Covariant Polarization Analysis of Spin 1 Particles

Covariant Polarization Analysis of Spin 1 ParticlesAuthor(s): Arif-Uz-ZamanSource: Proceedings of the Royal Society of London. Series A, Mathematical and PhysicalSciences, Vol. 268, No. 1334 (Jul. 24, 1962), pp. 371-389Published by: The Royal SocietyStable URL: http://www.jstor.org/stable/2414153 .

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Covariant polarization analysis of spin 1 particles

BY ARIF-UZ-ZAMAN

Department of Physics, Imperial College, London

(Communicated by A. Salam, F.R.S.-Received 30 January 1962)

A covariant theory of polarization analysis of spin 1 particles based on the Duffin-Kemmer equation is developed. The 16-dimensional covariant scattering equation is reduced to the ordinary three-dimensional scattering equation. Lorentz transformations and space rotations giving rise to relativistic corrections (when one applies the ordinary scattering theory to multiple scattering experiments) are obtained. These are found to be the same as those obtained by Stapp for spin I particles. The spin and energy projection operators for the Duffin-Kemmer equation are obtained in appendix A and a covariaflt density matrix for spin 1 'particles' is defined in appendix B.

1. INTRODUCTION

A covariant formulation of the polarization analysis of the scattering of spin - particles has been developed by Stapp (I956). The fundamental equation describing the final density matrix p' in terms of the S matrix and the initial density matrix p, that is p'tr p' = SpSt tr p,

has been obtained in a covariant form and it has been shown that this equation containing 4 x 4 Dirac matrices can be reduced to the usual non-covariant two- dimensional form in which p and p' are given in the rest frames of the initial and final particles respectively and the S matrix is given in the centre-of-mass frame. The Lorentz transformations which connect the two forms give rise to certain relativistic corrections which have to be applied when one uses the usual non-covariant scattering theory.

The present work is an attempt at obtaining a covariant polarization analysis of the scattering of spin 1 particles by spinless targets. Starting from a 16-dimensional representation of the Duffin-Kemmer equation a scattering theory is developed. The invariant spin projection operator is introduced which selects the spin 1 component and the covariant scattering equation is reduced into a three- dimensional form which can be compared with the equations of the non-covariant polarization analysis of spin 1 particles developed by Lakin (I954), Biedenhern (I958), Stapp (I957) and others.

The 16-dimensional representation of the Duffin-Kemmer equation for a free particle of non-vanishing mass 'in' Roman (i960)

{i1?( -i + n)(x) = 0 (1)

with pa,t satisfying the relation

lVEA1+ AA fa - I= + ?,, (1.2)

contains a spin 1 particle and a spin 0 particle. The 16-dimensional representation can be broken up into one 10 dimensional, one five-dimensional (describing particles

24 [ 371 ] Vol. 268. A.



372 Arif-Uz-Zaman

of spin 1 and spin 0, respectively) and one one-dimensional representation which is the trivial component. On account of the difficulties associated with the algebra of 10-dimensional /? matrices obeying (1.2), the following 16-dimensional representation a

= -(I x yzii + y,, x l ) (1.3)

is chosen here even though it contains both the spin 1 and spin 0 particles. The Dirac matrices , are all Hermitian and x stands for the direct or Kronecker product. For ready reference the following well-known properties of such products are given:

A. x B1A2 x B2= A1,A2 x B1B2, ( l4)

(A xB) = AO xxBO, (1 4')

where AO denotes the operation of Hermitian conjugation A * or inversion A-' (if it exists) or the transpose AT.

trA x B = trAtrB. (1.4")

Following Stapp (I956) we define the following quantities

Y(u) = ( ' (1.5)

y(it,W) U( 1 W L)' (1 5')

U.U11+ IW.W I

,8(u) (- u l, (1.5") (I~~~~~~~~~~~~~(

5 /)

with the convention that in (u. u)i- positive or positive imaginary sign is to be taken. For any two 4-vectors u and w,

(u)y (u) = 'Y(u, w)y (u, w) 1, (16)

yr(u)y (u, w) = Y(uw) (w). (1. 6')

The 16-dimensional Kemmer wave function 0 may be looked upon as a spinor of rank 2 which transforms under Lorentz transformations as a direct product of two Dirac spinors. If a Dirac spinor transforms under a Lorentz transformation as

(Belifante I939) Vf -?L> fl(. (1.7)

Then 0 transforms as -fl La sLf 067_ (LqLT)a4

-(L x LO),,

-(LK 0),4(1.8) The Lorentz transformation defined by (see Stapp I956)

x12 1lo(t) = xV(t = 0), (1-9)

XI = Y, V(t)XV(t = 0), (19')

where x,(t = 0) is the value of x, in the Lorentz frame in which the space part of 4-momentum t vanishes, is represented in the Dirac space by L(t), with the properties

L(t)yL-1(t) = 2'Y(t)yV, (1.10)

L-l(t)y7pL(t) = ySv(t). (1 - I10')



Covariant polarization analysis of spin 1 particles 373

Let f and f' denote the initial and final 4-momenta of the particle and t, the 4- momentum of the centre of mass. Stapp (1956) has shown that

