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Urs Schreiber
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In 2017, it is still a public secret that perturbative quantum field theory is mathematically well defined. The common perception, spread by bad textbooks, is that "There is no mathematically sound formulation of realistic QFT" [1]. This is wrong! See: https://physics.stackexchange.com/a/362428/5603

[1] https://physics.stackexchange.com/a/337431/5603

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My paper on using homotopy type theory to represent 'the', in particular as used in 'the structure of', has at last been accepted by the journal Synthese. The latest preprint is available at the Philsci Archive, having become considerably longer than an earlier version under the prompting of successive referees.

If Bertrand Russell in 1905 could kick-start analytic philosophy with an analysis of 'The present King of France is bald' by means of the then new predicate calculus, perhaps a return to the realm of 'definite description' but this time with HoTT can achieve something.

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I am preparing an introduction A first Idea of Quantum Field Theory -- From first principles to the derivation of the causal propagator:

http://ncatlab.org/nlab/print/geometry+of+physics+--+A+first+idea+of+quantum+field+theory

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Severin Bunk's PhD thesis on "higher pre-quantum geometry" [1] is out:

Severin Bunk:
Categorical Structures on Bundle Gerbes and Higher Geometric Prequantisation
https://arxiv.org/abs/1709.06174

[1] ncatlab.org/schreiber/print/Higher+Prequantum+Geometry
Categorical Structures on Bundle Gerbes and Higher Geometric Prequantisation

Severin Bunk

We present a construction of a 2-Hilbert space of sections of a bundle gerbe, a suitable candidate for a prequantum 2-Hilbert space in higher geometric quantisation. We introduce a direct sum on the morphism categories in the 2-category of bundle gerbes and show that these categories are cartesian monoidal and abelian. Endomorphisms of the trivial bundle gerbe, or higher functions, carry the structure of a rig-category, which acts on generic morphism categories of bundle gerbes. We continue by presenting a categorification of the hermitean metric on a hermitean line bundle. This is achieved by introducing a functorial dual that extends the dual of vector bundles to morphisms of bundle gerbes, and constructing a two-variable adjunction for the aforementioned rig-module category structure on morphism categories. Its right internal hom is the module action, composed by taking the dual of higher functions, while the left internal hom is interpreted as a bundle gerbe metric. Sections of bundle gerbes are defined as morphisms from the trivial bundle gerbe to a given bundle gerbe. The resulting categories of sections carry a rig-module structure over the category of finite-dimensional Hilbert spaces. A suitable definition of 2-Hilbert spaces is given, modifying previous definitions by the use of two-variable adjunctions. We prove that the category of sections of a bundle gerbe fits into this framework, thus obtaining a 2-Hilbert space of sections. In particular, this can be constructed for prequantum bundle gerbes in problems of higher geometric quantisation. We define a dimensional reduction functor and show that the categorical structures introduced on bundle gerbes naturally reduce to their counterparts on hermitean line bundles with connections. In several places in this thesis, we provide examples, making 2-Hilbert spaces of sections and dimensional reduction very explicit.

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This is the beginning of a series that gives a rigorous introduction to perturbative quantum field theory:
https://www.physicsforums.com/insights/paqft-idea-references/

This includes the theories of quantum electrodynamics, quantum chromodynamics, and perturbative quantum gravity — hence the standard model of particle physics — on Minkowski spacetime (for particle accelerator experiments) and on cosmological spacetimes (for the cosmic microwave background) and on black-hole spacetimes (for black hole radiation).

This first part introduces the broad idea and provides a commented list of references. The next part will start with general discussion of a pivotal part of the theory: the “S-matrix” in causal perturbation theory.

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+Domenico Fiorenza has been preparing expository proceeding notes on T-Duality in Rational Homotopy Theory for his last talk at "Higher Structures in Lisbon" http://hsl2017.campus.ciencias.ulisboa.pt/ and his upcoming talk at the LMS meeting https://www.lms.ac.uk/content/lms-midlands-regional-meeting-0

The link to his pdf file is here:

https://ncatlab.org/schreiber/print/T-Duality+from+super+Lie+n-algebra+cocycles+for+super+p-branes#RationaTopologicalTDualityFiorenza

Abstract:
What in string theory is known as topological T-duality between K0-cocycles in type IIA string theory and K1-cocycles in type IIB string theory or as Hori's formula can be recognized as a Fourier-Mukai transform between twisted cohomologies when looked through the lenses of rational homotopy theory. Remarkably, the whole construction naturally emerges and is actually derived from noticing that the (super-)Chevalley-Eilenberg algebra of the super-Minkowski space R^8;1|16+16 carries two distinct 2-cocycles, whose product is an exact 4-cochain with an explicit trivializing 3-cochain. The super-form components of the RR-fields in type IIA and IIB string theory are then realized as rational K-theory cocycles twisted by these two 2-cocycles, and the trivializing 3-cochain induces rational topological T-duality between them.

