Gauge invariance and canonical transformations in Dirac generalized mechanics

Ricardo A. Tello-Llanos

Kuo‐Ho Yang

Anonymous, Matthias Schmidt

We provide an extended account of the recent statistical mechanical theory of gauge invariance against operator shifting in quantum many-body systems [J. Müller and M. Schmidt, Phys. Rev. E 114, L022104 (2026)10.1103/dzz3-m28k]. The gauge transformation is enacted by a shifting superoperator that displaces the fundamental position and momentum degrees of freedom. The shifting superoperator constitutes a map between Hilbert space operators, and it features Lie algebra commutator structure. Averages of general observables remain invariant under the shifting both in and out of thermal equilibrium, as well as in ground states. The gauge invariance induces exact sum rules that interconnect global observables and associated locally resolved correlation functions. In particular we describe the resulting one-body force, hyperforce, product, and two-body sum rules. We relate the shifting superoperator to a previously formulated quantum canonical transformation and present the generalization of quantum shifting to multicomponent systems. The gauge theory respects fundamental fermionic and bosonic particle properties, as we demonstrate by proving the compatibility of operator shifting and exchange symmetry. We formulate the quantum version of hyperdensity functional theory to provide formal access to hyperforces as well as to general averaged quantum observables via universal density functionals. For time-dependent situations, we describe quantum dynamical gauge invariance and prove exact dynamical sum rules for nonequilibrium situations, as generated by Hamiltonian time dependence. We argue for the fundamental status of statistical mechanical gauge invariance based on the compliance of the underlying geometry with canonical quantization according to Dirac's correspondence principle. Analogies and differences of the quantum mechanical sum rules with their classical counterparts remain indicative of the respective levels of description.

Emmanuel Rousseau, Didier Felbacq

This note is a reply to the paper arXiv:1801.05590: "The equivalence of the Power-Zineau-Woolley picture and the Poincar{\'e} gauge from the very first principles" by G. K{\'o}nya, et al. In a recent paper [2], we have shown that the Power-Zienau-Woolley Hamiltonian does not derived from the minimal-coupling hamiltonian with the help of a gauge transformation. This result has been challenged by G. K{\'o}nya, al. in a comment 1 where the authors claim the equivalence between the Power-Zienau-Woolley hamiltonian and the minimal-coupling hamiltonian in the Poincar{\'e} gauge. They claim that we have made one error and one wrong emphasis in our paper: The error as summarized by G. K{\'o}nya al. would be: "The canonical field momentum is not gauge invariant. Equivalent transformations of the Lagrangian do change the momentum. In field theories, gauge transformations are special cases of such transformations. The electric field E is gauge invariant, but its capacity of being the canonical momentum is not. " The wrong emphasis as summarized by G.K{\'o}nya al. would be: "The use of the canonical coordinate/momentum pair A p and E in Poincar{\'e} gauge is presented as mandatory in Rousseau and Felbacq paper, whereas as there is a certain freedom of choice in selecting this pair. Also in Poincar{\'e} gauge it is possible to use A c as canonical coordinate, in which case the conjugate momentum will be D. This is the most convenient choice in terms of the set of nontrivial Dirac brackets. Cf. Table 1 in G. K{\'o}nya al. paper 1 for possible choices." We do not share these conclusions and show in this reply that these statements are incorrect. Specifically, we show that under a gauge transformation, the canonical momentum $\pi$(x,t) conjugated to the vector potential A(x,t) is given by $\pi$(x,t) = --$\epsilon$\_0 E(x,t). This happens because the Lagrangian does not contains terms proportional to $\partial$\_t $\phi$ (x,t) where $\phi$ (x,t) is the scalar potential. Moreover our choice of canonical variables was challenged. Actually, our set of independent variables is exactly the same as in G. K{\'o}nya al. except that we do not write explicitly the dependent variables in term of the independent ones. This is one great advantage of the Dirac procedure for constrained hamiltonian.

P. Nattermann

Emmanuel Rousseau, Didier Felbacq

This note is a reply to the paper arXiv:1801.05590: "The equivalence of the Power-Zineau-Woolley picture and the Poincar{é} gauge from the very first principles" by G. K{ó}nya, et al. In a recent paper [2], we have shown that the Power-Zienau-Woolley Hamiltonian does not derived from the minimal-coupling hamiltonian with the help of a gauge transformation. This result has been challenged by G. K{ó}nya, al. in a comment 1 where the authors claim the equivalence between the Power-Zienau-Woolley hamiltonian and the minimal-coupling hamiltonian in the Poincar{é} gauge. They claim that we have made one error and one wrong emphasis in our paper: The error as summarized by G. K{ó}nya al. would be: "The canonical field momentum is not gauge invariant. Equivalent transformations of the Lagrangian do change the momentum. In field theories, gauge transformations are special cases of such transformations. The electric field E is gauge invariant, but its capacity of being the canonical momentum is not. " The wrong emphasis as summarized by G.K{ó}nya al. would be: "The use of the canonical coordinate/momentum pair A p and E in Poincar{é} gauge is presented as mandatory in Rousseau and Felbacq paper, whereas as there is a certain freedom of choice in selecting this pair. Also in Poincar{é} gauge it is possible to use A c as canonical coordinate, in which case the conjugate momentum will be D. This is the most convenient choice in terms of the set of nontrivial Dirac brackets. Cf. Table 1 in G. K{ó}nya al. paper 1 for possible choices." We do not share these conclusions and show in this reply that these statements are incorrect. Specifically, we show that under a gauge transformation, the canonical momentum $π$(x,t) conjugated to the vector potential A(x,t) is given by $π$(x,t) = --$ε$\_0 E(x,t). This happens because the Lagrangian does not contains terms proportional to $\partial$\_t $ϕ$ (x,t) where $ϕ$ (x,t) is the scalar potential. Moreover our choice of canonical variables was challenged. Actually, our set of independent variables is exactly the same as in G. K{ó}nya al. except that we do not write explicitly the dependent variables in term of the independent ones. This is one great advantage of the Dirac procedure for constrained hamiltonian.