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

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.

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Yang–Mills theory and fermionic path integrals

Modern Physics Letters A ◽

10.1142/s0217732316300044 ◽

2016 ◽

Vol 31 (04) ◽

pp. 1630004 ◽

Cited By ~ 1

Author(s):

Kazuo Fujikawa

Keyword(s):

Field Theory ◽

Path Integral ◽

Integral Formula ◽

Path Integrals ◽

Recent Analysis ◽

Yang Mills ◽

Fundamental Particles ◽

Important Field ◽

Quantum Anomalies ◽

Mills Field

The Yang–Mills gauge field theory, which was proposed 60 years ago, is extremely successful in describing the basic interactions of fundamental particles. The Yang–Mills theory in the course of its developments also stimulated many important field theoretical machinery. In this brief review I discuss the path integral techniques, in particular, the fermionic path integrals which were developed together with the successful applications of quantized Yang–Mills field theory. I start with the Faddeev–Popov path integral formula with emphasis on the treatment of fermionic ghosts as an application of Grassmann numbers. I then discuss the ordinary fermionic path integrals and the general treatment of quantum anomalies. The contents of this review are mostly pedagogical except for a recent analysis of path integral bosonization.

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Finite BRST–anti-BRST transformations in generalized Hamiltonian formalism

International Journal of Modern Physics A ◽

10.1142/s0217751x14501590 ◽

2014 ◽

Vol 29 (27) ◽

pp. 1450159 ◽

Cited By ~ 11

Author(s):

Pavel Yu. Moshin ◽

Alexander A. Reshetnyak

Keyword(s):

Dynamical Systems ◽

Path Integral ◽

Hamiltonian Path ◽

Hamiltonian Formalism ◽

Lagrangian Formalism ◽

Yang Mills ◽

Gauge Dependence ◽

Field Dependent ◽

Gauge Fixing ◽

Constrained Dynamical Systems

We introduce the notion of finite BRST–anti-BRST transformations for constrained dynamical systems in the generalized Hamiltonian formalism, both global and field-dependent, with a doublet λa, a = 1, 2, of anticommuting Grassmann parameters and find explicit Jacobians corresponding to these changes of variables in the path integral. It turns out that the finite transformations are quadratic in their parameters. Exactly as in the case of finite field-dependent BRST–anti-BRST transformations for the Yang–Mills vacuum functional in the Lagrangian formalism examined in our previous paper [arXiv:1405.0790 [hep-th]], special field-dependent BRST–anti-BRST transformations with functionally-dependent parameters λa= ∫ dt(saΛ), generated by a finite even-valued function Λ(t) and by the anticommuting generators saof BRST–anti-BRST transformations, amount to a precise change of the gauge-fixing function for arbitrary constrained dynamical systems. This proves the independence of the vacuum functional under such transformations. We derive a new form of the Ward identities, depending on the parameters λaand study the problem of gauge dependence. We present the form of transformation parameters which generates a change of the gauge in the Hamiltonian path integral, evaluate it explicitly for connecting two arbitrary Rξ-like gauges in the Yang–Mills theory and establish, after integration over momenta, a coincidence with the Lagrangian path integral [arXiv:1405.0790 [hep-th]], which justifies the unitarity of the S-matrix in the Lagrangian approach.

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p-Adic Path Integrals for Quadratic Actions

Modern Physics Letters A ◽

10.1142/s0217732397001485 ◽

1997 ◽

Vol 12 (20) ◽

pp. 1455-1463 ◽

Cited By ~ 21

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.

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Tendon Geometry Optimization Using Path Integrals

Journal of the International Association for Shell and Spatial Structures ◽

10.20898/j.iass.2020.007 ◽

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.

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Euclidean path integrals and quantum mechanics (QM)

Quantum Field Theory and Critical Phenomena ◽

10.1093/oso/9780198834625.003.0002 ◽

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.

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The 1+1SU(2) Yang–Mills path integral

Journal of Physics G Nuclear and Particle Physics ◽

10.1088/0954-3899/30/10/003 ◽

2004 ◽

Vol 30 (10) ◽

pp. 1361-1370

Author(s):

Mark S Swanson

Keyword(s):

Path Integral ◽

Yang Mills

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

International Journal of Modern Physics A ◽

10.1142/s0217751x89002326 ◽

1989 ◽

Vol 04 (20) ◽

pp. 5433-5451 ◽

Cited By ~ 3

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.

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Path Integrals, Spontaneous Localisation, and the Classical Limit

Zeitschrift für Naturforschung A ◽

10.1515/zna-2019-0251 ◽

2020 ◽

Vol 75 (2) ◽

pp. 131-141 ◽

Cited By ~ 1

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.

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Edge dynamics from the path integral — Maxwell and Yang-Mills

Journal of High Energy Physics ◽

10.1007/jhep11(2018)080 ◽

2018 ◽

Vol 2018 (11) ◽

Cited By ~ 14

Author(s):

Andreas Blommaert ◽

Thomas G. Mertens ◽

Henri Verschelde

Keyword(s):

Path Integral ◽

Yang Mills

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ON THE PATH INTEGRAL QUANTIZATION OF CHIRAL BOSONS

International Journal of Modern Physics A ◽

10.1142/s0217751x89000091 ◽

1989 ◽

Vol 04 (01) ◽

pp. 249-255 ◽

Cited By ~ 2

Author(s):

F. A. SCHAPOSNIK ◽

J. E. SOLOMIN

Keyword(s):

Path Integral ◽

Degree Of Freedom ◽

Lagrangian Formalism ◽

Proper Treatment ◽

Gauge Transformations ◽

Path Integral Quantization ◽

Gauge Degree ◽

Chiral Bosons

We show that, in the covariant Lagrangian formalism, a proper treatment of the gauge degree of freedom in a model of chiral bosons proposed by Siegel uncovers the presence of a Jacobian (a "Wess-Zumino action"): the group of gauge transformations gets quantized and the anomaly is absorbed.

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