Discussion of Path Integral Quantization, and its Limitations
Jay R. Yablon
action, Mark Hopkins and I have been having a discussion at
sci.physics.research which has migrated over to issues about path
integration, and specifically about how it may (or may not) be limited
as a useful method for quantization. I would like to reproduce my most
recent post below, so that we can also discuss these questions here.
Thanks. Jay
"Rock Brentwood" <mark...@yahoo.com> wrote in message
news:94d92ab5-a4c1-4451...@g6g2000vbr.googlegroups.com...
Hi Mark,
I have some preliminary questions regarding the end of your point (4)
and point (5), as excerpted below:
> Some comments on your article and reply:
. . .
> (4) On the inequivalence issue:
. . .
>
> In the Hamiltonian dynamics of gauge theory (both classical and
> quantized), the selection of an appropriate Lagrangian is not only a
> major issue; it's THE major issue.
>
> (5) Finally, the answer to how far you can get to not know the
> explicit form of the Lagrangian for the path integral method is -- not
> very far. In fact, the assumption of the specific form of the
> Lagrangian is built right into the method. The moment you employ the
> gaussian trick, you're already fixing the Lagrangian to be quadratic
> homogeneous in the field strengths (i.e. the Maxwell-Lorentz
> Lagrangian or its non-Abelian generalization, the Yang-Mills
> Lagrangian). At the very least you're doing something analogous to a
> "mean field" expansion about a quadratic homogeneous Lagrangian.
>
> The path integral method, itself, is not well-founded: the
> Osterwalder-
> Schrader Theorem (sp?) - which is the basis of the path integral
> formalism - only applies to flat-spacetimes. There is no known
> formulation of path integrals to curved spacetimes. More general, more
> robust formalisms should be used that don't suffer this defect.
>
> You may endeavor to limit your exercise to flat spacetime (and further
> limit it to only those foliations of flat spacetime that have flat
> foliation layers), but then the question is: "To what end, other than
> self-edification? In particular, to what end that is not already
> superseded by more robust approaches?" There are already methods that
> generalize both perturbation theory and renormalization theory (and,
>thus, also effective field theory) to curved spacetimes.
>
Hi Mark,
Let's write the path integral in the general form ($=integral, == means
"defined as"):
Z = $D psi exp i$[d^4x L] == C exp i[W] (1)
Here, psi is the field, L is the Lagrangian density, and W is an
"amplitude expression." (Perhaps there is a more precise name that W is
given.) S=$[d^4x L] is the action.
This, in effect, is a *definition* of W. If one has some Lagrangian
density L (recognizing your statement that "the selection of an
appropriate Lagrangian is not only a major issue; it's THE major
issue"), then the only question left is to do the mathematics (which can
be formidable if not daunting) to obtain an exact expression for W as
defined in (1).
For example, suppose with used as our Lagrangian density the Einstein
Hilbert action:
S = $ [(.5/kappa)R + L_m] sqrt(-g) d^4x (2)
where L_m == L_matter, see
http://en.wikipedia.org/wiki/Einstein%E2%80%93Hilbert_action. This
corresponds wholly to the Einstein equation, it is non-linear, it takes
account of gravitational fields and accounts for curved spacetime, and I
assume you would regard this as an "appropriate Lagrangian" and if not
would explain why not.
IF one was able to then use (2) in all of the gory non-linear detail
that is hidden in (2), and be able to do the mathematics so as to
ascertain an EXACT expression for W in (1), then W would be the exact
amplitude expression for gravitational integrations and would inherently
impose quantum constraints upon gravitational theory which would enable
us to understand the manner in which gravitation is quantized. Or, at
least this seems to me to be so, in principle.