7(t1)y(t:L,f1) =L(f1,(11 ,Y(fl , tl),y(t1) = L-1(f),(11'

L(t) y(t) y(x, t) L-1(t) = y(t,) y(x,, t,), (1.12)

L-1(t)y (t,) y(x,, t,) L(t) = y(t) y(x, t), (P112')

where the subscript 1 denotes the values in the centre-of-mass frame given by t = 0, x is eitherf orf'. The Lorentz operators in the space of spinors of rank 2 are given by (1 -8). y(u) and y(u , w) for time like u and w are self-adjoint

)yt('u) -Y4Y*(U)Y4 = 7(u), (1.13)

yt(uw) -747 (U5,w)y4 = /(Uw). (1.13')

Also P = '4L*Y4 = L'. (1.14)

In the space of spinors of rank 2, the adjoint is defined by

qt -0*4 Xqy 4 - 04, (1.15)

rt =y4* (1.15')

2. THE S-MATRIX

The 16-dimensional S matrix can be written down in terms of a sum of Kronecker products of two 4 x 4 matrices

, _ X Y(2); (2.1) 1=1.

s is some finite integer. Even though the S and p matrices will be multiplied by the spin-projection operator 0(1) to select spin 1 part of the matrices only,J it is found that the calculations are much simplified if these matrices obey a certain symmetry condition which arises from the fact that solutions of Kemmer equation describing spin 1 particles are symmetric and those describing spinless particles are anti- symmetric. This means that for spin 1 particles (Belifante I939)

Ua(f)_ U(f) (2.2)

and for spinless particles Ub(f) =-U (f). (2.2')

The transition probability is proportional to

]Uoat,(f ) o 3(f ', t, 0, 3(f)1 2, (2 3)

where S is a matrix element in the momentum space and a matrix in the space of spinors of rank 2. For symmetric U(f ) and U'(f ) we can assume without loss of generality that S satisfies S X;,B' (2.4)

=sxx,;,8fl (2.4')

Sala; fA dt (2.4")

: The projection operators yi (f ), y0(f ), 0(1)(f ), 00(f ), z? (f) z0(f) are derived in appendix A of this paper.

24-2



374 Arif-Uz-Zaman

Writing t9() =Al+ iB1, (2.5)

19(2) = Al-iBi, (2.5')

we have from (2.1) S = .(A, xAAl+B xBl)+i(BtxAl-A, xB1). (2.6)

The first term within the brackets above obeys the symmetry condition (2.4") while the second terms obeys the same type of symmetry condition with a negative sign. Therefore the S matrix can be written in the form

S(f', tf) = (f',tf) X (',tf)

ESI(f',t,f). (2-7)

Consider a collision process in which the initial and final spin 1 particles (of 4- momentafandf',respectively) are either 'particles' or 'antiparticles' and the target and recoil particles are of spin zero. The S matrix for such a process has only one term in its idempotent expansion in terms of y- and yO. That is, S is either given by

(f', tf) = y+(f')S(f',tf)#+(f) (2 8)

for particles, or by S(f', t,f) =-(f') S(f', t,f) -(f) (2d8')

for antiparticles. By (A 14), S satisfies in both of these cases the condition

8(f')S(f',tf),8(f) = S(f',tf). (2.9)

This corresponds to Stapp's (1956) 'Hole theory condition' for Dirac particles. It is found that the calculations are much simplified if we utilize another condition on the S matrix. This follows from the application of (A 22) on (2 8') and (2 8)

#,(f')S(f',tf) = S(f',tf), (2X10)

S(f',tf)(f') = S(f',t,f). (2.10')

Putting in the expression (2 7) for S and going over to the representation (1.3), we can write down (2.9) in the form

9Y(f',tf) x9 %(f',tf) = 1{ Ey{(f')>ry(f) x?+y(f')9Y'x9? y(f) l l

+ 95^y(f ) x y(f') 9A+ 9Ax y(f') Sl'y(f)} (2 1 1) and (2-10) and (2.10') as

/x') X y(f)xY5f E XMI, (2.12)

E YX Yry(f) xM E7 = E Xx -97 (2312') 1 1

Let k and k' denote the initial and final relative 4-nmomenta. With Stapp we define a quantityYI'(k', t, k) by

) (f', t)9'(k', t l) y(f, t). (2.13)




Using (16) and (1.6'), we can write (2.11), (2.12) and (2.12') in terms of 9,'(k', t, k).

S'(k', t, k) E9' x = X {y(t)91y(t) xY J +y(t)59" x 9Yy(t) 1 a

+?9y(t) x ( x y(t)97(t)}, (2.14)

Aoy(t) x y(t)Y x9y" = EX x 9", (2.15) l l

1i X Y"Y(t) x Y(t) _ X Y. (2115')

By multiplying the last two equations by 1 x y(t), y(t) x 1, etc., one can easily show that the four terms on the right-hand side of (2.14) are equal hence we can write (2.14) in the form

(WI9 - y(t)l "y(t)) x J = 0. (2.16)

Assuming that each term in the summation above vanishes we can put the following necessary conditions on9'I

y(t)Y'(k', t, k) y(t) = Y(k', t, k). (2.17)

