Unstable systems and measurement processes

IL NUOVO CIM:ENTO VOL. 24A, N. 2 21 Novombre 1974 Unstable Systems and Measurement Processes ('). S. TWAREQUE ALI International Centre /or Theoretical Physics - Trieste G.C. GmRA~Dr Istituto di .Fisica Teorica dell' Universit~ - Trieste (ricevuto il 16 Maggie 1974) - - The results of some earlier work on the time evolution of unstable quantum-mechanical systems have been carried further. Under the assumption that the observation of the decay of an unstable elementary particle necessarily involves making randomly repeated localization measurements on it, leading each time to a collapse of its wave packet, certain experimentally realizable collapse processes have been studied. In particular, the earlier problem of encountering infinite mean energies in the collapsed state has been overcome. Summary. 1. - Introduction. I n recent years the problem of the time evolution of q u a n t u m - m e c h a n i c a l systems which are subjected to repeated measurements, occurring at r a n d o m times, has been investigated b y various authors (1-3). The most interesting application of these investigations, from our point of view, is the s t u d y of the time evolution of an unstable q u a n t u m system. I n fact, a critical analysis of a n y actual experiment performed to determine (*) (1) (2) (a) 220 Supported in part by the Istituto Nazionale di Fisica Nucleare, Sezione di Trieste. E. B. DAVIES: Comm. Math. Phys., 15, 277 (1969). P. A. BENIOFF: Journ. Math. Phys., 13, 1347 (1972). L. FONDA, G. C, GHIRARDI and A. RIMINI: N~OVO Cimento, 18 B, 1 (1973). UNSTABLE SYSTEMS AND MEASUREMENT PROCESSES 221 the decuy law of such a s y s t e m leads to the conclusion t h a t the s y s t e m cannot be considered as evolving undisturbed b u t t h a t , on the contrary, it is subjected to r e p e a t e d r ~ n d o m m e a s u r e m e n t s which ascertain whether the s y s t e m has decayed or not ('.s). The practical consequences of this fact h a v e been fully exploited in ref. (5), while its theoretical implications oll the possibility of using a semi-group law for the description of the decay h a v e been analysed in ref. (6). I n particular, it has been proved in ref. (~) t h a t , within the f r a m e w o r k of this approach, t h e n o n d e c a y p r o b a b i l i t y F(t) for an unstable s y s t e m turns out to be a pure exponential (for all pructical purposes) (1.1) F(t) ~ exp [-- t/T] , where t h e p a r a m e t e r v depends u p o n the n o n d e c a y p r o b a b i l i t y P(t) for undisturbed evolution and the m e a n frequency 2 of the me.~surements. The m a n n e r in which T is influenced b y 2 has been discussed in detail in ref. (7). The whole procedure developed in ref. (5) rests upon the identification of a particular state IFv::', which can be assumed to describe the unstable state. I t s definition appears n a t u r a l l y once one takes into account the particular dynamics of the p r o b l e m and identifies (us m u s t be (lone on physical o'rounds) the repeuted m e a s u r e m e n t s with localization procedures. These localizations ascertain whether the separation between the f r a g m e n t s constituting the unstable s y s t e m is greater or smaller t h a n a given distance r0, characteristic of the measuring apparutus. The m a i n purpose of this p a p e r is to investigate in greater detail the meusu r e m e n t process referred to above. We shall show t h a t the a s s u m p t i o n of sharp localization of the d e c a y products, besides being unphysical because of its discontinuous character, leads to un infinite nle~n energy for the unstable state. I t will t h e n be necessary, for the logical consistency of the theory, to modify this description in order to eliminate this u n p l e a s a n t feature. We shall show t h a t this can be done in at leust two different ways, b o t h of which m a i n t a i n t h e point of view t h a t the m e a s u r e m e n t is a localization b u t account b e t t e r for the physical elements involved. Moreover, we shall p r o v e thut, after these modifications h a v e been t a k e n into account, the derivation of ref. 0) can be r e p e a t e d step b y step within the new formalism, so t h a t all the conclusions reached there r e m a i n valid. A consideration of the second m e t h o d for overcoming the difficulties coming f r o m a sharp localization turns out to be interesting also f r o m a general theo- (4) H. EKSTEIkN and A. J. F. SIEGEI4T: Ann. o] Phys., 68, 509 (1972). (5) L. ]7ONDA, G. C. GttIRARDI, n . RIMINI and T. WEBER: NUOVO Cimento, 15 A, 689 (1973). (6) S . T . ALl, L. FONDA and G. C. GI~IRARDI: ICTP, Trieste, preprint IC/74/24. (7) A. DEGASPEI~IS, L. FONDA and G. C. GHIRARDI: ICTP, Trieste. preprint IC/74/1. 