Implications of Parity and Time-Reversal Symmetries in Atoms

CIIINESE JOURNAL OF PHYSICS VOL. 33, NO. 2 APRIL l’J94

Implications of Parity and Time-Reversal Symmetries in Atoms

Hsiang-Shun Chou’

Instttute of Atomic and Molecular Sczences, Academza Sznzca P.O. Box 23-166

Taipei, Tatwan 106, R.O.C. Keh-Ning Huang

Institute of Atomic and Molecular Sciences, Academza Sinica P.O. Box 23-166

Taipei, Taiwan 106, R.O.C.

and Department of Physics, Nattonal Taiwan Unaverszty Tazpei, Taiwan 106, R.O. C.

(Received November 29, 1993)

We have investigated the implications of parity and time-reversal symmetries in atoms. The atomic wave function is expressed as a linear combination of configuration wave functions. If parity symmetry is violated, an atomic state no longer has a definite parity such that both electronic orbitals and configuration weight coefficients contain parity-nonconserving components. Time-reversal symmetry guarantees that the radial wave functions of the large and small components of the Dirac orbital differ in phase by &n/2, and the parity-conserving and parity-nonconserving components of the Dirac orbital differ in phase by &x/2 as well. In addition, time-reversal symmetry implies that the relative phases between parity-conserving configuration weight coefficients are 0 or H, while the relative phases between parity-conserving and parity-nonconserving configuration weight coefficients are &n/2. The absence of permanent electric dipole moments of atoms with definite angular momentum is demonstrated. In addition, the Kramers theorem is elucidated explicitly in the relativistic context. In the multipole expansion of the photon field, the relative phases between the expansion coefficients for the transverse electric multipole potentials and for the magnetic or longitudinal electric multipole potentials are f7r/2. Finally, we show that the relative phases between competing transition amplitudes are 0 or s, which leads to constructive or destructive interferences between competing atomic transitions.

PACS. 11.30.Er - Charge conjugation, parity, time reversal, and other discrete symmetries.

PACS. 31.10.+2 - General theory of electronic structure, electronic transitions, and chemical binding.

PACS. 32.80.-t - Photon interactions with atoms.

1 7 1 @ 1994 THE PHYSICAL SOCIETY OF THE REPUBLIC OF CHINA

172 IMPLICATIOIiS OF PARITY AND T I M E - R E V E R S A L S Y M M E T R I E S VOL. 32 I. INTRODUCTION

Parity and time-reversal symmetries imply that physical laws are invariant under the parity and time-reversal transformations. Traditionally, the dynamics of atoms are described by the coupled Dirac and lvIaxwel1 equations, which are parity and time-reversal invariant. The implications of parity symmetry in atoms were first investigated by Wigner [l], who demonstrated that the Laporte selection rule [2] for atomic transitions was a consequence of parity symmetry. Lee and Yang [3] hs owed that parity and time-reversal symmetries prohibit the existence of permanent electric dipole moments of atoms with definite angular momentum. In 1967 Weinberg, Salam, and Glashow [4,5] proposed the unified theory of weak and electromagnetic interactions, according to which, there is a parity-nonconserving (PNC) potential between electrons and nucleons. The discovery of CP violation [6] in 1964 also predicted time-reversal-violating (TRV) interactions within atoms [7-111. During the past, decades, many experiments have been performed to test parity and time-reversal symmetries in atoms [12-161. At the same time, several theoretical approaches have been proposed to calculate PNC and TRV effects in atoms [17-211.

In this paper, we investigate some aspects of the implications of parity and reversal symmetries in atoms. In Sec. II, we explore the implications of parity and time-reversal symmetries in the atomic wave functions. The atomic wave function is expressed as a linear combination of configuration wave functions. If parity symmetry is violated, both electronic orbitals and configuration weight coefficients contain PNC components. Time-reversal symmetry guarantees that the radial wave functions of the large and small components of the Dirac orbital differ in phase by &w/2, and the parity-conserving (PC) and PNC components of the Dirac orbital differ in phase by f7r/2 as well. In addition, time-reversal symmetry implies that the relative phases between PC configuration weight coefficients are 0 or K, while the relative phases between PC and PNC configuration weight coefficients are f7r/2. The Lee-Yang theorem [3] arises naturally as a consequence of the preceding assertions. In addition, the Kramers theorem [22] is elucidated in the relativistic context.

