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arXiv:2107.03982 · Journal of Physics A Mathematical and Theoretical

The Schwinger action principle for classical systems*

A D Bermúdez Manjarres

Published Sep 2, 2021• 5 citations
arXivHTMLPDF

Abstract

Abstract We use the Schwinger action principle to obtain the equations of motion in the Koopman–von Neumann operational version of classical mechanics. We restrict our analysis to non-dissipative systems. We show that for velocity-independent forces the Schwinger action principle can be interpreted as a variational principle.
DOI
10.1088/1751-8121/ac2321

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Abstract Chapter 4 introduces the Schwinger Action Principle, along with associated particle and potential sources. While the methods described here originally arose in the relativistic quantum field theory of elementary particle physics, they have also profoundly advanced our understanding of non-relativistic many-particle physics. The Schwinger Action Principle is a quantum-mechanical variational principle that closely parallels the Hamilton Principle of Least Action of classical mechanics, generalizing it to include the role of quantum operators as generalized coordinates and momenta. As such, it unifies all aspects of quantum theory, incorporating Hamilton equations of motion for those operators and the Heisenberg equation, as well as producing the canonical equal-time commutation/anticommutation relations. It yields dynamical coupled field equations for the creation and annihilation operators of the interacting many-body system by variational differentiation of the Hamiltonian with respect to the field operators. Also, equations for the development of matrix elements (underlying Green’s functions) are derived using variations with respect to particle and potential “sources” (and coupling strength). Variational calculus, involving impressed potentials, c-number coordinates and fields, also quantum operator coordinates and fields, is discussed in full detail. Attention is given to the introduction of fermion and boson particle sources and their use in variational calculus.
Abstract We construct an operational formulation of classical mechanics without presupposing previous results from analytical mechanics. In doing so, we rediscover several results from analytical mechanics from an entirely new perspective. We start by expressing the position and velocity of point particles as the eigenvalues of self-adjoint operators acting on a suitable Hilbert space. The concept of holonomic constraint is shown to be equivalent to a restriction to a linear subspace of the free Hilbert space. The principal results we obtain are: (1) the Lagrange equations of motion are derived without the use of D’Alembert or Hamilton principles, (2) the constraining forces are obtained without the use of Lagrange multipliers, (3) the passage from a position–velocity to a position–momentum description of the movement is done without the use of a Legendre transformation, (4) the Koopman–von Neumann theory is obtained as a result of our ab initio operational approach, (5) previous work on the Schwinger action principle for classical systems is generalized to include holonomic constraints.
We present the variational action principle for initial value problems in classical, conservative-force point particle mechanics. We rigorously derive this formulation by taking the classical limit of the Schwinger-Keldysh expression for the time dependence of the expectation value for operators in quantum mechanics. We clarify the connection between the variation of the position and the variation of the velocity of a particle when implementing Hamilton's Principle in deriving the Euler-Lagrange Equations. We show that both the plus and minus Keldysh paths (of the average and difference of the forward/backward paths) have classical paths and fluctuations -- unlike the common perception that the minus path provides the fluctuations around the single classical solution given by the plus path -- and that the fluctuations of both paths are crucial for the correct normalization of the classical limit. The classical limit yields "initial conditions" and equations of motion for the minus paths such that the unique classical solution for the minus paths is that they are identically zero, and, fascinatingly, that the minus paths' solution propagates backwards in time; thus one does not need to set the minus paths to zero by hand when taking the classical limit of the Schwinger-Keldysh formalism. We note implications for the classical and quantum mechanics of non-holonomic constraints and quantum field theories with gauges dependent on the derivatives of the fields.
The Schwinger quantum action principle is a dynamic characterization of the transformation functions and is based on the algebraic structure derived from the kinematic analysis of the measurement processes at the quantum level. As such, this variational principle, allows to derive the canonical commutation relations in a consistent way. Moreover, the dynamic pictures of Schrödinger, Heisenberg and a quantum Hamilton-Jacobi equation can be derived from it. We will implement this formalism by solving simple systems such as the free particle, the quantum harmonic oscillator and the quantum forced harmonic oscillator©by the Sociedade Brasileira de Física. Printed in Brazil.
We begin with a presentation of the Schwinger variational principle in quantum mechanics. In the second part, after a re-assessment of the problem of motion in quantum mechanics, it is shown that the Green kernel, which represents the probability amplitude of finding a particle in space at a given time, once its location at another time is known, can be represented by a sum over ‘space–time paths’. According to this interpretation, one integrates the exponential of i times the classical action divided by ħ, with a (formal) measure over all space–time paths matching the initial location x i at time t i and the final location x f at time t f . Such a way of evaluating the Green kernel is applied to any quadratic Lagrangian for a generic quantum system including, in particular, the harmonic oscillator and the free particle. Lastly, we outline a formalism which involves ordinary functions of commuting variables, and exactly reproduces ordinary quantum mechanics. It works with a phase space endowed with commuting coordinates, so that one is dealing with quantum mechanics in phase space. The Schwinger formulation of quantum dynamics In both the Schrödinger formulation in terms of wave functions (with associated state vectors) and the Heisenberg formulation in terms of noncommuting matrices (with corresponding linear operators) the dynamical evolution is specified in terms of the Hamiltonian. This has motivated our presentation of classical dynamics, but is in marked contrast with the view according to which the Lagrangian and the action specify the dynamics. Hamilton's principle is formulated in terms of the action, the time integral of the Lagrangian.
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.