Space and time reflections in relativistic theory

Nuclear Phys=cs 15 (1960) 1--12; (~) North-Holland Pubhshsng Co, Amsterdam Not to be reproduced by photoprmt or mmrofflm vnthout written pertmsston from the pubhsher

S P A C E AND TIME REFLECTIONS IN RELATIVISTIC T H E O R Y Yu. M. S H I R O K O V

P N. Lebedev Physzcal Institute, Moscow Received 22 June 1959 A complete classffxcataon of representations of the mhomogeneous Lorentz group is obtained including space reflectxons and time reflections of the Wzgner type. The problem of the squares of varzous reflectmu operations for partmles of senn-mteger spin xs investigated m detaxl. The general results thus obtained are compared with those obtained when superimposing addltaonal reqmrements on the theory, notably, the reqmrements of the locahty of the fmld operators.

Abstract:

1. Introduction

The purpose of the paper is to obtain all the laws of transformations for the wave functions (the state vectors) for space and time reflections using only requirements of the relativistic invariance without resorting to the properties of the locality of the field operators. The following propositions are taken as a starting point a) Rdatwistic invariance. The mathematical formulation of these requirements consists 1) in that the theory should contain the operators May, Pa, satisfying the commutation relations IMam, Ma~] = i(Oj,¢Ma~+O/,aM~+OwMt,a+Oa~M~,),

[ M ~ , Pal = ~(P.O~a--Pl,0.a),

[Pt,, P~] = O.

(1)

b) The Wigner /ormulation o/the law o/ t, me reflect, on 8) is accepted, according to which the transformation of the state vector ~u = J t 7/'

(2)

corresponds to the operation t : --t', where J t as a certain hnear operator and K is the non-hnear operation of complex conjugation K~-":

~'*.

(3)

The Schwinger rule of transpositon of matrix elements n) m a y be obtained from the Wigner formulation by substituting (3) in (2). The formulation in the framework of the conventional theory of representations is consxdered in ref. 4) February (1) 1960

1

")

Yu.

M.

SHIROKOV

c) The operations o/reflections are not assigned explicitly and are determined by thezr geometrical properties. Thus, the inversion J 6 is determined as an operator transforming the state vector in reflecting the space coordinates x = - - x : ~' = J , ~'.

(4)

The operator of time reflection is determined in ('2). It is convenient to introduce another operator .fstK corresponding to the reflection of all four axes X~ ~- --X'~,

t// = JstK~r/,.

(5)

= .1,t.

(6)

It Is obvious that

Speaking figuratively the inversion is defined as an operator connecting the state vector with that of the same physical system observed through a mirror. From the above definitions and from the fact that the sequences of coordinate transformations ,

,

X p-~= X p

and

,

p'q- ~"pv x v,

X p -~- - -

. . . . . . . . . . Xp "~ --~p

,

Xl,

~

(7)

Xp --~p-q-~poX, v

lead to the same resulting system one m a y obtain the commutation relations for the operator J s t

PJ,

= 0,

(8)

J/[ #v,f st-q-JstM:, = O.

(9)

Here the star designates complex (not hermitian) conjugation t. Similarly for J s the following relations m a y be obtained:

[fs, M] = o,

[.fs, Po]

= 0,

J s p + p ~ f s = 0,

. f s N + N . f 8 = 0,

(10)

where

N, = --iM,4.

(tl)

It is important to emphasise that each of the operations of reflection determined b y relations (2)--(5) may or m a y not contain charge conjugation. According to (8) and (10), on the other hand, these operations should be mvariant for all interactions. d) For particles o/~nteger spin the squares o/all re/lect~ons are equal to unity. For parttcles o/hall integer spin the squares o[ each o[ the rellections d~s, J t K, , f st K may be equal to unity or minus unity, but must be the same for all particles wzth hall znteger sp,ns. In other words the squares of reflections determine the properties of space-time itself and not a property of individual particles. There* T h e o p e r a t i o n of c o m p l e x c o n j u g a t i o n does n o t a f f e c t t h e x m a g m a r y u m t y m t h e f o u r t h c o m p o n e n t s of v e c t o r s a n d tensors, so t h a t P4* = tPo*, for e x a m p l e T h i s f a c t is n o t e s s e n t i a l s i n c e t h e use of c o n t r a v a r l a n t v e c t o r s w~tt-_ t h e c o n v e n t m n a l d e h m t m n of c o n j u g a t i o n l e a d s t o t h e s a m e result~

