The Microcanonical Functional Integral. I. The Gravitational Field

The gravitational field in a spatially finite region is described as a microcanonical system. The density of states $\nu$ is expressed formally as a functional integral over Lorentzian metrics and is a functional of the geometrical boundary data that are fixed in the corresponding action. These boundary data are the thermodynamical extensive variables, including the energy and angular momentum of the system. When the boundary data are chosen such that the system is described semiclassically by {\it any} real stationary axisymmetric black hole, then in this same approximation $\ln\nu$ is shown to equal 1/4 the area of the black hole event horizon. The canonical and grand canonical partition functions are obtained by integral transforms of $\nu$ that lead to "imaginary time" functional integrals. A general form of the first law of thermodynamics for stationary black holes is derived. For the simpler case of nonrelativistic mechanics, the density of states is expressed as a real-time functional integral and then used to deduce Feynman's imaginary-time functional integral for the canonical partition function.Comment: 29 pages, plain Te

General U(N) gauge transformations in the realm of covariant Hamiltonian field theory

A consistent, local coordinate formulation of covariant Hamiltonian field theory is presented. While the covariant canonical field equations are equivalent to the Euler-Lagrange field equations, the covariant canonical transformation theory offers more general means for defining mappings that preserve the action functional - and hence the form of the field equations - than the usual Lagrangian description. Similar to the well-known canonical transformation theory of point dynamics, the canonical transformation rules for fields are derived from generating functions. As an interesting example, we work out the generating function of type F_2 of a general local U(N) gauge transformation and thus derive the most general form of a Hamiltonian density that is form-invariant under local U(N) gauge transformations.Comment: 36 pages, Symposium on Exciting Physics: Quarks and gluons/atomic nuclei/biological systems/networks, Makutsi Safari Farm, South Africa, 13-20 November 2011; Exciting Interdisciplinary Physics, Walter Greiner, Ed., FIAS Interdisciplinary Science Series, Springer International Publishing Switzerland, 201

Finite Temperature Path Integral Method for Fermions and Bosons: a Grand Canonical Approach

The calculation of the density matrix for fermions and bosons in the Grand Canonical Ensemble allows an efficient way for the inclusion of fermionic and bosonic statistics at all temperatures. It is shown that in a Path Integral Formulation fermionic density matrix can be expressed via an integration over a novel representation of the universal temperature dependent functional. While several representations for the universal functional have already been developed, they are usually presented in a form inconvenient for computer calculations. In this work we discuss a new representation for the universal functional in terms of Hankel functions which is advantageous for computational applications. Temperature scaling for the universal functional and its derivatives are also introduced thus allowing an efficient rescaling rather then recalculation of the functional at different temperatures. A simple illustration of the method of calculation of density profiles in Grand Canonical ensemble is presented using a system of noninteracting electrons in a finite confining potential.Comment: 13 pages 3 figure

A Covariant Approach To Ashtekar's Canonical Gravity

A Lorentz and general co-ordinate co-variant form of canonical gravity, using Ashtekar's variables, is investigated. A co-variant treatment due to Crnkovic and Witten is used, in which a point in phase space represents a solution of the equations of motion and a symplectic functional two form is constructed which is Lorentz and general co-ordinate invariant. The subtleties and difficulties due to the complex nature of Ashtekar's variables are addressed and resolved.Comment: 18 pages, Plain Te

Dirac Constraint Quantization of a Dilatonic Model of Gravitational Collapse

We present an anomaly-free Dirac constraint quantization of the string-inspired dilatonic gravity (the CGHS model) in an open 2-dimensional spacetime. We show that the quantum theory has the same degrees of freedom as the classical theory; namely, all the modes of the scalar field on an auxiliary flat background, supplemented by a single additional variable corresponding to the primordial component of the black hole mass. The functional Heisenberg equations of motion for these dynamical variables and their canonical conjugates are linear, and they have exactly the same form as the corresponding classical equations. A canonical transformation brings us back to the physical geometry and induces its quantization.Comment: 37 pages, LATEX, no figures, submitted to Physical Review