In Sec. III, we explore the implications of parity and time-reversal symmetries in the photon field. In the multipole expansion of the photon field, the relative phases between the expansion coefficients for the transverse electric multipole potentials and for the magnetic or longitudinal electric multipole potentials are &r/2, provided time-reversal symmetry holds.

In Sec. IV, we explore the implications of parity and time-reversal symmetries

i n

atomic transitions. Parity symmetry leads to the parity selection rule for atomic transitions. If parity symmetry is violated, however, transitions between two atomic states cont,ain both PC and PNC components. The interference of these competing transition amplitudes

VOL. 32 HSIANG-SHUN CHOU AND KEH-NING HUANG 173

depends on their relative phases. We show that the relative phases between competing transition amplitudes are 0 or K, which leads to constructive or destructive interferences between competing atomic transitions. Conclusions are made in Sec. V. In the Appendix, we present the parity and time-reversal operators for the coupled electron-positron and photon fields.

II. IMPLICATIONS OF PARITY AND T I M E - R E V E R S A L S Y M M E T R I E S I N T H E A T O M I C W A V E F U N C T I O N S

II-l. The atomic wave function

The atomic wave function Q may be expressed as a linear combination of configuration wave functions. If the atomic Hamiltonian does not conserve parity, the wave function contains configurations of both parities:

(2.1)

Q a

where o and 6 are the configuration indices for the PC and PNC configurations, respectively. The configuration wave functions are built up from Dirac orbitals. We can construct the Dirac orbital as a simultaneous eigenfunction of

HD, Ii', J2 ,j,,

and Z2 with eigenvalues E, -6, j(j + l), m, a n d l/2(1/2 + l), respectively, where

HD

is the one-electron Dirac Hamiltonian for a central field, and li is defined as

(2.2) Here y” and c’ are the Dirac matrices

y” =

(

)

1 0 0 -1 ’ IL ;; ) ( )

(2.3)

(2.4)

w h e r e 5 are the Pauli matrices, and the unit entries in y” represent 2 x 2 u n i t m a t r i c e s . Relativistic units are employed in this paper. The total angular-momentum operator 7 is given by

I=

i+z,

where the orbital and spin angular-momentum operators are defined as

i=efi,

P-5)

(2.6)

----174 IMPLICATIONS OF PARITY AND TIME-REVERSAL SYMMETRIES VOL. 32

l-s’=

--c.

2

(2.7)

For a definite K., the total momentum quantum number j and the orbital angular-momentum quantum number I of the large component, which determines the parity of the Dirac orbital, are given as

j = IKI - + )

(2.8)

l=

“-JE-1

_i

_>

_n-co.

K

>o,

(2.9)

Dirac orbitals with definite Ercrn take the explicit form

UC,, = ( ;;;;:;m ) , (2.10)

Here G,, and F,, are the radial wave functions of the large and small components. The angular functions R,, in Eq. (2.10) are normalized spherical spinors defined as

(2.11)

where Y!M is the spherical harmonics, and X, is the spin eigenfunction with s = l/2 and s, = p, given, for example, by the two-component Pauli spinor.

II-Z. The parity and time-reversal symmetries

It can be shown that the parity operator for the electron-positron field is given as (see the Appendix)

P, = yOP, (2.12)

where P is the inversion operator for the spatial coordinates. For an N-electron atom, the parity operator is given as

(2.13)

j=l

where j is the index for the j-th electron. It follows from Eqs. (2.10) and (2.12) that the parity of the Dirac orbital is (-)‘. It is interesting to note that because of y” parity symmetry requires the large and small components of the Dirac orbital to be of opposite parities in the spatial coordinates. If parity symmetry is violated, a Dirac orbital c o n t a i n s a PNC component:

VOL. 32

where

HSIANG-SHUN CHOU AND KEH-NING

UPNC _

E)Srn - ( ;$;rn ) .

