arXiv subjects
Publications and source records attributed to Guido Cognola.
The one-loop contributions to the entropy for a massive scalar field in a Kerr black hole are investigated using an approximation of the metric, which, after a conformal transformation, permits to work in a Rindler-like spacetime. Of course, as for the Schwarzschild case, the entropy is divergent in the proximity of the event horizon.
Using standard coordinates, the Maxwell equations in the Reissner-Nordström geometry are written in terms of a couple of scalar fields satisfying Klein-Gordon like equations. The density of states is derived in the semi-classical approximation and the first quantum corrections to the black hole entropy is computed by using the brick-wall model.
The analytic properties of the zeta-function for a Laplace operator on a generalised cone are studied in some detail using the Cheeger's approach and explicit expressions are given. In the compact case, the zeta-function of the Laplace operator turns out to be singular at the origin. As a result, strictly speaking, the zeta-function regularisation does not ``regularise'' and a further subtraction is required for the related one-loop effective potential.
The one-loop effective action for a massive self-interacting scalar field is investigated in $4$-dimensional ultrastatic space-time $ R \times H^3/Γ$, $H^3/Γ$ being a non-compact hyperbolic manifold with finite volume. Making use of the Selberg trace formula, the $ζ$-function related to the small disturbance operator is constructed. For an arbitrary gravitational coupling, it is found that $ζ(s)$ has a simple pole at $s=0$. The one-loop effective action is analysed by means of proper-time regularisations and the one-loop divergences are explicitly found. It is pointed out that, in this special case, also $ζ$-function regularisation requires a divergent counterterm, which however is not necessary in the free massless conformal invariant coupling case. Finite temperature effects are studied and the high-temperature expansion is presented. A possible application to the problem of the divergences of the entanglement entropy for a free massless scalar field in a Rindler-like space-time is briefly discussed.
The functional determinant of Laplace-type operators on the 3-dimensional non-compact hyperbolic manifold with invariant fundamental domain of finite volume is computed by quadratures and making use of the related terms of the Selberg trace formula.
The off-shell entropy for a massless scalar field in a D-dimensional Rindler-like space-time is investigated within the conical Euclidean approach in the manifold $C_\be\times\M^N$, $C_\be$ being the 2-dimensional cone, making use of the zeta-function regularisation. Due to the presence of conical singularities, it is shown that the relation between the zeta-function and the heat kernel is non trivial and, as first pointed out by Cheeger, requires a separation between small and large eigenvalues of the Laplace operator. As a consequence, in the massless case, the (naive) non existence of the Mellin transform is by-passed by the Cheeger's analytical continuation of the zeta-function on manifold with conical singularities. Furthermore, the continuous spectrum leads to the introduction of smeared traces. In general, it is pointed out that the presence of the divergences may depend on the smearing function and they arise in removing the smearing cutoff. With a simple choice of the smearing function, horizon divergences in the thermodynamical quantities are recovered and these are similar to the divergences found by means of off-shell methods like the brick wall model, the optical conformal transformation techniques or the canonical path integral method.
At the very early Universe the matter fields are described by the GUT models in curved space-time. At high energies these fields are asymptotically free and conformally coupled to external metric. The only possible quantum effect is the appearance of the conformal anomaly, which leads to the propagation of the new degree of freedom - conformal factor. Simultaneously with the expansion of the Universe, the scale of energies decreases and the propagating conformal factor starts to interact with the Higgs field due to the violation of conformal invariance in the matter fields sector. In a previous paper \cite{foo} we have shown that this interaction can lead to special physical effects like the renormalization group flow, which ends in some fixed point. Furthermore in the vicinity of this fixed point there occur the first order phase transitions. In the present paper we consider the same theory of conformal factor coupled to Higgs field and incorporate the temperature effects. We reduce the complicated higher-derivative operator to several ones of the standard second-derivative form and calculate an exact effective potential with temperature on the anti de Sitter (AdS) background.
The first quantum corrections to the free energy for massive fields in $D$-dimensional space-times of the form $\R\times\R^+\times\M^{N-1}$, where $D=N+1$ and $\M^{N-1}$ is a constant curvature manifold, is investigated by means of the $ζ$-function regularization. It is suggested that the nature of the divergences, which are present in the thermodynamical quantities, might be better understood making use of the conformal related optical metric and associated techniques. The general form of the horizon divercences of the free energy is obtained as a function of free energy densities of fields having negative square masses (absence of the gap in the Laplace operator spectrum) on ultrastatic manifolds with hyperbolic spatial section $H^{N-2n}$ and of the Seeley-DeWitt coefficients of the Laplace operator on the manifold $\M^{N-1}$. Furthermore, recurrence relations are found relating higher and lower dimensions. The cases of Rindler space, where $\M^{N-1}=\R^{N-1}$ and very massive $D$-dimensional black holes, where $\M^{N-1}=S^{N-1}$ are treated as examples. The renormalization of the internal energy is also discussed.
The zeta function associated with higher-spin fields on the Euclidean $4$-ball is investigated. The leading coefficients of the corresponding heat-kernel expansion are given explicitly and the zeta functional determinant is calculated. For fermionic fields the determinant is shown to differ for local and spectral boundary conditions.
The first quantum corrections to the free energy for an eternal 4-dimensional black hole is investigated at one-loop level, in the large mass limit of the black hole, making use of the conformal techniques related to the optical metric. The quadratic and logarithmic divergences as well as a finite part associated with the first quantum correction to the entropy are obtained at a generic temperature. It is argued that, at the Hawking temperature, the horizon divergences of the internal energy should cancel. Some comments on the divergences of the entropy are also presented.
The finite temperature one-loop effective potential for a scalar field defined on an ultrastatic spacetime, whose spatial part is a compact hyperbolic manifold, is studied. Different analytic expressions, especially valuable at low and high temperature are derived. Based on these results, the symmetry breaking and the topological mass generation are discussed.
Free and self-interacting scalar fields in the presence of conical singularities are analized in some detail. The role of such a kind of singularities on free and vacuum energy and also on the one-loop effective action is pointed out using $ζ$-function regularization and heat-kernel techniques.
We consider a self-interacting scalar field theory in a slowly varying gravitational background field. Using zeta-function regularization and heat-kernel techniques, we derive the one-loop effective Lagrangian up to second order in the variation of the background field and up to quadratic terms in the curvature tensors. Specializing to different spacetimes of physical interest, the influence of the curvature on the phase transition is considered.
The heat coefficients related to the Laplace-Beltrami operator defined on the hyperbolic compact manifold $H^3/\Ga$ are evaluated in the case in which the discrete group $\Ga$ contains elliptic and hyperbolic elements. It is shown that while hyperbolic elements give only exponentially vanishing corrections to the trace of the heat kernel, elliptic elements modify all coefficients of the asymptotic expansion, but the Weyl term, which remains unchanged. Some physical consequences are briefly discussed in the examples.
The one-loop effective potential for a scalar field defined on an ultrastatic space-time whose spatial part is a compact hyperbolic manifold, is studied using zeta-function regularization for the one-loop effective action. Other possible regularizations are discussed in detail. The renormalization group equations are derived and their connection with the conformal anomaly is pointed out. The symmetry breaking and the topological mass generation are also discussed.
The problem of Bose-Einstein condensation for a relativistic ideal gas on a 3+1 dimensional manifold with a hyperbolic spatial part is analyzed in some detail. The critical temperature is evaluated and its dependence of curvature is pointed out.