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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.
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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
-5-
-
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
-7 -
-
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 -
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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
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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
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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-
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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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