# Are field theories special?

Our best descriptions of the microscopic world, that satisfy many fundamental requirements (as we know them today), are field theories.

Is there something fundamental about field interactions, or are they simply one tool that is flexible enough to offer a good description of fundamental physics?

I'm not asking whether the standard model is complete, but rather how unique our current framework is. If for example advanced life elsewhere has a mathematical description of fundamental physics - is it a field theory, or possibly a completely different approach?

• Have you read this en.wikipedia.org/wiki/Field_%28physics%29#Field_theory ? you are asking how unique the mathematical systems we have developed . I do not think there exists an answer in physics. – anna v Oct 13 '15 at 18:04
• I think this is an interesting question. It might be a little less "pathological" if slightly re-worded to ask something more like "what constraints on the physics are imposed by describing things via field theory" or something like that :D – DanielSank Oct 13 '15 at 18:17
• @annav yes - what about that page exactly? I don't know how you are defining uniqueness when talking about mathematical systems. I thought there might exist alternative systems already described, or maybe a strong statement about why all other such systems that exhibit Lorenz inv., causality, etc., are isomorphic to some field description. – anon01 Oct 13 '15 at 18:44
• @DanielSank that's a good thought. I am fishing for something a bit broader, so I may leave it as-is for a little while. – anon01 Oct 13 '15 at 18:47

## 3 Answers

Comments to the question (v2):

1. There are various speculations that spacetime (and fields on spacetime) are not fundamental concepts/objects but rather emerging/effective properties of an underlying theory of everything (ToE) yet to be discovered. See e.g. many talks online by Nima Arkani-Hamed, who often stresses this point.

2. A popular candidate for the ToE is string theory (ST) rather than quantum field theory (QFT). [Strictly speaking, ST is a non-linear sigma model (NL$\Sigma$M) on a 2D world-sheet rather than spacetime, and hence a field theory (FT) in that sense.] There are efforts to develop a second quantized version of ST known as string field theory (SFT).

3. A related issue is: if a FT does exist, does it have a Lagrangian formulation? See e.g. this Phys.SE post and links therein.

Why we use QFT? Locality, firstly. There is no initely fast propagation of information. For this you got to have some field through which signal spreads at speeds less or equal to speed of light. On the pther hand, STofR and QM imply that number of particles is not conserved. Ordinary QM can not reproduce this fact but QFT can. So QFT is just quantization of classical field. Also, in Lagrangian formalism thery is manifestly inariant to Lorentz transformations. Is there some other theory, better than QFT? I dont know. Also, QFT somewhat resolves wave particle duality introduced by QM.

• Yes, there are many good reasons why we have chosen field theories - not least of all many of the reasons you mention. The question is getting at whether there are alternative descriptions that also - at least on observed scales - obey such properties. – anon01 Oct 13 '15 at 18:52
• O, I get it. Well, as far as I know, there is no such theory yet. Most succesfull and accurate theory for now is QFT. But of course you can think of particles as fundamental. But QFT has proven its worth, so as far as know, no, there is no such theory that would do it. – Žarko Tomičić Oct 13 '15 at 19:48

Not only is there a room for an alternative approach, one will have to be found because neither of the two existing ones is overall consistent. The standard model as we know it is supposed to be a perturbative approximation to a yet unknown master theory. The most popular particle interpretation of it is in terms of a Fock space that actually involves no fields, and has particle configurations and creation/annihilation operators instead. It is good enough for perturbative computations with Feynman diagrams, but can't work non-perturbatively because as Malament showed no position operator can be defined even in free QFT. No positions - no particles.

It was originally thought that the field, a.k.a. wavefunctional, interpretation can escape this problem because it can make do with non-point localization operators. Instead of particle configurations it describes QFT states as superpositions of classical fields (the wavefunctionals), and observables as operator-valued fields. This is analogous to the Schrodinger interpretation of quantum mechanics. However, since the wavefunctional space is mathematically equivalent to the Fock space, it can't work either. More explicitly the problem is the following. On one hand, spacetime symmetries are supposed to produce physically equivalent descriptions. On the other hand, rotations of so-called coherent states produce unitarily inequivalent representations in the wavefunctional space, so "equivalent" states have physically inequivalent field content. See Baker's Against Field Interpretations of Quantum Field Theories.

The plurality of unification schemes show how non-unique the current framework might be. String theory alone offers vastly different reinterpretations of it, while loop quantum gravity tries to preserve at least some flavor of the field interpretation.

• "creation/annihilation operators instead" are they not Operator fields?analogous to vector, tensor fields at a point (x,y,z,t) ? – anna v Oct 14 '15 at 4:29