Of course the designation of an orbital V, by the quantum somewhat artificial since it contains components of opposite p a r a g r a p h s w e a d o p t t h e c o n v e n t i o n t h a t a n o r b i t a l v, is n u m b e r s CY E (E,, IC,, m,) of its PC component.

HUANG 175

(2.15)

n u m b e r s Ed, K,, and m, is values of K. In the following referred to by the quantum

The time-reversal operator for the electron-positron field is given as (see the Ap-pendix)

T, = iy’T3CT, (2.16)

where C is the complex conjugate operator, and T stands for the inversion operator for the temporal coordinate. For an N-electron atom, the time-reversal operator is given as

N

Ta =

iN

n

y,‘yQCjT,. (2.17)

j=l

It follows from Eq. (2.16) that the time-reversed state of the Dirac orbital u,,, (2.14) is given as

(2.18)

where we have made use of the identities

(jlmlj2m2lj3m3) = (-)j1+jz-j3 (‘jl - VLlj2 - TfL2lj3 - 77X3) . (2.20)

11-3. Phase relations

Symmetry considerations reveal a phase relation between the Dirac orbital and its time-reversed partner. If the interactions are invariant under time-reversal, the Mamiltonian commutes with the time-reversal operator, i.e.,

[h,Tel =

0. (2.21)

It is also straightforward to show that

178 IMPLICATIONS OF PARITY AND TIME-REVERSAL SYMMETRIES VOL. 32

for atomic states wit,h no degeneracy other than that implied by rotational invariance. Com-parison of Eqs. (2.37) and (2.38) reveals that the relative phases between PC configuration weight coefficients are 0 or r, while the relative phases between PC and PNC configura-tion weight coefficients are &r/2. We may therefore choose the phase for the configuraconfigura-tion weight coefficient such that the PC and PNC components are real and purely imaginary, respectively.

The preceding assertions concerning the relative phases of the configuration weight coefficients may be established from an alternative point of view. According to the vari-ational principle, the expectation value of the atomic Hamiltonian should be stationary with respect to variations in the configuration weight coefficients; this leads to a system of secular equations:

H C = E C . (2.39)

Here H stands for the matrix {HPq} with elements

.

HP, = PPwwIww~)) I (2.40)

and C = { C , } is its eigenvector, whose components in the configuration basis are the configuration weight coefficients. The phases of the matrix elements Hpq are relevant to

time-reversal symmetry.

It is convenient to introduce an anti-unitary operator 0, as:

0, = cJ&(~),

(2.41)

where Ry(7r) stands for the rotation operator associated with a rotation through an angle 7r about the y-axis. For an ,V-electron system, we can define

0, = fi 0,j. (2.42)

J=l

By making use of the identity (2.20) and

KM@ - Q,r - 4) = (-)‘+%-M(Q, 4)) (2.43)

%W, =

(

-)p+1’2x_p

) (2.44)

it follows that

R&-)tk,, =

(-)J+m?lc,_, .

Consequently, Eqs. (2.18) and (2.45) yield oezi,,, = (-)1+%,,, .

(2.45)

(2.46)

VOL.32 HSIANG-SHUN CHOU AND KEH-NING HUANG 179

For an N-electron system, we have therefore

w&(JM) =

(-)N”4T,$p14) i

(2.47)

If the interactions are invariant under rotations and time-reversal, the atomic Hamiltonian H commutes with 0,. Accordingly, it follows from Eq. (2.47) that

HP, = (%(JWWIWW

= (~~(JM)lohHo,l~l,(JM))

= "p~P(~'p(JfM)IHl~~(jM))+ (2.48)

=

_{vQHp*q *}

Inspection of Eq. (2.48) hs ows that Hap and H,p are real, while H,p and H,p are purely imaginary.

It is convenient to define a matrix S = {S,,} with

i

1,

p=q=a,

S,, = i, p=q=cy,

.

