Large N phase transition in the heat kernel on the U (N) group

The large N phase transition point is investigated in the heat kernel on the U(N) group with respect to arbitrary boundary conditions. A simple functional relation is found relating the density of eigenvalues of the boundary field to the saddle point shape of the typical Young tableaux in the large N limit of the character expansion of the heat kernel. Both strong coupling and weak coupling phases are investigated for some particular cases of the boundary holonomy.     

Strong coupling expansion for the mass gap in SU(2) lattice gauge theory with mixed action

We perform a strong coupling expansion up to O(β⁷) for the mass-gap in SU(2) lattice gauge theory with mixed action. A novel feature of the strong coupling expansion is discussed. The strong coupling series appears to approach the scaling region more smoothly and Padé approximants become more stable than in the case with simple Wilson action. The region of validity of a recently proposed resummation of perturbation theory as applied to the determination of the asymptotic scaling behavior is investigated. Results of a strong coupling calculation for the heat kernel action, which is related to the mixed action for a special choice of parameters, are also reported.     

Asymptotic expansion of the heat kernel trace of Laplacians with polynomial potentials

It is well known that the asymptotic expansion of the trace of the heat kernel for Laplace operators on smooth compact Riemannian manifolds can be obtained through termwise integration of the asymptotic expansion of the on-diagonal heat kernel. The purpose of this work is to show that, in certain circumstances, termwise integration can be used to obtain the asymptotic expansion of the heat kernel trace for Laplace operators endowed with a suitable polynomial potential on unbounded domains. This is achieved by utilizing a resummed form of the asymptotic expansion of the on-diagonal heat kernel.     

Mass gap of lattice O(3) non-linear σ-model with naive and improved actions

We evaluate the strong coupling expansion for the mass gap of the euclidean lattice O(3) non-linear σ-model in two dimensions with the naive action and with the tree level improved action. For the naive action the expansion series exhibits scaling behaviour in the weak coupling region. For the tree level improved action, our series is consistent with Monte Carlo data in the intermediate coupling region but is too short to reproduce the scaling.     

Heat Kernel Coefficients for Laplace Operators on the Spherical Suspension

In this paper we compute the coefficients of the heat kernel asymptotic expansion for Laplace operators acting on scalar functions defined on the so called spherical suspension (or Riemann cap) subjected to Dirichlet boundary conditions. By utilizing a contour integral representation of the spectral zeta function for the Laplacian on the spherical suspension we find its analytic continuation in the complex plane and its associated meromorphic structure. Thanks to the well known relation between the zeta function and the heat kernel obtainable via Mellin transform we compute the coefficients of the asymptotic expansion in arbitrary dimensions. The particular case of a d-dimensional sphere as the base manifold is studied as well and the first few heat kernel coefficients are given.     

An action function for a higher step Grushin operator

The purpose of this paper is to discuss how we can construct the heat kernel for (sub)-Laplacian in an explicit (integral) form in terms of a certain class of special functions. Of course, such cases will be highly limited. Here we only treat a typical operator, called Grushin operator. So, first we explain two methods to construct the heat kernel of a “step 2” Grushin operator. One is the eigenfunction expansion which leads to an integral form for the heat kernel, then we treat the formula by a method called, complex Hamilton–Jacobi method invented by Beals–Gaveau–Greiner. One of the main result in this paper is to construct an action function for a higher order oscillator. Until now, no explicit expression of the heat kernel for higher order cases have been given in an explicit form and we show a phenomenon that our action function will play a role toward the construction of the heat kernel of higher step Grushin operators.     

The SU(2) chiral model in an external field: A complex stochastic process on a non-abelian group

The principal chiral model in an external field has a complex action. Conventional methods for performing Monte Carlo simulations are therefore useless for strong fields or weak coupling (i.e. large lattices). We apply the Langevin equation to this action and generate a complex stochastic process on a non-abelian group. Nice agreement with the pertubative expansion is observed for strong fields. A clear verification of the second-order phase transition for weak fields proved by Polyakov and Wiegmann seems to require larger lattices and more computer time than we have available, but should be straightforward using the complex Langevin equation.     

