What is generalized free field?

Question

I came across the term generalized free field in a paper recently but I don't know its definition. Google leads to other papers which take it for granted and use it without defining it. It appears that O.W. Greenberg introduced the term in the paper Generalized free fields and models of local field theory, Ann. Physics 16. Unfortunately I can't access this paper. Can someone please explain it to me or give me a link to that paper? Thank you.

Answer 1

A generalized free field is one for which (modulo field redefinitions) the connected $n$-point functions $G_n(x_1,...,x_n)$ vanish whenever $n > 2$. This means, basically, that a generalized free field is one for which the Euclidean functional measure is Gaussian. The field is completely specified by its 2-point functions $G_2(x,y)$.

Generalized free fields are usually discussed using the parametrization given by the Kallen-Lehmann decomposition of the 2-point function $G_2(x,y)$, which says that (for scalar fields)

$G_2(x,y) = \int_0^\infty d\rho(m) \Delta_m(x-y)$,

where $\Delta_m(x-y)$ is the real-space propagator for a free real field of mass $m$ and $\rho$ is a positive measure.

This parametrization makes it easy to write down examples of generalized free scalar fields. Just pick a positive measure $\rho$ on the mass line $[0,\infty)$. The simplest example is the free field of mass $M$, which corresponds to $\rho(m) = \delta_M(m)$, the Dirac delta function supported at $M$. You get other examples by picking other measures. For example, you can take a purely continuous measure, like $d\rho(m) = \Theta_M(m)dm$ or $d\rho(m) = m^2dm$. (Here, $\Theta_M(m) = 0$ if $m <M$ and $1$ otherwise.)

These examples where the measure has continuos support are somewhat difficult to think of in the usual Lagrangian formalism; it's as if you have a continuum of fields of different mass which are all constrained to move together. But giving the correlation functions is enough to define a field theory; you can use Wightman's reconstruction theorem to recover the Hilbert space and the field operators.

If you're working in axiomatic field theory, then you usually impose some sort of growth conditions on your correlation functions. In the case of generalized free fields, these growth conditions translate into conditions on $G_2$, or equivalently on $rho$. For example, if the free field's Euclidean measure obeys the Osterwalder-Schrader axioms, then $\rho$ must be a tempered measure of polynomial growth at $\infty$ (and not too unreasonable behavior near $0$).