222 s.T. xL~ and a. c. GHIRARDI retieM point of view. I n fact, it implies the use of the general formalism of ref. (1), and thus yields an explicit expression for the interaction operator appearing there. This allows for an interesting physical interpretation of t h a t operator. 2. - R e v i e w o f p r e v i o u s f o r m a l i s m s . As shown in ref. (3), the evolution of a q u a n t u m system subjected to repeated measurements of an observable A, occurring at r a n d o m times with mean frequency A, can be described b y means of an integral equation for the density matrix, i.e. t (2.1) @(t) = exp [-- At] exp [-- iHt] @(0)exp [iHt] + A_ld5 exp [-- Ab]. o where P . is the projection operator associated to the n-th eigenmanifold of A. We note t h a t in this equation the results of the measurements of A have been left unspecified. E q u a t i o n (2.1) implies conservation of probability d ~ T r e(t) = 0 , (2.o) as can immediately be verified. Suppose now t h a t one is interested in studying the evolution of the same system b u t under the condition that, in all measurements before t, the same eigenvalue al of A has been found. Physically this amounts to assuming t h a t , e v e r y time a measurement has been performed, the measuring apparatus absorbs those parts of the collapsed state which correspond to eigenvalues of A different from al. I n this way, even t h o u g h during the undisturbed evolution of the state the @-matrix can develop components in the manifolds of P~, for n ~- 1, and even t h o u g h during this t i m e regeneration of the component in t h e Pl-manifold from these other components does occur, such regeneration from the collapsed part of the state is not allowed to take place once a measurement has been performed. This point has been discussed in detail in ref. (6). I t is clear t h a t under this assumption eq. (2.1) is modified to (2.3) @(t) = exp [-- At] exp [-- iHt] @(0) exp [iHt] -tt ÷ Afd6exp [-- A6] exp [-- ill6] P1 @(t -- ~i) P1 exp [ill6]. 0 UNSTABLE SYSTEMS A~;D MEASUREMENT PROCESSES 223 I t is obvious t h a t eq. (2.3) implies a loss of probability, i.e. (2.4) Tr e(t) < T r ~(0). I n ref. (6) it has also been proved rigorously t h a t eqs. (2.1) and (2.3) represent the integrated forms of the differential equations de(t) -- i[e(t), H I - - 2e(t ) ~- 2 ~ / ) , e ( t ) P ~ dt ,~ and de(t) -- i[e(t), H] -- he(t ) + 2Ple(t)P1, dt respectively. E q u a t i o n (2.3) is the basic equation for our description of the decay process of an unstable q u a n t u m system. Suppose now t h a t it is possible to define a unique unstable system state vector I ~ ) ~nd let the measurement to which the system is subjected be of the yes-no type, i.e. one which simply ascertains whether the system has decayed or not. This is physically reasonable, since in all actual cuses only such measurements take place. One can think for instance of bubble chamber experiments where each bubble indicates whether the system has decayed or not. I n such an experiment the assumption t h a t rescattering is forbidden after a measurement has found the state to have decayed is also perfectly justified since it corresponds to the fact t h a t one never expects two tracks which initially diverge (the tracks of the decay products) to come back again and regenerate the unstable state. F o r a detailed discussion of these points the reader is referred to ref. (4.6). The above assumptions lead us to identify P , in eq. (2.3) with I ~ ) ( ~ v l Let us assume t h a t the state has been prepared at t ~-- 0 in the pure state ly~). :For the probability F ( t ) = (y¥]~(t)]F~), t h a t in a measurement at t the state ]F~) be found again, we get the integral equation t (2.5) F(t) = exp [-- 2tiP(t) + exp [-- 2 b J P ( 5 ) F ( t - b), o where (2.6) P(t) = I(y~I exp [-- iHtJ ly~)l ~- is the nondecay probability for undisturbed evolution. Using (2.5), one can prove t h a t for almost all times (5) P(t) = e x p [ - t / r ] , so t h a t we get a purely exponential decay law. 224 s.T. ALI ~nd G. C. GH~RARD~ The above derivation is quite simple and straightforward, but it contains two unpleasant features. First of all, one must assume the existence of a welldefined unique wave function for the unstable system, a fact which is not at all obvious, and, secondly, one must also assume t h a t the measnring apparatus in a certain sense knows what this state is, since it acts as a projection operator for it. I n ref. (~) these difficulties have been overcome in an elegant m a n n e r b y analysing in detail the actual physical situation. A critical analysis of the possible experimental set-up has led to an identification of the measurement process with a localization procedure