Mathematically-Rigorous Path Integration??
Jay R. Yablon
Following up some recent discussions here with such luminaries as Dr.
Neumaier, Peter, J. Thornburg, X-Phy, P. Helbig, and of course, the
irrepressible Igor K., ;-) I have tried rolling up my sleeves and diving
into the problems that have been pointed out about the ill-defined
nature of the path integral, to see if I could make some headway in
cleaning things up. I have posted my efforts for review and feedback
at:
http://jayryablon.files.wordpress.com/2009/12/rigorous-path-intergation.pdf
For sake of this discussion, I have also excerpted two pages from each
of Zee's QFT in a Nutshell, and Sakurai's Modern Quantum Mechanics, and
posted these in a single PDF file at:
http://jayryablon.files.wordpress.com/2009/12/sakurai-and-zee.pdf.
In summary, and seconding what Dr. Neumaier and Igor in particular have
been pointing out, it appears from my vantage point that the calculation
of the path integral in the form:
Z = ${-oo to +oo}Dq exp [iS] (1)
is really only "half" a calculation, in which the "ugly" terms are
gathered up and "swept under the rug" in Dq, and not ultimately dealt
with, including the mathematically-undefined infinite-dimensional
integral:
$...$$$ dq_0 dq_i dq_2 ... dq_oo, (2)
the pathology of which Igor has highlighted in prior discussion. In
particular, it seems very clear that Dq is a "faux" element of
integration, which really is a "rug" under which the ills of path
intergation are swept, and which does not have the rigorous calculus
meaning of, say, the usual integration element dq. The "handwaving"
which Dr. Neumaier has earlier referred to, appears to me, to occur when
one treats "D" as if it was "d" when doing integration, when is simply
is not a true, rigorous "d."
In essence, what I have attempted here, is to take everything back out
from under the Dq "rug," and complete the other "half" of this
calculation without sweeping anything "under the rug" into Dq, in a
mathematically rigorous fashion consistent with the limit-based
definition of Riemannian integration, and then redefined the transition
amplitudes W(J) in a way that places them as on a firm mathematical
footing of real integration based on properly taking limits and
resolving the nasty infinite products.
To summarize the "new" development, after taking everything "out from
under the rug" in Section 5, it is section 6 in which I carry through
the calculation with all of the "ugly" stuff from Dq included, and show
by a careful consideration of the infinitesimal limit, that in fact,
$Dq=1. Given that, a slight adjustment to the definition of the
transition amplitude W(J) is required, to place this as well on a
rigorous foundation. Section 1 is introductory, section 2 and 3 focuses
on integration in finite and infinite dimensional spaces based on
Sakurai's treatment, to ensure that even the single integral $dq in the
completeness relationship
I = ${-oo to +oo} dq |q><q| (3)
is introduced on a rigorous foundation. Section 4 carries through the
"customary" development of path integration.
I look forward to your comments, and to further discussion of these
foundational questions.
Happy holidays!
Jay
____________________________
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
Jay R. Yablon
good discussion, which is worth reviewing for anyone interested in this
topic. I'd like to transplant some of that over here, below:
"Jay R. Yablon" <jya...@nycap.rr.com> wrote in message
news:7pcu43...@mid.individual.net...
>
> "Robert Israel" <isr...@math.MyUniversitysInitials.ca> wrote in
> message
> news:rbisrael.20091222002353$0d...@news.acm.uiuc.edu...
>> "Jay R. Yablon" <jya...@nycap.rr.com> writes:
>>
>>> Dear Friends,
>>>
>>> Following up some recent discussions in sci.physics.reseacrh with
>>> such
>>> luminaries as Dr. Neumaier, Peter, J. Thornburg, X-Phy, P. Helbig,
>>> and
>>> of course, the irrepressible Igor K., ;-) I have tried rolling up my
>>> sleeves and diving into the problems that have been pointed out
>>> about
>>> the ill-defined nature of the path integral, to see if I could make
>>> some
>>> headway in cleaning things up. I have posted my efforts for review
>>> and
>>> feedback at:
>>>
>>> http://jayryablon.files.wordpress.com/2009/12/rigorous-path-intergation.pdf
>>
>> Rather than re-inventing the wheel, why don't you look at what
>> mathematical
>> physicists have already done? You might look at
>>
>>
>> Glimm and Jaffe, "Quantum Physics: A Functional Integral Point of
>> View",
>> Springer-Verlag 1981, and
>>
>> Simon, "Functional Integration and Quantum Physics", Academic Press
>> 1979.
