- #36
bhobba
Mentor
- 10,796
- 3,664
vanhees71 said:Most of the fundamental fields in the standard model are not representing observables directly:
Got it
Thanks
Bill
vanhees71 said:Most of the fundamental fields in the standard model are not representing observables directly:
strangerep said:Why the change from "configuration" to "history"?
strangerep said:And why not call it simply "field on spacetime"?
strangerep said:I guess that's to make a distinction between smooth/rough fields?
strangerep said:But do we ever need rough fields in QFT?
strangerep said:In "aspects of the concept of fields" appears the notation ##\delta_{EL}=0##. Later, under Remark 3.2. (possible field histories), it reappears as ##\delta_{EL}{\mathbf L}=0##. Is that a typo?
vanhees71 said:but the choice of words, which may mislead a beginner to think that the fundamental field operators are observables.
bhobba said:But as to why it happens
I'd propose to use ''quantity'' for any functional on the space of field histories, and ''observable'' for those quantities that are actually observable.[URL='https://www.physicsforums.com/insights/author/urs-schreiber/']Urs Schreiber[/URL] said:There are various further conditions on observables which we will eventually consider, forming subspaces of gauge invariant observables (def. 11.2), local observables (def. 7.35 below), Hamiltonian local observables (def. 8.12 below) and microcausal observables (def. 14.5). While in the end it is only these special kinds of observables that matter, it is useful to first consider the unconstrained concept and then consecutively characterize smaller subspaces of well-behaved observables. In fact it is useful to consider yet more generally the observables on the full space of field histories (not just the on-shell subspace), called the off-shell observables.
bhobba said:I am sure you have read, as have I, the following:
https://www.amazon.com/dp/0387493859/?tag=pfamazon01-20
I don't know if you have read Ballentine:
https://www.amazon.com/dp/9814578576/?tag=pfamazon01-20
A number on this forum, me included, tend to think of Ballentine as our bible on QM.
A. Neumaier said:I'd propose to use ''quantity'' for any functional on the space of field histories
I'm not so much concerned about the use of the term "observable". That's anyway finally defined as something that's indeed measured in the lab, and the (far from trivial!) task of any quantum-field theory is to map the formalism to this operational definition of "observable". I'm a bit quibbled, why you use the term "field histories". A "history", as I understand the term, is a sequence of observed facts, but as we seem to agree upon, the fundamental fields are usually not directly observables in the formalism but are used to construct observables (or more carefully stated the corresponding representing operators of observables) via a local realization of the Poincare algebra.[URL='https://www.physicsforums.com/insights/author/urs-schreiber/']Urs Schreiber[/URL] said:Besides the condition of gauge invariance on observables that you mention, there are various further conditions which need to be considered. In the end the quantum algebra is based only on those observables which in addition are also local and microcausal. Also "on-shell" will have to be added as a qualifier, since it is useful also to consider off-shell observables.
For this reason, it is natural to say just "observable" for any functional on the space of field histories, and then eventually to add adjectives as one specializes. Citing from the lead-in paragraphs of chapter 7:
There are various further conditions on observables which we will eventually consider, forming subspaces of gauge invariant observables (def. 11.2), local observables (def. 7.35 below), Hamiltonian local observables (def. 8.12 below) and microcausal observables (def. 14.5). While in the end it is only these special kinds of observables that matter, it is useful to first consider the unconstrained concept and then consecutively characterize smaller subspaces of well-behaved observables. In fact it is useful to consider yet more generally the observables on the full space of field histories (not just the on-shell subspace), called the off-shell observables.
This may be the current state of the art in 4D relativistic quantum theory, but this is only because we still lack the right mathematical tools. A nonperturbative mathematical construction of any QFT will necessarily produce a representation of the (bounded part of the) quantum algebra on a Hilbert space of physical states. Each positive linear functional provides such a Hilbert space, and inequivalent representations are accounted for by taking a direct sum.[URL='https://www.physicsforums.com/insights/author/urs-schreiber/']Urs Schreiber[/URL] said:epresenting the latter on a Hilbert space may be a convenient tool for ensuring positivity of such functionals, but it is neither necessary nor in general possible or helpful.
Well, then I respectfully challenge you to derive the usual quantum angular momentum spectrum without using a Hilbert space (directly or indirectly). Cf. Ballentine section 7.1 where he derives that spectrum from little more than the requirement to represent SO(3) unitarily on an abstract Hilbert space.[URL='https://www.physicsforums.com/insights/author/urs-schreiber/']Urs Schreiber[/URL] said:When looking closely at field theory, it turns out that this is unhelpful, that other concepts are prior, and that a Hilbert space of states is an addendum to be considered after the theory has been constructed algebraically. States are a priori positive linear functionals on the abstractly defined algebra of quantum observables, and representing the latter on a Hilbert space may be a convenient tool for ensuring positivity of such functionals, but it is neither necessary nor in general possible or helpful.
OK, that's fine. But perhaps you could insert something like the last part of this sentence near the place where you define "field history"?[URL='https://www.physicsforums.com/insights/author/urs-schreiber/']Urs Schreiber[/URL] said:Because it's more appropriate in the Lorentzian setup, where a section of the field bundle is a field configuration over every spatial slice together with its change in time, hence a history.strangerep said:Why the change from "configuration" to "history"?