which ascertains whether the decay products have a relative separation which is sm~]ler or greater t h a n ~ given distance to. The result (~yes the relative distance r is smaller t h a n r0 >>is ~hen identified with the experimental statement ((the system has been found u n d e c a y e d >). Such a description of the measurement t h e n eliminates the need for the measuring apparatus to actually know which state it is measuring. Further, the particular dynamics of the unstable system t h e n essentially brings the state, when it has been found undecayed, back to the unique state I~g}. I n this way there also emerges a natural definition of the unstable-state wave function. L e t us describe this in greater detail. Using eq. (2.3) and identifying P~ with the projection operator P(0.~,) on the linear manifold r < ro, defined in the r-representation b y (2.7) [-P~o.,.) ~](r) = O(ro - r) ~(r), the evolution equation is (2.s) @(t) = exp [-- 2t] exp [-- iHt] @(0) exp [iHt] -~- t + afdO exp [-- 2~] exp [-- iItd]P(o,~°) @(t -- d) Pio,,°)exp [illS]. o I n ref. (5) it has been shown that, due to the particular dynamical structure of the resonance phenomenon, the application of P(0.,0) to any state whose energy form factor overlaps only one resonance leads to a state which differs v e r y slightly from (2.9) k]~(k)k~) I~v} = f d3k J~(k, l~k-}' where the F~'s are the improper eigenstates of the t t a m i l t o n i a n of the system, ]zR(k) is the Jost function for the resonant wave and Jz~(k, k~) is a smoothly varying function of k in a neighbourhood of the resonance m o m e n t u m kR. I t is t h e n natural to assume (2.9) as the definition of the unstable-state wave function. Assuming t h a t we have initially prepared the system in the unstable ~NSTABLE SYSTEMS AND MEASUREMENT 225 PROCESSES state IVy}, we c~m rewrite (2.8) as t <~pvl@(t)l,fv} = exp [ - 2t]P(t) + (2.10) 2fd,)exp [-- 26]. 0 "I<Vvl exp [-- iH6]P,o.,°)IFv) [o-(y,vl@(t- 6)lFv} + t + 2fd~ exp [-- 2~] {I(Y'v] exp [-- iHS]P(o.ro)l~p±)']~(~zl@(t -- b)l~±} + 0 -~ 2 Re (~v] exp [-- iH~]P(o.~o)[~ v} (y¥10(t -- (5)1~±} (~±]P(o.,.) exp [iH~] [~v}}, where ]~f:: is short-hand for a complete set in the manifold orthogonal to [~v}. W h a t has actually been proved in ref. (5) -~mouats to saying t h a t one can p u t (2.11) { I~1 exp [- iH~]P<o,~.,l~S:>[~= l(~l exp U- illS] ]Vu}] ~-=-:P(b), I(Yv]exp [--iHS]P(o.,o)ly~±)l~-=O, thereby introducing no more t h a n a very small error in (~vl@(t)]Fo),. Therefore (2.10) coincides for all practical purposes with (2.5) and all previous calculations remain valid. 3. - D i s c u s s i o n o f t h e definition o f t h e u n s t a b l e state. The analysis of the previous Section rests upon the recognition that, owing to the dynamical structure of the system and the high frequency of the measurements, there exists a state ly~v) such that, in practically all measurements in which the system is found undecayed, the reduction process of the wave function leads back again to this state. Let us remark, however, t h a t the definition (2.9) of ly~v} exhibits some unpleasant features. In particul,~r~ if one takes into account the fact t h a t (5) (a.~a) (3.]b) /~(k) ~ 1, Jz(k, kn) ~,~ 1/k , one immediately sees t h a t the expectation value of the energy in the state I~v} is divergent. In fact co (3.2) ; ]J~,(k, kn)l ~ (y~vIHl~) o c j 0 --[/~,(k)[2 ~kdk = ~ . 226 s.T. AL~ and G. C. GHIRARDI This is r a t h e r unsatisfactory. The origin of this result, however, is quite simple. I t is due to the sharp cutting of the w a v e function as a result of the action of ~Pm.%) in r-space. I n fact, such a sharp cut introduces a (~-function b e h a v i o u r of the derivative of [P(o.~.)~v](r), and, f r o m general properties of the Fourier transform, this implies the slow convergence, (3.1b), at infildty in k - - h e n c e the divergence of (3.2). I t is obvious t h a t the discontinuity in r-space introduced b y P(0.%) m u s t be unphysical, for, as is well -known in the q u a n t u m t h e o r y of measurements, when one has an observable possessing a continuous s p e c t r u m it is impossible to discriminate sharply between a range of eigenvalues and its c o m p l e m e n t w i t h o u t going into conceptual difficulties. I n fact such unphysicul m e a s u r e m e n t s introduce infinite energies which m u s t be supplied b y t h e app a r a t u s to t h e system. To get an idea of how to avoid this difficulty, we observe t h a t in real cases the a p p a r a t u s will not cut the wave function s h a r p l y but, on the contrary, it will lead to a smoothing of it. The p r o b a b i l i t y density of finding a given separation b e t w e e n the f r a g m e n t s should s m o o t h l y go to zero. E v e n if it is