I've just acquainted myself with 'effective field theories', and have found myself thinking about the following question. In the effective field programme we generate low-energy theories from high-energy theories by 'integrating out' the high-energy modes, and this process is said to be irreversible on the grounds that too much information is lost in this process for us to be able to reconstruct the original theory from the low-energy theory that results from this integration process. May I ask: does anyone know of a low-energy theory that can be shown to be a low-energy approximation of two (or more) distinct high-energy QFTs? I ask only because I think it would be very interesting if we could show that a given quantum field theory could be derived from a plurality of more fundamental theories. For then the irreversibility is not just a reflection of our inability to reconstruct some one unique theory from the integrated-out version but more a fact about how the world might be structured. Thanks folks!
I think all proposed models for a GUT are examples - they all give the SM at low energies (well, at least that is expected), but different results for high energy.
Thanks mfb! If you happen to know of a reference where this is discussed explicitly I'd love to hear of it -- or even somewhere where the still-in-contention candidate GUTs are enumerated would be useful. No worries if not though, of course.
It is pretty much the entire point, and certainly the power, of effective field theory that many high energy theories can correspond to the same low energy theory. As another example aside from GUT models, you can explore the space of models that lead to neutrino masses by systematically "opening up" (un-integrating in some sense...) the various effective operators that lead to neutrino masses, as this paper does: http://arxiv.org/abs/1212.6111
I actually had a similar thought a little while back... I am a lattice field theoriest, and also fond of the LQG approach, and when I found out that many continuum theories can be formulated as distinct theories on the lattice and renormalize to the same conimuum limit. It got me thinking about how nature handles ambiguity. From a historical perspective, whenever there is ambiguity, nature usually demands that it is quite important (guage, probability amplitudes, etc...) and so if it is possible that at extremely high energies, distinct theories all yeild the same low energy effictive limits, perhaps nature thinks the same way, and does not even perfer a high energy theory, but rather an equivilence class of theories which all renormalize to the correct limit.
No problem. FYI this line of reasoning quickly leads into conversations about naturalness; the hierarchy problem is a problem because from the effective field theory perspective the Standard Model Higgs sector doesn't make a lot of sense: if you stick in by hand any generic new physics at any scale then the weak scale is not "protected" from it. The generic effect of going from the high scale theory to the low scale theory is *not* to reproduce the Standard Model, rather the effects of the high scale theory cause the electroweak scale to stay up at the scale of new physics. If the Standard Model was "well behaved" from this perspective you would expect practically any new physics to be ok; instead things have to be very carefully arranged or you don't get our low energy world. Generally it is expected that this is a sign that there is some new symmetry like SUSY that saves us and makes the world as it exists less surprising (thus a lot of those GUTs are SUSY GUTs). There are some dubious assumptions about probability theory floating around in these arguments, but I think the spirit of them is solid.
@jfy4: I can't follow you. Do you mean theories with different high-energy behavior? In that case, we can (at least in theory) distinguish them with very high-energetic processes, right? How can the universe be in some equivalence class then?
Here is an example. Suppose I have a low-energy theory phi-fourth theory ##\mathcal{L} = \frac{1}{2}\partial_\mu \phi \partial^\mu \phi - \frac{1}{2} m \phi^2 - \frac{1}{24} \lambda \phi^4## We can add pretty much any particles we want with masses ##\gg m## and we will get the the same low-energy theory, as long as the new high-energy physics doesn't break any of the symmetries of the above phi-fourth theory. For instance we could add another scalar with mass ##M \gg m##: ##\mathcal{L} = \frac{1}{2}\partial_\mu \phi \partial^\mu \phi - \frac{1}{2} m \phi^2 - \frac{1}{24} \lambda \phi^4 + \frac{1}{2}\partial_\mu \sigma \partial^\mu \sigma - \frac{1}{2} M \sigma^2 - \frac{1}{24} \lambda' \sigma^4 + \lambda'' \phi^2 \sigma^2## But this is just the simplest possibility; we can add any sort of high-energy particle content and interactions we want [for instance, ten new high-mass scalars instead of one] and still get the same low-energy theory as long as we respect the symmetry under ##\phi \to -\phi##. This has to be the case because the low-energy theory must be a renormalizable theory of a single scalar with a symmetry under ##\phi \to -\phi##, and the only possibility is the original phi-fourth theory. The only low-energy effect of the high-mass particles is to shift the mass and self-coupling of ##\phi## through loop effects.
Thanks very much The_Duck. The concrete example is really helpful. But can I ask why shifting the mass and self-coupling doesn't count as an important difference between the two low-energy theories? Thanks again!
If you want to get the exact same low-energy physics in both cases, all you have to do is fiddle with the underlying bare couplings. When you add a new high-mass particle, you'll change the mass and couplings of your low-mass particle. But you can restore their original mass and couplings by adjusting the bare mass and couplings to cancel out the shift. Then once you've made this adjustment to the bare parameters of the high-energy theory, it has the exact same low-energy physics as the original theory.
Great. Final question: I appreciate from your comment that we may -- from a mathematical point of view -- fiddle with the parameters in order that they come out the same in the two theories. But can we expect the parameters of the low-energy limit of each theory to be numerically identical from a physical point of view? Or given that we have to match the parameters to experiment anyway, is this question perhaps not well-defined from the point of view of the physics?
If you write down two different high-energy theories that have the same low-energy particle content and symmetries, then the resulting two low-energy effective Lagrangians will have the same form. But the numerical values of the parameters in the low-energy Lagrangian will be different, unless you specifically tune the numerical parameters of the high-energy theories in order to make the two low-energy theories match. You could do this matching by requiring the two high-energy theories to produce the same numerical predictions for a few well-chosen low-energy scattering processes.
well, the idea I suggested implies that when we could probe those energies, we could not distinguish which theory is the correct one by different expirements. A situation where different theories predict different phenomena, some of which may be mutually exclusive, yet we observe phenomena from both.