Stapp (I956) has shown that the covariant matrix Y9 satisfying (2. 17) is of the form

(k%', t, k) = A+(t) (FJ+ + iGiy5y. n') + A-(t) (FI + ?iy5y.n'), (2-18)

where n' is the unit pseudo-vector which has the properties

n'.t = n'.k = n'.k' = (1-n'.n') = 0 (2.19)

and reduces in the centre-of-mass frame to the three-dimensional vector

'-'N '-tkAk' 1~~~~~~ IkA kI ) (2 20)

However, if we take the above form for tYJ(k', t, k), we find that the S matrix does not contain a large number of terms which it possibly could. The reason for this is that while applying the condition (2.17) to the expansion (dropping 1 for the moment)

5Y(k', 3t~k) = A +Baya+ 12Cav Tav + Daui75Y, W75' (2.21)

Stapp has discarded all terms which are not invariant with respect to parity operation. For example, Do, is determined by the condition

D =t0l O (2.22)

and Stapp has selected only the first of the following three independent vectors with this property

D(-) = NMDM(- i) ev^AXkvk'to- = D(I)n,' (2.23)

D12) = N(2)D(2)( -i) e, Itxkvn0tc = (2(223)

D(3) - N(3)D(3)(-i)e ,, k'n, t' = .D(3)n/[, (2.23")

normalized in such a way that

D5i)D5M ) = D(1WDWi) (i = 1,or 2 or 3). (2.24)



376 Arif-Uz-Zaman

In the present case however, the S matrix contains the Kronecker squares of 5J and the terms non-invariant with respect to space reflexion in $4 will give rise to parity conserving as well as parity non-conserving terms in S. Writing D,1 as a linear combination of four vectors n', n' and n', we have

t D(1)nu + D(2)n + D(3)n.. (2.25)

Again C,, -CV, is determined by

CmtV= 0, (2.26)

C,V can be expanded in terms of four independent vectors tl, k,), lo' and n' instead of in terms of k1o, k> tu alone as has been done for the case of Dirac particles. Such an expansion satisfying (2.26) is

rn2 - m/22 - J2 = ae kkkv k t + (t 1C c t kfl)- (t1 kv-k;ltv)

fkdkttk~tII-i + d (ni. knv k/ +e(nz k'-nv Okf)- (n"/I V - V I It /. t el (' V t) (2 27)

rn and rn' are the rest masses of the initial and final spin 1 particles and M and 1' are those of the target and recoil particles which are of spin 0. It has also been shown by Stapp that for CQv obeying (2.26)

2 ltvclt, = iY(t)y5ycp,, (2p28)

where cp is given by Cp = 1( -its) 6pO1v/ clt I to t (2.28')

Substituting for Cp from (2.27) it is readily found that ep is given by

c = G(Vn + C(2)np?+ C(3)n.' (2.29)

W and B,1 are of the same form as in Stapp's work, i.e.

W = 0, (2 30)

BA = Bt/11, (2-30f)

Collecting the terms and writing 1 = A+(t) + A-(t), y(t) A+(t) - A-(t), we can put 9?(k', t, k) in the form

=?1E 1&Ai(t) {Fi ? iys~75(H?

no + Ken; + En7')}. (2.31) A,i

It should be noted that on account of (2.22), A'(t) commutes with iy5y,,,n>, etc., and n> n, n/' obeying (2.22) reduce to 3-vectors in the centre-of-mass frame. For the sake of conciseness we define G- nl by

n/ =H, n' + K- n' + El n" (2.32)

so that Y'9(k', t, k) can be written

Yf (k ', t, A) ? (t) (FX + G+ iy5 y. n). (2.33)

Writing Q-' Fl' + Giy5y.n (2.34)




we have Sl(k', t, k) = (A+(t) Qj + A-(t) Qr) x (A+(t) Qt + A-(t) Q+)

= y+(t) Qt+ X Qj+ + -it) Qj x Q- / +?y? (t) Qt+ x QF + (t) Q x Qt. (2.35)

The S matrix must obey (2 8) or (2 8') hence either all the invariants F1, Hi, K , EF vanish giving S' =+(t) Q+ x Q+, (2.36)

or all Fj+, Hl+, Kj+, Ej+ vanish and

q=y-(t)Q- xQ . (2-38)

The matrix S(f', t,f) is given by

S(f', t,f) = E y(f' t) y(t) x y(f', t) y(t) St(k', t, k) . 7(t) y(f, t) x y(t) y(f, t). (2.39) I

The matrix Sl(f', t,f) describing the scattering of spin 1 particles only is obtained by multiplying the matrix given above by O(l(f ) from right and by O(l)(f') from left. Using the results (A 28) and (A 28') we get for S,

S1(f, tf = 'Y(f', t) y(t) X y(f', t) y(t) . O(1)(t) A?(t) E (Fl' + Gl i757 . n)

x A?(t) (F: + G i'y5Y . n) O(1)(t) . y(t) y(f, t) x y(t) y(f, t). (2-40)

Density matrices

A covariant density matrix pl(f) for particles of spin 1 has been defined in appendix B. Writing P1(f) =-O )(f)P(f ) 0(l)(f) (2.41)

and making p(f) obey equation (2.4"), it can be expanded in terms of the Kronecker product form ro

