QED with double cutoff or in low dimension

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Marc Nardmann

Posted: Sat Feb 24, 2007 4:58 am

Guest

In my spare time, I'd like to learn some mathematically rigorous quantum
field theory. John Baez wrote in an old message from May 2003:

Quote:

The fact is, the logical foundations of QED are poorly understood
unless we water it down by treating it either

1) with an infrared cutoff, perturbatively

(... this is the approach I have seen before, in several variations...)

Quote:

or

2) with both infrared and ultraviolet cutoffs

Can you or someone else here on the group tell me a reference where this
second approach is carried out? From other comments of John Baez, it
seems that one can prove Borel summability of the resulting formal power
series. I would like to see the precise statement and the proof.

Another question: Is there a version of QED in two or three spacetime
dimensions which has been rigorously constructed (without unremoved
cutoffs, with provably [Borel?] convergent power series), satisfying,
say, the Wightman axioms? Reference?

-- Marc Nardmann

Igor Khavkine

Posted: Sun Feb 25, 2007 10:33 am

Guest

On 2007-02-24, Marc Nardmann <Marc.Nardmann@bigfoot.de> wrote:

Quote:

In my spare time, I'd like to learn some mathematically rigorous quantum
field theory. John Baez wrote in an old message from May 2003:

The fact is, the logical foundations of QED are poorly understood
unless we water it down by treating it either

1) with an infrared cutoff, perturbatively

(... this is the approach I have seen before, in several variations...)

or

2) with both infrared and ultraviolet cutoffs

Can you or someone else here on the group tell me a reference where this
second approach is carried out? From other comments of John Baez, it
seems that one can prove Borel summability of the resulting formal power
series. I would like to see the precise statement and the proof.

I believe that's the usual euphemism for the lattice formulation of QFT.
The minimal spacing between lattice points provides the ultraviolet
cutoff, while the finite lattice size gives an infrared one. There are
many references, including textbooks, on how to simulate lattice field
theories on a computer. One of the common reference texts is

Montvay & Munster, _Quantum Fields on a Lattice_ (CUP, 1994).

I'm sure that people have also looked at the question of whether lattice
field theories give continuum field theories in a well defined limit,
but I'm not familiar with that literature.

Quote:

Another question: Is there a version of QED in two or three spacetime
dimensions which has been rigorously constructed (without unremoved
cutoffs, with provably [Borel?] convergent power series), satisfying,
say, the Wightman axioms? Reference?

Don't know about a version of QED. I think scalar field theory is the
favorite testing ground for constructive quantum field theory. The case
of 2 space-time dimensions is special, since there are many models that
are exactly solvable (for example, using bosonization or conformal field
theory techniques). I believe that 1+1D QED, aka the Schwinger model, is
amenable to this kind of treatment. A classic text on exactly solvable
2-dimensional field theories is

Lieb & Mattis, _Mathematical Physics in one dimension; exactly
soluble models of interacting particles_ (Academic Press, 1966)

and its successor

Mattis, _The Many-body problem: an encyclopedia of exactly solvable
models in one dimension_, (World Scientific, 1993).

The phi^4 scalar field theory in 2D can be constructed as a perturbation
of the free theory by introducing Wick ordering (short distance
regularization) and an adiabatic interaction switching function
(infrared regularization). The word perturbation is used here in a
different sense than usual. Instead of computing a perturbation series,
one proves that the Hamiltonian H = H_0 + V is well defined and
self-adjoint by starting with a self-adjoint H_0 and treating V is a
"perturbation". This is done at the end of section X.7 of

Reed & Simon, _Methods of Modern Mathematical Physics, vol.2:
Fourier analysis, self-adjointness_ (Academic Press, 1975).

Of course, the classic reference for constructive field theory is

Glimm & Jaffe, _Quantum Physics: a functional integral point of view_
(Springer-Verlag, 1981).

They base their constructions on the rigorously defined Wick rotated
path integral. The Lorentz-signature field theory (with Wightman axioms)
is obtained through Osterwalder-Schrader reconstruction from the
Euclidean field-correlation functions. I believe Glimm & Jaffe carry out
this construction for some 2D field theories in their book. They also
outline the 3D construction in Chapter 20 and give further references.
There is also a discussion of Borel summability for some field theories
in the same chapter.

Finally, a disclamer. I'm only superficially familiar with most of these
results. I hope someone corrects me if I've made any obvious
misstatements. So, I may have misrepresented some of them. In any case,
these references are a good place to start reading about the subject.