[URL='https://www.physicsforums.com/insights/author/urs-schreiber/']Urs Schreiber[/URL] said:The QFT textbook to recommend, as I did before, which gets the concepts right, is
- Katarzyna Rejzner, Perturbative Algebraic Quantum Field Theory, Mathematical Physics Studies, Springer 2016 (web)
strangerep said:perhaps you could insert something like the last part of this sentence near the place where you define "field history"?
strangerep said:derive the usual quantum angular momentum spectrum without using a Hilbert space (directly or indirectly). Cf. Ballentine section 7.1 where he derives that spectrum from little more than the requirement to represent SO(3) unitarily on an abstract Hilbert space.
vanhees71 said:Well, then I've big trouble making any sense of QFT at all since then I don't know, how to define states and how to interpret the S-matrix physically, or how do you define states and probabilities in a theory without an underlying Hilbert space.
Well,... no, I don't notice that. (Do you have a copy of Ballentine in front of you?)[URL='https://www.physicsforums.com/insights/author/urs-schreiber/']Urs Schreiber[/URL] said:You'll notice that what this computation (in Ballentine or elsewhere) really uses is the commutation relations of the angular momentum observables together with one ground state they act on.
Nanni's section 12 (wherein he searches for a suitable factorization of the Hamiltonian) relies on results back in sections 6,7,8 concerning angular momentum properties of wave functions found from the usual Schrodinger equation.For amplification see for instance also the quantization of the hydrogen atom in the Heisenberg picture, e.g. section 12 of Nanni 15.
Well, I guess that refers to free particles in the Heisenberg picture (which is anyway the most natural picture to use; it's only due to overemphasizing wave mechanics rather than the canonical approach, why in QM 1 we usually start with the Schroedinger picture). Then we construct the Fock space as the occupation-number eigenstates (usually in the single-particle momentum representation). Now, isn't his a Hilbert space (at least for a finite quantization volume imposing periodic boundary conditions)? I'll have a look at the mentioned lecture notes by Fredenhagen et al.[URL='https://www.physicsforums.com/insights/author/urs-schreiber/']Urs Schreiber[/URL] said:But you are secretly very familiar with this Heisenberg-picture perspective, because this is precisely what traditional perturbative QFT is based on. There we have the ##n##-point functions or scattering amplitudes defined as linear functionals on the collection of products of field observables, with respect to a "vacuum state" which is implicitly defined thereby, and no Hilbert space needs to be mentioned to speak about these. In fact for pQFT in curved spacetime no such Hilbert space needs to exist, and still the standard machine applies.
strangerep said:Then he uses positivity of the Hilbert space inner product
strangerep said:Nanni's section 12 (wherein he searches for a suitable factorization of the Hamiltonian) relies on results back in sections 6,7,8 concerning angular momentum
vanhees71 said:Well, I guess that refers to free particles
vanhees71 said:in the Heisenberg picture (which is anyway the most natural picture to use; it's only due to overemphasizing wave mechanics rather than the canonical approach, why in QM 1 we usually start with the Schroedinger picture).
vanhees71 said:Then we construct the Fock space as the occupation-number eigenstates (usually in the single-particle momentum representation).
vanhees71 said:Now, isn't his a Hilbert space
[URL='https://www.physicsforums.com/insights/author/urs-schreiber/']Urs Schreiber[/URL] said:There are issues of dense sub-domains of unbounded operators, of self-adjoint extensions of symmetric operators on a Hilbert space etc. that need to be dscussed if one is really speaking Hilbert spaces.
bhobba said:Whats your view of resolving it with Rigged Hilbert Spaces and the Generalized Spectral Theorem?
[URL='https://www.physicsforums.com/insights/author/urs-schreiber/']Urs Schreiber[/URL] said:On the one hand there are intrinsic problems with applying the Hilbert space Schrödinger picture to QFT even in principle (Torre-Varadarajan 98).
[URL='https://www.physicsforums.com/insights/author/urs-schreiber/']Urs Schreiber[/URL] said:No, this applies generally, also to interacting theory. In pQFT what changes as one turns on the interaction is that the Wick normal ordred product on observables gets deformed into the "retarded products" (the infinitesimal version of Bogoliubov's formula) and then it's still the vacuum state (or more generally Hadamard state) which serves to send such products of observables to their actual expectation value, which is the corresponding correlator.
Absolutely. And in the Schrödinger picture a Hilbert space of states needs to be assumed from the outset, while in the Heisenberg picture it is an afterthought, if it exsists at all. What matters in the Heisenberg picture is that we know one state, the vacuum state, given as a positive linear non-degenerate function which sends observables to their expectation value. In optimal situation the GNS construction allows to reconstruct a Hilbert space from this, but not generally (notably not in interacting pQFT).
