Volume 93A, number 8 PHYSICS LETTERS 7 February 1983

THE EXACT EQUIVALENCE OF n-WAVE REGULARIZATION TO THE

RENORMALIZATION PROCEDURE FOR THE SPINOR FIELD IN ISOTROPIC SPACE-TIME

S.G. MAMAYEV and V.M. MOSTEPANENKO Department o f Mathematics, Leningrad Institute o f Onema Engineering, 191126, Leningrad, USSR

Received 18 October 1982

Contrary to the assertion of Kharkov the exact equivalence of the n-wave procedure and the renormalization of the physical constants is demonstrated with the help of dimensional regularization. The vacuum energy density of the spin-l/2 field in the Friedmann universe is obtained and its mass dependent part is shown to be negligibly small compared to the background.

In the late 70s the vacuum polarization of massless quantum fields in isotropic space- t imes was calculat. ed by a number of authors (e.g., refs. [ 1 -5 ] , where various methods of regularization of vacuum expecta. tion values of the stress-energy tensor are employed; see also the review papers [6,7] and the monographs [8,9]). For the massive case the total s tress-energy tensor (SET) of quantized fields, including the con- tribution of particles created from vacuum by the gra- vitational field was obtained [5]. The most practical method was found to be the n-wave regularization, proposed in ref. [10] and applied for isotropic space - times in refs. [5,7]. According to refs. [5,7], the total SET equals the sum of mass-independent purely local terms (leading to conformal anomaly) and massive terms as in QED in the strong external field [ 11 ]. For the confonnal scalar field in isotropic space- t ime the vacuum SET has only one type of divergence, while for the spin-l/2 field two out o f three subtractions which are presecribed by the n-wave procedure prove to be inf'mite [5].

It was DeWitt [12] who first made the general conjecture about the equivalence of the n-wave meth- od to renormalization o f the physical constants in the bare gravitational lagrangian. However, this is not rea- dily evident in the framework of the formalism of refs. [5,10], since the momentum cutoff results in the appearance of noncovariant terms in the subtract- ed parts o f the SET (similar noncovariance arises when adiabatic regularization is employed [13]). This

led the author of refs. [14,15] to the conclusion that the n-wave procedure cannot be justified in terms of renormalizafion. Here we shall show why this conclu- sion is false and demonstrate that the n-wave method is exactly equivalent to renormalization of the con- stants. Also we obtain here the correct expressions for the mass-dependent terms of the total SET for the spin-l/2 field, which were computed erroneously in ref. [14].

Vacuum expectations of the SET of the spin-l/2 field of mass m in an isotropic spatially fiat metric are (see ref. [5])

(0IT0010) = (2/Tr2a 4) f d X ~k26o( - ½ + Sh) , 0

(01Ta ~ 10) = - (2/37r2a 4)

× fox ~k4¢.o -1 [-- ½ + Sh -- (ma/2X)ux] • (1) 0

Here a is a scale factor, 602 = ~2 + m2a2; the quanti- ties s x, u x together with o x are bflinear forms of time- dependent parts o f the solutions of the Dirac equa-. tion and obey the first-order equations

~x=½wux, bx=2~u~ ,

t~ x = w(1 - 2sx) - 2wvx, w = mltX/6o 2 , (2)

with the initial conditions sx0?o) = Ux(r/0) = ox070) = 0

0 031-9163/83/0000-0000/$ 03.00 © 1983 North-HoUand 391

Volume 93A, number 8 PHYSICS LETTERS 7 February 1983

[the dot denotes a derivative with respect to the~ conformal time *7 = f dt/a(t)]. Sufficient smoothness of the behaviour of the scale factor at *7 = *70 is sup- posed.

To eliminate ultraviolet divergencies in (1) one should, according to the n-wave procedure, subtract from the integrands the first three terms of their asymptotic expansions in powers of 6o -1 . The first and the second subtractions amount to dropping the ( - 1 [ 2 ) terms and subtracting from s x and u x the terms

1 2 s 2 = ig (w/co) , u 2 = (4w) - l(d/d*7)(w/co). (3)

The third subtraction yields finite mass-independent polarization terms to the total SET

(Tik) 0 = (2880rr2) -1 (11 (3)Hik - (1)Hik), (4)

where (1)Hik , (3)Hik are tensors, quadratic in the cur- vature tensor [ 1 - 9 ] . Note that the result of ref. [14] does not contain the generally accepted terms (4).

In order to fit the first two infinite subtractions into a renormalization scheme we shall now examine their geometric structure using dimensional regulari- zation. This will allow us to avoid noncovariance, which arises if the momentum cutoff is employed. In the space- t ime with the (3 - 2e)-dimensional spatial section the first subtraction is finite:

(TOO) ~) = - ½ B e ( M a ) 2 e a - 4 J" d;k~2-2eco ,

0

(Taa)~) = [2 (3 - 2 0 ] -1Be(Ma)2ea-4

X f dX X4-2eco -1 , (5) 0

where B e = 22ene-3/2/P(~ - e), P(z) is the gamma function; an arbitrary parameter M with the dimen- sion of mass is introduced to ensure that the SET has the usual dimension for e 4: 0.

