Fortschritte der Physik / Progress of Physics: Band 16, Heft 7 1968 [Reprint 2021 ed.] 9783112500521, 9783112500514

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FORTSCHRITTE DER PHYSIK 1IK Ii AUSGEGEBEN IM AUFTRAGE DER PHYSIKALISCHEN GESELLSCHAFT IN DER DEUTSCHEN DEMOKRATISCHEN R E P U B L I K VON F. KASCHLUHN, A. LÖSCHE, R. RITSCHL UND R. ROMPE

B A N D 16 • H E F T 7 • 1968

A K A D E M I E

- V E R L A G

•

B E R L I N

Prof. Dr.-Ing. W. R E I C H A R D T , Dresden

Grundlagen der Technischen Akustik 1968. 664 Seiten mit 463 Abbildungen 16,7 X 24,0 cm. K u n s t l e d e r 58,— Mark Das B u c h stellt eine Weiterentwicklung der vorangegangenen drei Auflagen „ G r u n d l a g e n der E l e k t r o a k u s t i k " dar. Es ist eine E i n f ü h r u n g in die Grundlagen der Technik der Schallübert r a g u n g u n d Schallaufzeichnung u n d b e h a n d e l t a u c h F r a g e n der L ä r m a b w e h r . E s ist in erster Linie f ü r d e n L e r n e n d e n u n d den j ü n g e r e n P r a k t i k e r in der I n d u s t r i e , aber a u c h als Nachschlagehilfe f ü r den Ingenieur g e d a c h t . Die elektromeclianischen u n d

elektroakustischen

Analogien bilden die Grundlage, v o n der aus die elektrischen und akustischen Probleme einheitlich b e t r a c h t e t werden können.

Akademische Verlagsgesellschaft Geest & Portig K.-G., Leipzig

BEZUGSMÖGLICHKEITEN Sämtliche Veröffentlichungen unseres Verlages sind durch jede B u c h h a n d l u n g im In- u n d Ausland z u beziehen. Falls keine Bezugsmöglichkeit v o r h a n d e n ist, wende man sich in der D e u t s c h e n D e m o k r a t i s c h e n R e p u b l i k a n d e n A K A D E M I E - V E R L A G , G m b H , 108 Berlin, Leipziger S t r a ß e 3—4 in der D e u t s c h e n Bundesrepublik a n K U N S T U N D W I S S E N , E r i c h Bieber, 7 S t u t t g a r t 1. Wilhelmstraße 4 - 6 in Osterreich an d e n G L O B U S - B u c h v e r t r i e b , W i e n I, Salzgries 16 in Nord- u n d S ü d a m e r i k a a n G o r d o n a n d Breach Science Publishers, Inc., 150 F i f t h Avenue, New Y o r k , N. Y. 100 11 U.S.A. bei W o h n s i t z im übrigen nichtsozialistischen Ausland a n d e n D e u t s c h e n B u c h - E x p o r t u n d - I m p o r t G m b H , 701 Leipzig, L e n i n s t r a ß e 16. I m sozialistischen Ausland k ö n n e n Bestellungen ü b e r die B u c h h a n d l u n g e n f ü r f r e m d s p r a c h i g e L i t e r a t u r bzw. d e n zuständigen P o s t z e i t u n g s v e r t r i e b erfolgen. Auf W u n s c h sendet der A K A D E M I E - V E R L A G I n t e r e s s e n t e n bei B e k a n n t g a b e der Anschrift u n d F a c h g e b i e t e unverbindlich I n f o r m a t i o n e n ü b e r lieferbare u n d k o m m e n d e Veröffentlichungen u n d gibt a u c h Bezugsquellen im In- u n d Ausland b e k a n n t .

Fortschritte der Physik 16, 373-418 (1968)

Dirac Formalism for Arbitrary Spin in S-Matrix Theory B. 0.

Institute

of Theoretical Physics, and B. E.

Institute

of Theoretical Physics, and Brandeis

University,

ENFLO

University

of Stockholm,

Sweden

of Stockholm,

Sweden

LAURENT

University Waltham,

Mass.,

U.S.A.

Contents Introduction I. Dirac formalism II. Spinor states III. Special representations IV. The Dirac equation for arbitrary spin V. Crossing and discrete symmetries VI. The invariant structure of the M-function VII. Unitarity VIII. The limit -> 0 IX. Comparison with the field-theoretic formalism for spin 1/2 Appendix 1. The matrices A and F Appendix 2. The function D(p) Appendix 3. The operators Y, Z, A and F for spin 1 Appendix 4. Calculation of an inner product when m — 0

373 375 378 382 388 391 394 400 402 409 412 414 416 417

The /S-matrix formalism is built up from the beginning, for particles with and without mass, through systematic use of the representation Ds>0 © Do s of the homogeneous Lorentz group. Only concepts referring to space and time are considered. Within the framework of the formalism, the close interdependence between CPT, spin and statistics, crossing symmetry and unitarity is explicitly emphasized. The concepts of intrinsic parity, spin-flip and particle conjugation are re-examined. A comparison is made with conventional field-theoretic formalism. Introduction During t h e last few years, /S-matrix t h e o r y has become t h e object of g r e a t current interest a n d m a y now be regarded as competing w i t h relativistic q u a n t u m field theories in describing elementary particle phenomena. Since t h e work of S T A P P 27

Zeitschrift „Fortschritte der Physik", Heft 7

374

B . O. ENFLO a n d B . E . LAURENT

[1] and the review article of B E R E S T E T S K I I [2] in 1 9 6 2 , several authors have been working on the problem of formulating analyticity assumptions and extracting information on the singularities of the ¿'-matrix from these. References to this work are found in the books of C H E W [3] and of E D E N , LANDSHOFF, O L I V E and POLKINGHORNE [4]. Although there is a strong evidence that all calculations in elementary particle physics may equally well be made in terms of ¿-matrix concepts as in terms of field theory, the formalism of the latter has hitherto been almost exclusively dominant. Examples of calculations made with use of ¿-matrix concepts are given for instance in the book of B A R U T [5] and in the works on $-matrix theory of electromagnetic interactions by H A U G [6] and by CHOU and D R E S D E N [7]. BARTJT andHAXJG use, for spin 1 / 2 , the two-spinor formalism, which has dominated in /S-matrix calculations since the paper of STAPP [7], Because most people are used to the Dirac four-spinor formalism, this may perhaps be a barrier which provents wider use of the ¿(-matrix formalism in calculations. The work of CHOU and D R E S D E N takes over the field-theoretical apparatus of formulae, with the usual four-spinor formalism for spin 1/2, unaltered to ¿'-matrix theory. One may, however, ask whether this apparatus is the most natural and appropriate from the view-point of the ¿-matrix-theoretic assumptions. We hope that the present paper fills a pedagogical need of developing a unified view on elementary particle reactions, always through particle concepts, in terms of a formalism similar to the Dirac formalism of field theory. As will be seen, the formalisms are, however, not identical. B y particle concepts, we shall mean the momentum, spin and intrinsic parity of a particle: we shall deal exclusively in this paper with these. This means that we separate the space-time symmetries and conservations from other symmetries and conservations. The former include Lorentz-, principle C- (i.e. change of fermion intrinsic parity), P- and T-invariance, the Pauli and crossing symmetry, the latter e.g. charge, baryon and lepton conservation and isospin symmetry. These latter symmetries can very easily, almost trivially, be imposed on the scattering amplitude after the application of the space-time symmetries. Fermion number is in an intermediate position connected to the space-time properties through the intrinsic parity. To illustrate the use of fermion number conservation we have applied it in a few examples. In order to include the spacetime symmetries, a Dirac formalism, somewhat different from the field-theoretic one and easy to generalize to arbitrary spin and to massless particles, has proved useful. I n this context we mention a work by W E I N B E R G [ 1 1 ] , who investigates the ¿-matrix in the zero mass limit for bosons in a manner similar to ours. For the field-theoretic Dirac formalism, we shall refer to [S] throughout. In Chapter I the algebraic properties of the matrices corresponding to the Dirac y-matrices and related matrices are established for arbitrary spin. Chapter I I deals with the generalization of the well-known Dirac u and v spinors. We also define an M-iunction, corresponding to that of S T A P P [ 1 ] . Chapter I I I gives examples of the formalism, applied to spin 0,1/2 and 1. For spin 1 the 6-component antisymmetric tensor representation is used. The related representations for higher spins are given. In Chapter I V the Dirac equation is generalized to arbitrary spin. I n analogy to the Dirac equation, these equations are invariant under ¿-transformations. Chapter V deals with the discrete symmetry properties of the AT-function. Among these we include crossing symmetry, which is assumed as a fundamental principle, and the Pauli principle. In Chapter VI we give a systematic scheme for finding the invariant structure of the ili-function, apply it on a few simple cases and give an example of a search for kinematic singularities. Chapter V I I deals with unitarity,