Hope this helps,

Igor

Arnold Neumaier

Posted: Mon Feb 26, 2007 5:11 am

Guest

Igor Khavkine schrieb:

Quote:

On 2007-02-24, Marc Nardmann <Marc.Nardmann@bigfoot.de> wrote:
In my spare time, I'd like to learn some mathematically rigorous quantum
field theory. John Baez wrote in an old message from May 2003:

The fact is, the logical foundations of QED are poorly understood
unless we water it down by treating it either

1) with an infrared cutoff, perturbatively
(... this is the approach I have seen before, in several variations...)

or

2) with both infrared and ultraviolet cutoffs
Can you or someone else here on the group tell me a reference where this
second approach is carried out? From other comments of John Baez, it
seems that one can prove Borel summability of the resulting formal power
series. I would like to see the precise statement and the proof.

I believe that's the usual euphemism for the lattice formulation of QFT.
The minimal spacing between lattice points provides the ultraviolet
cutoff, while the finite lattice size gives an infrared one. There are
many references, including textbooks, on how to simulate lattice field
theories on a computer. One of the common reference texts is

Montvay & Munster, _Quantum Fields on a Lattice_ (CUP, 1994).

I'm sure that people have also looked at the question of whether lattice
field theories give continuum field theories in a well defined limit,
but I'm not familiar with that literature.

They do in 2 and 3 dimensions; the standard reference (for d=2)
is the book by
J. Glimm and A. Jaffe,
Quantum physics. A functional integral point of view
New York, 1981
The case d=4 is a famous unsolved problem; the special case of 4D
Yang-Mills SU(2) gauge theory is one of the Clay Millenium problems with
a 1 Million Dollar prize attached to its solution.

(Yang-Mills SU(2) gauge theory is asymptotically free, in contrast to
4D QED, which is widely believed - though I don't share this belief -
not to exist as an exact theory, because of the Landau pole.)

Arnold Neumaier

Igor Khavkine

Posted: Wed Feb 28, 2007 1:23 pm

Guest

On 2007-02-26, Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote:

Quote:

Igor Khavkine schrieb:

I'm sure that people have also looked at the question of whether lattice
field theories give continuum field theories in a well defined limit,
but I'm not familiar with that literature.

They do in 2 and 3 dimensions; the standard reference (for d=2)
is the book by
J. Glimm and A. Jaffe,
Quantum physics. A functional integral point of view
New York, 1981
The case d=4 is a famous unsolved problem; the special case of 4D
Yang-Mills SU(2) gauge theory is one of the Clay Millenium problems with
a 1 Million Dollar prize attached to its solution.

I must confess that I never got far enough in Glimm and Jaffe to figure
out what the "trick" in their construction was. It's nice to know that,
at least conceptually it's rather simple.

Thanks.

Igor

Arnold Neumaier

Posted: Thu Mar 01, 2007 3:36 am

Guest

Igor Khavkine schrieb:

Quote:

On 2007-02-26, Arnold Neumaier <Arnold.Neumaier@univie.ac.at> wrote:
Igor Khavkine schrieb:

I'm sure that people have also looked at the question of whether lattice
field theories give continuum field theories in a well defined limit,
but I'm not familiar with that literature.
They do in 2 and 3 dimensions; the standard reference (for d=2)
is the book by
J. Glimm and A. Jaffe,
Quantum physics. A functional integral point of view
New York, 1981
The case d=4 is a famous unsolved problem; the special case of 4D
Yang-Mills SU(2) gauge theory is one of the Clay Millenium problems with
a 1 Million Dollar prize attached to its solution.

I must confess that I never got far enough in Glimm and Jaffe to figure
out what the "trick" in their construction was.

Let me explain some aspects of the construction given in
Glimm and Jaffe.

First one needs to understand that the construction breaks the Lorentz
symmetry. This is (although they don't draw this connection) because in
irreducible Poincare representations, one can construct only three
commuting coordinates, and their construction is observer-dependent,
i..e, dependent on singling out a preferred time. Of course, the final
theory is again Lorentz invariant.

To motivate construction, one therefore needs to choose a time
coordinate, then one makes analytical continuation to Euclidean time
(i.e. it in place of t), and shows that one gets an SO(4) symmetric
field theory in place of the Lorentz symmetry. The advantage gained is
that the functional calculus over a space with definite metric is
well-defined mathematically (via a limit approach through lattices, or
via Wiener measures) - this is just classical stochastic calculus.

