(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

I have the following task:

In quantum free scalar field theory find commutators of creation and anihilation operators with total four-momentum operator, starting with commutators for fields and canonical momenta. Show that vacuum energy is zero.

2. Relevant equations

[itex]\hat{p}^\nu=\int d^3\vec{k} k^\nu \hat{a}^\dagger(k)\hat{a}(k)[/itex]

[itex][\hat{a}(p),\hat{a}^\dagger(q)] = \delta^{(3)}(\vec{p} - \vec{q})[/itex]

3. The attempt at a solution

I managed to find commutators:

[itex][\hat{a}(q),\hat{p}^\nu] = q^\nu \hat{a}(q)[/itex]

[itex][\hat{a}^\dagger(q),\hat{p}^\nu] = - q^\nu \hat{a}^\dagger(q)[/itex]

Then I used this result to show that [itex]\hat{a}(q)|p>[/itex] and [itex]\hat{a}^\dagger(q)|p>[/itex] are eigenstates of the four-momentum operator [itex]\hat{p}^\nu[/itex] with eigenvalues [itex](p^\nu - q^\nu)[/itex] and [itex](p^\nu + q^\nu)[/itex] respectively.

However i seem to be stuck now as I have no ide how to show that there is a lower boundary for energy values and that this value is indeed 0.

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# Homework Help: Vacuum energy of free scalar field

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