Dirac Formalism for Arbitrary Spin in iS'-Matrix Theory

375

which is brought into manageable form by using the CPT invariance of the Mfunction. The invariance of the unitarity condition itself under CPT is explicitly shown. I n Chapter V I I I we apply the formalism to zero mass particles b y investigating the limit m -> 0 systematically. We do not need to introduce potentials, using quantities corresponding to the field strength throughout. Cf. W E I N B E R G [11]. Finally, in Chapter I X , we make a comparison between the conventional field-theoretic formalism for spin 1/2 and the ^-matrix formalism developed in this paper. I n particular we discuss the particle conjugation operator which is simpler here t h a n it is conventionally. As a consequence of the pedagogical program mentioned above, and of our strict restriction to space-time structure, we have found t h a t some points can be analyzed quite clearly. One example is the concept of antiparticle. We consider this concept as a secondary one, deduced from crossing symmetry. From this one finds t h a t bosons are their own antiparticles and t h a t antifermions are the particles with opposite intrinsic parity to t h a t of the fermions. We t h u s need not refer to any negative-energy solutions of field equations in order to introduce antiparticles. Another example is the operation of spin-flip. I t s fundamental importance in e.g. time reversal and crossing symmetry is explicitly emphasized in this paper but seems to be obscured in the field-theoretic formalism. A third example is the CPT operation, which is easier to make use of t h a n in field theory because of our simpler particle conjugation operator. We have used CPT invariance to write the unitarity condition in a form which explicitly contains the imaginary parts of the invariants of the Af-function. I t is also seen in the present formalism t h a t there is a close connection between the Pauli principle, unitarity and CPT invariance. A fourth example is the covariant normalization and completeness conditions of mass zero u and v spinors, which are deduced in the present formalism by a most natural limiting transition. I n most textbooks these conditions are postulated for neutrinos and their connection with the corresponding conditions for particles with mass is not clear. Our approach does not contain any strong assumptions of the analyticity of the M-function. I n the terms of S T A F F [I] we assume „minimal analyticity" and a „physical connection", which we have formulated as our postulate of crossing symmetry in Chapter V. Our results are thus largely independent of what special dynamical models may be invented in the future. We emphasize t h a t our aim is not to „prove" the CPT theorem, the connection between spin and statistics or any other theorem. Our aim is rather to present a clear, manageable and unified formalism, in which these and other comparatively well-established facts find their natural place. Instead of trying to prove the connection between spin and statistics we have pointed out where a relaxation of the Pauli principle would cause difficulties for us in developing the formalism in a simple and direct way. Our metric is gr00 = — 1, g11 = g22 = g33 = 1. Four-vectors and spinors are indicated with Greek indices, space vectors with Latin indices. The Levi-Civita symbol e 0123 is equal to 1. I. Dirac formalism Our aim in this chapter is to define a set of matrices which, for any spin, can play the part t h a t the y-matrices play in the case of spin 1/2. We introduce the square matrices Uk (k = 1, 2, 3), Y and Z of dimension 2 {2s -j- 1), where s is the spin. We 27*

376

B . O . ENFLO a n d B . E . LAURENT

require them to fulfil the following equations (£1)2 +

\Eh,El] = it*lmEm (£2)2 + (¿73)2 = + 1)

(1) (2)

[7, £*] = &,£*] = 0 {Y,Z}= 2

(3)

0 2

Y = z

(4) = 1.

(5)

In the case of spin 1/2 the matrices Z k = - \e M m YiYm Y = iy5=

(6)

¿70717273

(7)

Z = iyo

(8)

satisfy (1)—(5). The equations (6)—(8) can be solved with respect to the y-matrices and give (9) y o = - i Z 7* = —2i Y ZEh.

(10)

It may be noted that our set of matrices bear a close relationship to those introduced originally by D I R A C [9], and so we call this a Dirac formalism. Returning to the general case we see that the following matrices

HL:) z

"(:,>)•

>



\p> «3.

sz>

f)-

(3)

We have here added the intrinsic parity, f , to the characteristics of the physical state. Sometimes nature does not seem to allow both | p, s3, + 1 ) and | p, s3, — 1). This fact need not bother us here; we can take care of it later on. The idea with the spinor states is that the coefficients (p, s3, £) should be so chosen that | p, «} transforms with a Ds 0 0 Do s representation (this means that oc should run over 2 • (2s -(- 1) different values). We therefore have equally many spinor components and components in the physical state vector. Thus L(A)\p,*)=Z\P/l,«')D.M), at'

(4)

where (A) is the representation mentioned. In what follows we shall suppress the spinor indices. Equation (4) then runs L(A)\p)

= \pA)D(A).

(5)

This equation together with (3) shows that w must have the following property W (r

4)

1

, « „ S) D ( A ) =

We choose it in the 2-direction.

2 »;

K s i

( ¿ r o t ) ® (P,

i) .

(6)

D i r a c F o r m a l i s m for A r b i t r a r y S p i n in S - M a t r i x T h e o r y

379

Specializing A to a three-dimensional rotation and putting p = jp this gives w (p\ s3 , f ) 2?

= 2 Si

Stsi

w $

£).

(7)

We have here used (1.23) and (1.25). Specializing A (in (6)) to a special Lorentz transformation along p and chosing p = p we obtain w(p,ss ,^

^w&s^^DiAp™).

(8)

Remembering that Z corresponds to space inversion we obtain from (3): w(p,

s3 , £)Z

= £w(pp ,

s3 , £ ) ,

(9)

where pp is p with the opposite sign of all space components. As Y and Z anticommute Y must change £ when applied to w. Choosing a relative phase factor we can therefore write W(p,aa ,S)Y

= w{p,aa ,-S).