SPACE AND TIME REFLECTIONS IN RELATIVIST|C THEOKY

3

fore particles with different squares of reflection cannot exist simultaneously s) since that would lead to the existence of systems with an integer angular momentum and a negative square of reflection. As was indicated in refs. 4,e), the above considerations m a y be given a geometric interpretation if a group of rotations is changed so that the element J 2 . of a rotation of angle 2zr around an arbitrary axis is introduced into this group (This extension naturally arises in considering the topological properties of parameter space of a rotation group.) The square of each reflection m a y be then equal either unity or a rotation of angle 2n, which leads to the following eight groups: TABLE 1 j2

G4

Gs

Gs

G7

Ga

Jr#

J,n

J*n

1

I

J*~t [ 1 ] ".¢=n

•,¢*n

1

J2n

1

J*u I Jzu[

JzTr

1

1

J~v

Gt

G~

[

l

l

( J t K ) 2 = "Jct'Jft* I

1

(JstK) * = J s t J ~

1

j

i

G3 [Jtt

1

Table 1 holds both for integer ( J , , = 1) and half integer ( J 2 , ---- -- 1) spm~ Real space-time is transformed according to one of these o g h t groups, which fixes the squares of reflections for all particles at once. Thus there are eight structures of space-time non-equivalent to each other. Below it will be shown that the difference between these structures is in principle accessible to experimental check. e) No assumptzons are made as to the locality o/the field operators and the kz nd o/ equations o] motion in general. The above-hsted basic propositions have a number of features different from those accepted in the investigations of this kind (see, for example, ref. 7) which as a review contains references to the principal previous papers) The adoption of a) and e) allows for an extensive generahzatlon and makes It possible to obtain a number of transformations of reflection which satisfy the requirements of relativistic invariance and yet cannot be reduced to local transformations of the Dirac field operators. Point (c) is in fact not new but rather imparts to the operations of reflection their original geometrical meaning. This point, however, involves a certain change of the very ideology of research of the reflection operations. Usually (as in ref. 7)) certain operators (P, PC, T, CT, etc ) are assigned to act upon the field operators and a study is made oi their conservation properties. The present author, on the other hand, assigns the requirement of the existence of conserved operators connected with certain coordinate transformations and satisfying relations (6), (8), (9), (10) and one

4

Y u . M. SHIROKOV

of the columns of table 1. (The non-fulfilment of this requirement is t a n t a m o u n t to the rejection of Euclidian space-time.) The task of the present investigation is to find the explicit form of these operators. The requirements of d) are not new s) (though they contain a somewhat unusual geometric interpretation), but are investigated in detail for the first time. These requirements impose rather rigid limitations (different for each of the eight groups) on the possible form of the reflection operators.

2. Particles with Non-Zero Rest Mass The methods of the field theory are unsuitable for research which is not based on the locality of the theory. Therefore we shall make use of the mathematical technique employed in ref. s) for obtaining an explicit form of the representations of the inhomogeneous Lorentz group. These representations were first obtained by Wignerg). According to ref. s), the state vector of the free relativistic particle of mass K and spin s m a y always be reduced to the form

~ 7 (P)

(12)

where the kinematic variables are the three-dimensional m o m e n t u m p and the projection of the spin m,. Apart from mass and spin, the particle m a y possess other variables (such as charge) invariant with respect to the non-dimensional rotations and translations; to designate these the index ~ is introduced. The relativistic invariance is here ensured by the fact that for the state vector 02) the operators M~,~,Px satisfying relations (1) m a y be determined s) as follows: =

P,

Po =

e, -

~

N~ie,~p

M=--ip×~+S, rSl, s2] = s3 . . . .

S×p e~+~'

s" = s ( s + l).

