Dimensions and the Generating Functional

  1. 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. jcsd
  3. Avodyne

    Avodyne 1,335
    Science Advisor

    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 :smile:).
     
  4. 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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