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Journal of Mathematical Physics

Covariant path integrals on hyperbolic surfaces

Joe Schaefer

Published Nov 1, 1997• 1 citation
Publisher

Abstract

DeWitt’s covariant formulation of path integration [B. De Witt, “Dynamical theory in curved spaces. I. A review of the classical and quantum action principles,” Rev. Mod. Phys. 29, 377–397 (1957)] has two practical advantages over the traditional methods of “lattice approximations;” there is no ordering problem, and classical symmetries are manifestly preserved at the quantum level. Applying the spectral theorem for unbounded self-adjoint operators, we provide a rigorous proof of the convergence of certain path integrals on Riemann surfaces of constant curvature −1. The Pauli–DeWitt curvature correction term arises, as in DeWitt’s work. Introducing a Fuchsian group Γ of the first kind, and a continuous, bounded, Γ-automorphic potential V, we obtain a Feynman–Kac formula for the automorphic Schrödinger equation on the Riemann surface Γ\ℍ. We analyze the Wick rotation and prove the strong convergence of the so-called Feynman maps [K. D. Elworthy, Path Integration on Manifolds, Mathematical Aspects of Superspace, edited by Seifert, Clarke, and Rosenblum (Reidel, Boston, 1983), pp. 47–90] on a dense set of states. Finally, we give a new proof of some results in C. Grosche and F. Steiner, “The path integral on the Poincare upper half plane and for Liouville quantum mechanics,” Phys. Lett. A 123, 319–328 (1987).

DOI
10.1063/1.532158

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In this paper we shall introduce a generalized equi-continuity of a family of semigroups and prove a new type of Trotter-Kato Theorem, applicable to the weak convergence of semigroups.In [13], we prove the existence of non-unitary solutions to formally self-adjoint Schr\"odinger equations.In that paper, we need the Trotter-Kato Theorem for the weak convergence.However, various versions of the Trotter-Kato Theorem in locally convex spaces already published are not applicable to the weak convergence as far as the authors knows.Therefore we shall give a generalized form of the Trotter-Kato Theorem in Yosida [18].Next we shall define a kind of operator-valued integration and define the Feynman path integrals of Riemann integral type.It seems that it is one of the best possible conditions of the existence of the path integrals of Riemann integral type for Schr\"odinger equation with singular potentials.Our class of potentials is wide enough;theenough;theenough;the real measurable potential UUU should be continuous except a closed set of measure zero.それどころか，内積を取る相手が完全系を構成するかどうかについても，あまり厳密に考えて いません (実際に，そうした論文を読んだことがあります). 証明が厳密でなくても，物理的な 直観に照らして正当 (と感じられる) ならば充分であり，そのうち数学の得意な人がちゃんとし た証明をしてくれると期待している訳です．J \S 2. Trotter-Kato Theorem for Weak Covergence We shall introduce a generalized equicontinuity of a family of semigroups and prove a new type of Trotter-Kato Theorem, applicable to the weak convergence of semigroups.We begin by introducing some terminology and notation and present those aspects of the basic theory which are required in subsequent subsections.\S 2.1.Filter Definition 2.1.Given a set EEE , a partial ordering ⊂\subset⊂ can be defined on the powerset P(E)\mathcal{P}(E)P(E) by subset inclusion.Define a filter F\mathcal{F}F on EEE as a subset of P(E)\mathcal{P}(E)P(E) with the following properties:Definition 2.4.We say that filter base B\mathcal{B}B converges to xxx , denoted by Barrowx\mathcal{B}arrow xBarrowx , if for every neighborhood UUU of xxx , there is a B∈BB\in \mathcal{B}B∈B such that B⊂UB\subset UB⊂U .In this case, xxx is called a limit of B\mathcal{B}B and B\mathcal{B}B is called a convergent filter base.Lemma 2.5.For every neighbourhood base U(x)\mathcal{U}(x)U(x) of xxx , it follows that U(x)arrowx\mathcal{U}(x)arrow xU(x)arrowx .Lemma 2.6.XXX is a Hausdorff space ifand only ifevery filter base on XXX has at most one limit.For details concerning the filter, we refer to Bourbaki [1].\S 2.2.Locally Convex Topologies Definition2.7.A linear topological space XXX over the complex number field C\mathbb{C}C is called a locally convex linear topological space, or, in short, a locally convex space, if and only if its open sets ∋O\ni O∋O contains a convex, balanced and absorbing open set.Let M⊂XM\subset XM⊂X .Then
In this paper rigorous path integral treatments are presented of free motion on the Poincaré disc, the Poincaré upper half-plane and on the hyperbolic strip, three spaces which are analytically equivalent to each other. Whereas the path integral treatments on the disc and on the strip are new, two further path integral treatments are discussed for the Poincaré upper half-plane to the existing one. All the calculations are mainly based on Fourier-expansions of the Feynman kernels which can be easily performed. The remaining path integrals on D, U and S can be reduced to the path integral problems on the pseudosphere Δ2, Liouville quantum mechanics and the modified Pöschl-Teller potential problem, respectively, where the results by means of path integrals are known. The corresponding normalized wave-functions and the energy-spectrum are derived. The energy-spectra are the same in all three spaces and read E = (1/2m) (p2 + 1/4) (p -momentum). The “zero-point” energy E0 = 1/8m is discussed which can be interpreted in terms of the Heisenberg uncertainty relation. The equivalences between the Feynman kernels on Δ2, D, U and S are also discussed.
This conference talk elaborates on a recently discovered mapping procedure by which classical orbits and path integrals for the motion of a point particle in flat space can be transformed correctly into those in curved space. This procedure evolved from well established methods in the theory of plastic deformations where crystals with defects are described mathematically by applying nonholonomic coordinate transformations to ideal crystals. In the context of time-sliced path integrals, there seems to exists a quantum equivalence principle which determines the measures of fluctating orbits in spaces with curvature and torsion. The nonholonomic transformations produce a nontrivial Jacobian in the path measure which in a curved space produces an additional term proportional to the curvature scalar canceling a similar term found earlier by DeWitt from a naive formulation of Feynman's time-sliced path integral. This cancelation is important in correctly describing semiclassically and quantum mechanically various systems such as the hydrogen atom, a particle on the surface of a sphere, and a spinning top. It is also indispensible for the process of bosonization, by which Fermi particles are redescribed in terms of Bose fields.