R!IC/68/50

IETEHHAL REPORT

(Limited distribution)

SPECTRAL REPRESENTATION OP THE COTARIAKT CTO-POIHT IOTCTIO1? AITD

INFUHTB-COMPONENT FIELDS WITH ARBITRARY MASS SPECTROTI *

I.T. Todorov

and

R.P, Zaikov **#

TRIESTE

June 1968

* This was the subject of a seminar given at the InternationalSymposium on Contemporary Physics, Trieste, June 1968,

** International Centre for Theoretical Physios, Trieste.

On leave of absence from Joint Institute for Nuclear Research,Dubna, USSR, and from Physical Institute of the BulgarianAcademy of Sciences, Sofia, Bulgaria.

*** Joint Institute for Nuclear Research, Itobna, USSR.

ABSTRACT

The general form of a Lorentz covariant two-point function

ie written down in momentum space as an expansion in terms of the

total spin eigenfunctions. It leads naturally to a local expression

for finite-component fields but incorporates non-local infinite-

component fields. Explicit examples of such fields with increasing

mass spectrum are constructed. The two-point function is shown to

fall exponentially for large space-like separations provided that

the lowest mass in the theory is positive.

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SPECTRAL REPRESENTATION OP THE COVARIANT TWO-POINT FUNCTION AND

INFINITE-COMPONENT FIELDS ¥ITH ARBITRARY MASS SPECTRUM

INTRODUCTION

Usually, -when one is dealing with fields describing particles

of definite spin, the spectral representation of the two-point

function is given by an integral in the total mass [l,2J . If,

however, a field contains more than one spin, then it is necessary

to decompose the two-point function with respect also to the spin

variable (i.e. with respect to the second Casimir operator of the

Poincare group). This is always the case for infinite-component

fields. We find the form of such a representation for a field

transforming under an arbitrary irreducible representation of

SL(2,C) .(the result applies also to more general fields which can

be decomposed into irreducible components). The clue to the

solution of the problem is the use of the formalism of homogeneous

functions of two complex variables (instead of the tensor formalism)

in the description of the irreducible representations of the Lorentz

group (see e.g. [3J or Appendix A to [4] ).

For finite-component fields, Poincare invariance and spectral

conditions imply the equivalence between TCP and weak local

commutativity [5] . For infinite-component fields this is not the

case [6] . In Sec.3 we construct explicit examples of non-local

(TCP-invariant) generalized free infinite-component fields with an

increasing mass spectrum (with respect to spin). If the lowest

mass in the theory is positive then the two-point function (and

hence both the commutator and the anticontmutator) decrease

exponentially for large space-like separations.

1, GENERAL FORM OF THE COVARIANT TWO-POINT FUNCTION

A field V(x) transforming according to the irreducible

representation 170 l^Q, - O of SL(2,C) can be written down as a

homogeneous function y(x,z) of the complex (Lorentz) spinor

(zv z2) (cf. [8] )

(1.1)

where the degree of homogeneity (y ,

the representation IiL, i . 3 by

related to the number of

(we recall that the single-valuedness of the ~\U implies that

v - V «'2i is an integer). For the special case of finite-

dimensional representations (for i/-, Vp ~ non-negative integers)

0(x, z) is "by definition a polynomial of z and 2", its coefficients

"being the ordinary (spinor or tensor) field components. In particular

the Pauli two-component spinor f and the vector field A correspond

in our notation to the polynomials

2,

3

, z) = V AM{A(x

(further, the> sum sign in similar expressions with repeated upper

and lower indices will be omitted).

The relativistic transformation law for ̂ has the form

U(a,A) u"X(a,A) x + a ; -z A " 1 )

where A€SL(2,C) and A = A(A) is the proper Lorentz transformation

defined by

ACT A*v V

( (1.4)

- 3 -

where

cd _ rac ,. bd

All non-vanishing scalar products of these vectors are proportional

to the product of the complex conjugate invariants Vc « z£w and Tc :

£*7 = - XX = 2 \K I2 , £X = rjX = fX = 17X = 0 . (l.ll)

More generally, the tensor indentity holds

XX" + U =?T7 + r) I - g |rj; (1.12)

it implies in particular that

- ( x P ) ( x P ) = f p2 f n = P 2 | « | 2 . ( 1 . 1 3 )

Assuming -^0^. IXQ! (which can be achieved without loss of

generality by a possible interchange z*->w)and taking into account

the above identities, we find the following general form for the

invariant kernel |(_ (see Appendix A) :

\\(Piz,w)= IC ° (Xp) ° (p?) (prj) h(cosG;p2) , (1.14)

win ere

cose - 1 -

(e is the'angle between J, and j? in the rest frame of p). The

kernel (1.14) is one-valued if and only if i Q ±i' are integers

which will be always assumed.

