Heidi said:
Merry Christmas to all Physics forumers.
Let me explain why i am not entirely satisfied with this article.
I opened a previous thread about a simple situation: a charged particle in magnetic fiels (no time dependence, no electric field)
I saw that there two main gauge choices: the Landau gauge and the symmetric gauge
one is with the vector potential (0,xB,0) and the othe (-yB/2,xB/2)
the first is weird with its dagneracy usign plane waves and the second seems more physical (two oscillators)
The point is not in which gauge you work but to find a physically easily interpretable set of energy eigenvectors. It's crucial to understand that the observable quantities and probabilities for the outcome of measurements should refer to gauge-invariant quantities and to prove that the corresponding probabilities, expectation values, etc. are indeed gauge independent.
For the particle in a static, homogeneous magnetic field you can choose the Hamiltonian, which can be shown to be gauge invariant (see also the papers by Yang and Kobe quoted above). Now the usual treatment is to choose the compatible momentum components to get a uniquely defined complete orthonormalized system (CONS) of energy eigenstates, but these do not have a gauge-invariant meaning, and it may be complicated to interpret their physical meaning, although of course you can use them to calculate any meaningful probability or expectation value related to gauge-invariant quantities.
Alternatively you can choose a CONS of simultaneous eigenvectors of only gauge-invariant observables, leading to a easily interpretable set of energy eigenvectors. Instead of using the gauge-dependent canonical momenta ##\hat{\vec{p}}## you can consider the "kinetic momentum"
$$\hat{\vec{\pi}}=\hat{\vec{p}}-q \vec{A}(\hat{\vec{x}})=m \hat{\vec{v}}=\frac{m}{\mathrm{i} \hbar} [\hat{\vec{x}},\hat{H}],$$
where ##\vec{A}## may be chosen in any gauge you like. The commutation relations of the kinetic momenta are also gauge invariant of course,
$$[\hat{\pi}_j,\hat{\pi}_k]=\mathrm{i} q \epsilon_{jkl} B_l.$$
For the following I make ##\vec{B}=B \vec{e}_3## (##B=\text{const}##, because we consider a homogeneous magnetic field). Since
$$\hat{H}=\frac{1}{2m} \hat{\vec{\pi}}^2.$$
We can choose as a complete compatible set of (gauge-invariant!) observables to get a unique gauge-invariant CONS of energy eigenvalues: ##\hat{H}##, ##\hat{H}_{\text{perp}}##, and ##\hat{\pi}_3##, where
$$\hat{H}_{\text{perp]}=\frac{1}{2m} (\hat{\pi}_1^2+\hat{\pi}_2^2).$$
The solution can be obtained completely algebraically with appropriate ladder operators as for the harmonic oscillator.
For a complete treatment, see my QM manuscript. I hope that's understandable although it's written in German (the formula density is high and many intermediate step of the calculation are worked out explicitly):
https://itp.uni-frankfurt.de/~hees/publ/theo3-l3.pdf
Sect. 3.11.5 (p. 88ff).
Heidi said:
when i read that in the paper:
Taking this point of view we wish to formulate a mathematical description of this system using only operators which have gauge invariant expectation values and whose physical interpretation is also gauge invariant. We formalize our rules of interpretation by postulating a physical correspondence principle. "A quantum mechanical operator can represent a physical quantity with with classical analogue only if there exists a Newtonian quantity whose equation of motion is formally identical to the equation of motion of the operator, and if such a correspondence exists, the operator is interpreted as the quantum equivalent of the Newtonian quantity
i thought that i could find in the paper why one the two gauge seems more physical.
But no. this simple case is not dimply treated, just the precessing magnetic field and in that case he writes: Rather than provide a complete solution, which has been presented elsewhere...
Neither gauge is more physical than any other. All are equivalent, but one must keep in mind that nothing gauge-dependent can be taken as an observable, although you can of course use a gauge-dependent basis to calculate gauge-independent physical quantities.
Heidi said:
is it in the second paper? and will i find in it the answer of my question?
I hope, the section in my manuscript answers your question. Maybe I'll translate it as another FAQ article :-).
A very good source is also the textbook by Cohen-Tannoudji and Laloe. He has a very concise section and appendix concerning these questions about gauge invariance and all that. It's of course a bit obscured in its application to the non-relativistic case, but it can be worked out carefully of course, since gauge invariance is consistent with the non-relativistic approximation for the matter part.
Also note that all these problems do not occur when using time-dependent perturbation theory, because the time-dependent Schrödinger equation is of course gauge covariant, and an expansion in powers of ##\hbar## for gauge-invariant observable quantities leads to gauge-invariant results about physical observables (as in standard QED or NRQED).