VNU J OU RN AL O F SCIENCE, Nat Sci,, t.x v , n ° - 1999 T H E RELATIVISTIC O PE R A T O R Q U A NTIZATIO N OF LINEARIZED G R A V ITY TH EO RY Nguyen Suan Han Fiicnlty of Physics - College of Na.tural Sciences - VNU Quantization of the linearized gravity gives an evidence about the importance of physically motivated assumptions for the small metric - tensor components to be neglected, which concerns with the existence of gravitational waves in the conventional understanding of this problem Giavitational wave in gxavity theory are considered as quantum excitations of weak classical fields In this context, the construction of a gravity quantization scheme which is adequate to the problem of elementary excitations is important From such a point of view the relativist,ic operator quantization method(*) with an explicit, solution of the constraint equation [1, 2] is distinguished among the large variety of gravitational field quantization approaches [3, Consider the action for Einstein gravity theory S= Ịự ^d^x (1 ) Where R is scalar curvature g = d e t ự ^ ụ.v{^)- is the inverse of the tensor In a weak field approximation hụ,Vi I- (2 ) The variables h^,^{:r) discribe the linearzed gravitational field, and fj.L' is the Minkovski metric tensoi with diagonal ( , - , - , - ) The Lagrangian then takes the form (up to { h ^ ) - tprmc;) i/a (3) = -2 d o h o ,{d kh kt - d ,h jj)-d k h o ,{ở ,h o i, - dkho,) This action contains constraints which intioducp a transverse structure ốh 00 = 0^ d ,A Ĩ = 0, A Ị = drhị,, - (4 with the corresponding oquatioii of motion = => d o A l = dr { d , h o , - d,h o r) (5) ỗ h 0? (*) For brevity this m e t h o d can be called "mini mai " bec au s e it is concerned with t h e q ua nt i z at i o n only of minimal n u m b e r of physical degrees of freedom remaining after t h e explicit solution of a c on s tr a in t on the classical level tl].[2] Oft T h e R e l a ti v i s H c O il O p e r a t o r Q u a n t i z a t i o n tlir so h itiííii ( j f t l u ' coDistraiiits (4) = l^d>‘h , ,A ( ih A(//> ỈÌĨÌỊ 27 o f, (5 ), L agraiiK iaij (3) ro a d s hn.)fLhi,n, d , d i , A ^kĩìì — í^íA-^/rỉí (6) dk dm íl * wlicic iii(‘ proji 'cti on o p e r a t o r A ị i k ị l ì ì i ) r a n be^ co nsidered as defining th e d i s t a n c e in th e SỊ)HC(' of (l\ jiHinical Hold ỉì,ị,- o rb it s with m s p e c t t o infinitesimal gauge transforrnatioiis ỉhk — ỉ>ik + + ỡklh^ (7) F r o m L a ^ i a n g i a n (6) caiioniral nioiueiita is o b ta in e d Prs ^ Ẩ \{ r s \h ìì) d o h ir n { -r ) (8) which ot)('\- ĩh(’ foUowin^ {■oiiiniutation relationy (■/')./>r.s(//)] = M ỉ w \ r s ) { r ) ố ( r ~ y ) Tli(' - I i i o n i n i t u m te n so r is o b t a i n e d t o he svinnietric aiiid - ịt)^,u^"ha■^i^^^ụ'7ì)^alll,n■ '-/'/u.* = (9) invariant (10) It