2. DECOMPOSITION OF THE IF7ARIAM1 KERNEL WITH EESPECT TO SPIN

The kernel (1.14) describes in general propagation of

particles of different spins (provided that the representation of

SL(2,C) for the field is not of the form [& Q>\& J + 1 1 ). We shall

now find the invariant kernels which are eigenfunctions of the spin

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square operator

S4M M M M 5 i = i 2 + i2 Ĵ— 2 crp cr/u 2 0 1 2 CT

P P(2.1)

Here M are the generators of one of the representations [•£„,-£-,]o r [i , i ] under consideration. In terms of z, I^" can be

written as

2\

where 7^ are the Dirac matrices in a basis in which

5 ('

(see C4J )• As far as we are looking for Lorentz invariant

solutions of the equation

[s2 -s {S + l)] f((p;z,w) = 0 (2.3)

(of the form (1,14)) it is convenient to go to the rest frame

(p B 0) in which

S2 = M2 = i M . . M « 4 ( z a . f - §= a.zf-~» — 2 i j 4 j oz 0% 1

= i(z3l-zaf} "2 {(zfa^)(z£aT) + ( z € ^ K z f a f )

(2.4)

(Acting on a homogeneous funotion the first term in the right-hand

side of (2.4) gives Jl2). Substituting (2.4) and (1.14) (with p = 0)

in (2.3) we obtain the following equation for h :

(dcoser " ° 9x h(cos6;pz) = 0 .

(2.5)2,

The general solution of this equation, regular for cos & « 1, is

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2 2 { W Wh (cosS; p^) = p (p^) P , (cos6) (2.6)s s s-JiQ

where P^a (x) are the Jacobi polynomials (^n>UJ)«

2¥e would cone to the same result starting with an _S_ in

which Jr are the generators of the representation ^ n ' ^ l

expressed in terms of w and 3/9w(this is a consequence of the spin

conservation and can he checked direct).

In general, h(cos$, p ) is a superposition of the spin-

eigenfunctions (2.6)j s takes a finite number of values if at

least one of the representations [$- , JL A and [J?'J?!] is

finite dimensional; otherwise i t assumes all values of the form

£Q+ n where n is a non-negative integer.

In the important special case where y>= \p* (and, hence,

A= "£Q 2 0, Jl^Xi ) the corresponding decomposition of the invariant

kernel reads

&^Xt3 (2 .7)

We point out that the kernels K, defined "by (2.7) are positive-

definite in this case "because

6

U S ( . (P , I ) ^s^fp^) (2.8)

where A is a positive constant and

p°-t- nn -r Oj p 1

( O ^ ( B ^ B (2.9)

f * are the canonical basis vectors (see Appendix B).

For finite-dimensional representations of SL(2,C) £{'£Q

is a positive integer and K & is a polynomial with respect to

p. In that case the (weak) locality condition for the two-point

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function is equivalent to TCP invariance [5] which implies [6 ]

For instance, for a local tensor field y ({XQJIA = [o,n + l] )

eq.. (2.7) reduoes to

- (2.n)

where K is a homogeneous polynomial of p of degree 2n;

(2.12)

and "g and In are defined "by (1.8).

For infinite-component fields the locality of the two-point

function is rather an exception. It implies that 1C is a poly-

nomial with respect to p^ , piy and pX (or p"X ).-[lCQ . As an

example of a local infinite-component field we take the free field

z) of mass 171- , transforming under some unitary representation

i

(2.14)

Loth, examples (2.13) and (2.14) correspond to infinite-mass

degeneracy with respect to spin. This is not accidental, Grodsky

and Streater showed recently [ill that an infinite-component

field with polynomially bounded two-point function in momentum

space can be local only for infinite mass degeneracy.

