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Some consequences of time-reversal symmetry Maroun, David Peter 1964

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SOME CONSEQUENCES OF TIME-REVERSAL SYMMETRY • by DAVID PETER MAROUN B;Sc.(Hon.), St. Francis Xavier U n i v e r s i t y , 1963 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n the Department of PHYSICS We accept t h i s t h e s i s as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September, 1964 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree at the U n i v e r s i t y of • B r i t i s h Columbia, I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study„ I f u r t h e r agree that per-m i s s i o n f o r extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s r e p r e s e n t a t i v e s . I t i s understood that,copying or p u b l i -c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n permission* Department of Physics The U n i v e r s i t y of B r i t i s h Columbia, Vancouver 8 5 Canada Date September &. 1964 ABSTRACT The purpose -of t h i s work i s to discuss the symmetry, or lack of i t , under rever s a l of motion i n p h y s i c a l objects, states and processes. Considerations of such symmetry are made i n both c l a s s i c a l and quantum physics, notably i n the problem of r e c o n c i l i n g the :assumed time-reversal symmetry of microscopic processes with the -observed asymmetry of macroscopic processes. In the case of c l a s s i c a l mechanics, a simple -model of a free p a r t i c l e c o l l i d i n g with a se r i e s . o f almost stationary or stationary p a r t i c l e s of smaller mass i s introduced i n order to show how a f r i c t i o n - l i k e phenomenon can a r i s e from processes a l l of which have symmetry under, r e v e r s a l of motion. It i s maintained throughout that symmetry under r e v e r s a l of motion i s a property of a l l fundamental states and.processes i n nature. i v ACKNOWLEDGEMENTS I would l i k e to express my gratitude to Dr.'F,A Kaempffer, who • o r i g i n a l l y suggested to me the subject of t h i s thesis., and who provided much guidance throughout the work done on i t . Chapter I I I i s l a r g e l y taken from Dr. Kaempffer's book, "Concepts i n Quantum "Mechanics", which i s to be published. Also, I would l i k e to thank the National Research Council f o r providing f i n a n c i a l assistance i n the form .of a bursary. i i i TABLE OF CONTENTS 1 PAGE ABSTRACT i i ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i v PREAMBLE . • .. 1 CHAPTER I : INTRODUCTION "1. • Terminology .................. .- ... 2 2. Considerations-Permitting One In P r i n c i p l e "To-Check"Whether Reversality, Reciprocity Or Detailed Balance Holds ............. ... . :k 3« '.Further "Definitions ......... ......... . . . 7 CHAPTER I I : TIME-REVERSAL SYMMETRY IN CLASSICAL MECHANICS 1. Newton' s "Mechanics , .......................... 8 2. - 'Classical'Electrodynamics . . . . . .. ... . .. .12 CHAPTER I I I : TIME-REVERSAL IN QUANTUM MECHANICS -1. The Need To Represent Reversal Of Motion By. An Anti-Unitary Operator ........... . . . 19 2. Proof Of- Reci p r o c i t y From Time-Reversal 22 •Symmetry ............... . ... 3- Conditions'Under Which Detailed Balance 26 Holds ....•'. ....... ... .' . ' . . CHAPTER IV: EXPERIMENTS CHECKING OR INVOKING TIME-REVERSAL INVARIANCE OR DETAILED BALANCE 30 CHAPTER V: ARE THERE-PARTICLES WHICH ARE THE TIME-REVERSED -COUNTERPARTS OF OTHERS? . 35 DIAGRAMS Figure 1 . . ..... . ......... following page .......... 6 Figure 2 . fol l o w i n g page .......... 6 Figure 3 • • •• .... page i , following page . . . . . . . . . 28 Figure h ... . page i i , f ollowing page ............ 28 Figure 5 • - • • page . i i i , following page . . . . . . . . . 28 BIBLIOGRAPHY ... . '.. . 37 1. PREAMBLE Suppose that while some p h y s i c a l process occurs, the en t i r e process, from the i n i t i a l state -of the system to the f i n a l state i n c l u s i v e , i s photographed by a motion picture, camera and the f i l m l a t e r projected onto a motion p i c t u r e screen. Suppose that t h i s f i l m i s then run backwards so to •show a process that i s the reverse of the o r i g i n a l one. I f the .original process was -an elementary one in v o l v i n g few bodies.- such as .the c o l l i s i o n of two b i l l i a r d b a l l s - - .then the -observer w i l l l i k e l y experience no surprise .: upon seeing the reversed process, and (as has been v e r i f i e d by actual-experiment) .may not be-able to t e l l which process was the - o r i g i n a l and which was obtained by a p r o j e c t i o n i s t ' s t r i c k , ©n the other hand, i f the process was a complex one i n v o l v i n g macroscopic phenomena - such as a body sl i d i n g . o n a rough -surface, or an explosion - then the -observer w i l l experience a f e e l i n g of surprise upon seeing the reversed process; the l a t t e r w i l l , seem to him unnatural, and he -will be able to t e l l the o r i g i n a l from the -reversed process. I t i s a curious f a c t that while f o r elementary processes the reverse process can u s u a l l y be found i n nature, t h i s i s not so f o r processes i n which s t a t i s t i c a l considerations play, a r o l e . The general b e l i e f i s that the s t a t i s t i c a l laws are themselves responsible f o r t h i s asymmetry with respect to the d i r e c t i o n of time. An i n t e r e s t i n g and open question i s : what consequences would r e s u l t on the macroscopic l e v e l i f elementary processes were not symmetrical with respect to the d i r e c t i o n of time? In.any case, thermodynamics and s t a t i s t i c a l mechanics do-not give any information as to whether such time-reversal•symmetry e x i s t s f o r elementary processes; thus i t i s i n t e r e s t i n g to inquire into the question of i t s existence i n these cases. 2. CHAPTER I: INTRODUCTION 1. Terminology Terms that w i l l be used i n t h i s t h e s i s w i l l now be introduced and defined. F i r s t of a l l , i t i s necessary to d i s t i n g u i s h between r e v e r s i b i l i t y , as understood i n thermodynamics, and r e v e r s a l i t y . A process having r e v e r s i b i l i t y i s one f o r which the -entropy of the enti r e closed p h y s i c a l system concerned (where -a closed system i s one that does not i n t e r a c t with other systems) remains i •constant; that i s , the degree of randomness o f the enti r e closed system remains constant. 'Such- a process cannot a c t u a l l y be found i n nature, though i t may be approximated by some natural processes. An example of a r e v e r s i b l e process would be the motion of a p i s t o n separating two gases at equal pressures; a c t u a l l y , i t . would not move, but by an i n f i n i t e s i m a l a l t e r a t i o n of the pressure of e i t h e r gas, one could make the p i s t o n move i n ei t h e r d i r e c t i o n so that the system passes through a.continuous series of states of equilibrium, and so the o r i g i n a l degree of randomness of the system i s not destroyed. We s h a l l r e f e r to such; a process as a r e v e r s i b l e process. On the other hand, r e v e r s a l i t y may be defined as follows: ;A process or state r e f e r r i n g to an object or objects has r e v e r s a l i t y i f , and o n l y - i f , r e v e r s a l . o f - a l l motion i n that process or - state y i e l d s - a p h y s i c a l l y p o s s i b l e -process-'or state i n v o l v i n g the same-actual objects. For example, the motion of two bodies m^ " and mg under mutual g r a v i t a t i o n a l a t t r a c t i o n i s a process having r e v e r s a l i t y , since the process having i n i t i a l l y -mj. and at p o s i t i o n s r^ _ and rg with momenta p^ _ and p^ re s p e c t i v e l y , and; having f i n a l l y m-^  and mg. at "rj_ -and r ^ with momenta "p*^  and p^ r e s p e c t i v e l y , - i s j u s t as consistent with the equations of motion as •is the process which has i n i t i a l l y m-j_ and mg at p o s i t i o n s r ^ and r ^ with momenta -pj_ and -p*g r e s p e c t i v e l y and has f i n a l l y m-^  and mg at "r^ -and rvj .i'.