>> --
>> Robert Israel isr...@math.MyUniversitysInitials.ca
>> Department of Mathematics http://www.math.ubc.ca/~israel
>> University of British Columbia Vancouver, BC, Canada
>
> Thank you for these references.
>
> However, from what I understand after some extensive discussion in
> sci.physics.research with some folks who have extensive knowledge
> about
> this (you should check those threads), there remains a mathematical
> problem which has not yet been solved despite years of effort by many,
> of giving a rigorous calculus limit foundation to the path integral,
> because of the infinitely-dimensional $...$$$dqdqdq...dq integral that
> is swept into the "integral over paths" Dq.
>
> I strongly suspect that this problem is not solved by these
> references,
> otherwise people would not still be writing textbooks and papers 25 or
> 30 years later with this problem still not resolved, and the
> knowledgeable folks at sci.physics.foundations would not be talking
> about how path integration, while useful, "couldn't be made logically
> consistent, in spite of many attempts by some of the best
> mathematicians
> and physicists." (quote from A. Neumaier)
>
> Jay
>
Since the above initial exchanges, what has become apparent to me from
some private exchanges and further development in that spr thread is
that Glimm and Jaffe is something of the leading edge reference on exact
two-dimensional solutions to path / functional integrals, but that the
calculation is very involved even in 2D including renormalization by
subtraction of an infinite constant, and there is no clear or easy
generalization to higher dimensions.
I am hoping to get my hands on this book during the next week, but what
everything I have read so far lead me to infer, albeit with Glimm and
Jaffe sight unseen at this point, that my own 2D derivation at
http://jayryablon.files.wordpress.com/2009/12/rigorous-path-intergation.pdf
(which is Euclidean insofar as nothing there enforces a t,q=+,-
signature), if it is correct (and it is in section 6 that I suspect this
will either pass or fail), is different than what has been done to date,
because it does retain the intuitive property of a sum over all paths,
and because it gives a closed exact expression for the integral over
path by the straightforward application of calculus limits and does not
require any sort of renormalization or laborious mathematics.
While one might doubt that there is a simple shortcut of this sort, do
keep in mind, that nature at bottom is simple and elegant (or at least
that has been the running hypothesis for the last century), even if we
humans often have difficulty finding the simplest way to mirror nature's
inherent simplicity in our symbolic formulations.
I look forward to further discussion of everyone's thoughts on path
integration, how far it has been made rigorous to date, whether those
rigorous derivations which exist to date in limited contexts seem
natural or unnecessarily labored, where its it still on a shaky
foundation, and why.
Jay
Arnold Neumaier
[[Mod. note -- 65 excessively-quoted lines snipped. -- jt]]
> Since the above initial exchanges, what has become apparent to me from
> some private exchanges and further development in that spr thread is
> that Glimm and Jaffe is something of the leading edge reference on exact
> two-dimensional solutions to path / functional integrals, but that the
> calculation is very involved even in 2D including renormalization by
> subtraction of an infinite constant, and there is no clear or easy
> generalization to higher dimensions.
>
> I am hoping to get my hands on this book during the next week, but what
> everything I have read so far lead me to infer, albeit with Glimm and
> Jaffe sight unseen at this point, that my own 2D derivation at
> http://jayryablon.files.wordpress.com/2009/12/rigorous-path-intergation.pdf
> (which is Euclidean insofar as nothing there enforces a t,q=+,-
> signature),
Euclidean path integrals are essentially Wiener integrals and are
well-understood as limits of sums over paths, or by a variety of
other constructions.
