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Canonical commutation relation, from QM to QFT.

  1. Feb 18, 2017 #1
    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. jcsd
  3. Feb 23, 2017 #2
    Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.
     
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