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Reply to " The equivalence of the Power-Zineau-Woolley picture and the Poincaré gauge from the very first principles " by G. Kónya, et al.

Emmanuel Rousseau, Didier Felbacq

Published Apr 6, 2018• 0 citations
OpenAlex

Abstract

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.

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Reply to "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

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.

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This note is a reply to the paper (arXiv:1801.0559v2): " The gauge-invariant Lagrangian, the Power-Zienau-Woolley picture, and the choices of field momenta in nonrelativistic quantum electrodynamics" by G. Kónya, et al.See also our first reply (arXiv:1804.07472v1) in response to their initial comment "The equivalence of the Power-Zineau-Woolley picture and the Poincaré gauge from the very first principles " ( arXiv:1801.05590v1) ————————————– In a recent paper, 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. Kónya and al. They first claim an error [arXiv:1801.05590v1] followed by a subsequent allegation that the Power-Zienau-Woolley hamiltonian can be derived starting from the electromagnetic action.We do not share their conclusions and show in two back-to-back replies [arXiv:1804.07472v1 followed by this note] that their successive statements are not correct.

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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 π\piπ(x,t) conjugated to the vector potential A(x,t) is given by π\piπ(x,t) = --ϵ\epsilonϵ_0 E(x,t). This happens because the Lagrangian does not contains terms proportional to ∂\partial∂_t ϕ\phiϕ (x,t) where ϕ\phiϕ (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.
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 πππ(x,t) conjugated to the vector potential A(x,t) is given by πππ(x,t) = --εεε\_0 E(x,t). This happens because the Lagrangian does not contains terms proportional to ∂\partial∂\_t ϕϕϕ (x,t) where ϕϕϕ (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.
This note is a reply to the paper (arXiv:1801.0559v2): " The gauge-invariant Lagrangian, the Power-Zienau-Woolley picture, and the choices of field momenta in nonrelativistic quantum electrodynamics" by G. Kónya, et al.See also our first reply (arXiv:1804.07472v1) in response to their initial comment "The equivalence of the Power-Zineau-Woolley picture and the Poincaré gauge from the very first principles " ( arXiv:1801.05590v1) ————————————– In a recent paper, 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. Kónya and al. They first claim an error [arXiv:1801.05590v1] followed by a subsequent allegation that the Power-Zienau-Woolley hamiltonian can be derived starting from the electromagnetic action.We do not share their conclusions and show in two back-to-back replies [arXiv:1804.07472v1 followed by this note] that their successive statements are not correct.
This article deals with the fundamental problem of light-matter interaction in the quantum theory. Although it is described through the vector potential in quantum electrodynamics, it is believed by some that a hamiltonian involving only the electric and the magnetic fields is preferable. In the literature this hamiltonian is known as the Power-Zienau-Woolley hamiltonian. We question its validity and show that it is not equivalent to the minimal-coupling hamiltonian. In this article, we show that these two hamiltonians are not connected through a gauge transformation. We find that the gauge is not fixed in the Power-Zienau-Woolley hamiltonian. The interaction term is written in one gauge whereas the rest of the hamiltonian is written in another gauge. The Power-Zienau-Woolley hamiltonian and the minimal-coupling one are related through a unitary transformation that does not fulfill the gauge fixing constraints. Consequently, they predict different physical results. In this letter, we provide the correct quantum theory in the multipolar gauge with a hamiltonian involving only the physical fields.
It is shown that Mandel's argument that the electric dipole interaction in quantum electrodynamics is inconsistent with Maxwell's equations relies on the use of the wrong canonical momenta in the multipolar form of the Hamiltonian; the field conjugate to the vector potential in the Coulomb gauge is proportional to the transverse displacement vector and not, as in the minimal-coupling form of the Hamiltonian, to the transverse electric field, and similarly the canonical momenta of the charged particles differ in the two cases. Some consequences of this for time-dependent perturbation theory are discussed.
Based on (i) the spectral resolution definition of the momentum operator, (ii) the linearity of correspondence between physical observables and quantum Hermitian operators, (iii) the definition of conjugate coordinate-momentum variables in classical mechanics, and (iv) the invariance of the classical Hamiltonian to canonical transformations (transformations that change from one set of conjugate variables to another), we show that the $[\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{x},\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{p}]$ coordinate-momentum commutator must have the value ihihih where hhh is a real nonzero number (which we can identify with the experimentally determined $\ensuremath{\hbar}$). The results are then extended to include all generalized coordinates and their conjugate momenta as well as the Cartesian special-relativistic case.