Similarly the second subtraction is

(T00)~O = Be(Ma)2~a-4 f d X k 2 - 2 e w S 2 ,

0

(Taa)(2e) = ( 2 e - 3)- l Be(Ma)2ea-4

o o

x fdX x a - 2 e ] c o - 1 Is 2 -- ( m a ] 2 X ) u 2 ] . (6) 0

Expanding (5) and (6) in powers of e and dropping the terms which vanish as e ~ 0, we find

(Tik)~) = (m 4/16rr 2) (e -1 + b)g}~ ) ,

(Tik)~ 0 = - (m2/48r r 2) (e -1 + b)G~ ) , (7)

where g)~) and G}~ ) are the metric tensor and the Einstein tensor in (4 - 2 0 dimensions, b = ~ - C - ln(m2/47rM2), C = 0.577... (Euler's constant).

The geometric structure of eqs. (7) shows that the two subtractions are exactly equivalent to renormali- zations of the cosmological term and the gravitational constant, respectively, in the bare Einstein equations or in the effective lagrangian. In particular, the renor- realization condition for the gravitational constant is

Gren(e) = G(be L [1 - (m2/6rr)G(bOare(e-1 + b ) ] - I . (8)

The total renormalized SET is the sum of (4) and the mass-dependent part

(To°) m = (2/lr2a 4) f d X ~.2co(s x - s2 ) , (9) 0

(spatial components may be obtained using the con- servation equation V k (Ti/C) m = 0).

For the epoch t - t o ,~ m - 1 using the solutions of eqs. (2) [7], we f'md

(T00) m = (m2/4rrZa 4) a 2 [ ln(ma) + 5 + C]

7/

+ f (a(*7) in (*7 - .7') no

t

-fd,f'~(*7")tn(*7'-,7") . (10) rio

If the expansion law is a = a 1 rip = aotq, q = P/(P + 1) and *70 = O, eq. (10) yields

(TOO) m = -(q2m2/81r2t2){ln(1/mt) + ~ [q/(1 - q ) ]

+ l n ( 1 - q ) 4 _ ( 1 _ q ) ] [ 2 ( 2 q _ l ) ] } , (11)

where ~0(z) = I"(z)/I'(z). Note that the corresponding result of ref. [14] (eq. (5)) is incorrect. It is evident already from the fact that it contains the term In *7 depending on the choice of the unit of the proper time (in ref. [14] our *7 is denoted by t).

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I fq = ½ (i.e. if the background is radiation domi- nated), (TOO) m has a pole. This divergence, however, is not of a field-theoretic nature and is due to the in- sufficient smoothness ofa(rT): ~ ~ 0 when ~ ~ 0. In fact for t ~ tpl = G 1/2 it is necessary to take into ac- count the reaction of the mass-independent part (4) of the total vacuum SET upon the metric of the space-time. The results obtained in this direction [16,17] make it plausible that the conjecture that the initial state of the universe may be described by a de Sitter metric holds. That is why the conclusion of ref. [14] that (TOO) m may be of the same order of magnitude as the background ifq is taken sufficiently close to ~ (q - ~ -~ Gm 2) is invalid. In order to ob- tain a physically reasonable result one may, as the simplest model, consider the de Sitter space-time which at t = t 1 ~ tpt smoothly goes over to the Friedmann expansion:

a 07) = 2 t 1 / [2 (2 t 1/al)l/2 -rl], --~ < rl <. (2t 1/al)I/2,

= a 177, (2 t 1/a 1 )1/2 ~< 77 <~oo, (12)

(the exact value of t I is determined by the curvature of the de Sitter space when the contributions of all quantized fields are summed up).

Putting in eq. (10) r/0 = -0% we fred

(TOO) m = (m21327r2t2)[-ln( I/Zmt) + ~ + C

+ ½ In ( t / t l ) ] . (13)

The energy density (13) is small compared to that of the background for the masses of all the known ele- mentary particles (since Gm 2 ,~ 1).

Thus, contrary to the assertion of ref. [14], the massive part of the total SET cannot account for all the matter in the Friedmann cosmology created as a result of quantum explosion. Finally we wish to point out that eqs. (11) and (13) are analogous to a similar

result for a strong electric field [11]: (TOO) ~ E 2 (inE 2 + const). In gravitation the role of the field strength E is played by the ratio H/m, where H = a -1 da/dt is the Hubble parameter. This shows that for t "~m ~1 the massive part of the SET cannot be ascribed to the created particles only and contains mass-dependent polarization terms. One must, how- ever, bear in mind, that no unique splitting of the vacuum SET into a polarization part and a contribu- tion of the created particles exists.

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