3. EXAMPLES OF NON-LOCAL INFINITE-COMPONENT' FIELDS KITH

INCREASING MASS SPECTRUM

In agreement with the above-mentioned general result

all examples of inf inite-ooraponent fields with, a non-trivial mass

spectrum, obtained by specializing the coefficients P (p ) in (2.7)»

correspond to non-local theories. However, if the lowest mass of the

theory is positive, i.e., if all o (p ) vanish for p < m Q (> 0),

then, just as in conventional theory of finite-component fields, the two-

point function (l«5) goes to zero as r* e"1^0** (with some real X ) for

r = -(x-y) —#» (see Appendix C). The same is true (as a con-

sequence) for the vacuum expectation values of both the commutator

and the anticornmutator of

_

( 3 # 2 )

where

R - (1 - 2ccos^ e-ia(nz) + c2

( Znw)2^ ( z n z ) ^ " 1 ( w n w / 1 ^ - 1 , (3.3)

and the range of integration is the upper unit hyperboloid

n = V 1 + n~ * All other two-point functions (as well as all

truncated Wightman functions) are zero "by definition. The field

f(x,z) so defined is TCP-invariant hut not weakly local. For the

mass degenerate limit a a 0, if we put c = 1 , m,. = m , £Q** 0, ij = -g-

we find a local field of the type (2.14).

It would be interesting to analyse the implications of

locality on the two—point function in the case when it is a Jaffe

type generalized function [12] and to find whether the Grodsky-Streater

••No-go theorem" [llj can he extended to this case also. We hope that

a further investigation of the examples provided "by the representation

(1.5) and (2.7) may give some hint for the solution of this problem.

ACKNOWLEDGMENTS

The authors would like to thank ProT. D.I. Blokhintsev and Dr. D. Tz. Stoyanov for interestingdiscussions and Prof. R. F. Streater for a pre-publication copy of his paper.

The first-named author (I.T.TJ is grateful to Professors Abdus Salam and P. Budini and to the IAEAfor the hospitality kindly extended to him at the International Centre for Theoretical Physics, Trieste,where the present paper was completed.

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APPENDIX A

DERIVATION OF THE REPRESENTATION (l.H) FOR THE INVARIANT KERNEL

We first determine the general form of the invariant monomial

al a2 bl - b2 c d

M = K K (xP) V P ) z(gp)c(np)a (A.i)

of degree of homogeneity (v.,v?) (1.2) with respect to z and

iyl. , Vp) with respect to w. Recalling the definition of K , g , r\, X

in terms of z and v (see (l. 8), (1.9)) we find the following set of

equations for the exponents in (A.I):

i

(A.2)

The general solution of the system (A,2) may be expressed in terms of

two arbitrary parameters k and X

c o £ - £ - 1 - X f d - l ' - i - -1 -A. .1 0 l ° (A.3)

Substituting (A.3) in (A.l) and using (1.13) and (1.15 ) we find

fl+fl' 0 -Q[ 0 - $ -" * i * -0 0 fc , 1 0

io+io,Y.Vii.,e,

irV1K (Xp) (gp)

{1 - cos6) k (1 + cos0) X " k (A.4)

The homogeneous invariant kernel K is a superposition of such

monomials and hence has the form (1.14).

We mention that the scalar products' p£ and prj are always

positive (for ? n = I § 1 , Hn = 1 1 , P > ] p | ) s o *ha-t any (complex)

u i — I u ' -̂ - W

- 1 1 -

. + .*,• .I

power of these products is well defined and regular in the domain

of integration in (l»5)» This is not the case for the first two

factors in (A.4) (or (1.14)). That is the reason why we ask the

ejcponents of X and ^"p to "be positive and apply (1.14) only for

Jo - 1 U©! • If we had instead, for instance, ,̂ > j £J we shouldput A= to" to+/l' an(i rewrite (A.4) in the form

M = K (Xp) ($p) (i?p) - \, o J- (1 + cos0)2^ (P

2)k

(A. 5)implying a corresponding modification of (1.14).

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APPENDIX B

EXPLICIT EXPRESSIONS FOE THE CANONICAL BASIS AND THE SPIN EIGENFUNCTIONS

AND PROOF OF THE SIMM ATI OE FOKTULA (2. 8)

The transition from the continuous variable z = (z,, z?) to

the discrete indices s£ is given "by the expressions of the vectors

= f (z) of the canonical basis. We shall present them in a

form which exhibits their relation to the jnatrix elements of the

irreducible representations of SU(2) and allows automatic summation

over the spin projection to be performed (in particular, to prove

(2.8)).

To do this we introduce generalized polar co-ordinates in the

two—dimensional complex space putting

= yr sin—- e l n ... = \r cos— e , z = yr sin— e

( r > . 0 , 0 < e < T T , 0 < < p < 2 i r , 0 < a < 4 j r ) . (B.l)

It is easy to express the generators of SL(2,C) in terras of these

variables. The infinitesimal generators of the SU(2) subgroup (the

"compact generators") depend only on the angles

M 3 * - i — M = M X + iM 2 = e±i(p (± — + i cte —M ^ ' M ± M -liVi e \*Qe g 9c sin da

the generators of the pure Lorentz transformations have the form

q 3 9N = i(cos0 r r - - sin0 r r ) ,

or od

± .