assumed t h a t an idealized counter with efficiency 1 is used, there will always exist a certain shell such t h a t only after this region has been crossed will the efficiency h a v e reached the value 1. I n other words, there exists a region (to, ro ~- zJ) such t h a t , even if t h e answer (( yes, the particle is within the counter ~>is obtained, there is a certain p r o b a b i l i t y t h a t the particle is within ro and ro ~ ~. As one goes f r o m ro to ro ~- LI this p r o b a b i l i t y m u s t decrease to zero. I t is our t a s k therefore to schematize b e t t e r the effect of the localization m e a s u r e m e n t s , so t h a t we shall be nearer to t h e actual physical situation. Before coming to this point, however, we m u s t m a k e some p r e l i m i n a r y considerations on the structure of our equations and on their possible generalizations. 4. - General formulation o f an equation for the density matrix. Before discussing the modifications which m u s t be introduced into the description of t h e reduction process, let us discuss some properties of our basic equations. F o r simplicity, let us assume t h a t t h e observable being m e a s u r e d is a t w o - v a l u e d observable, so t h a t the possible answers can be indicated with <~yes )> or (( no ~). I n the m e a s u r e m e n t the density m a t r i x of the s y s t e m is affected. L e t us indicate b y @ye. ~ and @no the density matrices obtained f r o m a given initial @, w h e n t h e answer (( yes ~> or <(no )) is obtained, respectively. I t is obvious t h a t if we do not discriminate between the two answers, after the m e a s u r e m e n t we shall h a v e a density m a t r i x , to be denoted b y @~, which is the sum of @g and ~ yes The evolution equation for the density m a t r i x o b t a i n e d under the assumption t h a t the observable under consideration is r e p e a t e d l y m e a s u r e d is t h e n UNSTABLE SYSTEMS AND MEASUI~EMENT 227 rROCESSES identical to (2.1) with QH(t--5) substituted for ~P,~@(t--5)P,,: (4.1) ~(t) = exp [-- ~t] exp [-- iHt] @(0) exp [iHt] jr t ~- ~fd5 exp [-- ~ ] exp [-- iHb] @~(t-- 5) exp [iHS]. 0 Since we do not discriminate between the different answers, we must impose, at this level, conservation of probability, i.e. eq. (2.2). B y taking the derivative with respect to t, integrating b y parts and again using (4.1) we see t h a t eq. (2.2) is satisfied if and only if the measurement process is such t h a t (4.2) Tr @--~ Tr @~. E q u a t i o n (4.2) is v e r y i m p o r t a n t ; it tells us t h a t if we want to describe a system through eq. (4.1) or even through an analogous equation where we accept only the answer <<yes }>(obtained from (4.1) b y putting @~ in place of @~), we must yeB give a description of the effect of the measurement not only when the answer <~yes >> is obtained, b u t also when tile alternative answer <(no >) is obtained, and this description must be such t h a t eq. (4.2) is satisfied. We could h o w e v e r - - a n d a particular case will be considered b e l o w - - b e interested in describing the measurement process in such a way t h a t (4.2) would not be satisfied. If we wanted to maintain conservation of probability, this would m e a n t h a t eq. (4.1) itself should be modified. This can be done following ref. (1), and we m a y write t (4.3) ~(t) ---- exp [Zt] @(0) exp [Z ¢ t] -}- 2_(d6 exp [Z5] @~(t-- b) exp [W 6]. o If one chooses R (4.4) Z = -- i H - - -2' eq. (4.3) becomes (4.5) @(0)exp iHt----Rt 2 @(t) = exp [-- i I I t - - ~ t [. ~- t 0 which, when one chooses (4.6) R = 2, becomes eq. (4.1). Therefore (4.5) is a generalization of (4.1). Let us now see under which conditions Tr Q(t)----const. B y evaluating d@/dt through (4.5), 228 s . T . AL~ and G. C. GHIRARDI performing an integration b y parts, using (4.5) again and finally t a k i n g t h e traces, we get t h a t ( d / d t ) T r @(t)= 0 if (4.7) Tr R~ ---- 2 Tr ~ for all t's. E q u a t i o n (4.7) is therefore a generalization of (4.2) which m u s t be used when t h e t r a c e is not preserved in t h e m e a s u r e m e n t . E q u a t i o n (4.5) as it stands a n d t h e related equation obtained b y s u b s t i t u t i n g ~ for ~ are those which in our scheme correspond to t h e evolution equations (in the usual f o r m a l i s m of the t h e o r y of unstable systems (s-~o)) written, respectively, in the whole H i l b e r t space of the s y s t e m and in t h e manifold of t h e unstable states alone. A v e r y i m p o r t a n t fact which has been p r o v e d in ref. (~) is t h a t , while in the usual description a semi-group law for the evolution within the unstable manifold is unacceptable, b o t h our equations h a v e t h e n a t u r e of semi-groups in the t i m e p a r a m e t e r . I t is also f r o m this point of view t h a t t h e present a p p r o a c h is more satisfactory. We now proceed with the modifications to be m a d e in t h e reduction process. 