What is Quantum Field Theory?
Charles Francis
------- Forwarded message follows -------
In article <8pmgj3$sh2$1...@nnrp1.deja.com>, thus spake alfred_einstead@my-
deja.com
>The S-matrix expansion is supposed to provide a perturbation expansion
>of the exact solution.
>
>The exact solution of what?
I always thought it was the solution of the interacting Dirac equation
in the case of QED, and some equation with a different interaction term
more generally, but I seem to have got into trouble recently for
thinking that.
Surely the real point of S matrix theory as it used to be taught was
that you did not have to know what the process was that you were
solving, and instead looked to general constraints such as analyticity.
>I've never seen a clear formuation of what precisely the actual
>equations of quantum field theory are, that are supposedly being solved,
>which the perturbation expansion is supposedly an expansion of.
>
>What do the actual equations look like and (MUCH more importantly!) in
>which mathematical space are the expressions and solutions supposed to
>reside?
I think that is the unsolved issue. I fear too that it must remain
unsolved, because it seems to have been decreed that the solution must
satisfy the Garding-Wightman, Haag-Kastler or Osterwalder-Schroder
axioms, and I don't find it reasonable to suppose that that is
necessarily the case, or even to think that any structure does satisfy
any of these axioms. So if anyone does claim to have found a space in
which the perturbation expansion is meaningful as a solution of an exact
equation, it can immediately be said that they have solved the wrong
problem.
{actually since I originally wrote that and it vanished into the ether I
have found that the terms of the millenium prize are "axioms at least as
strong as" GW or OS, which is rather more reasonable}
>
>It's supposed to be a system of some kind of partial differential
>equations over a non-numeric mathematical structure (which brings to
>mind the question of exactly what a 'derivative' is in this structure).
>
>In the classical theory, which is the pre 2nd-quantized approximation of
>the actual theory, a set of equations are derived by the principle of
>least action. However, in the 2nd quantized theory, no such principle
>can be clearly stated since the definition of 'least' is not clear
>without an ordering structure on the underlying mathematical space. If
>there's a Lagrangian, then what exactly is the action which is being
>minimized?
Quite. Even in the classical mechanics of the 19th Century the action
principle was not taken seriously (by many) as a piece of metaphysics,
but was mainly just a step on the way to a neater way of calculating
solutions to the many body problem for the purpose of astronomical
tables. This only gained favour after the advent of qm, when it was
noticed that there was a similarity between the Hamiltonian formulation
of classical mechanics and the Schrodinger equation. But apart from the
fact that this formulation has sometimes given clues about quantisation
(which in my view can be done much more rationally by other arguments)
there really seems to be no logical reason for giving it the priority it
has in modern physics.
--
Regards
Charles Francis
cha...@clef.demon.co.uk
--
Regards
Charles Francis
cha...@clef.demon.co.uk