Now, let's look at the "Gaussian trick," which uses the mathematical
Gaussian identity:
$ d psi exp[-V(psi) - .5 K psi^2 + J psi]
= C exp[-V(delta/delta J)] exp[.5 K^-1 J^2] (3)
Of course, this contains quadratic terms homogeneous in the field
strengths and so may be akin to the drunk looking for the quarter under
the lamp post because "that's where the light is." But the use of the
terms V(psi) and V(delta/delta J) do provide the mathematical means to
solve the path integral (1) for W *exactly*, even if we cannot put the
solution into closed forms and have to deal with a double infinite
series from which we extract our Greens functions and take our Feynman
diagrams. Thus, these terms do not restrict us to any particular
Lagrangian form *other than* that we have the Lagrangian density terms
.5 K psi^2 + J psi off of which to do an expansion. Once we have .5 K
psi^2 + J psi in our Lagrangian density, we can solve the path integral
exactly, in principle, no matter what other terms may be in there.
In gravitational theory:
T_uv = -2 (delta L_m / delta g^uv) + g_uv L_m (4)
so that multiplying through by g^uv:
4 L_m = g^uv T_uv + 2 g^uv (delta L_m / delta g^uv) (5)
and the g^uv T_uv gives us the J psi term needed for the Gaussian trick.
Certainly, there are terms in L with psi^2 ~ g_uv ^2.
Thus, unless you can argue that having at least the terms .5 K psi^2 + J
psi in a Lagrangian density, plus any and all other terms of any order
in the fields, is so badly limiting as to exclude real Lagrangian
densities of real physical interest which do not have the terms .5 K
psi^2 + J psi, is seems to me that (1)-(3) above do provide a useful
vehicle -- in principle -- to perform a path integral quantization of
all known theories of physical interest, including gravitational theory
in curved spacetime, even if *calculating* the precise mathematical
solution or consolidating the solution into a closed form is still a
daunting challenge.
I'd appreciate your thoughts on these reflections.
Thanks,
Jay
PS: You wrote elsewhere in the thread "I've written this up as a PDF as
well. Drop me a note and I'll send a copy; you're free to archive it."
Did you get my email?
____________________________
Jay R. Yablon
Email: jya...@nycap.rr.com
co-moderator: sci.physics.foundations
Weblog: http://jayryablon.wordpress.com/
Web Site: http://home.roadrunner.com/~jry/FermionMass.htm
Ken S. Tucker
(I've been following the discussion in sp research)
On Oct 11, 9:54 am, "Jay R. Yablon" <jyab...@nycap.rr.com> wrote:
> Out of the thread below about Integration by Parts of the Maxwell QED
> action, Mark Hopkins and I have been having a discussion at
> sci.physics.research which has migrated over to issues about path
> integration, and specifically about how it may (or may not) be limited
> as a useful method for quantization. I would like to reproduce my most
> recent post below, so that we can also discuss these questions here.
> Thanks. Jay
>
> "Rock Brentwood" <markw...@yahoo.com> wrote in message
I think our pal P.G. Bergmann in "Intro to ... Relativity" has done
a nice readable effort beginning on pg.140, "Sommerfeld's theory
of the hydrogen fine structure", which is a bit hilarious, then onward
past the 'second quantum condition' to "De Broglie waves", let me
quote from pg.144, "Let us now transform the wave function (psi).
We shall assume that (psi) is a scalar."
Well that assumption seems to be quite a time saver ;-).
Anyway I think the articles are worth a glance.
Regards
Ken S. Tucker
PS: Bergmann is one of my fav authors.
...
Ken S. Tucker
> I think our pal P.G. Bergmann in "Intro to ... Relativity" has done
> a nice readable effort beginning on pg.140, "Sommerfeld's theory
> of the hydrogen fine structure", which is a bit hilarious, then onward
> past the 'second quantum condition' to "De Broglie waves", let me
> quote from pg.144, "Let us now transform the wave function (psi).
> We shall assume that (psi) is a scalar."
Let me do a brief quote of the hilarity I mention, from pg.141,
"But it so happens that the two errors - the neglect of wave mechanics
and the neglect of spin - cancel each other in the case of the
hydrogen
atom, and so Sommerfeld's treatment which was based on two errors,
led to the correct result".
Isn't theoretical physics fun!
Regards
Ken