation for the probability amplitude of nondecay of the unstable system at rest, e.g., see [Fonda et al., 1978]. Eq. (8) was obtained by Stefanovich 1996) in a different way. He defined TP as Upoo where Up represents Lorentz transformation from the rest frame of unstable particle to the frame where its momentum is p. Here XPp is defined by Eq. 2) and only one relativistic formula, Eq. 6), is used. 3 3. DILATION OF MOVING UNSTABLE PARTICLE DECAY Let us calculate and compare the nondecay law Ap(t) of moving unstable particle and nonclecay law Ao(t) of the particle at rest. For this purpose jc(A)j is needed, see Eq. 9). It is possible to calculate Ic(p)1' using solvable models, e.g., see [Levy, 19591, [Alzetta, d'Ambrogio, 1966], [Gi-Chol Cho, 1993], [Horwitz, 19951. Here the known simplified representation is used (cf. [Stefanovich, 1996]) r ICUL) I' = [( - m 4] (10) 27r In general r' andm are functions of jL, see [Messiah, 1961], [Goldberger, Watson, 19641. Here F and ?7i are supposed to be constants such that F/m I (this is true for all observable unstable systems). I hope that using Eq. (10) allows us to obtain a faithful qualitative notion on the relation of A,(t) and Ao(t). 3.1. In the Appendix the main contributions to Ap(t) and Ao(t) are calcu- [ated using Eq. (10) for all times except extremely small ones t < 1/rn. AO () = rf d1i [(tL M)2 + 2 /41 exp(-ittt) 27r0 - -itrn.- Ft/2 tM > 1 (I ) 27r m Tnt [( _ M2 + 2 Ap (t) = 0c' dit /41 exp(-it V1p2 + p2) 27r 0 - exp[-i VpT-+m2(1 - ,)l exp[-I't( + a/2-y] (12) I (27r) - 12 (Pt) - 1,2 exp(-ipt + i7r/4). 2 Tn Eq. 12) is true if t > 1/m and t 1. The latter condition does not make it possible to get Eq. (I 1) from Eq. 12) by putting p = ; VP-2+ M2/M = _ V2/Cl)-1/2, 1 1,2 P2 1 T,2 ly 2 - i (13) Cf = - - 8 2 + n2 P2 + M2 8 M2 ^/4 Eqs. (11) and 12) contain terms proportional to exp(-rt/2) and exp(-rt(I + a)/2-y). Let us call them exponential terms (ET). Besides ET the equations include terms containing small factors r,/,m and proportional to the inverse powers of t. They are small as compared to ET during several decades of lifetimes. However, they dominate at sufficiently large t. For their discussion see, e.g., [Fonda et al., 1978], [Norman et al., 19881 and references therein. 4 3.2. The exponential term in Eq. 12) is close to exp(-itm-y - t/2-y) because a < 1. Therefore, JAp(t) 12 almost coincides with A(t/-Y)12 at the times t when ETs dominate.