The difficulty is in the Wick-rotation to the Minkowski space.
This involves analytic continuation. It is precisely this
- to show that the domain of analyticity is large enough to
permit the rotation - that requires the bulk of the work.
Arnold Neumaier
Jay R. Yablon
news:4B37CE3...@univie.ac.at...
So, to start this off simply, is the path integral developed up to the
Dq by Zee in
http://jayryablon.files.wordpress.com/2009/12/sakurai-and-zee.pdf (third
and fourth pages of the PDF) Euclidean or Minkowskian, and why? How, in
that context, would one convert between one and the other? What would
be the obstacles with making a t-->it continuation to go back and forth,
and why? Jay.
carlip...@physics.ucdavis.edu
[...]
> Euclidean path integrals are essentially Wiener integrals and are
> well-understood as limits of sums over paths, or by a variety of
> other constructions.
> The difficulty is in the Wick-rotation to the Minkowski space.
> This involves analytic continuation. It is precisely this
> - to show that the domain of analyticity is large enough to
> permit the rotation - that requires the bulk of the work.
There are some cases in which the Lorentzian path integral can be
defined via promeasures. There's a discussion of this, and of other
related mathematical issues, in the new book by Cartier and DeWitt-
Morette, _Functional Integration_ (though the relevant bits are
scattered around a little).
Steve Carlip
Mike
<Arnold.Neuma...@univie.ac.at> wrote:
>
> The difficulty is in the Wick-rotation to the Minkowski space.
> This involves analytic continuation. It is precisely this
> - to show that the domain of analyticity is large enough to
> permit the rotation - that requires the bulk of the work.
>
The Wiener measure was developed from the physical considerations of
Brownian motion. But I seem to have developed the path integral by
simply using a recursion relation for the dirac delta function,
http://hook.sirus.com/users/mjake/delta_physics.htm
The dirac delta function is also a measure, and when the gaussian form
of the dirac delta function is used, it appears that the
exponentiation of the action integral is actually part of the dirac
delta measure. So the Wiener measure is really the dirac delta
measure. Note, I used a complex variance to get Feynman's path
integral, but if the variance were real the Wiener measure results.
My question is does the gaussian form of the dirac delta still remain
a measure even if its shrinking variance parameter is purely
imaginary? If it does, then the recursion relation applied to it
results in a well defined measure for the Feynman path integral.
Jay R. Yablon
. . .
> I am going to get my hands on the Glimm and Jaffe, and Cartier and
> DeWitt-Morette references this week, and review them while I am away
> next week, to see where my calculation (and yours) fits in the context
> of prior leading edge work in this area.
>
> Best,
>
> Jay
I just today got Glimm and Jaffe (second edition, 1987). At the link
below, I have posted a two page excerpt pertinent to this discussion:
http://jayryablon.files.wordpress.com/2010/01/glimm-and-jaffe.pdf
It is my understanding from some private communications that Glimm and
Jaffe presents what is even today, our leading edge understanding of
path integration and of efforts to develop path integration with
mathematical rigor. While I will be devouring this book on my vacation
next week, based on the excerpt above and some other initial perusal of
this book, as well as my having been advised that this book still does
largely represent the "state of the art" on path integration, I am of
the belief that my work posted at
http://jayryablon.files.wordpress.com/2009/12/rigorous-path-intergation.pdf
does advance path integration beyond this state of the art, and shows
how to make *Minkowski space* path integrals in four spacetime
dimensions, mathematically rigorous. Here is my preliminary
articulation of why I believe this is so:
1. Equation (3.1.4) Glimm and Jaffe (G&J) is the equivalent of what is
equation (4.7) in my derivation above. I work from the same Hamiltonian
operator (3.1.2), see the third line of my section 4, and have the same
action as G&J (3.1.3), in my (4.10). (I believe the mass m is implicit
in G&J (3.1.3).) The key statement by G&J, after (3.1.4), is as
follows: "The complex measure [the integrand in (3.1.4)] has not been
given a satisfactory mathematical meaning, *and for this reason
Feynman's formula has not played a larger role in the mathematically
rigorous treatment of quantum mechanics.*" This may be taken, IMHO, as
another way of saying what A. Neumaier and I. Khavkine have said
elsewhere about path integration.
