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Temporal component of the normal ordered momentum operator

  1. Dec 8, 2014 #1
    1. The problem statement, all variables and given/known data

    Consider the real scalar field with the Lagrangian [itex]\mathcal{L}=\frac{1}{2}\partial_\mu \phi \partial^\mu \phi - \frac{1}{2} m^2 \phi^2[/itex]. Show that after normal ordering the conserved four-momentum [itex]P^\mu = \int d^3x T^{0 \mu}[/itex] takes the operator form

    [tex]P^\mu = \int \frac{d^3p}{(2 \pi)^3} p^\mu a_p^\dagger a_p.[/tex]

    I have already showed that the three spatial components of the momentum operator satisfy the above. I'm left with showing that the temporal component of the normal-ordered momentum operator also satisfies the above.

    2. Relevant equations

    The classical temporal component, [itex]T^{00},[/itex] of the energy-mometum tensor is

    [tex]T^{00}=\frac{1}{2} \dot{\phi}^2+\frac{1}{2} \left( \nabla \phi \right)^2 + \frac{1}{2}m^2 \phi^2.[/tex]

    To quantize this we use the following expansion for the fields

    [tex]\phi(x) = \int \frac{d^3p}{(2 \pi)^3 \sqrt{2 E_p}} \left[ a_p e^{i \vec{p} \cdot \vec{x}} + a_p^\dagger e^{-i\vec{p} \cdot \vec{x}} \right]. [/tex]

    Note that [itex]a_p[/itex] and [itex]a_p^\dagger[/itex] satisfy the typical commutation relations for creation and annihilation operators.

    3. The attempt at a solution

    After taking the appropriate derivatives, expansion, simplifications using delta functions, commutation relations, and imposing that everything lies on the mass-shell I have been able to show that

    [tex]P^0 = \frac{1}{4} \int d^3p \left[ \left( a_p a_{-p} + a_p^\dagger a_{-p}^\dagger \right) \left( \frac{-2 \vec{p}^2}{E_p} \right) + \left( a_p a_p^\dagger + a_p^\dagger a_p \right) 2E_p \right], [/tex]

    where [itex]E_p = p^0.[/itex] I've been over this calculation twice, and am fairly confident that it is correct thus far, though I may still be wrong on that fact. If I only had the last two terms then this would be exactly what I was looking for. However, I can't see how to make the first two terms disappear in this case. For the spatial part I also had four terms, but instead of the first two terms being multiplied by [itex]p^2[/itex] they were only multiplied by [itex]p[/itex], making the first two terms odd, and thus disappear when integrated over the reals. In this case my first two terms are even, and so I'm a bit lost as to how to make them go away.

    Any help would be appreciated.

    Thanks.
     
  2. jcsd
  3. Dec 9, 2014 #2
    Unfortunately, it seems that your mistake was made in deriving the equation you've given, because the first two terms should cancel out.

    I think your issue might be in neglecting that the exponents in the field expansion should contain time-dependence. For example, I would expect that your term [itex]a_pa_{-p}[/itex] should be multiplying a factor [itex]e^{-2i\omega t}[/itex] (if I got the sign in the exponent right).
     
  4. Dec 9, 2014 #3
    I wanted to explicitly do this in the Schrodinger picture. A friend and I were working on this together, and he decided to do this in the Heisenberg picture while I did it in the Schrodinger picture. I don't see where the exponentials would come from in the Schrodinger picture. I can post the gory details if you think that would be helpful.
     
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