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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.


2d
comment Is General Relativity based on a Symmetry?
By the way, GR is not "the" theory invariant under diffeos. All covariant theories are invatiant under diffeos, in particular all topological theories are.
Oct
27
comment Why quantum mechanics?
Yes, this is called "geometric quantization". It's standard. (I was just offering a motivation for existing theory, not a new theory.) The geometric quantization of the standard examples (e.g. harmonic oscillator) is in all the standard textbooks and lecture notes, take your pick here: ncatlab.org/nlab/show/…
Mar
25
comment What are type system examples of local gauge transformation- and field strength-like objects?
To answer your question: yes, in HoTT there are gauge transformations, which are not present in an ordinary topos. For instance in cohesive HoTT there is a type BU(1)_conn of electromagnetic gauge fields. This does not exists in any plain topos (1-topos). For X a manifold, then a function X --> BU(1)_conn is an electromagnetic gauge field on X and a homotopy between two such maps is a gauge transformation between these field configurations. This gauge/homotopy theoretic aspect is not present in an ordinary topos.
Mar
20
comment The physics community's take on noncommutative geometry
I just see that an earlier reference that highlights that the particle limit of the superstring's SCFT is a spectral triple are these lecture notes: Jürg Fröhlich, Krzysztof Gawędzki, "Conformal Field Theory and Geometry of Strings", extended lecture notes for lecture given at the Mathematical Quantum Theory Conference, Vancouver, Canada, August 4-8 arxiv.org/abs/hep-th/9310187 . They explicitly spell out a bunch of examples.
Mar
7
comment Why am I wrong about how to view gauge theory?
@drake: locally every gauge transformation is small (for a connected gauge group), so already quotienting out small gauge transformations destroys locality.
Mar
7
comment Why am I wrong about how to view gauge theory?
@Jia Yiyang, I see, right, I have edited the reply above to include now a point (5) that makes the relation to the construction that the OP considers more explicit.
Feb
5
comment Did Hilbert publish general relativity field equation before Einstein?
...listed here: ncatlab.org/nlab/show/Einstein-Hilbert+action#History
Feb
5
comment Did Hilbert publish general relativity field equation before Einstein?
This is an excellent collection of quotes, thanks! It does seem to me, though, that Hilbert was quite fond of his way of deriving the equations and behind all the politeness was a feeling of superiority. Because in one of his "Lectures on the Foundations of Physics" that Hilbert gave a few days after Einstein's article surfaced he speaks of the work of Civita, Weyl, Schouten, Eddington and culminating in Einstein as a "colossal detour" and concludes that Einstein's equations confirm his variational computation is a "nice consistency check" (German: "schöne Gewähr"). See the citations listed..
Feb
4
comment Hilbert, Gödel, and “God equations” - a 19th century lesson for 21st century physicists?
Thanks Xiao-Gang Wen for this translation of the famous paragraph from the Dao-De-Jing. Re-reading it now it is striking how similar this is to Hegel's metaphysics in his "Science of Logic": ncatlab.org/nlab/show/Science+of+Logic
Feb
3
comment String theory in the context of quantization prescriptions
I suppose what you are after may not be in the literature yet. But another among articles that study spaces of choices in producing string worldsheet QFTs (for the rational case, e.g. WZW) is Runkel et al.'s "Uniqueness of open/closed rational CFT with given algebra of open states" arxiv.org/abs/hep-th/0612306 .
Feb
2
comment String theory in the context of quantization prescriptions
One exception is Klaus Pohlmeyer and a few of his followers to this day. Pohlmeyer had been hoping that if one changes something about the string quantization prescription that then the critical dimension would go away, see the review here: arxiv.org/abs/hep-th/0403260 (together with an argument for why it does not). The latest work inspired by Pohlmeyer is arxiv.org/abs/1204.6263 , which gives a rigorous but perturbative quantization of the 2d Nambu-Goto action, claiming that the anomaly is not seen there.
Feb
2
comment String theory in the context of quantization prescriptions
I see what you are after here. Right, so I am not aware of a decent discussion of the space of geometric quantization choices on the string worldsheet itself, the usual Kähler polarization looks maybe too canonical to have inspired many to look for something else. One exception is..
Jan
23
comment Superspace as the Hilbert Space for Quantum Gravity
Take that as an exercise, to see that metrics don't form a vector space. But even if you considered field species that did form a linear space, it would in general not naturally be a Hilbert space, and crucially it wouldn't be the Hilbert space of quantum states of the theory.
Jan
22
comment Superspace as the Hilbert Space for Quantum Gravity
Yes, you can imagine building a Hilbert space of wavefunctions on Wheeler superspace. But it is not possible to "interpret superspace as a Hilbert space" (it's not even a vector space, to begin with), which is what you were asking.
Jan
22
comment Superspace as the Hilbert Space for Quantum Gravity
Wheeler superspace is the configuration space of "the cosmos" for a given cosmological model. The Hilbert space of states is instead formed by wave functions on that superspace. Gravitons are hard to see in this perspective, which is non-perturbative if done right (which is a big "if").
Jan
11
comment about the Atiyah-Segal axioms on topological quantum field theory
Hey Trimok, what you write is just wrong and mixing things up. Please check first if you understand what a question is about before makeing statements like this.
Jan
11
comment How to treat differentials and infinitesimals?
On the other hand one can choose to build concrete models for the axioms in which notably the textbooks by Kock are written, hence for toposes that validate the Kock-Lawvere axioms. In the typical such models the category of smooth manifolds is enlarged somewhat by objects known as "smooth loci", which include for instance the space formally dual to the "ring of dual numbers", which is just the ring embodying the equation "epsilon^2 = 0". This more concrete incarnation of SDG can be phrased entirely in classical logic and hence shows which classical notions embody the idea of infinitesimals.
Jan
11
comment How to treat differentials and infinitesimals?
There are two complementary aspects to this. On the one hand the categorical logic of toposes allows to formally speak of the subset of the real line of elements that square to 0. This is just what people following Leibniz intuitively did anyway, but categorical logic shows that and how exactly this is consistent. This is then how notably Anders Kock (home.imf.au.dk/kock) wrote his two textbooks on synthetic differential geometry (home.imf.au.dk/kock/SGM-final.pdf): he speaks "synthetically" of the subset D of R on the elements that square to 0 and derives all of diff geometry.
Jan
9
comment What are the practical applications of quantum foundations?
Yes, I agree, one needs to be careful and I think the main message is that it is good to keep an open mind, either way. As in the last sentence of the quote above: "So it is a problem whether or not to worry about philosophies behind ideas"
Dec
25
comment Is it expected that all stellar black holes will be spinning near the maximum allowed $\omega$-velocity?
Nice comment, but for emphasis, let's notice that this is an argument in favor of the answer "yes, black holes will tend to have maximum angular momentum". Because that maximum angular momentum ("extremal Kerr black hole") is J=M^2, hence q=1. So under black hole formation a star may generally have more than this critical angular momentum and black hole formation sets in right at the critical value.