P(f) = E Pr(f ) X Pr(f) (2.41') r=l

for the initial particle. For the final particle we can write

pV(f') = E Pr(f ) x p(f') (2.41") r=O

It should be noted that p(f) is also a covariant matrix because O(')(f ) being Lorentz invariant commutes with all the generators of Lorentz transformations. As these matrices must describe states of particles or of antiparticles only, they satisfy the conditions /(f )P(fW)(f ) = p(fM) (2.42)

y(f)P(f) = P(f)y(f) = p(f)M (2.42')

and as before we can derive from them the necessary condition

Y(f )r(f )Y(f ) = Pr(f ) (2.43) The covariant matrix p,(f ) obeying this equation has been shown by Stapp to be of the form

Pr(f) = YrA+ (f ) (1 + i5 P+p) + Yr -(f ) (1 + i5p7) (244)



378 Arif-Uz-Zaman

where the 4-vectors pi are such that

P1 * f = O. (2.44')

In the same way as for the S matrix the matrix p1(f) describing the states of the particles (or antiparticles) with spin 1 only can be written

PI(I) t= (f [O()(f) E Cr {A?(f) (1 +i'y5Y *pP) r

?A?(f) ( 1 + iy5y7*pl}O? ()(f1)] (2e45)

T is the trace of the whole expression on the right-hand side of trp1(f)/T. If in the rest frame of the particle, pi is denoted by p, p is by (2.44') a 3-vector. The Lorentz invariant quantity T is easily calculated in the rest frame and is found out to be

T - AC'(3 + pi? 'i.), (2.46) r

which is invariant with respect to three-dimensional rotations. The same equations with Cr, f, pi and T replaced by Cr, f', pa and T' hold for the final density matrix

PI'(f).

3. SCATTERING THEORY

Using the abbreviation [A](2) = A x A (3.1)

and substituting the expressions forp'(f'), pl(f), SI and St in

pi(f') = S1(f t,f) P(f) St(f, t,f) (3.2) we obtain

1' trp'f C') Ol)(f') E Cr [(1 + iy5yP? +) A?(f')](2) 0(1) (f' T' tr pl(f ) r

= [y(f', t) y(t)](2) 0(1) (t) E [A?(t) (Fj1 + G?iy5y. n)](2) O(1)(t)

. [y(t) y(f, t)](2) (1/T) O(l)(f) E Cr [A?(f) (1 + iy5y Pr?)](2) 0(l) (f) r

* [y(f, t) 7(t)](2) 0(1) (t) E [A?(t) (E* - + G*iy5y . I'

*O(1)(t) [7(t) y (f t)](2), (3.3)

trp'(f')/trp(f) can be replaced by w, the total differential scattering cross-section. Except for the occurrence of O(1)'s and the Kronecker squares (3 3) is similar to equation (3.2) in Stapp's (I956) work and using the same method it can be cast into the form

(w/T') O(l)(O) E C}.+ [At?(o) (1 + ji5Y. p'?)](2) O(1)(O) r

O(l)(O) E [A?(O) (Fj? + CA iy5y . N)](2) 0(1)(0)

* (]IT) E O(1)(O) Cr[A?(O) (1 +i'y5y. p)](2)o(l)(o) r

Om 0(1(0) E [A (0) (F,+ + G*? i5y Y. N) ](2) 0(l) (0). (3-4) I'




The vectors (Pr,

0), (P', 0) are related to P', Pr respectively through the pure space rotations (Stapp I956) (35

Pi = Pi7yd(fi), (3.5') and the Lorentz transformations

Pr4 = P VY(f'), (3.6)

P, = Pv Yvj(f ) (3.6')

ni,- (N, 0) is the value of n/. in the centre-of-mass frame

4,/t = nVV,(#)- (3 7)

p'?, p?, and n are purely space-like. The rotations 7yi(xl) are defined by

R(x1) ys1?I(x1) = y'sX(x1) Y;, (3. 8) where R(x,) is defined by

R(xl) = L(x1)L(t)Lt(x). (3 8')

xI is the value of the 4-momentum x in the centre-of-mass frame. From now on we specialize to the case of 'particles' only so that only the super-

script + should be taken. Noting that

A+(0) = 2(I+ 74) ( ) 0 (39

i YYP =7iiP ( P 0 )39 i~5Y.P=Y4~.P= G.4

we have A+(0) (1 +iy5y.P) Y (+PP 0 ()-( 2)' (3 10)

A+(0)(Fi+ + GI+?75y.N)= (F 0 0) ( ) ) (3*10')

The matrix element of a matrix operator A between two Kemmer wave functions U and V is VtAU = VtVA4 '

,,. U (3.11)

If A = AlxA2,

VtAU = Vtfl(A1UAT)M, = tr VITA1 UAT

=tr VtTA1 x A2 U. (3.11')

Now consider a quantity X of the form

X [/~~ar+ ? x ar+ ? n1p7t ?\ n7t ?8 [bs+ ? 8 bs+ ? 0

-tErLs -J X J ~ JxV } V s S (3.12)

