C Z E C H O S L O V A K J O U R N A L O F P H Y S I C S Volume 43 �9 1993 Number 8 �9 pp. 777-856

O N T H E R E L A T I V I S T I C S P I N P R O J E C T I O N O P E R A T O R S

M. BEDN)~I~, P. KOL.~I~

Joint Institute for Nuclear Research, Dubna, Russia and

Institute of Physics, Czech Acad. Sci., Na Slovance 2, 18040 Praha 8, Czech Republic

Received 28 January 1993

In the Rarita-Schwinger formalism, the relativistic spin projection operators are dis- cussed with the help of the Pauli-Lubanski four-vector. It is shown that this approach is equivalent to the conventional one, but moreover, if enables one to derive recurrence relations for the spin projection operators. Such relations can be useful in practical ap- plications.

1. I n t r o d u c t i o n

Increasing experimental evidence for particles with higher spins and the possi- bility of investigating their interactions lead to the problem how to construct ef- fectively the wave functions of free relativistie particles with integer or half-integer spins. This problem is a subject of constant interest for more than thirty years.

Recently the techniques appropriate for describing the wave functions of higher spin particles were used in:

1. the theory of relativistic wave equations for particles with higher spins [1];

2. the description of decays of higher-spin particles [2];

3. the calculation of transition probabilities with the presence of higher-spin particles [3].

There are many formalisms for describing free relativistic particles of higher spins (Weyl and van der Waerden [4], Bargmann and Wigner [5], Rarita and Schwinger [6], Weinbers [7]).

The Rarita-Schwinger formalism is most frequently used in practical applica- tions. Within this formalism a relativistic particle of integer or half-integer spin j is described by a tensor function of rank j or j - 1/2, respectively. (Note that the components of the latter tensor are bispinors.)

The method of spin projection operators (SPO) yields a very useful tool for constructing higher spin wave functions. In the Rarita-Schwinger formalism the

Czech. J. Phys. Vol. 43 (1993) 777

M. BednS2 and P. KoM~

explicit expressions for these operators were obtained previously using the symme- try properties of the corrœ spin wave functions (Behrends and Fronsdal [8], Fronsdal [9]). The SPO for integer spins j, P(J), are defined in this approach by the following properties:

P(Jl!..y171171 = P(Jl!..ce,~...y ...a i;/31.../3i ,

pUp~�99 = O,

g"~'P(™ = O, (1) p(J) . p(J) = p(J),

where g~U = d i a g ( 1 , - 1 , - 1 , - 1 ) is the metric tensor and p denotes the four- momentum of a free particle with spin j, For half-integer spin s = j + 1/2 we have the additional condition

7~P (') = O, (2) /~a2""aS ;/31--./3j

where 7 ~ are the Dirac matrices. It was shown by Fronsdal [9] that these conditions determine uniquely the corresponding SPO.

An alternative approach was suggested by Weinberg [7, 11]. He obtained the SPO in terres of derivatives of the reduced (contracted) projection operator whieh can be expressed through Legendre's polynomials.

In this paper we present another method for the explicit construction of SPO. In out approach the SPO are defined in terms of the Pauli-Lubanski spin operator (see, e.g., [10]). It enables us to obtain useful relations for SPO.

2. P ro jec t ion opera to rs for integer spins

The Pauli-Lubanski four-vector operator W(%, representing the relativistic spin operator of a particle with spin s, four-momentum p and mass m, is defined by

W(,)(p) = -~e�87 pe, (3 )

where M(s)~u are the generators of the homogeneous Lorentz group acting in the space of spin states.

Let W be the space of spin-1 vector wave functions e(p) = {e~(p)} satisfying the subsidiary condition

p~e~,(p) = O.

The generators of the Lorentz group are represented on Vp by the matrices N~v with the elements

Then, in the space ~ | v,, |174187 (4)

j times

778 Czech. J. Phys. 43 (1993)

On the relatlvistic spin projection operators

the matrix elements (M0)~~)y of the Lorentz group generators M(j)#v are given by

(M(~),~),l...y = (N~~)c,~,~c~~~...gc,0~ + gy g-~z,

�9 .

O)

In the space, whieh is the direct product of space (4) and the Dirac bispinor spaee, the Lorentz group generators Mo)u�87 are given by

(M(o,~), , . . .% y226 = (M 0),~)c,...y226 '"ZJ 8,b + tic,Z, ac,:z~" "gcsas (E~,,,)y (�8

where s = j + 1/2 , and 6ab is the Kronecker symbol. The Lorentz group generators E~~ in the bispinor space are expressed in terms of Dirac's 7V-matrices as

x,~ = 4~[7,,-r,].