-i with momenta —p*-^  and -pV, res p e c t i v e l y . Since the motions of m-j_ and mg are completely determined by the equations-of motion and the i n i t i a l conditions, the notion of randomness does not apply to t h i s s i t u a t i o n and hence neither does the notion of r e v e r s i b i l i t y . There -are also processes which are not r e v e r s i b l e but have r e v e r s a l i t y . For-example, the escape of a gas from a container into a vacuum i s a process having r e v e r s a l i t y , since return of the gas into the container I s p h y s i c a l l y p o s s i b l e ; however, the escape involves passage of the system to a much more random state, so that the process ,is not r e v e r s i b l e . Next, the d i s t i n c t i o n w i l l be made between the reverse process and the inverse process. I f a p h y s i c a l process involves taking a system from a state A to a state B;, then the inverse process takes the system from B to A , while the -reverse process takes the system from -B to -A, where -B i s the time-reversed .state of B and -A i s the time-reversed state of A. For Instance, suppose that a process .involves m p a r t i c l e s with p o s i t i o n s i ^ , momenta p*^  and .spins "Sj_ i n the i n i t i a l state, - and ;has n p a r t i c l e s at p o s i t i o n s rj_ with momenta P*[ and spins "II • i n the f i n a l state. Then the inverse process has n p a r t i c l e s with p o s i t i o n s :tM, momenta pV'and spins sV i n the i n i t i a l state, and i n the f i n a l state has m p a r t i c l e s with p o s i t i o n s r ^ , momenta pT and.spins s^. However, the reverse process begins with n p a r t i c l e s having p o s i t i o n s r | , momenta -pj and spins -s*[ , and ends with m p a r t i c l e s at p o s i t i o n s r ^ with momenta -p*. and spins — S*. . I t w i l l be said that r e c i p r o c i t y holds f o r a process i f , and only i f , the process and i t s "corresponding reverse process have equal p r o b a b i l i t i e s of occurence. S i m i l a r l y , i t w i l l be said that d e t a i l e d balance holds f o r a process i f , - a n d only i f , the process and i t s corresponding inverse have equal p r o b a b i l i t i e s of-occurence. 2. Considerations'Permitting One i n P r i n c i p l e To Check Whether Reversality, Reciprocity Or Detailed Balance Holds Consider any p h y s i c a l system together with the functions and equations which describe i t s behaviour, and..let t be the time. Then r e v e r s a l i n time of the motion of the system i s formally equivalent to replacing t by - t I n the functions. ' This i s evident when one considers that any instant may a r b i t r a r i l y be chosen as t = 0, and by considering the reversed motion to begin at t •= 0, one considers motion i n the negative d i r e c t i o n of time. Consider, f o r example, a system having, a coordinate x and the corresponding v e l o c i t y dx/dt given by the equations x(*) - A Sir, t>t ( I . l ) W = A h C ~ ^ ( 1 . 2 ) where A. and b are constants. Then the reversed motion i s given by XC-t) ~ A siv b(-^> - ' — A SIT? ht ( 1 - 3 ) provided one chooses x(0) - 0 f o r both motions. Functions obtained by s u b s t i t u t i n g - t f o r t w i l l be r e f e r r e d to as time-reversed functions. I f the time-reversed functions s a t i s f y the same equations of motion as do the o r i g i n a l functions, then the reverse motion i s p h y s i c a l l y p o s s i b l e , and so the motion has r e v e r s a l i t y . Returning to the example shows that both x ( t ) and x ( - t ) s a t i s f y f t * = ~ h X . "(1.5) so that the motion described by (1.5) has r e v e r s a l i t y . Evidently, i f the time-reversed functions do not s a t i s y the same equations of motion as do.the o r i g i n a l functions, then the. motion does not have r e v e r s a l i t y . For example, i f a motion i s described by the equation then the equation f o r y(-'t) i s (1 .6 ) (1-7) - + r<->": • ( ,s , Hence the set of values through which y ( t ) and dy(t)/dt pass i n going from the values y^, ( dy/dt ) 1 to the values y 2 , (" dy/dt ) 2 ' r e s p e c t i v e l y i s not "'• the same as the set of values through which y ( - t ) and-dy(-t)/dt pass i n going from yg, -( dy/dt )g to •-( dy/dt )^ r e s p e c t i v e l y . Thus the motion described by (1.6) lacks r e v e r s a l i t y . In order to make an experimental check on whether a process e x h i b i t s time-reversal symmetry, the process must have -a d e f i n i t e sequence of states i n time. An experimental check on whether r e c i p r o c i t y held would involve observing, a system whose components passed through such a d e f i n i t e sequence*,-and then observing whether the process occurred as frequently as the reverse process. I f i t did, one could conclude that the p r o b a b i l i t i e s f o r the two processes were equal and hence that r e c i p r o c i t y held; i f i t did not, then one could conclude that the p r o b a b i l i t i e s f o r the two processes were not the same and hence that r e c i p r o c i t y d i d not hold. A t e s t f o r d e t a i l e d balance could be c a r r i e d out s i m i l a r l y , but one would consider the inverse process instead of the reverse process. An example of a s i t u a t i o n which - i f i t existed - would v i o l a t e r e c i p r o c i t y i s shown i n Figure 1. A p a r t i c l e decays, giving o f f an electron and gamma ray, both of which are at r i g h t angles to the spin of the p a r t i c l e , so that the d i r e c t i o n of t r a v e l of the gamma ray and.the spin form a r i g h t -handed system. Reversal of the state y i e l d s a left-handed system. The objects i n the o r i g i n a l state cannot be put into this.time-reversed state by/a rot a t i o n nor.by a r e f l e c t i o n , so that the reverse state cannot be obtained without the intervention of other processes. This s i t u a t i o n has never been observed experimentally. An example of a case i n which d e t a i l e d balance would not hold i s shown i n Figure 2. I n i t i a l l y B i s at re s t while A moves to the right'with momentum p; a f t e r c o l l i s i o n , • B moves to the r i g h t with momentum p.while A i s at rest. For the inverse process, B.must move r i g h t with momentum p*, c o l l i d e with A (which i s i n i t i a l l y at r e s t ) and then remain at re s t while A moves r i g h t with momentum pT. Due to the t r i a n g u l a r shapes of A.and B, the.inverse process cannot take place. Hence d e t a i l e d balance w i l l not hold; other processes are required before even an o v e r a l l balance can occur. Such a s i t u a t i o n , i n which d e t a i l e d balance -does not hold,., occurs when one. has a c l a s s i c a l gas of non-spherical molecules. The notion of o v e r a l l balance, which was mentioned previously, w i l l now be made more preci s e . Consider a system of p a r t i c l e s occupying various states. Let the p r o b a b i l i t y of f i n d i n g a p a r t i c l e i n the i t h state be P^ and l e t the p r o b a b i l i t y per unit time of a t r a n s i t i o n from state i to state j be L,- a (LJ • = 0). . I t i s found .that the i r r e v e r s i b l e approach of such a system ' X ( J . . -L. JL to s t a t i s t i c a l equilibrium i s given by the so - c a l l e d master equations Ifi = -ST f P. i .. - D l ) (1.9) At equilibrium, f o r every .1 and j,.P^ = Pj and.dP^/dt - = 0. In order that these two conditions be consistent with (1.9), one must have eit h e r of two •conditions: .In.one case, L^j = Lj-p which y i e l d s d e t a i l e d balance; i n the other case, %. L.. = 4^  