Writing u11 u12 u13 U14

U21 U22 U23 U24 U1 U2 (3.13) U31 U32 U33 U34 U3 U4

U41 U42 U43 U44 /



380 Arif-Uz-Zaman

we have for the matrix element VtXU

[VtT E a,+ x a,+x nTb+ x b,'+ul 01 VtXU = tr t,r,s

VtT E a x a,+7T1 x irtb+ x b1+u1 ?0 t, r,s

= tr VtT E a+ x ar+irTt x 7Ttb+ x b'+ut tr, s

= VtXre ucec Up. (3.14)

Therefore X operating in the space of 4 x 4 second-rank spinors can be reduced to the form in which it operates in the space of 2 x 2 second-rank spinors. Using the result (3. 14) and equations (3. 10), (3. 10') we can extract the following equation from (3 4)

(w1T) 0(11)(0) , C5r. 7T X t' 0v(1) (0)

r

= 0(1)(0) x gOvl)(0)(1/T) Ov(o)E C?-Tr X Tr OV( ) E? x f '(N0O), I r r

(3.15)

where (o'(0))2 - O(1)(o) - 4(3 + o x o) (3.16)

has also been used. Now consider Preduced given by

PR= (1/T) 4(3 + ? Xi xO ) E Cr7Tr X 7Tr 4 (3 + G', X Gj) r

= I I(T) (3 + o-, X o-) E C, I + (1X U + a X 1) Pr r

3 + 21 E (O'i X O'j + O'j x Oi) E1., iPr, i + E O'i x 'i P2,r i) P. (3 + 0S7XG k-Ac (3 17)

Let us define the spin operator

-2( x c+ asx :L.(3.18)

Then Ji Jj1+J)>i= 2(ax 1j+Oj x ) (i + j) (3.19)

Ji J = (I + o- x o-j) no summation overi (3.19')

and J2JJ = 1(3 +? xa) (3 19 ) Thus, PR can be written

- =+2J.P E(JiJj+JjX)PriPrnj+ E2J!Pri-Pr.Pr) 2 (3.20)

PR~ is of the samne form as PR with Cr and Pr replaced by Cq and P' and the reduced form of S matrix is

5R = 4(3 +, X- x o) EY5 Ix,!l4(3 +S X' x o) (3-21) - 4

= 2E{F+ F+ + 2F,+ I+ J .N + G+ 0+ (JqJj JjJi)NiNj-S+Gl+ I. N 1J2.

(3.22)

It is easily shown that the Ji's obey the three-dimensional Duffin-Kemmer relation

Ji Ji Jk + Jk Jj Ji = 6ij Jk + 6k i ji ((3.23)



Covari ant polarization analysis of spin 1 particles 381

and they also obey the anticommutation relation

JiJh JhJi= 6iklI'J (3.23')

These two relations follow from the usual properties of the Pauli matrices Oi. Equations (3.23) and (3.23') are just the two algebraic relations obeyed by the

three-dimensional matrices 01, 02203 of the spin 1 representation of the three-dimensional rotation group (Roman i960). The carrier space of the Ji's are the spinors

=(U11 1 (3.24) U21 b22!

UT = u and u1T = - u define a three-dimensional and a one-dimensional invariant subspace respectively because uT- + u, leads to (Ji u)T= +- i u. Hence there exists a similarity transformation which reduces all the J2's to a direct sum 6id aci. Since Ji and 6i are traceless matrices we must have ac = 0. J2 is reduced to 602? 0

and since E*2 = 200, where 00 is the three-dimensional unit matrix, the equation (3.15) is reduced by the similarity transformation into the same matrix equation with the J2's replaced by 6i's and unit four-dimensional matrices replaced by unit three-dimensional matrices. Writing

PriPrj = QriQrj+36i3Pr Pr _YQrij +~ 3 Gj Pro P Pr (3.25)

and Tii E60,Xi + j i)- 38ij o0 (3.25')

(2/T) E Cr P P. (3.26) r

(1IT) E CrQrij- = 2 I Ai (3.26') r

we obtain for the reduced equation,

wp' = SOp* (3.27)

with p = (0o+P?+T1Qtj) (3 28)

= -(006+0?.P'+TQtj). (3 28')

The polarization tensor Q jis traceless and trTj = O for all i,j. These density matrices are exactly the same as those obtained by Biedenhern (I958), Stapp (I957) and others from non-relativistic considerations.

The S matrix in (3.17) is of the form

SE{OOF + F= G 1 . N + 2G GITj N Nj + 100 GIG N . NJ, (3.29)

where the superscripts + have been omitted. N is given by

GI N Hi N'+ K1 N" + A N"', (3.30)

with N'=kAk'/IkAk'j, (3.31)

N" kAN'/IkAN'|, (3.31')

N" = k' AN'/I k'AN'|. (3.31")



382 Arif-Uz-Zaman

Hence G1 N can be written

G1N =H1N'+k{Kl Ek Ik k'I

tJklI 1 kI jkAk'j -k'( Jkl + El Ikl I kAk'j (3 32)

_H, N' + Xl k - Xl k. ( 3 32 ) Therefore S can be written in terms of the vector N', k, k' and 6i and T2j as

S = E [(X12 + G2N. N) 00 + ?F1HI O. N + ?I X 1O *k- FX O. k'

+ 2Tj{H2N Nj + HIX1(NY kj + N1 k1) - H X'(N kg + N kE) + X12 k1 kc + X2 ki kJ -XIXl(k1 v + k k)}] (3.33)

In the centre-of-mass frame (4.33) is the most general rotation invariant S matrix that can be formed with the help of the Oi matrices and the initial and final momenta k, k' of the spin 1 particle.