We shall now discuss the SPO for a partiele of integer spin j, mass fa and momentum p. We define

j-1 [(_WŠ + 1)] P(J)(p) - 1-[,=0 [ ~ u 2 3 9 2 3 9 2 3 9 ' (r)

where Wi�8 ) = ~ ) W 0 ) p. If follows immediately f roc (7) that pU)(p) projects spaee (4) on the subspace characterized by the maximal spin j.

We now prove the equivalence of (7) and (1). For this purpose i t i s convenient to rewrite (7) in the equivalent form:

(') . [':~3/_(-W~') (p)/m2)-l(l+l)] e~~~ z , z , = 7~u239

l 2 Otl ...Os j

x P(J- O'r~"TJ ;a~...,a ~ (8)

valid for j > 1. To prove relations (1), we shall use the induction over j . It is readily verified that relations (I) are valid for the operator

1 1 1 P(~)cO;~~ = "~Qc'rQ~6 + ~Qy - -~Qc~Q-~6,

where PcPfl

QOEZ = gcZ m~ .

Czech. J. Phys. 43 (1993) 779

M. BednM and P. Koldf

We suppose the relations true for j - 1 and prove them true for j . However, if relations (1) are valid for j - 1, then it follows from (3) and (5) that (8) can be rewritten into the form

J ~(j) o 1 X"' ~ ri(j-l)

2 J J

+ j ( 1 2j) ~ Z ~ p(j-1) - - " ~ O l n O t m O t l . . . O t ~ t _ l O ~ r , . [ . 1 , . . O t m _ l O t r a . + l , . . O t j ~ l ; [ ~ 2 . . . ~ j (9) n = l m = l

n < r n

for j > 1, and p(1) = QOE~. Recurrence relation (9)readily implies properties (1) for j, and the equivalence of (1) and (7) is proved:e Furthermore, from (9) we obtain the following properties of P(J):

(J) D(J) P~I...ai;Pl...~~ - r/~l'"~~;al'"aj'

gy - 2j + 1 p.!j.-1), y 2 j - 1 '

( - W ~ ) / m 2 ) P ( ~ ) = j ( j + 1)P(~).

Moreover, from (9) one can find the Behrends-Fronsdal formula

p(j) 1 2 J J a~...aj;~~...~j = (~) E ( H Qa~~~ .-b a~J)Qy 2 H Qad~~

p ( a ) l----1 1=3

{ (j) y140140171140 j even, (10)

+ ' " + _(J) Qy171 . . . . . . Qy171171 J odd, a(j_ 1)/2

where

The sum in (10) is taken over all permutations of y and fl [8, 9].

3. P r o j e c t i o n ope ra to r s for ha l f - in teger spins

For a half-integer spin s = j + 1/2 we define the projection operator by

P(')(p) =_ ( - W ~ , ) ( p ) / m 2 ) - s(8 - 1) s(s + 1) - s(s - 1) P(J)(P)" (11)

Repeating the considerations of Sec. 2, we obtain from (3) and (6) the following relation:

780 Czech. J. Phys. 43 (1993)

On the relativistic spin projection operators

pO) (J) O l l . . . ` = PoE1...ai;fll...flj

1 2j + 1 ~ I '~OekO I "~ ` 1 4 0 " ' ' ] ~ j

k=l (12)

for s > ~, and

p(3/2) 1 2 . ; ~ = g . a - ~ ' r ~ ' r ~ - ~ - ~ m ~ p . v f - ( v . ~ , , f - v ~ ' r . ) ( ' r � 8 7

f o r s = 3_ 2" Besides (1) and (2) we obtain from (12) the following relations:

pO) = pO) �9 . . O l n . . . O t m . . . ; [ ~ 1 . . . f l j . . ,Ofm. . . ` 1 �9 �9 .]~j '

p(') = pO) ̀ ""` ; f i l "" " f l j f l l ' " f l j ; a l ' " O f j '

[(7. p), P(')] = o, gy p(~!.;f.., j + 1 y f �87

= 2 j + 1 7 7 ~y ....

_ (W~s)/m ~) pO) = s(s + 1) pO).

The Behrends-Fronsdal relation for P('), i.e.,

p(8) = j + 1 p ( j + l ) y 2j + 3 7OETf y171171

also follows from (12).

4. C o n c l u d i n g r e m a r k s

We now turn to a comparison between our approach and the one presented by Behrends and Fronsdal. While Behrends and Fronsdal obtained the explicit expressions for SPO as a consequence of properties (1) and (2), in out approach these properties follow from definitions (7) and (11). Although our relations (9) and (12) do not exhibit the symmetry properties so directly like the Behrends-Fronsdal formulae, they can be useful in practical applications. In particular, formula (12) expresses the SPO for half-integer spin pO) in terms of P(J) for integer spin j = s - 1/2, in the most effective way. Indeed, if P(J) contains only independent terms, then we obtain immediately, from (12), the corresponding pO) with the same property.

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M. Bedn~~ and P. KoM~: On the relativistic spin projection operators

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