5. - The p h y s i c a l r e d u c t i o n process. W e observe first t h a t f r o m the discussion of Sect. 3 it is clear t h a t , since the reduction is due to the interaction of the state with the a p p a r a t u s , which is a macroscopic object, t h e m o s t n a t u r a l a s s u m p t i o n is t h a t , e v e n if t h e s y s t e m is in a pure state before t h e m e a s u r e m e n t , it should go over into a statistical m i x t u r e after the m e a s u r e m e n t . I f we w a n t to adhere to the description of t h e a p p a r a t u s as something which localizes the two f r a g m e n t s within or outside a certain distance ro we m u s t require t h a t the modification induced in the probability density b y the m e a s u r e m e n t h a v e the following characteristics: a) W h e n the ~nswer is <,yes ~>(*) "r' ~ 'r~ (5.1) and t h e transition f r o m S I~,1 , ~-- (rI~[r) { (rl~Ir) 0 for for r<ro, r > r o ~ LJ, to 0 should t a k e place continuously. (8) D. N. WILLIAMS: Comm..Math. Phys., 21, 314 (1971). (9) K. SI~HA: Helv. :Phys. Acta, 45, 619 (1972). (1°) L. P. HO~WITZ, J. A. LA V~TA and J.-P. MARCIIAND: Journ. Math. Phys., 12, 2537 (1971). (*) Throughout this Section we shall only use the matrix elements of q in the r, r' representation, ignoring the vector character of the position variables. This can be done since our localization procedure is assumed to be spherically symmetric, so that our considerations refer actually to the radial part of the 0-matrix (see the Appendices). The measurement does not mix the various waves. 229 U N S T A B L E SYSTEMS AND MEASUREI~IENT PROCESSES b) W h e n the answer is (~no *~ <rlg#=olr) > = (5.2) {o for r < ro, <r]eIr: :' for r > r o ~ - A, and again the two values should be continuously connected. I f we w a n t to use eq. (4.1) we should also require (5.3) <rio#Jr:> ~ <rle~o, lr[::,-[- <rl~#olr::, = <r]~]r), for all r, which ensures (4.2) and therefore conserves probability. Finally, to avoid unphysical discontinuities which could again bring in divergences in the mean energy, we must require t h a t the nondiagonal m a t r i x elements in configuration space, (r]o~,lr'i> and (r]9#olr'}, also be continuous functions of both variables. I n place of the operator P(o.,°) used previously, let us introduce a family of operators which we denote b y P~, and whose effect in configurution space is (5.4) [p~ ~](_r) ]~(r) ~(r), = where ]~(r) is 1 for r < x, 0 for r > x - ~ A/2, while it decreases continuously with continuous first and second derivatives from 1 to 0 as r goes from x to x + 3/2. I t t h e n has the shape shown in Fig. 1. f,(r)' 1 1* •% ro÷~ r Fig. 1. - The function Ix(r) ~ppcaring in cq. (5.4). Using these operators let us assume the following effect of the measurement, when the answer (~yes ~>is obtained: to+A/2 (5.5a) -~ P¢~ePf.dx . ra Similarly, when the answer (<no )) is obtained we assume ~'o+A/2 (5.5b) oo=2 f ro P~gPg~ dx , 230 s.T. where P ~ is defined in complete analogy with Psi. lated t h r o u g h the equation (5.6) g~(r) = ~ AL~ and G. e. GttIRARDI The two functions are re- ]~(r) , so t h a t gx(r) is 1 for r > x @ A/2 and vanishes for r < x, while it increases continuously with continuous first derivative f r o m 0 to 1 on going f r o m x to x ~ A/2. F r o m (5.5a) and (5 5b) we get (5.7a) <r]@~ j r ') = <rl@]r')F(r , r'), (5.7b) (r]@eolr' ) = <r]@lr' ) G(r, r ' ) , where %+z112 (5.Sa) F(r, r') = -A J~(r) /~(r')dx ra and to+A/2 (5.8b) G(r, r') :- -~ g~(r)g~(r')dx. ro F r o m (5.8a), (5.8b) and the definitions of ]x(r) and gx(r) one easily sees t h a t (5.1) and (5.2) are satisfied, while (5.6) ensures t h a t (5.3) is also satisfied. One can also easily verify t h a t the first and second derivatives of F(r, r') and G(r, r') with respect to b o t h variables are finite for a n y r, r', so t h a t b o t h (rl@e [r '} and (rl@~olr' ) h a v e this p r o p e r t y , p r o v i d e d the same is t r u e for (rl@lr'). Finally, we r e m a r k t h a t for / ] - ~ 0 (5.9a) (5.9b) ~o ~ PI,..~I oP( .... ) , so t h a t in this limit we regain the formMism of ref. (5). Dividing the (r, r')-plane into the regions for which r or r' are smaller t h a n to, belong to the (r, r o S A ) range, or are greater t h a n r0+gl, we h a v e t h a t F(r, r') and G(r, r') t a k e the values indicated in Fig. 2. H a v i n g satisfied the basic r e q u i r e m e n t of (5.3), we see t h a t (4.2) is satisfied so t h a t we can safely use eq. (4.1) without introducing t h e more general forrealism s u m m a r i z e d in (4.5). Coming to the t r e a t m e n t of the evolution of an unstable q u a n t u m system, since we w a n t to select those systems for which the answer ~ yes ~> has been obtained in all m e a s u r e m e n t s preceding t, we m u s t use eq. (4.1) with ~,~,(t~-- b), UNSTABLE SYSTEMS AND given b y (5.5a), substituted in place of @~(t - (5.