L.., and t h i s w i l l be c a l l e d o v e r a l l balance-•J . i j .3 J i spin •y -ray electron O r i g i n a l ' S t a t e e l e c t r o n <r r-ray r spin Reverse' State Figure 1. A hypothetical process which would v i o l a t e time-reversal symmetry. Figure 2. A process f o r which d e t a i l e d balance would not hold. 7-.As a p a r t i c u l a r example of a system f o r which d e t a i l e d balance does not hold but o v e r a l l balance does, the case w i l l be considered i n which there are p a r t i c l e s i n three states. The master equations, with values chosen f o r the L's to give .overall balance, are i 5 = X /? + 2 P3 - JL P2 - X pz 2 i £ (1 .10} I t i s r e a d i l y seen that i f P^ = Pg = P^, then the right-hand side of each of equations (.1.10) vanishes i n agreement with the desired equilibrium conditions-3- Further D e f i n i t i o n s I t i s convenient to introduce d e f i n i t i o n s of terms describing the nature of q u a n t i t i e s with respect to time-reversal. It w i l l be said that a quantity i s even under time-reversal i f , and.only i f , s u b s t i t u t i o n of - t f o r t (where t i s time) leaves that quantity unchanged. Also, a quantity i s odd under time-r e v e r s a l - i f , and only i f , that quantity changes sign upon s u b s t i t u t i o n of - t f o r t , becoming the negative of what i t was o r i g i n a l l y . For example, the quantity given by f ( t ) = t ^ i s even under time-reversal, since f ( - t ) = (-t)^ = f ( t ) ; and the quantity given by g(t) = t ^ i s odd under time-reversal, since g(-t) = ( - t ) 3 = - t 3.=. - g ( t ) . CHAPTER I I . TIME-REVERSAL SYMMETRY IN CLASSICAL PHYSICS 1. Newton's Mechanics. For a p a r t i c l e of mass m and p o s i t i o n r which i s acted upon by a force F, the Newtonian equation of motion i s This ecjuation determines ~r as a function of time. I f t i s replaced by - t , — » and. i f F i s inv a r i a n t under time-reversal, then since the left-hand side i s a second d e r i v a t i v e , therefore the time-reversed function "r(-t) obeys the same equation as does "?(t),.so that the system described has r e v e r s a l i t y . Thus i t i s seen that there i s no inherent asymmetry with respect to r e v e r s a l .of motion i n Newtonian mechanics;.any such .asymmetry a r i s e s only i n s p e c i a l cases. When f r i c t i o n a l forces are present i n a system, the equations-of motion involve terms p r o p o r t i o n a l to f i r s t d e r i v a t i v e s with respect to time. In t h i s case, time-reversal symmetry i s destroyed. However, i t i s generally bel i e v e d that f r i c t i o n i s a purely macroscopic phenomenon r e s u l t i n g from the a p p l i c a t i o n of s t a t i s t i c a l laws to fundamental processes which themselves have -reversality. An-attempt w i l l be made here to show how t h i s may be the case. F i r s t - o f a l l , consider a system of p a r t i c l e s having coordinates x^, Xg., xn., masses.m^_, mg, mn ( f o r convenience, the mass corresponding to coordinate Xj_ i s w r i t t e n as m^,, although, a given p a r t i c l e may have more than one coordinate-and hence some of the m^ '-s may be i d e n t i c a l to each other) and described by the Newtonian equations of motion (II-2) where the F's are invar i a n t with respect to the • s u b s t i t u t i o n of - t f o r t. It might be supposed that i f Xg, x^, • ••, x n were 'eliminated from the f i r s t of equations (II.2), thus leaving a l l terms i n that equation as e x p l i c i t functions of x-j_, i t s time d e r i v a t i v e s , and t, then there might r e s u l t an equation f o r x^ lacking time-reversal invariance. That t h i s cannot be so can be shown as follo w s : The equations (.11.2) are invariant under s u b s t i t u t i o n of - t f o r t; hence the motion described by these equations hat r e v e r s a l i t y , and consequently any other equations which c o r r e c t l y describe the motion must also have time-*reversal invariance. Another approach w i l l now be t r i e d . Suppose there i s a free p a r t i c l e o r i g i n a l l y having v e l o c i t y V i n the x- d i r e c t i o n , and - while continuing to :move i n the x - d i r e c t i o n - c o l l i d i n g at i n t e r v a l s -of distance 8 with successive p a r t i c l e s each having, a mass m :and. a small v e l o c i t y v which w i l l be l e f t undetermined. V and v are here defined to be p o s i t i v e i f the corresponding p a r t i c l e t r a v e l s i n the p o s i t i v e x - d i r e c t i o n , and negative i f i t t r a v e l s i n the negative x - d i r e c t i o n ; the p o s i t i v e x - d i r e c t i o n i s taken as the i n i t i a l d i r e c t i o n of t r a v e l of the free p a r t i c l e . Let M be the mass of the free p a r t i c l e and suppose that the c o l l i s i o n s are - a l l e l a s t i c . Taking x = 0 as the i n i t i a l p o s i t i o n of the p a r t i c l e , i t s p o s i t i o n p r i o r to the f i r s t collistLon i s given by x = V t . (11,3) Suppose that the f i r s t c o l l i s i o n takes place.. Since the c o l l i s i o n i s e l a s t i c , both l i n e a r momentum and k i n e t i c energy are conserved. Thus + M y = yyjv* + M.\r' ( H - 4 ) 2 2n>vz + M : \ r 2 = ? n V ' 2 + M.\r' ( I I . 5) 2 . 2 T T~ 10. where the primed qua n t i t i e s y' a n d V are the v e l o c i t i e s a f t e r the c o l l i s i o n of m and M r e s p e c t i v e l y . Solving '(II.U) and (II .5) f o r V y i e l d s V ' = £ - M - tr>) V Zmv_ (II.6) Assume that /v/ i s much l e s s than Ivl and that m i s -less than M. ' Then V i s i n the same d i r e c t i o n as "V and i s given by V = CM; - Y . ' (11.7) •Since the p a r t i c l e -of mass M t r a v e l l e d a distance 8 before the f i r s t c o l l i s i o n , i t s p o s i t i o n between the f i r s t and.second c o l l i s i o n s i s given by Continuing i n t h i s manner y i e l d s f o r the p o s i t i o n between the nth.and the (n + l ) th c o l l i s i o n s : * = (TtiZTV + - J . + -f - ' . 4 Jz— 1 ) , _ r- (II.9) \ M +77? ' Using the formula f o r the sum of a geometric progression y i e l d s ^ L J _ (£±jt2l) J / The v e l o c i t y between c o l l i s i o n s i s (~—— ) Y and i t decreases as time goes on. Thus the free p a r t i c l e of mass M slows down as does a body moving i n the presence of f r i c t i o n ; the p a r t i c l e s .of mass m and small speed arranged r e g u l a r l y and close to each other are l i k e p a r t i c l e s i n , a s o l i d . However, 11. a l l the processes involved i n the slowing-down have r e v e r s a l i t y . A reverse process i s conceivable but u n l i k e l y due to the d i f f i c u l t y of getting a l l the 'particles of mass m to behave i n the exact reverse manner to that obtained i n the o r i g i n a l process. Indeed, i f these p a r t i c l e s behave l i k e those i n a s o l i d , they w i l l return a f t e r c o l l i s i o n to t h e i r approximate o r i g i n a l p o s i t i o n s and v e l o c i t i e s so that i f the p a r t i c l e M i s sent i n the negative x - d i r e c t i o n with a given v e l o c i t y , i t w i l l simply slow down, as i n the o r i g i n a l process. The r e l a x a t i o n time can be found by evaluating the time elapsed before the second term on the right-hand.side -of (II.6) i s no longer n e g l i g i b l e . Suppose that t h i s happens at the (n.+ 'l) t h c o l l i s i o n . Then the v e l o c i t y . -Al - iv v of M previous to t h i s c o l l i s i o n i s \ yv<i -f ^  ) ^ > and i t s v e l o c i t y a f t e r t h i s c o l l i s i o n i s , by analogy with (II . 6 ) , V Y ( 1 1 . 