# How to derive the scaling transformation?

+ 3 like - 0 dislike
373 views

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?

+ 2 like - 0 dislike

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.

answered Aug 3, 2014 by (6,240 points)
edited Aug 3, 2014 by Dilaton

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

 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\varnothing$ysicsOverflowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.