No abstract

By carefully analyzing the relations between operator methods and the discretized and continuum path integral formulations of quantum-mechanical systems, we have found the correct Feynman rules for one-dimensional path integrals in curved spacetime. Although the prescription how to deal with the products of distributions that appear in the computation of Feynman diagrams in configuration space is surprising, this prescription follows unambiguously from the discretized path integral. We check our results by an explicit two-loop calculation.1 This research was supported in part by NSF grant no PHY9309888. 2 e-mail: deboer@insti.physics.sunysb.edu 3 e-mail: peeters@insti.physics.sunysb.edu 4 e-mail: kostas@insti.physics.sunysb.edu 5 e-mail: vannieu@insti.physics.sunysb.edu One-dimensional path integrals are interesting for a variety of reasons. They are the simplest toy models of higher dimensional path integrals, and since they describe quantum mechanics, which is a finite theory, it should be possible to understand them completely and rigorously. Indeed, it is relatively straightforward to reproduce some standard quantum mechanical facts for the harmonic oscillator and other simple systems from the corresponding path integrals. However, to do the same for a point particle in curved space turns out to be a much more delicate problem [1,2]. The corresponding path integrals can be used to compute one-loop anomalies of n-dimensional quantum field theories. This was first shown by Alvarez-Gaumé and Witten for various chiral anomalies [3], and later for trace anomalies in [4]. In the first case, only one-loop worldline graphs contribute, but in the second case one needs to go to higher loops, and details of the action and measure can no longer be neglected. This is yet another reason to properly understand one-dimensional path integrals. Inspired by this application we will in this article consider the path integral for a point particle in curved space, defined on a finite time interval [−β, 0]. Since the corresponding HamiltonianĤ appears as a regulator when we write anomalies A as the regularized trace of a JacobianĴ , A = Tr(Ĵ exp(−βĤ)), the Hamiltonian and its operator ordering are completely fixed by the properties of the original quantum field theory. Therefore, we will in this article only focus on the path integral computation of the transition element 6 T =< z| exp(−βĤ)|y >, given a fixed quantum HamiltonianĤ. The supersymmetric generalization, a full-fledged path integral treatment of (Majorana) fermions and internal symmetry ghosts and the explicit evaluation of various anomalies will be discussed in a future publication [5].At first sight it may seem surprising that one can find a fundamental new result in such an arcane subject as path integrals for a point particle. However, most of 6 We work in Euclidean space. For loop calculations, there is no essential difference with Minkowski space, and we ignore instantons, since we are interested in the short time expansion of T only.final result (4) discr...

We consider a series of duality transformations that leads to a constant shift in the harmonic functions appearing in the description of a configuration of branes. This way, for several intersections of branes, we can relate the original brane configuration which is asymptotically flat to a geometry of the type adS k × E l × S m . The implications of our results for supersymmetry enhancement, M(atrix) theory at finite N, and for supergravity theories in diverse dimensions are discussed.

We construct the path integral for one-dimensional non-linear sigma models, starting from a given Hamiltonian operator and states in a Hilbert space. By explicit evaluation of the discretized propagators and vertices we find the correct Feynman rules which differ from those often assumed. These rules, which we previously derived in bosonic systems [1], are now extended to fermionic systems. We then generalize the work of Alvarez-Gaumé and Witten [2] by developing a framework to compute anomalies of an n-dimensional quantum field theory by evaluating perturbatively a corresponding quantum mechanical path integral.