Mathematical Quantum Field Theory - Fields - Comments

[bhobba]

Well, I'm only a naive theoretical physicist, but I think that point of view doesn't make sense.

When one quantizises a quantum field at an intuitive level its the same as what's going on with a classical field as you would find in a book on classical field theory such as the one I have by Soper - Classical Field Theory. What you do is think of the field as a lot of small blobs interacting in some way. When it's quantisized the properties of those small blobs - whatever they are - momentum, position, electric field strength, or whatever are quanatizised and become operators. Then the blob size is taken to zero so you get a field of operators. That's heuristically what's happening. Urs is just making it mathematically rigorous.

I thought this was very well known. In fact many seem to think that taking that blob size to zero is the cause of much of the issues in QFT.

I know you don't like Zee's book - neither do I actually - but he explains all this in the first couple of chapters - he uses a mattress analogy.

Or maybe I am missing something?

Thanks
Bill

 

[bhobba]

I like to caution against this habit of organizing people into camps and then declaring what they do and do not think. I keep hearing what a) Mathematicians, b) Physicists, c) Mathematical physicsts etc. allegedly a) think, b) do or don't understand, and c) do or do not care about.

I agree.

But as to why it happens 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. But the difference in style between the two books is enormous. That's what I think tends to foster this false categorization you correctly identify. It shouldn't be like that - but - sigh - it is.

Thanks
Bill

 

[vanhees71]

When one quantizises a quantum field at an intuitive level its the same as what's going on with a classical field as you would find in a book on classical field theory such as the one I have by Soper - Classical Field Theory. What you do is think of the field as a lot of small blobs interacting in some way. When it's quantisized the properties of those small blobs - whatever they are - momentum, position, electric field strength, or whatever are quanatizised and become operators. Then the blob size is taken to zero so you get a field of operators. That's heuristically what's happening. Urs is just making it mathematically rigorous.

I thought this was very well known. In fact many seem to think that taking that blob size to zero is the cause of much of the issues in QFT.

I know you don't like Zee's book - neither do I actually - but he explains all this in the first couple of chapters - he uses a mattress analogy.

Or maybe I am missing something?

Thanks
Bill

That's all clear to me. In QFT in a field operator $\hat{\phi}(t,\vec{x})$ the arguments $\vec{x}$ are labels of continuously many degrees of freedom, analogous to the discrete label $i$ in the generalized configuration variables $q^i$ in the Hamilton formalism. That was not my point of criticism, which was not about the use of mathematics but the choice of words, which may mislead a beginner to think that the fundamental field operators are observables. Most of the fundamental fields in the standard model are not representing observables directly: Either they are (Dirac) fermions, which as fermionic field operators are not observable, because they anticommute at spacelike distances rather than commute (microcausality!) or they are gauge-boson fields and as such not gauge invariant and thus cannot be representing observables either. The observables are built by these field operators and determined via the Noether theorem, defining the true observables like energy, momentum, angular momentum, charges, and currents etc.

 

[vanhees71]

I'm going back and re-reading some stuff. In the "Fields" installment, you (@[URL='https://www.physicsforums.com/insights/author/urs-schreiber/']Urs Schreiber[/URL]) say: However, the nLab entry says: Why the change from "configuration" to "history"?

And why not call it simply "field on spacetime"? I guess that's to make a distinction between smooth/rough fields? But do we ever need rough fields in QFT?

Indeed, using the term "field configuration" makes it much better!

 

[A. Neumaier]

In fact many seem to think that taking that blob size to zero is the cause of much of the issues in QFT.

In nonrelativistic QFT this causes no problems. The problems are intrinsically relativistic - in preserving Poincare invariance.

 

[bhobba]

Most of the fundamental fields in the standard model are not representing observables directly:

Got it

Thanks
Bill

 

[Urs Schreiber]

Why the change from "configuration" to "history"?

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.

And why not call it simply "field on spacetime"?

Because this is too ambiguous and leads to confusion. For instance when we say "consider the electromagnetic field" we are not referring to a specific field history, but to the type of possible field histories.

I guess that's to make a distinction between smooth/rough fields?

No.

But do we ever need rough fields in QFT?

If you mean non-smooth field histories, then: No.

 

[Urs Schreiber]

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?

That's a typo, yes. I am fixing it.

 

[Urs Schreiber]

but the choice of words, which may mislead a beginner to think that the fundamental field operators are observables.

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.

 

[Urs Schreiber]

But as to why it happens

Allow me to suggest that it (namely the suggestion that researchers sit in camps that constrain their ability to think about the total nature of the problem at hand) happens out of intellectual laziness. It takes little effort to copy quotes from Wikipedia that make fun of people in what is perceived a different camp, while it takes effort to learn all aspects of the problem and transcend the camp spirit.

