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Keference International Centre for Theoretical KEFERENCE INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS IC/86/342 INTERNATIONAL THE REGULAR EFFECTIVE ACTION OF GAUGE FIELD THEORY AND QUANTUM GRAVITY ATOMIC ENERGY AGENCY Nazir S. Baaklini UNITED NATIONS EDUCATIONAL, SCIENTIFIC AND CULTURAL ORGANIZATION IC/86/342 International Atomic Energy Agency 0. INTRODUCTION and The effective action of quantum field theory, defined by the functional United Nations Educational Scientific and Cultural Organization integral , is an elegant and potentially, very powerful framework for compu- INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS ting quantum effects, in a manner that preserves underlying fundamental symmetries. However, this framework had been utilized mainly in computing effective potentials, and in certain discussions of the divergent counter-terms needed in gauge field THE REGULAR EFFECTIVE ACTION OF GAUGE FIELD THEORY theory and quantum gravity. Whereas it should ~ae very fruitful fo develop expre- AND QUANTUM GRAVITY ssions for the effective action which could have general applicability, computa- tions are still performed using conventional Feynman rules. What is needed is a perturbative formalism for the effective action vhicii applies to a general field Nazir S. Baaklini theory containing Bosonic as well as Fermionic fields, and vhich could be used to International Centre for Theoretical Physics, Trieste, Italy address fundamental issues of quantum field theory and quantum gravity. and Dahr el Chir Science Centre, Dhour el Choueir, Lebanon. On the other hand, our study of the effective action makes impact on two fundamental issues. One of these concerns the treatment of gauge field theory and other singular systems. The other concerns the ultraviolet divergences of quantum field theory and quantum gravity. Quite remarkably, both issues may be ABSTRACT tackled by the same concept, that of a properly metricizea functional space. This concept is realized, in one hand, by using the correct functional space metric We present a perturbative formalism for the effective quantum action which is determined by the Poisson brackets of the canonical field theory. For of a general Femi-Bose field theory. We propose a unitary alternative to the singular systems, the correct functional metric is determined by the modified 11) conventional virtual ghost prescription for handling the functional integral brackets in Dirac's treatment of constraints. This implements unitarity in of gauge fields, "based on the functional space metric that is determined by the perturbative expansion of the effective action. Our approach provides an the Dirac 'brackets of the Canonical theory, A gauge-invariant Gaussian cutoff elegant alternative to the conventional virtual ghost prescription * . is introduced by extending the functional space metric into a regular counter- Moreover, a fundamental gauge-invariant Gaussian cutoff may be implemented part. The regular one-loop contributions of a scalar field to the photon and through a regulating functional metric. The latter replaces the singular Dirae the graviton self-energies are computed. The otherwise logarithmically diver- delta-function which metricizes the virtual functional space by a regular, and gent contributions are found, in our scheme, to be independent of cutoff. gauge-covariant, counterpart. In See.l, we begin by reviewing the definition of the effective action in terms of the functional integral. The perturbative formalism, for purely Bosonic fields, begins in Sec.2. In Sec.3, we derive expressions for expectation values on the basis of the Gaussian functional integral. These are utilized in HIRAMAHE - TRIESTE deriving the perturbative expressions of the effective action. The latter is October 1986 given to two-loop order In section t. We also prescribe graphical rules for cons- tructing the higher orders. In Sec.5> we extend these results to Fenr.i-Bose theories, through a unified treatment of Fermionic and Bosonic fields. * To be submitted for publication. -2- We begin our discussion of vector gauge field theory in Sec.6. Our This gives upon functional differetiation, and using Eq,(2), novel scheme of treating the virtual gauge invariance of the functional integ- ral, by utilizing the correct functional metric, is presented in Sec.7. In Sec.8, ve introduce the regulating functional metric. In Sec,9> we apply our functional metric regularization method to the Eqs.