How to derive the scaling transformation?

Question

The question arose while reading  the Book of Subir Sachdev, Quantum Phase transition (second edition), p46. 

Given a partition function of the form 

\({\cal Z}= \int{\cal D}\phi_{\alpha}(x)\exp(-\frac{1}{2}\int d^{D}x[(\nabla_{x}\phi_{\alpha})^{2}+r\phi_{\alpha}(x)^{2}]) \)

where \(r\) is some parameter in the theory.  How one could guess the form of scaling transfromation which does not change  correlations associated with this partition function. Does it depends on the field theory or on the physical system for which the theory has been constructed?

Answer 1

As I understand it, scaling transformation are universally defined by the scaling parameter ($\lambda > 0$) such that spacetime transforms as

\(D(\lambda): \mathbb{R}^n \rightarrow \mathbb{R}^n\)

\(x \rightarrow \lambda x\)

and (components of) fields present in the theory transform according to their scaling dimension $d_{\phi}$ as

\((D(\lambda)\phi) (x )= \lambda^{-d_{\phi}} \phi(\lambda^{-1}x)\)

As Arnold said, the scaling dimension of the fields (components) can be determined by the requirement that the exponent of the path integral has scaling dimension zero.

Comments

And $d_\phi$ is determined by the action through the requirement that the exponent of the path integral has scaling dimension zero.