(13)

(14,)

The argument is complete because only one irreducible representation of the inhomogeneous Lorentz group corresponds to the given values of mass and spin, which m a y always be reduced to the form (13). Now we have to find for the state vector (12) operators J a , J t , Jmt satisfying (6), (8), (9), (10) and one of the columns of table 1. Let us represent these operators in the form -fe=~P,

drt----- ~t T,

Jst=~st

PT,

(15)

where P is an operator acting only on the variable p and changing the sign of the m o m e n t u m : BWana(p) = i2*ko~*~(--p), (16)

SPACE AND TIME REFLECTIONS IN RELATIVISTIC THEORY

5

and T is an operator acting on p, m, (but not on ~) and changing the sign of the momentum and the spin projection: T~,'~(p) = (-- 1)°-=. ~'_~,'~(--p).

(17)

From (16), (17) it follows that p-xpp T-lpT

= _p,

P-1SP

= --p,

p 2 = T ~. = JA, =

T-1ST

= S,

(18)

= --S*,

(19)

{ 11 (integer spin) _ (half integer spin).

(20)

Determined through (16) and (17), the operations P, T are a natural generalization, for arbitrary spin, of the quantities used in the spinor field theory. Substituting (13) and (15) in (6), (8), (9) and (10) and using (18) and (19) we obtain ,;tt = &t, (21) [&,mm] :0,

[~,pw]=0,

[)~t, P a ] = 0 ,

D~t, M s , ] = 0 -

(22)

According to (22) the operators ~ , At, ~t do not act on the kinematic variables p, m a since they are either numbers or matrices with respect to the invariant variable 0¢. For single-valued representations, according to (11) and (20), (integer spin) (23) and all factors ~ may be numbers. Two non-equivalent representations differentiated by parity correspond here to every value of mass and spin: ~2

=

AtAt*

=

1) / , =

~tAs* t =

1

1, At----1

2) / s = - - l ,

(integer spin).

At= 1

(24)

(The representations distinguished by a phase factor for it are not different, since this factor, according to (2) and (3), may be eliminated by multiplying by the corresponding phase factor of the state vector.) To obtain double-valued representations the formulae (15) and (20) should be substituted in (2) and JA, put equal to --1. As a result we obtain ~t*

Gx

G2

Gs

G,

G6

G6

GT

G,

;tB2

--1

--1

1

1

1

I

--1

--1

~t'lt*

-- 1

1

-- 1

1

1

-- 1

1

-- 1

~jt~st*

-- 1

1

1

1

1

--

1

-- 1

1

(25) --

If At (/,t) is a number, the quantity 2tAt* (2,tA*t) is necessarily positive definite.

6

YU M SHIROKOV

Therefore the factors 2 m a y be numbers only in the groups Gz, Gs. In the remaining six groups these factors must be two-row matrices acting on the additional independent variable so that the dimension of their reducible representation is double that of the corresponding representation of the proper group. It m a y be directly verified that the following irreducible sets of factors satisfy relations (21) and (25):

Gz

Gz.

'.

,

--z

z

I Gs --z

Pl

G= I ,oz.

I

G5

1

--1 [

[ GT

Ge

1

--1

~*t

zp=

1o.

-- zpt

*

--*

~P8

o.t, --~Ps[

Loz [ ,pt

(~6)

]

i

o. lo.

Gs

o. to. 1

-- 1

P' [ - P '

P' I - p'

In (26) the matrices Pl, Pz, P8 have the form of the corresponding Pauli matrices and act upon an addihonal variable (of the charge type) of the state vector. The matrices Pl, P2 correspond to an operation of the type of charge conjugation. As a result of a presence of the operation of complex conjugation in (2) and (5) the matrices pl, P2 cannot be transformed into each other through the transformation of equivalence if they enter in ~t or &t. Table (26) exhausts the irreducible representations for each of the groups G1--G8 for a half integer spin. 3. P a r t i c l e s

with

Zero

Rest

Mass

For zero rest mass two irreducible representations of the proper group, one-dimensional with respect to the spin variables and different by the sign of the spin projection on the m o m e n t u m (spirality), correspond to each absolute value of the spin 9). The change of sign of the momentum changes the sign of splrality so that the representations of the group containing space reflections should be two-row with respect to the spin variable of the sign of spirahty. The operators pa, Ma, for these representations m a y be written in the form fi = P,

Po ---- IPl,

M1 = --~ p~. ~

-- p~

+ sa~ sin--'O (27)

M z = --i N 1 = ~p ~

P3 ~ 1

3

-- Pl

+ Sff3 ----O'sIn

--scr3 sin 9 ctg O,

M3=-i

Pl~-P2

N 2=~p~+sa~cosq0ctg0,

, N 3---ip~p3.

