Dimensions and the Generating Functional

[TriTertButoxy]

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...

 

[Avodyne]

Both Z and W are dimensionless. This is obvious for W, since you couldn't put it into the exponential if it wasn't. As for Z, it's usually defined as the vacuum-to-vacuum transition amplitude in the presence of the source j, and this equals one if there is no source, so Z[0]=1. Thus Z[j] must be dimensionless. To get Z[0]=1, a normalization factor must be implicitly included in the measure over the fields.

None of this is specific to field theory. Similar statements apply to path integrals in NRQMOP (non-relavitistic quantum mechanics of one particle ).

 

[TriTertButoxy]

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!