1 2 ) and the condition that the second term i s appreciable i s (H-13) where some-suitable kind.of average value i s used f o r |v| . Using the fa c t that V ' i s p o s i t i v e , , M i » ] " « (M -tlx) V As can be seen from the de r i v a t i o n used to get ( i l . l l ) , the time required f o r n + 1 c o l l i s i o n s i s ' ". V L T ^ J L 1 ^M-™J J • (11.15) From (II.11+) and ( I I . I 5 ) , T ^ I r M ~ 7 V 1 I X (M+2»l _ 1 1 - . ^ . L 2-?vJLlvl 2. y» S J (II.16) i s the r e l a x a t i o n time.. 12. Suppose that the p a r t i c l e s of mass m constitute- a s o l i d , and suppose that the s o l i d has a simple cubic c r y s t a l l i n e structure. Then ( l l . l 6 ) can be put into a more convenient, form by expressing £ i n terms of the density of the s o l i d . This can be seen as follo w s : Since the p a r t i c l e s of mass m are r 3 separated by a distance z> , there i s one p a r t i c l e per £ . D f volume. Hence the mass per u n i t volume — and so the density - i s _ j222_ £ 3 (11.17) Hence & = \7 -f- -(H.18) and (II.16) becomes T ^ J- fjZL''f M-m I T V (M + Ty) - i 1 y V e L z T v J L i v / — ^ J . (11.19) The discussion of t h i s section, which was i n s p i r e d by B l a t t , 1959; merits furt h e r i n v e s t i g a t i o n , since i t seems somewhat unfamiliar to p h y s i c i s t s , and yet i t provides an elementary and simple means of seeing how apparent time-r e v e r s a l asymmetry can a r i s e from processes having r e v e r s a l i t y . 2. C l a s s i c a l Electrodynamics In r a t i o n a l i z e d MKS u n i t s , Maxwell's equations are c u r l H = 'J*.+ ? D (11.20) c u r l E = - (II. 21) d ivT? = e (11.22) div B •= 0 (11.23) where I? i s the magnetic f i e l d i n t e n s i t y , D* i s the e l e c t r i c displacement, E . -» —» i s the e l e c t r i c f i e l d , B i s the magnetic f i e l d , J i s the current density, and 13--> £ i s the charge density. In a f a i r l y general case, the components of D can be taken as l i n e a r functions of the components of E, and the components of B can be taken as l i n e a r functions of the components of H. In-order to f i n d out the properties of "these equations under time-reversal, t can be replaced by - t i n them. Assume that £ i s an even function under t h i s — * operation. Since-J w i l l behave-as a time d e r i v a t i v e of the charge density under t h i s transformation, i t i s an odd function of t. Then Maxwell's equations remain inv a r i a n t under time-reversal provided that D, and hence E, i s an even function, and that H*, and hence B, i s an odd function. However, the choice of £ as even i s quite a r b i t r a r y as f a r as 'Maxwell'.s equations are concerned; f could also be chosen as an odd function of t. In that case, E would have to be an odd function and B an even function i n order to preserve the form of these equations. Further information beyond Maxwell's equations i s required to decide whether ^ i s odd or even. At any rate, i t i s evident that Maxwell's equations do not require any general asymmetry between the past and the future. Nevertheless, i r r e v e r s i b i l i t y i s observed i n the phenomenon of r a d i a t i o n reaction. When- a charged p a r t i c l e i s accelerated, i t loses energy byr.sending out electromagnetic r a d i a t i o n . This l o s s i s regarded as due to a force -acting on the p a r t i c l e given by 7?/u J (II.2U) where e i s the e l e c t r i c charge, F ,, i s a f i e l d tensor, a 1 1 i s a space-time coordinate on the world-line of the p a r t i c l e -(a*4 i s the product of the v e l o c i t y of l i g h t and the time elapsed between a c e r t a i n zero time and the moment of observation),.and-a dot ind i c a t e s d i f f e r e n t i a t i o n with respect to the proper cotime oC , .which i s given by the product of the v e l o c i t y of l i g h t c and the 2 2 2 2 proper time, so that ( doc) = c ( time i n t e r v a l ' ) - (space i n t e r v a l ) ih. Lowering of indices takes place according to the convention that space components of a four-vector do not change at a l l under t h i s operation, while time components h t simply change sign. d<^ has the same sign as da . Equation (II.2U) gives the n-component of the force of r a d i a t i o n reaction; to get the equation of motion, the expression i n (II . 2U) must be added to the n-components -of a l l other forces that act on the p a r t i c l e and t h i s sum must then be equated to the product of the mass of the p a r t i c l e - a n d the second d e r i v a t i v e of a n with respect to In the case of a p a r t i c l e moving slowly r e l a t i v e to the speed of l i g h t , (ll.2k) can be s i m p l i f i e d . For t h i s case., d can be •approximated-as c dt. The f i r s t space component of (II . 2U) can then be evaluated by noting that a n i s of the order of the -ratio of the p a r t i c l e ' s v e l o c i t y to c and so i s n e g l i g -» • Its • • « _ Q i b l e ; that -a^-a =1; and that i s the product of c J and the time rate of •change of the given component of the ac c e l e r a t i o n A. Thus the right-hand side of. (II.2h) reduces to -3 V 7t (n-25) f o r the damping force. The force of r a d i a t i v e reaction v i o l a t e s .time-reversal symmetry, due to the odd-order d e r i v a t i v e s which appear i n i t . Wheeler and Feynman (19^5) .used the 'absorber theory o f r a d i a t i o n to reco n c i l e the existence of the r a d i a t i o n reaction with the view that a l l the fundamental processes involved are symmetric with respect to the d i r e c t i o n of time. Their argument runs as follows: It i s assumed that ( l ) An accelerated point charge i n otherwise charge-free space does not radiate electromagnetic energy. Hence the problem -of the- existence of an i r r e v e r s i b l e process i s disposed of i n the absence of an absorber, and one concludes immediately that the i r r e v e r s i b i l i t y of the r a d i a t i o n reaction a r i s e s from the presence of an absorber. 15-(2) The f i e l d s which act on a given p a r t i c l e a r i s e only from other p a r t i c l e s ; that i s , no p a r t i c l e i n t e r a c t s d i r e c t l y with' i t s e l f . (3) These f i e l d s - a r e represented by one-half the retarded plus one-half the advanced Lienard-Wiechart solutions of Maxwell's equations. This law of force i s symmetric with respect to the past and the future. (k) . S u f f i c i e n t l y many p a r t i c l e s are present to absorb completely the r a d i a t i o n given o f f by the source. Then the source of r a d i a t i o n i s taken to be an accelerated charge located i n the absorbing system. Since the absorption i s to be complete, a te s t charge outside the absorbing medium w i l l not be a f f e c t e d by the f i e l d s produced by the source. Thus, outside the absorber - a t ' a l l times f t (Jk) t J, N A (11.26) CA) /ji) where F D „ and F- are r e s p e c t i v e l y the retarded and the advanced RET ApV v J f i e l d s due to the >feth p a r t i c l e i n the system. CM) (A) At large distances, 5" F „ represents an outgoing wave while F represents an incoming wave. Since these two waves.cannot i n t e r f e r e destruct-i v e l y , each of these two sums must vanish separately i n order that "(II.26) be s a t i s f i e d . Consequently, outside the absorber at a l l times, i < $ f " r > ) - $ C f F ^ ' ) =0 ^ ( -9 RET 4 rAPV J u ' .Now the left-hand side of (II.27) has no s i n g u l a r i t i e s within the absorber and i s a s o l u t i o n of Maxwell's equations i n free space. Thus, since i t always vanishes outside the absorber, i t must also be always zero inside i t . Let the-ath. p a r t i c l e be the source. From the assumptions made previously, the t o t a l f i e l d a c t i n g on i t i s (11.28 16. Rewriting t h i s expression y i e l d s 2 F ( A > + ( ± _ 1 F<*> ) RET ^ ( _ 'f«rr -g- r/)p^ > S r i F<*> - i r U J ) ( I I ' 2 9 ) From (II .27), t h i s "becomes w f f i ^ f ^ J - ( I I . 