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The generating functional for scalar field theories

Lorenzo Ricci

Published Sep 1, 2016• 0 citations
OpenAlex

Abstract

The path-integral is a formulation of quantum mechanics equivalent to the standard one, offering a new point oh view of the subject which is more intuitive than the usual approach. In this work we firstly present the Dirac orginal idea and then we describe how Feynman developed it. Therefore we apply the path-integral approach to the case of a scalar field theory, introducing the concept of generating functional. Finally we illustrate two alternative representations of the generating functional, dual to the Schwinger formalism, discussing how they lead to a realation with the Hermite polynomials. We also introduce the Schwinger-Dyson equation and express it through the representations introduced.

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The path-integral is a formulation of quantum mechanics equivalent to the standard one, offering a new point oh view of the subject which is more intuitive than the usual approach. In this work we firstly present the Dirac orginal idea and then we describe how Feynman developed it. Therefore we apply the path-integral approach to the case of a scalar field theory, introducing the concept of generating functional. Finally we illustrate two alternative representations of the generating functional, dual to the Schwinger formalism, discussing how they lead to a realation with the Hermite polynomials. We also introduce the Schwinger-Dyson equation and express it through the representations introduced.
Operator calculusThe operator calculus developed by Feynman [Fey51] makes it possible to represent functions of (noncommuting) operators as path integrals, with the integrand being the path-ordered exponential of operators, the order of which is controlled by a parameter that varies along the trajectory.This procedure is termed Feynman disentangling.It is also applicable to functions of matrices (say, γ-matrices which are associated with a spinor particle).When applied to the evolution operator, this procedure results in the standard path-integral representation of quantum mechanics.In this chapter we first demonstrate the general technique using the simplest example, a free propagator in Euclidean space, and then consider the path-integral representation of quantum mechanics, as well as propagators in an external electromagnetic field. Free propagatorLet us first consider the simplest propagator of a free scalar field which is given in the operator formalism by the vacuum expectation value of thewith ϕ being the field-operator.The T -product (1.1) obeys the equationwhere d = 4 is the dimension of space-time, however the formulas are applicable at any value of d.In the operator formalism, Eq. (1.2) is a * The ordered products of operators were introduced by Dyson [Dys49].This paper and other classical papers on quantum electrodynamics are collected in the book edited by Schwinger [Sch58].
Feynman's path integral formulation arose from his attempt to incorporate the Lagrangian framework into quantum mechanics, offering what he regarded as a more fundamental perspective than the Hamiltonian approach, particularly in the context of quantum electrodynamics. Inspired by an analogy proposed by P. A. M. Dirac, Feynman showed that Schrödinger's equation could be recovered by promoting this analogy to an equality, up to a constant factor. This insight laid the foundation for the path integral formalism. In this article, we first outline the historical development of dynamical frameworks in physics and then trace Feynman's reasoning as he constructed his formulation, beginning with the double-slit experiment and its interpretation. We subsequently derive the Schrödinger equation from the path integral and demonstrate the conservation of probability. These final sections aim to provide students with a clear link between this elegant yet less commonly presented approach and the more standard methods typically taught in undergraduate courses.
Starting from the earlier notions of stationary action principles, we show how Schwinger's Quantum Action Principle descended from Dirac's formulation, which independently led Feynman to his path-integral formulation of quantum mechanics. The connection between the two is brought out, and applications are discussed. The Keldysh-Schwinger time-cycle method of extracting matrix elements is described. Part II will discuss the variational formulation of quantum electrodynamics and the development of source theory.
In this chapter we show that it is possible to describe quantum mechanics by using path integrals. This idea was depeloped by Richard Feynman on the basis of an earlier intuition of Paul Dirac. We discuss the path integral approach starting with the problem of a single quantum particle in the presence of an external potential. We introduce the concept of the saddle point and the stationary phase approximation, which is solved by using Fresnel integrals. We show that Gaussian integrals are crucial in the imaginary time formulation of the path integral, which is the appropriate framework for studying quantum systems at thermal equilibrium. Finally, we analyze the quantum tunneling problem, deriving an elegant formula for the tunneling probability based on the simple, but meaningful, saddle point approximation.
This book is a self-contained and concise introduction to the techniques and applications of path integral quantization and functional techniques, aimed at students and practitioners. The first half of the text focuses on quantum mechanics, including a review of the action formulation of classical mechanics and quantum mechanics in the Dirac operator and state formalism, and further examination of the path integral. The second part examines relativistic field theories, reviewing special relativity, as well as derivation of the path integral representation of the vacuum transition element for quantized scalar, spinor, and vector fields from the coherent state representation of the respective field theories
One of the alternative approaches used in quantum mechanics and quantum field theory, which takes the most advanced developments in the area, such as string theory, is to use the formalism of path-integrals . A path-integral is an efficient way to calculate scattering amplitudes of a transition from one quantum state to another. In this work are presented some essential ideas about the path-integrals, applications in ordinary quantum mechanics (nonrelativistic) and then to base the rules of Feynman for interactions between particles in quantum field theory, with some deductions to generating funcionals for scattering amplitudes in free field and in the presence of interactions fields