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arXiv:physics/9904003

Gauge Invariance and Canonical Variables

Iosif B. Khriplovich, Alexander I. Milstein

Published Apr 2, 1999• 2 citations
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Abstract

We discuss some paradoxes arising due to the gauge-dependence of canonical variables in mechanics.
DOI
10.48550/arxiv.physics/9904003

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The distinction between gauge independence and gauge covariance in electrodynamics is considered for quantities of physical interest. The gauge independence of directly observable quantities, such as the spectra of operators and energy conserving transition rates, is discussed and contrasted with the gauge covariance of other quantities such as scalar products. The quasigauge-invariance of the classical Lagrangian and the invariance of the Euler-Lagrange equations under the addition of a total time derivative to the Lagrangian are discussed. The consequent inherent nonuniqueness of the Lagrangian and Hamiltonian formulations of classical mechanics is pointed out. A physical interpretation of the explicit gauge dependence of classical canonical momenta and of the expectation values of the corresponding quantum mechanical operators is presented. The gauge independence of energy conserving transition rates calculated in the conventional finite-order time dependent perturbation theory is discussed and illustrated. The gauge dependence of the conventional time dependent transition amplitudes in the presence of electromagnetic fields is then discussed, and gauge independent transition amplitudes are constructed. An alternative formulation of the quantum mechanics of charged particles is obtained in terms of a new gauge independent Hamiltonian, which is, as might be expected, unique only within an arbitrary canonical transformation.
Gauge fields frequently used as an independent construction additional to so-called wave fields of matter. This artificial separation is of course useful in some applications (like Berry's interactions between the "heavy" and "light" sub-systems) but it is restrictive on the fundamental level of "elementary" particles and entangled states. It is shown that the linear superposition of action states and non-linear dynamics of the local dynamical variables form an oscillons of energy representing non-local particles - "lumps" arising together with their "affine gauge potential" agrees with Fubini-Study metric. I use the conservation laws of local dynamical variables (LDV's) during affine parallel transport in complex projective Hilbert space CP(N−1)CP(N-1)CP(N−1) for twofold aim. Firstly, I formulate the variation problem for the ``affine gauge potential" as system of partial differential equations \cite{Le1}. Their solutions provide embedding quantum dynamics into dynamical space-time whose state-dependent coordinates related to the qubit spinor subjected to Lorentz transformations of "quantum boosts" and "quantum rotations". Thereby, the problem of quantum measurement being reformulated as the comparison of LDV's during their affine parallel transport in CP(N−1)CP(N-1)CP(N−1), is inherently connected with space-time emergences. Secondly, the important application of these fields is the completeness of quantum theory. The EPR and Schrödinger's Cat paradoxes are discussed from the point of view of the restored Lorentz invariance due to the affine parallel transport of local Hamiltonian of the soliton-like field.