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I am a postdoc in mathematics, but have my degree in theoretical physics. My work is about mathematical structures motivated from quantum field theory and string theory. For more see here.


Oct
10
comment Quantum Field Theory from a mathematical point of view
I didn't mean to be polemical at all. Where do you sense polemics?
Oct
9
answered Quantum Field Theory from a mathematical point of view
Oct
9
comment Formalizing Quantum Field Theory
This December (2011) AMS publishes the volume "Mathematical Foundations of Quantum Field Theory and String Theory". The introduction with further links is at arxiv.org/abs/1109.0955
Oct
9
awarded  Supporter
Sep
27
answered The Role of Rigor
Sep
24
comment Status of local gauge invariance in axiomatic quantum field theory
By the way, there are no local degrees of freedom at all in TQFT.
Sep
24
comment Status of local gauge invariance in axiomatic quantum field theory
Kelly, this is simple: the fact that full TQFT state spaces are finite dimensional is because cap and cup cobordisms induce a trace on the state spaces, so traces must exist. In other words, the state space is a dualizable object. If you remove either the cap or the cup, it no longer needs to be a dualizable object, just a "Calabi-Yau object". The two references that I pointed to, by Costello and by Lurie, are foundational for the field of TQFT. If you are interested in this field, you need to take a look.
Sep
24
comment Status of local gauge invariance in axiomatic quantum field theory
Kelly, no, that's not what I said. You should try the references that I gave if what I say remains unclear. But the simple idea is that there is a non-full subcategory (or sub-n-category) of the usual cobordisms category (or cobordisms n-category) that only contains the cobordisms with non-empty incoming boundary. Representations of this are TQFTs with possibly non-finite state spaces, such as, in 2d, the A-model and the B-model topological strings and more generally, every 2d QFT defined by a Calabi-Yau A-infinity algebra.
Sep
24
comment Status of local gauge invariance in axiomatic quantum field theory
Kelly, the statement for (T)QFT axiomaized as cobordism representations (compact or non-compact) is similar to that of QFT axiomazized as local nets of observables: only in a small number of cases have examples of the axioms been constructed from actual quantization of an action functional. A general proof of existence or even an intrinsic characterization of those models of the axioms that arise from quantizing a gauge-theoretic Lagrangian has not been written out. This is an open problem that people are working on. See eg the FHLT reference that I provided.
Sep
24
comment Status of local gauge invariance in axiomatic quantum field theory
Kelly, it is easy to have the axiomatics so that it admits non-finite degrees of freedom: simply disallow cobordisms with no outgoing (alternatively: ingoing) boundary. That's sometimes called "non-compact" TQFT. See section 4.2 of Lurie's www-math.mit.edu/~lurie/papers/cobordism.pdf from def. 4.2.10 on. This is certainly axiomatic TQFT. But it is more general than Turaev's definition. See also the article by Costello arxiv.org/PS_cache/math/pdf/0412/0412149v7.pdf that it makes the relation to.
Sep
24
comment Status of local gauge invariance in axiomatic quantum field theory
As I said in reply that we are commenting on, in that class of examples the dimension is not the same. The original AQFT question only makes sense for a fixed dimension. The old problem in AQFT is: given a local net of observables on n-dimensional spacetime, did it arise as the quantization of a Yang-Mills-type theory on that spacetime? It is (another) open question to even formulate what AdS/CFT might mean in terms of the AQFT axtioms. There was once an attempt (ncatlab.org/nlab/show/Rehren+duality) but that didn't live up to its promise, I think.
Sep
24
comment Status of local gauge invariance in axiomatic quantum field theory
But the question is: can we invariantly tell from a QFT if it is a gauge theory at all? Both the su(n1) and the su(n2) theory are, so this example would still be consistent with the answer "yes". (The answer may still be "no", but not for this reason, as far as I can see.)
Sep
24
comment Status of local gauge invariance in axiomatic quantum field theory
Since Seiberg- and Montonen-Olive and other S-dualities relate two gauge theories with each other, that does not provide an argument that quantum gauge theories don't have an intrinsic characterization. For that argument you need that one side of the duality is not a gauge theory. And of the same dimension.
Sep
24
comment Status of local gauge invariance in axiomatic quantum field theory
Whether one can "even formulate" what it means for a theory to be a quantum Yang-Mills-type theory or a quantum Chern-Simons theory and so on is the (open) question here. It is not true that nothing about the gauge group is invariantly encoded. The invariant assigned by CS theory of course depend on this. So the fact that physical states are gauge invariant is not an argument that quantum YM does not have an intrinsic characterization.
Sep
24
answered Status of local gauge invariance in axiomatic quantum field theory
Sep
20
comment Do Gauge Theories (CFTs) Have Phase Transitions as the 't Hooft Coupling is Varied?
I've made it appear.
Sep
19
comment Instantons and Non Perturbative Amplitudes in Gravity
Re the first comment above asking "What is "the perturbative expansion" of gravity?". Something that's often forgotten is: there is no problem with treating gravity as an effective QFT and computing perturbative effects at relatively low energies with that. A nice review is Introduction to the Effective Field Theory Description of Gravity arxiv.org/abs/gr-qc/9512024 This is independent of your belief about the UV completion of gravity.
Sep
17
comment Kerr Geometry, Separability and Twistors
If you do follow the references given on that p. 4, you find all the details. For ingtance in [30] (arxiv.org/PS_cache/math/pdf/0204/0204322v1.pdf) around prop. 2.2 .
Sep
17
comment Kerr Geometry, Separability and Twistors
That's effectively what it says on that page 4 that I pointed to, after noticing that the sum of the Dirac operators on the two spinor factors is the Dirac-Kähler operator d + d^*.
Sep
17
answered Kerr Geometry, Separability and Twistors