0,

P#C

The similarity transformation

(2.49)

H'= SHS-'

(2.50)

will convert

H

to a real symmetric matrix. In the meantime, the eigenvectors transform according t%,

C’=

SC. (2.51)

It can be shown that the components of the eigenvectors of a real symmetric matrix differ in phase by 0 or n; this leads to the preceding assertions concerning the relative phases of the configuration weight coefficients.

11-4. The Lee-Yang theorem

As an application of the preceding assertions concerning the properties of the atomic wave functions, an alternative demonstration is given for the Lee-Yang theorem [3]. The permanent electric dipole moment. of an atom is given by

(WI‘E) 7 (2.52) where N r’= Ti,_c t=l (2.53) ____L I , .

180 IMPLICATIONS OF PARITY AND TIME-REVERSAL SYMMETRIES VOL. 32

and ‘J! is the atomic wave function in the absence of the electric field.

For an atomic state with magnetic quantum number M, the only nonvanishing com-ponent of the electric dipole moment is along the symmetry axis,

NJI~PI) > (2.54)

where the spatial coordinate operator r’has r’ = zz = y3 + zk

= z+g+z.

A justification of the above assertion goes as follows. A rotation through an angle T about the z-axis will convert the atomic state !P to 9’. Here

Q’ = R,(x)\lr

= e -iMT

(2.56) qI,

where R,(T) stands for the rotation operat,or. Accordingly, .

(!qZpq = (~lRr(~)RZ(~)~R~(~)~*(~)~*(~)~~

= -(qFpIJ) (2.57)

= _6:

where we have made use of the unitarity of the rotation operator and

{R&r), Z} = 0. (2.58)

Similarly, the y-component of the electric dipole moment vanishes as well. Consequently, axial symmetry permits of merely the electric dipole moment along the z-axis.

Moreover, an atomic state has a definite parity, provided parity symmetry holds. Accordingly, insertion of PJPa = 1 into Eq. (2.54) yields

(!P~z’~!P) = (!qP,tPaZP,tPal@)

= -(!qZpq (2.59)

1

0,

where we have made use of the identity

{P&Z} = 0. (2.60)

We therefore conclude that parity symmetry alone prohibits the existence of atomic per-manent electric dipole moments for states with axial symmetry.

If the parity symmetry is violated, however, an atomic state no longer has a definite parity. Even so, the atomic permanent electric dipole moment is strictly forbidden, to the extent that time-reversal symmetry holds. A justification of the preceding assertion goes as follows. It follows from Eqs. (2.1) and (2.47) that

VOL.32 HSIANG-SHUN CHOU AND KEH-NING HUANG 181

= (-)N&,

(CC% -

p3cl)

(2.61)

P

a = (-) Nu&, XP

where x, is the parity of the atomic state in the absence of PNC interactions. In arriving the configuration weight coefficient such that

purely imaginary, respectively. Accordingly, at Eq. (2.61), we have chosen the phase of

the PC and PNC components are real and insertion of @LO, = 1 into Eq. (2.54) yields

(*]z’/q) = (~]o~o,z@;o,]~) = -(Q/z]*)’

= --(Q]z]*)

.

= _0,

where we have made use of the identity

{O,,Z} = 0)

(2.62)

(2.63) Consequently, we arrive a.t the important result that time-reversal symmetry alone pro-hibits the existence of permanent electric dipole moments of atoms with definite angular momentum. Accordingly, a simultaneous breakdown of parity and time-reversal symme-tries will manifest itself as permanent electric dipole moments of atoms with definite angular momentum, which will give rise to the linear Stark effect in atoms [13-151.

11-5. The Kramers degeneracy

Another consequence of time-reversal symmetry is the Kramers degeneracy [22]. Al-though this is a well-known fact, for completeness we elucidate the Kramers theorem in the present relativistic formalism. It is evident that the energy eigenstate X@ and its time-reversed state T,Q are of the same energy eigenvalue, provided time-reversal symmetry holds. If !IJ and T,Q’ stood for the same state, they would differ at most by a phase factor, namely,

T,IE = eis’D. (2.64)

A repeated application of T, would yield

T,2’$=fL (2.65)

Nevertheless, this relation never holds for atoms with odd number of electrons, since it follows from Eq. (2.17) that

182 IMPLICATIONS OF PARITY AND TIME-REVERSAL SYMMETRIES VOL. 32

T,2 =

(-)“; .

