In an interacting theory I expect there to be caustics, resonances, and other situations in which some observables would give an infinite experimental result. Of course, these are idealized states and observables -- if a real device's measurement results are quite accurately modeled by such an observable, and we created a state in which its expected value is large enough, the device would be destroyed.

An idealized world in which models never predict infinite results seems somehow different from the world we live in. Even though experiments never return the measurement result "infinity", they do return the measurement result "I'm so sorry, I'm breaking now, it's bigger than you thought it could be". The Wightman axioms do not require operators to be bounded, and to me seem much better for it. QFT as it's used in practice isn't constructed within the Wightman axioms, but it's much less constructed within the Haag-Kastler axioms, and, it seems to me, particularly for this reason.

I take boundedness to mean that all the eigenvalues of an operator in its action on a Hilbert space of idealized Physical states are required to be finite. This is much stronger than requiring the expected values of an operator to be finite for a dense subset of the Hilbert space (or other, relatively weaker, requirements). It's certainly mathematically more convenient to use bounded operators (because we don't have to keep track of for which states we get a finite result for a given measurement, because they all do), but is that enough? At least, is this acceptable as part of a major attempt to axiomatize theoretical Physics? Is it obvious enough to be an axiom?

I'm prompted to ask this question in this way by a comment in Doplicher's "The principle of locality: Effectiveness, fate, and challenges" J.Math.Phys. 51, 015218 (2010), http://arxiv.org/abs/0911.5136, which I'm reading this morning, where he sets out on the first page (2nd page in the arXiv) that "In quantum mechanics the observables are given as bounded operators on a fixed Hilbert space", which seems a specially sanitized version of QM, insofar as it rules out position, momentum, and energy observables.

Finally, this question is asking, as always, have I got something (very) wrong?

  • $\begingroup$ I don't know enough about the Haag-Kastler axioms so I'll keep this as a comment for now. A simple way to get bounded observables from unbounded ones is to exponentiate them. For instance $(p,q)$ become $(U,V)$, where $U = e^{ip}$ and $V = e^{iq}$ etc. The resulting commutator algebra between $U$ and $V$ is known as the Weyl-Heisenberg algebra (I think). $U$ and $V$ are manifestly bounded operators! $\endgroup$
    – user346
    Jan 31 '11 at 19:17
  • $\begingroup$ Hi Space Cadet, yes. I'm pretty certain that's effectively the content of Marcel's answer, which I just accepted. In retrospect I'd say this was a wild question, although I've nonetheless learned quite a lot from both answers. $\endgroup$ Jan 31 '11 at 20:47

If you have an algebra of unbounded observables, the idea is to take the algebra of all bounded measurable functions of these observables.
In quantum mechanics this leads to the Weyl form: e.g. http://en.wikipedia.org/wiki/Stone%E2%80%93von_Neumann_theorem

Edit: A short comment on the title. In two-dimensions certain models were constructed in the algebraic framework which interact, models with so called factorizing S-matrix. Lechner showed in his thesis for a class of S-matrix that there is an algebra of local observables and that the theory is asymptotic complete etc. The existence in the Wightman setting is harder to establish, it goes under the name form factor programm.

Edit: Sorry still not able to comment: Can one measure something infinite in finite time. I don't think so?!

  • $\begingroup$ That's definitely a useful reminder, and perhaps it's an answer to my last question. One of my worries is that engineers should eventually be able to work with a fundamental theory. "Infinity in a model"="stuff may break" is not so hard to get hold of, at something like the level of the mathematics of signal analysis, say, but "the algebra of all bounded measurable functions" is a step into more abstraction which is not clearly necessary, except to make it easier to do certain mathematics. What statement in terms of bounded measurable observables corresponds to "stuff may break"? $\endgroup$ Jan 31 '11 at 18:03
  • $\begingroup$ If I have anything to do with it, you'll have commenting privileges pretty soon. 1+1D is obviously a counter-example, though I have not yet managed to fathom those models. Intuition fails again. I think one never observes "infinite", but if {0,1,2,3,bigger} are the measurement results we can report in Phys. Rev., then 4 is, in this rather weak sense, in the same equivalence class as infinity. Probably not a useful way to go about constructing mathematical models, however. $\endgroup$ Jan 31 '11 at 20:29
  • $\begingroup$ Except that a bounded measurable observable of some original unbounded measurable observable precisely maps -infinity..infinity to a bounded region. Unless we report measurement results with infinite accuracy, that must make infinity experimentally equivalent to at least some other measurement results. As an idealization, of course, we can report measurement results to infinite accuracy. Feel free not to reply to this, however! I'm jumping around a bit. $\endgroup$ Jan 31 '11 at 20:47

