On Oct 30, 12:10 pm, "Jay R. Yablon" <jyab...@nycap.rr.com> wrote:
> In the file linked below, I have laid out an exercise upon which I would
> appreciate your feedback:
You're looking at everything completely the wrong way. But it would be
impossible to explain in detail just what or how until you first get
thorough background in the field, itself.
First rule of research: (1) try a problem for yourself for a while
until you get a proper feel of the layout of the territory. This will
hone your search and focus your issues. Second rule (2) don't waste
any time beyond this step; but instead to an EXHAUSTIVE survey of
what's out there (as you're supposed to do when conducting research)
until you've gotten enough to even do an authoritative survey of
what's out there if you wanted to.
Some places to start.
Path Integral Methods and Applications
quant-ph/0004090v1 24 Apr 2000
A lucid account that gets away from the dumb textbook presentations
and explains clearly the link between the path integral the time
ordering operator (a.k.a. the Greens functions).
An Introduction into the Feynman Path Integral
hep-th/9302097v1 20 Feb 1993
Closer to the description I gave in an article a short while back when
discussing LNP 107 (whose section 7 may be thought of as a link of
"classical" version of the path intetegral formalism).
This one (and to a lesser degree the other articles) makes mention of
the general idea:
(A) Splitting formula: K(q+,t+, q-,t-) = integral K(q+,t+,q0,t0) K
(q0,t0,q-,t-) d(q0)
(The recursion, when intervals |t+ - t-| are large
(B) Base Formula "Classical Limit": K(q+,t+, q-,t-) = integral exp((i/
h-bar) S(q+,t+, q-,t-)) up to a factor
The function S is the total action over the interval (t-,t+) expressed
in terms of the boundary coordinates (q+) on the future segment W(t+)
of the hypersurface W's boundary Bdy(W) and the boundary coordinates
(q-) on the past segment W(t-) of Bdy(W).
This ties directly into the description I gave of W in terms of a
timelike foliation in which the layers W(t) are tied off on a common
boundary Bdy(W(t)) = H = Horizon, so that the boundary of W is Bdy(W)
= W(t+) - W(t-).
The base formula is the actual sense in which you have a classical
limit; not the sense you see described in textbooks (of the "classical
path dominating"; although this sense may be thought of as a
corollary).
A New Perspective on Functional Integration
funct-an/9602005v1 8 Feb 1996
(It may surprise you that one of the pioneers in this field is a
woman; and one who worked closely with a well-known female
mathematician who specialized in differential geometry).
Feynman's Path Integral
Definition Without Limiting Procedure
Commun. math. Phys. 28, 47-67 (1972); available on-line.
An earlier 1972 paper by the (female) co-author of the 1996 paper.
Path integral (Scholarpedia)
http://www.scholarpedia.org/article/Path_integral
Path integral formulation (Wikipedia)
http://en.wikipedia.org/wiki/Path_integral_formulation
Partition function (Wikipedia)
http://en.wikipedia.org/wiki/Partition_function_(quantum_field_theory)
Functional integration (Wikipedia)
http://en.wikipedia.org/wiki/Functional_integration
One of the main reasons the path integral formulation was so prevalent
is that because until recently it was the only way to approach the
quantization of gauge theory in the context of perturbation theory
(i.e. renormalization theory) and the only real way to get access to
the issues of the renormalization group, effective Lagrangian,
anomalies (and other issues related to global topology), etc.
So, some other related links:
In Landsman's book (his 1998 "Mathematical Topics Between Classical
and Quantum Mechanics", I think it was, Landsman deals indirectly in
his section II with quantization of particle dynamics on a curved
background. In the process he employs a curved-space form of Fourier
analysis. So add this to the list of references to examine).
Gauge invariance (Scholarpedia)
http://www.scholarpedia.org/article/Gauge_invariance
Gauge theories (Scholarpedia)
http://www.scholarpedia.org/article/Gauge_theories
You should also look up Slavnov-Taylor, BRST, Renormalization Group
Flow, etc.
However, things have changed. For one, as I've spelled out on numerous
occasions in various articles here in the recent past, many of the
relevant issues actually have nothing per se to do with quantum theory
at all, but are grounded firmly in CLASSICAL field theory
(particularly the issues of effective Lagrangians and renormalization
groups and other issues related to these).