(2.66)

Consequently,

T,2Q = -‘$, (2.67)

for atoms with odd number of electrons. It is crucial to note that the result is independent of the phase convention of the time-reversal operator. As a result, there is an inherent two-fold degeneracy for atoms with odd number of electrons, provided that there are no magnetic fields present to remove time-reversal symmetry. As a matter of fact, any system with odd number of electrons shares this attribute.

III. IMPLICATIONS OF PARITY AND TIME-REVERSAL SYMMETRIES IN THE PHOTON FIELD

. 111-I. The photon field

The photon field is described by the four-potential

A” = (c&/i).

_(3.1)

Its components satisfy the Maxwell equation

V2Ap _ - =a2Ap 0

at2 ’

with the subsidiary condition

(3.2)

v.i+*=o

_at

_’ (3.3)

Consider the photon field with the harmonic time-dependence exp(-iwt) such that

V2AP + w’A” = 0 , (34

with the subsidiary condition

V.,$--ik4=0, (3.5)

where k = LJ in the relativistic unit.s. The solutions of Eq. (3.4) can be expressed in terms of mult.ipole potentials:

(3.6)

VOL.32 HSIANG-SHUN CHOU AND KEH-NING HUANG 183

where Cj;l! are expansion coefficients depending on the specific form of A’. The scalar potential Qjm is

(3.8)

and the two transverse and one longitudinal multipole potentials are

it,-1) _

1m -

-$oojm,

where j,(kr) is the spherical Bessel function, and L’ = -iFx V. These potentials are normalized such that

Jd3?$;,,, . c#+,, = +(k’ - k)63+,,~m ,

with X = 0 or 51. Further manipulation leads to the expressions [24]:

i(j) = ij

2

If2

1m

( )

_7r

3j(kT)Fjjm(F)

7 A’(.4 =

1m iJ_’

(a)

1’2 [

(&)

1’2ji-l(kT)~(l-,)m(i)

112

-1

.ij+l(kT)p~(j+l)m(r)] 7 where PI,,,, are the vector spherical harmonics defined as

q=O,il

Here E, are spherical unit vectors defined in terms of Cartesian unit vectors by

(3.9)

(3.10) (3.11) (3.12) (3.13) (3.14) (3.15) (3.16) (3.17)

184 IMPLICATIONS OF PARITY AND TIME-REVERSAL SYMMETRIES VOL. 32

6+1 =

-wdm, +

_ii,) )

eo =

6= , (3.18)

P-1 = _{(U&)(6, - @,).}

111-2. The parity and time-reversal symmetries

It can be shown that the parity operator for the photon field is given as (see the Appendix) Py = (gyp = i 1 0 0 0 o-1 0 0 o o-1 0

1

p ’ 0 0 0 - 1 (3.19) and .

_{wjm =}

_(+$jm

) (3.20) (3.21) Consequently, the photon states of definite angular momentum fall into two types of oppo-site parities:

A& =

A;&IJ) + A&#3),

where the magnetic and electric multipole four-potentials are

(3.22)

A” (E) =

(C(-‘)&,&(‘)~(.‘) +

c(-‘)x(.-‘1)

.lm .lm .lm .Jm .Jm &lm ’

(3.23) (3.24)

which have parities (-)j+i and (-)j, respectively. Note that the magnetic multipole four-potential in (3.23) includes only the transverse multipole four-potential, while the electric mul-tipole four-potential in (3.24) includes the scalar, transverse, and longitudinal mulmul-tipole potentials.

The time-reversal operator for the photon field is given as (see the Appendix) 1 0 0 0

T7 = (gpV)CT = CT. (3.25)

VOL.32 HSIANG-SHUN CHOU AND KEH-NING HUANG 185

0, = TyRy(7r).

