Fortschritte der Physik / Progress of Physics: Band 27, Heft 4 1979 [Reprint 2021 ed.] 9783112522660, 9783112522653

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FORTSCHRITTE DER PHYSIK H E R A U S G E G E B E N IM A U F T R A G E D E R P H Y S I K A L I S C H E N

GESELLSCHAFT

DER DEUTSCHEN DEMOKRATISCHEN

REPUBLIK

VON F. K A S C H L l i H N , A. L Ö S C H E , R. R I T S C H L U N D R. ROM PK

H E F T 4 • 1979

A K A D E M I E -

• B A N D 27

V E R L A G

E V P 10,— M 31728

.

B E R L I N

BEZUGSMÖGLICHKEITEN Bestellungen sind zu richten — in der DDR an das Zeitungsvertriebsamt, an eine Buchhandlung oder an den AKADEMIE-VERLAG, DDR - 108 Berlin, Leipziger Straße 3 - 4 — im sozialistischen Ausland an eine Buchhandlung für fremdsprachige Literatur oder an den zuständigen Postzeitungsvertrieb — in der BRD und Westberlin an eine Buchhandlung oder an die Auslieferungsstelle KUNST UND WISSEN, Erich Bieber, 7 Stuttgart 1, Wilhelmstraße 4—6 — in Österreich an den Globus-Buchvertrieb, 1201 Wien, Höchstädtplatz 3 — in den übrigen westeuropäischen Ländern an eine Buchhandlung oder an die Auslieferungsstelle KUNST UND WISSEN, Erich Bieber GmbH, CH - 8008 Zürich/Schweiz, Dufourstraße 51 — im übrigen Ausland an den Internationalen Buch- und Zeitschriftenhandel; den Buchexport, Volkseigener Außenhandelsbetrieb der Deutschen Demokratischen Republik, DDR - 701 Leipzig, Postfach 160, oder an den AKADEMIE-VERLAG, DDR - 108 Berlin, Leipziger Straße 3—4

Zeitschrift „Fortschritte der P h y s i k " Herausgeber: Prof. Dr. Frank Kaschluhn, Prof. Dr. Axtur Lösche, Prof. Dr. Rudolf Ritsch], Prof. Dr. Robert Rompe, im Auftrag der Physikalischen Gesellschaft der Deutschen Demokratischen Republik. Verlag: Akademie-Verlag, D D R - 108 Berlin, Leipziger Straße 3 - 4 ; F e m r u f : 22 36221 und 22 36229; Telex-Nr. 114420; B a n k : Staatsbank der D D R , Berlin, Konto-Nr. 6836-26-20712. Chefredakteur: Dr. Lutz Rothkirch. Anschrift der Redaktion: Sektion Physik der Humboldt-Universität zu Berlin, D D R - 104 Berlin, Hessische Straße 2. Veröffentlicht unter der Lizenznummer 1324 des Presseamtes beim Vorsitzenden des Ministerrates der Deutschen Demokratischen Republik. Gesamthcrstellung: VEB Druckhaus „Maxim Gorki", D D R - 74 Altenburg, Carl-von-Ossietzky-Straße 30/31. Erscheinungsweise: Die Zeitschrift „Fortschritte der Physik" erscheint monatlich. Die 12 Hefte eines Jahres bilden einen Band. Bezugspreis je Band 180,— M zuzüglich Versandspesen (Preis für die D D R : 120,— M). Preis je H e f t 15,— M (Preis für die D D R : 1 0 , - M) Bestellnummer dieses H e f t e s : 1027/27/4. © 1979 by Akademie -Verlag Berlin. Printed in the German Demooratic Republic. AN (EDV) 57618

Fortschritte der Physik 27, 169—190 (1979)

The Algebraic Structure of the Thirring Model LYDIA

HECK

Fachbereich Physik, Universität Kaiserslautern, Kaiserslautern,

BRD

Abstract We summarize the representation theory of the group 8U( 1,1) as needed for the massless Thirring model. Representations of the current operator algebra are given taking account of conformal covariance. The conformal covariance transformation behaviour of the Thirring field is investigated. The Haag-Araki-Kastler observable algebra of the Thirring field is reconstructed from the Wightman theory of this model. 1. Introduction