2. Fast forward to 2006. The Cartier and DeWitt-Morette (CDM)
reference provided earlier in the thread by S. Carlip (which can be
viewed in part online at
http://www.amazon.com/Functional-Integration-Symmetries-Monographs-Mathematical/dp/0521866960/ref=sr_1_1?ie=UTF8&s=books&qid=1262752075&sr=8-1#reader_0521866960)
arrives at the same juncture. Immediately on pages 3-4, it is noted
that "the Wiener integral [1.1] . . . becomes the Feynman integral [1.2]
if one sets tau=it. Kac concluded that, *because of the i in the
exponent, Feynman's theory is not easily made rigorous.*" Yet another
restatement of A. Neumaier and I. Khavkine in this newsgroup. My
conclusion is that the current state of the art, is that to this day it
is still not known how to do rigorous calculations with Feynman path
integrals, especially in four dimensions.
3. The best approach that is presently known, as far as I can tell, is
to use analytic continuation by replacing t with -it, see just before
(3.1.5) in G&J. I also note that in this context, one can also take q
to represent three space dimensions, see G&J after (3.1.6) and also
between (3.1.7) and (3.1.8). However, the mathematics of doing the
analytic continuation, and converting back and forth, is extremely
complicated, and in essence, becomes the main topic of at least G&J, and
perhaps both of these books. Or, more to the point, and to avoid any
overstatement, if it were possible to give (3.1.4) in G&J, or (1.6) in
CDM, or (4.7) in my post a "satisfactory mathematical meaning," then the
reams of calculation following in both G&J and CDM would be rendered
unnecessary, because the need to even do these calculations in the first
place is conditioned on the fact that nobody knows how to calculate the
Feynman integral in a rigorous fashion and so needs to do a "sidestep,"
find other ways to do a rigorous calculation. And, from what I can
tell, the rigorous calculations that have been done to date, still do
not teach us how to ascribe a rigorous meaning to a four-dimensional
path integral in Minkowski space -- which is the space of the real,
physical world.
4. In my post at
http://jayryablon.files.wordpress.com/2009/12/rigorous-path-intergation.pdf,
my expression (4.7) a.k.a. (3.1.4) in G&J or (1.6) in CDM, is readily
manipulated into my (5.11). Here, I have simply taken out all of the
ugly parts of Dq which are ordinarily swept under the rug, and put them
back into the expression for <q_N|exp[-iHT]|q_0>. I have retained the
imaginary exponent, and so am not trying to solve the problem through
analytic continuation but rather am staying seated "on the bronco" and
trying to calculate through the path integral in Minkowski space rather
than Euclid space. Of course, (5.11) is still an ugly expression, and
it appears to me that nobody has solved this exactly, which would
require among other things figuring out how to correctly take the limit
where the dimensionality of the dqdqdq...dq in Dq is infinite and the
delta t limit is set to zero.
5. A key step in my derivation is (6.3), which finds that that with
delta t = 0, we have (p=m dq/dt):
<q|q> = 1 = $ i i^.5 (1/hbar) (p/q)^.5 dq (my 6.3)
This is the Minkowski space equivalent of G&J's ii. on page 45 in the
linked excerpt above, *but it is taken in Minkowski space rather than
Euclid space.* This means, following G&J's equation iii. on page 45,
that
i i^.5 (1/hbar) (p/q)^.5
may be interpreted as the probability density in Minkowski space, as a
given fixed time t. Because sqrt(2)i^.5=1+i, this means we have to
accept, mathematically, a complex probability density. But, keep in
mind as well, that when we use operators, [q-hat,p-hat]=ihbar, so we
ought not get too exercised about imaginary or complex factors showing
up at this stage, in an expression which involves q and p eigenvalues.