I think it is plain obvious that to understan QFT you need all of it: A good idea of its physical meaning as well as the mathematical tools that it takes not to get confused (say about what "field" means in field theory...)

 

[A. Neumaier]

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.

I'd propose to use ''quantity'' for any functional on the space of field histories, and ''observable'' for those quantities that are actually observable.

 

[Urs Schreiber]

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.

Ballentine's book as well as Varadarajan's both having emphasis on QM over QFT, put the concept of a Hilbert space of states in the center of attention. 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. We'll get to that in the series in chapters 14 and 16.

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)

 

[Urs Schreiber]

I'd propose to use ''quantity'' for any functional on the space of field histories

That seems overly unspecific and also unconventional

It is completely conventional and useful to speak of "local gauge-invariant on-shell observables" and we couldn't do that if we defined "observanle" to already mean "local gauge invariant on-shell observables".

To relativize concern about the choice of terminology remember that even with plenty of qualifiers added, the mathematical concept of "observable" is necessarily still a highly idealized formalization of what happens to our sensory system as we make an observation in nature (for one we haven't even touched general covariance yet, or noise or coarse graining, not to speak of biological and psychological aspects), so it seems misguided to be pedantic about naturalistic linguistic here over having a useful crisp technical terminology.

 

[vanhees71]

Another trouble is caused by taking the infinite-volume limit. Haag's theorem is related to it. See, e.g.,

A. Duncan, The conceptual framework of QFT, Oxford University Press (2012) Sect. 10.5

 

[vanhees71]

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.

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.

 

[A. Neumaier]

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.

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.

 

[strangerep]

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.

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.

 

[strangerep]

Why the change from "configuration" to "history"?

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.

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"?

 

[bhobba]

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)

Amazon has one copy left - may snap it up as a Christmas gift.

But it occurs to me, and this is something I have been meaning to investigate for some time now, it looks related to the latter work of Von-Neumann on C*algebras and QM. Would that be correct?. Of course I have read his classic Mathematical Foundations which is done entirely in Hilbert Spaces. In fact it was one of the first proper books on QM I ever read. Having studied Hilbert spaces as part of my degree it was a piece of cake so to speak - the other one I read - Dirac - was a big problem and I had to investigate RHS's to finally get a grip on it. When I did it was the other way around - I preferred Dirac to Von-Neumann.

Would the following be a good primer for the book you mentioned, as well as your whole series?
http://www.math.uchicago.edu/~may/VIGRE/VIGRE2009/REUPapers/Gleason.pdf

It has more formal math than I am used to these days - like Varadarajan it looks a bit of a 'slog' but if valuable will persevere.

Thanks
Bill

 

[Urs Schreiber]

perhaps you could insert something like the last part of this sentence near the place where you define "field history"?

Good point. Will do.

 

[Urs Schreiber]

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.

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. For amplification see for instance also the quantization of the hydrogen atom in the Heisenberg picture, e.g. section 12 of Nanni 15.

In passing from quantum mechanics to quantum field theory, it is important to notice that it is the Heisenberg picture (or interaction picture for perturbation theory) that generalizes well, not the Schrödinger picture (see also Torre-Varadarajan 98 for issues with the Schrödinger picture in QFT). Those quantum field observables $\mathbf{\Psi}(x)$ that we have been discussing elsewhere, they are the hallmark of the Heisenberg picture. And 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 usally an afterthought, not key for the core computations.

If one looks at old articles such as the original Epstein-Glaser article on causal perturbation theory, they always carry around phrases like "assuming that all operators involved can be chosen to have joint dense domain of definition" etc., which comes from insisting that the quantum observables be represented on a Hilbert space. But in fact this is an unnecessary asumption for the results of causal perturbation theory, everything goes through directly with considering just the "abstract" algebra of observables. It's easier, less conceptual baggage.

That is not to say that having a Hilbert space representation is not useful. But it's not conceptually primary for the development of QFT.

 

[vanhees71]

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. Of course, the general state is uniquely represented by a Statistical Operator rather than a Hilbert-space vector (or ray), but to make sense of the Statistical Operator you need at least a trace operation. Can this be defined without relying on a Hilbert-space concept?

I always thought the success of perturbative QFT, despite being mathematically quite undefined, is that the physicists intuitively take the limits in their calculations in a correct way, even although they are not able to rigorously define them in a mathematically satisfactory strict sense. Of course, I have in mind things like using a finite quantization volume (with periodic boundary conditions), adiabatic switching a la Gell-Mann and Low and various regularization procedures for divergent perturbative integrals (loops and Feynman diagrams), or appropriate counterterm-subtraction techniques without regularization (a la BPHZ) to make sense of the ill-defined divergent integrals.

