constraint on scaling dimension

Question

How can we show that for any scalar operator $\Delta\geq1$ (where $\Delta$ is the scaling dimension)? Where can I find a reference for reading where it comes from?

Answer 1

This is a consequence of the Lehman spectral representation for a physical scalar operator. The two point function of this operator (the expected value of the operator with it's conjugate) can be written as an integral over propagators:

$$ \langle \bar{\phi}(p)\phi(p') \rangle = (2\pi)^d\delta^d(p-p') \int_0^\infty {\rho(s)\over p^2 - s + i\epsilon} ds $$

Where each propagator falls off as ${1\over x^2}$ at short distances in 4 dimensions, and $\rho(s)>0$ for all s (because of Hilbert space positivity--- this is the norm of a state, namely $||\phi(p)|0\rangle||$).

A superposition of positive propagators falling off as ${1\over p^2}$ with positive coefficients annot produce a falloff at large p which is faster than ${1\over p^2}$. This means that the asymptotic scale dimension of the scalar operator can't be less than 1 in 4 dimensions, it can't be less then 1/2 in 3 dimensions, and it can't be negative in 2 dimensions.

This is not exactly mathematically true, because you can engineer a spectral weight which is growing near s=0 as a power law, to produce faster than 1/p^2 falloff. But it is physically true anyway, because such a growth requires an infinite number of particle species at p=0, which is inconsistent with the usual idea that a quantum field theory has a finite number of elementary fields, with a finite thermal entropy. The way to understand this is that superposing any finite tower of particles with positive spectral weights always leads to 1/p^2 falloff or slower, and a ${1\over x^a}$ propagator with $a\le 2$

The Kallen-Lehman spectral representation is a standard field theory result, it is found in most standard textbooks. The original paper is reprinted in Schwinger's reprint volume "Quantum Electrodynamics".