SPACE AND TIME REFLECTIONS IN RELATIVISTIC THEORY

7

Here a 3 is the Pauli matrix acting on the variable of the sign of spirality, ~0 and 0 are the polar angles of the momentum vector. For integer spin the operators oc,, oct, ocst satisfying (6), (8), (9), (10) and (11) are equal to ocs = asP,

oct = P,

(integer spin).

ocst = al

(28)

Only one irreducible representation here corresponds to each absolute value of the spin. For half integer spin we shall seek the operators ocs, OCt and ocst in the form J s = 2'sP',

!

oct = 2'tP',

ocst = --2st

(28a)

The operation P ' acts on the wave funchon as P, i.e. changes the momentum sign and its square is also equal to --1: p, ~rt(p) __ ~rt(_p),

(p,), ___ _ 1,

(29)

since the eigenfunctions of the operators (27) contain a factor exp (+½t~o), multiplied by i if ~oreplaced by ~0+~t. Substituting (28) in (6), (8), (9) and (10) we shall obtain ¢

2's2' t = 2 st,

(30)

2'saa+aa2' s = O,

(31)

2'taz--az2' t = 0,

(32)

2'staa+aa2'st = 0

(33)

Relations ( 3 1 ) - (33) express the fact that the spin projection on the momentum does not change under time reflection and changes sign under inversion Substituting (28) in (11 ) and using (29) we obtain the following table for the squares of 2' G4 m 1 ~.'t2't*

I--1

I

1 I --1

°.l II

1

1

1

--1

I I

Gs

G~

1

--1

--1

1

1

1

G8

I-1 I-1 I-'

(34)

According to (32) the factor 2' t with respect to the spin variable is either u m t y or proportional to a 3. Therefore, in the groups G1, Ga, Ge, Gs, where the values (2't2~*) are negative another discrete variable has to be introduced so that the

8

YU M. SHIROKOV

state vector of the representation must consist of four rows. In the remaining groups this doubling is not needed. Relations (30)--(34) are satisfied by the following sets of the factors exhausting the irreducible representations of the state vectors of the particles with zero rest mass and half integer spin: I'

Gz I

G,

G=

G,

G6

Gs

Gz

Gzp8

G7

Gs

1

p=

(35) t

p=

i" st

p:

1

~:

Gzp=

Gz

--tG=

G:

plOZ

0z

4. Dirac

p=

--zplOzl [ 2~ z

~Gzpz

Equation

The requirement of the locality of the field operators imposed on the theory leads to an essential change of the final results which may be traced on the example of a Dirac spinor field. These differences arise from three reasons. First, the Dirac equation is always a four-component one, while two-component spinors are possible in the above discussion for the groups G~ and Gs (see eq. ( 26) ). Second, certain representations in the local formulation prove forbidden. Third, certain representations equivalent within the framework of the general theory of representations may prove to be non-equivalent with respect to local transformations. Before proceeding to the investigation of the Dirac equation let us enumerate all physically non-equivalent four-component representations allowed by table (26) for the particles of a spin ½: No

G

1 2

Gz

t8 ~ --=

At

1st

p= p=

~p= --zp=

l

1

G= ~P3

10 11 12 13

Pz Px p1

--z zpz - - zpx

Pt

zpz 1

G:

Pz P*

Pa P=

~Ps --~Pl

G4

pz p3

Pa pz

--~p= ~p=

(36)

SPACE AND TIME REFLECTIONS IN RELATIVISTIC THEORY

No

G

~*

1

1

Pz

--1 1

1

-I

16

Pl

17 18 19 20

Gs

21 22

Ge

1

24

P: Pz

Pz

I

Pt

Pl

--1

P*

--P2

1

Pz

*Pz

Ps

P*

GT

~Pa

Pz

--Ps

Ga

zPz zPa

P2 P~

--P8 Pi

25

26

Pz --Pl 1

Pi Pz

--I

23

~st

~t

1

14 15

9

(36)