(10)

Apart from a normalization and an over all phase factor w completely. Introducing 5 )

the equations

(7) —(10)

define

w^A^W

(11)

we choose the invariant normalization w(p,

s3 = s,

f

1) w(p,

=

s3 = s, (

= 1) = 1

(12)

It then follows from the defining equations that

w(p, s3 , |) w(p, sj, £') = ¿d SsS ; da-.

(13)

There also exists a completeness relation corresponding to (13). £ £ w ( p , s 3 , £)w(p,

s 3 , £) = 1 .

(14)

Combining equations (7) and (8) we obtain Z

S* t w (P, s'3 , () = w (P, 8» f ) 9*

(P)

(15)

where =

D' 1

It is important to notice that it is operator

in spinor

Hf*

£* D (Av™,). (and

not e.g. E k)

(16)

that

constitutes

the

spin

space.

Combining equations (8) and (9) we obtain where

w(p,ss ,Z)2)(p) 3

5

(p) =

= Zw(p,s3 ,$)

D~ l ( A ^ ) Z D ( A ^ ) .

) All o u r e q u a t i o n s f o r w can a l t e r n a t i v e l y b e w r i t t e n a s e q u a t i o n s f o r w using ( I . 14).

(17)

(18)

380

B . O. ENFLO and B . E . LAURENT

3> thus constitutes the operator of intrinsic parity in spinor space. I t is going to play an important part in the following. We call it the Dirac operator because (17) will turn out to be the Dirac equation in the case of spin 1/2. From (18) and (5) we see immediately that 2=1.

(19)

We also deduce from (17), (14) and (19) Z V> (p, ss, £) w (p, sz, £) =

i) F =

- W *

(p, s3, $) F

(p).

(24)

To derive this we have also used (1.15), (16) and (23). I n the same manner we obtain from (17) £ w* (p, s3, £)F

= w* (p, s3, £) F 3 (p).

(25)

These two equations show that w* (p, s3, £) F = a (s3, £) • w (p, -

£),

(26)

where a(s3, £) are phase factors 6 ), which may be determined up to a common factor. The dependence of s3 is obtained by multiplying (26) from the right with y i _j_ 2 using (1.15), (15) and (10). The dependence of £ is obtained b y multiplication with Y. The result is w*(p,s3,£)F

= £(-l)«°w(p, - « „ £ ) ,

(27)

where the undeterminable common phase factor has been chosen in a particular way. This fixes the last indeterminacy in w, the over all phase factor 7 ). We shall refer to (27) as the spin flip equation. ) The absolute value of a(s 3 , ( ) must be unity due to (13). ') Strictly speaking the sign of w is still undetermined. This will be of no concern to us.

6

Dirac Formalism for Arbitrary Spin in S-Matrix Theory

381

In a way similar to that in which the well-known T-function is defined in terms of the physical states (p', S3,

... IS — 11 p, s 3 , £ . . . ) = (phase space factors) • T(p', s'3,

...; p, s3, f

...), (28)

where /Sis the scattering operator, we may define the M-function in terms of spinor states (p') A'1 (p' ... 1S - 11 p ...) = (phase space factors) • M(p'

p ...).

(29)

The ,phase space factors' which we shall use are the following (phase space factors) =

idi(p' y N/2

...—p..)

(30)

where V is the reaction volume, N the total number of particles and np = |/pfijm (for fermions) np = |/2p° (for bosons). The particular choice is, however, unimportant and the reader can easily replace this with his own conventions. Equations (28) and (29) give the following connection between the T- and If-functions M(p-,)=Z

^w(p,s3,Z)

T(p, s3, f ; )

M{-p) = Z T(-,p,s3,£)w(p,s3,£). «3,1

(31) (32)

We have here suppressed all arguments in the T- and ill-functions except the ones belonging to one particular particle. By a semicolon we indicate if this particle is ingoing, (T(;p, s3, £)) or outgoing (T(p, s3, f;)). Using the orthonormality equations (13) one can solve equations (31) and (32) for the T-function obtaining T(p,s3,£-,)

= P°+1w(p,s3,$)M(p-,)

(33)

T{-p,s3,S)

= £ M(-,p)w(p,s3,£).

(34)

We have seen that the coefficients w are defined (up to an over all factor) by the equations (7)—(10). Alternatively the equations (10), (15) and (17) can be used for this purpose. However, they determine w only up to an arbitrary factor which may be a function of p. Using (A42) the matrices Sf k and appearing in these equations can be written as ^(p)

= D~x (p) Zk D(p)

®(p) = D~1(p)ZD(p).

(35) (36)

(see eq. (16) and (18)). This provides us with a natural possibility to define w for negative energies through a continuation of (10), (15) and (17). In view of (A.45) we see that w (p, s3, i) ei7lY£ * satisfies the same equations as w (— p, s3, £). This shows that w(-p,

s3, f) = a(p) w(p, s3, f) e«»*^*,

(37)

B. 0. Enflo and B. E. Laurent

382

where a(p) is a c-number function. Equation (37) gives, using (8)

w(-p,s3,£)

= a(p)w(p,s3,£)D(p)e^Y^

= a(p)w(p, s3, £) D(-p).

(38)

If we regard this as a continuation of (8) we must put a = 1 and obtain w(-p,

8» S) = W(p, «3, £) e f » T * - * .

(39)

The equations of normalization and spin flipping (13) and (24) cannot be continued due to their non-analytic form. They must actually be slightly generalized to be valid for negative as well as positive energies. This is easily done using (39) but we shall not write the equations down here. Using (A 47) and (1.10) we may write (39) in the alternative form

w ( - p, s3, s) = w (p, s3, ( - 1 )2*|)

*.

(40)

I f we use helicities, that is if we choose the z axis in the direction of p, einZ'v becomes just a phase factor and we see that w for reversed p is essentially the same as the original w in case of integer spin and the same as the original w with reversed intrinsic parity in case of half integer spin.

III. Special representations 1. S p i n O Since in the representation D0 0 every homogeneous continuous Lorentz transformation is represented by unity, it follows from (11.18) that the Dirac operator 3/ is independent of p and equal to Z. The Dirac equation then takes the form: Zw(f) = £»(£) where the w's now only depend on

In the standard representation we obtain:

27* = 0

Y

(1)

(2) (3)

-Co

(4)

w(£ = 1) =

1

/I

V2 U

(5) (6)

Dirac Formalism for Arbitrary Spin in » + 27S.p). From (1.6) —(1.8) it follows that mp)

can be written:

( - *yo)

=

(8)

+ j

75 eHm y,ymp*)-

(9)

After using the anticommutation rules {r,y} = 2ir,

(io)

which follow from (1.6) —(1.8), we obtain finally ®(P) = ~~

¡it

r^P"

(H)

and the Dirac equation for spin 1/2: (iy^

+ £m)w(p,s3,

£) = 0 .

(12)

3. S p i n 1 For the description of spin 1 we use a second rank antisymmetric tensor, which has 6 components. As is well-known an arbitrary such tensor F ^ is reducible into the representations Dl a and DQ l . For this tensor representation , Z and Y are derived in appendix 3. They are = » V " {Pag.,iL

+ d3ageld*m -

e x (e, a))

(13)

Z'^a = g^gpa — g(seg*a

(14)

Y*

(15)

It can be directly verified that these operators fulfil (1.1) —(1.5). Now we can evaluate the Dirac operator by means of the last expression in (7)8). Since this is somewhat lengthy, we shall instead deduce 3>{p) by using (11.18) directly. This is 8

) See also (IV. 13).

