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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
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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
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HU ISSN 0368 5330 ISBN 963 372 258 6 -
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.
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
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]
(4)
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
(5)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)
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
(7)
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 ) ) )(8)
follows. The obvious relations for <ф|(5(5с) - c(íc))... and
<ф
|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) .(10)
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
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>
(14)
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.
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)
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