Has the free Klein-Gordon quantum field theory been fitted into the Haag-Kastler algebraic framework? (Actually, John Baez told me "yes", and he should know.) If so, can you describe the basic strategy and/or give pointers?

The same question for the free Dirac field theory.

  • 1
    $\begingroup$ Both responses were helpful, but it will take a while to digest them. It is interesting to note that the method does not involve beginning with the QFT and generating algebras from bounded functions of smeared fields. I can't help but wonder about that approach. But, hey, "Theoretical Physics" is shutting down anyway. $\endgroup$
    – Greg Weeks
    Apr 25 '12 at 18:28

Yes, these are the standard examples. Some references are colleced here.

For a quick review/survey see for instance slides 11-17 in

  • Edison Montoya, Algebraic quantum field theory (2009) (pdf)

Discussion of the free scalar on Minkowski and curved spacetime is around section 3.2 of

  • Romeo Brunetti, Klaus Fredenhagen, Quantum field theory on curved spacetimes (arXiv:0901.2063)

Discussion of the Dirac field and its deformations is for instance in

  • C. Dappiaggi, Gandalf Lechner, E. Morfa-Morales, Deformations of quantum field theories on spacetimes with Killing vector fields, Commun.Math.Phys.305:99-130, (2011), (arXiv:1006.3548)

There are many more, just chase references. If any example at all is discussed, it is the free scalar field. That's what motivated and instructed much of the theory. The art is to go beyond that example.

  • $\begingroup$ Isn't there an exception for the massless scalar field in 2D? $\endgroup$ Apr 19 '12 at 12:10
  • $\begingroup$ Exception of what? In fact, the application of AQFT to 2d CFT has been most succesful, see ncatlab.org/nlab/show/conformal+net for references. I like to think of this as being a bit ironic: many AQFT textbooks start out being motivated by understanding 4d YM and disliking string theory, but then most hard and important results are obtained in classification of 2d CFT, and string theory has probably profited more from this effort than 4d QFT has. $\endgroup$ Apr 19 '12 at 13:21
  • $\begingroup$ Dear Urs, I agree that 2D CFT has been more successful as an example of the axiomatic treatment that may be reconciled with rigorous mathematicians. And it's ironic from a sociological viewpoint. But I disagree with your implicit suggestion which is written at least in between the lines that the AQFT school of thought itself represented a significant part of the 2D insights. See e.g. BPZ (4000 citations) www33.atwiki.jp/_pub/sakurazemi/reference/BPZ1984.pdf which is a paper that matters here. No AQFT references... $\endgroup$ Apr 19 '12 at 15:53
  • $\begingroup$ Arnold, the exception you may have been thinking about is Coleman's proof that there are no Goldstone bosons in 2D, projecteuclid.org/… $\endgroup$ Apr 19 '12 at 15:54
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    $\begingroup$ I don't do science by counting citations, but if forced to prove a point here by pointing to the authority of citations I might point to Longo-Witten's arxiv.org/abs/1004.0616 which uses AQFT to study issues that remained with Witten's old "Some computations in background independent Open-String Field Theory". $\endgroup$ Apr 19 '12 at 16:22

As Urs mentioned in his answer, free fields really constitute the guiding examples for the various axiomatic systems of QFT. The construction of free fields in Minkowski space is a standard part of the theory, though it may take some digging to find the precise reference where it is checked that such a construction satisfies the desired set of axioms. In particular, for the Haag-Kastler axiom system, once the theory is built on all of Minkowski space, one must show injectivity and isomorphism of the algebras localized in subsets of Minkowski space, with respect to appropriate inclusions of these subsets.

Constructions of various free bosonic and fermionic fields, including the specific cases of scalar and Dirac fields can be found in these classic references, in somewhat varying degrees of detail:

  • Baez, J. C., Segal, I. E., Zhou, Z., Introduction to Algebraic and Constructive Quantum Field Theory (Princeton, 1992)

  • Wald, R. M., Quantum field theory in curved spacetime and black hole thermodynamics, (Chicago, 1994).

