How is QED solved non-perturbatively, basic outline.

[Spinnor]

On a recent thread about virtual particles I think Tom pointed out that QED can be solved non-perturbatively. Would anyone in the know please sketch out the basic steps, introductory papers, or Google searches. It sounded like we can dispense with virtual particles?

Thanks for any help!

 

[The_Duck]

Lattice gauge theory is one way to get nonperturbative results. Here you discretize spacetime and restrict yourself to a finite volume, which turns the path integral into a regular integral over a large but finite number of variables. Then you can do this integral on a computer. The result has errors due to the discretization of spacetime, the finite volume, and the fact that you end up having to estimate the integral stochastically. But you can make these errors arbitrarily small (at the cost of more computer time) by making the spacetime lattice finer, increasing the spacetime volume, and spending more to time get a better stochastic estimate of the integral. So in this fashion you can solve the theory to arbitrary precision in a completely non-perturbative way.

 

[A. Neumaier]

On a recent thread about virtual particles I think Tom pointed out that QED can be solved non-perturbatively. Would anyone in the know please sketch out the basic steps, introductory papers, or Google searches. It sounded like we can dispense with virtual particles?

Thanks for any help!

Enter the words
nonperturbative QED
into the field Title of the interface at
http://xxx.lanl.gov/multi?group=physics&/find=Search
to get a number of research papers on this.

 

[A. Neumaier]

Lattice gauge theory is one way to get nonperturbative results. Here you discretize spacetime and restrict yourself to a finite volume, which turns the path integral into a regular integral over a large but finite number of variables. Then you can do this integral on a computer. The result has errors due to the discretization of spacetime, the finite volume, and the fact that you end up having to estimate the integral stochastically. But you can make these errors arbitrarily small (at the cost of more computer time) by making the spacetime lattice finer, increasing the spacetime volume, and spending more to time get a better stochastic estimate of the integral. So in this fashion you can solve the theory to arbitrary precision in a completely non-perturbative way.

Arbitrary precision is an euphemism. All high precision calculations in QED (8-12 significant digits of accuracy) are done by perturbation theory in either the fine structure constant or the inverse speed of light.

With todays large computers, lattice gauge theory only produces low precision results (perhaps 5% error)

 

[tom.stoer]

Lattice gauge theory is somehow complementary to perturbation theory. It is suitable for large coupling, close to the QCD energy scale where perturbation theory breaks down. But for small coupling it is rather useless.

One problem with lattice gauge theory is that you model a finite, small portion of space, whereas for typical scattering experiments you would need a rather large region to be put on the lattice. I do not know whether lattice gauge theory has been proven to exist mathematically. I mean it exists for finite number of lattice points, but I do not know whether there is a proof that it exists in principle for the continuum limit (I doubt that such a proof exists b/c otherwise a millenium problem would have been solved - at least partially)

 

[A. Neumaier]

I do not know whether lattice gauge theory has been proven to exist mathematically. I mean it exists for finite number of lattice points, but I do not know whether there is a proof that it exists in principle for the continuum limit (I doubt that such a proof exists b/c otherwise a millenium problem would have been solved - at least partially)

The rigorous construction of interacting quantum field theories in 2 and 3 dimensions proceed just by taking the continuum limit of lattice theories.

In 4D, lattice simulations indicate that a simple continuum limit of lattice phi^4 theory and lattice QED apparently do _not_ give results in agreement with perturbative phi^4 theory and perturbative QED, whereas for nonabelian gauge theories, the limit seems to be achieved.
This is blamed on the existence of a Landau pole in 4D phi^4 theory and QED, but not in (asymptotially free) nonabelian gauge theories.

Thus for QED, it is not even clear how to take the correct lattice limit.

 

[stevendaryl]

The rigorous construction of interacting quantum field theories in 2 and 3 dimensions proceed just by taking the continuum limit of lattice theories.

In 4D, lattice simulations indicate that a simple continuum limit of lattice phi^4 theory and lattice QED apparently do _not_ give results in agreement with perturbative phi^4 theory and perturbative QED...

But is the disagreement to be interpreted as a flaw in the lattice calculations, or a flaw in the perturbative theory?

 

[A. Neumaier]

But is the disagreement to be interpreted as a flaw in the lattice calculations, or a flaw in the perturbative theory?

Since perturbative QED is known to be in excellent agreement with experiment, it is on the surface a problem of the lattice approach to QED. But in fact it is a problem of how to take the limit correctly. That we do not know how is just a symptom of the lack of a rigorous mathematical understanding of QED.