# Dimensions and the Generating Functional

by TriTertButoxy
Tags: dimensions, functional, generating
 P: 194 Something seems a little weird to me: What are the dimensions of a generating functional, $Z[j]$ -- say for real scalar field theory? $$Z[j]=\int\mathcal{D}\phi\,\exp\, i\!\int d^4x\left(\frac{1}{2}\partial_\mu\phi\partial^\mu\phi-\frac{1}{2}m^2\phi^2+j\phi\right)$$ Also, what about mass dimensions of the generating functional for connected Green's functions, $W[j]$? This is defined in terms of the log of the generating functional, $Z[j]$. $$Z[j]=e^{iW[j]}$$ This seems a little pathological...
 P: 194 Ah, so you mean in order for Z[0]=1, the integration measure, $\mathcal{D}\phi$, must be normalized such that it is unitless. I understand now. thanks, Avodyne!