*0) 231 MEASURE~,IENT PROCESSES @(t) = exp [-- ~tt] exp [-- - ~). Thus iHt]@(O)exp [iHt] -~- t ro+Zl d- Xf dS exp [-- XSJexp [-- iHSJ [~ f P~x~(t-- 5)P,xdx] exp [iHS] . This equation is the analogue of eq. (2.8), with the continuous character of the localization having been introduced. For A -> 0 this equation goes over into (2.8). r0+d / / /// % \ \ , ~ x ,,\ %+~ \ ) C(r, 2 f g~(r)g,(r')d-x % % Fig. 2. - Values gssumed by the functions in the (r, r')-plane. Q+a _F(.r.r') of eq. (5.7a) and G(r, r') of eq. (5.7b) At this point we should prove t h a t the same conclusions as drawn from (2.8) still follow from (5.10) and, moreover, t h a t a reduction does not introduce an infinite energy. B u t these facts should be obvious since we have eliminated the discontinuity which was responsible for the infinite energy, while, on the other hand, the effect of the measurement for all practical purposes is the same as t h a t produced b y P(0.~o). I n fact, apart from the smoothing of the @-matrix in a narrow region (ro, rot A), where the Q is almost zero, the two reduction processes coincide. A complete proof t h a t the reduction process does not introduce infinite energies is given ill Appendix I, while in Appendix I I it has been 232 s . T . XnI and G. C. GIIIRARDI shown terize rather before t h a t it is still possible, to use t h e p r e s e n t f o r m a l i s m to u n i q u e l y charact h e u n s t a b l e state. O b v i o u s l y this s t a t e will n o w be a d e n s i t y m a t r i x t h a n a v e c t o r s t a t e r e p r e s e n t e d b y a w a v e function, as was t h e case (el. ref. (5)). 6. - A n o t h e r possible r e d u c t i o n process. W i t h reference to t h e possible use of t h e m o r e general f o r m a l i s m s u m m a r i z e d in eq. (4.5), let us suppose t h a t t h e effect of t h e m e a s u r e m e n t c a n be described in t h e following w a y : (6.1a) 9 .......... . . . . . . y~ ~ ~" @y,B ~ = p,,o@P,,o , (6.1b) ~ = ( I - - PI,o)@(I-- P*,o), 9 . . . . . . . . ... t * 9no P,,o is defined according to (5.4). where N o t e t h a t w i t h t h e a b o v e defiuitions @#yeBhas all t h e desired p r o p e r t i e s , i.e. <rl@~oB]r'> coincides w i t h (r]@]r'i for r, r ' < r0, it is zero for r, r' > ro-~ A/2 a n d it varies c o n t i n u o u s l y f r o m <r]@]r'> to zero in t h e r e m a i n i n g regions. A n a l ogous considerations hold for @#no" Again, for A -~ 0 we r e c o v e r t h e old f o r m a l i s m , since PI,o -+ P(o.%)" This r e d u c t i o n process is t h e r e f o r e r e a s o n a b l e ; h o w e v e r , t h e r e are t w o m a i n differences w i t h r e s p e c t to t h a t discussed in t h e p r e v i o u s Section. F i r s t of all, we n o w h a v e (6.2) T r @ve T r @#. I n fact (6.3) T r @#= T r [Pf,o @Pso-F (] -- P,,o) 5(1 -- P,,o)] ---- T r {[p~2 _]_ (1 -- Ps,o) 2] @}, w h i c h in r - r e p r e s e n t a t i o n reads (6.4) f<rldi'> dr =f[l~%(r) -F (] -- f,o(r))~]<rlQIr> dr +f<rlel > dr. As discussed in Sect. 4 we m u s t t h e n use eq. (4.5), i n t r o d u c i n g an o p e r a t o r R for which (4.7) is satisfied. F r o m (6.3) we see t h a t we can satisfy (4.7) b y choosing a d i a g o n a l o p e r a t o r in r-space for R, i.e. (6.5) R = ~[p~,.+ ( 1 - p,, p]. T h e second difference comes f r o m t h e fact t h a t if we h a v e a p u r e s t a t e UNSTABLE SYSTEMS AND MEASUREMENT PROCESSES 2]~ before the measurement, the state remains pure when a definite answer is obtained (*), for instance This is the reuson for which we consider such an approach less physical. I t does not take into account the statisticul nature of the reduction process in the region where the wave function overlaps the apparatus. However, this case has an intrinsic interest from a general point of view, since it gives a simple example of the general t h e o r y of (4.3) a.nd (4.5). We have here an explicit form of the so-called ((interaction operator )) R appenring in {4.4) und (4.5). I n r-representation (6.6) (r]RIr'~ ~ G(r) ~(r -- r ' ) , where G(r) is as represented in Fig. 3. The physical meaning of the appearance of R should be clear. I n the language of the old formalism it means t h a t the mean frequency of the reductions ehunges according to the relative distance r of localization, as is reasonable. G(r)' L/ I I Fig. 3. - The function G(r) of eq. (6.6). We m a y again discuss the decay law for the unstable system. Since we wish to select only the answers (<yes ,), we have to use eq. (4.5) with Q~,, in place of e~: t 0 (*) Let us remark, however, that the evolution for the present case, which is (4.5) with Qre, ~ in place of QM,yields for t > 0 a mixed state even though Q(0) is a pure state. 