6. The next key step, is to take advantage of (6.3) in solving the
entire path integral. In particular, with L being the Lagrangian
density, we have to somehow put the exp[i L dt] back into (6.3). The
trick here, especially since we want L to apply to *interacting fields*
(i.e., with a potential V <> 0), and so need for L to be a function of
both q and q-dot, i.e., L = L(q,q-dot), is to convert over to a variable
of integration -- time -- which allows this exponential to become part
of the integrand. I show in (6.4) to (6.7) how to do this, with the
result being (6.8). In essence, section 6 contain the heart of the
"difficult" part of my derivation.
7. From my (6.7), it is s short hop to obtain the overall result, for a
Minkowski path integral, in my (6.11), that:
<q_0|exp(-iHT)|q_0> = exp[i$dt L] = $Dq exp[i$dt L] (6.11)
from which we extract the overall result that (see also my (4.8), which
shows the usual $Dq):
$Dq = 1
[= lim(N-->oo)(-i 2 pi m/delta t)(PI{0 to oo}$dq)]
I have seen it asserted that the integral over paths "ought to be" equal
to 1. But this derivation rigorously shows it to be equal to 1. And,
for the same reasons that one can regard q to be three-dimensional in
the Weiner integral, see again G&J after (3.1.6) and (3.1.7), I believe
that one can regard q here, to also be three dimensional. (If there are
no problems with the foregoing, my next update of the linked file above
would seek to show the full four-dimensional calculation explicitly.)
So, perhaps this is an exact, rigorous solution of the four-dimensional
Feynman path integral in Minkowski space, which appears to have not been
heretofore obtained. Or, at least a two-dimensional Minkowski space
solution which is far simpler and much more direct than anything yet
obtained, relying on nothing more complicated than ordinary Riemannian
integration and Newton-Leibnitz limits.
Jay
Jay R. Yablon
news:a7cbc7b4-eb6b-415e...@u41g2000yqe.googlegroups.com...
Mike,
On cursory look, the calculation seems OK, but a couple of questions
still
lurk:
1) You employ the Dirac delta as a "function" starting at (3). I
wonder if
the math majors here will regard this as a problem, since technically,
the
delta is not a function.
2) Your (6) is one way to define the delta from a nascent delta. But,
it
is not exclusive. There are a number of different ways to do so, so you
need to explain why your (6) is the way we have to go. Many road lead
to
Roam, and here, a number of different nascent deltas lead to the same
impulsive delta.
Nonetheless, it is an interesting calculation, and as you know, I think
that
infinite products of deltas are worth exploring.
In
http://jayryablon.files.wordpress.com/2009/12/rigorous-path-intergation.pdf,
see (6.11) and (7.5), I ditch the Dq entirely, by explicitly calculating
it
out, which AFAIK has not been done before. I previously said,
incorrectly,
that my path integral was Euclidean. It is Minkowski, because it has an
exp(iS), and the action is taken to be real. Of course, one can then
use
analytic continuity to introduce an imaginary component to the action,
but
my basic calculation gives a precise, well defined mathematical
expression
for a Minkowski / Lorentz path integral, in two dimensions, which I
believe
can be extended to more dimensions.
Mike
> "Mike" <mj...@sirus.com> wrote in message
> > The Wiener measure was developed from the physical considerations of
> > Brownian motion. But I seem to have developed the path integral by
> > simply using a recursion relation for the dirac delta function,
>
> >http://hook.sirus.com/users/mjake/delta_physics.htm
> On cursory look, the calculation seems OK, but a couple of questions
> still
> lurk:
>
> 1) You employ the Dirac delta as a "function" starting at (3). I
> wonder if
> the math majors here will regard this as a problem, since technically,
> the
> delta is not a function.
I'm not the first to use this product of dirac deltas in QM. See for
example,
http://users.physik.fu-berlin.de/~kleinert/public_html/kleiner_reb3/psfiles/pthic04.pdf
However, I think the author here is using this rather lightly without
regard to rigor.
