Heat Kernel Approach Research Articles

Entanglement entropy for the n-sphere

We calculate the entanglement entropy for a sphere and a massless scalar field in any dimensions. The reduced density matrix is expressed in terms of the infinitesimal generator of conformal transformations keeping the sphere fixed. The problem is mapped to the one of a thermal gas in a hyperbolic space and solved by the heat kernel approach. The coefficients of the logarithmic term in the entropy for 2 and 4 spacetime dimensions are in accordance with previous numerical and analytical results. In particular, the four-dimensional result, together with the one reported by Solodukhin, gives support to the Ryu–Takayanagi holographic ansatz. We also find that there is no logarithmic contribution to the entropy for odd spacetime dimensions.

A heat kernel approach to the stability of exponential equation in distributions and hyperfunctions

Let f,g,h:G×(0,∞)→C with G an Abelian group. We consider the stability of the Pexiderized exponential equation, |f(x+y,t+s)−g(x,t)h(y,s)|≤H(t,s),x,y∊G, t,s>0, where H:(0,∞)2→(0,∞) is a decreasing function. As an application we establish the stability problem of the Pexiderized exponential equation, u∘A−v⊗w∊D′L∞(R2n), in some spaces of generalized functions, where D′L∞(R2n) denotes the space of bounded distributions.

Spectral multipliers for Schrödinger operators

We prove a sharp Hormander multiplier theorem for Schrodinger operators $H=-\Delta+V$ on $\mathbb{R}^n$. The result is obtained under certain condition on a weighted $L^\infty$ estimate, coupled with a weighted $L^2$ estimate for $H$, which is a weaker condition than that for nonnegative operators via the heat kernel approach. Our approach is elaborated in one dimension with potential $V$ belonging to certain critical weighted $L^1$ class. Namely, we assume that $\int(1+|x|) |V(x)|\,dx$ is finite and $H$ has no resonance at zero. In the resonance case, we assume $\int(1+|x|^2) |V(x)|\, dx$ is finite.

Casimir Energy of 5D Electromagnetism and New Regularization Based on Minimal-Area Principle

We examine the Casimir energy of 5D electromagnetism in the recent standpoint. The bulk geometry is flat. Z$_2$ symmetry and the periodic property, for the extra coordinate, are taken into account. After confirming the consistency with the past result, we do new things based on a {\it new regularization}. In the treatment of the divergences, we introduce IR and UV cut-offs and {\it restrict} the (4D momentum, extra coordinate)-integral region. The regularized configuration is the {\it sphere lattice}, in the 4D continuum space, which changes along the extra coordinate. The change (renormalization flow) is specified by the {\it minimal area principle}, hence this regularization configuration is string-like. We do the analysis not in the Kaluza-Klein expanded form but in a {\it closed} form. We do {\it not} use any perturbation. The formalism is based on the heat-kernel approach using the {\it position/momentum propagator}. Interesting relations between the heat-kernels and the P/M propagators are obtained, where we introduce the {\it generalized} P/M propagators. A useful expression of the Casimir energy, in terms of the P/M propagator, is obtained. The restricted-region approach is replaced by the weight-function approach in the latter-half description. Its meaning, in relation to the {\it space-time quantization}, is argued. {\it Finite} Casimir energy is numerically obtained. The compactification-size parameter (periodicity) suffers from the renormalization effect. Numerical evaluation is exploited. Especially the minimal surface lines in the 5D flat space are obtained both numerically using the Runge-Kutta method and analytically using the general solution.