We call representations such as No. 7 and No. 8 "physically" non-equivalent if they are mathematically equivalent but correspond to different physical conceptions: in No. 7) the particle transforms into itself under inversion (parity conservation), in No. 8 it turns into the antiparticle under inversion (conservation of combined inversion). Now let us consider the possible transformations for the operators of the Dirac field which will correspond to the addition of the requirement of locality to the invariance conditions of (1), (8), (9) and (10). In the local formulation the following requirements are equivalent to these conditions: 1. Invariance of equations for the field operators. 2. Invariance of commutation relations. 3. Invariance of definition of the vacuum. Taking into account these requirements it m a y be established that only the following non-equivalent sets of transformations of the Dirac field operators ~(x, t) m a y correspond to the coordinate transformations of reflections:

group

No

n u m b e r oi corresponding representatmns m (36)

operators correspondmg to c o o r d i n a t e reflectzon x ~ --X'

t = -- t' x / , = - - x ' #

GI (37)

Gz

*P

T

*PT

Ga

PC

*CT

,PT

10

PC

zT

--z.PCT

12

PC

--*T

3 G4

I - -

I

4

,PCT

tI

12

10

Yu

group

No

M SHIROKOV

operators corresponding to coordinate reflectaon x = --x'

G5

t = --t"

Xp=--z'p

n u m b e r of corresponding representatlons In (36)

5

P

T

PT

14

6

--P

T

--PT

15

7

P

CT

PCT

17

8

P

--CT

--PCT

17

9

-- P

CT

-- PCT

18

10

--P

--CT

PCT

18

(37)

PT

19

(continued)

11

PC

CT

12

PC

T

PCT

20

13

PC

--T

--PCT

20

14

P

~CT

,PCT

21

15

--P

aCT

--,PCT

22

'

G, 16

*P

CT

17

*P

--CT

18

,P

aCT

19

~P

--aCT

~PCT

24

G, --,PCT

24

--PCT

26

G8 t

PCT

26

In (37) the operations P, T and C m a y be determined in the standard manner (with the assigned phase factors):

P (x, t) =

t),

(38)

Tv/(x, t) = 7x73yi~(x, --t),

(39)

CW(x, t) = y ~ y ~ ( x , 0.

(40)

It m a y be shown that with the above choice of the phase factors, transformation (38) and (39) upon the operators of the Dirac field are equivalent to transformations (16) and (17) upon the corresponding state vector. These operations are therefore designated by the same letters. This equivalence m a y be established with the aid of a Foldy-Wouthuysen's transformation lo) of the solution of the Dirac equation for the field operators and further transition to the configuration representation according to ref. n). Table (37) exhausts all possible laws of local transformation of reflections for Dirac particles. It will be remembered that here only those particles m a y exist simultaneously whose operators are transformed by the representations entering the same group, and that all operators of reflection should be conserved.

