Let a QM system be described in the Heisenberg picture by position variables [tex]q_j[/tex] with corresponding conjugate momenta [tex]p_j[/tex]. We have the equal-time commutators(adsbygoogle = window.adsbygoogle || []).push({});

[tex][q_j(t),p_k(t)]=i\hbar \delta_{jk}[/tex]

In quantum field theory, for the Dirac spinor field we have the equal-time commutator

[tex]\left\{\psi_j(\vec{x},t),\psi^\dagger_k (\vec{x}',t)\right\}=\delta_{jk}\delta^3(\vec{x}-\vec{x}')[/tex]

(subscripts refer to spinor components) Depending on how we scale the Lagrangian, the conjugate momentum to psi is

[tex]\pi_j=\frac{\partial \mathcal{L}}{\partial \dot{\psi_j}}=i\hbar\psi^\dagger_j[/tex]

So the commutator in terms of the conjugate momentum has a similar form to that its QM counterpart:

[tex]\left\{\psi_j(\vec{x},t),\pi_k (\vec{x}',t)\right\}=i\hbar \delta_{jk}\delta^3(\vec{x}-\vec{x}')[/tex]

A similar result holds (I think) for the real scalar field:

[tex]\left[\phi(\vec{x},t),\pi(\vec{x}',t)\right]=i\hbar\delta^3(\vec{x}-\vec{x}')[/tex]

Is this a general rule? Do we always have commutators between a quantity and its conjugate momentum proportional to i*hbar (times the delta function if continuous)? If so, what is the deeper meaning? Is there a principle this can be derived from?

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# Is there a general prescription for commutators?

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