The Thirring model is known since about twenty years. I t is an exactly solvable quantum field theory with interaction. The first investigations were made by T H I R R I N G followed by G L A S E R , J O H N S O N and S C A R F and W E S S [ 4 . 1 ] . Apart from attempts to study currents and fields as well-defined quantum operator fields, the most important result of this early work was a proof of triviality: Despite the interaction the ¿'-matrix is that of a free field. Since then the interest in the Thirring model was more concentrated on its formal aspects. B. K L A I B E R [4.2] summed up this early work and decisively influenced all later research by explicitly constructing an operator solution in a Hilbert space with positively defined metric. D E L L ' A N T O N I O et al. [4.3] and L O W E N S T E I N and S C H R O E R [4.6] recognized that the Thirring field has dilationally covariant transformation behaviour while a few years later K U P S C H et al. [4.4] found conformally covariant solutions of the Thirring field equation. During the last years the conformal covariance of quantum fields has been examined by many people [1.3] —[1.9], [4.4] and [4.5]. In the case of two-dimensional Minkowski space it is based on the representation theory of the group SU{1,1) [1.2]. In section two some essential results of this theory which have been used in the investigation of the Thirring model are summarized. In their paper, D E L L ' A N T O N I O et al. [4.3] reconstructed the Thirring field from the currents. They treated the currents as free massless vector fields. Exploiting the conformal covariance of massless fields in section three, representations of the current operator algebra are given taking account of conformal covariance. Irreducible representations are shown to correspond to different charge sectors. The generators of the representations of the conformal group are given explicitly for each sector. The Thirring field acts as intertwining operator from one charge sector to another. Restricted to one charge sector it is conformally covariant [4.5] and the representation is in general different in different charge sectors. 15

Zeitschrift „Fortschritte der Physik", Heft 4

170

L.

HECK

To formulate this in an elegant manner one can regard the Thirring field as an operator field over the universal covering space M2UC of two-dimensional space-time M2. These results on the transformation behaviour of the Thirring field are represented in section four. From the WiGHTMAjSr theory [2.1] — [2.4] of the Thirring field one can construct the Haag-Araki-Kastler observable algebra [3./] — [3.6] of this theory, if the Thirring field has integral or half-integral spins. This is displayed in section five. In an appendix we present some facts on the structure of a lattice of charge sectors that carries one Thirring field. 2. The conformal group and representations of it The group SL(2, R) is defined by

SL(2, R) =

or, - I t = 1,

•"-is 3

The group ££7(1, 1) =

a, f, r, r, €

\ß\ 2 = i,

rJ.

/Sec

is isomorphic to the group SL{2, R). A certain unitary matrix establishes this isomorphism by [ / . / ] ugu~1 = v , g € SL(2, R), t>e£L7(l,l). It is sometimes advantageous to use SV( 1, 1) instead of SL(2, R). An element g of the group SL(2, R) acts in R as the transformation:

R3 x

xg

=

ax + r ^ 6 R• fz + V

(2.1)

R is the Alexandroff compactification of R, R = R u {oo). The representation spaces of this group are the Gelfand spaces [2.2] 2t>(S.e)> s assumes any real value and e, the "parity", takes the values 0 or + 1 . The functions / £ ¿¿(S,e) fulfill the following conditions: (1) / : R C, and / £ C°° (All derivatives of / do exist and are continuous). (2) |a;|s-1 (sign cc)e /(I/a;) possesses also all derivatives. In the case when s is an integer and if s — 1 A e mod 2 it is sufficient to give s alone: The representation of the group SL(2, R) in the space

is defined by

(

G3C ~4~ T \

(2.2)

Now let s be an integer and s — 1 A s mod 2. In view of the conditions (1) and (2), a function has the asymptotic expansion [4.4], [1.2]

m

1,1-»+» > ^(«o

+

+ 0(x-*)).

(2.3)

When s 1 then Q>a contains a space i s of polynomials of degree s — 1. The quotient space 3>SI$3 can be decomposed into two subspaces: (^ s / and

x±2 = n

2

y - _ n

l t

n2 € Z

x i ^ x l + x*.

(3.15.1)

(3.15.2)

Then the lattice Jf** is defined by =t>i,27 «„ez.0 In an appendix we present the results on a number theoretic analysis of such lattices. We shall see that each lattice determines a Thirring field. Two pairs ( h°(x+)

h°(x+')

u>J-x(x„

x_) == IQ, jto(x+)

j2°(x_)

co^Hx.,

x+)

m

[Q, j2°(x-)

co^(x_,

xJ) =

(Q, j2°(x.)

