Continued Discussion about Path Integral and Canonical Formulations of Quantum Field Theory
Jay R. Yablon
Picture?" which I earlier initiated, an excellent discussion emerged,
largely involving Dr. Neumaier, Igor, X-Phy, and a few other folks,
about path integration and its virtues and limits.
It seems to me that a side-by-side exploration of the Path Integral
formulation and the Canonical formulations of QFT, how they are
interrelated, how they complement one another, what each one is good at
and bad at, etc., is a very worthwhile discussion to be having, so I am
bringing this over to start a separate thread.
I will start the discussion with a remark made by Dr. Neumaier following
some lengthy exchanges with X-Phy, where he finally said of path
integration:
"It is a big, heuristic, and useful picture. But one that couldn't be
made logically consistent, in spite of many attempts by some of the best
mathematicians and physicists. Let us not confuse usefulness and
logical consistency."
Since we at least have an agreement on the usefulness of path
integration, let's stick with that, and explore where path integration
is "useful," as well as where the "blocks" and "circumstances" are which
cause it to have logical inconsistencies.
Without going back to find all of the exact quotes, I also recall it
commented by Dr. Neumaier on several occasions that there is a problem
making path integration work -- logically -- for *sources*, but that it
does work for source-free fields. I would like to pinpoint that issue a
bit more.
I also saw a statement from Juan R. that "For some important theories
the simplest version of the path-integral method is, in Weinberg own
words, "simply wrong". Weinberg cites the case of the nonlinear sigma
model. This is the reason which Weinberg prefers to *obtain* path
integrals (including vertices corrections to Feynman's original version)
from the canonical formalism." I will make my own comment, perhaps
acerbic, that if Weinberg doesn't like something, it seems that most
other physicists won't like it either. We should explore why he doesn't
like it, and not stop short simply at asserting THAT the doesn't like
it. I also sort of wonder what Feynman might say about all of this, if
he was still around today. Maybe we could get him and Weinberg to "duke
it out"? ;-)
And, I'd like to throw into the mix, the Wiki article, with the
derivation based on Zee's book,
http://en.wikipedia.org/wiki/Relation_between_Schr%C3%B6dinger%27s_equation_and_the_path_integral_formulation_of_quantum_mechanics#From_Schr.C3.B6dinger.27s_equation_to_the_path_integral_formulation.
I do this, because this link makes the connection from the canonical
formulation to the path integral formulation, and is in some sense an
exploration of QFT that straddles both formulations. It also starts
with a Hamiltonian operator and ends up with a Lagrangian, so it
straddles those two major approaches as well. Perhaps this will give us
a vehicle to explore the logical limitations of the path integral
formulation.
Using the notation in the above link, the above derivation seeks to
obtain the transition amplitude:
Z == <F|psi(t)> = <F|exp(-(i/hbar)H-hat T)|0> (1)
where H-hat is the Hamiltonian operator, T is the total time, and
exp(-(i/hbar)H-hat T) is the unitary time evolution operator. After
slicing up the time T into a large number of tiny delta t intervals and
having those approach zero and the number of slices approach infinity,
and following that same calculation that Zee uses in section 1.2 of his
book but including the potential energy V all the way through, including
using the complete state set:
I = $ dq |q><q| (2)
and Gaussian integration, this link show how one determines that:
Z = <F|exp(-(i/hbar)H-hat T)|0> = $Dq exp [(i/hbar) S] (3)
where the action (q-dot=dq/dt):
S = ${0 to T} dt L (q,q-dot) (4)
and the classical Lagrangian density is just:
L (q,q-dot) = .5 m q-dot^2 - V(q) (5)
We sweep the "ugly" stuff into:
$Dq(t)
=limN-->oo(-i2pi m/delta t)^(N/2)(PI{j=1 to N-1}$dq_i) (6)
We also generally "ignore" the coefficient (-i2pi m/delta t)^(N/2),
which perhaps is some of the "handwaving" that Dr. Neumaier referred to
at several points in the discussion.
Stopping here for a moment, I'd like to know if there is anything
wrong with (3)-(6), logically or otherwise, or if this is just another
way of viewing the unitary (1)? I suspect that at this juncture, there
is no logical problem, but would like to hear otherwise if this is not
so.
