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124
A path integral approach to the Kontsevich quantization formula
 Comm. Math. Phys
"... Abstract. We give a quantum field theory interpretation of Kontsevich’s deformation quantization formula for Poisson manifolds. We show that it is given by the perturbative expansion of the path integral of a simple topological bosonic open string theory. Its Batalin–Vilkovisky quantization yields a ..."
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Cited by 238 (18 self)
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Abstract. We give a quantum field theory interpretation of Kontsevich’s deformation quantization formula for Poisson manifolds. We show that it is given by the perturbative expansion of the path integral of a simple topological bosonic open string theory. Its Batalin–Vilkovisky quantization yields a superconformal field theory. The associativity of the star product, and more generally the formality conjecture can then be understood by field theory methods. As an application, we compute the center of the deformed algebra in terms of the center of the Poisson algebra. 1.
Topological Open pBranes
, 2000
"... By exploiting the BV quantization of topological bosonic open membrane, we argue that flat 3form Cfield leads to deformations of the algebras of multivectors on the Dirichletbrane worldvolume as 2algebras. This would shed some new light on geometry of Mtheory 5brane and associated decoupled ..."
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Cited by 36 (1 self)
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By exploiting the BV quantization of topological bosonic open membrane, we argue that flat 3form Cfield leads to deformations of the algebras of multivectors on the Dirichletbrane worldvolume as 2algebras. This would shed some new light on geometry of Mtheory 5brane and associated decoupled theories. We show that, in general, topological open pbrane has a structure of (p + 1)algebra in the bulk, while a structure of palgebra in the boundary. The bulk/boundary correspondences are exactly as of the generalized Deligne conjecture (a theorem of Kontsevich) in the algebraic world of palgebras. It also imply that the algebras of quantum observables of (p − 1)brane are “close to ” the algebras of its classical observables as palgebras. We interpret above as deformation quantization of (p − 1)brane, generalizing the p = 1 case. We argue that there is such quantization based on the direct relation between BV master equation and Ward identity of the bulk topological theory. The path integral of the theory will lead to the explicit formula. We also discuss some applications to
The Small Scale Structure of SpaceTime: A Bibliographical Review
, 1995
"... This essay is a tour around many of the lesser known pregeometric models of physics, as well as the mainstream approaches to quantum gravity, in search of common themes which may provide a glimpse of the final theory which must lie behind them. 1 ..."
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Cited by 19 (0 self)
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This essay is a tour around many of the lesser known pregeometric models of physics, as well as the mainstream approaches to quantum gravity, in search of common themes which may provide a glimpse of the final theory which must lie behind them. 1
A Specimen of Theory Construction from Quantum Gravity, in The Creation of Ideas in
 Physics, Leplin, J. (Ed
, 1995
"... I describe the history of my attempts to arrive at a discrete substratum underlying the spacetime manifold, culminating in the hypothesis that the basic structure has the form of a partialorder (i.e. that it is a causal set). Like the other speakers in this session, I too am here much more as a wor ..."
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Cited by 19 (1 self)
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I describe the history of my attempts to arrive at a discrete substratum underlying the spacetime manifold, culminating in the hypothesis that the basic structure has the form of a partialorder (i.e. that it is a causal set). Like the other speakers in this session, I too am here much more as a working scientist than as a philosopher. Of course it is good to remember Peter Bergmann’s description of the physicist as “in many respects a philosopher in workingman’s* clothes”, but today I’m not going to change into a white shirt and attempt to draw philosophical lessons from the course of past work on quantum gravity. Instead I will merely try to recount a certain part of my own experience with this problem, explaining how I arrived at the idea of what I will call a causal set. This and similar structures have been proposed more than once as discrete replacements for spacetime. My excuse for not telling you also how others arrived at essentially the same idea [1] is naturally that my case is the only one I can hope to reconstruct with even minimal accuracy.
Quantum Measure Theory and its Interpretation
 Quantum Classical Correspondence: Proceedings of the 4 th Drexel Symposium on Quantum Nonintegrability, held Philadelphia, September 811
, 1994
"... We propose a realistic, spacetime interpretation of quantum theory in which reality constitutes a single history obeying a “law of motion ” that makes definite, but incomplete, predictions about its behavior. We associate a “quantum measure ” —S — to the set S of histories, and point out that —S — f ..."
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Cited by 17 (3 self)
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We propose a realistic, spacetime interpretation of quantum theory in which reality constitutes a single history obeying a “law of motion ” that makes definite, but incomplete, predictions about its behavior. We associate a “quantum measure ” —S — to the set S of histories, and point out that —S — fulfills a sum rule generalizing that of classical probability theory. We interpret —S— as a “propensity”, making this precise by stating a criterion for —S—=0 to imply “preclusion ” (meaning that the true history will not lie in S). The criterion involves triads of correlated events, and in application to electronelectron scattering, for example, it yields definite predictions about the electron trajectories themselves, independently of any measuring devices which might or might not be present. (In this way, we can give an objective account of measurements.) Two unfinished aspects of the interpretation involve
Kac's Moment Formula and the FeynmanKac Formula for Additive Functionals of a Markov Process
, 1998
"... Mark Kac introduced a method for calculating the distribution of the integral Av = R T 0 v(X t )dt for a function v of a Markov process (X t ; t 0) and a suitable random time T , which yields the FeynmanKac formula for the momentgenerating function of Av . We review Kac's method, with emphasis ..."