There are two main stages to the construction. One has to build the linear space of solutions as a symplectic manifold (that's the classical phase space). Then one has to turn the algebra of functions on this space into a non-commutative $C^*$-algebra of quantum observables (that's quantization). Since the theories in question are linear, once the necessary functional analysis is in place, this is done by an infinite dimensional version of how it is done for a simple harmonic oscillator. For fermions it's fairly straightforward. For bosons one has to use the intermediate trick of working with the algebra of bounded functions generated by exponentiated smeared fields (that's the Weyl algebra). The actual unbounded operators representing smeared fields are constructed by taking derivatives of the elements of the Weyl algebra, once a representation has been chosen.

The same steps appear also in the work on QFT on curved spacetime, where different references described the individual steps in varying levels of detail. To get something like the Haag-Kastler axioms out of the latter constructions, one simply has to restrict onself to spacetimes consisting of causal-diamond shaped subspaces of Minkowski space. Classic references for particular field theories include:

The biggest technical sticking point in these references is how they treat (or don't treat) non-compact Cauchy surfaces.

The modern generalization of the Haag-Kastler axioms to arbitrary globally hyperbolic Lorentzian spacetimes are the Brunetti-Fredenhagen-Verch (or Locally Covariant Quantum Field Theory) axioms. Here's a couple of modern references that give the construction of free fields (subsuming most of the above particular examples), in great mathematical detail, that fits directly into this framework:

(Note for the cognoscenti: in this restricted situation, quantization does happen to be a functor!)

Of course, much more literature can be found by digging backward and forward in the citation network starting with the above references.

  • $\begingroup$ Thanks. Two notes: The Baez/Segal/Zhou text is in "reslib.com". And the Streater & Wightman textbook promptly presents the noninteracting theories as examples of the text's axioms, while the algebraic textbooks of Haag and of Araki do not (based on my limited digestion of the latter texts). $\endgroup$
    – Greg Weeks
    Apr 19 '12 at 21:10
  • $\begingroup$ Regarding your sketch of the procedure: What about the anticommutivity of fermions? Phrased othewise: IIUC, classical fermion solutions do not exist. (To anyone who asks "Then how do we integrate over fermionic solutions in path-integrals?", my reply is "The integration is formal integration of formal polynomials in many formal variables".) $\endgroup$
    – Greg Weeks
    Apr 21 '12 at 17:57
  • $\begingroup$ What about it? The issue of classical fermion solutions is very interesting, but is completely circumvented in the linear case using standard treatments. One uses the dual space (i.e. test functions) to the space of classical (real or complex valued) solutions to build a Clifford algebra using the CAR. A norm on this algebra is used to complete it to a C* one. The fermionic field itself can be seen as a map of the test functions onto the generators of the abstract Clifford algebra. $\endgroup$
    – Igor Khavkine
    Apr 22 '12 at 8:53
  • $\begingroup$ Reading from "Remarks and references" in reslib.com/book/…, I see that Haag, Araki, and friends were not using algebras generated by unobservable fields after 1964. And that is what I had in mind in my question. For the free Dirac theory, the vacuum sector would contain only states with zero lepton number. Is that the case for the contruction that you outlined? $\endgroup$
    – Greg Weeks
    Apr 22 '12 at 17:41
  • $\begingroup$ The construction I outlined is the algebraic version of the one used in every QFT book. So it does use unobservable fields. If you want to transition to only observable fields, all you have to do is restrict to the subalgebra of fields of even fermion number. Any representation of the larger algebra then splits, when restricted to the even subalgebra, into superselection sectors. I believe the superselection charge here is the parity of the fermion number, rather than just the number. Though I could be wrong about that detail. $\endgroup$
    – Igor Khavkine
    Apr 22 '12 at 19:10

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