It seems to me what happens next, is that we make a leap of
abstraction, which physicists are prone to do. Although S in (3) is the
particular action given in (4) and (5), we generalize this to any and
all actions whatsoever, including actions with sources such a mass m and
current J^u. Further, while the action in (4) is established by taking
the integral over dt from t=0 to t=T, we generalize the integral to
include t,x,y,z in the form of d^4x, and we generalize the range to go
from negative to positive infinity. And, we generalize the state q to
the field psi. Thus, we abstract (3) through (6) into:
Z = $D psi exp [(i/hbar) S] (3a)
S = ${-oo to +oo} d^4x L (4a)
L (psi, d^u psi) = any Lagrangian, even with sources (5a)
$D psi = swept under the rug after integration (6a)
That, it seems to me, is path integration in a nutshell. The rest is
making it work wherever possible, and finding tricks to make it work,
and asserting that it is not logically consistent in those places where
despite concerted effort to date it has not been made to work. Is it
the abstracting from (3)-(6) over to (3a)-(6a) that raises the logical
concerns we have been discussing? Or, are these logical problems
already nascently present in some way in (3)-(6)?
I'll stop for now, and start the discussion here.
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
Peter
> � � In the separate thread "Quantum Field Theory: The Big, Simple
> co-moderator: sci.physics.foundations
> Weblog:http://jayryablon.wordpress.com/
> Web Site:http://home.roadrunner.com/~jry/FermionMass.htm
Hello Jay,
The answer could be in the physics. In his original (1948) paper,
Feynman observed that the transition probability amplitudes,
P_ab = <a|b>
where |b> is a solution to the time-dependent Schr�dinger eq., obey
P_ab = Sum(c) P_ac P_cb
This is one form of the Chapman-Kolmogorov eq. It applies, because the
time-dependent Schr�dinger eq. is of *1st* order in time. Thus, if you
insert entities the equation of motion of which are of *2nd* order in
time, you loose that ground. - The way back to this ground consists in
a physically sound decomposition into 2 eqs. of 1st order and to work
with matrices instead of scalar transition probability amplitudes.
Good luck!
Peter
Igor Khavkine
> Using the notation in the above link, the above derivation seeks to
> obtain the transition amplitude:
>
> Z == <F|psi(t)> = <F|exp(-(i/hbar)H-hat T)|0> (1)
>
> where H-hat is the Hamiltonian operator, T is the total time, and
> exp(-(i/hbar)H-hat T) is the unitary time evolution operator. After
> slicing up the time T into a large number of tiny delta t intervals and
> having those approach zero and the number of slices approach infinity,
> and following that same calculation that Zee uses in section 1.2 of his
> book but including the potential energy V all the way through, including
> using the complete state set:
>
> I = $ dq |q><q| (2)
>
> and Gaussian integration, this link show how one determines that:
>
> Z = <F|exp(-(i/hbar)H-hat T)|0> = $Dq exp [(i/hbar) S] (3)
>
> where the action (q-dot=dq/dt):
>
> S = ${0 to T} dt L (q,q-dot) (4)
>
> and the classical Lagrangian density is just:
>
> L (q,q-dot) = .5 m q-dot^2 - V(q) (5)
>
> We sweep the "ugly" stuff into:
>
> $Dq(t)
> =limN-->oo(-i2pi m/delta t)^(N/2)(PI{j=1 to N-1}$dq_i) (6)
>
> We also generally "ignore" the coefficient (-i2pi m/delta t)^(N/2),
> which perhaps is some of the "handwaving" that Dr. Neumaier referred to
> at several points in the discussion.
Neglecting this overall constant is not a big deal. It can be factored
out and does not affect any amplitude calculations.
> Stopping here for a moment, I'd like to know if there is anything
> wrong with (3)-(6), logically or otherwise, or if this is just another
> way of viewing the unitary (1)? I suspect that at this juncture, there
> is no logical problem, but would like to hear otherwise if this is not
> so.
At (6) is precisely where logical problems start. This limit is not
defined. Go ahead, dig out any elementary (or even not so elementary)
calculus or analysis book where limits are defined. There, you'll find
a bunch of theorems on when limits do exist and when the limit itself
can serve as the definition of a mathematical object. None of these
theorems are applicable to (6). This limiting process is
mathematically undefined and hence cannot serve appear in the
mathematical definition of the path integral. The issue is as simple
as that. It already appears in ordinary quantum mechanics. The
problems only get worse in field theory where integration over the
intermediate dq_i is itself an "integration" over an infinite
dimensional space of field configurations at a fixed time.
The failure of giving the generic path integral a sound mathematical
definition comes back to the problems of identifying the correct
limiting procedure in (6) and proving that such a limit actually
exists. Until such a sound mathematical foundation exists, the right
hand side of (3) can only have the mathematical meaning of a shorthand
for the right hand side of (1). If you want to talk about foundational
issues, this is the core of them.
