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KFKI-1984-70

A, M É S Z Á R O S

GUPTA-BLEULER QUANTISATION OF THE FREE MASSLESS SPIN 2 FIELD

*H u n g a ria n ‘A ca d em y o f ‘S c ie n c e s

C E N T R A L R E S E A R C H

I N S T I T U T E F O R P H Y S I C S

B U D A P E S T

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KFKI-1984-70

GUPTA-BLEULER QUANTISATION OF THE FREE MASSLESS SPIN 2 FIELD

A. MÉSZÁROS

Central Research Institute for Physics H-1525 Budapest 114, P.O.B.49, Hungary

к

к

HU ISSN 0368 5330 ISBN 963 372 258 6 -

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ABSTRACT

Using eight pseudoparticles the precise procedure of Gupta-Bleuler quantisation is given for the free massless spin 2 field.

АННОТАЦИЯ

Дана точная процедура квантования Гупта-Блейлера дгя свободного беэмзс- сового поля со спином два. Используется восемь псевдочаст:1Ц .

KIVONAT

Nyolc pszeudorészecskét felhasználva megadjuk a Gupte-Bleu]er-féle kvantálási eljárás pontos formáját a szabad kettes spinű tömegnélküli mező esetére.

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As far as it is known, the p recise procedure <pf Gupta-Bleuler quant i s a t i o n for the free m a s s l e s s spin 2 field w a s not g i ven yet.

Indeed, already G u pta formulated the procedure a long time ago [1], b u t he used ad hoc nine pseudoparticles. This is in fact an uncomprehensible choice. The potential of the spin 2 field is a symmetric U ^ - ^ U 31 tensor, and therefore there s hould obviously be two physical and eight unp h y s i c a l polarisations. Of course, from the physical point of v i e w the choice of ni n e or eight p s e u ­ doparticles is n o t essential, because in any case there are only two physical polarisations. Nevertheless, the p r e c i s e formulation of the Gu p t a - B l e u l e r q u antisation p r o c edure is all the same u s e ­ ful. As it is well-known, in the case of self-interacting gauge fields the pseudoparticles determine the behaviour of ghost p a r t ­ icles in Lorentz-gauge [2],[3].

Thi s paper gives the p recise procedure of quantisation. Of course, the questions that are identical to the case of spin 1 field - opposite sign of commutators, indefinite metric, etc...

(for details see, e . g . , [4]) - are n o t considered here.

In the Lorentz-gauge the e q u a tions of the free mass l e s s spin 2 field are g i v e n by [5]

0 U lj = 0 2 U 13 ,. = U ' 1 U7 5 U,

l (1 )

where an index after a c o m m a denotes partial derivatives,

and the Latin indices take the values 0,1,2,3. T h e indices are m o ved b y the т)13 = п. , =diag (1 ,-l ,-l,-1) Mink o w s k i a n m etric tensor of flat space-time. We have still the following gauge freedom:

6 ij = U ij + V ( i 'j ) ; □V = 0, (2)

where ( ) d enotes symmetrisation w i t h o u t the factor and V"*- are functions of x 1 coordinates. In the m o m e n t u m space we have

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2

U 1^ = (2 П ) 3

'2 ik xn

dk (Böjtje n + 6 13''t'- n /2k

-ik x n

(*)e n ), (3) О

where к* = [k°,le] is the wave vector. Of course, all this is not new.

In order to introduce the eight p s e u d o p o l a r i s a t i o n s we shal]

use the orthonormal e , f , n , m vectors, w h ich are wel l - k n o w n [4].

The y fulfil the f o l l o w i n g relations:

m^m^ i j

- е е - f ifj - n^n^ = nij ; i

e e, = f 1. i

f i = n n i - i , -m m . ’ = -1;

l

e if .

l = e 1n i = i e m^ =

f l n i = f^n = 1

n m^ =i 0; k ik i = (k°) 2 - i h 2= o ;

F- II 1 0 , i , i 4

к (n +m ) i _

; e = [ 0 , e 3; f 1 5 [0,?]? i

n и [0, •— ];

m 1 = [1,0,0,0]

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One m a y write:

+ j

= a (3c) (e1e-) - f i fj ) + b(3c)e(ifj) + c í í c j e ^ n 15) +

+ , . ..

+ d (3c) e ^ m - 1

+ . . 4“

. ,

. + .

+ g (3c) f ^ п

- 1'

+ h(3c)f'1m-1^ + p(3c) (m’Sn^+nJ n * ) + , . ..

