How Are Commutation Relations Derived in Quantum Field Theory?

[Chopin]

In Srednicki's book, he discusses quantizing a non-interacting spin-0 field $$\phi(x)$$ by defining the KG Lagrangian, and then using it to derive the canonical conjugate momentum $$\pi(x) = \dot{\phi}(x)$$. Then, he states that, by analogy with normal QM, the commutation relations between these fields is:

$$[\phi(x), \phi(x')] = 0$$
$$[\pi(x), \pi(x')] = 0$$
$$[\phi(x), \pi(x')] = i\delta^3(x-x')$$

Can this be derived from anything we know so far, or does it simply have to be taken on faith? These relations are used to derive the commutation relations for the creation/annihilation operators, which in turn allow us to derive the spectrum of the Hamiltonian, so it looks like they form the basis of pretty much everything that follows.

 

[ccesare]

You can treat these, formally, as postulates. However, they are physically motivated postulates. For example the first two are related to causality (with the fields all taken at the same time), and the last one is analogous to QM.

 

[tom.stoer]

You can motivate these commutation relations by looking at momentum space. If you have finite volume in position space you get discrete momenta and for each momentum k_n you find a pair of creation and annihilation operators like for the harmonic oscillator.

Another way to see that is to look at classical field theory and Poisson brackets in the canonical formalism. For a field and its canonical momentum the Poisson bracket reads

$$\{\phi(x),\pi(y)\} = \delta(x-y)$$

Quantizing the fields i.e. replacing them by field operators means just introducing the "i", just like in ordinary QM for the operators x and p.

 

[The_Duck]

Zee's discussion of this is nice, I think: he considers a QM system like a mattress with a bunch of particles that each have one degree of freedom that are arranged in a discrete grid with nearby particles coupled together with springs or something. Then you take the limit where the grid spacing goes to zero and you get the quantum mechanical description of a continuous field, with those commutation relations emerging as the limit of the commutation relations in the discrete case.

 

[tom.stoer]

Zee's discussion of this is nice, I think: he considers a QM system like a mattress with a bunch of particles that each have one degree of freedom that are arranged in a discrete grid with nearby particles coupled together with springs or something. Then you take the limit where the grid spacing goes to zero and you get the quantum mechanical description of a continuous field, with those commutation relations emerging as the limit of the commutation relations in the discrete case.

Yes, this is an excellent "derivation" of the commutation relations for quantum field theories.