Igor
Jay R. Yablon
news:7ec62f56-a5a9-4640...@r40g2000yqn.googlegroups.com...
***>The issue is as simple
> as that. It already appears in ordinary quantum mechanics. The
> problems only get worse in field theory where integration over the
> intermediate dq_i is itself an "integration" over an infinite
> dimensional space of field configurations at a fixed time.***
I read in this that you are pointing to two problems. One of them
involves the infinite product PI{j=1 to N-1}$dq_i. This was something
of a red flag for me when I first studied the path integral a couple of
years ago, though I pushed it to the back of my mind and did not realy
hone in on it.
But if I read you correctly, you are also pointing to an even more
fundamental even with the simple 'intergation':
$dq_i
for a single, infinitesmal transition. And, it seems to me the problem
you are pointing to really seems to reside in (2) above:
I = $ dq |q><q| (2)
For, in this, we have taken the completeness / closure relationship
Sigma{q'}|q'><q'| = I (3)
which involves a "sum" over an outer product of an eigenket and an
eigenbra, and tried to convert it over into an "integral" (2).
Integrals in mathematics are defined using infinite sums; but not every
infinite summation is necessarily an integral. Am I on the right track
to perceive that the real root of the problem is turning the summation
(3) into the intergal (2) without really stating, in a mathematcially
tight fashion, how the integral in (2) is defined using some sort of
Newton / Liebnitz limiting principle?
That is, are we looking most elementaly and fundamentally, at the need
to state what it means, exactly, to go from
Sigma{q'}|q'><q'| = I to $ dq' |q'><q'|
and even more exactly, from:
Sigma{q'} to $dq' ?
If so, we need some new calculus understanding of what might be dubbed
"bra-ket calculus." And, as you point out this problem "already appears
in ordinary quantum mechanics."
Thanks,
Jay
Jay R. Yablon
news:bf492715-0483-490d...@r24g2000yqd.googlegroups.com...[[Mod. note -- 163 excessively-quoted lines snipped. -- jt]]
> The answer could be in the physics. In his original (1948) paper,
> Feynman observed that the transition probability amplitudes,
>
> P_ab = <a|b>
>
>
> P_ab = Sum(c) P_ac P_cb
>
> This is one form of the Chapman-Kolmogorov eq. It applies, because the
> insert entities the equation of motion of which are of *2nd* order in
> time, you loose that ground. - The way back to this ground consists in
> a physically sound decomposition into 2 eqs. of 1st order and to work
> with matrices instead of scalar transition probability amplitudes.
>
> Good luck!
> Peter
>
It looks to me like the problem may be with the outer product
completeness relation:
Sigma{q'}|q'><q'| = I
and how one turns this into a true calculus integral
$ dq' |q'><q'| = I
See also most forthcoming reply to Igor. Your thoughts?
Jay.
CarlB
> We also generally "ignore" the coefficient (-i2pi m/delta t)^(N/2),
> which perhaps is some of the "handwaving" that Dr. Neumaier referred to
> at several points in the discussion.
I sent a paper into Foundations of Physics on spin path integrals.
The idea is to make path integrals that apply only through
spin space (ignore position completely), and then see what
happens. To make this non trivial, you have to expand the
basis for spin using a concept that the quantum information
theory people call "mutually unbiased bases".
But anyway, when you do this, even though you're working
entirely inside of finite Hilbert spaces (i.e. spin-1/2), you still
end up with those coefficients. You have to set them so as
to preserve probability. See the discussion at the beginning
of section 4:
http://www.brannenworks.com/Gravity/spinpath.pdf
There are some papers that do similar calculations and
which interpret the arbitrary constant has having something
to do with the volume of the phase space that is being
integrated over. See equation (25) and discussion below
about the "peculiar factor" in this paper by Foster and
Jacobson:
http://arxiv.org/abs/hep-th/0310166v2
Carl
Igor Khavkine
> "Igor Khavkine" <igor...@gmail.com> wrote in message
> news:7ec62f56-a5a9-4640...@r40g2000yqn.googlegroups.com...
> > On Dec 9, 7:56 am, "Jay R. Yablon" <jyab...@nycap.rr.com> wrote:
> >> $Dq(t)
> >> =limN-->oo(-i2pi m/delta t)^(N/2)(PI{j=1 to N-1}$dq_i) �(6)
> > At (6) is precisely where logical problems start. This limit is not
> > defined. Go ahead, dig out any elementary (or even not so elementary)
> > calculus or analysis book where limits are defined. There, you'll find
> > a bunch of theorems on when limits do exist and when the limit itself
> > can serve as the definition of a mathematical object. None of these
> > theorems are applicable to (6). This limiting process is
> > mathematically undefined and hence cannot serve appear in the
> > mathematical definition of the path integral.