(l), (3), and <M give computation of the one-loop scalar field contribution to the photon self-energy. Likewise, we examine the regular one-loop contribution of a scalar field to the (5) graviton self-energy, in Sec.10. Conclusions are presented in Sec.11. In Appendix A, ve develop a formalism for constructing the renormalized Making the shift * •+ f + + in Eq. (5), obtain effective action. Appendices B, C, and D provide supporting formalism and pedagogic applications to the systems of a self-interacting scalar, and quantum electrodyna- (6) mics. In Appendix E, we apply Dirac's Hamiltonian treatment to gauge fields, and provide pointers needed for computing the effective action of vector gauge field theory. Appendices (F, G, H, I, J) provide prerequisite mathematical material 2. ITERATIVE EXPANSION FOR EOSONIC THEORY especially developed for handling the computations of sections 9 and 10. We shall be concerned here with Bosonic field theory. The extension to Ferai-Bose systems will he given in a later section. The effective action will be expanded iteratively in Planck's constant h, 1. THE EFFECTIVE ACTION OF QUANTUM FIELD THEORY: DEFINITION The generating functional for connected Green'n functions Z(J) is r(*} = rU) + hrU) + « r(+) + ... tT) defined by the functional integral Reinstating h into Eq.(6), we have eiZ(J> = (1) e<i/*>rf*) = /d,e t8) where f represents a quantum field, J. a corresponding external source, and W($) is the classical action functional. Here, we use a compact notation where We shall obtain an expression for rt*) to order h , and then prescribe graphic the index i represents all labels and spacetime arguments, and la = 1. The field rules for higher orders. To this end, we write the Taylor expansion $., being functionally integrated over, will be termed the virtual field. On the other hand, the effective field is defined by (2) 19) 3J. The generating functional for proper vertices or the effective action Here (VT, W., W , V , ...) are the classical action functional W($) and its 1 1J 1J id is defined by the Legendre transformation totally symmetric functional derivatives, (10) rU) = z(J) - (3) -3- -U- T This expression is valid provided W is a nonsingular matrix. Me shall return From Eq. (8), obtain immediately that f( if) = W( <p), and write to the case where W Is singular, and which concerns gauge theory. (11) By taking successive derivatives of Eqs.(lfi) and (17) with respect to W , obtain Hence, the term proportional to W. cancels in Eq.(8). The term proportional to T. contributes to r, and so on. (16) I1 2 Utilizing Eos.(9) and (11) in Eq.(8), obtain w w i J (19) (i/h)r e (12) Here, [(1/3!) v ] + (13) " Vj*k ijk-M-i . (20) and we have defined the expectation value Here, W~ is the symmetric inverse of W (21) From Eq.(12), obtain to order and we have used = W - (15) fi(TrlnW) = (22) To evaluate this expression, we need to compute the expectation values of (23) polynomials in $,. The expectation values vanish for odd powers of . This is shown by taking the expectation value iW < > which vanishes due to the 3. GAUSSIAN INTEGRAL AND EXPECTATION VALUES J-J convergence of the Gaussian integrand. But since W is nonsingular <$ > = 0 The Gaussian integral follows. Then taking successive derivatives of the latter with respect to W , obtain that all odd powers of $. give vanishing expectation values, (16) <*i»,?k ..-> = 0, (odd power) (2lt) may be expressed by a functional determinant 1/2 <1> = |detw|" = (17) -5- -6- 5. EXTENSION TO FERMI BOSE THEORY THE TWO_L00P EFFECTIVE ACTION AND HIGHER ORDERS In this section, we extend the preceeding results to theories Now back to the computation of Eq.(15), dropping all odd powers of containing Bosonic (commuting) fields + , and fermionic (anticommiting) fields and using Eqs.(18)-(2O), obtain * . Without loss of generality, both types are taken to be real. We shall give a unified treatment. Consider the hyperfield $A = $ ). This obeys the gene- r = U + h(l/2)TrlnW - ralized commutation and differentiation rules (2 ) (28) 5 +A*B At thi3 point, use the first order result (29) r = (i/2)Trmw ; C26) (30) in Eq. (25), and o*btain Here, a(AB) is a sign factor defined by 1 if AB = ij or ig f = C27) -1 if AS = a6. (31) The resulting first and second order quantum contributions in Eq.(27) may be represented, respectively, by irreducible one and two-loop graphs. The We shall utilize the following definitions. three and four-leg vertices correspond to V.,. and W , respectively, while ijk ijkl = o(A). the Joining lines (effective propagators) correspond to W,-1 . Notice that the a(AB + CD) = o(AB)o(CD); (32) —1 —1_ —1 term W..(WT;¥, .,.)(W~ W .) which can be represented by a reducible graph (two IJ K± Kli mn Binj Hote that d(2A) = o (A) = 1. closed loops joined by a line) cancels in the derivation of Eq.(2T). This corres- ponds to the important fact that the effective action should only generate We shall employ totally hypersymmetric tensors T^j, ,. These have one-particle-irreducible vertices. the property of being multiplied by a(AB) under a permutation of any two indices The extension of the preceeding results to higher orders is straight- A and B. To construct a totally hypersymmetric tensor 'r^gf. \ from a general forward. However, the intermediate expressions becoae very lengthy. As a short- tensor 1HOC , construct the totally symmetric tensor and multiply each term by cut, we may turn to graphical rules.
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