384

B . 0 . E n t l o and B . E . Latjbbnt

written in the following form given in (7) : =

(16)

W\ are well-known: The matrix elements of D{AVJ%!)

=

where

- ex(/i, v)

(17)

po

•o o _—m

(.18)

Jj/C

(19)

m

c\ = ff1 +

tfVi . m(p° + m)

(20)

The transformation coefficients of D2,(Ap^.p) are given by {Di(Ap^)f\v

where

= (c2)% (c*)\ - ex(n, v)

(21)

(e'r„ = C V V

(22)

Prom (18) —(20) and (22) follow: (P0)2

m

= 2^

- 1

(23)

(c*)\=(c2)\=

(24)

( c ^ = «5f+ 2 ^ - ' . (25) Tib2 If the sign is changed of all components of (c Y„ for which fi = 1, 2, 3,, a tensor d", is obtained : (26) TO2 The Dirac operator now follows : {^(P)}»^ = 1/2 ^

{ J D 2 (/l^ > )}"V= 1 / 2 ( M ; -

(c2)%(c2)\ - ex (u, v) =

= 1/2 d « ^ , - ex («, fi) - exfci, v) = i ^0 and D o s of the homogeneous Lorentz group are irreducible under space rotations, they must be given by tensors F t . J.,

lr = 1 , 2 , 3

(36)

with the same symmetry properties as the T' s. The indices lr are all transformed according to a Z>10 representation for F^.,.1, and a D 0>1 representation for -fiT-.-i,- For space rotations lT is still transformed as a space vector index. All indices in F+ or F~ must transform according to the same spin 1 representation in order t h a t the relation M ok = T i £ t

(37)

be true in Ds0 and Do s respectively (cf. 1.26). In order to find a particular representation D s 0 0 B o s we first double the number of dimensions of each index and start with the symmetric tensor -Fro,...™,'

m

r = 1...6.

(38)

386

B . O. ENFLO a n d B . E .

LAURENT

This tensor is transformed in each mT according to a representation Di Representations F+ and F~ are obtained by the conditions: F k - . m ^ I 1/2 ( ± r + l ) m i m ; . . . 1/2 ( ± 7 + 1 ) ^ ^ . . . ^ .

m[...me'

0

0

D01. (39)

Now it is necessary to find a condition on Fmi±rni, which is valid for both F+ and F~ but for no other representations contained in Fmi m j . It follows from (39) and (1.5) that the condition £

^m^mt Y mimi F m i tt

m^' mi

„ mi .. ma ~ -^mi.. ms

(40)

is fulfilled both for F+ and F~. Because of the symmetry of Fmi mi it is irrelevant on which of the indices m 1 . . . ms the Y's work in (40). I t is also easy to verify that no representation without the same sign of all Y's in (39), fulfils (40). As for (35) we must also have a contraction relation for the tensor (38). For F± taken from (39) with Y diagonal, the contraction conditions (with all combinations of and — allowed) i

l)mtmi Fm1..mic..m)..m, — 0

(41)

are obvious. A contraction condition suitable for the direct sum of F+ and F~ is thus the two relations: fim/ctni -^mi.. mi.. m;.. m8

nth, mi

^ ^mtmi -^»ii.. mi, mi

=

^

(42)

m4 ~

(43)

If we, instead of the indices mr use an antisymmetric four-vector index pair /urvr, the equations (40), (42) and (43) obtain the manifestly covariant forms with use of (A52) : e fk'k C* I ' l p i n *' FPi'i..Pk , >k , -Pl'l , , ••= F — 1/4 I cm*k c

PPiti:nvk..

1

_— o"

ffkfk Pl'n' -W , ,'k-m , , >t ..fi'i =0 1 ni>,-to

1

d'i

...Pi's

(441

(45} (461 \*v>

The tensor is symmetric in all index pairs. The equation (44) and the symmetry conditions on F are of course independent of the Lorentz metric, which is introduced through the conditions (45) and (46). For half odd-integer spin s + 1/2 a Dirac spinor index is simply added to The new spinor-tensor fulfils (44) —(46). We obtain one more set of equations of the same type as (40) by noticing that one of the Y ' s can work on the spinor index. This gives

S (iy-oU (-y

W*v)

=

(47)

Dirac Formalism for Arbitrary Spin in (S-Matrix Theory

387

There is one more contraction: F t . * . , =

p

o.

(48)

We need not separate strictly between the _D1/2 0 and D01/2 representations, because we allow similarity transformations on the y's. B y using the fact that (47) eliminates half of the components of F„ ff , iyi .. ^ Vj , it is easy to verify by (48) and an induction proof that there remain 2(2« -j- 2) components. The operators Ek, Y, Z and 3) are obvious generalizations of (A49), (A52), (A53) and (27): ( ^ • " • • " " V = (^»'"¿riiW'^i

••• (1 )"'VWr/ +

+ - + (1)"%;,; • • • (1)"'-""*'-,..'-. m - . ; > = =

(49)

••• (1)"%.'./.

(50)

- ff^,; g,1/t;) ••• (g^.- g„„' ~ 9 W

(51)

^ " • • " " V ; , ; . . ^ ' , , ' = ^"•"V;,; •••

(52)

The unity operator is given as (1

=

(53)

Finally we shall work out a generalization of the divergence relation (33). The spin 2 Dirac equation is according to (52) and (27): (k * +

$

+ 2sf-Jf}

tf

+ 2 6 ^

+ 2%

w ^ ) = (54)

The momentum and spin component dependences of waPxd are suppressed. If the positive intrinsic parity and the rest system momentum are chosen, then (54) becomes 2 pPp0wa0>p0w*P*a + p"p0ufPa?- -) (pPp>-p0p9w*°x0 + + pP p" p0p0waOOA + p* p* PqPqwPP*0 + pxpxp0p0w°Pm) Putting

= 0

(55)

/? = 0 and simplifying yields pO^lM

'¿ .pi-u^O

3 px yyxW». = o

(56)

and thus ufm = 0

(57)

w«oto = 0

(58)

Equations (57) and (58) are the rest system form of the equation = 1) = 0.

(59)

388

B . O . E N F L O a n d B . E . LAURENT

I n order to generalize (59) we notice t h a t the following equation easily follows from (52) and the fact t h a t 22 = 1: 2

2

{2

1 ) . . . ( 2 + 1).F = f 11 ( 2 + 1 ) . . . ( 2 + 1 ) J

+

(60)

where all tensor indices are suppressed. Thus, for £ = 1, s—2 pairs P.»

(1 );,; (2 + !)••••

+ !)•••;]

7T7 = 0

(61)

s—2 factors follows from (54) and (59) and s—1 pairs 2V. [fl

{2>± 1)"..

+ + l)"..,] ^ s—1 factors

0

(62)

follows b y multiplying (61) by 2 ± 1 • Because [ 7 T 1

....1]F

=

F

(63)

according to (44) and jY,

2 }

= 0

(64)

according to (7) and (1.4), it is obvious t h a t we can obtain both signs in each of the s — 1 factors in (62). This means, however, t h a t the relation (f = 1) = 0

(65)

is fulfilled. For £ = — 1 follows as usual F

p

M

=

— 1) = 0 .

