DSpace at VNU: The Relativistic Operator Quantization of Linearized Gravity Theory

Quantization of the linearized gravity gives an evidence a b o u t the importance of physically motivated assumptions for the small metric - tensor components to be neglected, which conc

VNU J O U RN AL O F SCIENCE, Nat Sci,, t.x v , n ° 2 - 1999

T H E R ELATIVISTIC O PE R A T O R Q U A N T IZ A T IO N

OF L IN E A R IZED G R A V IT Y TH EO R Y

Nguyen Suan Han

Fiicnlty o f Physics - College of Na.tural Sciences - V N U

1 Quantization of the linearized gravity gives an evidence a b o u t 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, 4

2 Consider the action for Einstein gravity theory

The variables h^,^{:r) discribe the linearzed gravitational field, and fj.L' is the Minkovski

metric tensoi with diagonal ( 1 , - 1 , - 1 , - 1 ) The Lagrangian then takes the form (up to

0 { h ^ ) - tprm c;)

i/a

(3)

= - 2 d o h o , { d k h k t - d , h j j ) - d k h o , { ở , h o i , - d k h o , )

This action contains constraints which intioducp a transverse structure

with the corresponding oquatioii of motion

= 0 => d o A l = d r { d , h o , - d , h o r ) (5)

ỗ h0 ?

( * ) For brevity this m e t h o d can be called " mi n i mai " b e c au s e it is con c e r n e d wi t h t h e q ua nt i z at i o n only

of minimal n u m b e r o f physical degrees o f f reedom remaining aft er t h e explicit solution o f a c o n s t r a i n t on t he classical level tl].[2]

Oft

T h e R e l a t i v i s H c O p e r a t o r Q u a n t i z a t i o n o f , 2 7

O i l t l i r s o h i t i í í i i ( j f t l u ' c o D i s t r a i i i t s ( 4 ) ( 5 ) , L a g r a i i K i a i j ( 3 ) r o a d s

2

d , d i, d k d m

í l

wlicic iii(‘ p r o j i ' c t i o n o p e r a t o r A ị i k ị l ì ì i ) r a n be^ c o n s i d e r e d as defining t h e d i s t a n c e in t h e SỊ)HC(' of (l\ jiHinical Hold ỉì,ị,- o r b i t s wi th m s p e c t t o infi nites imal g a u g e t ra n s f o r rn a t i o i i s

F r o m L a ^ i a n g i a n (6) c ai i on ir a l ni oiue iita is o b t a i n e d

P r s ^ Ẩ \ { r s \ h ì ì ) d o h i r n { - r ) ( 8 )

which ot)('\- ĩ h (’ foUowin^ {■oiiiniutation relationy

(■/' )./>r.s(//)] = M ỉ w \ r s ) { r ) ố ( r ~ y ) ( 9 ) Tli(' - I i i o n i n i t u m t e n s o r is o b t a i n e d t o he s vi nni et ri c aiiid i n va r ia n t

It <1(K'S not r('Ị)r(\soní a full (iorivative a n d ^iv(\s riso to a set of P o i n c a r é - gxoup

^(MK'raturs in wliicli b o o s t ^ e n e i a t o i s i n d uc e a n d a d d i t i o n a l g a u g e t r a n s f o n n a t i o n of t h e

<l\iiaiuical iii'lcls

i [ \ I , o i i r s i - r ) ] = { r o d , - v , d o ) M r s \ l i n ) } i i r „ { r ) + { ở r ~ P i s + ( 1 1 )

riiis additional ^ a iig í'traiisfonnatioii leads to a time - axis rotation th at ensure the

i elativisti( ccA'ariaiKT o f t h i s inanifostly n on -c ova ria iit q u a n t i z a t i o n p r o c e d u r e ,

i n t l i i r : : p a r c r n n h o í U' f i i ì í ' ( Ị a n

e : , A { i k \ h n ) e l , = 0 ^ ” ^ (12)

a , b = l , 2

wIk'io /*/, is tlu' i h r v v - (liuHuisional p ro j e c t i o n o p e r a t o r

p, k = e f 4 , e‘^ P , , 4 - a , 1 3 = 1 , 2

T h u s th(‘ r e l e v a n t p o l a r i z a t i o n s a re fo und t o be

aiul for tli(' i i u l r p p i u k ' u t p h y s i c a l variables

the free two - component scalar field Lagrangiaii is obtained

hoiuT plaiK' wav(’s an' pi'('S('iit ill tli(' rxcitalioii ^pi'ctniiii of tlir liiU'Hi izf‘<l j;ravitv V.