Heat-Kernel Approach to UV/IR Mixing on Isospectral Deformation Manifolds

We work out the general features of perturbative field theory on noncommutative manifolds defined by isospectral deformation. These (in general curved) ‘quantum spaces’, generalizing Moyal planes and noncommutative tori, are constructed using Rieffel’s theory of deformation quantization by actions of $$\mathbb{R}^l .$$ Our framework, incorporating background field methods and tools of QFT in curved spaces, allows to deal both with compact and non-compact spaces, as well as with periodic and non-periodic deformations, essentially in the same way. We compute the quantum effective action up to one loop for a scalar theory, showing the different UV/IR mixing phenomena for different kinds of isospectral deformations. The presence and behavior of the non-planar parts of the Green functions is understood simply in terms of off-diagonal heat kernel contributions. For periodic deformations, a Diophantine condition on the noncommutivity parameters is found to play a role in the analytical nature of the non-planar part of the one-loop reduced effective action. Existence of fixed points for the action may give rise to a new kind of UV/IR mixing. Communicated by Vincent Rivasseau

Cartan–Weyl Dirac and Laplacian Operators, Brownian Motions: The Quantum Potential and Scalar Curvature, Maxwell’s and Dirac-Hestenes Equations, and Supersymmetric Systems

We present the Dirac and Laplacian operators on Clifford bundles over space–time, associated to metric compatible linear connections of Cartan–Weyl, with trace-torsion, Q. In the case of nondegenerate metrics, we obtain a theory of generalized Brownian motions whose drift is the metric conjugate of Q. We give the constitutive equations for Q. We find that it contains Maxwell’s equations, characterized by two potentials, an harmonic one which has a zero field (Bohm-Aharonov potential) and a coexact term that generalizes the Hertz potential of Maxwell’s equations in Minkowski space.We develop the theory of the Hertz potential for a general Riemannian manifold. We study the invariant state for the theory, and determine the decomposition of Q in this state which has an invariant Born measure. In addition to the logarithmic potential derivative term, we have the previous Maxwellian potentials normalized by the invariant density. We characterize the time-evolution irreversibility of the Brownian motions generated by the Cartan–Weyl laplacians, in terms of these normalized Maxwell’s potentials. We prove the equivalence of the sourceless Maxwell equation on Minkowski space, and the Dirac-Hestenes equation for a Dirac-Hestenes spinor field written on Minkowski space provided with a Cartan–Weyl connection. If Q is characterized by the invariant state of the diffusion process generated on Euclidean space, then the Maxwell’s potentials appearing in Q can be seen alternatively as derived from the internal rotational degrees of freedom of the Dirac-Hestenes spinor field, yet the equivalence between Maxwell’s equation and Dirac-Hestenes equations is valid if we have that these potentials have only two components corresponding to the spin-plane. We present Lorentz-invariant diffusion representations for the Cartan–Weyl connections that sustain the equivalence of these equations, and furthermore, the diffusion of differential forms along these Brownian motions. We prove that the construction of the relativistic Brownian motion theory for the flat Minkowski metric, follows from the choices of the degenerate Clifford structure and the Oron and Horwitz relativistic Gaussian, instead of the Euclidean structure and the orthogonal invariant Gaussian. We further indicate the random Poincare–Cartan invariants of phase-space provided with the canonical symplectic structure. We introduce the energy-form of the exact terms of Q and derive the relativistic quantum potential from the groundstate representation. We derive the field equations corresponding to these exact terms from an average on the invariant state Cartan scalar curvature, and find that the quantum potential can be identified with 1 / 12R(g), where R(g) is the metric scalar curvature. We establish a link between an anisotropic noise tensor and the genesis of a gravitational field in terms of the generalized Brownian motions. Thus, when we have a nontrivial curvature, we can identify the quantum nonlocal correlations with the gravitational field. We discuss the relations of this work with the heat kernel approach in quantum gravity. We finally present for the case of Q restricted to this exact term a supersymmetric system, in the classical sense due to E.Witten, and discuss the possible extensions to include the electromagnetic potential terms of Q

Chiral Anomaly with MIT Bag Boundary Conditions

After a brief review of the heat kernel approach we obtain a chiral anomaly for local MIT bag boundary conditions.

