# References on $C^{*}$-algebras, $W^{*}$-algebras and Quantum Theories

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I would like to know some references regarding $C^{*}$ and $W^{*}$-algebras and quantum theories.

I'm interested in concrete physical applications, models and problems.

Here it is the list of references I already know:

• Dixmier: $C^{*}$-algebras

• Dixmier: $W^{*}$-algebras

• Pedersen: $C^{*}$-algebras and their automorphic groups

• Landsman: Lecture notes on $C^{*}$-algebras and quantum Mechanics

• Araki: The mathematical theory of quantum fields

This post imported from StackExchange Physics at 2014-04-13 14:32 (UCT), posted by SE-user Ilcapitano

recategorized Apr 24, 2014

This post imported from StackExchange Physics at 2014-04-13 14:32 (UCT), posted by SE-user Qmechanic

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If you are interested in physical applications you could also include:

Bratteli-Robinson: Operator algebras and quantum statistical mechanics

It is a two-volume quite complete book, mathematically minded, discussing lots of applications of operator algebras theory to several physical systems, especially arising from statistical mechanics.

Haag: Local quantum physics

It is an, in my opinion, important book on modern mathematical physics (although very often mathematical proofs are only sketched) discussing the local operator algebras formulation of quantum theories, especially, quantum field theory (relying on the well known Haag-Kastler theory). The second edition is considerably better than the first one.

Sewell: Quantum Mechanics and its emergent macrophysics

It is a relatively recent book containing several applications of operators algebras especially to quantum statistical mechanics. The style is less mathematical than the one of the previous pair of books.

As general references, in addition to those you already mentioned, I also suggest the classical mathematical books on the subject:

Kadison-Ringrose: Fundamentals of the theory of Operators Algebras

Takesaki: Theory of Operator Algebras

This post imported from StackExchange Physics at 2014-04-13 14:32 (UCT), posted by SE-user V. Moretti
answered Apr 10, 2014 by (2,025 points)
Thank You. In any case i did not mention that I'm interested particularly in applications in gauge theories.

This post imported from StackExchange Physics at 2014-04-13 14:32 (UCT), posted by SE-user Ilcapitano
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Regarding gauge theory: one of the well-kept secrets of algebraic quantum field theory is that despite all the original motivation from quantization of Yang-Mills theory, very little regarding the quantization of gauge theories has actually been accomplished, or even considered. (There is a kind of perturbation theory in AQFT spirit which does make the connection to BV-BRST formalism for perturbative quantization of gauge theories, but that is explicitly not dealing with $C^\ast$-algebras...)

One recent article that at least starts addressing this question is

• Marco Benini, Claudio Dappiaggi, Alexander Schenkel, Quantized Abelian principal connections on Lorentzian manifolds, Communications in Mathematical Physics 2013 (arXiv:1303.2515)

But notice that the article observes that they run into a problem: they start with ordinary field bundle formalism and then find that... for gauge theory the result of AQFT quantization is not in fact a local net of observables, in that locality breaks.

see here: for gauge theories the "field bundle" is stacky, is a "2-bundle" and if one does not take this into account, then one will not find that quantization of Yang-Mills gives a local net of $C^\ast$-algebras...

answered Apr 13, 2014 by (5,805 points)

Considered, yes, but the axioms need modifications. There is lots of work by Strocchi and collegues, including a recent book (An introduction to non-perturbative foundations of quantum field theory, Oxford Univ. Press, 2013).

Thanks for the pointer, I wasn't aware of Strocchi's work. Could you give me a pointer to page and verse of an online available article where I can have a quick look at the "modified axioms"? Thanks!

There is not yet a valid set of modified axioms, only partial insights into what would need to change. Maybe you'd formulate a specific questions; then I'll give some more details than a comment warrants. It does not fit the context of the OPs question.

The op specified in his comment here http://www.physicsoverflow.org/14915/references-on-%24c-algebras-%24w-algebras-and-quantum-theories?show=14918#c14918 that he is interested in the application of $C^\ast$-algebras to the quantization of gauge theories. I stated that little to nothing is known or has even been considered regarding algebraic quantum field theory (AQFT) of gauge theories. Above you said that something has been considered. So I am asking you to please provide me with a pointer to online available details of what you are thinking of.

I added a separate answer, replacing my comment here a few minutes ago.

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Not quite C^*-algebras, but closely related is my online book

Arnold Neumaier and Dennis Westra, Classical and Quantum Mechanics via Lie algebras, 2008, 2011. http://lanl.arxiv.org/abs/0810.1019. (Though the only piece directly related to gauge theories is in Example 18.3.1 of the 2011 version.)

answered Apr 13, 2014 by (12,425 points)
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I don't have a complete picture on what has been done on gauge theories in the context of axiomatic QFT.  Some time ago, there was some discussion in the PhysicsForums thread

How does a gauge field lead to charge superselection?

which also gives specific references. Some of what I said there is a
bit inaccurate, due to my then still limited understanding.