Under time reversal -i -Si, (3?34) k-> -k', (3.34')

'- -k. (3.34") Hence if time reversal invariance holds

E XI HI = -E X'HI, l l

E XF, - -EXIFS l l

EX2 =EX12. (3.35) l l

If only parity conserving terms are present

EFIXI = EFIXl-0O(3.36) l l

E H1XI = E HIX = O. (3.36') l l

The S and p matrices can be expressed in terms of the spherical haromonics YJ(k), etc. and the tensor momenta TM. The form (3.33) of the S matrix with (3.35), (3.36), (3.36') is the same as that given in non-relativistic treatments quoted above.

Comparing equations (3-5) to (3.8') with the corresponding equations in Stapp's (I956) work, shows that the relativistic corrections are exactly the same in the two cases. However, the 'rotational corrections' are to be applied to each index of the polarization tensor Qja. A discussion of these corrections is to be found in Stapp's paper (1956).

I am extremely grateful to Professor Abdus Salam, F.R.S., for suggesting this problem to me and for this continued guidance throughout this work. I wish to thank Dr Kibble for going over the first draft of this paper and for valuable sugges- tions. I am also indebted to Dr I. Saavedra, Mr L. Castell and Mr Y. Neenian for many helpful conversations.



Covariant polarization analysis of spin I particles 383

APPENDIX A. ENERGY AND SPIN PROJECTION OPERATORS

Writing q(x) = (2I)4s(f) eif xd4f (A 1)

and applying (1 1), we get (i,? fa + M)(f) = 0. (A 2)

Let us define the following operators:

yT(f) = _ 2l.2 {i(/3.f)2 + m.f}, (A3)

(f )= _1 2{-2im2- 2i(,.f )2}, (A 3')

so that q+(f ) +y-(f ) +y0?(f) - 1 (A4)

From the Duffin-Kemmer commutation relation (12) it can easily be proved that

(V3.f)3 =f2/3.f, (A5)

i.e. [3(f)]3 = /3(f). (A 5')

From this relation it follows that

f ) -f ) =

2i 2im2

( f)2 (f2 +

m2), (A 6)

(fimqD)D {i(/ *.f )2 + mf.fj}( f2 + m2). (A 7)

In view of (A 6), (A7) and the fact that

-(f) = (iM f+m) 2if 2 (A 8)

the solution of A 2 can be written as

?(f) = 8(f 2 + m2) 3 .fyq+(f) x(f ) + 8(f2 + m2) /3 .fq0(f) x0(f)f (A 9)

where x(f) and Xo(f) are arbitrary spinors of rank 2 in momentum space. Sub- stituting (A 9) into (A 1) one obtains easily

O(x) = I - {eif. x/3.fy+(f) x(f) + e-if x(- -/3 .f) y+(-f) x(-f)

(2-T) 2f0 2wt

+ eif'x7 .fry0(f) x0(f) + e-if X( - / *.f) r7?(-f) Xo(- f)}. (A 10)

The 4-momentum f occurring in this equation lies on the mass shell

f = w= +V(f2+m2) (AI 0')

hence Xf =3(f) =X'f (A10") (f -f)A im

therefore yj (f) and y0(f) can be written

y?(f) = j{[/3(f)]2+?l(f)}, (A l1)

y 0(f)= I-[/3(f)]2. (All')



384 Arif-Uz-Zaman

These satisfy the usual properties of the projection operators

V(f) yi(f) = &jyi (ij = +, -, 0) (A 12)

as can be easily verified. Also we have from the definitions

7+(-f) = 8-(f), (A 13)

8O( -f) = yO(f) (A 13') and from (A 5), (A 5'),

/. fyi(f) = ?iyl(f ) Mn= + (f*f)2 V? (f) (A 14)

'fl-f?(f)= 0. (A 14')

Hence the Fourier decomposition of 0(x) becomes

55(t) =27T fJ dw {eif x+(f ) x(f ) + e-if Y i-(f ) X(-f )} im. (A ) 5()=(2n)2J 2w ( 5

If we write U?(f) = imy1(f)x(?f), (A 16)

Ui (f ) are seen to satisfy the particle and antiparticle equations

(?i/+.fd-)m)U(f) = 0

or IAf) U?(f) +? U(f) (A17)

while UO =i(f ) x0(f) satisfies A8.fUo(f) = 0. (A 17')

Introducing the projection operators for a Dirac particle

A+(f) - 1 ? 7(f)) (A 18)

we can write qy?(f), y0(f) in the representation 1 3

91(if) =l(f) x Al(f), (A 19)

(f = A-(f ) x A+(f ) + A+(f) x A-(f) (A 19')

Vol #(f ) + 42 (f )( 19")

where 0A(f)970(f) = A70 LA, = 1,2, (A 20)

and '0(f ) yq(f )=. (A20')

