Spontaneously broken symmetry and cosmological constantM. Y. Wang Citation: Journal of Mathematical Physics 17, 704 (1976); doi: 10.1063/1.522965 View online: http://dx.doi.org/10.1063/1.522965 View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/17/5?ver=pdfcov Published by the AIP Publishing Articles you may be interested in The Cosmological Constant Problem, The Spontaneously Broken Symmetry, And The Generalized Rest Energy AIP Conf. Proc. 624, 42 (2002); 10.1063/1.1492152 Path integrals in polar variables with spontaneously broken symmetry J. Math. Phys. 36, 2675 (1995); 10.1063/1.531360 Conformal symmetry inheritance with cosmological constant J. Math. Phys. 33, 4002 (1992); 10.1063/1.529850 All spontaneously broken symmetries for noncovariant currents J. Math. Phys. 20, 2596 (1979); 10.1063/1.524021 Selfadjointness and spontaneously broken symmetry Am. J. Phys. 45, 823 (1977); 10.1119/1.11055

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Spontaneously broken symmetry and cosmological constant M. Y. Wang*

Center for Theoretical Studies, University of Miami, Coral Gables, Florida (Received 12 May 1975; revised manuscript received 30 July 1975)

A solution of the Einstein equation with cosmological term produced by spontaneously broken symmetry is presented. The solution implies that the universe will recontract.

In those years the theories of spontaneously broken symmetry and Higgs phenomena have been a topic of active investigation in elementary particle physics. 1

The crucial pOints are that spontaneously broken sym­metry requires a nonzero vacuum expectation value of scalar meson, and the vector meson acquires mass from Higgs mechanism. These mechanisms have been applied to unify the theories of weak, electromagnetic and strong interactions. 1 Recently the question of possi­ble relationships between the spontaneously broken symmetry and the cosmological constant was raised by several authors. 2,3 Their arguments, based on the con­jecture of Zeldovich and Novikov4 are that the vacuum value of energy momentum tensor T", v appears in the form of a cosmological term in the vacuum field equations.

In this paper, a solution of the Einstein equation with cosmological term produced by spontaneously broken symmetry is presented. The solution is shown to be consistent with the conjecture of Zeldovich and Novikov. The implications of the result are discussed.

Let us consider a system of the triplet scalar and 80(3) gauge fields coupled with the gravitational field. The action of the system can be written as5

13 = - ~ f r-g(D"'QaD", Qa) - ~ f..L 2Q~ - tA(Q~)2.

where A is the cosmological constant and

(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

Wa and Qa are a triplet of vector fields and scalar fields respectively. We choose the parameter f..L 2 to be nega­tive so that field Q yields a nonzero vacuum expectation value:

(9)

Now we ask for a solution of the field equations that is static and spherically symmetric, i. e.,

(10)

704 Journal of Mathematical Physics, Vol. 17, No.5, May 1976

Following the ansatz of Wu and Yang, 6 we can write Qa

and Wla as

Qa(X, t) = x"Q(r), (11)

WI.(X, t) = E lab xbW(r),

where Elab is the usual E symbol. The most general static and spherically symmetric tensor g",v in the cartesian coordinate is shown to be of the form 7

goo = - QI(r), gOI = 0, (12)

glj = I)lj - (1 - f3)x l x j /r, where QI and f3 are function of r only. After some alge­bra, the Lagrangian L becomes

L = _ 47Tf r2{QI'1/2(~~~) ()1/2 + ~QI'1/2(~;) ()1/2

_ ~ Ql1/2(df3)f3'3/2 +!. Ql1/2 f3'1/2 _!.QI'3/2(dQl)2 f3'1/2 r dr r2 2 dr

x(~~r +4(j -1) W 2 +%Q

2

+rQ(~~)+2er2WQ2 +}(~ -1) Q2 +r(~ -l)Qe~)

+ r2(dQ )2 + !.(..!. _1) r2 fd Q )2 2 dr 2 f3 \dr

+ e2r4W2Q2 _ A:2 r2Q2 + ~r4Q4 _ 2A ]}.

The field equations can be obtained from Eq. (13) by varying QI, f3, Q, and W. The final forms are

Copyright © 1976 American Institute of Physics

(13)

704

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+4W 2 +-+rQ - +- - a Q2 (dQ) r2 (dQ) 2] 2 dr 2 dr '

~[2J3-'0(~~)+4J3-' y3W] a ' / 2 J31/ 2}

= [4r2W+6er4W 2 +2e2r SW3 +4J3-1r3(~~)

+8J3-1r2W +2er4Q2 +2e2r SWQ2]a 1/2 13' / 2,

d~ {fl y3Q + 13- 1 r4e~)Ja1/2 13' /2} = [2Qr2 + 4er4WQ + J3-1Qr2 +y3 t:r1(~~)

+2e2r 6 W 2Q _ iI.[2 r4Q +} rSQ3] a'/2 13' /2.

It is easily verified that

W(r)=-I/er2,

Q(r)=F/r,

a(r) = J3-1(r) = 1 - 2m/r + 1/4e2y2

705 J. Math. Phys., Vol. 17, No.5, May 1976

(14)

(15)

(16)

(17)

(18)

is a solution of Eqs. (9) and (14)-(17). The cosmologi­cal constant is found self-consistently to be

A = - te Ar (19)

The following remarks on the above solution are in order:

1. Solution (18) reduces to the t'Hooft's magnetic monopoles in flat space -time.

2. The cosmological constant A is consistent with that of Dreitlein. 3 Thus, if the universe at the present epoch is isotropic, the result indicates that the universe will eventually contract, as has been pOinted out by Dreitlein. 3

Recently Coleman and Weinberg9 have investigated the possibility that radiative correction may produce spontaneous symmetry broken down. In that case, the radiative correction can be viewed as the dynamical origin of cosmolOgical term. This problem is under investigation.

ACKNOWLEDGMENTS

I am indebted to Dr. Arnold Perlmutter for his cal reading of the manuscript. I would also like to thank Professor Behram Kursunoglu for his hospitality at the Center for Theoretical Studies, Universities of Miami, Carol Gables, Florida.

*Present address, Nuclear Engineering Lab., University of Illinois at Urbana-Champaign, Urbana, Illinois 61801.

1G.8. Aber and B. W. Lee, Phys. Rep. C 9, 1 (1973) and ref­erences cited therein.

2A.D. Linde, Pis. Zh. Eksp. Teor. Fiz. 19, 320 (1974) [JETP Lett. 19, 183 (1974)].

3J. Dreitlein, Phys. Rev. Lett. 33, 1243 (1974). 4Ya. B. Zeldovich and I. D. Novikov, Relativistic Astrophysics (University of Chicago Press, Chicago, Illinois, 1971), Vol. 1, pp. 28ff.

5In the context, we have taken the unit 167fG = 1. 6T. T. Wu and C. N. Yang, Properties of Matter under Unusual Condition, edited by Mark and Fernback (Interscience, New York, 1969).

7J. L. Anderson, Princij)le of Relativity Physics (Academic, New York, 1967).

8G. t'Hooft, Nucl. Phys. B 79, 276 (1974). 98. Coleman and E. Weinberg, Phys. Rev. B 7, 1888 (1973).

M.Y. Wang 705

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