>
> ***>The issue is as simple
>
> > as that. It already appears in ordinary quantum mechanics. The
> > problems only get worse in field theory where integration over the
> > intermediate dq_i is itself an "integration" over an infinite
> > dimensional space of field configurations at a fixed time.***
>
> I read in this that you are pointing to two problems. �One of them
> involves the infinite product PI{j=1 to N-1}$dq_i. �This was something
> of a red flag for me when I first studied the path integral a couple of
> years ago, though I pushed it to the back of my mind and did not realy
> hone in on it.
You have done it, and likely every physicist who has practically used
path integrals had to do the same at some point. However, this is
where mathematics and logic really get stuck. And this is the
mathematical problem that prevents path integrals from being well
defined.
> But if I read you correctly, you are also pointing to an even more
> fundamental even with the simple 'intergation':
>
> $dq_i
>
> for a single, infinitesmal transition. �And, it seems to me the problem
> you are pointing to really seems to reside in (2) above:
>
> �I = $ dq |q><q| � (2)
>
> For, in this, we have taken the completeness / closure relationship
>
> Sigma{q'}|q'><q'| = I � (3)
>
> which involves a "sum" over an outer product of an eigenket and an
> eigenbra, and tried to convert it over into an "integral" (2).
I don't see how you could have read this from what I've written. My
exact quote is given above, but let me highlight the relevant part:
The problems only get worse in field theory where integration
over the intermediate dq_i is itself an "integration" over an
infinite dimensional space of field configurations at a fixed time.
When q_i is a single real parameter (or even a finite number of real
parameters) equation (2) poses absolutely no problem at all. A problem
appears when q_i lives in an *infinite dimensional* space, such as the
space of all continuous field configurations at a fixed time. To
integrate over such a large space, you'd need to coordinatize it with
infinitely many coordinates (q_i0,q_i1,q_i2,...) and then integrate
over all of them
dq_i = dq_i0 dq_i1 dq_i2 ...
This kind of integral is as ill defined as the entire path integral.
The pathology is the same: lack of well defined notion of integration
over an infinite dimensional space. Now, I've just repeated what I
said earlier. But I hope it's clearer now.
Igor
Peter
...
> Hi Peter,
>
> It looks to me like the problem may be with the outer product
> completeness relation:
>
> Sigma{q'}|q'><q'| = I
>
> and how one turns this into a true calculus integral
>
> $ dq' |q'><q'| = I
>
> See also most forthcoming reply to Igor. Your thoughts?
I agree
May be, we can learn something from the transition
chain of coupled oscillators -> waves in a continuum
considered in many textbooks on solid state physics (and in my
book ;-)?
Looking forward,
Peter
Mike
> When q_i is a single real parameter (or even a finite number of real
> parameters) equation (2) poses absolutely no problem at all. A problem
> appears when q_i lives in an *infinite dimensional* space, such as the
> space of all continuous field configurations at a fixed time. To
> integrate over such a large space, you'd need to coordinatize it with
> infinitely many coordinates (q_i0,q_i1,q_i2,...) and then integrate
> over all of them
>
> dq_i = dq_i0 dq_i1 dq_i2 ...
>
> This kind of integral is as ill defined as the entire path integral.
> The pathology is the same: lack of well defined notion of integration
> over an infinite dimensional space. Now, I've just repeated what I
> said earlier. But I hope it's clearer now.
Thank you, Igor, for that statement. Let me suggest a different
approach. It seems the path integral can also be obtained by
interating a recursion relation for the Dirac delta function. See
details at:
http://hook.sirus.com/users/mjake/delta_physics.htm
And I'm told that the delta function IS a well defined measure. The
question is: are we using the wrong measure? Should we instead be
trying to look at the path integral as a functional integral of an
infinite products of a dirac delta measures?
Viewed as an interation process, any number of integration of any
number of deltas (you know what I mean) is still equal to a delta and
has a measure of a delta, which is well defined.
Jay R. Yablon
Yes it is.
Now to the next step:
Can you pinpoint exactly what it is about an infinite dimensional space
which makes it impossible to define integration (or at least has so far
defied a well-defined notion of integration), versus what it is about a
space with, say, 10^100 dimensions which presumably still allows a
proper definition of integration? Also, does a real versus a complex
(Hilbert) vector space make a difference?