(66)

We notice t h a t the sign in (51) is conventional, and in principle we may have a separate convention for each spin. We make the convention so t h a t (65) and (66) have the same form for all spins (cf. A56). IV. The Dirac equation for arbitrary spin I n order to evaluate the Dirac operator for arbitrary spin we start with the general form given in (III.7): ifz-v 2

(p) =

Z

m

iPl)

(1)

Dirac Formalism for Arbitrary Spin in /S-Matrix Theory

389

If £ • p and Y are chosen diagonal, the matrix for 3> (p) can be written exphcitly: 1 m

(P°+\P\)

1

~|2S—2

-TO ( ? " + =

IpDJ

Z

•

~

(2)

IP I )

(P°

m

( P ° - \ P \

Since every diagonal in 2(2s-(~ 1) dimensional space can be written as a linear combination of the diagonal matrices 1, 2 - p , ... (2 -p)2*, Y , Y 2 • p, ... Y(S • p) 2s , the Dirac operator can also be written as ®(P)

-I + + + +

-

(P°+

! 1 - ( 1 *

IPl)

2s—2 1 (P° +

~ ( P m

0

+

IP I)

•s 2-S "1

The coefficients each k) :

akt(s)

I

J

( s )

(

Y ) 2 a i=i

i t

S

.

+

••

( s )

2s+1 (1 +

- ( 1 * -2s

( P » ~ \ P \ )

+

—2s t -

v

28+1

- ( 1 *

\P\)

- ( P ° - \ P \ ) m

m

2« + l Y ) Z a i=i

+

Y)2 (p) in the crossing equation for it to be invariant10). The space inversion operator, P op , has the following effect on the physical states =

(8)

Using (II.3), (II.9) and (11.29) this gives Mp(p';p)

(9)

= ZM(p'p-,pp)Z

as the space inversion equation for the Jf-function. We notice that the crossing relations (1) and (5) are invariant under space inversion. This would of course not have been true had (1) contained an extra factor Y on the right hand side for example. When one considers the possible structure of the M-function because of invariance under homogeneous Lorentz transformations (see chapter VI) one notices that it must necessarily fulfil M(p';p)

= Ms(p';p)=

Y26' M(—p'; — p) 7 2 e .

(10)

One says that it is invariant under strong reflection. In view of (1.14) and (1.15) the crossing relation is obviously invariant under strong reflection. Using the crossing relation in (10) we obtain another invariance of the Ji-function. For reasons which will become clear later it is called CPT invariance. M(p';p)

= MCFT (p' ;p) = B Y2* M(p;p')

Y*°'

B ~ ( 1 1 )

There are a few comments regarding this formula: Whenever a formula of this type appears we shall always understand that the (unwritten) arguments which go together with the primed p shall be compared on the left and right hand sides 10 )

We have here chosen the alternative closest to the conventional treatment.

394

B . O . ENFLO a n d B . E . LAURENT

of the equation. Likewise for the unprimed p. There should actually be arguments for more than two particles in (11) although they are not written out. It is understood that the particles which go together with the particle with momentum p' appear in the same order everywhere in (11), and likewise for p. The rule which we gave in connection with the crossing relation together with the Pauli principle and the fact that the total number of fermions always is even leads effectively to a sign ( — l) 28 ' for each particle in the right hand group of particles on the right hand side of (11). This cancels an identical sign which arises due to the equations (1.17) and (1.18) which give fl~ = ( - l ) 2 s B .

(12)

We thus see that it is the Pauli principle which gives (11) its simple form. Combining CPT, C and P we obtain a new transformation MT(p' -,p) = rjBZM(pp;p'p)

(13)

ZB^r,'.

Translated to a transformation of the T-function this gives TT(p', «'» ('; P, a„ () = f]'v

i2* (-1Y'-8'

T(pP,

£; P't» - «*> f')-

(14)

This transformation changes the sign of all three-momenta, flips all the spins and changes 'out' to 'in' and vice versa, but it does not change the character of the different particles. It is natural to name such a transformation time reversal, and it is easy to show that our crossing relation is invariant also under this transformation. VI. The invariant structure of the ^/-function

In this chapter we shall first draw some general conclusions on the properties of the If-function. Because of the invariances, which the M-function of a given reaction must fulfil, there must be an interdependence between its components. It will in fact turn out, that a limited number of scalar invariant functions completely determines the M-function. We shall now make some remarks on the method of searching for these functions and give some examples for a few simple reactions. To this end we choose, for spin greater than 1/2, the special tensor and spinortensor representations treated in Chapter III. An M-function, which is transformed covariantly under the proper Lorentz group, can then be written M(p',r'\p,r),

(1)

where r, r' is shorthand for the tensor indices of (III.44), with a spin 1/2 four-index added, if the particle is a fermion. The number of spin 1/2 indices must be even. We have at our disposal for the construction of this covariant spinor-tensor quantity the momenta p>* of the particles in the reaction and the /-matrices for spin 1/2. Thus the task is to find all expressions in y a n d pP, which have the same transformation property as the M-function under proper Lorentz transformations. The il/-function must then be a linear combination of these expressions, where the coefficients are scalar invariant functions.

Dirac Formalism for Arbitrary Spin in S-Matrix Theory

395

From the general structure of the M-function it is easy to see that it must be invariant under strong reflection, which was anticipated in (V.10). This is trivial for integer spin, so we assume that r' and r in (1) are spinor-tensor indices a, ... fisvs. Because of (111.47) there are two equivalent expressions for Y: î

and

^

/

'

(iyB)»'«(1 ^ " V » ; • • •

=

(2)

= ô.'a ( - i e ^ y ^ ) (1 )„;,; . . (1 )

(!)

All arguments except those belonging to one particle on each side in the reaction are suppressed. The summation is performed over all intermediate states. By Nn we mean the product of normalization factors and j/2 p° for that intermediate state containing n particles. The first aim of this chapter is to write (1) in terms of M--functions. After use of (11.33) and (11.34), relation (1) becomes: i[{®(p, «3. f ) a%s+1M(p-, p') r W(p', s'3, £')}* - W(p', s'3> ?) H'^M{p'-,p) X £ w(p, s3, £)] = (2n)*Z

DHP

+ ''

V

~

VN)

\w{n)

fr*M(pn;

X

p') X

X £'w(p', 4 , f )]* [w(n) %*n+iM(pn; p) £ w(p, s3, |)].

(2)

The symbol w (n) stands for the direct product of w's for each particle in the intermediate state. After elimination of w(p, s3, w(p', s3, £'). £ and f by means of (11.11) and (11.17) we obtain: i[A*$**°(p)

M* (p; p') -

A9"(p')

M(p'-,p)]

= (2nfZ

^

X

X [w* ( » ) M* (pn; p')] [w (») M (pn; p)].

(3)

Now we use the GPT invariance formula ( V . l l ) on the first term on the left-hand side and the first factor on the right-hand side of (3). The result is: i [A F* F* 2s ' M* (p'; p) F* 28 F-1* S>28 (p) = (2ti)* 2 n

6i{P

\T2 j/n~ Vn

Vn)

A QP*' (p') M (p'; p)]

=

LAF* Y* 28 ' M* (p'; p„) Y*2'« F-1* w(n)] X

X [w{n) M(p„ ;p)].

