If I write the basic scalar field as $$\phi(x)=\int\frac{d^3k}{(2\pi)^3}\frac1{\sqrt{2E}}\left(ae^{-ik\cdot x}+a^\dagger e^{ik\cdot x}\right),$$ this would seem to imply that the creation and annihilation operators carry mass dimension -3/2. That's the only way I can get the total field ##\phi## to have mass dimension 1, since the ##d^3k## carries three dimensions of mass and the ##1/\sqrt{2E}## factor carries dimension -1/2. This seems contradictory to the fact that ##a^\dagger a## is usually defined as the number operator, which should be dimensionless. Am I missing something here? (My question at first was just how to get the dimensions of the Fourier-expanded field to come out right.)(adsbygoogle = window.adsbygoogle || []).push({});

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# Units of Fourier expanded field

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