Let put this into the context of a Riemann-type (standard) definite
integral from A to B, defined as:
$ dx F(x) = Sum {i=1 to N} delta x_i F(x_i)
lim (N-->oo, delta x_i -->0) (1)
with x_1=A, x_oo=B
Or, if you have a better / more concise definition than (1), I am happy
to go with that.
Let us perhaps do this in four steps:
1) Let's take a plain old real four-dimensional spacetime, with the
volume element d^4x. Or even a three-dimensional physical space with
d^3x. Pick either or both. What would you posit as the calculus
"definition" of the integral, using some extension of (1)?
2) Now make those spaces complex, doubling the degrees of freedom. Any
change from step 1)?
3) Now give either of step 1) or 2) a very large number D of
dimensions, as large as you like. 10^100, for example. What now is the
definition of the integral? What differences would there be, for real
versus complex spaces?
4) Now, we want to have D-->oo. What precisely is it in 3) that hits a
wall and causes the definition of the integral to break down?
Jay
>
> Igor
>
Peter
...
> Hi Peter,
>
> It looks to me like the problem may be with the outer product
> completeness relation:
>
> Sigma{q'}|q'><q'| = I
>
> and how one turns this into a true calculus integral
>
> $ dq' |q'><q'| = I
>
> See also most forthcoming reply to Igor. �Your thoughts?
Let me add what I have discussed with a colleague (a mathematician
with extremely rigorous thinking).
The following is mathematically sound. For fixed x_a, x_b,
G_n(x_a,x_b) = Int(-oo,+oo) Int(-oo,+oo) ... Int(-oo,+oo)
G(x_a,x_1) G(x_1,x_2) ... G(x_n,x_b) dx_1 dx_2 ... dx_n
is a number; it's essentially Feynman's P_ab = Sum(c) P_ac P_cb in
ordinary space (I have omitted the time variables in the propagators,
G). Since it describes the physical propagation of something from x_a
to x_b (or conversely), it should be finite for all n, 0<=n<oo. Iow,
the sequence
G_1, G_2, ..., G_n, ...
should converge; then,
G_oo = lim(n->oo) G_n
is well defined.
Try to translate this construction of a well-defined sequence of
numbers to your problem - good luck!
Peter
Mike
> And I'm told that the delta function IS a well defined measure. The
> question is: are we using the wrong measure? Should we instead be
> trying to look at the path integral as a functional integral of an
> infinite products of a dirac delta measures?
>
> Viewed as an interation process, any number of integration of any
> number of deltas (you know what I mean) is still equal to a delta and
> has a measure of a delta, which is well defined.
In this case the measure includes the entire integrand. And the
integral in the exponent can be anything as long as it breaks down to
dirac delta functions.
Lou Pecora
"Jay R. Yablon" <jya...@nycap.rr.com> wrote:
> Now to the next step:
>
> Can you pinpoint exactly what it is about an infinite dimensional space
> which makes it impossible to define integration (or at least has so far
> defied a well-defined notion of integration), versus what it is about a
> space with, say, 10^100 dimensions which presumably still allows a
> proper definition of integration? Also, does a real versus a complex
> (Hilbert) vector space make a difference?
I think the main road block is trying to define a measure on an infinite
dimensional space of paths (or functions, if you like). Even in 10^100
dimensions you still have the case that a simple spatial element (e.g. a
parallelopiped) has finite volume. In infinite dimensions, it's
divergent. What measure do you use? I recall (maybe incorrectly) that
for real integrands, a gaussian kernel can be used to define a weight
and thereby a measure that is finite in infinite dimensions (I'd like to
hear from knowledgable others here), but in the quantum case you're
dealing with complex quantities. The measure problem rears its head,
again.
Remember, 1D or 10^100D are both the "same" distance from infinity.
--
-- Lou Pecora
Arnold Neumaier
>
> Let me add what I have discussed with a colleague (a mathematician
> with extremely rigorous thinking).
>
> The following is mathematically sound. For fixed x_a, x_b,
>
> G_n(x_a,x_b) = Int(-oo,+oo) Int(-oo,+oo) ... Int(-oo,+oo)
>
> G(x_a,x_1) G(x_1,x_2) ... G(x_n,x_b) dx_1 dx_2 ... dx_n
>
> is a number; it's essentially Feynman's P_ab = Sum(c) P_ac P_cb in
> ordinary space (I have omitted the time variables in the propagators,
> G). Since it describes the physical propagation of something from x_a
> to x_b (or conversely), it should be finite for all n, 0<=n<oo. Iow,
> the sequence
>
> G_1, G_2, ..., G_n, ...