(4)

After use of the relation (11.14) for w(n) and multiplication by A-1 on the left the following result is obtained: i[F* F* 28 ' M* (p'; p) F* 28 F-1* ^2s (p) - ^>28' (p') M(p'; =

(2^2

n

dHv

+ ' •f -"n V

Pn)

F* F* 28 ' M*(p';

p)] =

pn) YF~™®(pn)

M(pn;

p).

(5)

The symbols F*28» and 3> (p„) stand for the product of the corresponding operators for each particle in the intermediate state. I t should be noticed that in this formula

401

Dirac Formalism for Arbitrary Spin in S-Matrix Theory

we have assumed that particles with $ = — 1 as well as £ = + 1 occur as intermediate particles. If one wishes to avoid this one must replace Q) (p„) by the proper projection operator (see (11.21)). I n practice, the intrinsic parities of the in- and outgoing particles in the reaction are often known. If the parities are f and respectively, this fact is taken into account by multiplying (5) with 1 /2 (p') + £') on the left and 1 ¡2 (3! (p) + f) on the right. The result is: Ap(p') i[F* Y*28' M*(p';p) = A? (p') (27i)*£

n

'

6H P

y*2s F~l* p

+ '' ~ J V

JI

Vn)

"

- r28' M{p'; p)] As(p)

F* Y™' M* (p'; pn) Y*^

= &{Pm)

x

X M(p„;p)A((p).

(6)

The amplitude M* (p'; p) contains the complex conjugate of the invariant amplitudes and the y matrices, which may be present. Because of (A62) and (VI.2) the effect of the operators F and Y on the left hand side in (6) is restricted to the Dirac spinor indices. The dependence of M (p'; p) on these indices can be expressed with help of the usual y-matrices for spin 1/2. The following relation is valid for spin 1/2: (7) p* Y*y>1* y*F~™ = y,JL. Thus the two terms on the left-hand side of (6) are identical with the exception that the first contains the complex conjugate of the invariant amplitudes. If these are appropriately chosen, the real parts on the left-hand side of (6) cancel and we are left with the imaginary part of every amplitude. (To that end, we must, for instance replace B by iB in (VI.29).) We will now prove that the unitary condition (5) is invariant under the CPT transformation given in (V.ll). In order to make some crucial points clearer, we make the transformation in two steps, i. e. first a strong reflection, then move every particle to the other side by crossing symmetry. If we apply the strong reflection symmetry (V.10) on all amplitudes in (5), we obtain after multiplying with Y2s' from the left and Y2s from the right: i{F* y*2s' M* (— p'; -p) =

n

Si(P

y*28 F-1*

t'Pn" -"n '

Pn)

(-p)

F*

- ®2s'(-p')

M*(-p';-pn)

M( — p'; -p)} Y**°nF-i*&(-pn)

xM(-pn;-p).

= X (8)

The sign of the argument in the Dirac operators changes because of the relation Y*@(p)

Y2S =3>(-p).

(9)

Because of the non-analyticity of the unitary condition, we cannot consider (8) as an analytic continuation of (5). The next step is to use the crossing relation M(-p';

-p)

= A~FM(p\

p')

(A-F)-1

(10)

402

B . O. E n t l o and B . E . L a u r e n t

on all M-functions in (8). The relation (10) follows from (V.l), (V.12), the Pauli principle and the rules for moving particles in the Tli-function. The combination of (8) and (10) gives i{3>~2'(-p)

Y~2°A~M*(p;p')A~-1Y~2s'-A~FM(p;p')(A~F)-19h'*°'(-p')}

= ( 2 ^ 2 n

di(P

~t;v~ - " » '

X M* (p„; p') A y -

Pn)

2 5

A~FM(p;

pn) (A~F)~i

=

®~(-pn)

A~ X

' .

(11)

After multiphcation with 2d~2s(-p) on the left and S ~ 2 s ' ( - p ' ) on the right, taking the complex conjugate, and using (9) we obtain: i{F* Y*2sM*{p; = (2nfZ

p') F*28' F-1* @»'(>p') -

8i(P

+ n

;v7 ^

Y-2*'A~

M(p; p')} =

* Y**M*(p;

pn) Y™»F-™2(pn)

F

M(pn; p'),

(12)

i.e. the unitary condition (5) is CPT invariant. We see that without the Pauli principle, one factor of the type (—l) 2 8 would enter on the left-hand side of (12) and two such factors would enter on the right-hand side. The invariance of the unitary condition under a CPT transformation would then be destroyed. This constitutes and example of the intimate connection between the Pauli principle, crossing symmetry, CPT invariance and unitarity. VIII. The limit m/p°

0

For zero mass particles, the ^-matrix and thus also the T-function cannot be defined because of the non-existence of asymptotic plane wave states. Since, however, physically meaningful results can in many cases be derived by ascribing a small mass to the particle considered, then letting the mass tend to zero at the end of the calculation, we shall consistently treat the zero mass particles as the limiting case m/p0 0. Thus we are going to investigate the w's in that limit. There is an equivalent possibility of introducing limit functions directly by a procedure similar to that followed in chapter I I . Until otherwise stated, it is assumed that p° > 0. I t turns out that it is suitable to investigate the limit of certain linear combinations of the w's, namely eigenfunctions of Y, rather than of the w's themselves. We denote the functions for which we shall take the limit m/p° = 0, by Q(p, .s3, rj), defined for spin s by:

(

m\s 1 2 /

Y

?

S

3

'

^

=

+

N

W

(

~

P

'

=

~

^

=

±

L

From the properties of Y and Z in chapter I I follow: Y Q{p,s3,rj)

=

rj Q{p, s3, rj)

ZQ{p, s3, rj) = D(pv, ss,

-rj).

(2) (3)

403

Dirac Formalism for Arbitrary Spin in ¿/-Matrix Theory

From (1) also follows:

(

m\2t ô

2 )

(4)

i\-i-

Whenp is in the -positive z direction the orthonormalization of Û (p, as follows, by use of (II.8), (3) and (4):

Q(p,

s's,

co»

Q(p\s3,r,)

V

' ) Z Q(p,

s3, n)

, ai,

=

n -(P°+|PI) I 7ÏI I

=

r1 =

r,')

n \ - ( I Tit

fi I

Jfl

+ |p|)

(jfi

P ° + \ P

Q(P

s3, rj)

~

.

«3. v')

is deduced

( P ° + \ P \ )

' S3>

Q(P

—v)

l 2 '*' /m\2s

1~(P°

+

\PI)\

(

T

(5)

)

From the fact that the left hand side of (5) can be written Q+D in a representation, where A = Z, it follows from (5) that the functions 0(p, s3, rj) tend to zero for m/p° -> 0, unless rjs3

=

s.

