# Canonical commutation relation, from QM to QFT.

1. Feb 18, 2017

### annihilatorM

1. The problem statement, all variables and given/known data

This is a system of n coupled harmonic oscillators in 1 dimension.

Since the distance between neighboring oscillators is $\Delta x$ one can characterize the oscillators equally well by $q(x,t)$ instead of $q_j(t)$. Then $q_{j \pm 1}$ should be replaced by $q(x \pm \Delta x , t)$. Show that the quantization condition

$$[\dot{q}_j(t), q_k (t)] = - \frac{i}{\mu} \delta _{jk}$$

can be expressed as

$$[ \dot{q} (x,t), q (x',t)] = - \frac{i}{\mu} \int_{ \frac{- \Delta x}{2}}^{\frac{ \Delta x}{2}} dy \delta (y-x'+x)$$

Hint: Distinguish the two cases $x=x'$ and $| x-x' | = j \Delta x$

2. Relevant equations

3. The attempt at a solution

Making the substitution of $q(x,t)$ in place of $q_j (t)$ I obtain

$$[ \dot{q} (x,t), q (x',t)] = \delta (x-x')$$

which is the expected commutation relation i've seen from other sources, however, I don't understand why the integration with respect to y is introduced or where the integration bounds come from. What is wrong with simply writing down what I have above?

2. Feb 23, 2017