>
> should converge;
Not necessarily. The sequence could turn out to behave like
G_k=k or G_k=(-1)^k,
and in both cases, it does not converge. For convergense, one needs to
show that the G_n ultimately stay arbitrarily close to some number (the
limit). This is the real difficulty in this problem.
Arnold Neumaier
Jonathan Thornburg [remove -animal to reply]
> which makes it impossible to define integration (or at least has so far
> defied a well-defined notion of integration), versus what it is about a
> space with, say, 10^100 dimensions which presumably still allows a
> proper definition of integration? Also, does a real versus a complex
> (Hilbert) vector space make a difference?
The usual notions of integration are closely tied to measure theory,
and the very first google hit for "infinite-dimensional measure theory"
just now is a Wikipedia article with a rather interesting url:
http://en.wikipedia.org/wiki/There_is_no_infinite-dimensional_Lebesgue_measure
The article includes a proof of the theorem in question.
It may be that what you are trying to do actually differs from what
is proved to be impossible by the proof given in that Wikipedia article.
But given the comments by various other people in this thread, some of
whom know a lot about this subject, I think the burden is on *you* to
demonstrate that you have a mathematically sensible notion of integration
on $R^\infty$, and (given the standard equivalence between integration
and measure theory) that you've somehow found a way around the theorem
I referred to above.
> Let put this into the context of a Riemann-type (standard) definite
> integral from A to B, defined as:
>
> $ dx F(x) = Sum {i=1 to N} delta x_i F(x_i)
> lim (N-->oo, delta x_i -->0) (1)
> with x_1=A, x_oo=B
If you're going to call something an "integral", it should deserve
that name. That is, if your limit does indeed exist, does it still
have the various nice properties that finite-dimensional Riemann,
Lebesgue, etc, integrals have? For example, does the fundamental
theorem of calculus (a.k.a. Stoke's theorem) still hold for your
infinite-dimensional integrals? How about Green's theorem? Gauss's
theorem, a.k.a. the divergence theorem? (How) does integration by
parts work? Can you prove that these things work the way you think
they do?
Of if you'd prefer something more concrete, how about demonstrating
the calculation of each of the following for the $n \to \infty$ case
using your infinite-dimensional integrals:
(a) What is the volume of a unit sphere in $R^n$?
(b) What normalization is needed for a Gaussian to have unit integral
over the entire space $R^n$?
(c) What is the integral of a Gaussian-normalized-this-way over the
unit sphere in $R^n$?
(d) What is the ratio of the real number (c) divided by the real
number (a)?
I would want to see reasonable answers to all the above questions
(or very convincing arguments why it's ok for them not exist) before
I called something an "integral".
--
-- Jonathan Thornburg <jth...@astro.indiana.edu>
Dept of Astronomy, Indiana University, Bloomington, Indiana, USA
"I'd like a large order of Fibonachos, please."
"Okay, sir...that will be the cost of a small order, plus the cost of
a medium order." -- from sci.math
Jay R. Yablon
Peter Enders in an earlier reply in this thread stated the following:
[begin Enders]
"Let me add what I have discussed with a colleague (a mathematician
with extremely rigorous thinking).
The following is mathematically sound. For fixed x_a, x_b,
G_n(x_a,x_b) = Int(-oo,+oo) Int(-oo,+oo) ... Int(-oo,+oo)
G(x_a,x_1) G(x_1,x_2) ... G(x_n,x_b) dx_1 dx_2 ... dx_n
is a number; it's essentially Feynman's P_ab = Sum(c) P_ac P_cb in
ordinary space (I have omitted the time variables in the propagators,
G). Since it describes the physical propagation of something from x_a
to x_b (or conversely), it should be finite for all n, 0<=n<oo. Iow,
the sequence
G_1, G_2, ..., G_n, ...
should converge; then,
G_oo = lim(n->oo) G_n
is well defined.
Try to translate this construction of a well-defined sequence of
numbers to your problem - good luck!"
[end Enders]
Would you concer with Peter about this being a good approach to this
problem?
Jay.
Igor Khavkine
What Peter says is, for any finite n, is perfectly justified. However,
even if the limits of the form G_oo illustrated above exist and are
finite, that is not enough to define a path integral. The problem is
once again with the interchange of limits and integrals.
In schematic notation mirroring Peter's, for finite n, we are
perfectly justified in writing
G_n = Int(1) ... Int(n) G(1) ... G(n).
At this point, one could consider taking the limit G_oo = lim_n->oo
G_n. That's fine to do, but that limit is not what people mean by a
path integral.