Heat kernel approach in quantum field theory

We give a short overview of the effective action approach in quantum field theory and quantum gravity and describe various methods for calculation of the asymptotic expansion of the heat kernel for second-order elliptic partial differential operators acting on sections of vector bundles over a compact Riemannian manifold. We consider both Laplace type operators and non-Laplace type operators on manifolds without boundary as well as Laplace type operators on manifolds with boundary with oblique and non-smooth boundary conditions.

Temperature in fermion systems and the chiral fermion determinant

We give an interpretation to the issue of the chiral determinant in the heat-kernel approach. The extra dimension (fifth dimension) is interpreted as (inverse) temperature. The $(1+4)$-dimensional Dirac equation is naturally derived by the Wick rotation for the temperature. In order to define a ``good'' temperature, we choose those solutions of the Dirac equation which propagate in a fixed direction in the extra coordinate. This choice fixes the regularization of the fermion determinant. The $(1+4)$-dimensional Dirac mass (M) is naturally introduced and the relation $|\mathrm{f}\mathrm{o}\mathrm{u}\mathrm{r}\ensuremath{-}\mathrm{d}\mathrm{i}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{a}\mathrm{l}\mathrm{ }\mathrm{electron}\mathrm{ }\mathrm{momentum}|\ensuremath{\ll}|M|\ensuremath{\ll}\mathrm{ultraviolet}$ cutoff naturally appears. The chiral anomaly is explicitly derived for the two-dimensional Abelian model. Typically two different regularizations appear depending on the choice of propagators. One corresponds to the chiral theory, the other to the nonchiral (Hermitian) theory.

New index formulas as a meromorphic generalization of the Chern-Gauss-Bonnet theorem

Laplace operators perturbed by meromorphic potential on the Riemann and separated-type Klein surfaces are constructed and their indices are calculated in two different ways. The topological expressions for the indices are obtained from the study of the spectral properties of the operators. Analytical expressions are provided by the heat kernel approach in terms of functional integrals. As a result, two formulae connecting characteristics of meromorphic (real meromorphic) functions and topological properties of Riemann (separated-type Klein) surfaces are derived.

On the chiral anomaly in non-Riemannian spacetimes

The translational Chern-Simons type three-form coframe torsion on a Riemann-Cartan spacetime is related (by differentiation) to the Nieh-Yan four-form. Following Chandia and Zanelli, two spaces with non-trivial translational Chern-Simons forms are discussed. We then demonstrate, firstly within the classical Einstein-Cartan-Dirac theory and secondly in the quantum heat kernel approach to the Dirac operator, how the Nieh-Yan form surfaces in both contexts, in contrast to what has been assumed previously.

ZETA-FUNCTION REGULARIZATION APPROACH TO FINITE TEMPERATURE EFFECTS IN KALUZA-KLEIN SPACE-TIMES

In the framework of heat-kernel approach to zeta-function regularization, the one-loop effective potential at finite temperature for scalar and spinor fields on Kaluza-Klein space-time of the form [Formula: see text], where MP is p-dimensional Minkowski space-time is evaluated. In particular, when the compact manifold is [Formula: see text], the Selberg trace formula associated with discrete torsion-free group Γ of the n-dimensional Lobachevsky space Hn is used. An explicit representation for the thermodynamic potential valid for arbitrary temperature is found. As a result a complete high temperature expansion is presented and the roles of zero modes and topological contributions is discussed.

MASSLESS SCALAR CASIMIR EFFECT IN A CLASS OF HYPERBOLIC KALUZA-KLEIN SPACE-TIMES

In the framework of heat-kernel approach to zeta-function regularization we calculate the one-loop effective potential (Casimir effect) massless scalar field on Kaluza-Klein space-time of the form RD−n×Hn/Γ(2≤n<D). In addition the Selberg trace formula associated with discrete torsion-free group Γ of the n-dimensional Lobachevsky space Hn is used. A negative Casimir effect related to trivial line bundle with character χ=1 is found. A comparison of the results obtained and Casimir effect for massless field on torus backgrounds is also presented.