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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Some consequences of time-reversal symmetry Maroun, David Peter 1964
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Title | Some consequences of time-reversal symmetry |
Creator |
Maroun, David Peter |
Publisher | University of British Columbia |
Date Issued | 1964 |
Description | The purpose -of this work is to discuss the symmetry, or lack of it, under reversal of motion in physical objects, states and processes. Considerations of such symmetry are made in both classical and quantum physics, notably in the problem of reconciling the assumed time-reversal symmetry of microscopic processes with the observed asymmetry of macroscopic processes. In the case of classical mechanics, a simple model of a free particle colliding with a series of almost stationary or stationary particles of smaller mass is introduced in order to show how a friction-like phenomenon can arise from processes all of which have symmetry under reversal of motion. It is maintained throughout that symmetry under reversal of motion is a property of all fundamental states and processes in nature. |
Subject |
Quantum theory Time reversal Symmetry |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2011-09-29 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0085950 |
URI | http://hdl.handle.net/2429/37699 |
Degree |
Master of Science - MSc |
Program |
Physics |
Affiliation |
Science, Faculty of Physics and Astronomy, Department of |
Degree Grantor | University of British Columbia |
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UBCV |
Scholarly Level | Graduate |
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