Ostensibly, the path integral would be defined as
PathInt = [lim_n->oo Int(1) ... Int(n)] [lim_n->oo G(1) ... G(n)]
The second term on the right is a limit of a product of numbers and it
might well converge to a finite value. The first term on the right, on
the other hand, is different. Its limit should give a way to integrate
functions on an infinite dimensional space of paths. However, this is
where the major problems appear, as there is no known mathematically
rigorous way to take this limit (or, depending on the context, it is
known that one cannot take this limit at all). The problems of
mathematical definitions of path/functional integration start here.
However, even if the limit is successfully taken, it still remains to
show that G_oo and PathInt represent the same mathematical object, in
order to establish the equivalence of path integral formulation of
quantum mechanics/field theory to operator based formulations.
Igor
Jay R. Yablon
One other question about all of this, pertaining to when one takes the
limit:
In R. P. Feynman, Space-Time Approach to Non-Relativistic Quantum
Mechanics, Rev. Mod. Phys. 20 (1948) 367-387; reprint in: J. Schwinger
(Ed.), Selected Papers on Quantum Electrodynamics, Dover (1958), paper
27, Feynman says on 371, top of second column: "the limit epsilon->0
*must be taken at the end of a calculation.*"
I read this to say that first one takes the time separation delta t to
be small but finite when subdividing the path into N segments with T = N
delta t, and N a finite number. Then, the final step of the
calculation, *after all others have been done and all other possible
reductions and simplifications have been made*, is to take delta t->0
and N->oo.
Jay
Igor Khavkine
> [[Mod. note -- 173 excessively-quoted lines snipped. -- jt]]
>
> One other question about all of this, pertaining to when one takes the
> limit:
Jay, what is your question exactly?
> In R. P. Feynman, Space-Time Approach to Non-Relativistic Quantum
> Mechanics, Rev. Mod. Phys. 20 (1948) 367-387; reprint in: J. Schwinger
> (Ed.), Selected Papers on Quantum Electrodynamics, Dover (1958), paper
> 27, Feynman says on 371, top of second column: "the limit epsilon->0
> *must be taken at the end of a calculation.*"
>
> I read this to say that first one takes the time separation delta t to
> be small but finite when subdividing the path into N segments with T = N
> delta t, and N a finite number. �Then, the final step of the
> calculation, *after all others have been done and all other possible
> reductions and simplifications have been made*, is to take delta t->0
> and N->oo.
I'm not sure why you felt the need to go to Feynman's original papers.
Essentially the same information is given in any modern treatment,
often with more details and more attention to mathematical subtleties.
Given that I do not see a particular question, I'll just comment again
that there is no problem with approximating the time evolution
operator for interval T with a product of N simpler evolution
operators each for interval T/N. In fact, a variant of this technique
is used in the so-called "split operator method" for numerically
solving the time-dependent Schroedinger equation.
Another comment about the "after reductions and simplifications"
remark. It is absolutely devoid of mathematical content. The number 5
is still the same number if I write it as 2+3. The existence of a
limit is independent of the specific form the mathematical object in
question is expressed in. If the object in question is the procedure
of naive integration over the infinite dimensional space of paths,
then this limit simply does not exist, no matter how you write the
formulas.
Igor
Mike
> The existence of a
> limit is independent of the specific form the mathematical object in
> question is expressed in. If the object in question is the procedure
> of naive integration over the infinite dimensional space of paths,
> then this limit simply does not exist, no matter how you write the
> formulas.
I'm not sure the "infinite dimensional space of paths" is not just an
alternative interpretation of the infinte dimensional volume element,
dx1dx2dx3...dxn,
where n goes to infinity. This infinite dimensional Lebesque measure
is not well defined in and of itself, but if one includes the
exponential of the action integral including the normalization factor
(which together becomes the dirac delta measure/distribution), then it
is called the Weiner measure and the infinite dimensional Weiner
measure is well defined, right?
Jay R. Yablon
> On Feb 15, 6:07 am, "Jay R. Yablon" <jyab...@nycap.rr.com> wrote:
>> [[Mod. note -- 173 excessively-quoted lines snipped. -- jt]]
>>
>> One other question about all of this, pertaining to when one takes
>> the
>> limit:
>
> Jay, what is your question exactly?
>
>> In R. P. Feynman, Space-Time Approach to Non-Relativistic Quantum
>> Mechanics, Rev. Mod. Phys. 20 (1948) 367-387; reprint in: J.
