scholarly journals Geometry and complexity of path integrals in inhomogeneous CFTs

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Paweł Caputa ◽  
Ian MacCormack
Keyword(s):  
Path Integral ◽  
Path Integrals ◽  
Optimal Path ◽  
Dependent Manner ◽  
Minimal Path ◽  
Inhomogeneous Systems ◽  
Universal Classes ◽  
General Optimization Problem ◽  
Hill’S Equations ◽  
Integral Complexity

Abstract In this work we develop the path integral optimization in a class of inhomogeneous 2d CFTs constructed by putting an ordinary CFT on a space with a position dependent metric. After setting up and solving the general optimization problem, we study specific examples, including the Möbius, SSD and Rainbow deformed CFTs, and analyze path integral geometries and complexity for universal classes of states in these models. We find that metrics for optimal path integrals coincide with particular slices of AdS3 geometries, on which Einstein’s equations are equivalent to the condition for minimal path integral complexity. We also find that while leading divergences of path integral complexity remain unchanged, constant contributions are modified in a universal, position dependent manner. Moreover, we analyze entanglement entropies in inhomogeneous CFTs and show that they satisfy Hill’s equations, which can be used to extract the energy density consistent with the first law of entanglement. Our findings not only support comparisons between slices of bulk spacetimes and circuits of path integrations, but also demonstrate that path integral geometries and complexity serve as a powerful tool for understanding the interesting physics of inhomogeneous systems.

scholarly journals p-Adic Path Integrals for Quadratic Actions

1997 ◽  
Vol 12 (20) ◽  
pp. 1455-1463 ◽  
Author(s):  
G. S. Djordjević ◽  
B. Dragovich
Keyword(s):  
Quantum Mechanics ◽  
Path Integral ◽  
Path Integrals ◽  
Probability Amplitude ◽  
Exact Form ◽  
Feynman Path Integral ◽  
Feynman Path ◽  
One Dimensional ◽  
Ordinary Quantum

The Feynman path integral in p-adic quantum mechanics is considered. The probability amplitude [Formula: see text] for one-dimensional systems with quadratic actions is calculated in an exact form, which is the same as that in ordinary quantum mechanics.

Tendon Geometry Optimization Using Path Integrals

2020 ◽  
Vol 61 (4) ◽  
pp. 247-254
Author(s):  
Jack Lehrecke ◽  
Juan Pablo Osman-Letelier ◽  
Mike Schlaich
Keyword(s):  
Topology Optimization ◽  
Objective Function ◽  
Path Integral ◽  
Path Integrals ◽  
Geometry Optimization ◽  
Structural Behavior ◽  
Spatial Structures ◽  
Simply Supported ◽  
Mathematical Background ◽  
Doubly Curved

The implementation of post-tensioned elements in concrete structures offers a multitude of benefits with regards to the overall structural behavior, with the efficacy of the applied tendons depending heavily on their geometry. However, the derivation of an optimal tendon geometry for a given structure is nontrivial, requiring engineering experience or the use of complex and often computationally demanding methodologies, e.g.the use of topology optimization strategies. This paper aims to investigate the possibility for optimizing tendon geometries using a path integral based objective function developed at the TU Berlin. For this purpose, the mathematical background is first presented to illustrate the proposed concept. Beginning with a tendon geometry optimization of a simply supported beam and progressing to more complex systems, a generalized approach for doubly curved spatial structures will be presented.

Euclidean path integrals and quantum mechanics (QM)

2021 ◽  
pp. 18-41
Author(s):  
Jean Zinn-Justin
Keyword(s):  
Quantum Mechanics ◽  
Path Integral ◽  
Correlation Functions ◽  
Path Integrals ◽  
Continuous Phase ◽  
Functional Integral ◽  
Smooth Transition ◽  
Path Integral Representation ◽  
The Matrix ◽  
Functional Measure

Functional integrals are basic tools to study first quantum mechanics (QM), and quantum field theory (QFT). The path integral formulation of QM is well suited to the study of systems with an arbitrary number of degrees of freedom. It makes a smooth transition between nonrelativistic QM and QFT possible. The Euclidean functional integral also emphasizes the deep connection between QFT and the statistical physics of systems with short-range interactions near a continuous phase transition. The path integral representation of the matrix elements of the quantum statistical operator e-β H for Hamiltonians of the simple separable form p2/2m +V(q) is derived. To the path integral corresponds a functional measure and expectation values called correlation functions, which are generalized moments, and related to quantum observables, after an analytic continuation in time. The path integral corresponding to the Euclidean action of a harmonic oscillator, to which is added a time-dependent external force, is calculated explicitly. The result is used to generate Gaussian correlation functions and also to reduce the evaluation of path integrals to perturbation theory. The path integral also provides a convenient tool to derive semi-classical approximations.