(p

in positive

z

direction)

(6)

Thus effectively only two components are left. Because, for small m 12«

1 ~ ( P °

+

\P\)

= fi

1 / m2 m \2(p°)2

- IP I

+

y "'/

(?)

we obtain for the limit functions m (p, A), 12 ] = s: w{p,i')Zm(p,X)

(p0)28^,

=

(8)

where % is the helicity. The coordinate system, in which (8) is evaluated, is special only in the sense that the z-axis is chosen in the p direction. Since p° as well as Z is invariant under space rotations, the normalization (8) is valid in all Lorentz coordinate systems, i.e. we have an invariant normalization of co(p, X). We conjecture that the extension of (8) to a manifestly covariant form implies use of the y's mentioned at the end of chapter IV. In order to be able to sum over intermediate polarizations in the unitary condition and in calculations of cross sections, we must also have a completeness relation corresponding to (11.14). This relation is deduced as follows, with the spinor indices suppressed: 2

Q(p,s3,7ì)Q(p,s3,rì)

Z

Sj= -s

- = i (™Y ~ 2\2 J

£

1=±l

X

w{p,

S3,

f

=

1)

. »s

n

® ( P ) Z i 1 + v Y ) v -ÏYZ

w(p,

T

s3,

f = 1) X

( l + r i L

Y ) Z

=

p

(9)

404

B . 0 . ENFLO a n d B . E . LAURENT

If Y and 2 • p are chosen diagonal, then according to (IV.2) all the matrix elements except two tend to zero for m/p° 0. The sum in (9) then has contributions only from the two terms with r j s = s . The completeness relation for the limit functions m (p, A) is then: 3

£ (p, A).

(29)

406

B . 0 . ENFLO a n d B . E . LAURENT

From (27) and the obvious relation (30)

A~F & Z = Z~0>~ A~F

we see that the M(\mp) function satisfies the same crossing relation (25) as the M(;np) function. As the crossing relation is the same as before, the GPT transformation is also the same. Using (10) we can easily solve (28) and (29), obtaining the M-function expressed in terms of the T function. In very much the same way as (VII.5) was derived we may derive the following relation in the case m = 0 J [ ( _ 1)2«'

X 2 n

F*M*(mp';mp)

F-1* - M(mp'\mp)] F*M*(mp'-mpn)

y

=

( 2 T E ) 4 ( - 1 )2S'

Y™» F~»

X

M(mpn-mp). VPn) "

(31)

We have here assumed that all the particles involved have zero mass which may not be very realistic. One may, however, freely mix particles with and without rest mass in the unitary condition, just picking the relevant factors from (VII.5) and (31) respectively. Examples 1. Spin 1/2 The projection operator 3 s for spin 1/2 is according to (11), (1.8) and (III.11) ^ = ¿0

(32)

According to (12) this gives v*r

r° = 2P° ">(P>

(33)

Multiplying this by p^y*1, remembering p2 = 0, we obtain p^y" w(p, X) = 0.

(34)

The massless spin 1/2 particles which occur in nature, the neutrinos, have the property that, though their intrinsic parity cannot be determined, they have a definite fermion number. More precisely the neutrino with A = 1/2 has the fermion number —1 and the neutrino with X = —1/2 has the fermion number -j-1. We thus formally assign to them the intrinsic parities f = — 1 and £ = + 1 respectively. The crossing rule (26) now follows immediately from (V.5). The T-function for m = 0 may be regarded as the limit of the T-function for m =)= 0 if we choose in the latter case, the following jW-function: M(p'-,p) = {A+(p'){i

-

Y)yA+(p)-A-(p')(i

+ Y)y A-{p)} •

(35)

This ^/-function corresponds to the well-known weak interaction given by F E Y N MAN and G E L L - M A N N [12]. Notice that the presence of the A operators, which is necessary for fermion number conservation, makes the M function itself undeter-

407

Dirac Formalism for Arbitrary Spin in iS-Matrix Theory

mined when m -> 0. The neutrinos do not admit separate C and P transformations. The CP invariance has the following form where M(mp';mp)

M(mp';mp)

= Mcp (mp' ;mp) = — Z M (mp'p-,mpp) Z

=p'°0>Z[-

1/2(1 -Y)y»

+ 1/2(1 -+- Y) yf] & Z(~p°)

(36) •

(37)

2. Spin 1 The projection operator 0> for spin 1, is, according to (11) and (111.27) =

W

Pv - ex («, fi) - ex(fi, v)} Z*"

(38)

where Z is given in (III.14). Remembering p2 = 0 we see that (12) gives p,a**{p,X)= As

0.

(39)

commutes with Y we obtain from the same equation e ^ p f o ^ p , A) = 0.

(40)

In (39) and (40) we recognize the Maxwell equations. The photon is the only massless spin 1 particle observed in nature. Being a massless particle, it has no definite intrinsic parity and in contrast to the neutrino it has no fermion number which makes one or the other intrinsic parity seem natural. One may, without physical consequence, assign e.g. negative intrinsic parity to the photon. We feel, however, that it is more elegant and more englightening to regard the photon as the limit (when m 0) of a superposition \V, s3,rj) =

{\p, s3, £ = 1) - 7)\p,s3,$ = -

1)}.

(41)

This has the consequence that the M (; np) function can be taken to be the same as the M(\p) function apart from a factor 2/m which must be inserted to compensate for the corresponding factor in (1). We see that with this convention we obtain the crossing relation (26) for photons, using (V.5). We also see that the invariance under space inversion and charge conjugation must take the same form as for the case m =(= 0. 3. The N — N — y t r i o d e This example deals with the M-function M(k /iv, pz \ px), where k is the photon momentum, k2 = 0 pa outgoing nucleon momentum pl ingoing nucleon momentum. 29*

408

B . 0 . ENFLO a n d B . E . LAURENT

With the substitution p = 1/2 ( P l + p2)

(42)

and suppression of Dirac spinor indices, the most general fermion number conserving M-function satisfying G and P invariance is Mimk/iv^^pj)

= A+(p2)(P^yU(Pi)

+ A-(p2) i y5&"* i y6 A-(pt)

(43)

where (Pf = A s^ + B(yu J? - y'pf) + C(yi*k" — yvk") + D(p^kv — pvk").

(44)

Because the photon has zero mass, we cannot, as in the deuteron example, project out any definite intrinsic parity by using a vector representation. In fact, we find it simpler to stick to the antisymmetric tensor, which gives us no gauge troubles. Having C and P invariance, it remains for us to use equations (28) and (29) solved, which tell us that the iW-function (of type m) must be divergence-free in the same manner as the co function. The equation corresponding to (40) tells us that the Mfunction must have the following form M{mk fiv;) = «/"A? — J"kf

(45)

and the equation corresponding to (39) tells us that must satisfy k^J" = 0.

(46)

4 = 5 = 0

(47)

Equation (45) is satisfied if we put

and equation (46) is satisfied if we moreover require i Cijn^ - m2) + Dp- k = 0.

(48)

Using the identity p • k = 1/2 (m2 + mj) (m2 — mj)

(49)

we see that (48) is automatically satisfied if mx — m2 but if Wj 4= m2 w e must put •

(50)

Thus we have one single amplitude if the masses are unequal. Physical examples of such reactions are ¡x e y, A-^-n-f-y. This result is derived also in [5] p. 267. The identity ^ | i yfy J p> A+ (p2) A+(pl) = i 1 ^ a ( f t ) y*A+(Pi) - 2" ¿ + ( f t ) « " ( f t - ft). MPi) (51) can be used in the case when = m2 to express the charge and the anomalous magnetic moment in terms of the constants C and D. In the case mx =(= m2 the yf terms cancel due to (50) and the entire interaction is of the Pauli type. This

Dirac Formalism for Arbitrary Spin in (VII.5) and (31), the right-hand side contains a term with one intermediate photon. That term contains an inner product of (53) and (52) of t h e type in appendix 4. From this appendix we conclude t h a t an intermediate photon effectively gives a vector coupling. I t is essential to notice t h a t we have not used any quantity corresponding to the vector potential in deriving this result. The vector potential is not needed in a pure 8 matrix theory. I t is the requirement of local interactions which necessitates the use of a potential in field theory. Cf. S. WEINBERG

[li].

