Path Integral Methods From the Generalized Displacement Operator, and Some of Their Applications

2021 ◽  
Vol 63 (1) ◽  
Author(s):  
A. Benkrane ◽  
H. Benzair ◽  
T. Boudjedaa
Keyword(s):  
Path Integral ◽  
Displacement Operator ◽  
Integral Methods

scholarly journals Heat capacity estimators for random series path-integral methods by finite-difference schemes

2003 ◽  
Vol 119 (23) ◽  
pp. 12119-12128 ◽  
Author(s):  
Cristian Predescu ◽  
Dubravko Sabo ◽  
J. D. Doll ◽  
David L. Freeman
Keyword(s):  
Heat Capacity ◽  
Finite Difference ◽  
Path Integral ◽  
Difference Schemes ◽  
Finite Difference Schemes ◽  
Random Series ◽  
Integral Methods

scholarly journals COORDINATE-INVARIANT PATH INTEGRAL METHODS IN CONFORMAL FIELD THEORY

2006 ◽  
Vol 21 (17) ◽  
pp. 3525-3563 ◽  
Author(s):  
ANDRÉ VAN TONDER
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Path Integral ◽  
Direct Calculation ◽  
Conformal Anomaly ◽  
Operator Formalism ◽  
Conformal Field ◽  
Integral Methods ◽  
Change Of Variables ◽  
Definition Of

We present a coordinate-invariant approach, based on a Pauli–Villars measure, to the definition of the path integral in two-dimensional conformal field theory. We discuss some advantages of this approach compared to the operator formalism and alternative path integral approaches. We show that our path integral measure is invariant under conformal transformations and field reparametrizations, in contrast to the measure used in the Fujikawa calculation, and we show the agreement, despite different origins, of the conformal anomaly in the two approaches. The natural energy–momentum in the Pauli–Villars approach is a true coordinate-invariant tensor quantity, and we discuss its nontrivial relationship to the corresponding nontensor object arising in the operator formalism, thus providing a novel explanation within a path integral context for the anomalous Ward identities of the latter. We provide a direct calculation of the nontrivial contact terms arising in expectation values of certain energy–momentum products, and we use these to perform a simple consistency check confirming the validity of the change of variables formula for the path integral. Finally, we review the relationship between the conformal anomaly and the energy–momentum two-point functions in our formalism.

Path Integral Methods for the Probabilistic Analysis of Nonlinear Systems Under a White-Noise Process

2020 ◽  
Vol 6 (4) ◽  
Author(s):  
Mario Di Paola ◽  
Gioacchino Alotta
Keyword(s):  
Path Integral ◽  
Path Integration ◽  
Integral Method ◽  
Degrees Of Freedom ◽  
Kolmogorov Equation ◽  
Integral Methods ◽  
Alternative Approach ◽  
Path Integration Method ◽  
Path Integral Method ◽  
Barrier Problems

Abstract In this paper, the widely known path integral method, derived from the application of the Chapman–Kolmogorov equation, is described in details and discussed with reference to the main results available in literature in several decades of contributions. The most simple application of the method is related to the solution of Fokker–Planck type equations. In this paper, the solution in the presence of normal, α-stable, and Poissonian white noises is first discussed. Then, application to barrier problems, such as first passage problems and vibroimpact problems is described. Further, the extension of the path integral method to problems involving multi-degrees-of-freedom systems is analyzed. Lastly, an alternative approach to the path integration method, that is the Wiener Path integration (WPI), also based on the Chapman–Komogorov equation, is discussed. The main advantages and the drawbacks in using these two methods are deeply analyzed and the main results available in literature are highlighted.

Blip-summed quantum–classical path integral with cumulative quantum memory

2016 ◽  
Vol 195 ◽  
pp. 81-92 ◽  
Author(s):  
Nancy Makri
Keyword(s):  
Path Integral ◽  
Degrees Of Freedom ◽  
Quantum Memory ◽  
Integral Methods ◽  
Acceleration Techniques ◽  
Classical Path ◽  
Forced Oscillator ◽  
Influence Functional ◽  
Harmonic Bath ◽  
Explicit Enumeration

The quantum-classical path integral (QCPI) offers a rigorous methodology for simulating quantum mechanical processes in condensed-phase environments treated in full atomistic detail. This paper describes the implementation of QCPI on system–bath models, which are frequently employed in studying the dynamics of reactive processes. The QCPI methodology incorporates all effects associated with stimulated phonon absorption and emission as its crudest limit, thus can (in some regimes) converge faster than influence functional-based path integral methods specifically designed for system–bath Hamiltonians. It is shown that the QCPI phase arising from a harmonic bath can be summed analytically with respect to the discrete bath degrees of freedom and expressed in terms of precomputed influence functional coefficients, avoiding the explicit enumeration of forced oscillator trajectories, whose number grows exponentially with the length of quantum memory. Further, adoption of the blip decomposition (which classifies the system paths based on the time length over which their forward and backward components are not identical) and a cumulative treatment of the QCPI phase between blips allows elimination of the majority of system paths, leading to a dramatic increase in efficiency. The generalization of these acceleration techniques to anharmonic environments is discussed.

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