The Wightman axioms are known not to cover any of the fundamental quantum fields theories (QED, QCD, and the standard models), as these are gauge theories and the Wightman axioms are not
applicable to these. All realistic 4D field theories are either nonrenormalizable (so that not even perturbation theory defined them well), or gauge theories (for which the Wightman axioms are
inappropriate as they don't allow gauge-dependent charged states). There have been attempts to repair this (notably by Strocchi); some of this can be found in the recent book

F. Strocchi,
An introduction to non-perturbative foundations of quantum field theory,
Oxford Univ. Press, 2013.

http://www.researchgate.net/publication/38330367_Euclidean_formulation_of_quantum_field_theory_without_positivity/file/9c96052834f1359310.pdf

W.G. Ritter,
Description of noncommutative theories and matrix models by Wightman functions,
J. Math. Phys. 45, 4980 (2004)
http://scitation.aip.org/content/aip/journal/jmp/45/12/10.1063/1.1775874 ;
(preprint version, significantly different:
http://arxiv.org/pdf/math-ph/0404027.pdf)

But the current understanding is partial only. Thus we currently have no adequate system of axioms for QFT, only a number of ideas how it could possibly look like.

The main reason why modifications are needed is that gauge fields  are unobservable, but the standard Wightman axioms cover only the (expectation values of products of generators of the) C^*-algebra of bounded observable fields, which is a small subalgebra  of the algebra generated by the gauge fields. It does not even contain charged operators.

This small algebra does not contain enough local observables to give a Wightman theory.  The reason is that gauge invariant fields made from gauge fields are necessarily nonlocal, while the Wightman axioms assume implicitly that the algebra is generated by local fields.

The gauge fields themselves are local but on the bigger algebra generated by them it is impossible to define a positive definite inner product, and hence a C^* algebra. The  Hilbert space is replaced by an indefinite inner product space, whence the bounded operators of  the bigger algebra do not form a C^*-algebra structure.

This makes it difficult to figure out how to model a gauge theory within the Wightman framework. Partial work involves the classification of superselection sectors, which defines charged representations, but only leads to global gauge transformations, not to local gauge theories.

Note that among the algebraic tools, closest to the standard model are in fact not the nonperturbative Wightman axioms but the perturbative Epstein-Glaser approach to quantum field theory; see http://en.wikipedia.org/wiki/Causal_perturbation_theory and the book

G. Scharf,
Quantum Gauge Theories: A true ghost story

But this is mathematically incomplete as it is just renormalized perturbation theory, without well-defined observables.

answered Apr 14, 2014 by (12,425 points)
edited Apr 16, 2014

Well, as I mentioned, for perturbation theory there is plenty in AQFT-spirit, involving all you may desire: gauge theory BV-BRST, renormalization and everything. The commented list of references that I mentioned is here: http://ncatlab.org/nlab/show/perturbation+theory#ReferencesInAQFT . (This is work by Klaus Fredenhagen and collaborators during the last years.)

But this mostly does NOT WORK WITH $C^\ast$-algebras anymore, which is what the question was about.

The book by F. Strocchi which you keep pointing to is, naturally,behind a paywall. I still don't know what it says in that book and how that is an answer to the question here. But I would really like to know. Probably Strocchi has some online available publications where he develops the theory in peer review before writing the monograph? That's what I was kindly asking for. Could you point me -- or else just recall for us -- what Strocchi's axioms actually are and how they are relevant here?

Please look at the PhysicsForums link, where there are a number of additional references. I don't have more details - there is no clean set of axioms but a discussion of possibilities and assumptions, scattered in a number of papers, which you can find through scholar.google.

a sample paper by Strocchi:

http://www.researchgate.net/publication/38330367_Euclidean_formulation_of_quantum_field_theory_without_positivity/file/9c96052834f1359310.pdf

and 24 papers citing it:

another relevant paper:

W.G. Ritter,
Description of noncommutative theories and matrix models by Wightman functions,
J. Math. Phys. 45, 4980 (2004)
http://scitation.aip.org/content/aip/journal/jmp/45/12/10.1063/1.1775874
(preprint version, significantly different:
http://arxiv.org/pdf/math-ph/0404027.pdf)

Okay, thanks, I followed the PhysicsForum link. What I found is discussion of the fact that quotienting out gauge equivalence is not compatible with locality. Somewhere in the discussion somebody luckily remarks that this is not news.

I still think the recent work by Diapaggi et. al. which I mentioned makes this issue most clear: they attempt to actually quantize (abelian) Yang-Mills in the AQFT framework, starting from fields which are sections of a field bundle. Then they find failure of locality.

Spelled out this way this is not surprising: gauge fields are not sections of a field bundle. They are sections of a stacky bundle (a 2-bundle). The language of stacks is precisely what combines the gauge principle with the principle of locality.

Ways to deal with this in an AQFT-spirit are well known. That's for instance what Costello's factorization algebras do. However, one sticking point remains: this is no longer using $C^\ast$-algebras...

The link to factorization algebras does not work; I get the error message: This webpage is not available

But I found Castillo's book here: http://math.northwestern.edu/~costello/factorization.pdf

The problem with factorization algebras is that they only encode half of a quantum field theory, namely the algebra of observables. The theory is silent about the other half: interesting states (positive linear functionals) that would give the observables a probability interpretation and with it physical content.

The Wightman axioms provide physically interesting states: the vacuum state via the GNS construction, and many other states by the general folium construction. The problem Strocchi and others are struggling with is to find the right set of propoerties of correlation functions that would permit to do the same for gauge theories.

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