3 0 ) The second term represents a force on the source p a r t i c l e having as i t s n-component F f° - I F ^ ) e ~ ( — 'r)0i fief \J At>v ' '3-ot \ v Z ' »« 2- n a c ™v (11.31) where e i s the charge on the p a r t i c l e . Dirac has shown that t h i s reduces to -f e * ( ~ (H.32) which i s the force of r a d i a t i v e reaction (Dirac, 1938). Thus i t i s the second term i n (II.30) which y i e l d s t h i s force. Hence, using (II.30) and ( l l .32) , : the equation of motion of the source p a r t i c l e i s ' ; n a % r- A a ) a* ( H - 3 3 ) where ma i s the mass of the source p a r t i c l e . This r e s u l t i s i n accord with experience. Contrary to what might at f i r s t be thought, i t i s not the property of complete absorption, which was assumed to characterize the absorber, but rather s t a t i s t i c a l mechanical considerations which explain the i r r e v e r s i b i l i t y of the r a d i a t i o n reaction. The condition f o r complete absorption, (II .27), i s symmetrical i n advanced and retarded f i e l d s . By interchanging the r o l e s of the advanced and retarded f i e l d s i n the derivation following (II .27), the equation of motion of the source p a r t i c l e i s obtained as IT-This equation i s j u s t as va l i d - a s (II.33) and- i s consistent with i t , but the force of r a d i a t i v e reaction has a d i f f e r e n t sign i n front of i t . Thus the paradox of I r r e v e r s i b i l i t y remains. However, i n the case being considered, the absorber consisted of p a r t i c l e s that were e i t h e r at rest or i n random motion at the i n i t i a l time of a c c e l e r a t i o n of the source; thus t h e i r retarded f i e l d s were zero or of n e g l i g i b l e e f f e c t at the source -at that time, and i n (II.33) "the r a d i a t i o n reaction term i s dominant. But the s i t u a t i o n d i f f e r s when advanced f i e l d s are considered. The absorber p a r t i c l e s begin to move,, i n response to the s- (A) a c c e l e r a t i o n of the source, j u s t i n time to contribute to F Equating the right-hand sides of (II.33) an& (II-3*0 a t the i n i t i a l time of C-k) CA) a c c e l e r a t i o n , when the F. are n e g l i g i b l e but the F „ are not, shows that t h i s c o n t r i b u t i o n has twice the magnitude of the r a d i a t i o n reaction. Thus the second term on the right-hand side -of (11.3^) i s cancelled out at t -.= 0, and a force of the expected sign and magnitude remains. I f the absorber p a r t i c l e s undergo some sort of acc e l e r a t i o n which i s not due to the motion of the source and which takes-place p r i o r to t = 0, t h i s may have some resultant e f f e c t on the source p a r t i c l e , but t h i s e f f e c t i s neglected i n t h i s c a l c u l a t i o n . I f the motion of the absorber p a r t i c l e s i s s u f f i c i e n t l y random, t h i s e f f e c t w i l l be very small anyway. In order to see that i t i s indeed p r o b a b i l i t y considerations which determine the i r r e v e r s i b i l i t y of r a d i a t i o n , one can imagine the reverse of the process j u s t described. In-this case, chaotic motion i n the absorber causes each p a r t i c l e to receive -at the proper moment j u s t the r i g h t impulse to generate a disturbance converging on the source at the instant of i t s acc e l e r a t i o n . The source receives energy and the p a r t i c l e s of the absorber 18. lose v e l o c i t y . This s i t u a t i o n i s j u s t as consistent with the equations of motion as I s the o r i g i n a l process, and only the small p r o b a b i l i t y of the i n i t i a l conditions serves to exclude i t . ( T I I . l ) 19. CHAPTER I I I : TIME-REVERSAL IN QUANTUM MECHANICS 1. The Need To Represent Reversal Of Motion By/An Anti-Unitary Operator Let 0 be the operator of rever s a l of motion so that i f |b> i s a quantum mechanical state vector, then 0 Jb/'ds the state vector f o r the " time-reversed state. Since the descriptions by /b? and 0 j \>? must be equivalent, 0 must s a t i s f y one of two conditions: E i t h e r i t i s a unitary operator, i n which case i t s a t i s f i e s 0 f a . , Ih> + 4 Z U? ) = 4 a 6 lbs T 1 Z f o r any numbers a-j_ and ag and any state vectors \~b? and I c ? ; or i t i s anti-u n i t a r y , and then i t s a t i s f i e s 9 ( / * > +• 42-lc? ) ••(III.2) = aj 9 l*> + a* e lc> f o r any numbers a^ and a^, and any state vectors I b ^ and | c^> . The symbol •|*,f represents the operation of taking the complex conjugate. I t i s necessary that e i t h e r ( i l l . l ) or (I I I . 2 ) be s a t i s f i e d i n order that the •expectation values and t r a n s i t i o n p r o b a b i l i t i e s not be affe c t e d by the transformation by 0. It i s i n t u i t i v e l y evident that since '0 representes r e v e r s a l of motion, therefore the ^ Hamilton operator H, the coordinate operator Q^, the operator f o r the conjugate l i n e a r momentum component P^, and the angular momentum operator J must transform as OH & - H 9 ?h 9'1 * -p{ 9 Qi & ' 1 - a i (HI-3) 9 f e "3 - - T 20. Furthermore, these operators s a t i s f y the equations ( i n natural u n i t s ) : n ?t . ( i i i . i i ) i Pi > $ j ] = 1' * ' (111,5) J x J - i f (m-6) where t^±> Q±] ~~ ^ i ^ i " ' S i ^ i ^ i s "the unit operator. Suppose that 9 i s a unitar y operator. M u l t i p l y i n g (III.U) on the l e f t hy 9 gives 7) t 9 H B ' 1 9lb> =• iJL & - - i 3_ ( 9 I b>) , •Using :(lII . - 3 ) , H(9lb>) = -i2_(elb>) . ( I I I . 7) Thus 9 | b ,> does not s a t i s f y the same equation as does | b ^ T. Furthermore, a p p l i c a t i o n of 9 to the l e f t and 9 to the r i g h t of both sides of ( i l l . 5) and ( i l l . 6 ) y i e l d s L pi .>«i J " " i 1 (111.8). J x J = - i J ( i n .9) which are inconsistent, with ( i l l . 5) and ( i l l . 6 ) . This s i t u a t i o n can be remedied by l e t t i n g 9 be an ant i - u n i t a r y operator, so that 9 9 7 k (III.10) where T i s a un i t a r y operator and K i s the operator of complex conjugation. A p p l i c a t i o n of 9 to (37) then y i e l d s 21. eHe'1 & ib> = c-i) 3. & ib> U ( 9 lb> ) - i ___ (9/ h>) <P* - ( i l l . - i i ) so that 9 s a t i s f i e s the same equation as does / b)> .. Furthermore, a p p l i c a t i o n of 9 on the l e f t and 9~^ on the r i g h t leaves ( i l l . 5') and ( i l l . 6 ) i n v a r i a n t . . . Thus i n order to incorporate symmetry between past and future into the formalism of quantum mechanics, i t i s necessary to represent r e v e r s a l of motion by an' a n t i - u n i t a r y operator. The argument given here has not shown that quantum mechanics has symmetry with respect to time-reversal; rather, t h i s i s given here as a hypothesis needing experimental v e r i f i c a t i o n . However, t h i s assumption i s consistent with the r e s t of quantum mechanics, and no exception to i t has been discovered up to the present time. Hence quantum mechanics • i t s e l f does not require any asymmetry with respect to r e v e r s a l of motion. A discussion of t h i s problem from a group-theoretical point Of view may be found i n Wigner, 1959-Since 9 i s an a n t i - u n i t a r y operator, i t i s meaningless to speak of i t s eigenstates as f a r as single p a r t i c l e states.are concerned. Indeed, m u l t i -p l i c a t i o n of the state vector of such a state by an a r b i t r a r y phase f a c t o r w i l l a f f e c t the r e s u l t of a p p l i c a t i o n of 9 to i t , although i t w i l l not a f f e c t the p h y s i c a l s i g n i f i c a n c e of the vector. However, something analogous to an eigenstate of a l i n e a r operator can a r i s e i n the case of a state which i s a d i r e c t product of s i n g l e p a r t i c l e states and t h e i r reverse states. Thus i f a state |c> can be expressed as / C > S ' l k> * lh> Tj (III.12) 22. where l h ? 7 " i s the time-reverse of /b^ , then m u l t i p l i c a t i o n of \ b,> by •a phase exp ( i ^  ) r e s u l t s i n | b/' 7'being m u l t i p l i e d by exp (-!£.), so that \ c y remains the same and the a p p l i c a t i o n of 9 to j cj> gives tfce.