Conversely, and this is the constructive part, given an SO(4) symmetric
field theory, one can choose a direction as Euclidean time and obtain
(via a fairly simple construction detailed in Chapter 7) within that
theory a well-defined Hamiltonian on a suitably constructed Hilbert
space of 3-dimensional fields. This Hamiltonian defines a time
evolution as in ordinary quantum mechanics. The nontrivial part (which
is the Osterwalder-Schrader reconstruction theorem stated in Chapter 7
but proved much later in the book - the forward references in Glimm
and Jaffe are, unfortunately, quite confusing) is to show that the
resulting theory is Lorentz invariant.

Thus the construction reduces to constructing the Euclidean field
theory. This is done via a Lattice regularization; indeed, all lattice
field theory and computation is based on the Euclidean formulation
rather than the Minkowski formulation.

In 2D and 3D, the existing analytic error estimation techniques are
sufficient to prove the existence of the limit with suitably
renormalized operators. In 4D, there are additional technical
problems that have not been overcome so far. But neither has it been
proved that any of the 4D field theories cannot exist. There are some
informal arguments suggesting this or that, but none of them is
conclusive in the sense of having paved the way towards a construction
or a no-go theorem.

Arnold Neumaier

Marc Nardmann

Posted: Thu Mar 08, 2007 8:38 pm

Guest

Thanks to Igor and Arnold Neumaier for your replies. Igor Khavkine wrote:

Quote:

On 2007-02-24, Marc Nardmann <Marc.Nardmann@bigfoot.de> wrote:

In my spare time, I'd like to learn some mathematically rigorous quantum
field theory. John Baez wrote in an old message from May 2003:

The fact is, the logical foundations of QED are poorly understood
unless we water it down by treating it either

1) with an infrared cutoff, perturbatively

(... this is the approach I have seen before, in several variations...)

or

2) with both infrared and ultraviolet cutoffs

Can you or someone else here on the group tell me a reference where this
second approach is carried out? From other comments of John Baez, it
seems that one can prove Borel summability of the resulting formal power
series. I would like to see the precise statement and the proof.

I believe that's the usual euphemism for the lattice formulation of QFT.
The minimal spacing between lattice points provides the ultraviolet
cutoff, while the finite lattice size gives an infrared one.

But if one sends the lattice spacing to 0 and the limit turns out to
exist, then one has removed the ultraviolet cutoff. I got the impression
that John Baez meant something different. He went on:

Quote:

2) means we artificially impose a "cutoff" that makes the electric
charge of every particle goes to zero far from here, as well as in
the distant future and past; we also impose a "cutoff" that rules out
virtual particles with energy or momentum bigger than a certain amount;
we get well-defined answers to questions, which however depend on these
cutoffs.

He also wrote in another thread from April/May 2003:

Quote:

Aaron appears to be claiming that QED
exists in a rigorous way when we put in both an infrared
and an ultraviolet cutoff. This is true. But I don't know
if anyone has tried to rigorously estimate the Lamb shift
or the magnetic moment of an electron as predicted by this
cutoff theory, to demonstrate that it gives "extremely
accurate predictions".

Aaron Bergman replied:

Quote:

I don't know about rigor, but the standard RG type arguments
give you that the predictions of the cutoff theory and what we
normally write down are the same.

John Baez answered:

Quote:

This is the conventional wisdom, and it has the
potential to be made rigorous, because the cutoff
theory can be rigorously constructed - so
what you're saying, unlike most claims about
interacting quantum field theories, is a
claim about a well-defined mathematical structure!

It's not just a murky claim about something that
may not exist. It means something fairly definite,
so I could explain it to a mathematician, and they
could set out to actually prove it!

However, I don't think I've seen anyone actually
try to prove it.

And, it's probably not too easy to prove. It's easy
to understand how the cutoff theory works perturbatively,
but it's a lot harder to get control over it at the
nonperturbative level - i.e., to understand how well the
"sum over all Feynman diagrams" is approximated by perturbation
theory, where you sum over finitely many diagrams. If
we could do this, we'd be a lot closer to understanding
what happens when we remove the cutoff!

(I put "sum over all Feynman diagrams" in quotes, because
this sum probably diverges even with a cutoff, requiring
Borel summation. Also, the only cutoff theory I know how
to rigorously construct in a nonperturbative way uses
canonical quantization, not a path integral.)

That was the complete context, as far as I can see. I might be wrong,
but I don't think that he talked about convergence of lattice models. I
guess he meant that some formal power series, given somehow by doubly
cut off integrals over continuous spacetime, is Borel summable. Such a
result is not mentioned in any of the books I have seen so far.
Comparing this theorem (whatever it says precisely) to the standard
approach should be really interesting.

Anyway, thank you for the references!

-- Marc Nardmann

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