But in more recent times, the lid was blown off the top, so to say,
when the Zurich school (i.e. Scharf's people) in the Epstein-Glaser
crowd not only reconstructed "already-renormalized" perturbation
theory for gauge fields almost literally from scratch (this formed a
series of papers in Nuovo Cimento in the mid 1990's which may be
regarded as the launch point of the enterprise, along with Scharf's
19891995 "Finite Electromagnetism" book) but even got as far as being
able to DERIVE the relevant Lagrangians from more fundamental
principles -- which one-up's the effective Lagrangian approach and
Path integral formulation. Scharf even dealt with Slavnov-Taylor in
his 1995 book in the context of the causal approach.
The flurry of activity after the mid 1990's surrounding Scharf
included results such as deriving the symmetry breaking sector of
symmetry breaking theories WITHOUT the need for symmetry breaking;
deriving an enveloping generalization of the standard model
Lagrangian; and going one step beyond the effective Lagrangian
approach even in quantum gravity (such as there exists such a thing).
Around the end of the 20th century Nakanishi and Ojima did a treatment
of quantizing gauge fields from an operator point of view (I don't
have the reference on hand, but the book is titled something along the
lines of "gauge theory from an operator point of view"). No path
integrals involved.
Brunetti and Fredenhagen, as I mentioned, toward the early '00's
expanded the Zurich school's Epstein-Glaser approach by bringing in
the microlocal analysis routines. That's strictly outside the path
integral formulation.
A comparison of path integrals vs. the "time ordering operator
renormalization" that's tacit in Epstein-Glaser-Bogoliubov causal
approach is this:
Path integral: int Dq exp(iS_{Free}/h-bar) (...) = <0| T[...] |0>
-- the T[...] is the UN-renormalized T[].
Causal: the already-renormalized T'[...] defined directly in place
of the Wick T[].
The fact that path integrals give you Wick T[] is directly connected
to their essential quadratic nature (the Wick ordering only handles 2-
point clusters directly and crashes as soon as you get to non-trivial
3-point clusters). That's the genius of the Bogoliubov formula that
heralded the T'[] operator. It correctly handles 3-point and higher-
point clusters at the get-go without need for further
"renormalization" or "infinity-extracting".
In effect, it does both what the path integral is trying to do and yet
gets you the effective Lagrangian all in one step. Bogoliubov's
account of the effective Lagrangian is just as lucid as that arrived
above:
Small interval K(t+,t-) = integral (iS/h-bar) base case
Large interval K(t+,t-) defined recursively, produces integral(iS_
{effective}/h-bar)
One of the arXiv papers I think completely does away with the Fourier
analysis part of the textbook treatment.
On the more general note: the real deal about what's going on with
Wick rotation is that you're transforming from a Lorentzian to a
Euclidean signature. The problem with curved backgrounds is that
unlike the case of flat spacetime, there is no canonical one-on-one
correspondence between spacetimes of the two signature types. The best
you can do (which is what Hawking did) was restrict focus to a SUB-
class of curved backgrounds (e.g. the Schwarzschild metrics) and link
these one-on-one to locally 4-Euclidean geometries.
What lies at the bottom of this problem has nothing whatsoever to do
with the question of extending Fourier analysis to curved spaces.
Rather, the key question is how do you get from Lorentzian to
Euclidean signatures. In the absence of a one-on-one link between
spaces of the two signature types, what you really need is a way that
effects a CONTINUOUS transition from Lorentzian theories (i.e. the
whole infrastructure underlying field theory, gravity, geometry, etc.)
to Euclidean theories.
Of necessity, this involves passing through a transition which is
neither of these signature types. If all 3 spatial dimensions are
treated on the same footing, then there are only two ways to undergo
this transition: either
(a) c -> infinity -- the Galilean limit. The in-between geometries are
those whose flat-space cases respect the invariants dt^2 and Del^2.
These are the spacetimes of non-relativistic theory.
(b) c -> 0 -- the Archimedean limit. These in-between geometries are
those whose flat-space versions respect the invariants dr^2 (Euclidean
distance in 3-D) and (d_t)^2 (time flow; i.e. motion). In these
spaces, there is a notion of absolute motion and absolute rest; as
well as absolute spatial distance. These are the geometries of
Hellenistic philosophers of old -- the Archimedean signature.
So, to consistently round out a curved space formulation for path
integrals that goes the route of trying to generalize the Wick
rotation to curved settings; you're directly confronting head-on the
issues of the Galilean Limit (c -> infinity) and Archimedean Limit (c -
> 0).
These are also representative, respectively, of the boundary
thresholds corresponding to Big Bang type cosmological singularities
(in which in the metric ds^2 = A dt^2 - B dr^2, the ratio B/A goes to
0, so that effectively c^2 -> infinity); and the thresholds known as
"horizons" (where in the metric ds^2 = A dt^2 - B dr^2, the ratio A/B
goes to 0, so that c^2 -> 0 in effect).
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