(B.3)

The canonical basis for any irreducible representation f^ \. J of

SL(2,C) is defined by the set of properly normalized eigenvectors |

of Sjf and M3 (see [7], M )

M 2 | s ? ) = s ( s + l ) | s S ) , M 3 ] s ? ) = S i s ? ) ,

- s < ? < s . (B-4)

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The conventionally normalized [4] solution of eqj. (B,4) has theform

£ ^n -8,-1 -.(s) , o .

I . O - f . ^ W A . r 1 Dioc

cos© = cosO cos© + sin6 sin6 003(9., - cp )

( B . 9 )

Comparing ( 2 . 7 ) w i t h ( B . 8 ) and ( B . 9 ) we check ( 2 , 8 ) .

- 1 5 -

APPENDIX C

ASYMPTOTIC BEHAVIOUR OP THE TWO-POIBT FUflCTICOf FOR LARGE SPACE-LIKE

SEPJLRATIOHS

We shall only outline the derivation of the exponential decrease

of the two—point function (1.15) (in a theory with a lowest positive

mass m^) without going into mathematical details.

To state the problem properly we have to pass from the

continuous variables 2 and w to vector variables f and g belonging

to the representation space (the function F ^ (zj a,w) is in general

a distibution of z and w - it rcay not be defined for some special

values of these parameters). Namely, let the field y(xtz) be trans-

forming under the irreducible representation f= l^o/^ii &nd let

f (z) € D, T , where D T is the set of homogeneous function of

degree (1.2), infinitely differentiable in the -whole complex space C

except the origin. Then, in general, only the linear functional

(C.I)

has a meaning (as an operator valued dis t r ibut ion in S(Rr)).

In these terms the two-point function can be written in the

following way:

F A (x; f / g}=

Now we put x m 0 and choose the third axis along x (so that

x - (O,0,r), r > 0 ).

We see from (C.3) that for k sufficiently large

is absolutely integrable^ This permits «s(after a repeated integration

l^y part with respeot to n^and a subsequent integration in n, and n~)

to rewrite (C.2) in the form

with

t K s

REFERENCES

1. S, Kamefuchi and H. "TJmezawa, Progr. Theoret. Phys. (Kyoto)

J5, 543 (1951 )j

G. Kallen, Helv. Phys. Act a 2^t 417 (1952);

H. Lehmann, Huovo Cimento ll_f 342 (l954)j

0. Steinmann, J. Math. Phys. 4_, 583 (1963).

2. I. Rasziller and U.H. Schiller, Huovo Cimento A48, 617, 635,

645 (1967).

3. I.M, Gel'fand, M.I. Graev and IT, Ya. Vilenkin, Generalized

functions, rv.5: Integral geometry and representation theory (Academic

Press, New York,1966); see also Appendix B to v.l:Properties and

operations (Academic Press, New York, 1964) .

4. D. Tz. Stoyanov and I.T. Todorov, J. Math. Phys, £ (1968 )j

or ICTF, Trieste, preprint IC/67/58 (1967).

5. R. Jost, Helv. Phys. Acta ^0, 409 (1957); see also

R.F, Streater and A.S. Wightraan, PTC %i_sp.in-iand- stati sties and

all, tvig.t,T (W.A. Benjamin, New York, 1964).

6. A.I. Oksak and I.T. TodojovTInvalidity of TCP theorem for

infinite-com-ponent fields, Buhna, JINR, preprint (1968).

7. ¥e are using the notation of I.M. Gel'fand, R.A. J'inlos and

Z. Ya. Shapiro, Re-presentations of the rotation and Lorentz groi.rp_..j'-rid

their a-p-plicationa (Pergaraon Press, London, 1963) for the irreducible

representations of SL(2,C). In H.A. Haimark, Linear representations

of the Lorentz group (Pergamon Press, London, 1964) they are denoted

8. Hao Vong IXic and Nguyen Van Hieu, J. Nucl. Phys. (USSR) 6.,

1861 (1967).

9, Actually, this requirement is not independent. We shall prove

else.where that it follows rigorously from the invariance assumption.

-18-

10, N.IT* Bogolutiov and V.S. Vladimirov, Naucbnye Dolclady Vyschei

Shkoly No.3, 26 (l958).and No.2, 179 (1959).

11, I.T. Grodsky and R.F. Streater, Phys. Rev. Letters 2(3, 695 (1968).

12, A.M. Jaffe, Phya. Rev, 1^8, 1454 (1967).

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