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.