Undergrad Mass Dimension of Fields (Momentum space)

[thatboi]

Hi all,
We know from requiring the action be invariant that the mass-dimension of a scalar field $\phi$ is $\frac{d-2}{2}$ where $d$ is the space-time dimension. But what is the mass-dimension of $\phi(p)$? I ask because free-theory 2-pt correlation function (in Euclidean space) is written as $\langle\phi(p)\phi(q)\rangle = (2\pi)^{d}\delta^{d}(p+q)\frac{1}{p^{2}+m^{2}}$. The dirac delta seems to contribute a mass dimension $-d$ and then the fractional component contributes another mass-dimension of $-2$? I'm not sure if this makes sense.

Thanks.

 

[Orodruin]

Indeed, and therefore …

Another way of deriving it is to look at the action expressed in the momentum variables or just the definition of $\phi(p)$ in terms of $\phi(x)$.

 

[thatboi]

Indeed, and therefore …

Another way of deriving it is to look at the action expressed in the momentum variables or just the definition of $\phi(p)$ in terms of $\phi(x)$.

Ah, so the mass dimension is $\frac{-d-2}{2}$. The fourier transform is indeed significantly simpler to see this.