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Journal of Physics A Mathematical and General

Supersymmetry and point canonical transformations in the path integral

M O'Connor

Published Jun 21, 1990• 6 citations
Publisher

Abstract

The author shows that in the path integral quantisation procedure for free nonrelativistic particles, point canonical transformations can be performed by a naive change of variables in the path integral, provided one uses the corresponding supersymmetric path integral.
DOI
10.1088/0305-4470/23/12/039

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It is often stated in the literature on path integrals that naive changes of coordinates might in general give inequivalent theories. The discrepancy is presumably connected to subtleties in the discretization, to the stochastic nature of quantum paths, or to operator order ambiguities in the canonical quantization. Here we argue that in order to define a path integral one needs not only a Lagrangian but also a set of paths that join succesive points in the discretized paths. If the set of paths is maintained when performing a point canonical transformation the path integral does not change. We explicitly show this with the calculation of the free particle kernel in polar coordinates.
We discuss how the Hamiltonian changes in quantum canonical transformations. To the operator $\stackrel{^}{\mathcal{H}}(\stackrel{^}{p},\stackrel{^}{q})$ one can associate (in a given ordering rule) a ccc-number function H(p,q)\mathcal{H}(p,q)H(p,q). It is this function that appears in the action of the phase-space path integral. A quantum canonical transformation $\stackrel{^}{\mathcal{H}}\ensuremath{\rightarrow}{\stackrel{^}{\mathcal{H}}}^{\ensuremath{'}}$ can now be expressed as an integral transformation ${\mathcal{H}}^{\ensuremath{'}}(\overline{p},\overline{q})=\ensuremath{\int}\mathrm{dp}\mathrm{dq}\mathcal{T}(\overline{p},\overline{q};p,q)\mathcal{H}(p,q)$. The kernel T\mathcal{T}T is constructed explicitly for point transformations and for the $p=\ensuremath{-}\overline{q}$, q=p‾q=\overline{p}q=p​ reflection by studying changes of variables in the path integral. The ordering dependence of T\mathcal{T}T is displayed. The invariance of commutation rules is also discussed.
The behavior of Hamiltonian path integrals under canonical transformations defined by a generator is investigated. The exact form of the kernel of the unitary operator which implements the quantum transformation is indicated. Equivalence rules are found (for the Hamiltonian formalism in the one-dimensional case) allowing one to get rid of the nonstandard terms in the action. It is shown that under canonical transformations the Hamiltonian path integral changes its form: in the transformed expression one cannot get along only with the corresponding classical Hamiltonian function.
The path integral formulation of quantum mechanics constructs the propagator by evaluating the action S for all classical paths in coordinate space. A corresponding momentum path integral may also be defined through Fourier transforms in the endpoints. Although these momentum path integrals are especially simple for several special cases, no one has, to my knowledge, ever formally constructed them from all classical paths in momentum space. I show that this is possible because there exists another classical mechanics based on an alternate classical action R. Hamilton's Canonical equations result from a variational principle in both S and R. S uses fixed beginning and ending spatial points while R uses fixed beginning and ending momentum points. This alternative action's classical mechanics also includes a Hamilton-Jacobi equation. I also present some important points concerning the beginning and ending conditions on the action necessary to apply a Canonical transformation. These properties explain the failure of the Canonical transformation in the phase space path integral. It follows that a path integral may be constructed from classical position paths using S in the coordinate representation or from classical momentum paths using R in the momentum representation. Several example calculations are presented that illustrate the simplifications and practical advantages made possible by this broader view of the path integral. In particular, the normalized amplitude for a free particle is found without using the Schrodinger equation, the internal spin degree of freedom is simply and naturally derived, and the simple harmonic oscillator is calculated.