I thought, particularly the Epstein-Glaser causal approach, is just another particularly physical way to take these limits in using "smeared" field operators, which makes a lot of physical sense, particularly in view of the Wilson approach to renormalization and his physical interpretation of the renormalization-group equations, and I also thought that this is at least some step towards a mathematically more satisfying foundation in providing the correct rules of mutliplying distribution-like operators or Green's and vertex functions.

 

[Urs Schreiber]

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.

This is the old concept of state in AQFT, I recommend the lecture notes Fredenhagen 03 (section 2) for the non-perturbative ($C^\ast$-algebraic) version and Fredenhagen-Rejzner 12 (around def. 2.4) as well as Rejzner 16 (starting with section 2.1.2) for the perturbative QFT version (just dropping the $C^\ast$-condition).

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.

We will come to this in the series in chapters 14. Free quantum fields and 16. Quantum observables.

Regarding Epstein-Glaser 73, what they achieved is to axiomatiize the properties of the perturbative S-matrix, thereby making sense of the would-be path integral, and proving from these (very simple) axioms the construction of pQFT via renormalization by splitting/extension of the distributions given by the Feynman diagrams (here). Hence they managed to make good sense of (Lorentzian) pQFT.

We come to this in the series in chapter 15. Scattering.

 

[strangerep]

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.

Well,... no, I don't notice that. (Do you have a copy of Ballentine in front of you?)

I see (on p160) that he starts by recognizing $J^2$ and $J_z$ as commuting self-adjoint operators. Hence if they act on a Hilbert space, that Hilbert space is spanned by common eigenvectors parameterized by the eigenvalues of these operators ($\beta,m$). Then he uses positivity of the Hilbert space inner product to derive an inequality $\beta \ge m^2$ in eq(7.4). Then he use the ladder operators $J_\pm$ and the inequality (7.4) to show that repeated application of either ladder operator eventually gives 0, and for any given value of $\beta$ there is a maximum value "$j$" of $m$. Thus he deduces the dimension of the Hilbert space for any given value of $\beta$, and also that $\beta = j(j+1)$ -- eq(7.10).

For amplification see for instance also the quantization of the hydrogen atom in the Heisenberg picture, e.g. section 12 of Nanni 15.

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.

 

[vanhees71]

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.

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.

 

[Urs Schreiber]

Then he uses positivity of the Hilbert space inner product

Right, one uses the positivity of a given state (one of the axioms on a state on an algebra of observables).

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

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.

You can tell that most constructions in quantum mechanics don't necessarily need the Hilbert space concept by the simple fact that standard textbooks and most students don't even know the functional analysis involved when really considering Hilbert spaces, and not just commutation relations. 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 discussed if one is really speaking Hilbert spaces. The reason that students and textbooks get away without even mentioning this (such as Ballentine's book) is due to the fact that from just the algebra structure (commutators) of the observables and the basic properties of states (linearity, normalization, positivity) as functions from observables to complex numbers, one can obtain most results.

 

[Urs Schreiber]

Well, I guess that refers to free particles

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.

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).

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).

Then we construct the Fock space as the occupation-number eigenstates (usually in the single-particle momentum representation).

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.

Now, isn't his a Hilbert space

The Fock space is a Hilbert space, yes. In good cases it happens to exist. In general it does not, and even if it does, it is not actually necessary to do any and all of pQFT.

 

[bhobba]

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.

Too true.

Whats your view of resolving it with Rigged Hilbert Spaces and the Generalized Spectral Theorem?

Thanks
Bill

 

[Urs Schreiber]

Whats your view of resolving it with Rigged Hilbert Spaces and the Generalized Spectral Theorem?

These are good tools if one wants to be serious about Schrödinger-picture quantum mechanics.

There is good stuff to be found in the Hilbert space Schrödinger picture, if done with due care (as amplified in texts like "Self-adjoint extensions of operators and the teaching of quantum mechanics").

One just has to exercise care with regarding this good QM stuff as indication that the Schrödinger picture is a robust foundation for quantum physics in a generality that includes QFT. It turns out that it is not. On the one hand there are intrinsic problems with applying the Hilbert space Schrödinger picture to QFT even in principle (Torre-Varadarajan 98). On the other hand the Heisenberg/interaction picture without a choice of Hilbert space (just with the option to find one, if possible) works wonders and is in fact, more or less secretly, precisely what everyone uses in practice anyway, even if it superficially seems as if Hilbert spaces are being used.

 

[bhobba]

On the one hand there are intrinsic problems with applying the Hilbert space Schrödinger picture to QFT even in principle (Torre-Varadarajan 98).

Dirac always disliked re-normalization. But I did read somewhere by using the Heisenberg picture he did obtain the same results without it. Evidently they were long but it did work.

Thanks
Bill