From a Hamiltonian for the Dirac Equation, we can add a potential term to it simply by adjusting the momentum operator so that p^(nu)->p^(nu)−A^(nu), where A^(nu) is the relevant potential. But how do you calculate A^(nu)? For example, what would A^(nu) be for an electron in an electromagnetic field given by the tensor F^(alpha beta)?

Matias Aguirre 2022-07-23 Answered
From a Hamiltonian for the Dirac Equation, we can add a potential term to it simply by adjusting the momentum operator so that p μ p μ A μ , where A μ is the relevant potential. But how do you calculate A μ ? For example, what would A μ be for an electron in an electromagnetic field given by the tensor F α β ?
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Answers (1)

dtal50
Answered 2022-07-24 Author has 10 answers
The field tensor can be derived from the vector potential like so:
F μ ν = μ A ν ν A μ
If F is simple enough, you can usually construct an appropriate A without too much difficulty. Otherwise you're stuck inverting this with a bunch of indefinite integrals.
Note that A is not uniquely determined by this relation. If A μ is a valid vector potential, then for any analytic function ϕ
A μ = A μ + μ ϕ
will give equivalent results.
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