# About states, observables and the wave functional interpretation in QFT with gauge fields

First of all, I'm a mathematician, so forgive me for my possible trivial mistakes and poor knowledge of physics.

In a QFT, we just start with a field (scalar, vectorial, spinorial, gauge etc), so I would like to know what are the observables and the states in this context.

In QFT, the general approach would be by using the Fock space (for the free field case, since I don't really know if this would be true for the interacting one) and getting down, by using the particles associated to the operators $a$ and $a^{\dagger}$, to QM particles (I don't really know if this is true, because the number of particles is not constant and depends on the observer) or by using the wave functional interpretation (a functional on the space of field configurations satisfying Schrödinger equation), though I've heard that this functional is not Lorentz covariant (by the way, any proof?). However, according to this article (David John Baker, Against Field Interpretations of Quantum Field Theory, http://core.ac.uk/download/pdf/11921990.pdf) the wave functional interpretation is equivalent to the Fock space, so, in any case, this interpretation is not physically reasonable.

In AQFT, in contrast, the operators are already given (so we already have the observables). Furthermore, if the Lorentzian manifold is globally hyperbolic, a Cauchy hyper surface would be a possible interpretation for a state.

In other aspect, are the quantized fields of a given QFT really observables in the sense that they measure?

Now, adding gauge fields, everything will be groupoid valued and observables would be defined on quotients by the gauge group. In this context, I haven't really seen anything written about states and I have no idea on how the Fock space would be. The naive approach would be to consider the wave functional interpretation with domain in a groupoid.

Furthermore, if we restrict ourselves to TQFT, CFT or other specific class of field theories, would all this problem be solved?

• An historical remark: the pdf reference you cite seems to be quite out of date concerning the references...the interpretations of QFT provided there and the related discussions/problems are known since the end of the fifties of the last century ;-) – yuggib Apr 17 '15 at 11:07
• I gave a very detailed answer at physicsoverflow.org/30642 – Arnold Neumaier May 1 '15 at 14:11

The algebraic approach gives the better idea of what the states and observables of a quantum theory are, and this holds in infinite dimensional systems as well.

In the modern mathematical terminology, observables of quantum mechanics are the elements of a topological $*$-algebra, and states are objects of its topological dual that are positive and have norm one. The most usual case is to take the $*$-algebra to be a $C^*$ or $W^*$ (von Neumann) algebra; however with such choice unbounded operators are not, strictly speaking, observables (but they can be "affiliated" to the algebra if their spectral projections are in the algebra). The advantage of this abstract approach is that, by the GNS construction, one can immediately associate an Hilbert space to the given $*$-algebra (and a particular state), where the elements of the algebra act as linear operators, and the given state as the average w.r.t. a specific Hilbert space vector.

In usual physical terms, only self-adjoint operators are considered to be observables, for an observable should have real spectrum (and could be associated to a strongly continuous group of unitary operators). The quantum field is, usually, considered to be an observable in a QFT (it is self-adjoint but unbounded, so often it would be affiliated to the $W^*$ algebra generated by its family of exponentials, the Weyl operators); and it is perfectly possible, theoretically, to measure its average value on states (to do it really in experiments, that is all another problem).

Quantum field theories are almost always represented in Fock spaces. However, since the Heisenberg group associated with an infinite dimensional symplectic space is not locally compact, the Stone-von Neumann theorem does not hold and there are infinitely many irreducible inequivalent representations of the Weyl relations, the Fock space being only one of them. To complicate things more, the Haag's theorem states that, roughly speaking, the free and interacting Fock representations are unitarily inequivalent (but that is a problem mostly for scattering theory, not at a fundamental level).

The "wave functional interpretation" (never heard this terminology) is just the functorial nature of the second quantization procedure that can associate to each Hilbert space the corresponding Fock space. This is due to Segal and you may also consult Nelson. The idea is that to each Hilbert space $\mathscr{H}$ one can associate a Gaussian probability space $(\Omega,\mu)$ such that the Fock space $\Gamma(\mathscr{H})$ is unitarily equivalent to $L^2(\Omega,\mu)$, and the map between $\mathscr{H}$ and $\Gamma(\mathscr{H})$($L^2(\Omega,\mu)$) is a functor in the category of Hilbert spaces with self-adjoint and unitary maps as morphisms. The $L^2(\Omega,\mu)$ point of view becomes very natural if one is interested to study QFTs by means of the stochastic integral approach (Feynman-Kac formulas) in euclidean time.

• Thanks for your answer. I've never heard about interacting Fock space, is there any reference? About the wave functional, I don't really know how can I get an Hamiltonian to construct a Schrödinger equation to this functional. Furthermore, in the case of gauge fields, do you know how observables and states would be defined? Actually, I've never seen Wightman axioms for the case of a gauge fields (any reference?), so I don't really know what's a QFT with gauge fields. – user40276 Apr 17 '15 at 10:26
• The interacting Fock space cannot be rigorously constructed in most interesting QFTs; however you may take a look to the second book of Bratteli-Robinson to get an idea (applied on a different context) of the Haag's theorem and the inequivalent vacuum/ground-state representations associated to different QFTs. Also the book by Derezinski and Gerard gives some detail (in the end) on quantization of interacting theories. Finally, you may also try to take a direct look at the original works by Haag himself. – yuggib Apr 17 '15 at 10:38
• Concerning the wave functional, the Hamiltonian in that case would be, roughly speaking, the same as in the Fock representation but with the field replaced by the multiplication by the gaussian functional, and the momentum replaced by the derivative w.r.t. to the aforementioned functional. In general the Hamiltonian has to be a self-adjoint operator on the $L^2(\Omega,\mu)$ space. Anyways I am not completely familiar with this type of description, so take these informations with benefit of the doubt ;-) – yuggib Apr 17 '15 at 10:41
• Finally, gauge theories are not different, in principle, to other field theories. I am not an expert on this context either, but I suggest you to take again a look at the second volume of the Bratteli-Robinson where gauge fields are studied in the language of AQFT, even if the application they have in mind are mostly in statistical mechanics (anyways this should be not so different from what you look for). – yuggib Apr 17 '15 at 10:49
• Sorry, but what do you mean by the Fock representation. There is no sympletic space at the beginning of the construction, so ,given a QFT, how can you associate a Fock representation? – user40276 Apr 17 '15 at 11:12