## Re: Normal ordering and VEV subtraction & simpler definition of normal

> I'm looking at Wald's book on QFT in curved spacetime.

[which works within the algebraic approach].

> In a few simple cases, the normal ordering and the point splitting
> prescriptin can be applied explicitly and the results agreee. But this
> is still somewhat mysterious to me. Wald claims that "for physically
> reasonable states in the standard Fock space, the singular behavior of
> the bidistribution $<\phi(x)\phi(y)>$ is the same as for
> $<0|\phi(x)\phi(y)|0>$" (paraphrasing slightly).

Reasonable assumption is that the state space is
${ A|0>: A$ from a class of well-behaved operators }
Then for any state $A|0><\phi(x)\phi(y)> = <0| A* \phi(x) \phi(y) A |0>$
and the A's should have no effect on the singularity behavior.

A simpler prescription which subsumes Wald and everyone else -- but
which apparently eluded Wald and others -- is to just define
$N(\phi(x)\phi(y)) =$ unique (up to constant) operator Z
such that Ad(Z) $= Ad(\phi(x)\phi(y))$.
Note that even though $\phi(x)\phi(y)$ may be singular, it's *adjoint* is
generally not. Various authors try to state this point by pointing out
that the infinity is "only" a c-number, meaning that it isn't even
there under the Ad(). If ths spectrum of $\phi(x)\phi(y)$ is bounded, the
constant is fixed by convention of setting the lower bound to .

In an irreducible representation, the only operator A with Ad(A) = is
a c-number. Therefore, the above prescription does indeed uniquely
specify the operator ... assuming that a solution exists.

The motivation for the foregoing is simply that nearly the only place
(at least in QFT) the stress-energy tensor is used is in the context of
the Lie bracket and the (generalized) Heisenberg equations:
$[P_a, O] = i$ h-bar $dO/dx^a$.

In general terms, the concept of adjoint-equivalence allows you to
consider a larger class of operators which are all equal to the
original class of bounded, well-behaved operators, modulo the
equivalence. For these operators, normal ordering defined via
adjoint-equivalence then specifies ... up to a constant ... a unique
bounded operator which behaves the same as the original operator in all
contexts involving the commutators.