= - J L (-fo -

Q) Q)

=

x+')

+ io)-«

(3.28.1)

0

(3.28.2) 0 ) = 0

j2°{xJ)

Q ) ^

- - J L {-x_

- x J ) +

t0)-».

(3.28.3)

But these distributions define just the scalar products in the spaces &^(x ± )l(xi)l!S'1(xi), i = 1, 2, with respect to these scalar products. The one-boson Hilbert space J f is therefore isomorphic to [S>1M(x+)l S\(x+) © (x~)\S\(a;_)]COmpletion where the completion is performed with respect to the two-point functions co2iA(Xi, »;')> i = 1, 2. The test-function space @1M(x+)l e2(x_)

e

0 0

(x

+

, x_)

+ 003^+» »-) = 2jt -

e

0 3

(x

+

, *_) =

2 n

f\

(3.31.1)

- U x ^ f - . .

(3.31.2)

-.ji(x

+

The generators of the Weyl group are given by H + P = 2n f :?',(£C+)2: dx+ H

-

P

=

2 n

J

: j

D

+

M

=

2n

j

x+

D - M = 2n f z7)

2

( x _ f :

dx_

:j1(x

) 2 : dx+

+

dx_

(3.32.1) (3.32.2) (3.32.3) (3.32.4)

Energy momentum tensors t h a t are bilinear in the currents have been studied in general bySuGA[2.7]. Such energy-momentum tensor is also called Sugawara tensor. 8 ) The double points represent the usual Wick products. WARA

180

,

L . HECK

where H represents the total energy operator and P is the total momentum operator, D generates the dilations andilf the Lorentz transformations. The commutation relations of these operators with the currents can be found in [4.3]. One can also look for the generators of the special conformai transformations Us :

u

'-'«

°-' -

r n ^ T f »> ( | ¿ t r ) •

'

'

,3 3S 2)

Here setting ^ = 0 = | 2 gives the identity transformation. The generators Sit i == 1, 2, of the special conformai transformation must fulfill the relations [4.9] : »[«.,)>W] =

(3.34.1)

= 8_(x_*j2(xj)

•i[S2,

(3.34.2)

i[8„ Ji(® + )] = i[8u j2{xJ)] = 0 .

(3.34.3)

If the operators S-t have the form: 81 = n J dx+ x+2 :)\(x+)2:

(3.35.1)

f dx_ xJ :j2(x.)2:

(3.35.2)

S2=n

the relations (3.34) are satisfied. If one decomposes ?;(£{), i = 1, 2, in the canonical way into positive and negative frequency parts and uses the definitions (3.11) and (3.13) one gets in the general form for the Wick product of two currents, i = 1,2 :/«(*») H(xi)'- = Qi2 1"^(x+, »_) produces the chiral charge and the field component ip2"-P(x+,x_) the chiral charge The field i/ja^(x+, «_) acts as a mapping from the charge sector (nr — 1 ,n2 — 1, a, ft) to the sector (nu n2; a, ft) of Thus acts on the lattice of sectors and the use of the subscripts oc, ft at all field operators j and ip should clarify that we consider this special representation of the Thirring field. The decomposition of the charge eigenvalues performed in (3.15) now finds a natural explanation. The use of the parameters x, ft instead of y± is, however, only a historic convention [4.J]. One postulates, that the generators H(Q+Q-'^) and P(Q+Qj-P) defined by the representation of space-time translations of the currents j^P, i = 1, 2, are also the generators of space-time translations of the field yf-P, i.e. [4.3]: X_) = J [II(Q+"-P, Q_'-P) ± P(Q+*-e, Q_*% r-?(x+, *_)].

(4.4)

182

L . HECK

From the relations (4.1), (3.32.1) and (3.32.2), one gets 8iy)'^(x+, x.) = 2incfJ

:ji'-f(xi)

(4.5)

x_)\.

