Canonical Commutation Relation

Gauge Invariance

Canonical quantization is applied, by definition, on canonical coordinates. However, in the presence of an electromagnetic field, the canonical momentum p is not gauge invariant. The correct gauge-invariant momentum (or "kinetic momentum") is

(SI units) (cgs units),

where q is the particle's electric charge, A is the vector potential, and c is the speed of light. Although the quantity p_kin is the "physical momentum", in that it is the quantity to be identified with momentum in laboratory experiments, it does not satisfy the canonical commutation relations; only the canonical momentum does that. This can be seen as follows.

The non-relativistic Hamiltonian for a quantized charged particle of mass m in a classical electromagnetic field is (in cgs units)

where A is the three-vector potential and is the scalar potential. This form of the Hamiltonian, as well as the Schroedinger equation, the Maxwell equations and the Lorentz force law are invariant under the gauge transformation

where

and is the gauge function.

The angular momentum operator is

and obeys the canonical quantization relations

defining the Lie algebra for so(3), where is the Levi-Civita symbol. Under gauge transformations, the angular momentum transforms as

$\langle \psi \vert L \vert \psi \rangle \to \langle \psi^\prime \vert L^\prime \vert \psi^\prime \rangle = \langle \psi \vert L \vert \psi \rangle + \frac {q}{\hbar c} \langle \psi \vert r \times \nabla \Lambda \vert \psi \rangle \, .$

The gauge-invariant angular momentum (or "kinetic angular momentum") is given by

which has the commutation relations

$=i\hbar {\epsilon_{ij}}^{\,k} \left(K_k+\frac{q\hbar}{c} x_k \left(x \cdot B\right)\right)$

where

is the magnetic field. The inequivalence of these two formulations shows up in the Zeeman effect and the Aharonov–Bohm effect.

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Canonical Commutation Relation - Gauge Invariance