- #1

- 2,370

- 1,408

This is a topic I've mentioned a few times before. Essentially the structure of matter in quantum gauge field theories is unclear to me. I have no clear question here, just some initial discussion points.

So at the first level, it seems a particle based view of quantum field theory is difficult to maintain. In brief on the Hilbert Spaces of interacting field theories there is no well-defined number operator ##N##. In some texts it's said that states in the interacting theory are a superposition of different particle numbers, but in reality the fact that an ##N## operator doesn't exist means there are no states with a well-defined particle number. You can see this explicitly in Glimm, Osterwalder and Schrader's work on ##\phi^{4}_{3}##, where the number operator for the interacting theory diverges as the cutoff is removed.

In addition to this we have Malament's theorem where he shows that relaitivistic quantum theories don't possesses any states with the intuitive property of particles. In essence he shows that there are no local particle creation operators.

Then we have the usual picture of fields and particles being their excitations. Now if particle like states aren't elements of interacting field theory Hilbert spaces, I don't think one can say they are excitations of fields. What one can say is that most field theories can be shown to give rise to states that at late times can be proven to have a well defined number of "clicks" they cause in detectors, with the detectors being represented by specific local observables (details are to be found in the monographs of Araki and Haag, which we can go into, but I don't want to overload the initial post).

At this point we might turn to fields as the fundamental objects. However this remains tricky to me, let's look at gauge theories.

The problem with Gauge theories is that fields carrying the gauge charges cannot be local. If ##\Omega_{\Delta}## is a projector onto states confined to the region ##\Delta## of Minkowski spacetime and ##A## is an observable carrying the gauge charge, then it is always the case that ##\Omega_{\Delta}A \neq 0##. For this reason parameters like ##x## in the quark field ##\psi(x)## are formal, carried over from classical notation, but don't reflect actual localisation at a point.

If one wants ##\psi(x)## to be local, it can only be so as an operator on a Hilbert-Krein space, not on a Hilbert Space of physical states. If one does this there needs to be a condition selecting out the subset of the Hilbert-Krein space containing the physical states (BRST condition in non-rigorous language).

All the operators defined on the actual Hilbert space in gauge theories are necessarily loop like, such as Wilson operators.

All of this is to be found in Chapter 7 of Strocchi's "An introduction to Nonperturbative Foundations of Quantum Field Theory".

So in something like QED the electron and photon fields aren't well-defined local operators and thus actual physical states (which can be localised) can't be associated with them directly. In QCD it is even worse, where the quark and gluon fields, even ignoring this problem, would map one out of the physical Hilbert Space regardless as they carry color charge when all physical states are colorless.

Gauge symmetry itself is formulated in terms of these formal objects ##\psi(x), A_{\mu}(x)## carrying gauge charges and most properties of such gauge symmetries (e.g. conserved charges) do not survive quantization. The only aspect of them that does are local Gauss laws (see Nakanishi "Covariant Operator Formalism of Gauge Theories of Quantum Gravity" and Strocchi's book).

So really QCD's physical content to me appears to be there are states which at early/late times make detectors click in a fashion that corresponds to the properties we give nucleons, i.e. late time states will act like particles with nucleon properties. There is a relation between the scattering angles and amplitudes of these nucleon like states that can be encoded in differential forms expressing local Gauss laws.

However the entire structure of quarks, gluons and to some degree even their local fields, seems like a crutch we use so that we have objects that give nice integrals and implement the Gauss law in a way that is easy to use (as Guage "charges") since it reduces it to group theoretic calculations. However the price is that these objects are a long way from the physical content of the theory.

This seems to render statements like "

So at the first level, it seems a particle based view of quantum field theory is difficult to maintain. In brief on the Hilbert Spaces of interacting field theories there is no well-defined number operator ##N##. In some texts it's said that states in the interacting theory are a superposition of different particle numbers, but in reality the fact that an ##N## operator doesn't exist means there are no states with a well-defined particle number. You can see this explicitly in Glimm, Osterwalder and Schrader's work on ##\phi^{4}_{3}##, where the number operator for the interacting theory diverges as the cutoff is removed.

In addition to this we have Malament's theorem where he shows that relaitivistic quantum theories don't possesses any states with the intuitive property of particles. In essence he shows that there are no local particle creation operators.

Then we have the usual picture of fields and particles being their excitations. Now if particle like states aren't elements of interacting field theory Hilbert spaces, I don't think one can say they are excitations of fields. What one can say is that most field theories can be shown to give rise to states that at late times can be proven to have a well defined number of "clicks" they cause in detectors, with the detectors being represented by specific local observables (details are to be found in the monographs of Araki and Haag, which we can go into, but I don't want to overload the initial post).

At this point we might turn to fields as the fundamental objects. However this remains tricky to me, let's look at gauge theories.

The problem with Gauge theories is that fields carrying the gauge charges cannot be local. If ##\Omega_{\Delta}## is a projector onto states confined to the region ##\Delta## of Minkowski spacetime and ##A## is an observable carrying the gauge charge, then it is always the case that ##\Omega_{\Delta}A \neq 0##. For this reason parameters like ##x## in the quark field ##\psi(x)## are formal, carried over from classical notation, but don't reflect actual localisation at a point.

If one wants ##\psi(x)## to be local, it can only be so as an operator on a Hilbert-Krein space, not on a Hilbert Space of physical states. If one does this there needs to be a condition selecting out the subset of the Hilbert-Krein space containing the physical states (BRST condition in non-rigorous language).

All the operators defined on the actual Hilbert space in gauge theories are necessarily loop like, such as Wilson operators.

All of this is to be found in Chapter 7 of Strocchi's "An introduction to Nonperturbative Foundations of Quantum Field Theory".

So in something like QED the electron and photon fields aren't well-defined local operators and thus actual physical states (which can be localised) can't be associated with them directly. In QCD it is even worse, where the quark and gluon fields, even ignoring this problem, would map one out of the physical Hilbert Space regardless as they carry color charge when all physical states are colorless.

Gauge symmetry itself is formulated in terms of these formal objects ##\psi(x), A_{\mu}(x)## carrying gauge charges and most properties of such gauge symmetries (e.g. conserved charges) do not survive quantization. The only aspect of them that does are local Gauss laws (see Nakanishi "Covariant Operator Formalism of Gauge Theories of Quantum Gravity" and Strocchi's book).

So really QCD's physical content to me appears to be there are states which at early/late times make detectors click in a fashion that corresponds to the properties we give nucleons, i.e. late time states will act like particles with nucleon properties. There is a relation between the scattering angles and amplitudes of these nucleon like states that can be encoded in differential forms expressing local Gauss laws.

However the entire structure of quarks, gluons and to some degree even their local fields, seems like a crutch we use so that we have objects that give nice integrals and implement the Gauss law in a way that is easy to use (as Guage "charges") since it reduces it to group theoretic calculations. However the price is that these objects are a long way from the physical content of the theory.

This seems to render statements like "

*The proton is made of three quarks*" into shorthand for "*There is a state which at late times has localized properties x,y,z. The restriction the Gauss law imposes on its amplitudes can be modeled perturbatively by decomposing its creation operator into a product of three fields containing charges. Though the states corresponding to these fields are unphysical, lacking even positivity, we have conditions (BRST) to recover what we need for the late time particulate state*"
Last edited: