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One defines the "scaling dimension" (as opposed to "engineering dimension") of an operator $\cal{O}$ as $[\cal{O}]$ such that if $\cal{O}(t^{-1}x) = t^{[\cal{O}]}\cal{O}(x)$ then the Lagrangian in which $\cal{O}$ appears would be scale invariant.

  • Unlike for "engineering dimensions" it seems that the value of scaling dimensions (even classically!) can't be derived from just looking at the operator but one seems to need to know the Lagrangian in which it appears so that the "right" $[\cal{O}]$ can be assigned to preserve scale-invariance.

For example - how else does one explain that the "engineering dimension" of $m^2\phi$ is $3$ whereas its "scaling dimension" is $1$? (same as that of $\phi$) (..the above obviously follows if I think of the term to be occurring in a $2+1$-dimensional Lagrangian and ask as to what should the scaling dimensions be so that the Lagrangian is scale-invariant..but something doesn't look very intuitive..)

  • I would like to know what is the special difficulty that is faced in defining $2-$point correlation functions of $\cal{O}$ if it is real? (..as opposed to when they are complex like in the next question - thought not that obvious either!..)

  • For complex $\cal{O}$ it "follows" that $<\cal{O}(x)\cal{O}^*(y)> \sim \vert x - y \vert ^{-2[ \cal{O}]}$ It is clearly consistent with definitions of the scaling dimension but is there a "derivation" for this? I have often seen the statement that the above short-distance behaviour follows from "reflection positivity" (..ala Wightman axioms..) I would like to know of some explanations.

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Your definition of scaling dimension does not agree with the standard definition in quantum field theory and conformal field theory. The standard definition is that the "engineering" dimension is the dimension as determined by dimensional analysis of the Lagrangian, whether the Lagrangian is scale invariant or not, while the full "scaling dimension" is determined by the exact two point function (including quantum corrections). This is discussed in most QFT textbooks and a brief discussion is also available on Wikipedia under "anomalous scaling dimension." –  Jeff Harvey Jan 21 '12 at 17:24
    
@Jeff Delighted to see a reply from you! Let me give my references, in case I am misreading something. My definition of "scaling dimension" is what is discussed on the first page of this lecture by Witten, math.ias.edu/QFT/fall/wittn2.ps What he calls as just dimensions on the first page here is what he seems to also call scaling dimension in the discussion on the first two pages of the next lecture, math.ias.edu/QFT/fall/wittn3.ps (..as say very clearly stated just above remark 2 on the second page of my second link..)I thought my terminology is consistent with these. –  Anirbit Jan 21 '12 at 20:17
    
@Jeff Ofcourse all these definitions are classical and clearly when quantized one would get new definitions of dimension and hence the notion of "anomalous" dimension. As has been pointed out in the remark just above section 3.5 on page 8 of my second link. –  Anirbit Jan 21 '12 at 20:22

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