The Haag-Kastler axioms and the Wightman axioms are almost equivalent, AFAIK the equivalence has been proven with some additional technical assumptions that are believed to hold true for all "relevant" physical systems. (It's a matter of fact that all experts of axiomatic QFT work with some form of the Haag-Kastler axioms only, not with the Wightman axioms.)

See for example:

  • Borchers, Yngvason: "From Quantum Fields to Local von Neumann Algebras", Rev.Math.Phys. Special issue, 1992, p.15-47.

Edit: The problem is that taking the algebra of all bounded functions of Wightman fields may result in a net of algebras that does not fulfill the locality axiom, for more details the the Borchers/Yngvason paper.

Here is a quick and dirty pro/contra list:

  • Pro: Some observables do not have an upper bound, e.g. you can always increase the impulse of an electron, no matter how big that already is (in theory).

  • Contra: A detector has an upper bound for every observable it can detect, and that is what is actually measured in an experiment. The situation of the pro-bullet won’t arise because we measure observables in bounded regions of spacetime, and the energy contained in a bounded region is always finite (we are excluding artifacts like black holes from our consideration, of course).

  • Contra to even asking the question: Theories of bounded and unbounded operators/observables are physically equivalent modulo mathematical subtleties (see paper above), meaning they produce the same numbers that can be compared to experiments.

As for the construction of models with interactions: During the past two decades there has been some progress in the task of constructing QFTs with interactions within the Haag-Kastler axioms, see for example:

  • Romeo Brunetti, Michael Duetsch, Klaus Fredenhagen: "Perturbative Algebraic Quantum Field Theory and the Renormalization Groups" (arXiv)

The problems in constructing such a theory is not related to the fact that the Wightman axioms are about unbounded operators and the Haag-Kastler axioms are about bounded ones.

Hint: It is true that AQFT is vastly more complicated than QM and has some surprising and counterintuitive features. Example: The Reeh-Schlieder theorem implies that local algebras cannot contain a particle count operator, see the nLab page for more details:

Edit: The philosophy of the Haag-Kastler approach is very different from the Wightman axioms however, in the Wightman axioms the fields are put in by hand. The philosophy of the Haag-Kastler axioms is that the theory is fully specified by specifying the observables, that is by prescribing a local net of von Neumann algebras. Everything else should be constructed from this net, including "fields". This can indeed be done in certain situations, the relevant result is known as the "Doplicher-Roberts reconstruction theorem".

  • $\begingroup$ Lots to think about. Learned quite a lot from your answer. Thanks, Tim. To your "It's a matter of fact that all experts of axiomatic QFT work with some form of the Haag-Kastler axioms only, not with the Wightman axioms.", perhaps we need to remember that experts in QFT in practice work with a system that is rather closer to the Wightman axioms. To your "physically equivalent modulo mathematical subtleties", this doesn't mean, I think, that we can't work with the Wightman axioms and operator-valued distributions, just that Mathematicians find the Haag-Kastler approach more copacetic. $\endgroup$ Jan 31 '11 at 18:14
  • 1
    $\begingroup$ The experts I had in mind are the people that do active research in axiomatic quantum field theory, most of whom participated at this conference in Göttingen: uni-math.gwdg.de/aqft (most of these people consider the Haag-Kastler axioms as a more convenient, but equivalent, axiomatization compared to the Wightman axioms, AFAIK.) For any practical purposes (calculating numbers) that I know of neither axiomatization scheme is of any importance. $\endgroup$ Jan 31 '11 at 19:28

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