>> Schwinger
>> (Ed.), Selected Papers on Quantum Electrodynamics, Dover (1958),
>> paper
>> 27, Feynman says on 371, top of second column: "the limit epsilon->0
>> *must be taken at the end of a calculation.*"
>>
>> I read this to say that first one takes the time separation delta t
>> to
>> be small but finite when subdividing the path into N segments with T
>> = N
>> delta t, and N a finite number. Then, the final step of the
>> calculation, *after all others have been done and all other possible
>> reductions and simplifications have been made*, is to take delta t->0
>> and N->oo.
>
> I'm not sure why you felt the need to go to Feynman's original papers.
> Essentially the same information is given in any modern treatment,
> often with more details and more attention to mathematical subtleties.
Are you suggesting that when one is studying a subject, one should
bypass the seminal work upon which that subject is based, and just go on
to all of the later developments? Why are these mutually exclusive?
Besides Dirac and Feynman, I am am also studying Glimm and Jaffe
(recommended by A. Neumaier), Kleinert (see below, recommended by P.
Enders), Cartier and DeWitt-Morette (recommended by S. Carlip), and any
other path integral references I can get my hands on. If there are any
particular references not in this list which you suggest, please post
them and I will study them as well.
> Given that I do not see a particular question, I'll just comment again
> that there is no problem with approximating the time evolution
> operator for interval T with a product of N simpler evolution
> operators each for interval T/N. In fact, a variant of this technique
> is used in the so-called "split operator method" for numerically
> solving the time-dependent Schroedinger equation.
Agreed.
> Another comment about the "after reductions and simplifications"
> remark. It is absolutely devoid of mathematical content. The number 5
> is still the same number if I write it as 2+3. The existence of a
> limit is independent of the specific form the mathematical object in
> question is expressed in. If the object in question is the procedure
> of naive integration over the infinite dimensional space of paths,
> then this limit simply does not exist, no matter how you write the
> formulas.
With all due respect, Hagen Kleinert, "Path Integrals in Quantum
Mechanics, Statistics, Polymer Physics, and Financial Markets" in (2.72)
and (2.74), successfully completes the intergation over the infinite
dimensional space of paths. It does so in the simple case of a
potential V(q)=0, (and later for a harmonic oscillator), so your
repeated suggestions that the "limit simply does not exist" for
"integration over the infinite dimensional space of paths" would seem to
be refuted by the calculations in Kleinert. The question now becomes,
not *whether* one can do exact calculations taking the limit over the
infinite dimensional space of paths, but *which* exact calculations one
is able to do in this space, and which one cannot do, at least yet.
Your qualification of integration as "naive" is itself devoid of
mathematical content, and in the event you were making suppositions
about some calculation I may have been doing, I will state that I was
able to *independently* derive (2.72) and (2.74) in Kleinert on my own
in the past two weeks, before I become aware of his work, so I know that
at least this particular calculation was not naive but was correct, and
that I have now gotten past some earlier no-go calculations which I had
attempted for doing an exact integration over the infinite-dimensional
path space. I will post my correct calculation here or send it to if
you would like.
What I would agree is "naive" integration, if that is what you have in
mind, is using Dq in the path integral as if it was a dq, which is what
is done in QED, and is at least in part what I think of when, for
example, A. Neumaier refers to "handwaving" in path integration.
As for Kleinert, there is a link to this book at
http://users.physik.fu-berlin.de/~kleinert/kleinert/?p=loadbook&book=8
which allows one to download five chapters for free, and this book can
also be perused at
http://www.amazon.com/Integrals-Quantum-Mechanics-Statistics-Financial/dp/9814273562/ref=sr_1_1?ie=UTF8&s=books&qid=1266268505&sr=8-1#reader_9814273562.
Jay
>
> Igor
>
Arnold Neumaier
> With all due respect, Hagen Kleinert, "Path Integrals in Quantum
> Mechanics, Statistics, Polymer Physics, and Financial Markets" in (2.72)
> and (2.74), successfully completes the intergation over the infinite
> dimensional space of paths.
Only for a number of simple exactly solvable problems, which includes
the free fields. That these have a well-defined path integral is
well-known but useless - since applicatiomns require interactions.
The general case is handled by him through variational perturbation
theory, based on the usual handwaving arguments with the undefined
measure in infinte dimensions. The only systems where he goes beyond
that are those with a finite number of degrees of freedom (anharmonic
oscillators, etc.), but this is very far even from 2D field theory.
If one is content with the usual handwaving use of the path integral,
one can get very far by just using formal manipulations and crossing
fingers to hope for good results (sometimes they get wrong without
simple signs).
Only to give the path integral in interacting 4D field theory a
well-defined meaning is the unsolved problem.
Arnold Neumaier