Albanese Maps and Off Diagonal Long Time Asymptotics for the Heat Kernel

We discuss long time asymptotic behaviors of the heat kernel on a non-compact Riemannian manifold which admits a discontinuous free action of an abelian isometry group with a compact quotient. A local central limit theorem and the asymptotic power series expansion for the heat kernel as the time parameter goes to infinity are established by employing perturbation arguments on eigenvalues and eigenfunctions of twisted Laplacians. Our ideas and techniques are motivated partly by analogy with Floque–Bloch theory on periodic Schr?dinger operators. For the asymptotic expansion, we make careful use of the classical Laplace method. In the course of a discussion, we observe that the notion of Albanese maps associated with the abelian group action is closely related to the asymptotics. A similar idea is available for asymptotics of the transition probability of a random walk on a lattice graph. The results obtained in the present paper refine our previous ones [4]. In the asymptotics, the Euclidean distance associated with the standard realization of the lattice graph, which we call the Albanese distance, plays a crucial role. Received: 20 September 1998 / Accepted: 19 August 1999     

Critical bubbles and fluctuations at the electroweak phase transition

We discuss the critical bubbles of the electroweak phase transition using an effective high-temperature 3-dimensional action for the Higgs field ϕ. The separate integration of gauge and Goldstone boson degrees of freedom is conveniently described in the 't Hooft-Feynman covariant background gauge. The effective dimensionless gauge coupling g3 (T) z in the broken phase is well behaved throughout the phase transition. However, the behavior of the one-loop Z(ϕ) factors of the Higgs and gauge kinetic terms signalizes the breakdown of the derivative expansion and of the perturbative expansion for a range of small ϕ values increasing with the Higgs mass m_H Taking a functional S_z [ϕ] with constant Z(ϕ) = z instead of the full non-local effective action in some neighborhood of the saddle point we are calculating the critical bubbles for several temperatures. The fluctuation determinant is calculated to high accuracy using a variant of the heat kernel method. It gives a strong suppression of the transition rate compared to previous estimates.     

1/m c expansion of Ω b (*) →Ω c (*) semileptonic decay form factors

The higher derivative expansion of the one-loop effective action for an external scalar potential is calculated to order {ie111-1}, using the string-inspired Bern-Kosower method in the first quantized path integral formulation. Comparisons are made with standard heat kernel calculations and with the corresponding Feynman diagrammatic calculation in order to show the efficiency of the present method.Partially supported by funds provided by the Deutsche Forschungsgemeinschaft     

Perturbative contributions to the electroweak interface tension

The main perturbative contribution to the free energy of an electroweak interface is due to the effective potential and the tree level kinetic term. The derivative corrections are investigated with one-loop perturbation theory. The action is treated in derivative, in heat kernel, and in a multi local expansion. The massive contributions turn out to be well described by the Z-factor. The massless mode, plagued by infrared problems, is numerically less important. Its perturbatively reliable part can by calculated in derivative expansion as well. A self consistent way to include the Z-factor in the formula for the interface tension is presented.     

Zeta function regularization of path integrals in curved spacetime

This paper describes a technique for regularizing quadratic path integrals on a curved background spacetime. One forms a generalized zeta function from the eigenvalues of the differential operator that appears in the action integral. The zeta function is a meromorphic function and its gradient at the origin is defined to be the determinant of the operator. This technique agrees with dimensional regularization where one generalises ton dimensions by adding extra flat dimensions. The generalized zeta function can be expressed as a Mellin transform of the kernel of the heat equation which describes diffusion over the four dimensional spacetime manifold in a fith dimension of parameter time. Using the asymptotic expansion for the heat kernel, one can deduce the behaviour of the path integral under scale transformations of the background metric. This suggests that there may be a natural cut off in the integral over all black hole background metrics. By functionally differentiating the path integral one obtains an energy momentum tensor which is finite even on the horizon of a black hole. This energy momentum tensor has an anomalous trace.     