BY making use of the identities d& = (-)j+m&-m ) p”m’* = (-) 3+~+ww2J~~m , hd+bn = (-)i+“4,--m ) Ry( +i;xm) = ( -,j+-/qm ) it follows that (3.26) (3.27) (3.28) (3.29) (3.30) (3.31) 111-3. Phase relations

. Symmetry considerations reveal a phase relation between the expansion coefficients for multipole potentials. Consider a photon state AZ, which is the eigenfunction of the momentum operator p’and helicity operator y

P,,Fl = 9,

@‘/IT. $1. It is straightforward to show that (3.32) (3.33) where we have chosen the quantization axis in the z direction. We have therefore

O,Azq = eioA& , (3.34)

because a photon state is completely specified (3.34) shows that

by z and q. Comparison of Eqs. (3.31) and

(3.35) Consequently, the relative phases between the expansion coefficients Cjz for the transverse electric multipole potentials and ($2 for the magnetic multipole potential or C,‘;‘) for the longitudinal electric multipole potentials are &n/2. Again this is a well-known fact in classical electrodynamics; however, it arises naturally as a consequence of time-reversal symmetry.

1 8 6 IMPLICATIONS

IV. IMPLICATIONS

OF PARITY AND TIME-REVERSAL SYMMETRIES VOL. 32

OF PARITY AND TIME-REVERSAL SYMMETRIES IN ATOMIC TRANSITIONS

The transition matrix elements of atoms in the presence of the photon field A’“ , take

the form

kq

where 9; and 91f are the initial and final states of the atoms, respectively. E q s . (3.6), (3.7), and (3.22)-(3.24) into Eq. (4.1), we can express the

elements as

Tji =

C[Tfi(lQ) +

Tji(JJj)] >

(4.1)

After substituting transition matrix (4.2) . where Tji(Ej) = 5 Ti?pkAFm(E) k=l and Tji(Alj) =

are the electric and magnetic 2J-pole transition matrix elements, respectively. H e r e

is =

r”r’.

(4.5)

Parity symmetry requires

7rjiT;7rr = 1 , (4.6)

where xy refers to the parity of the photon field. This leads to the well-known parity selection rules for the multipole transitions:

VOL.32 HSIANG-SHUN CHOU AND KEH-KING HUANG 187

where X = 0 or 1 correspond to the magnetic or electric 2J-pole transitions, respectively. Transitions between two atomic states may contain competing components. Parity symme-try permits of mixed transition which contains both magnetic 2j-r-pole and electric 2j-pole components. If parity symmetry is violated, an atomic state no longer has a definite parity. Therefore, transitions between two atomic states contain both PC and PNC components. PNC interactions in atoms may be studied by investigating the interference between mag-netic dipole and PNC electric dipole transitions, which gives rise to the optical rotation effect in heavy atoms [16]. An alternative proposal is to investigate the electric dipole-electric quadrupole interference in muonic atoms, which leads to angular asymmetries of the emmitted photons [25].

The interference between competing transition amplitudes depends on their relative phases. The phase relations of the competing transition amplitudes have been explored by Lloyd [26] and Bouchiat and Bouchiat [27]. An alternative demonstration is given here in the relativistic context. Insertion of @LO, = 1 into Eqs. (4.3) a.nd (4.4) yields

q;(R) =

Tfi(“j) =

=

(4.8)

(4.9)

where we have made use of Eq. (3.35) and the identities

{YP,Y) = W” ,

(4.10)

(4.12) Comparison between Eqs. (4.8) and (4.9) hs ows that the relative phases between competing transition amplitudes are 0 or 7r, which leads to constructive or destructive interferences between competing atomic transitions.

