Why Do We Assume Ultraviolet Divergences are Physical

[maverick_starstrider]

Pardon me if this is a really silly question, my knowledge of field theory pretty much only comes from Condensed Matter. However, I know, before RG, QFT had a big problem with integrals blowing up unless you assume some cut-off frequency exists. My question is, what is WRONG with a cut-off frequency? Doesn't a cut-off frequency imply a quantization of space. Wouldn't QFT with a cut-off frequency be similar to a doubly-special view of relativity? Why do we think it is wrong to say that space and time are quantized (or why is imposing a cut-off wavelength not the same as saying space is quantized should I be wrong about that). Also, is there an experiment that can be performed that could actually put upper and lower bounds (should they exist) on the value of the cut-off frequency?

 

[Physics Monkey]

There is absolutely nothing with a cutoff. The Fermi theory of weak interactions has a cutoff. Many modern chromodynamic calculations use effective theories that are explicitly only valid up to some energy scale. Examples include heavy quark effective theory and soft collinear effective theory. Beyond this scale new degrees of freedom appear (it doesn't have to be a lattice or some ultimate cutoff). I think most field theorists now days take the cutoff quite seriously.

Since you mention condensed matter physics, you must be aware that at low energy all condensed matter systems are described by effective field theories. Of course, in this context there is usually an explicit cutoff provided by the lattice.

The only more serious concern is finding a cutoff that preserves a symmetry of interest. For high energy physics, one such symmetry is lorentz invariance. For simple field theories coming from lattice models one sometimes finds that lorentz invariance is automatically restored in the infrared i.e. all lorentz violating operators are irrelevant. However, the standard model is more complicated and would have to be much more finely tuned at the putative lattice scale (ignoring gravity) to realize low energy lorentz invariance, especially to the accuracy we have observed.

The challenge relevant for high energy physics would be to find a physical (not dim reg or pauli-villars) regulator that naturally preserves lorentz invariance (perhaps with very suppressed corrections) and includes gravity. This is of course a very complicated problem.

Hope this helps.

 

[meopemuk]

My question is, what is WRONG with a cut-off frequency? Doesn't a cut-off frequency imply a quantization of space. Wouldn't QFT with a cut-off frequency be similar to a doubly-special view of relativity? Why do we think it is wrong to say that space and time are quantized (or why is imposing a cut-off wavelength not the same as saying space is quantized should I be wrong about that). Also, is there an experiment that can be performed that could actually put upper and lower bounds (should they exist) on the value of the cut-off frequency?

The only problem(s) with space quantization is that nobody knows how it looks like, why it should be there, and what's the physics behind it. Moreover, there are no experiments that can probe "quantized space". Otherwise, this is a very respectable concept.

Eugene.

 

[maverick_starstrider]

The only problem(s) with space quantization is that nobody knows how it looks like, why it should be there, and what's the physics behind it. Moreover, there are no experiments that can probe "quantized space". Otherwise, this is a very respectable concept.

Eugene.

But is there fundamentally anything WRONG with saying that the universe is a quantized grid (like a computer screen with pixels) and, potentially, time is also similarily quantized and thus all integrals over wavenumber should be cut-off at the lattice size. The size of this lattice needing to be determined by experiment. Does this violate anything? Is there something wrong with this? I feel like this puts a preferred reference frame on things up to an order of our space quanta but would this deviation from the math of no preferred reference frame introduce profound deviations from standard results or tiny ones? If they are tiny could we experimentally detect them? Obviously this would all depend on what the actual SIZE of the space quanta is but can experiment tell us anything about that?

 

[meopemuk]

But is there fundamentally anything WRONG with saying that the universe is a quantized grid (like a computer screen with pixels) and, potentially, time is also similarily quantized and thus all integrals over wavenumber should be cut-off at the lattice size. The size of this lattice needing to be determined by experiment. Does this violate anything? Is there something wrong with this? I feel like this puts a preferred reference frame on things up to an order of our space quanta but would this deviation from the math of no preferred reference frame introduce profound deviations from standard results or tiny ones? If they are tiny could we experimentally detect them? Obviously this would all depend on what the actual SIZE of the space quanta is but can experiment tell us anything about that?

The idea of discrete/quantized space is very old. However, so far it hasn't led to any consistent physical theory and there is not a single experiment supporting it. You are welcome to think about this hypothesis, but in my personal opinion this would be a waste of time.

Eugene.

 

[Physics Monkey]

There is essentially nothing wrong with imagining that the non-gravitational part of the standard model actually lives on some very fine grained lattice. I say essentially only because of some issues about regulating chiral gauge theories on a lattice, but that is another discussion for another day.

Just as in condensed matter physics, the low energy theory may look Lorentz invariant, but the presence of the lattice can always be detected by experiments probing sufficiently high energies or short distances. In formal terms, there will be irrelevant operators in the low energy theory that break Lorentz invariance and dominate the physics at high enough energies. More dangerous are relevant or marginal operators that break Lorentz invariance at low energy and would have to be tuned very precisely at the lattice scale to recover low energy Lorentz invariance. For example, even keeping rotational invariance, the standard model would have on the order of tens of relevant and marginal operators that would have to be fine tuned. Physicists have looked very hard for violations of Lorentz invariance, and having found none so far, we are forced to conclude that if the lattice picture is correct, it is a highly fine tuned theory.

Of course, gravity is another story entirely.

You may find this paper quite interesting http://arxiv.org/abs/hep-ph/9812418