+ r ( k ) m U n D;

+ “b

+ u<*> + v(k) (mim j _ n in j, + /2

+ +

+ H W _ ^ L v U s l (eiej + fifjj.

n

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One has the conditions

(2 U lj (jc)kj - k 1 Ü (ít) ) I Ф > = О; <Ф I (2tj*j (it) к - 5 (it) к i) - 0, (6)

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3

where |ф> is the state vector of Fock-space. Hence

(d(ic) - c (ic) )

I Ф

> = (h(k) - g (Jc) ) |

Ф

> = (p(k) - r (

Jc)

) |

ф

> =

= (u (k) - v (íc) ) I Ф> = 0

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and

/ 2 (Je) I ф > = (a(ic) (e1e^ - f 1f j ) + b (1c) e (if3) c (k)

, О

e (ik j)

g(k) f (i j) ■ p(k)

O K 0 2

к (ки Г

k 1^ u (к ) /2 k C

( (m^-n^) k-'

k 1

(mJ - n D ) ) )

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follows. The obvious relations for <ф|(5(5с) - c(íc))... and

|Й±3 (ic) were not written down. Deno t i n g

+ + . + . + . ,

/2 V(ic) = с(5с)ех + g (ic) f x + p (ic) (m1+ n 1 ) + u(ic)

(mi- n i ) (9) one immediately sees that only the p o larisations given by o p e r a ­ tors a(ic), B(ic) have physical meaning; compare wi t h (2). The commutators are the f o l l o w i n g : ’

[ a (ic) , a (q) ] = [£(Íc), É(q)] = [c(jc), c(q)] = ~[d(íc) , S(q)] =

= [ g (íc) ,. g (q) ] = ~[h (ic) , Й (q) ] = [p(ic), p(q)] = -[r (ic) , r(q)] =

= [u (ic) , u (q) ] = — [ v (5c)

I

v (q) ] = 6 (íc - q) .

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A long but straightforward calculation leads to the relation:

2 [ (íc) , Öp r (q) ] = 6(í - 5 ) ( n i p njr + ni r njp - n± j np r ).

( I D

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4

In order to write down the dynamical invariants we proceed as follows. As Lagrangian one m a y use

L :U 1^ ,kU

ij'k* (12)

where z is an arbitrary real number; z ^ . For any z one o b ­ tains О и 1-1 = 0. In standard w a y from (12) one obtains the four- - m o m e n t u m :

P = dk.k (5=>m (£) U. (k) -

lm 5(£)) (13)

and hence

<ф I P |'ф> = <Ф die.к 1 (a (íc) 5 (íc) + É (5c)Б (ic) + (1 2 z ) v (к) v (к) ) I ф>

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Thus obvio u s l y z = is the right choice in (12).

Note still that (11) is not the only possible commutator, because as potential one+may use ^ U 1-1 + y n 1"1 U) too, w h ere у is real and i. Because U(je) = 2 v(5e) holds, (11) m a y be substi­

tuted by

2[Ui-’(ie), 5 p r (q)] = 6 (je-q) (ni p n^r + ni r n^P - t п±:*ПР Г ), (15) where

t + 4y + 1 > 1

2 ‘ (16)

We have seen: if one uses the vectors e>f,n,m, then the quantisation is straightforward but not trivial.

ACKNOWLEDGEMENT

Thanks are due to D r .Á.Sebestyén for valuable remarks.

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5

REFERENCES

[1] Gupta S.N. : P r o c . P h y s . S o c . , 6j>, 161 (1952)

[2] Bogolyubov N . N . , Shirkov D . V . : K v a ntoviye pólya, Nauka, Moscow (1980)

[3] Duff M.J.: in Q u a n t u m Gravity, An Oxford S y m p . , ed. by C.J.Isham, R.Penrose and D.W.Sciama, Clarendon Press (1975) [4] Bogolyubov N.N., Shirkov D . V . : Vved y e n y i y e V tyeoriyu

kvantovanikh poley, Nauka, M o s c o w (1973)

[5] Thirring W.E.: A n n a l s of Phys., _16 , 96 (1961)

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Kiadja a Központi Fizikai Kutató Intézet i'elelSs kiadó: Szegő Károly

Szakmai lektor: Sebestyén Ákos Nyelvi lektor: Banai Miklós Gépelte: Simándi Józsefné

Példányszám: 190 Törzsszám: 84-364 Készült a KFKI sokszorosító üzemében Felelős vezető: Töreki Béláné

Budapest, 1984. junius hó

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