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A reformulation of Schwinger's action principle via Gateaux variations : the classical analogue and the consistency of Lagrangian and Hamiltonian quantum mechanics on a Riemannian manifold

Shaharir Mohamad Zain

Published Jan 1, 1974• 0 citations
Publisher

Abstract

Thesis (Ph.D.) -- La Trobe University, 1974.; Submitted to the Dept. of Applied Mathematics.
DOI
10.26181/32283456

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The modified Hamilton-Schwinger action principle

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The c-number variation in the quantum-mechanical action principle of Schwinger is extended to a q-number variation which uses an action integral analogous to the classical modified Hamilton action integral. An explicit realization of the admissible q-number variation for the hamiltonian operator, H, given by H(q,p,t)= 1 / 2 p j g jk (q)p k + 1 / 2 (A j (q),p j )+W(q,t) is discussed in terms of Gauteaux variation. The action principle yields the correct Hamilton-Heisenberg equations and a relationship between the lagrangian L and the hamiltonian H. It is also shown that while the fundamental commutation relation is successfully derived by Schwinger in his c-number variational principle, the same argument cannot be used in the present q-number variation. A new method of quantization is suggested.

Journal of Physics A Mathematical Nuclear and General•Mar 21, 1974•2 citations
Using extended Schwinger's quantization approach quantum mechanics on a Riemannian manifold MMM with a given action of an intransitive group of isometries is developed. It was shown that quantum mechanics can be determined unequivocally only on submanifolds of MMM where GGG acts simply transitively (orbits of GGG-action). The remaining part of degrees of freedom can be described unequivocally after introducing some additional assumptions. Being logically unmotivated, these assumptions are similar to canonical quantization postulates. Besides this ambiguity that has a geometrical nature there is undetermined gauge field of hbar\\hbarhbar (or higher) order, vanishing in the classical limit hbarto0\\hbar\\to 0hbarto0.
The classical and quantum theories of dynamical systems whose Lagrangians are linear in the co-ordinate derivatives are studied, with a view to clarifying a n-uiber of points in relation to Schwinger's quantum-mechanical variational principle. In particular, linear variations of the dynamical variables are considered, and it is shown that they may be consistently included in Schwinger's formalism. It is also found to be possible to obtain Green's generalized commutation rules by excluding c number variations from the theory, for a suitably restricted class of Lagrangian.
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.
The c-number variation in the quantum-mechanical action principle of Schwinger is extended to a q-number variation which uses an action integral analogous to the classical modified Hamilton action integral. An explicit realization of the admissible q-number variation for the hamiltonian operator, H, given by H(q,p,t)= 1 / 2 p j g jk (q)p k + 1 / 2 (A j (q),p j )+W(q,t) is discussed in terms of Gauteaux variation. The action principle yields the correct Hamilton-Heisenberg equations and a relationship between the lagrangian L and the hamiltonian H. It is also shown that while the fundamental commutation relation is successfully derived by Schwinger in his c-number variational principle, the same argument cannot be used in the present q-number variation. A new method of quantization is suggested.