16 - T1 Nuovo Chnenio A. S.T. ALl and G. c. GHIRARDI 2~ I n the two Appendices we h a v e p r o v e d t h a t in the present case also the reduction process does not introduce infinite energies and t h a t t a k i n g into account t h e d y n a m i c s of t h e u n s t a b l e s y s t e m one can introduce an unstable-state @-matrix in a unique way. 7. - Conclusion. We h a v e p r o v e d t h a t the f o r m a l i s m of ref. (5) can be modified in such a w a y t h a t the a p p e a r a n c e of infinite energies due to the m e a s u r e m e n t process is no longer present. Moreover, the reduction mechanisms we h a v e introduced in Sect. 5 and 6 t a k e into account b e t t e r the actual physical situation. W e h a v e not p r o v e d in detail t h a t after we h a v e introduced the unstablestate density matrices all subsequent m e a s u r e m e n t s lead b a c k to essentially the same state, since this should be obvious f r o m the derivation given in Appendix I I . Moreover, it should be clear t h a t f r o m this viewpoint the situation is b y no means different f r o m t h a t of ref. (5), where a rigorous proof of this fact has been given. I n conclusion, we can say t h a t we have given a m o r e rigorous and physical foundation to the results obtained in ref. (5) concerning the dynamics of the decay processes of unstable systems. Useful discussions with Prof. L. FONDA are gratefully acknowledged. One of the authors (S.T.A.) wishes to t h a n k Prof. A. SALAM, the I n t e r n a t i o n a l Atomic E n e r g y Agency and UI~ESCO for hospitality at the I n t e r n a t i o n a l Centre for Theoretical Physics, Trieste. APPENDIX I Suppose t h a t the s t a t e of a q u a n t u m s y s t e m is specified b y u given density m a t r i x @. I n configuration space we can write (A-I.1) <ri@lr,) = ~ ~m,~'m' gz~.~,m'(r,r') y~(~)y.,(~,). rr! l~ow suppose we w a n t to e v a l u a t e t h e m e a n value of the kinetic-energy opera t o r T in the s t a t e @. W e h a v e (A-I.2) ( T ) = Tr T@ =fd~r <liT@If> • UNSTABLE SYSTEMS AND MEASUREMFNT PROCESSES 235 Using the fact t h a t in co-ordinate representation ]t,2 (A-I.3) <(rlTIF" -- r "~ ~m ~ (,_IF,, we get (A-I.4) /,T~>• - - 2m v ~+-"J co [ c,r" ~ r'r') ,':, _ r"- g~m.~(r,r') }. 0 With reference to the reduction processes (5.5a), (5.5b) and (6.1a), (6.1b), suppose t h a t we start with a ~-matrix before the measurement for whieh <T> is finite, or, in other words, for which g(r, r)/r ~- and 82g/~r2 are integrable over the positive real axis. As discussed in Sect. 5, after the measurement, the m a t r i x elements of e,+: ~ and Q=o H are simply those of ~ before the measurement times the function /Z(r, r') or G(r, r') (cf. (5.8a) and (5.8b)), respectively. F r o m the fact t h a t these functions are continuous in both their arguments, as are their first derivatives also, one easily sees that, when g,~: ~ (r, r') or g:o (r, r') is used lm, lm lm,lm in (A-I.4) in place of g, the integral is still convergent. The same is true for the corresponding ~-matriees in (6.1a) and (6.1b) for exactly the same reasons. Since the infinite energy which appeared in the formalism of ref. (~) had its origin in the expectation value of the kinetic-energy operator (in fact, the expectation value of the potential energy V is certainly finite, provided V converges sufficiently rapidly in configuration space--as is always assumed for the problem at hand). We have proved t h a t the modifications introduced in the reduction process in Sect. 5. and 6 actually ensure t h a t infinite energies disappear from the formalism. APPENDIX II In a m a n n e r completely analogous to (A-I) we can write the m a t r i x element of a density m a t r i x in the representation of the continuum eigenstates of the Hamiltonian H governing the evolution of the unstable system as (A41.1) CV~JOIV'_+~'.>+= ~ g~.,..,.+(k,kk,k') y~(~)y,~,(lp). gm,[trg r ? We use the partial-wave expansion for f,rIF~>: (A-II.2) <rl~'~> = ~ Yz*,~(/~)Y~,~(¢)iz Fz(k, r) ~m kr F~(k, r) being the /-th wave radial scattering wave function normalized to a &function in k-space co (A-II.3) f F ~ ( k , r)F~(k', r ) d r = 6 ( k - - k') . o 236 8. T. A L I a n d G. C. G H I R A R D I T h e function ~ ( k , r) can be expressed in t e r m s of the regular solutions of the radial equation b e h a v i n g at the origin like r~+1/(21+ 1 ) ! ! and the J o s t function ]~(k), according to (A-II.4) v (k, ~ ( k , r) ~ = + " F r o m (A-II.1) we know t h a t the density m a t r i x can be written as (*) (A-II.5) 0_IId3kd3k,l~,!)