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Dimensions and the Generating Functional 
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#1
Sep2008, 10:58 PM

P: 194

Something seems a little weird to me: What are the dimensions of a generating functional, [itex]Z[j][/itex]  say for real scalar field theory?
[tex]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)[/tex] Also, what about mass dimensions of the generating functional for connected Green's functions, [itex]W[j][/itex]? This is defined in terms of the log of the generating functional, [itex]Z[j][/itex]. [tex]Z[j]=e^{iW[j]}[/tex] This seems a little pathological... 


#2
Sep2208, 06:59 PM

Sci Advisor
P: 1,205

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 vacuumtovacuum 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 (nonrelavitistic quantum mechanics of one particle ). 


#3
Sep2308, 12:10 AM

P: 194

Ah, so you mean in order for Z[0]=1, the integration measure, [itex]\mathcal{D}\phi[/itex], must be normalized such that it is unitless.
I understand now. thanks, Avodyne! 


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