Editing Canonical commutation relation (section)

== Gauge invariance ==
Canonical quantization is applied, by definition, on [[canonical coordinates]].  However, in the presence of an [[electromagnetic field]], the canonical momentum {{mvar|p}} is not [[gauge invariant]].  The correct gauge-invariant momentum (or "kinetic momentum") is
: <math>p_\text{kin} = p - qA \,\!</math>  ([[SI units]]) {{spaces|4}} <math>p_\text{kin} = p - \frac{qA}{c} \,\!</math>  ([[Gaussian units|cgs units]]),
where {{mvar|q}} is the particle's [[electric charge]], {{mvar|A}} is the [[Magnetic vector potential|vector potential]], and {{math|''c''}} is the [[speed of light]]. Although the quantity {{math|''p''<sub>kin</sub>}} 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 (quantum mechanics)|Hamiltonian]] for a quantized charged particle of mass {{mvar|m}} in a classical electromagnetic field is (in cgs units)
<math display="block">H=\frac{1}{2m} \left(p-\frac{qA}{c}\right)^2 +q\phi</math>
where {{mvar|A}} is the three-vector potential and {{mvar|φ}} is the [[scalar potential]]. This form of the Hamiltonian, as well as the [[Schrödinger equation]] {{math|1=''Hψ'' = ''iħ∂ψ/∂t''}}, the [[Maxwell equation]]s and the [[Lorentz force law]] are invariant under the gauge transformation
<math display="block">A\to A' = A+\nabla \Lambda</math>
<math display="block">\phi\to \phi' = \phi-\frac{1}{c} \frac{\partial \Lambda}{\partial t}</math>
<math display="block">\psi \to \psi' = U\psi</math>
<math display="block">H\to H' = U H U^\dagger,</math>
where <math display="block">U=\exp \left( \frac{iq\Lambda}{\hbar c}\right)</math> and {{math|1=Λ = Λ(''x'',''t'')}} is the gauge function.

The [[angular momentum operator]] is
<math display="block">L=r \times p \,\!</math>
and obeys the canonical quantization relations
<math display="block">[L_i, L_j]= i\hbar {\epsilon_{ijk}} L_k</math>
defining the [[Lie algebra]] for [[so(3)]], where <math>\epsilon_{ijk}</math> is the [[Levi-Civita symbol]].  Under gauge transformations, the angular momentum transforms as
<math display="block"> \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 \, .
</math>

The gauge-invariant angular momentum (or "kinetic angular momentum") is given by
<math display="block">K=r \times \left(p-\frac{qA}{c}\right),</math>
which has the commutation relations
<math display="block">[K_i,K_j]=i\hbar {\epsilon_{ij}}^{\,k}
\left(K_k+\frac{q\hbar}{c} x_k
\left(x \cdot B\right)\right)</math>
where <math display="block">B=\nabla \times A</math> is the [[magnetic field]]. The inequivalence of these two formulations shows up in the [[Zeeman effect]] and the [[Aharonov–Bohm effect]].