If one defines g = 2 ayS-H

(4.6)

the equations (4.5) contain the differential equations (3.1) which define the Thirring field ([4.3], [4.5-]). K U P S C H et al. [4.4] have shewn that one can find solutions of these equations which are conformally covariant. From the commutation relations of the field ip"^ with the generators D and M of the dilations and Lorentz transformations respectively [4.3] we get the spin s and the dimension d of the field yp-P i[D,

i[M,

x_)] = (x+3+ + x_8J

Vl"f(x+,

^

»_)] = (x+8+ -

Vf-f(x+,

x_) + ~ ((7+)2 + (y_) 2 ) Vi-f(x+,

e«((y + ) 2 _ (y_)«)

x_8_) v,*Hx+> xJ) + j

(4.7)

Vt-.f(x+,

x_)

(4.8)

(no summation over double indices),

where e11 = + 1 , e22 = - 1 , i = 1, 2. The dimension of the field is then in accordance with (4.7)

< * = y ( ( y + ) 2 + (r-) 2 )

(4-9)

and for the spin we get from equation (4.8): *=j\(y+)2-

(y-) 2 l-

(4.10)

Recalling the definition of spin and dimension (see section two, equation (2.12)) we see that equations (4.9), (4.10) fix two out of four parameters that label irreducible representations of the universal covering group of the conformal group. The field component ipf-P transforms such as (7+1, T+1; jJ, TJ) and belongs to a representation (j + 2 , T+2; ?'_2, r_2), r±i, i = 1, 2 have still to be determined. W e have iused the notations: ?v =

?

v = j(-(y±)

2

+ 1).

(4.11)

W e define (y±Y

= (N±)ltl,

( y T ) 2 = ( N±)2,2.

(4.11')

I t has been found by K U P S C H et al. [4.4] that the representation, namely the parameters T ± *of the field intertwining a pair of charge sectors depends on the charge quantum numbers of these sectors. This feature can be expressed in a technically elegant form if we use the realisation of the Thirring field over the universal covering space-time M2MC. W e define [4.5]: f,( _ ) )

i =

1, 2

(4.12)

The Algebraic Structure of the Thirring Model

where

183

X_O,A,

X+0,D

®

X-°-D

=

(-?'+>

- R

+

)

x_o,d possesses an invariant subspace ^''^l.d (x) In view of the defiiition in [ 1 . 5 ] , the signs are exchanged. The two-point functions o){j{(p, 0) it is sufficient to consider the representations D(s, 0) or D(0, s) respectively. The corresponding unitarity conditions are D(«,0):

P-^VA^AJP)

D(0,a):

=0,

{i = l,...,2s)

(2.12)

p-oBÂWÀ1..Autp)=0.

To give an example we take s = 1/2, use the fact that the matrices a^ are the transpose of (1, —G), G the Pauli matrices, and obtain the unitarity condition p • o, p), a> = \p\. The transformation law, as obtained from (1.1), is

(U(a, A) 0) (x) = D{A) 0[A-\x

- a)].

(4.3)

Observe that'due to the manifest covariance of (1.1) the transformation law (4.3) is local and this is the reason why we insisted on manifestly covariant representations. The unitarity condition for the configuration space functions 0 is simply obtained through the replacement of p^ by id^ in (1.4) and is given by

MlrS l 0{x) = i(M + N) 8,0{x).

(4.4)

The irreducibility conditions, being ^-independent in momentum space, remain unchanged and the orbital condition p 2 = 0 is transformed into the familiar wave-equation

•

for massless fields.

0(x) = 0

(4.5)

5. Parity The representations considered so far are representations of the proper orthochronous Poincaré group and we now wish to extend SP^ by including space-reversal. It has aeen shown in [1] that space-reversal can be linearly implemented only if the representation D of the Lorentz group in (1.1) is pseudo-unitary, i.e. if there exists a pseudounitary metric r) with = D\A) rjD(A) = 7?. (5.1) The parity transformation P(P 2 = 1) is then represented by

( U(P) v) (p) =

W

(f>) ,

p = (p°, - p ) ,

(5.2)

ind the connection between P and Poincaré transformations is given by

P{a,A)P

= (»,/lt- 1 ).