And that's where it begins to be misleading. First of all the representation of observables as operators on a Fock space is not necessary for the construction of the Wick algebra and second it does not generally exist for QFT on curved backgrounds. Instead what one needs is the Hadamard propagator which allows to construct the Wick algebra of observables and at the same time defines a single state (positive linear non-degenrate functional on observables) on this.
bhobba said:Dirac always disliked re-normalization.
vanhees71 said:the physical interpretation of Heisenberg field operators is highly non-trivial
vanhees71 said:So the trick is that you only need the vacuum state and then reconstruct everything through the N-point functions, defined as "vacuum expectation values"? That's very interesting since it sounds intuitively to be sufficient to define S-matrix elements for definite scattering processes since for asymptotic free states you have a particle interpretation.
Yes, that's what "enters". But section 8 doesn't derive the quantum angular momentum spectrum. The output is important here, not just what "enters". The closest it gets is a passing mention of an eigenvalue equation for ##L^2## in terms of ordinary wave functions.[URL='https://www.physicsforums.com/insights/author/urs-schreiber/']Urs Schreiber[/URL] said:Section 8 that the discussion relies on is aptly titled "Commutation relations". That means: The algebra structure of the algebra of observables. That's what enters.
"Most" being the key word here. The point of my challenge to try and establish whether derivation of the quantum angular momentum spectrum is one of the cases in QM for which a Hilbert space is essential.You can tell that most constructions in quantum mechanics don't necessarily need the Hilbert space concept [...]
It's about physics. You have no particle interpretation of transient states. For practical purposes, it's a delicate issue. One example is the off-equilibrium production of photons in heavy-ion collisions. There was quite some debate due to these problems. We have investigated it for a simple toy model (photon production due to a time-dependent scalar background field):[URL='https://www.physicsforums.com/insights/author/urs-schreiber/']Urs Schreiber[/URL] said:Here I am not fully certain what this is arguing about. If you mean Heisenberg picture as opposed to interaction picture, hence non-perturbative as opposed to perrturbative, then there is a dearth of examples, sure, but no conceptual issue.
I don't understand what's the issue with angular momentum. It's a nice operator algebra of a compact semisimple Lie algebra and as such doesn't make any trouble at all in the standard Hilbert-space theory. You construct them algebraically via raising- and lowering operators or, even more convenient, using the fact that the 2D harmonic oscillator has SU(2) symmetry and construct everything with annihilation and creation phonon operators.strangerep said:Yes, that's what "enters". But section 8 doesn't derive the quantum angular momentum spectrum. The output is important here, not just what "enters". The closest it gets is a passing mention of an eigenvalue equation for ##L^2## in terms of ordinary wave functions.
"Most" being the key word here. The point of my challenge to try and establish whether derivation of the quantum angular momentum spectrum is one of the cases in QM for which a Hilbert space is essential.
You're not the first person to whom I have offered this challenge. But so far, no one has actually provided a satisfactory response (nor reference) to the point of the challenge, instead evading that point by giving references that don't actually address the point, and (eventually) by unhelpful denigration of other authors. I grow concerned that you seem to be sliding into the latter category.
I also notice that you ignored my question about whether you have a copy of Ballentine there to refer to. I guess your non-response means "no"?
The GNS construction always associates to the vacuum state a Hilbert space representing the algebra. The only requirement is that the vacuum state is a positive linear functional of the associated *-algebra; this basic property is a necessary physical requirement. Thus the Hilbert space is as relevant to the Heisenberg picture as it is to the Schrödinger picture. The only difference is that the Heisenberg picture is manifestly covariant, while the Schroedinger picture assumes a preferred frame (or foliation) .[URL='https://www.physicsforums.com/insights/author/urs-schreiber/']Urs Schreiber[/URL] said:in the Heisenberg picture the Hilbert space of states is secondary. What is primary is the algebra of observables together with one state on them (the vacuum or ground state). Of course, as Arnold recalled above, under good circumstances a Hilbert space of states may be reconstructed from this data, but this is usually an afterthought, not key for the core computations.
Well, there's one book, treating the Schrödinger picture in relativistic QFT (although I never understood, why I should use it for this purpose anyway):[URL='https://www.physicsforums.com/insights/author/urs-schreiber/']Urs Schreiber[/URL] said:Clearly the old Schwinger-Tomonaga-Feynman-Dyson renormalization is to be disliked. But I think this is unrelated to the issue of the Schrödinger picture that I just mentioned. Essentially nobody ever works or worked in the Schrödinger picture in QFT, it's only that people fall back to it when trying to conceptualize what they are doing
How can you say that? In your work you don't only discuss S-matrix elements but all the correlation functions, related by Kadanoff-Baym equations!vanhees71 said:Well, the physical interpretation of Heisenberg field operators is highly non-trivial, if not even one can say it basically doesn't exist. That's the reason why one finally only discusses S-matrix elements
And one has to give up the idea that ##\hbar## is a number - instead it is only a formal parameter! And one has to give up the idea that operators act on more than a compact part of space-time - to avoid all the infrared problems.[URL='https://www.physicsforums.com/insights/author/urs-schreiber/']Urs Schreiber[/URL] said:The point of "perturbative AQFT" is to notice that if one keeps everything about Haag-Kastler except the demand that the star-algebras of observables have a ##C^*##-algebra structure,