V. CONCLUSIONS

We have investigated some aspects of the implications of parity and time-reversal symmetries in atoms. If parity symmetry is violated, an atomic state no longer has a definite

--1 8 8 IMPLICATIONS OF PARITY AND TIME-REVERSAL SYMMETRIES VOL. 32

parity such that both electronic orbit& and configuration weight coefficients contain

P N C

components. Time-reversal symmetry guarantees that the radial wave functions of the large and small components of the Dirac orbital differ in phase by f7r/2, and the PC and PNC components of the Dirac orbital differ in phase by &r/2 as well. In addition, time-reversal symmetry implies that the relative phases between PC configuration weight coefficients are 0 or K, while the relative phases between PC and PNC configuration weight coefficients are &r/2. The Lee-Yang theorem arises naturally as a consequence of the preceding assertions. In addition, the Kramers theorem is elucidated in the relativistic context. In the multipole expansion of the photon field, the relative phases between the expansion coefficients for the transverse electric multipole potentials and for the magnetic or longitudinal electric multipole potentials are f7r/2. Finally, we show that the relative phases between competing transition amplitudes are 0 or K, which leads to constructive or destructive interferences between competing atomic transitions. Analogous considerations may also be applied to collision, multiphoton, and Auger processes.

.

A C K N O W L E D G M E N T S

One of the authors (Chou) would like to thank Dr. W.-Y. Cheng for invaluable comments and suggest,ions. This work was supported in part by the National Science Council of the Republic of China under Grant Nos. 0208-M019-002 and NSC83-0208-MOOl-007.

A P P E N D I X : P A R I T Y A N D T I M E - R E V E R S A L T R A N S F O R M A T I O N S O F T H E C O U P L E D E L E C T R O N - P O S I T R O N A N D P H O T O N F I E L D S

The equations of motion for the coupled electron-positron and photon fields are the Dirac and Maxwell equations

(iypi?a, - m)u(s) = -eypA,(z)u(z),

(Al)

dvd“Ap(z) -

d,apA"(z)

=

-eti(s)y@u(z),

w4

where u(z) and Afi( )z are the electron-positron and photon fields, respectively.

The space-time coordinates transform under the parity and time-reversal transfor-mations according to

(t, q

5

(t’,?) = (t, 4) ) 643)

(6 q

5

(t’,S) = (-t,q.

(A41

VOL.32 HSIANG-SHUN CHOU AND KEH-NING HUANG 189

The most general homogeneous Lorentz transformation between two coordinate systems is a linear transformation

Xf/J = a!J ”

.x

7 (A5)

with

apv

gppaPc = svo .

(A61

In Eq. (A6) gPy is the metric tensor with components 900 = -911 = -922 = -933 = 1,

(A7) 9PV = 0 f o r p # V.

Suppose the interaction is invariant under the Lorentz transformation, there is a transfor-mation law for the field quantities, so that the transformed fields ~‘(2’) and A”l(x’) satisfy the same equations of motion in the new coordinate system; i. e.,

(iypt$ - m)u’(x’) = -eypA;(x’)u’(x’), (A@

.

8;8vA’p(x’) - ,;~‘A’“(x’) = -eii(~‘)y~u’(x’), (W

where

u’(x’) =

Au(x) )

WO)

A’Q(x’) = Lap Ap(x).

_(All)

Furthermore, relativistic invariance requires the transformations to preserve the absolute values of scalar products between two states. Consequently, the transformation operators must be unitary or antiunitary. We first assume that both transformations (AlO) and (All) are unitary transformations. Substitution of Eqs. (AlO) and (All) into Eqs. (A8) and (A9) yields

(iap r”d, - m)Av(x) =

-eLpp ywAp(~)Au(~),

w4

&PLp~ Ap(x) - a/ upp d,@L”, A?(x) = -eu+(x)A+yOyPA~(x). (AI3) In addition, multiplication of Eqs. (Al) and (A2) from the left by A and Lpp yields

(iA-/“& - mA)u(x) = -eAyPA&)u(z), (AId)

t3,EYLp~ AP(x) - i3,@Lpp A?(x) = -ev+(x)L’o “f”ypu(x). VW

190 IMPLICATIONS OF PARITY AND TIME-REVERSAL’SYMMETRIES VOL. 32

Comparing Eqs. (A14) and (Al5) with Eqs. (A12) and (A13) shows that the coupled Dirac and Maxwell equations are invariant under the Lorentz transformations, provided unitary operators A and L can be found which ha,ve the properties

LPfl

y”yp = A+,OyL”A

7

(-w

(AW

These imply

Lap = a”p , (-419)

h+y’A = y” _, (AW

.