(5.3)

Since rj commutes with the operator M + N the transformation (5.2) is unitary with espect to the inner product (1.5). The parity transformation in configuration space is liven by (U(P) 0) (x) = V0(x). (5.4) To discuss the combined effect of parity and the unitarity condition we notice that the epresentation D of the Lorentz group is pseudo-unitarity if and only if its reduction into the irreducible constituents D(m, n) is symmetric in (m, n). The irreducible represenations of the extended group ^f' are therefore D(m, m) and D(m, n) @ D(n, m). The rst of these, due to (1.1), leads to the trivial spin-zero field and the second leads to a eld carrying both helicities — n\. The simplest irreducible representation of the arity-extended Poincaré group ^ carrying the helicities ± s ( s > 0) is thus obtained rom D = D(s, 0) @ D(0, .s) and in the Lemma of section 2 we have that seen this reresentation is already sufficient.

198

U.

NIEDERER

6. Quantization: Covariant commutators The behaviour of Poincaré covariant quantized fields has been studied in [7] for the massive case and we now analyze the massless case along similar lines. Let 0(.v) be a field carrying a unitary massless irreducible representation of the Poincaré group. Assuming

< v m -

-r

& W, (« = ± 1 )

(6.1 )

where star denotes adjoint in Fock space and complex conjugation in spinor space; the choice between commutator and anticommutator is yet left open. I n the present section we show t h a t the Poincaré representation properties of 0 imply that the function S is unique up to a real constant on each of the two orbits, while in the next section the physical requirements of causality and positive energy are used to prove the spin-statistics theorem and to determine S uniquely up to one positive multiple. The function S, due to the transformation law (4.3), transforms as

S(x, y) = D(A) S[A~Hx - a), A~\y - a)] I) i.l).

((5.2)

I t follows from the translational part of (6.2) that S satisfies

S(x,y)=S(x-y),

(6.3)

and from the orbital condition (4.5) that S is of the form S(z) = f ^ -

[ e - ^ S A v ) + ei,M)].

(P° = « = |p|).

(6.4)

The Lorentz part of (6.2) thus implies the property

S± (p) = D(A) S (A^p) l' i A ).

(0.5;

W e next turn to the unitarity condition. I t might be argued that in the quantized version the Hilbert space representation of the Poincaré group is defined on the states rather than on the fields and that the representation carried by the operators 0 need therefore not be unitary with respect to an inner product like (1.5). However, as was already point ed out in the introduction, the unitarity condition also plays the role of an irreducibility condition removing the unwanted helicity components, and as such it retains its necessity in quantization. The consequence of the unitarity condition is described by the follow ing lemma. Lemma: If D in (6.5) is the irreducible representation D(m, n) then both functions are unique up to a real multiple. Proof: With (6.5) we transform S±(p) 0, 1) and write

{p

to the massless standard frame [ / ] p = p = (1, 0)

S±{p)=D-^(A{p))S±D^(A[p)),

S==S±(p),

(6.6)

where A(p) is a Lorentz transformation w i t h / l ( ^ ) p = p. The unitarity condition (1.2) and (1.3) then implies

J3S±

= (m -n)S±,

KaS±

= i(m + n)S±,

where Jf = l/2« i J t ; ilf t i and = Mi0 are the Lorentz generators. I n the SU(2) (x; basis [10] of the Lorentz group, which is generated by the two commuting sets of

(6.7) SV(2) SU(2)

199

Massless Fields as Unitary Representations of the Poincaré Group

generators M = 1/2(J — iK), N = 1/2(J + iK), condition (6.7) reads M3S±

=mS±,

N3S±

= -nS±.

(6.8)

In this basis the matrix elements of S± can be written as S±(a,b)ta',b'i> where (a, b) run through the eigenvalues of {M3, Ns); hence (6.8) implies that the only nonvanishing components of S± are/S±(OT>-„)(0',i,'). Together with the hermiticity =- S±, which isa consequence of definition (6.1), it then follows that the only nonvanishing component is 8±(m.-n)tm.-n) a n d that it is real. Thus S ± are unique up to a real multiple and this property carries over to S±(p) by (6.6). Applying the lemma to the spinor representations D(s, 0) and D(0, s) we now show that the corresponding functions S±(p) necessarily are of the form

(6-9) = P±P • S + have no off-diagonal blocks and we may write S±(p)

= S±^(p)

(6.12)

In physical terms this means that fields of opposite helicity commute. B y (6.9) and (6.12) S±(p) are then uniquely determined by the four parameters a ± , /?±. W e now show that invariance under space-reversal, which was the motivation for the use of D(s, 0) '©Z>(0, .s), reduces the number of parameters from four to two. The transformation law (5.4) of parity implies the condition VS±

(p) rj = S+(P),

{> = (oj, - p ) .

