Does QED Originate from Non-Relativistic Systems?

[vanhees71]

I meant more the part that for QED the standard techniques of perturbation theory in the continuum is more applicable. Of course, I also agree with the statement that one needs to make contact with the continuum limit in lattice QCD and to achieve this one needs renormalized perturbative lQCD. As I stressed several times, I don't think that lattice QED is of any practical significance.

 

[atyy]

Yes, the Wilsonian point of view is about "coarse graining", but it's not different in content from the older RG approaches, which are just more practical in calculations leading to phenomenology. As I said, the running of the coupling tells you, where PT breaks down. For QED that's at high energies since the coupling becomes large at high energies.

Of course, I can only agree with the quote of Capitani, you cited in #171.

Ok, so we agree then.

 

[A. Neumaier]

so we agree then.

except on the part involving lattice QED; see #175 and #176!

 

[dextercioby]

As also privately mitgeteilt, thank you, Arnold, for providing references to the state of the art work on rigorous QFT.

 

[atyy]

Of course, I can only agree with the quote of Capitani, you cited in #171.

So if one agrees with Capitani, would one agree that it is alright to consider the high energy theory (at some high but finite energy cutoff) from which QED is derived to be non-relativistic?

Specifically, one starts with lattice QED at fine but finite spacing, which is a non-relativistic theory. Then the usual covariant perturbative continuum QED is derived as a low energy effective theory. Obviously, this is only in principle, as one would not use this method for practical calculations.

An analogy from condensed matter is the non-relativistic theory theory of the graphene lattice which gives rise to relativistic massless Dirac fermions as a low energy effective theory.

 

[vanhees71]

There's not the slightest hint of the violation of Poincare symmetry. Only recently the measurement of antihydrogen energy levels as well as the magnetic moment of the antiproton once more indicate the precise fulfillment of the CPT theorem, following from Poincare invariance and locality of QFT underlying the Standard Model.

Of course, this doesn't rule out possible violations of Poincare symmetry at higher energies, where the Standard Model becomes invalid. However, the low-energy effective theory, defined by perturbative continuum QED, is obviously a very accurate description which is Poincare symmetric for all practical purposes.

 

[atyy]

There's not the slightest hint of the violation of Poincare symmetry. Only recently the measurement of antihydrogen energy levels as well as the magnetic moment of the antiproton once more indicate the precise fulfillment of the CPT theorem, following from Poincare invariance and locality of QFT underlying the Standard Model.

Of course, this doesn't rule out possible violations of Poincare symmetry at higher energies, where the Standard Model becomes invalid. However, the low-energy effective theory, defined by perturbative continuum QED, is obviously a very accurate description which is Poincare symmetric for all practical purposes.

Here you are answering with real data. In real life, lattice QED will fail way below its cutoff because it doesn't incorporate the weak and strong interactions.

Would you agree that a theory with an energy cutoff cannot be truly Poincare invariant?

 

[A. Neumaier]

ne starts with lattice QED at fine but finite spacing, which is a non-relativistic theory. Then the usual covariant perturbative continuum QED is derived as a low energy effective theory. Obviously, this is only in principle

This is not even in principle, only in your imagination.

Please point to a paper where the in principle proof is given that the usual covariant perturbative continuum QED is derivable as a low energy effective theory.

An analogy from condensed matter is the non-relativistic theory theory of the graphene lattice which gives rise to relativistic massless Dirac fermions as a low energy effective theory.

This is only a hoped-for analogy - until someone proves your claim that one can actually construct low energy continuum QED is in 3 space dimensions from a lattice!

The starting point is not a discretized version of the theory one ends up with (as you propose it for getting low energy continuum QED from lattice QED), but a quite different lattice theory! We discussed this at length in https://www.physicsforums.com/posts/5294008/ and later posts there.

Note also that graphene a much simpler situation. Graphene and the resulting relativistic Dirac fermions are in 2 space dimensions only, and the Fermions resulting are massless. In comparison, relativistic interacting quantum field theories in 2 space-dimensions satisfying the Wightman axioms have been constructed, even in the massive case; the problem is here much simpler because of superrenormalizability. We discussed already much of this earlier: https://www.physicsforums.com/posts/5443402/ and other posts in that thread.

 

[A. Neumaier]

Would you agree that a theory with an energy cutoff cannot be truly Poincare invariant?

As long as the cutoff is fixed the theory is not Poincare invariant. But perturbative QED can be constructed without any cutoff at all!

Even in lattice QCD, which you so like, the extrapolation to the continuum limit must be done to compare with experiment, and Poincare invariance is believed to emerge in this limit. For some lower-dimensional lattice theories this can even be proved!

 

[A. Neumaier]

Specifically, one starts with lattice QED at fine but finite spacing, which is a non-relativistic theory. Then the usual covariant perturbative continuum QED is derived as a low energy effective theory.

You should first tell us how this can be done without performing the continuum limit (extrapolating for lattice spacing to zero)! This is needed if one wants to recover the theory with whose discretized action one started!

 

[Elias1960]

Lattice QED was specifically designed to have the right continuum limit. So it's not a good example if you're wanting to show that Lorentz invariance can arise natural as a continuum approximation to a non-invariant theory. To be convincing you would have to have an independent motivation for lattice QED that did not rely on having the right continuum limit.

What about a quite arbitrary atomic model which, in the large distance limit, gives a standard wave equation for its sound waves of type $(\partial_t^2 - c^2\partial_i^2) u = 0$? Such things exist everywhere in a quite natural way, without any humans inventing them. But the wave equation has Lorentz symmetry too.