Why does dilation invariance often imply proper conformal invariance?

Why does a quantum field theory invariant under dilations almost always also have to be invariant under proper conformal transformations? To show your favorite dilatation invariant theory is also invariant under proper conformal transformations is seldom straightforward. Integration by parts, introducing Weyl connections and so on and so forth are needed, but yet at the end of the day, it can almost always be done. Why is that?

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You can look for Polchinski's paper "Scale and Conformal Invariance in QFT" which discusses the issue in detail. Maybe there have been some developments since then, but that is a good starting point. – user566 Mar 5 '11 at 18:56
For 4d field theory it's still an open question whether scale invariance implies conformal invariance. There's been some recent work on this topic by Slava Rychkov and collaborators, see e.g. 1101.5385. – Matt Reece Mar 6 '11 at 23:06
By the way, given that scale invariance does not imply conformal invariance, maybe the question can be rephrased. – user566 Mar 9 '11 at 18:14
I'd like to point to a review by Yu Nakayama, available at arxiv.org/abs/1302.0884 – Siva Sep 20 '13 at 7:28

Maybe this does it:
$\begin{array}{rccl} \textrm{Translation:}&P_\mu&=&-i\partial_\mu\\ \textrm{Rotation:}&M_{\mu\nu}&=&i(x_\mu\partial_\nu-x_\nu\partial_\mu)\\ \textrm{Dilation:}&D&=&ix^\mu\partial^\mu\\ \textrm{Special Conformal:}&C_\mu&=&-i(\vec{x}\cdot\vec{x}-2x_\mu\vec{x}\cdot\partial) \end{array}$

Then the commutation relation gives:
$[D,C_\mu] = -iC_\mu$
so $C^\mu$ acts as raising and lowering operators for the eigenvectors of the dilation operator $D$. That is, suppose:
$D|d\rangle = d|d\rangle$
By the commutation relation:
$DC_\mu - C_\mu D = -iC_\mu$
so
$DC_\mu|d\rangle = (C_\mu D -iC_\mu)|d\rangle$
and
$D(C_\mu|d\rangle) = (d-i)(C_\mu|d\rangle)$

But given the dilational eigenvectors, it's possible to define the raising and lowering operators from them alone. And so that defines the $C_\mu$.

P.S. I cribbed this from:

http://web.mit.edu/~mcgreevy/www/fall08/handouts/lecture09.pdf

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Let's say you cited Mr. McGreevy. – mbq Mar 6 '11 at 13:04
The issue is that it just isn't true that scale invariance implies conformal invariance. The simplest counterexample is some self-interacting Levy field theory. – Ron Maimon May 4 '12 at 19:44
What are these $|d\rangle$ kets you refer to? Are they fields? And does $d$ correspond to the scaling dimension of the field? I realise this is an old thread, so I would appreciate comments from anyone. – CAF Jul 26 '14 at 16:36

The rule-of-thumb is that 'conformal ⇒ scale', but the converse is not necessarily true (some condition(s) needs to be satisfied) — but, of course, this varies with the dimensionality of the problem you're dealing.

PS: Polchinski's article: Scale and conformal invariance in quantum field theory.

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It's not a rule-of-thumb that conformal implies scale, it's just a fact. The conditions are mostly locality and low-order of derivatives, which is sometimes imposed by unitarity and renormalizability. – Ron Maimon May 4 '12 at 19:43
@RonMaimon: Conformal invariance requires scale invariance in a Poincare invariant theory simply because of the commutator $[K_\mu,P_\nu]=2i(\eta_{\mu\nu}D-M_{\mu\nu})$. The notation should be obvious. – AndyS Jun 15 '12 at 23:10
@AndyS: The very existence of D in the conformal group is enough to show conformal implies scale--- it's not a rule of thumb, it's an obvious implication, that's what the comment above was trying to say. You don't need the commutator business to show this, the dilatation is a conformal transformation all by itself. – Ron Maimon Jun 16 '12 at 6:51
@RonMaimon: What you're saying is not true; you need the commutator in order to prove what you call "an obvious implication". Also, there is a clear distinction between dilatations and special conformal transformations. – AndyS Jun 19 '12 at 3:23
@AndyS: What I am saying is true, and you are saying nonsense. Dilatations are to conformal invariance as rotations about the z-axis are to rotations. They are a special case. If you have rotational invariance, you have rotational invariance around the z-axis. If you have conformal invariance, you have dilatation invariance. This is not an arguable point, it is not a difficult point, and I don't know why you make the comment above. – Ron Maimon Jun 19 '12 at 15:48

As commented in previous answers, conformal invariance implies scale invariance but the converse is not true in general. In fact, you can have a look at Scale Vs. Conformal Invariance in the AdS/CFT Correspondence. In that paper, authors explicitly construct two non trivial field theories which are scale invariant but not conformally invariant. They proceed by placing some conformal field theories in flat space onto curved backgrounds by means of the AdS/CFT correspondence.

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Thanks for that, somehow I missed this one, it is really interesting. – user566 Mar 9 '11 at 18:14
I knew about this paper but reading this question, made it came to my mind again (fortunately, because it's very interesting) – xavimol Mar 9 '11 at 19:23