A COMPARISON OF THE PATH-INTEGRAL AND THE OSCILLATOR APPROACH TO COVARIANT LOOP CALCULUS

1989 ◽  
Vol 04 (20) ◽  
pp. 5433-5451 ◽  
Author(s):  
K. ROLAND
Keyword(s):  
Path Integral ◽  
Path Integrals ◽  
Bosonic String ◽  
Strong Indication

We consider the construction of the N-string g-loop vertex for the bosonic string. The sewing procedure of Ref. 1, based on path integrals, is reformulated in terms of operators. This makes it trivial to demonstrate complete equivalence with the oscillator approach of Refs. 2, 3, and 4. Also, the equivalence of two very different loop-sewing procedures for the g-loop vacuum diagram can be demonstrated, this being a very strong indication that the formalism is completely independent of the sewing procedure even in the case of loops.

scholarly journals Path Integrals, Spontaneous Localisation, and the Classical Limit

2020 ◽  
Vol 75 (2) ◽  
pp. 131-141 ◽  
Author(s):  
Bhavya Bhatt ◽  
Manish Ram Chander ◽  
Raj Patil ◽  
Ruchira Mishra ◽  
Shlok Nahar ◽  
...  
Keyword(s):  
Quantum Mechanics ◽  
Density Matrix ◽  
Path Integral ◽  
Classical Limit ◽  
Measurement Problem ◽  
Path Integrals ◽  
Path Integral Formulation ◽  
Von Neumann ◽  
Matrix Evolution ◽  
Grw Model

AbstractThe measurement problem and the absence of macroscopic superposition are two foundational problems of quantum mechanics today. One possible solution is to consider the Ghirardi–Rimini–Weber (GRW) model of spontaneous localisation. Here, we describe how spontaneous localisation modifies the path integral formulation of density matrix evolution in quantum mechanics. We provide two new pedagogical derivations of the GRW propagator. We then show how the von Neumann equation and the Liouville equation for the density matrix arise in the quantum and classical limit, respectively, from the GRW path integral.

The physics of meron pairs. I. Introduction, motivation, and formalism

1980 ◽  
Vol 58 (6) ◽  
pp. 845-858 ◽  
Author(s):  
David G. Laughton
Keyword(s):  
Saddle Point ◽  
Path Integral ◽  
Path Integrals ◽  
Field Configuration ◽  
Degree Of Freedom ◽  
Leading Order ◽  
Local Minima ◽  
Yang Mills ◽  
Ordinary Integral

The physics of meron pairs is considered in this series of papers. The first paper presents the motivation for focussing on this particular type of field configuration as an important degree of freedom in the SU(N) Yang–Mills theory. It also outlines the formalism for doing a saddle point expansion of path integrals about configurations which are constrained minima of the action (such as meron pairs) as opposed to local minima (such as instantons). The formalism is illustrated by the treatment of an ordinary integral which is analogous to the meron pair region of the Yang–Mills path integral. It is found that the expansion about constrained minima depends to leading order on the constraints chosen to partition the integral. This means that some criteria must be found for the choice of constraints. This problem is discussed. The actual meron pair calculations are partially described here and done in the second and third papers. Applications are to be considered in any subsequent papers.

scholarly journals Studies on Polyakov and Nambu–Goto random surface path integrals on QCD(SU(∞)): Interquark potential and phenomenological scattering amplitudes

2017 ◽  
Vol 32 (05) ◽  
pp. 1750030
Author(s):  
Luiz C. L. botelho
Keyword(s):  
Asymptotic Behavior ◽  
Scattering Amplitudes ◽  
Path Integral ◽  
Fourth Order ◽  
Path Integrals ◽  
Infrared Region ◽  
Two Dimensional ◽  
Long Distance ◽  
Random Surface ◽  
Interquark Potential

We present new path integral studies on the Polyakov noncritical and Nambu–Goto critical string theories and their applications to [Formula: see text] interquark potential. We also evaluate the long distance asymptotic behavior of the interquark potential on the Nambu–Goto string theory with an extrinsic term in Polyakov’s string at [Formula: see text]. We also propose an alternative and a new view to covariant Polyakov’s string path integral with a fourth-order two-dimensional quantum gravity, is an effective stringy description for [Formula: see text] at the deep infrared region.

Phase space path integral for the Weyl propagator

1992 ◽  
Vol 439 (1905) ◽  
pp. 139-153 ◽  
Keyword(s):  
Phase Space ◽  
Path Integral ◽  
Simple Form ◽  
Path Integrals ◽  
The Other ◽  
Weyl Representation

The Weyl representation of an operator  is a function A(x) in phase space. It is shown that a product  1 ...  2 n is represented by an integral over all (2 n +1)-sided polygons where the midpoint of one side is centred on x and the other midpoints take on the values A 1 ( x 1 ), ..., A 2 n ( x 2 n ). This leads to a new path integral for Û t = exp( - i ħ -1 Ĥt ) in the Weyl representation : U(x) is an integral over all the paths whose endpoints form a chord with x as its midpoint. No restriction is imposed on the form of the hamiltonian. Equivalence with previous path integrals generalizes these by substituting the Weyl hamiltonian for the classical hamiltonian when the latter does not have the simple form p 2 /2 + V(q) .

scholarly journals The origin of similarity fields in steady elastoplastic crack propagation under K–T loading

2010 ◽  
Vol 467 (2130) ◽  
pp. 1739-1748
Author(s):  
Jae Beom Park ◽  
Tapan Sabuwala ◽  
Gustavo Gioia
Keyword(s):  
Plastic Zone ◽  
Crack Propagation ◽  
Fracture Energy ◽  
Computer Simulations ◽  
Path Integral ◽  
Path Integrals ◽  
Mathematical Physics ◽  
Length Scale ◽  
Intrinsic Length ◽  
Specific Fracture

It has been inferred from computer simulations that the plastic-zone fields of a crack that propagates steadily under K–T loading are similarity fields. Here, we show theoretically that these similarity fields are but a manifestation of the existence of an invariant path integral. We also show that the attendant similarity variable involves an intrinsic length scale set by the specific fracture energy that flows into the crack tip. Finally, we show that where the crack is stationary there can be no similarity fields, even though there exists a (different) invariant path integral. Our results afford some new insights into the relation between similarity fields and invariant path integrals in mathematical physics.

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