AtyOy”A = a”D y”yP . W1)

Whereas if both transformations (AlO) and (All) are anti-unitary transformations, the operators A and L can be written in the forms

A=%, _(A22)

_

L”fl = L”pC, _(A23)

where A and z are unitary operators, and C stands for the complex conjugate operator. Substitution of Eqs. (A22) and (A23) into Eqs. (A8) and (A9) yields

(iap r@‘d, - m)h~*(~) = --eyP’ZPP A;(s)ilu*(z) , (AW

cY,6’“E”~

A’*(z) - a,* a’$ &@I”, A-‘*(z) =

-ezl+*(z)y’*il+y~ilzl*(s) .

_(A25)

Multiplication of Eqs. (Al) and (A2) from the left by AC and -& C yields

-A(iy’*d, +

m).u*(z) = -enrp*A;(z)u*(z), (A26) ava”Z”p A’*(Z) - 3,@L”p A’*(z) = -eL@p ,t*(,)yO*yfl*,*(,). (A27) Comparing Eqs. (A26) and (A27) with Eqs. (A24) and (A25) shows that

gY” IPp = a,” a”0 I”, , j+yp yo*y8* = pyO*yP‘~ )

(A281 C-429)

VOL. 32 HSIANG-SHUN CHOU AND KEH-NING HUANG &pi\- _{= -a, 7}P a = L,fl ya . These imply L”p = -aa0 , QyQ~ = +’ ,

li_+7u7aA = _a”p 7u’7P*

191

ww

WV

(A32) (A33) It can be shown that ,~+A is positive definite; therefore its trace is greater than zero

tr {AtA} > 0. (A34

For unitary transformations the following identities hold:

A+A = AtyoyoA = sop y"yp , _(A35)

. tr {A+A} = 4noo . (A36)

Inspection of Eqs. (A34) and (A36) reveals that

a00 > 0. _(A37)

Whereas for antiunitary transformations, the analogues of Eq. (A35) and (A36) are

A+A = A+y”yoA = -sop y’*yfl* , _(A38)

tr {AtA} = -4a’u. _ww

Inspection of Eqs. (A34) and (A39) reveals that

a00 < 0. _(A40)

In conclusion, under the orthchronous Lorentz transformations, the electron-positron and the photon fields transform as

r/(x’) = Au(z) , OW

A’a(z’) = a”0 A’(Z).

_(A4‘4

The unitary operator _A in Eq. (A41) satisfies the relations (A20) and (A21). One particular orthchronous transformation is the parity transformation (A3). The parity operator for the electron-positron field is given as

192 IMPLICATIONS OF PARITY AND TIME-REVERSAL SYMMETRIES VOL. 32

P, = ei@-yOP ? (Ads)

where P is the inversion operator for the spatial coordinates,

Pu(t,Z) = up, 4). (Jw

The quantity e’‘$ in Eq. (A43) is an arbitrary phase factor. It is customary to set this phase factor equal to 1. The parity operator for the photon field is given by

1 0 0 0

Py = (gyp = (A45)

On the other hand, under the antichronous Lorentz transformations, the electron-positron and photon fields transform as

?+‘) = Au*(,) ) (.446)

.

A’C’(d) = -u”~ A@*(.). (447)

The unitary operator A in Eq. (A46) satisfies the relations (A32) and (A33). One particular antichronous transformation is the time-reversal transformation (A4). The time-reversal operator for the electron-positron field is

T, = ez6y’y3CT, (A48j

where T is the inversion operator for the temporal coordinate?

Tu(t,Z) = u(-t,?). (-449)

The quantity e” in Eq. (-448) is again an arbitrary phase factor. It is customary to set, this phase factor equal to i. The time-reversal operator for the photon field is given as

1 0 0 0

T7 = (gp”)CT = CT. (A50)

R E F E R E N C E S

* Present address: Division of General Education, National Taiwan Ocean Universit,y, Keelung, Taiwan 202, R.O.C.

VOL. 32 HSIANG-SHUN CHOU AND KEH-NING HUANG 193

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