Non-canonical distribution and non-equilibrium transport beyond weak system-bath coupling regime: A polaron transformation approach

The concept of polaron, emerged from condense matter physics, describes the dynamical interaction of moving particle with its surrounding bosonic modes. This concept has been developed into a useful method to treat open quantum systems with a complete range of system-bath coupling strength. Especially, the polaron transformation approach shows its validity in the intermediate coupling regime, in which the Redfield equation or Fermi’s golden rule will fail. In the polaron frame, the equilibrium distribution carried out by perturbative expansion presents a deviation from the canonical distribution, which is beyond the usual weak coupling assumption in thermodynamics. A polaron transformed Redfield equation (PTRE) not only reproduces the dissipative quantum dynamics but also provides an accurate and efficient way to calculate the non-equilibrium steady states. Applications of the PTRE approach to problems such as exciton diffusion, heat transport and light-harvesting energy transfer are presented.     

Quintessential inflation from a variable cosmological constant in a 5D vacuum

We explore an effective 4D cosmological model for the universe where the variable cosmological constant governs its evolution and the pressure remains negative along all the expansion. This model is introduced from a 5D vacuum state where the (space-like) extra coordinate is considered as noncompact. The expansion is produced by the inflaton field, which is considered as nonminimally coupled to gravity. We conclude from experimental data that the coupling of the inflaton with gravity should be weak, but variable in different epochs of the evolution of the universe.     

On the low frequency electromagnetic response of strong coupling superconductors

In this paper we report about penetration depth measurements performed on strong coupling Pb–Bi alloys. The change of penetration depth with temperature is obtained from the frequency shift of a superconducting resonant cavity. The experimental results are compared with the low frequency electromagnetic response kernel calculated from the strong coupling theory and the scaled weak coupling theory respectively. A very good agreement between experiment and strong coupling theory is observed. The fit of the scaled weak coupling theory to the measured change of penetration depth yields values of the superconducting energy gap, which agree with the corresponding results of tunneling measurements.     

Finite size Wilson loops

A study of Wilson loop averages for finite size loops is initiated. Within the framework of euclidean four-dimensional lattice SU(2) gauge theory with elementary Wilson action we compute the expectation values of all rectangular loops to 12th order in the strong coupling expansion. The leading term for weak coupling is evaluated for loops up to 4 × 4. A comparison to Monte Carlo data is presented. Other related issues are also discussed.     

An effective action for a variable electromagnetic field

This work consists essentially of two parts. The first part is an analysis of the one-loop effective action using the zeta-function approach. This gives a simple expression for the effective action in terms of the background field propagator. The next-of-kin to the zeta-function, the heat kernel, is given in terms of B. DeWitt's proper time expansion (also known as P. B. Gilkey's theorem). It is calculated in the second part for fermions interacting with an external electromagnetic field to first nonvanishing order in the variations of the gauge field.     

Gravity, non-commutative geometry and the Wodzicki residue

We derive an action for gravity in the framework of non-commutative geometry by using the Wodzicki residue. We prove that for a Dirac operator D on an n dimensional compact Riemannian manifold with n ≥ 4, n even, the Wodzicki residue Res(D⁻ⁿ⁺²) is the integral of the second coefficient of the heat kernel expansion of D². We use this result to derive a gravity action for commutative geometry which is the usual Einstein-Hilbert action and we also apply our results to a non-commutative extension which is given by the tensor product of the algebra of smooth functions on a manifold and a finite dimensional matrix algebra. In this case we obtain gravity with a cosmological constant.     

Quantum noncommutative gravity in two dimensions

We study quantisation of noncommutative gravity theories in two dimensions (with noncommutativity defined by the Moyal star product). We show that in the case of noncommutative Jackiw–Teitelboim gravity the path integral over gravitational degrees of freedom can be performed exactly even in the presence of a matter field. In the matter sector, we study possible choices of the operators describing quantum fluctuations and define their basic properties (e.g., the Lichnerowicz formula). Then we evaluate two leading terms in the heat kernel expansion, calculate the conformal anomaly and the Polyakov action (as an expansion in the conformal field).