It is seen that in the momentum space representation, the second-rank spinor space splits up into three subspaces, y+, 9- and y0 subspace. A spinor belonging to the y0 subspace does not satisfy the positive or negative energy Kemmer equation. The momentum space representation of a particle obeying Kemmer equation (1.1) must belong to I+ if it is a particle or to q- if it is an antiparticle. The operator (f) =-2[fl(f)]2- 1 = y(f) x y(f) (A 21)

has the property q(f/)1+?(f) = r(f (A 22)

Y(f)nqo(f) = -y0(f). (A22')




Spin projection operators The Lorentz invariant spin projection operator for the case of Duffin-Kemmer

equation is given by (see Corson 1953)

0(f) =ff{ i2 [A,fl] [flA,5jff- i[fA, 8] [/3jf,] fv} (A 23)

In the representation (1.3) of /? matrices this reduces to

0(f) -

i2{- 2y+ yAy XyA7y/-Ay(f ) XyAY(f)} (A23')

In the rest frame of the particle, i.e. for f = O 0(f) is given by

O(f = 0) 0--(0)- (3 + 6i x (rs), (A 24)

where 0i are given in terms of the Pauli matrices oi by

i74757j = j = (; ). (A25)

The eigenvalues of 0(0) are zero and 2, and it follows from the Lorentz invariance of 0(f) that its eigen values are also 0, 2. The spin 1 and spin 0 projection operators are givenby 0(l)(f)= -'0(f), (A26)

0(1)(0) = 4 (3 + ?Ti x C), (A26')

0(O)(f ) = 1- 0(f), (A27)

0(?)(0) = '( (- (6i X (:i). (A27')

O(l)(f ) and 0(0)(f ) commute with each of y/: (f ), y0?(f ). This is easily proved in the rest frame and by Lorentz covariance it holds in any other frame also.

We now prove a result used in the text.

0(t) y(t) y(f, t) x 'y(t) y(f, t) = Lt (t) LK(t) 0(t) LjK(t) LK(t) y(t) y(f, t) x y(t) y(f, t) LjK(t) LK(t) = Lt (t) . 0(0) . 7(tl) y(f, t1) x y(t1) y(f1, tl_) Le(t)

= LK(t) 0(0) LK(fi) Le(t),

where (1.9), (1.10), (1.11), (1.12) have been used. Hence

O(t) y(t) y(f, t) x y(t) y(f, t) = LK(t) LK(f) L (fi) 0(0) LK(fl) LK(t) - L (t) LK(fl) 0(fi) LK(t)

= LP(t) LK(fl) L4(t) 14(t) 0(f1) LK(t).

Therefore using (1 11') and (1I12') we have finally

0(t)y(t)y(f,t) x y(t) y(f,t)= y(t)y(f,t) x y(t)7 (f,t) 0(). (A28)

It can be proved directly from (2 23') that O (f) = (f ), hence taking adjoint of the last equation and replacing f by f' we get

y(f', t) y(t) x y(f', t) y(t) O(t) = O(f') y(f', t) y(t) x y(f', t) 7(t). (A 28')



386 Arif-Uz-Zaman

Spin components The infinitesimal operators corresponding to spatial rotations are (see Corson I 953)

Ei -iik[fj, (A29)

- '(1 x iY+5; x 1). (A29')

If we define (f) by Y =(f) = ffi (A30)

it follows from Kemmer-Duffin relation (1 .2) that

[X(f)]3 = (f). (A 30')

Hence the eigenvalues of E(f) are 0 and + 1. The projection operators are

Z+(f) - [(E(f))2 + ?(f)], (A31)

ZO(f) = [1-(E(f))2]. (A 31')

These satisfy E(f)Z+(f) = ?Z?(f), (A32)

E(f)Zo(f) = 0. (A 32')

It can be shown that E(f) and hence Z (f), Z?(f) commute with O(l)(f), O(?)(f) and

Vy (f ) and p10(f ). The O(l)(f ) subspace splits up further into O(M)(f) Z+(f), O(l)(f) Z-(f) OZl)(f) Z(f).

APPENDIX B

The 16 spinors U1i(f) orthogonal in the sense

U(f ) Ub(f ) = Ur()4U1(f) O for r tj (B 1)

are classified according to the scheme

ZT ) U. [C(f) I r < 3 6 O(l)(f)+?() Ur(f ) r =4 6 O(O)(f )+(f )

UV (f){Ur(f) 5 s< r K 7 6 Oe V)(f U2.r(f r= 8 e O(?)(V) 'r V

UO(f){Ur(f) 9 ? r < 16 6 yW(f)

From (A 22) and (A 22') it follows that

4 U (f= 0) = y4 U? (0) = U (0), (B 2)

y4 U0(0) = -U0(0); (B 2')

hence for the invariant U ( f)4U (f )=-* (0) N4 U (0) we get

UjIf )4U(f) = U; (0)U (0) > 0 (B 3)

and Uo(f),qU0(f) =- UoO(?) Uo(0) < 0. (B 3')

Ur(f) are normalized by writing

all Uo(0) Uo(0)=1, (B4)

U* ()U () = 1. (B 4')