SPACE

AND

TIME

REFLECTIONS

IN

RELATIVISTIC

THEORY

11

5. D i s c u s s i o n The above s t u d y shows that the known properties of space-time should be fully determined in accordance with one of the rows of table 1, so that the question arises according to which group G, real space time is transformed. To solve this problem the following differences between representations of these groups m a y be used: a) Pseudoscalarity o/the ground state o/positonium. This argument does not depend on whether parity or combined parity is conserved, since the system of two photons has even charge parity. This requirement excludes from table (36), representations 1--6, 8, 9, 11, 13, 16, 23 and 25 which are just the same representations which are absent in the local variants (37). Thus, the requirements of pseudoscalarity of positonium eliminates the possibility of group GI in which the squares of all reflections are equal to unity. b) A two-component neutrino. If the neutrino is two-component, the groups G 1, G3, Ge, G 8 become impossible according to (35). c) Four components /or all known spinor particles w~th non-zero rest mass. This argument, taken together with the previous one, makes especially probable such representations in which the wave function of the particle of spin ½ m a y be two-component for zero mass but is necessarily four-component for particles of finite mass. This condition is satisfied in the groups G4, G~. The next set of arguments is connected with non-conservation of the operation of charge conjugation C(pl, P2). It stands to reason that what is meant here is the conservation in all interactions, since the problem as it is discussed involves the geometrical properties of space-time. The non-conservation of C leads to the fact that if, for example, the quantity PC corresponds to the transformation x = - - x ' and consequently is conserved, the quantity P will not conserve. On the basis of this remark the following additional restrictions on the representations m a y be considered. d) Conservation o/PCT. This requirement is hypothetical if the condition of locality is not imposed on the theory. The conservation of PCT leads to the non-conservation of PT, whereby representations 3, 4, 7, 9, 10, 14.... 16 and 19 in (36) are excluded. Taken together with a) this requirement leads to the exclusion of the groups G 2 and G3. e) The hypothes~s o/conservation o/combined panty 5, 12). The conservation of P C ehminates representations 1--7, 11, 13, 14 18, 21, 22, 24 and 26 from (36) Taken jointly with a) this point excludes all representations of the groups G 1, G 2, Ge, G~ and G 8. f) The hypothes~s o/the conservation o] T. It excludes representations 1, 2, 5, 6, 9--11, 13, 17--19, 21, 22, 24 26, i.e., jointly with a) it excludes all representations of the groups G1, Ga, G6, G~ and G s. The requirement of locahty is equivalent to the combination of conditions a) and d) Only representation 12 from (36) satisfies all the above-enumerated requirements. This representation

12

Yu M. SHIROKOV

belongs to the group G4. In (37) two locally non-equivalent representations 3 and 4 correspond to it. If requirement c) is rejected, the remaining requirements will also be satisfied by representation 20 of the group G5 from (36). In (37) two locally non-equivalent representations, 12 and 13, will correspond to it. In this case, however, the particles whose wave-functions transform according to other various representations of the group G5 become possible in principle, as a result of which the laws of conservation of d), e) and f) will lose their universal character. Thus, the presently available experimental data lead us to the conclusion that real space-time transforms either according to the group G4 or according to the group G5, the group G4, in which the squares of inversion and reflection of all four axes are equal to unity being more probable. (It will be noted that if the reflection is interpreted as a rotation at an angle n around a certain additional coordinate axis the square of reflection will be reduced to a rotation at 2n). It is noteworthy that such a subtle property of space-time connected with the topology of the parameter space of the rotation groups turns out to be accessible to experimental determination. In the group G4 all particles should transform according to representation 12 from (36). In the transition to the local formulation two representations, 3 and 4, non-equivalent with respect to local transformations, will correspond in (37) to this representation. A characteristic of these representations is that while the transformation properties of the state vectors of the particles described by them are perfectly indentical, the field operators are transformed differently on reflecting all four axes. The difference between these particles therefore will have no effect on the conditions of invariance of the S matrix, determined, for example, according to ref. is), but will have a marked effect in deriving the interaction Lagrangian converting one particle into another ~. t Prof A S Wightman informed me that some of the questions considered m this paper have also been investigated by Prof E. P. Wlgner (unpublished).

References 1 2 3 4 5 6 7 8 9 10 11 12

13)

P A. M. D1rac, Revs Mod P h y s 21 (1949) 392 E P Wlgner, Nachr Acad ~vVlss Gdtt. (1932) 546 I Schwmger, Phys Rev. 82 (1951) 914 Yu. M. Shlrokov, ) E T P 36 (1959) 879 E P. Wlgner, C G. Wick and A S Wlghtman, Phys Rev 88 (1952)101 ¥ u M Shtrokov, J E T P 34 (1958) 717 T D Lee and C N Yang, Elementary Particles and Weak Interaction. Brookhaven, 1957 Yu M Shlrokov, J E T P 33 (1957) 861, 1196, 1208 E P ~V1gner, Ann. of Math 40 (1939)149 L L Foldy and S A Wouthuysen, Phys Rev 78 (1950)29 Yu M Shlrokov, J E T P 24 (1953) 129, 135 T. D Lee and C N Yang, Phys Rev. 105 (1957) 1672, L D Landau, ) E T P 32 (1957) 405, V G Solovyev, Nuclear Physms 6 (1958) 618 Yu M. Shlrokov, J E T P 35 (1958) 1005