(6.13)

This condition is best analyzed in a basis where the usual convention D(0, s) = D*(s, 0) of spinor formalism is replaced by the convention D(0, s) = D _1 t(s, 0) because then the pseudounitary metric rj for D(s, 0) 0 D ( O , s) just interchanges D(s, 0) and D(0, s). In this basis the functions iS ± ( 0 , s ) (j)) of (6.9) are replaced by £±(0,»)i1.J1,B1..B„(iJ)

=

.

ffiIB1

p

. aAltBlt>

(6

14)

hence, due to the relation p • aAB = p •

• OacP • aBb = -^pa^tvtc

=

Q^ÀP),

(6.16)

hence the functions S± (p) are given by

(0, .s). We now show that this possibility generalizes to arbitrary integer helicities in the sense that the boson field carrying D(s, 0) © D(0, a) can be obtained from a potential car rying 7)(.s/2, .s-/2). Thi'oughout this section .s will be a positive integer. Let be a symmetric tensor field of a rank .s and define

Pv,An-JP)>

VW.-.f.»» =

(8-l)

where the permutation operator P s is defined by P. = 1 1 [« ; l

(,";'';)] -

p

o = 1>

(8.2)

with e the identity and (¡uv) the permutation which interchanges the indices ft ar.d v. It is easily seen that, due to the properties of P s and the symmetry of A, the field ip satisfies the unitaritv condition (3.9) and all of the irreducibility conditions (3.6) or (3.8) except the trace condition (14). The latter, in terms of A, is the condition + P.P^ti',:',-

= 0,

(* ^ 2)

(8.3)

which can also be written as

=0. S

=

(8.4)

."'-I -n...:, — 2

A tensor A satisfying (8.4) thus leads to a field y> which has all the necessary properties to carry a massless unitary representation of the Poincaré group. In the case ,9 = 1 (electromagnetisin) (8.4) is empty but the additional unitarity condition (3.10) demands P'AAP)

= o,

(8.5)

which is the Lorentz condition for the potential A^. For a = 2 the condition (8.4) is

piAtr - J PpA't = 0,

(8.6)

which is the analog of the Lorentz condition, sometimes called Hilbeit gauge, in the linearized theory of gravitation [9] where A^ is the deviation of the metric from the Minkowski metric. The potentials are not uniquely determined by the required properties of ip and without changing (8.4) we may subject A to a gauge transformation of the form A

„,.JP)

A

,,.JP)

+ : A , . . , . ( ? > ) + ••• + Pr.Bw.jP),

(8.7)

where B is an arbitrary symmetric tensor of rank s — 1. The simplest choice of a potential satisfying (8.4) is to let A satisfy the two conditions

=0,

p>A1„..Jp)= 0,

(8.8

203

Massless Fields as Unitary Representations of the Poincaré Group

and we adopt (8.8) for the remainder of the section. Total symmetry and tracelessness imply that the tensor A carries the representation D{sj2, sj2) of the Lorentz group; if it also satisfies the second condition (8.8) it has altogether 2s + 1 independent components, and the 2s — 1 components in excess over the two components of if (one for each of the helicities represent the gauge freedom. It is interesting to notice that the conditions (8.8) and total symmetry are exactly the conditions of the massive case [/] that an s-rank tensor A carries a unitary irreducible spin s representation of the Poincaré group. The transformation law for the potential, consistent with (1.1), is ( U(a, A) A),».,, (p) = eir-'A^

... A^A^A-ty

(8.9)

and is covariant. However, the inner product (1.5), which for potentials can be written as (Au A2) = ( - 4 ) ' f J

A\..K(p)

w

,

"

(8.10)

is no longer positive definite but positive semi-definite, i.e. (A,A)^0.

(8.11)

In fact, one can show that the potentials with zero norm (8.11) are those obtained from A = 0 by the gauge transformations (8.7). The absence of a positive definite inner product means that the potentials do not carry a unitary representation of the Poincaré group, a fact which had to be expected because a unitary representation based on D(sj2, s/2) would, by (1.2), necessarily carry helicity zero instead of ¿ s . One way to reinstate unitarity is to define the representation not on the potentials themselves but on the equivalence classes of potentials modulo zero-norm potentials [11]. The more customary method, however, is to force positive definiteness of (8.10) by demanding either of the gauge conditions ¿e..,.(?) = 0 ,

V k A k l L ^( V ) = 0 .