-same r e s u l t . as i f the m u l t i p l i c a t i o n by exp ( i £ ) had not been .carried out. Thus the equation £ / c > ^ £ / c > , (III-13) where £ i s a number, i s meaningful. A state jc^> f o r "which ( i l l . 12) holds may be c a l l e d a state of s p e c i f i c r e v e r s a l i t y . I t i s not c l e a r whether such states are r e a l i z e d i n nature. Some speculations have been made that there e x i s t s a second lepton number whose operator anticommutes with 9,.so that ( f o r example) a state c o n s i s t i n g of one ordinary neutrino and one muon neutrino would be.a candidate f o r having s p e c i f i c r e v e r s a l i t y . .Despite the -absence of obvious s e l e c t i o n rules analagous to those following from conservation of p a r i t y , time-reversal symmetry does lead to some experimentally accessible consequences whenever t r a n s i t i o n s take place involving, a d e f i n i t e sequence of events i n time. These consequences are connected with the so- c a l l e d p r i n c i p l e of r e c i p r o c i t y , which w i l l be discussed i n the following section. 2. Proof Of Reci p r o c i t y From Time-Reversal Symmetry •Consider the cases i n which.the p r o b a b i l i t y amplitude f o r a t r a n s i t i o n from a state / tr > to a state I t'/^ i s given by the matrix element < T / / S It > (I I I . I l l ) of the s c a t t e r i n g operator ' s , which i s defined as the l i m i t 5 = U ( + 0 0 ; - °° j (HI-15) 23-of a unitary operator t ^ t g , t-j_) which s a t i s f i e s i9_ UCtzy tx) _ : H l f l T C t J U ( t Z } tt) ( m i 6 ) with U(t^, t^) = 1, where H ^  ?-(^2^ i s 'the i n t e r a c t i o n Hamiltonian i n the in t e r a c t i o n p i c t u r e (see Jauch and Rohrlich, pages 117 ff>). I f i s the reverse state of t*> , and / *4 > i s the reverse of / ^  ^ , then the p r o b a b i l i t y amplitude f o r the process which i s the reverse of ( H I . l U ) i s <T T /5/Tr T ^ ijhe p r i n c i p l e of r e c i p r o c i t y then states that < r'/s/* > = ^ < r r • / S l z r > ' ( i n . 17)* The reverse process s ~* I^T^ should not be confused with the inverse process '\t'^ "* i n which only the sequence of states i s interchanged. In order to prove t h i s p r i n c i p l e , the an t i - u n i t a r y property of 9 w i l l be •represented by the operation of tr a n s p o s i t i o n i n the space of occupation numbers. Thus i f , with a suitable choice of the phase, the vacuum state transforms as e i o > = < o i . (HI . is) then any occupation state obtained from the vacuum by a p p l i c a t i o n of a given number of crea t i o n operators as | t > =• tfcvj) • • \0> (III.19) w i l l transform as 9 l t > = €^<ol*.r cv^)r] , zCc^XJ (111.20) where i s a phase f a c t o r +_ 1 which can be s p e c i f i e d a r b i t r a r i l y f o r a given number ^ of p a r t i c l e s without v i o l a t i n g the commutation ( a n t i -commutation) r e l a t i o n s between the operators a('t) and a+fo). In order to 2k. provide f o r the e f f e c t of 9 on a l l the dynamical v a r i a b l e s , the time-reversed quantum numbers i; have been substituted f o r the o r i g i n a l quantum numbers 7 . S i m i l a r l y , <r'l&-1 = < o / a £ < ) . . . ^Cr^) 9-1 (III.21) . = ltT > , Treating the e f f e c t of 9 i n t h i s way, one can transform ( i l l . I k ) as <z'!s/r > = <vI 9 ~x 9 s e'1 Q I ^ > = <^TI ^ t / - r r > , (III.22) where = 9 5 Q _ 1 ( i l l .23) i s the time-reversed s c a t t e r i n g operator. I t remains to prove that 5 T " $ . (111.2U) The adjoint of (M?) i s • -Since H 1 / V T ( t a ) i s Hermitean, H I N T ( t 2 ) ,= H I N T ( t 2 ) ; also, U * ( t 2 , t 2 ) = t 2 ) . Hence ( i l l .25) becomes (III.26) Interchange of t-j_ and t, 2 y i e l d s - i i u - C t , , t j - \ ) ( \ , \ ) ^ r c t i - ) . (ni.27) 31^ Adding (III . 1 6 ) and ( i l l .27) and se t t i n g t 2 = t, t x = - t y i e l d s • zi£<ju,-t) = • / / , , » ; ( / f * , - t ) ( I I I . 2 8 ) Time-reversal invariance requires that, from (ill.2 8 ) , + L n i H r C - t J ] T U T C t , - t ) ? . (111.29). where a time-reversed function i s indicated by the subscript y , so that ( f o r example) U r = 9U9 . The an t i - u n i t a r y property of 9 r e s u l t s i n the •reversal of the order of the f a c t o r s i n products on the right-hand side of the equation (ill;28) i n going to (ill.2 9 ) , since both U and H contain c r e a t i o n and a n n i h i l a t i o n operators. From the d e f i n i t i o n of time-reversal, [ V O ] T = u,NT c-o . ( m 3 o ) Hence ( i l l . 2 9 ) can be written a i j L u T C t , - t ) = / / / m f t ; oT(t,-t) Comparison of (ill.28) and (ill.31),shows that U :(t, -t) s a t i s f i e s the same equation as does U(t, - t ) . Hence Ur (t, -t) ' = u ( t , -t) , (III.32) Taking the l i m i t s of the functions i n (ill.32) as t goes to *a"°. , and using the d e f i n i t i o n of S i n terms of U, S r = 5 • (III.33) S u b s t i t u t i n g from (III.33) i n t o (ill.22) y i e l d s (III.I7), so the p r i n c i p l e of r e c i p r o c i t y holds. Experimentally, f o r a process characterized by one often does not observe the reverse process (which i s characterized by the matrix element 5/ > but rather the inverse process .'••26.. (characterized by ^C'/S / T>). An equality between the p r o b a b i l i t i e s f o r a process and i t s inverse i s u s e f u l i n the s t a t i s t i c a l analysis of states i n thermal equilibrium and the approach to equilibrium, since i t means that the t r a n s i t i o n s from-state % to a state can be balanced d i r e c t l y by the inverse t r a n s i t i o n s from the state T 'to'jstate 7f ..without.invoking .-any./intermediate states through which such a balance might be effected. Under some conditions such an e q u a l i t y - i n the form of the p r i n c i p l e of d e t a i l e d balance - i s implied by r e c i p r o c i t y , as w i l l be shown i n the next section. 3- Conditions Under Which Detailed Balance Holds The p r i n c i p l e of d e t a i l e d balance states that the p r o b a b i l i t y f o r a t r a n s i t i o n i s equal to the p r o b a b i l i t y f o r the -inverse t r a n s i t i o n : ./ < r ' / s y t - : > / A - 1 < * / s 1 %' >\2~ , (111.3^) .As was pointed out previously, t h i s p r i n c i p l e i s not generally v a l i d . However, there are s p e c i a l cases i n which the p r i n c i p l e does hold, and some of these w i l l be given here i n the form of theorems. • Theorem -I. I f S can be represented as the time i n t e g r a l of H ^ r ( a s can be done -in cases-of weak i n t e r a c t i o n wherein S can be given by the f i r s t approximation of a perturbation expansion), then the p r i n c i p l e of d e t a i l e d balance holds. • Proof: From the Hermitean property of H ^  r , < ^ ' / V r / r >* ' « I HINT W> • .(III -35) Since S i s . approximately the time i n t e g r a l of H^ 7- , <r'/5/T>* =. < r l 5 / t'> . .(111.36) Hence \<*'ISlZ>lz = I < 11 S IZ1 > ^  . ( i l l . 3 7 ) • 27-Theorem -II. I f the process i s invariant under coordinate inversion, and i f only those -quantum -numbers are -measured which change sign under time r e v e r s a l and also under coordinate inversion, then d e t a i l e d balance holds. Proof: Let the process occur between states characterized by momenta k-and spins "s. By the p r i n c i p l e of r e c i p r o c i t y , < £ c ' , 5CV • • • / 5 1 , SA , - - , > = + < -* A , -3J,, < IS I -*c, - X , • > , (III.38) .From'invariance under inv e r s i o n of coordinates, ( H I . 39) From (III.3 8 ) and ( i l l . 3 9 ) , ' < f<c > 5 c • - . / S / 12A ) S A •> ' * ' ^ (n i .Uo) Ignorance of the values of the spin v a r i a b l e s i s equivalent to having a process i n which a l l values -of the spins occur. Hence the p r o b a b i l i t y f o r the t r a n s i t i o n i s the sum over a l l values of a l l spins of the p r o b a b i l i t y f o r t r a n s i t i o n with given momenta and spins: ... $?ws (III.41; 28. From ( i l l . 