Obtaining the Dirac function from field operator commutation

[abivz]

TL;DR Summary

Need help obtaining the Dirac function from the commutation of two field operators

Hi everyone, I'm new to PF and this is my second post, I'm taking a QFT course this semester and my teacher asked us to obtain:
$$[\Phi(x,t), \dot{\Phi}(y,t) = iZ\delta^3(x-y)]$$
We're using the Otto Nachtman: Elementary Particle Physics but I've seen other books use this notation:
$$[\Phi(x,t), {\Pi}(y,t) = iZ\delta^3(x-y)]$$
In the book they give:
$$[\Phi(x,t)] = \int\frac{d^3k}{(2\pi)^3}\frac{1}{2\omega}e^{-ikx}(e^{i\omega t}a^{\dagger}(k)+e^{-iwt}a(-k))$$
$$[\dot\Phi(x,t)] = \int\frac{d^3k}{(2\pi)^3}\frac{i}{2}e^{-ikx}(e^{i\omega t}a^{\dagger}(k)-e^{-iwt}a(-k))$$

I'm confused as to how one can obtain this form from those definitions, specially because of the annihilation and creation operators, I haven't found a book that explains it, they just give the equation, does anyone have any tips on how to start or have any book or lecture notes that could help? Thanks in advance :)

 

[troglodyte]

This is a continuum approximation .The roots for this notion comes from the uncertainty principle where you cannot make explicit shape peaked statements about the position and momentum at the same time.So here you want to reconstruct a wavefuntion with position dependency to field formalism in such a manner that quantum mechanics and special relativity collaborate.So have to act in the momentum space to get sharp peaked position statements .This is the so called mode expansion,which involved the fouriertransform concepts but uses also the fact that the incoming plane waves anhilates and the outgoing plane waves creates particles.the whole new theory should be canonical,which means that the field operators should fulfill the canonical commutation relations.
Maybe you can check

https://en.m.wikipedia.org/wiki/Second_quantization

(In the field operator section) to get a better intuition /felling for this problem.

troglodyte

 

[vanhees71]

I'm not sure, whether I understand the question right. First of all from the Lagrangian for the free uncharged Klein-Gordon field
$$\mathcal{L}=\frac{1}{2} (\partial_{\mu} \Phi)(\partial^{\mu} \Phi) - \frac{m}{2} \Phi^2,$$
you get the canonical field momenta as
$$\Pi=\frac{\partial \mathcal{L}}{\partial \dot{\Phi}}=\dot{\Phi}.$$
Then the equation of motion gives you the mode decomposition you wrote above. The time derivative of the field is easy: you just differentiate the exponential functions in the mode decomposition.

Then you can derive the equal-time commutation relation between the field operator and the canonical-field-momentum operator using the commutation relations for the annihilation and creation operators.

Usually the argument is in the other direction, because the equal-time commutation relations between field and canonical-momentum operator are usually conjectured in what's known as "canonical quantization".