[Physics] Why do we use potential for quantizing the electromagnetic field

electromagnetismgauge-theorypotentialquantum-electrodynamics

For quantizing the electromagnetic field authors go to its potential and then find themselves facing to the problems of degree of freedom from gauge transformation.

Why we can't simply quantize electromagnetic field itself: decompose it to wave planes and promote normal modes to quantum harmonic oscillator?

Best Answer

The $\bf E$ and $\bf B$ fields viewed as independent quantum oscillators contain too many DOFs, if that's what you mean. But I'm getting ahead of myself. Here is one line of reasoning:

  1. It is reasonable to expect that for a consistent quantum theory of E&M, it should have a classical limit $\hbar\to 0$ where the classical electromagnetic fields are governed by a classical action $S$.

  2. In particular, in this classical action formulation, we demand that the Maxwell equations are (i) either automatically satisfied or (ii) arise as the Euler-Lagrange (EL) equations for $S$.

  3. It turns out to be very challenging to try to formulate the classical E&M action $S$ without the use of potentials, cf. e.g. this Phys.SE post.

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