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Time Evolution of the Complex Scalar Field

  1. Sep 26, 2014 #1
    1. The problem statement, all variables and given/known data

    Consider the Lagrangian, L, given by

    [tex] L = \partial_{\mu}\phi^{*}(x)\partial^{\mu}\phi(x) - m^2\phi^{*}(x)\phi(x) . [/tex]

    The conjugate momenta to [itex] \phi(x) [/itex] and [itex] \phi^{*}(x) [/itex] are denoted, respectively, by [itex] \pi(x) [/itex] and [itex] \pi^{*}(x) [/itex]. Thus,

    [tex] \pi(x) = \frac{\partial L}{\partial(\partial_{0}\phi(x))} = \partial_0\phi^{*}(x) [/tex]
    [tex] \pi^{*}(x) = \frac{\partial L}{\partial(\partial_{0}\phi^{*}(x))} = \partial_0\phi(x) .[/tex]

    Upon quantizing the system, [itex] \phi(x) [/itex] and [itex] \phi^{*}(x) [/itex] are promoted to operators which satisfy the equal-time commutation relations:

    [tex] [ \phi(x) , \pi(y) ] = i\delta^{(3)}(\vec{x} - \vec{y}) [/tex]
    [tex] [ \phi^{*}(x) , \pi^{*}(y) ] = i\delta^{(3)}(\vec{x} - \vec{y}) [/tex]

    (all others zero). In the Heisenberg regime, the time evolution of the operator [itex] \phi(x) [/itex], [itex] i \partial_0 \phi(x) [/itex], is given by

    [tex] i \partial_0 \phi(x) = \left[ \phi(x) , H(y) \right]. [/tex]

    The Hamiltonian may be derived from the Lagrangian, and we find that

    [tex] i\frac{\partial \phi(x)}{\partial t} = \int d^{3}y \left( \left[ \phi(x) , \pi(y)\pi^{*}(y) \right] + \left[ \phi(x) , \nabla\phi^{*}(y) \cdot \nabla\phi(y) \right] + m^2 \left[ \phi(x) , \phi^{*}(y)\phi(y) \right] \right). [/tex]

    Now here's my question. When we evaluate the commutators both my professor and a solution manual to Peskin and Schroeder claim that only the first commutator survives, because [itex] \phi(x) [/itex] commutes with everything except for the its conjugate momentum (by the canonical commutation relations). I don't see why. The canonical commutation relations only give us a relationship between [itex] \phi(x) [/itex] and [itex] \pi(y) [/itex], not e.g. [itex] \phi(x) [/itex] and [itex] \phi(y) [/itex]. The point is pressed by the fact that one can only show that the commutator [itex] \left[ \phi(x) , \phi(y) \right] [/itex] vanishes for space-like separation between the points x and y (this is the condition which preserves causality).

    I guess it would be resolved if the commutator were instead [itex] \left[ \phi(x) , H(x) \right] [/itex], but this doesn't seem to be how it's done.

    Thanks for any help!

    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Sep 27, 2014 #2

    Orodruin

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    The Hamiltonian really should be the Hamiltonian at time ##x^0 = y^0## (note that you are integrating over the spatial components of ##y##). As a result, ##x## and ##y## have space-like separation and the commutator ##[\phi(x),\phi(y)]## vanishes (they are even equal-time).
     
  4. Sep 27, 2014 #3
    Cool, thanks!
     
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