1+0), t h i s l a s t i s given by Kit:, • • < , < • >>!* = x j r , / < s . , . •./sis:, %, . • ->i\ (111.42) where the l a s t step follows from the f a c t that each takes on the same values as does -s*^ . ( i l l . 7 3 ) i s the p r i n c i p l e of d e t a i l e d balance f o r the case when only the momenta are known. Theorem I I I . Assume that a rea c t i o n involves two p a r t i c l e s i n the i n i t i a l state and i n the f i n a l state; that the spins of the p a r t i c l e s l i e i n the reaction plane; and that the reaction i s inva r i a n t under rotations i n space. Then, even i f p a r i t y i s not conserved i n the reaction, d e t a i l e d balance holds. Proof: Consider the diagram of the process i n Figure 3- None of the spins SA> SB> SC> SD have any components perpendicular to the plane formed by the - -* momentum vectors k^, kg. -By conservation of momentum, t h i s plane i s the - • > — » same as the plane formed by kg and k-p . Next consider the reverse process as drawn i n Figure 4. I f the process i s inva r i a n t under rotations i n space, -then the reverse process has an amplitude equal to the-amplitude f o r the process which i s obtained by r o t a t i n g every vector i n the reverse process by an angle ff around an axis perpendicular to the plane of the reaction. The r e s u l t of t h i s r o t a t i o n i s shown i n Figure 5' Comparison of Figure 5 with Figure 3 shows that Figure 5 gives the inverse process. Thus the amplitudes f o r the process and i t s inverse are equal, and therefore d e t a i l e d balance holds. This conclusion i s correct even i f p a r i t y i s not conserved i n the reaction, since invariance under inversion of coordinates has not been assumed. Figure '3- The o r ig ina l , process,. Final': State Figure U. The reverse process. Ill Figure 5. The reverse process.with vectors rotated through ff 29-Theorem -IV. I f a reaction i s such that both the i n i t i a l and f i n a l • s t a t e s are characterized only by the quantum numbers j , m of the t o t a l angular momentum and by sets of other scalar quantum numbers ^/), ^8 > • • • a n d T c , , . . . which change neither magnitude nor sign under 'time-reversal, and i f the r e a c t i o n i s inv a r i a n t under rotations i n space, then the p r i n c i p l e of d e t a i l e d balance holds. Proof: Under the conditions given, only the quantum number m changes i t s value upon r e v e r s a l of motion,,and t h i s change i s only a change i n sign. Then the p r i n c i p l e of r e c i p r o c i t y implies (111.U3) . I f S i s i n v a r i a n t under rotatio n s , the matrix .element on the right-hand side i s equal to the one obtained from i t by a r o t a t i o n of coordinates which transforms m into -m but leaves the scalars t, j invariant- Hence <(Cc , ; • • '>J> & I S l X A > ^ B > - ' ' > J, ft > (III.kk) Taking the square of the absolute value of both sides of (III.I+U) gives the p r i n c i p l e of d e t a i l e d balance. .30. • CHAPTER IV: EXPERIMENTS CHECKING OR INVOKING TIME-REVERSAL INVARIANCE OR DETAILED BALANCE A number of experiments have been performed which invoke, or t e s t the v a l i d i t y of, d e t a i l e d balance or r e c i p r o c i t y . For- instance, consider the reactions H + (** ^ P •+ TT+ f (IV.1) where H i s a hydrogen atom, p + i s a proton, D i s a deuteron, and fTTls a pion- These reactions are inverses of each other. For t h i s case, theorem II of the preceding section i s applicable, and d e t a i l e d balance may be •assumed to hold. Let pp be the momentum of the p.+ r e l a t i v e to the H, while p^. i s the momentum of the pion r e l a t i v e to the D. Then, using the center of mass coordinate system, the equality.of the p r o b a b i l i t i e s f o r the two r e a c t i o n s can be stated as - (ZXt + DCZTr-fLj a^ss , (IV.2) Here the f a c t o r ^ on the left-hand side i s due to the i n d i s t i n g u i s h a b i l i t y of the two protons•in the reaction producing the pion, 07. i s the cross-rR op -section f o r the'production of the pion, and err c Is the cross-section f o r the inverse process, while the I's are the quantum numbers of the i n t r i n s i c angular momenta f o r the various p a r t i c l e s . Thus ^ +i = L (^~) SI*™ (iv.3) .This r e l a t i o n p r e d i c t s the spin of the p o s i t i v e pion. Measurement of the momenta and cross-sections f o r the reactions then provides a means of determining experimentally t h i s spin .(see Elton, page 2^7). .•It was point out by Jackson ;et a l . (1957) that i n the decay of oriented n u c l e i a c o r r e l a t i o n of the form 31-1 +»(<J>/j)-(^/£e)x (fr A . ) { l Y h ) w i l l . e x i s t among the electron momentum p* ., :the antineutrino momentum p*~ , and the p o l a r i z a t i o n of the nucleus - i f , and only i f , beta decay i s the r e s u l t of an i n t e r a c t i o n that i s not invar i a n t under time-reversal. A search f o r t h i s c o r r e l a t i o n was made f o r the case of free p o l a r i z e d neutrons by Burgy et a l . (1958), who sought an upper l i m i t f o r D. Their r e s u l t s indicate that D i s zero o r ' l i t t l e d i f f e r e n t from zero. Thus the existence of time-reversal symmetry i s supported b y . t h e i r r e s u l t s . Tests have.also been made f o r strong i n t e r a c t i o n s , based on comparison of the po l a r i z a t i o n . P produced i n the scattering.of unpolarized-protons and the -asymmetry, (or depolarization) e produced when f u l l y p o l a r i z e d protons are scattered. I f p a r i t y conservation i s assumed, then invariance under time-reversal requires that -P = e. A l s o , . i n the case of proton-proton sca t t e r i n g , i f there i s any time-reversal asymmetry, then, at angles near U50 i n the center of mass coordinate system, JP - e/ i s a maximum and of the same order of magnitude as the r a t i o between the c o e f f i c i e n t s of the two parts of the sca t t e r i n g matrix which are re s p e c t i v e l y non-invariant and inva r i a n t under time-reversal. -Hillman et a l . (1958) obtained r e s u l t s f o r high energy s c a t t e r i n g from hydrogen,-lithium,.beryllium and aluminum which indicate that'time-reversal invariance holds to within a few percent. A f t e r similar, experiments on proton-proton scattering, Abashian and Hafner (1958) concluded that the term of the scattering, matrix which i s not time-r e v e r s a l i n v a r i a n t i s no more than a few percent of the average magnitude of the invariant terms. The p o s s i b i l i t y has also been considered that time-asymmetric events may occur i n the decay of strange p a r t i c l e s . Strange p a r t i c l e s may decay i n two ways: i n leptonic decay, they give o f f l i g h t p a r t i c l e s such as 32. electrons or neutrinos, while i n non-leptonic decay they do not give o f f l i g h t p a r t i c l e s . Sachs (1963) has suggested that t e s t s f o r time-reversal symmetry be made f o r the non-leptonic modes of decay of strange p a r t i c l e s independently of t e s t s i n other cases, and has presented the following considerations f o r such decay: The dominant decay mode of the A -hyperon i s the non-leptonic mode here p i s a proton and TT ~ i s a negatively charged pion. One can define the p o l a r i z a t i o n as the f r a c t i o n of the A -hyperons whose spins are oriented one way or the other, and one can measure p o l a r i z a t i o n q u a n t i t a t i v e l y as the diffe r e n c e P between the f r a c t i o n spinning clockwise and the f r a c t i o n spinning counterwise,.these being the only possible spins f o r a / ^ - p a r t i c l e . Then the rate of d i s i n t e g r a t i o n of a /I - p a r t i c l e with p o l a r i z a t i o n P into .a proton and a pion with c e r t a i n d e f i n i t e v e l o c i t i e s may be denoted by n(P). While n(.P) also i s a function of the v e l o c i t i e s of the products, t h i s dependence w i l l not be made e x p l i c i t here. Let P' be the time-reverse of P, while n 1 i s the time-reverse function corresponding to n. Then invariance under r e v e r s a l of motion requires that The rate n* (.P1 ) i s obtained from n(P) by expressing P i n terms of P'. A -» p + Tf ~ (IV.5) (iv.6) Because of the way i n which P depends on spins (which themselves are odd under time-reversal), i t changes sign under time-reversal,.so (IV.7) •33-Suppose that any non-invariance under time-reversal revealed i t s e l f by a dependence of the decay rate on P i n the manner ' » * 7?i t P y?