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 (1) , Didier Felbacq (1)

1 L2C - Laboratoire Charles Coulomb

This note is a reply to the paper 1 arXiv:1801.0559: " The equivalence of the Power-Zineau-Woolley picture and the Poincaré gauge from the very first principles " by G. Kónya, 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. Kónya, al. in a comment 1 where the authors claim the equivalence between the Power-Zienau-Woolley hamiltonian and the minimal-coupling hamiltonian in the Poincaré gauge. They claim that we have made one error and one wrong emphasis in our paper: The error as summarized by G. Kónya 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.Kónya al. would be: " The use of the canonical coordinate/momentum pair A p and E in Poincaré 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 Poincaré 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 ´ onya 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 ∂_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. Kónya 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.

Domaines: Physique Quantique [quant-ph]; Optique [physics.optics]

Emmanuel Rousseau, Didier Felbacq. Reply to " The equivalence of the Power-Zineau-Woolley picture and the Poincaré gauge from the very first principles " by G. Kónya, et al.. 2018. ⟨hal-01760460v1⟩