Dimensions and the Generating Functional

by TriTertButoxy
Tags: dimensions, functional, generating
TriTertButoxy is offline
Sep20-08, 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].


This seems a little pathological...
Phys.Org News Partner Physics news on Phys.org
Physicists design quantum switches which can be activated by single photons
'Dressed' laser aimed at clouds may be key to inducing rain, lightning
Higher-order nonlinear optical processes observed using the SACLA X-ray free-electron laser
Avodyne is offline
Sep22-08, 06:59 PM
Sci Advisor
P: 1,185
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 is offline
Sep23-08, 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!

Register to reply

Related Discussions
Generating EM radiation Classical Physics 5
Generating Functions Calculus & Beyond Homework 0
Generating functions Calculus & Beyond Homework 2
9 Space Dimensions 2 Time Dimensions General Physics 7