IX. Comparison with the field-theoretic formalism for spin 1/2 I n this chapter we shall briefly treat the differences between the formalism described in this paper and the field-theoretic formalism usually used to describe reactions with spin 1/2 particles involved. We restrict ourselves to these particles because of the lack of a field-theoretic formalism directly applicable to arbitrary spin and because the »S'-matrix formalism in this paper is developed for all spins in close analogy with the spin 1/2 case. For example, we have treated charge conjugation and crossing symmetry very differently from the usual way of handling these things. I n order to clarify the above-mentioned difference, we first establish the connection between our functions w (p, s3, £) and the generally used u and v spinors, f o r which we adopt the conventions in [#]. We will make the comparison in a coordinate system chosen so t h a t the momentum is in the z direction. The functions w(p\ ± 1/2, ± 1 ) are to be compared with the conventionally used functions u± (p), v± (p), where the notation is taken from [5]. I n the following the momentum variable is suppressed. With use of (1.7), (11.10), (11.27) and (A26) we obtain the following relations in t h e left column. The right column gives the corresponding relations in [8]. i y 5 w ( ± 1/2, + 1 ) = - u > ( ± 1/2, - 1 )

Y b

u± =

±v

iy5w(±

1/2, - 1 ) = - w { ± 1/2, + 1 )

y5«T=T«±

C*w*(±

1/2, + 1 ) = ±M>(T1/2, - 1 )

C * u l = v ±

C*w*(±

1 / 2 , - l ) = T w ( T 1/2, + 1 )

C*v%=u^.

T

(1)

(2)

410

B . O. ENFLO and B . E . LAURENT

The corresponding relations in (1) and (2) are identical if w ( + 1/2, + 1) = e-i"'* u+ w ( + l / 2 , — 1) = e-«' 4 (—i) vw ( —

1/2, + 1) = e-"' 4

u -

w { -

1/2, - 1 ) = e-«' 4

i v + .

(3)

Of course we can now, in conventional formalism, use the functions u±, v± to define an M-function, which we denote M.(p' \P)- Its connection with the Tfunction is: T ( ; p ± ) T

(

p

±

T ( ; p

= M ( - , p ) u • , ) = M {

± )

=

v

±

i

p ) v

±

±

( p ) t e ) -

0 ,

( p ) M ( P

(4)

where p indicates an antiparticle with momentum p. Comparison of (4) with (11.33, 34) and use of (3) and the spin-flip equation (11.27) give the following relations between the two types of M-function: M ( ;

M ( p

p )

= e*'4

M ( ;

;) = e -i "' 4

A +

p )

A + ( p )

(p)

M ( p

e-"' 4

+

A

~

F

;) - e!'*'4 M(;

A - ( p )

p )

M ( p

A - ( p )

;)

(5)

( A ~ F ) ~

l

.

(6)

The crossing relation follows from (5), (6) and (V.l): 1 1 ) K ( P - , )

=

(7)

M ( ; - P ) .

Recall that we have used a coordinate system in which the z axis is in the p direction. The equations (5), (6) and (7) are, and the equations (3) and (4) can be made independent12) of that choice of coordinate system. In (4) we can also use helicities. Of course it is quite possible to introduce creation and annihilation operators for free particles from the physical particle states, i.e.: | p,

a

( P >

s3, i )

=

«3. £)

a+

IP'

(p,

s

3>

s

i )

3

| vac)

(8)

= Ivac) •

(9)

, i )

For clarity, we introduce separate fields for the two fermions with different intrinsic parity rather than describe both fermions and antifermions by the same field. The free particle operator fields in coordinate space are then defined as: (¿v. $ f

d

'

p

y ^ °

(

p

'

w

(p

' *»i}

gipx

•

(io)

) This crossing relation is in accordance with the „substitution law" in [S], p. 162. ) In the conventional formalism we can work with u(p, s3) and v(p, s3) generally defined by (3). u

l2

Dirac Formalism for Arbitrary Spin in S-Matrix Theory

411

This field now fulfils the Dirac equation in coordinate space: ( » J V P " + £»)¥»(*; f ) = 0,

where

(11)

. 8

V =

—

•

dx„

The spinor fields generally used in field-theoretic formalism can now be constructed in accordance with [5], formula (3—86), with use of (3): *P(x) =ei»li[y>(x;£

= 1)-J?-1

f = — 1)].

(12)

I t is easy to verify t h a t these „conventional" fields W{x) fulfil the Dirac equation (iYllp(x; f). They are (cf. [±Z+

Y->±Y*,

(Al)

Z~^±Z*.

(A2)

Because of the uniqueness of Zk, Y, Z u p to a similarity transformation, the new representations given in (Al), (A2) must be equivalent with the old ones. Thus there must exist matrices A and F with the properties: AEkA~1

= Zh+, AY A-1 = - Y+, AZA-1

FEkF~i

=

FYF-1

= Z+

= - Y*, FZF-1

= Z*.

(A3) (A4)

From the relations (A3), (A4) follow: A+ = aA, 1

F* = fF- ,

a*a=

1

f* = / .

(A5) (A6)

From (A3) it is seen t h a t A is determined up to a multiplicative factor and from (A5) it is seen t h a t this factor can change the phase of a. We choose a to be equal to 1. I n the standard representation Ek, Y and Z are hermitian and A t h u s com14

) It is easy to verify that (20) applied to (12) with (10) inserted gives a change of the parity (without spin-flip) of the creation and annihilation operators.

Dirac Formalism for Arbitrary Spin in S-Matrix Theory

413

mutes with X k and Z and anticommutes with Y. Therefore A can be given as

*-n

«A»)

in the standard representation. It is also seen from (A4) that F is determined up to a multiplicative factor. This factor can change the magnitude but not the sign of /, as is seen from (A6). We choose |/| = 1. Under a similarity transformation on E k, Y and Z, F transforms as follows, in order that (A4) be invariant:

F' = g(S) S*F S- 1

(AS)

where g(S) is a c-number which depends on the (A6) we obtain: F'* = g* (S) g{S) fF'- 1

transformation. Instead of (A9)

and thus it is seen that the sign of / is invariant tinder an ¿'-transformation. In order to determine the sign of the constant / we can thus use the standard representation. There F anticommutes with Y and commutes with Z and thus can be written

Now /} must be determined so that F&F-1 = -Zh*.

(All)

In the standard representation for angular momentum K ± ia t ) K > = y (a T *») (« ±

+ 1) |«rí ± *>

(A12)

the following definition of /? is used, in order to give (All) 0ki> = (-1)04 !-•

(A13)

From (A13) there follow immediately yS2 = 1

(A14)

/Ja3/H=- = - K T

(A15) ki>-

(A16)

Because the o*'s are hermitian and cr3 and % ¿ ¿ff¡ are real, we can write (A15) and (A16):

p