Tlit‘i('foro ininiinal qiiaiitizatiun of weak ^lavitaiional fii'lds irpixKluci's th(' ladia-

t ioiial - K'sults To^(‘tli(‘r w i t h c o r r e s p o i u l i i i g a t ld i t i un a l c o i u l i t i o n s wliicli a i ( ‘ in fact geii(Maĩ(Ml hv tli(‘ (HỊuaíious o f m o t i o n for thí' I i o n d y n a i n i r a l Helds //(,() a n d //{),.

For tlie oiii;inal thf'ory (1) without any additional assum ptions alìoiiĩ the holds

Iiiiniiiial q u a n t i z a t i o i i c o n s i s t s o f e’xc’l uc l in g noiipliVvSical o f fn'iHloni i l i r o u g h tlio

exac t sohitions of tlieii ('qnations of motion (constiaints) Howovoi tho coincidf'nce of the

linearized expansion of the action obtainod with the one rousidered above (in the naivf'

linearization S(‘lionio) is by no means obvious, the reason l)riug tho (listin^uishod rolo of the Newton coinpoiH'iit (/00 Thus avSyuniiiig th at coiulitioii (2) coiu-orns only dynamical fields wo aro foirod to ronsidor also components hio as small vai ial)l('s hocause of thí' constlaints hut no rrstiiction is iinpospd on the XfnvtonCoiiipoiioiit //()() In th e iniiiiinal

quantization lurthod a coniponeut //()() is considi'ircl as a classic'al OIIO and assumption

potontial the gravitational wave is int('ractiii^ with

W ith tlir help of the K'lativistic opi'ratoi' quantization nii'thod thv th('ory of lin­

earized ^ravitatioiial fiold is foniiulatfHl in a manifest rolativistic - covariant fonii provid­ ing its straightforwaid quantization with saiiH’ transfoMuatioii pn)p(*i‘ti(\s of tli(' qiiaiitiz(‘(l fields with I(\sp('ct to tli(' Lonnitz - gro\ip action as claissical th(‘orv Tlio Lagiangiaii (16) obtaiiietl (IpsciitK'S an unconstrained hainiltonian syst('iii

Th(‘ author would like to thank Profs v x Poivushin L Litov and Ilieva N Fur stimulating (liscussioiis Tlio financial support of the National Basic H('S('Hifh PiOii^iani iii Natuial Scif'iicos KT-04 is hiiz,hly appreciatotl

R EFEREN CES

N g u v i ' T t S ' l i i i i H i v n V' \ P i ' i \ ' u R h i n K i n d P ỉ i ì j H L t ' f i A X ( , 2 ( l 0 ^ 7 ■ F f n t f u h l Pln/.s N,8(1989)()lỉ; Can.J Fhys 69( 1991 )rp.'()84 - 691.

2] NAimMi Suan Han ICTP, I C / 9 5 / n Trieste

3] RA.M Dirac Pror Roy 5or./l 246{ 1958)pp.333.

4 R Arncnvitt s Dosi'i c MisiHM PÌIÌỊS /?ír/'.117(196())pi>.15í)(>.

5] N Ilii^va L.Lilov V.X.Pnvushin JINR E2-90-507.

T A P CHI KH O A HOC ĐHQGHN, K H T N , t x v , - 1999

ÁP DỤNG P U Ư U S G PIỈÁP LU’QXG T Ư HÓA TOÁX T V i ncjNC; ĐOỈ

TÍXU CHO TRƯ ỜX G HAP DAX TU Y EX TÍXH

N g u y e n X u â n H ã n

Khoa Vật lý - Dại học K H Tựnhỉẻỉi - ĐHQG HàNội

Ap (lụng phưưng pháp Iưựiig từ hóa toán từ tirơiií’ (lối tínli cho tiirừỉi^ hấp (lẫn tuyốii tính ừ í>ần (lúng tnrừno yốu Bằng viộc giài chính xác phưưn*^ tiìiili lira kốt (Ir cho thành phần Nínvton qua các í hành phần vật lý khác chúng ta đ ã th u (lưực La^ian^iaii

(16) inò tà hô gồin nliửa» thành phần vật lý độc lập vứi nhau.