[ .I d g~.~,~,(k,k')yz~(fc)yzm([c. ' ~ )(]y ~ Llm, l'm' , • ~]~I The m a t r i x 0 is then completely specified b y assigning gz~.z,,,,,(k, k'). The evolution of ~ when no m e a s u r e m e n t takes place, i.e. the application of e x p [ - - i H t ] ~ exp [iHt], induces the following change in g~m.z,,~,(k,k'): (A-II.6) g~.z,,~,(k,k') ,~o,,t~o~exp [-- iE~t]g~.~,,~,(k, k') exp [iE~,t] . On the o t h e r hand, it is easy to see t h a t if one w a n t s J* k , k'), where has to replace g,~.zm(k, k') b y g,~.,,~,( co (A-IIT) 'k P~oPs,, one simply co =f fdk" 0 k",- 'k" k'). 0 I n eq. (A-II.7) we h a v e p u t 0o (A-II.8) ~,(k, k') =fdry~(k, r)],(r)y~,(k', r) ~ J{,(k, k') Mk)/~(k') " 0 F o r simplicity we h a v e introduced the q u a n t i t y to (A-II.4), is J{qk, k') which, according co (A-II.9) J{:(k, k') -- 2kZ+lk'l+lf dr ~ ( k , 7~ r)q3z(k', r)]x(r). 0 Suppose now t h a t before the first m e a s u r e m e n t we h a v e a density m a t r i x characterized b y a g~m,~,,,,(k,k') which overlaps the resonance m o m e n t u m k--~ kR and k'-----ka for the resonant wave, and no other resonances in the same or in other waves. Suppose, moreover, t h a t the answer ((yes ~) has been o b t a i n e d in the m e a s u r e m e n t . According to (A-II.7) the Q~.. m a t r i x will then (*) We are obviously assuming here that the states in which we are interested have no components on the possible bound states of H. 237 UNSTABLE SYSTEMS AND M E A S U R E M E N T PROCESSES be characterized b y roWzJ/2 (A-II.lOa) Ig~.~ ( k , k ' ) - z~.r~' - * k' co co t axt~ A],(k)]~,( ) J ~'o . J d 0 0 ..C.(k, • E')g~.,.~,m,(k", V)dp,(~", ~') /~(~")h'(~") if the reduction occurs as specified in eq. (5.5a), while it will be given b y co (A-II.10b) II~.~ co 1 /" "/'d " k ~* k' / d k / k • (k,k')=- 0 o • It! J,,,0(k, k )g~.,,~,(k, k )d~.o(k, k') /?(k")h,(k") i! tl H! .f if the reduction occurs as specified in eq. (6.1a). We can now follow the same arguments used in Sect. 3 of ref. (5). Obser~'ing t h a t 1/J~(k") and 1//z,(k") are v e r y peaked for l = l ' = l~ and k " = k " = kR while J{-(k, k") (J{,o(k, k")) and J{~(k", k') (J{,0(k", k')) are smoothly varying functions of k" and k", respectively, and writing (A-II.10a) as to+A/2 (A-II.11) f L~,o~ g(k, k') -co dxJ?(k, k~)J~o(k~, ~')- co d{,(k,k~) l~(k")J~,(k") d{~(k~, k')] ' o o and analogously for (A-II.10b), we see t h a t the expression within the square brackets is almost independent of k and k' and is appreciably different from zero only when 1 = l ' = l~. We can then define the unstable state as being characterized, apart from a normalization factor, b y a gzm.z'~' which is obtained from (A-II.11) or the analogous equation for II~ b y putting the square bracket equal to d,~b,,t,. In other words, for ~c.~t~b~o we shall have the expression ro+A (A-II.12a) g~.,~.~,m,)(k, k') oc 2bz~.d~'iR A]/k)j~(k') f dxJ{;(k, k,lJ{:(k,, k') ro if the collapse is as assumed in Sect. G, and (A-II.12b) h(k)t/k') if the collapse is as assumed in Sect. 6. " 238 s . T . ALI and G. e. GI-tlRARDI U s i n g ( A - I I . 1 2 a ) a n d ( A - I I . 1 2 b ) one c a n e a s i l y to p r o v e t h a t w i t h t h e s e d e f i n i t i o n s for t h e u n s t a b l e s t a t e t h e e x p e c t a t i o n v a l u e of t h e e n e r g y is finite, a f a c t w h i c h s h o u l d b e o b v i o u s b y now. • RIASSUNTO Si continua la ricerea iniziata in lavori preccdenti sull'evoluzione temporalc dei sistemi qunntistiei instabili arrieehendo i risultati ottenuti. Aecettando l'ipotesi ehe l'osservazione del proecsso di deeadimcnto di una partieella elementare instabile impliehi neeessariamcnte l'assoggettarla a ripetute misure di loealizzazione che portano a riduzioni del relative pacehetto d'onde, si studiano in dettaglio aleuni proecssi di riduzione sperimentMmentc possibili. In partieolarc si supera la diffieolt£, presente nel preeedente approecio, ehc traeva origine dall'apparire di energie medic divergenti per lo state ridotto. HeeTa6HabHMe CHCTeMbl H HpoHece H3MepeHHH. Pe3IoMe (*). - - rlpOBO,ZlVlTCn,~anblte~mee laccne;IoBarme pe3yYmTaTOBnpe~Ibl)lylUe~ pa6oxL~ no BpeMermo~ 3BOmOUnWHecTa6rxnbnb~x rBariTOBOMexann,-xecrrlxC~CTeM. B1,mn n3yqerr~I HeKOTOp~le 3KCtlepHMeHTaJIbHO peanrI3yeMble llpoIIeCcbI i<o:manca BOYlHOBblX rlaKeTOB, ~tpeRnonara~, HTOHa6mo~euue pacna~a HecTa6HJIbHO~I3JIeMeHTapHo~ qaCTHI~bIHe~3~eYKHO BK~IOqaeT rlpoBe~eHPIe XaOTHqeCKH HOBTOp~IIOIIII/IXCaH3MepeHB~ MeCTOHaXOX~eHBfl q a c THHbI, KoTopMe IIpHBO~flT Kax~MR pa3 K KOJInancy ee BOYIHOBOFO IIaKeTa. B qaCTHOCTI,I, 6blna pa3pemeaa npexu~ta npo6neMa noflBJIeHrI~ ~eCKoHeqHlalX cpe,armx 3Hepri,l~ B KO.II- .rlallCnpOBaHHblX COCTO~IHH$1X. (*) Ilepeeedeno peOat¢que&