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Cited by 16 (0 self)
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Mark Kac introduced a method for calculating the distribution of the integral Av = R T 0 v(X t )dt for a function v of a Markov process (X t ; t 0) and a suitable random time T , which yields the FeynmanKac formula for the momentgenerating function of Av . We review Kac's method, with emphasis on an aspect often overlooked. This is Kac's formula for moments of Av , which may be stated as follows. For any random time T such that the killed process (X t ; 0 t ! T ) is Markov with substochastic semigroup K t (x; dy) = P x(X t 2 dy; T ? t), any nonnegative measurable function v, and any initial distribution , the n th moment of Av is P A n v = n!(GMv ) n 1 where G = R 1 0 K t dt is the Green's operator of the killed process, Mv is the operator of multiplication by v, and 1 is the function that is identically 1. 1 Introduction Mark Kac [32, 33, 13], introduced a method for calculating the distribution of the integral A v = Z T 0 v(X t )dt (1) for a function v defined on...
Deformation quantization in the teaching of quantum mechanics
 AM. J. PHYS
, 2008
"... We discuss the deformation quantization approach for the teaching of quantum mechanics. This approach has certain conceptual advantages that make its consideration worthwhile. In particular, it sheds new light on the relation between classical and quantum mechanics. We demonstrate how it can be used ..."
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Cited by 11 (2 self)
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We discuss the deformation quantization approach for the teaching of quantum mechanics. This approach has certain conceptual advantages that make its consideration worthwhile. In particular, it sheds new light on the relation between classical and quantum mechanics. We demonstrate how it can be used to solve specific problems and clarify its relation to conventional quantization and path integral techniques. We also discuss its recent applications in relativistic quantum field theory.
Solution of the Cauchy problem for a timedependent Schrödinger equation
 J. Math. Phys
"... Abstract. We construct an explicit solution of the Cauchy initial value problem for the ndimensional Schrödinger equation with certain timedependent Hamiltonian operator of a modified oscillator. The dynamical SU (1, 1) symmetry of the harmonic oscillator wave functions, Bargmann’s functions for t ..."
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Cited by 10 (6 self)
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Abstract. We construct an explicit solution of the Cauchy initial value problem for the ndimensional Schrödinger equation with certain timedependent Hamiltonian operator of a modified oscillator. The dynamical SU (1, 1) symmetry of the harmonic oscillator wave functions, Bargmann’s functions for the discrete positive series of the irreducible representations of this group, the Fourier integral of a weighted product of the Meixner–Pollaczek polynomials, a Hankeltype integral transform and the hyperspherical harmonics are utilized in order to derive the corresponding Green function. It is then generalized to a case of the forced modified oscillator. The propagators for two models of the relativistic oscillator are also found. An expansion formula of a plane wave in terms of the hyperspherical harmonics and solution of certain infinite system of ordinary differential equations are derived as a byproduct. 1.
Risk Sensitive Path Integral Control
"... Recently path integral methods have been developed for stochastic optimal control for a wide class of models with nonlinear dynamics in continuous spacetime. Path integral methods find the control that minimizes the expected costtogo. In this paper we show that under the same assumptions, path i ..."
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Cited by 9 (0 self)
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Recently path integral methods have been developed for stochastic optimal control for a wide class of models with nonlinear dynamics in continuous spacetime. Path integral methods find the control that minimizes the expected costtogo. In this paper we show that under the same assumptions, path integral methods generalize directlytorisksensitivestochasticoptimalcontrol. Here the method minimizes in expectation an exponentially weighted costtogo. Depending on the exponential weight, risk seeking or risk averse behaviour is obtained. We demonstrate the approach on risk sensitive stochastic optimal control problems beyond the linearquadratic case, showing the intricate interaction of multimodal control with risk sensitivity. 1
The FeynmanKac formula and decomposition of Brownian paths
, 1997
"... this paper. In Section 3 we show how some refinements of the FeynmanKac formula may be understood in terms of a decomposition of the Brownian path at the time of the last visit to zero before time ` where ` is an exponentially distributed random time independent of the Brownian motion. We also show ..."
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Cited by 7 (5 self)
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this paper. In Section 3 we show how some refinements of the FeynmanKac formula may be understood in terms of a decomposition of the Brownian path at the time of the last visit to zero before time ` where ` is an exponentially distributed random time independent of the Brownian motion. We also show how D.Williams' decomposition at the maximum of the generic excursion under Ito's measure translates in terms of solutions of a SturmLiouville equation. Finally, Section 4 is devoted to explicit computations of the laws of