The objection seems to come from distribution theory where a the
product of distributions is not allowed. See,
http://en.wikipedia.org/wiki/Distribution_(mathematics)#Problem_of_multiplication
But there doesn't seem to be any use of anything in distribution
theory necessary in the definition of either the limits or the
integration process used in the definition of the dirac delta
function. All the distribution theory texts I've seen so far define
the dirac delta function even before they use it in rest of the text.
They define the dirac delta as an integral that equals one no matter
how small a limiting parameter gets. And there is nothing in
distribution theory that defines either the integration process or the
limiting process used in the definition of the dirac delta. All the
objections might prove is that a product of distribution is not a
ligitimate construction in the language of distribution theory. That
does not prove that the integration of a product of dirac delta
functions is not a ligitimate mathematical object outside distribution
theory.
For example, a Chapman-Kolmogorov equation shows how the integration
of two gaussian function equals a third. See
http://www.physicsforums.com/showpost.php?p=2499941&postcount=27
Proof of this equation can be found in "The Feynman Integral and
Feynman's Operational Calculus", by Gerald W. Johnson and Michael L.
Lapidus, page 38. And it's easy to see how if (t-s) and (s-r) were
each allowed to approach zero, we would have the integration of two
dirac delta function equal to a thrid. So the recursion relation (3)
that you mention does seem to be just as valid as the integration of a
single dirac delta which equals one. The same process is used in both,
namely, do the integration first and then take the limit as the
variance approaches zero.
>
> 2) Your (6) is one way to define the delta from a nascent delta. But,
> it
> is not exclusive. There are a number of different ways to do so, so you
> need to explain why your (6) is the way we have to go.
Obviously, an exponential form for the dirac delta suggests itself
since
e^a X e^b = e^(a+b),
and
$e^x dx = e^x.
And the Chapman-Kolmogorov equation proves that the gaussian form of
the dirac delta works in this situation. At this point, I'm not really
sure whether this recursion relation works for any other form of the
dirac delta function. Intuitively it seems to be valid for any form of
delta, but I have not tried every form of delta.
Using the recursion relation (3), seems to be one way to get to the
path integral. But I originally got to the path integral from more of
a direct process of logic. See,
http://hook.sirus.com/users/mjake/QMfromlogic.htm
There is it seen that the delta function is a mathematical expression
of the logical operation of material implication. "Paths" emerge from
a conjunction of facts, where each path is described by a infinite
conjunction of implications, which mathematically translates into an
infinite product of delta functions. When the deltas take the form of
a gaussian, the path integral results.
It seems natural to give the term "path" to a conjunction of
implications, where the conclusion of one implication is the premise
in the next implication. For what is a "path" except a description
that states that IF you are at this point, THEN the next point is
here, AND IF you are at that point, THEN the next point is there, AND
IF you are at that point, THEN the next point is here, etc, etc. Every
possible path is accounted for, but there is no preference on which
path is preferred. So each path is a Random Walk through all the
possible states. This is very similar to Brownian motion of a particle
influenced by thermal agitations, which is mathematically described by
the Wiener process. The Wiener process uses a gaussian distribution to
describe the probablity of going from one state to the next. And so
the Wiener process strongly suggests itself in describing the random
paths in my work as well.
to...@cc.usu.edu
> There are some cases in which the Lorentzian path integral can be
> defined via promeasures. �There's a discussion of this, and of other
> related mathematical issues, in the new book by Cartier and DeWitt-
> Morette, _Functional Integration_ (though the relevant bits are
> scattered around a little).
>
Here's another nice reference where certain instances of rigorous
definitions of the (real-time) path integral appear:
"Mathematical Theory of Feynman Path Integrals"
A. Albeverio and R. Hoegh-Krohn, (Springer, 1976).
charlie
Arnold Neumaier
Can any of the two be used to interpret quantum field theory?
At least in 1+1 dimensions? or are they restricted to ordinary
quantum mechnaics with finitely many degrees of freedom?
Arnold Neumaier