Notes for the talks given at Higher Structures Lisbon 2017 and at the Loughborough workshop on Geometry and Physics 2017. Based on joint work with Hisham Sati and Urs Schreiber (arXiv:1611.06536).

On PhysicsOverflow somebody says he is intrigued by the suggestion that quantization of (pre-)symplectic spaces is The most general prescription for quantization (www.physicsoverflow.org/14741/the-most-general-procedure-for-quantization)

Yours truly replies (www.physicsoverflow.org/14741/the-most-general-procedure-for-quantization?show=39426#a39426):

This is a surprisingly good question. "Good", because it is indeed true that there is this very general prescription for quantization; and "surprisingly" because, while the general idea has been around for ages, this has been understood in decent generality only last year!

Namely, on the one hand it is long appreciated in the context of quantum mechanics that what physicists sweepingly call "canonical quantization" is really this: the construction of the covariant phase space as a (pre-)symplectic manifold, and then the quantization of this by the prescription of either algebraic deformation quantization or geometric quantization.

In contrast, it has been understood only surprisingly more recently that established methods of perturbative quantization of field theories, especially in the guise of Epstein-Glaser's causal peruturbation theory (such as QED, QCD, and also perturbative quantum gravity, as in Scharf's textbooks) are indeed also examples of this general method.

For free fields (no interactions), this was first understood in

J. Dito. "Star-product approach to quantum field theory: The free scalar field". Letters in Mathematical Physics, 20(2):125–134, 1990.

and then amplified in a long series of articles on locally covariant perturbative quantum field theory
https://ncatlab.org/nlab/show/locally+covariant+perturbative+quantum+field+theory
by Klaus Fredenhagen and collaborators, starting with

M. Dütsch and K. Fredenhagen. "Perturbative algebraic field theory, and deformation quantization". In R. Longo (ed), "Mathematical Physics in Mathematics and Physics, Quantum and Operator Algebraic Aspects", volume 30 of Fields Institute Communications, pages 151–160. American Mathematical Society, 2001.

Curiously, despite this insight, these authors continued to treat interacting quantum field theory by the comparatively ad hoc Bogoliubov formula, instead of similarly deriving it from a quantization of the (pre-)symplectic structure of the phase space of the interacting theory.

That last step, to show that the traditional construction of interacting peturbative quantum field theory via time-ordered products and Bogoliubov's formula also follows from the general prescription of deformation/geometric quantization of (pre-)symplectic phase space was made, unbelievably, only last year, in the highly recommendable thesis

Giovanni Collini,
Fedosov Quantization and Perturbative Quantum Field Theory
https://arxiv.org/abs/1603.09626

Just read the introduction of this thesis, it is very much worthwhile.

(I learned about this article from +Igor Khavkine and +Alexander Schenkel for which I am grateful.)

In a similar spirit a little later appeared

Eli Hawkins, Kasia Rejzner, The Star Product in Interacting Quantum Field Theory arxiv:1612.09157

which disucsses the situation in a bit more generality than Collini does, but omitting the technical details of renormalization in this perspective.

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Video recording lecture2go.uni-hamburg.de/l2go/-/get/v/21898
of Super p-Brane Theory emerging from Super Homotopy Theory
at StringMath2017 ncatlab.org/schreiber/print/StringMath2017

It is a notorious open problem to determine the nature of the non-pertubative theory formerly known as Strings. I present results showing that, rationally, many of its phenomena emerge as stages of a Whitehead tower, invariant modulo R-symmetry, that emerges out of the superpoint regarded in super-geometric rational homotopy theory:

This includes super-spacetime as such, the bouquet of all Green-Schwarz super p-branes, D-brane charge in twisted K-theory, M-brane charge, double dimensional reduction, T-duality, Buscher rules for RR-fields, doubled spacetimes, F-theory fibrations, S-duality. The orbifold S^4/S^1 (familiar from the near horizon geometry of M5-branes at A-type singularities) appears in a surprising unifying role.

These results (arXiv:1611.06536 and arXiv:1702.01774) are joint with Domenico Fiorenza, Hisham Sati and John Huerta; also with Vincent Schlegel.
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Next week takes place Higher Structures Lisbon 2017 hsl2017.campus.ciencias.ulisboa.pt/description on "Higher Structures in Mathematics and Physics" (ncatlab.org/schreiber/print/Higher+Structures).

Among the highlights are

+Domenico Fiorenza speaking on "T-Duality in Rational Homtopy Theory" (ncatlab.org/schreiber/print/T-Duality+from+super+Lie+n-algebra+cocycles+for+super+p-branes)

+Christian Saemann speaking on "The self-dual String and the (2,0)-Theory from Higher Structures" (arxiv.org/abs/1705.02353)

+Alexander Schenkel speaking on "Homotopical Algebraic Quantum Field Theory" (ncatlab.org/nlab/show/homotopical+algebraic+quantum+field+theory)
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