& ; , ( i v -8 ) where n-^  and s a t i s f y (IV.6). By expressing the right-hand side of (IV.8) i n terms of P 1, .one obtains 7 3 ' = n 1 - P mz . (iv.9) So n(P) does not s a t i s f y (XV.6), and hence i s not i n v a r i a n t under time-r e v e r s a l . A c t u a l l y , one can e s t a b l i s h the lack of symmetry by showing that (IV.8) holds and that ng s a t i s f i e s (TV.6). The above argument can be modified to take account of the quantum mechanical nature of the problem. I t y i e l d s a means of t e s t i n g experimentally f o r a time-reversal invariance i n the decay of strange p a r t i c l e s . Cronin and Overseth (1962) have c a r r i e d out such experiments and have found that they d i d not observe the term ng, although the errors involved are comparable to the rather small value of 10.2 c a l c u l a t e d from data on pion-nucleon s c a t t e r i n g upon assumption of time-reversal invariance. Thus t h e i r r e s u l t s favor time-reversal symmetry i n the non-leptonic decay of strange p a r t i c l e s . Much work remains to be done to e s t a b l i s h time-reversal symmetry experimentally i n nuclear physics. As Henley and Jacobsohn (1959) pointed out, r e s u l t s of many of the experiments performed thus f a r have included rather large experimental errors, so that they have not c o n c l u s i v e l y established the invariance except as a rough approximation. Some authors (see Sachs, l o c . c i t . ) think that the problem of determining the d i r e c t i o n of time may be l i n k e d to an as yet undiscovered fundamental process which lacks time-reversal symmetry and which e x i s t s on 34. the microscopic l e v e l ; experimental evidence i s not yet s u f f i c i e n t to: rule out t h i s ' p o s s i b i l i t y . However, nobody has yet shown that i f such a process existed on the microscopic l e v e l , t h i s would imply any observable asymmetry on the macroscopic l e v e l . The problem of the macroscopic consequences of time-reversal 1 asymmetric microscopic processes i s s t i l l an open :one. 35-CHAPTER V: ARE THERE PARTICLES WHICH ARE THE TIME-REVERSED COUNTERPARTS OF OTHERS? One may wonder whether, i f the operation of r e v e r s a l of motion i s applied to a p a r t i c l e , the r e s u l t may be a p a r t i c l e of a d i f f e r e n t kind than the o r i g i n a l . Feynman (194-9) suggested that a po s i t r o n be regarded as an electron t r a v e l l i n g along a world-line that leads backward i n time. This further suggests the view that the a n t i p a r t i c l e of any lepton (or l i g h t p a r t i c l e ) i s i t s time-reversed counterpart. Since the c h a r a c t e r i s t i c which distinguishes a lepton from i t s a n t i p a r t i c l e i s the lepton number L, which i s -KL.for a lepton and -1 f o r an antilepton, t h i s means that L i s odd under time-reversal. However, discovery was l a t e r made of a lepton — namely.the neutrino - which has a specific.handedness while i t s a n t i p a r t i c l e has the opposing handedness. Since handedness does not change under r e v e r s a l of motion, one cannot regard L as the c h a r a c t e r i s t i c which removes the degeneracy with respect to 9 that a r i s e s i f the time-reverse of a p a r t i c l e i s the same as the o r i g i n a l p a r t i c l e . Thus the change I n lepton number required i n going from an electon state to a p o s i t r o n state cannot be obtained by the operation of time-reversal, and hence the p o s i t r o n i s not the time-reverse of an electron, contrary to Feynman's suggestion. Of course, the above argument would f a i l i f i t were shown that L did not have unique transformation properties under time-reversal, but t h i s would amount to abandoning lepton number as a genuine a t t r i b u t e o f . l i g h t p a r t i c l e s . Nevertheless, there e x i s t s a second kind of neutrino, the muon neutrino, which d i f f e r s from an ordinary neutrino by the i n t r i n s i c a t t r i b u t e L^,, known as the muon number; L ^ has the value +1. f o r one of these p a r t i c l e s , and -1 f o r the other. From the decay of (L mesons, i t i s known.that the muon neutrino has the same handedness as does -an ordinary neutrino (see Danby et a l . , 1962, and Feinberg and Gursey, I962), so i t i s not inconsistent with 36. present knowledge to .assume t e n t a t i v e l y that the muon neutrino state i s the time-reverse of the ordinary neutrino state. This would mean that i s the a t t r i b u t e which removes the degeneracy of neutrino and muon neutrino states, because L ^  i s odd under time-reversal. Then the d i r e c t product of these two states can be a state of s p e c i f i c r e v e r s a l i t y because the expectation value of L ^  i s zero f o r i t so that the f a c t that L ^ anticommutes with 9 does not i n t e r f e r e with the assignment of a s p e c i f i c r e v e r s a l i t y . One can speculate on the p o s s i b i l i t y of s e l e c t i o n rules a r i s i n g from the invariance under 9 of the d i r e c t product state, i n a manner s i m i l a r to what happens i n the analogous case of a positronium state having s p e c i f i c conjugality. 37-BIBLIOGRAPHY Abashian, A. and Hafner, E.M. Ph y s i c a l Review Let t e r s , 1, 255, (1958). B l a t t , J.M. Progress of Theoretical'Physics, , 2 2 , 745, (1959). B l a t t , J.M., and Weisskopf, V. Theo r e t i c a l Nuclear Physics. New York: John Wiley and Sons, 1952-Burgy, M;T., Krohn, V.E.,, Novey, T.B., Ringo, G.R., Telegdi, V.L. Phy s i c a l Review L e t t e r s , ' 1 , 324, (1958). Coester, F. Physical^Review, '84, 1259, (l95l)-Cronin, J.W., and Overseth,-Q.E. Proceedings of the 1962 International Conference on High Energy. Physics. CERN, Geneva, 1962. Danhy, G., G a i l l a r d , J.M., Goulianos, K.,-Lederman, L-M., Mistry, N., Schwartz, M., Steinberger, J. Ph y s i c a l Review Letters, £, 36, (1962). Dirac, P.A.M. Proceedings of the Royal Society, of London, AI67, 148, (1938). E l t o n , L.R.B. Introductory Nuclear Theory. London: S i r Isaac Pitman and Sons Ltd., 1959-Feinberg, G., and Gursey, F. Ph y s i c a l Review, 128, 378, (1962). Feynman, R.P. Physical• Review, 76*, 749, (1949)-38. Henley, E.M., and Jacobsohn, B.A. Ph y s i c a l Review, 113, 225, (1959)-Hillman, P., Johansson, . A., and T i b e l l , G-. Physical Review, 110, 12l8, (1958). Jackson, J.D., Treiman, S.B.. , and Wyld, H.W.Jr. Phys i c a l Review, 106, 517, (1957)• Jauch, J.M. , and Rohrlich,. F. The Theory of Photons and Electorons. Cambridge 42, Massachusetts: Addison-Wesley Publishing Company Inc., 1955-Sachs, R.G.. "Can the D i r e c t i o n of Flow of Time be Determined?" Science, ihO, 1284.,. .(1963)-Tolman, R.C. . . The 'Principles of S t a t i s t i c a l Mechanics. London: Oxford U n i v e r s i t y "Press, 1938. Wheeler, J.A., and Feynman, R.Po Reviews of Modern Physics, 17, 157, (19^5). Wigner, E.P. Group Theory and I t s Application to the'Quantum Mechanics of Atomis Spectra. . New York and London: Academic Press, 1959-Zocher, H., Torok, C Proceedings of the National Academy of Science, 39, 68l, (1953)-

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