Covariant polarization analys3is of spin 1 particles 387

Hence UV() yN UI(f UW(f) Uj,(f) 612Cr, (B 5)

where r =1 K (1r8)

6r =- 1 (9 --r K,1 6). The identity operator is thus

16 1 >UV() erUrt(f), (B 6)

16 or 6(X5 E r= rUr,~I(

and we define the traces by 16

tr Q(f) = Ur'(f )Q(f ) U?(f)Cr (B 7) r= 1

16

= Urt aQa,(f) Ur, ftCr

= =aQa Qaa. (B 7')

Unlike the case of particles obeying Dirac equations, the probability density P(x) for the Kemmer case given by (see Roman i960)

PWx = PtX) fl4qS(X) + X ()

It can be proved from equations A 17 that

U~t(f)/4U-TWf 0, (B 8)

UI-(f )f&tlU(f V ? (r/f) UZVf) U?(f) (B 8')

UotfVl4 UO(f = 0, (B 8")

Uot(f)/34U?(f) ? (M/2f0) UO*(f) U?(f).(B 8"'/)

Since Ut (f ) /4 U? (f ) transforms as the fourth component of a vector, it can be written as

Ut (f)flUI(f) = ?f0 f/r (f is an invariant) (B 9)

and for {Urt f)/UVf)}f=0 =+ invariant J. (B 10) But from B 8',

{Ut_(fY)f84 U-(f)}f0= ? {(M/fO) U1 jf) Uh(f)}f=O = U;(0) U?(0) (B II)

+ ?1. (BIlI')

The invariant J in (B 9) is thus equal to unity and we can write

Urtf)&r~f +o = +f UtVf) UV7(f

? f or I <r <4

- f or 5 <r <8. (B 12)

If the spin-component projection operators are written as

-+ Z(1), Z- = Z(2), ZO = Z() (B 13)

25 Vol. 268. A.



388 Arif-Uz-Zaman

U*(f ) for i = 1, 2, 3 can be written

U(f ) = -(f ) O(W)(f) Z(i)(f ) UV(f); (B 14)

hence we have for i, j = 1, 2, 3,

U9(f) )4 U*(f) = U(f) y+(f) O(l)(f)ZUj)(f) f4Z(4)(f) O(l)(f) y+(f) U*(f)

0 (itj) (B15)

Since fl4 commutes with Z(W)(f) and Z(W)(f) Z(i)(f) Zjz(i)(f). Finally we have

UV)(fl)4Ui(f) - foU(f) U(f) = f ai (ij = 1, 2, 3). (B 16)

Equation (B 16) is in accordance with the demand that the probability of finding the 'particle' in volume rn/fo be unity, i.e.

f 0mt(x)fl40(x)d3X= 1, (B 17) m/fO

with q(x) = U(f) eifx. (B 17')

The density of states in momentum representation is 1/(21T)3(rn/fo) d3f. One can define Kemmer spinors Wfr(f) by a linear unitary transformation of 1(m/fo) UV(f)

3

1f=(f) EbikUk(f)V(m/f0) (i = 1, 2,3), (B 18) k=1

3 3 where E bibkj = dij = E bbkb71 (i = 1, 2, 3). (B 18')

h=1 k=1

Wj(f ) are arbitrary solutions of the positive energy Kemmer equation belonging to spin 1 subspace and are orthonormal in the sense

ffl(f)64?(f) = ij

If the probability of finding free particles in the state Rfi(f) is Wi, the probability of finding it in the region d3f at f is

3 rn\2 1m wd ) =l

E +/go Ujj4fi(f)

i W(27T3 gm d3f (B 19) i~j=1 f0 fo

= E (g) 2u.(f)f4ugk(f)ut445(f)bik(bll)w Wi2f) (B 20) 3 M 2U f)m d~f (B1

= E U3-) (f ) A Ukt( V k) WA (U() j(f) bik(bil)*W (2) (B )

With the use of (B 5) and (B 7) the last equation can be written

w(df) = (27rn3 s1Ytrp1(f)T (B22)

M 2 f 2 (2m d f

where PI(f) = E Uk(f) UI(f)Wbik(bil)* W3 (B 23) i, 1, I 3

= E fr(f ) Wl4t/(f) (i = 1, 2, 3). (B 23') i=1



Covariant polarization analysis of spin 1 particles 389 3

p1 (f ) is obviously a covariant matrix with tr p (f ) =Z Wi and w(df) is invariant. i=1

The relevant formulae can now be derived as in the appendix of Stapp's work (I o6).

REFERENCES

Belinfante, F. J. 1939 Physica, 6, 849, 873. Biedenharn, L. C. 1958 Anacs Phys. 4, 104. Corson, E. M. 1953 Introduction to tensors, spinors and relativistic wave-

equations, pp. 96, 128. London: Blackie and Sons Ltd. Lakin, W. 1954 Phys. Rev. 98, 139. Roman, P. i960 Theory of elementary particles, chaps. ii, v; p. 429.

Amsterdam: North Holland Publishing Co. Stapp, H. P. 1956 Phys. Rev. 103, 425, 1956. Stapp, H. P. 1957 Phys. Rev. 107, 607.

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Covariant Polarization Analysis of Spin 1 Particles

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