(8.12)

The two conditions, called radiation gauge or Coulomb gauge in electiodynamics, are equivalent by (8.8). The integral (8.10) then becomes the positive definite inner product (A1,A2)=

4« f ^ L Ì J CO H...A,=1

(8-13)

However, since the gauge conditions (8.12) are not consistent with the Lorentz transormations (8.9) we have to find another transformation law for which we make the Anzatz {U[A) A)^,(p)

= B(A,

...R{A,

A^.u(A-*p),

(8.14)

vhere the quantity R has to be chosen such that (8.8) and (8.12) are consistent with 8.14), that (8.14) defines a unitary representation, and that the transformation law (1.1) or y> remains unchanged. All these conditions can be met if Ii is the zero-mass limit of the Signer rotation, R{A,p) =\imRm{A,p), (8.15) m-> 0

ehere the Wigner rotation Rm for finite mass is defined [3, 12] by Rm{A, p) = Bm(Ap)

ABm~1(p),

(8.16)

204

U.

NIEDERER

with Bm(p) the rotation-free boost which transforms p into the rest frame vector (m, o). Although the zero-mass limit does not exist for Bm it does exist for IIm. The way to define a unitary massless representation for helicities i s in terms of potentials is thus to take the zero-mass limit of the massive spin s Wigner representations [3] carried by the totally symmetric traceless tensors A ^ . ^ p ) . The 2s + 1 components of A are then reduced to the desired two components for helicity by the gauge condition PkA,ci2..iSp) = 0,

(8.17)

which in the massless ease, as opposed to the massive case, is consistent with the transformation law (8.14). The price to be paid for unitarity, of course, is the lack of manifest covariance in the transformation law. Work supported by the Swiss National Foundation.

Appendix A : Invariant spin-tensors

The following conventions are used throughout this paper: (i)

p„, = diag ( 1 , - 1 , - 1 , - 1 ) ,

£0123

= +!•

(A.l)

(ii) The summation convention applies both to tensor and to spinor indices. (iii) The complex conjugate of an object carrying spinor indices is denoted by dotting all undotted and undotting all dotted indices. (iv) The indices of spinors are raised and lowered by eAB

=

£

AB =

-

=

S a b

_

=

e A j s

(A.2) ZAB*BC

Thus T . . a . . =

eAB

T.B.., T-A-

•=

SCBSba

=

= d

c

A

,

eABT - £ " ; this rule does not apply to e itself.

A quantity T^.^.j.. of mixed tensoral and spinoral character is called an invariant spintensor if it remains unchanged under the proper orthochronous Lorentz transformations, i.e. if it satisfies A { s ) ; • • • *AC

T,..A..i.. =

••• H

h

• • • r>..c..b..,

(A.3)

where s 6 SL(2, C) and /l(s) is the corresponding Lorentz transformation [5, 10}. The only invariant spin-tensors (besides e) we shall need are the Pauli matrices

^

=

(o

l )

A i

'

=

=

( l

(i

^

=

(o

J )

A

i >

( A A )

and their commutators =

4"

(

f f

MCffrS

C

—

C.,.>.„ then its irreducible constituents are carried by similar tensors having certain symmetry properties called irreducibility conditions in the introduction. In this appendix we determine the irreducibility conditions for the representations D(s, 0) and D(0, s) and their direct sum D(s, 0) © D(0, s), which is an irreducible representation of the parity-extended Lorentz group. Our method is to construct the tensors from the corresponding spinors by means of the invariant spin-tensors a^ of appendix A. D(s, 0) and D{0, s): In the following the (+/—)-signs refer to the representations D(s, 0)/D(0, .s). These representations are carried by totally symmetric spinors of rank 2s and the corresponding tensors are defined as V&..K.V.

=

• • • °,>,*/'B'. Due to (B. 2), (B. 3), (B. 7), (B. 8) the tensors y> satisfy the relations W-.f.. = —V>..'t

(B.12)

W-f'.-ea.. = 1p..ea..i**..>

(B.13)

=