![]() ![]() 31 It is easy to obtain this factor by replacing V ( x ) with m 2 + λ ϕ 2 ( x ) ∕ 2 in Eq. Although one does not have to worry about wave function renormalization to first order in the ϕ 4 theory, the kinetic term is multiplied by a nontrivial functional of the field. The renormalization constants for the fields as well as for the other parameters in the theory should be defined in terms of the poles of ε and appropriate counterterms should also be introduced in order to end up with a finite set of parameters in the theory. ![]() What is known from the perturbation theory is that there is no one-loop wave function renormalization for the ϕ 4 theory, whereas one has to deal with that for the Yukawa theory even at one loop. Here, it is appropriate to make a comparison between the ϕ 4 theory and the Yukawa theory as far as the perturbation theory in terms of ε expansion is concerned. This combination, of course, arises from the fact that the scalar field is assumed to be inhomogeneous or in other words nonconstant while the determinant coming from integrating the N fermions out is being calculated. If one pays more attention to this effective action, it is easy to notice that there is a term involving gradients which is multiplied by a logarithm of the field. There are other ways of applying the symbol calculus essentially exploiting Wigner-type transformations 26 or utilizing a suitable representation of the logarithm as integral of a resolvent 27 however, they are harder to generalize to manifolds. 24 An important example of the evaluation of chiral Jacobians via the zeta function method and the symbol calculus is given in Ref. It is also possible to give a complete description of the zeta determinants for Dirac- and Laplace-type operators over finite cylinders using the contour integration method equipped with different boundary conditions. 9 Other exact solutions on homogeneous spaces can be found in Ref. 22,23 There are also other cases such as torus T N, sphere S N, and hyperbolic space H N in which it is possible to find exact solutions of the heat kernel equation and to give the zeta function for the Laplace-Beltrami operator in closed form. In the case of Laplace operators defined over a ball or over a generalized cone, one may actually evaluate the semigroup integral and find an exact result for the determinant. 20,21 In this work, we will apply this expansion to the zeta function via the semigroup integral representation. The proper mathematical tool for this is the symbol calculus for pseudodifferential operators. Thus, it is natural to look for a kind of derivative expansion. It is physically reasonable to assume that the contributions coming from the derivatives of the fields are becoming smaller as the order of the derivative increases. For such determinants, one should resort to an approximation method. 3,14,15 In general, for higher dimensional determinants, there is no analog of the Gelfand-Yaglom formula 16 (see, however, the recent attempts 17–19). 11–13 An alternative path is to evaluate the zeta function through the semigroup integral, which is used to define complex powers of elliptic operators. In general, the regularized determinants do not meet this last criterion there is, for example, by now the well known Kontsevich-Vishik multiplicative correction. The disadvantage of this approach is that the operator under consideration should be positive definite or its determinant should be related to the determinant of its square without any correction terms (i.e., without a multiplicative anomaly). ![]() 1,7–10 The advantage is that there is a systematic short time expansion of the heat kernel, the coefficients of which are all related to geometric invariants and especially suitable for theories which involve gauge fields. 3–6 In quantum field theory, the calculation of the zeta function through the use of heat kernel (or, its similar version, proper time regularization) is favored. The main tool is the introduction of a zeta function for the operator. The literature on regularized determinants is vast: we will not be able to do justice to all who has contributed to this area. 1,2 In this work, we present a derivative expansion for such regularized determinants which is especially suitable for nonpositive definite operators, such as the Dirac operator. ![]() From the one-loop effective action to instanton calculations, the main tool is the evaluation of such an infinite dimensional determinant. Calculation of functional determinants is very important in quantum field theories. ![]()
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