I am a researcher in the department Algebra, Geometry and Mathematical Physics of the Institute of Mathematics at the Czech Academy of the Sciences (CAS) in Prague.
Presently I am on leave at the Max Planck Institute for Mathematics in Bonn.
This is one chapter in a series on Mathematical Quantum Field Theory The previous chapter is: 15. Interacting quantum fields. 16. Renormalization In this chapter we discuss the following topics: Epstein-Glaser normalization Stückelberg-Petermann re-normalization UV-Regularization via Counterterms Wilson-Polchinski effective QFT flow Renormalization group flow Gell-Mann & Low RG Flow In the previous chapter we have…
This is one chapter in a series on Mathematical Quantum Field Theory. The previous chapter is 14. Free quantum fields. The next chapter is 16. Renormalization. 15. Interacting quantum fields In this chapter we discuss the following topics: Free field vacua Perturbative S-matrices Conceptual remarks Interacting field observables Time-ordered products (“Re”-)Normalization Feynman perturbation series Effective…
This is one chapter in a series on Mathematical Quantum Field Theory The previous chapter is 13. Quantization. The next chapter is 15. Interacting quantum fields. 14. Free quantum fields In this chapter we discuss the following topics: Wick algebra Time-ordered product Operator product notation Hadamard vacuum state Free quantum BV-differential Schwinger-Dyson equation In the…
The following is one chapter in a series on Mathematical Quantum Field Theory. The previous chapter is 12. Gauge fixing. The next chapter is 14. Free quantum fields. 13. Quantization In this chapter we discuss the following topics: Motivation from Lie theory Geometric quantization Moyal star products Moyal star product as deformation quantization Moyal star…
The following is one chapter in a series of Mathematical Quantum Field Theory. The previous chapter is 11. Reduced phase space. The next chapter is 13. Quantization. 12. Gauge fixing While in the previous chapter we had constructed the reduced phase space of a Lagrangian field theory, embodied by the local BV-BRST complex (example 11.21),…
The following is one chapter in a series on Mathematical Quantum Field Theory. The previous chapter is 10. Gauge symmetries. The next chapter is 12. Gauge fixing. 11. Reduced phase space In this chapter we discuss these topics: Global gauge reduction for strictly invariant functions (action functionals): Derived critical loci inside Lie algebroids Schouten bracket…
This is one chapter in a series on Mathematical Quantum Field Theory. The previous chapter is 9. Propagators. The next chapter is 11. Reduced phase space. 10. Gauge symmetries In this chapter we discuss these topics: compactly supported infinitesimal symmetries obstruct covariant phase space Infinitesimal gauge symmetries Lie algebra action and Lie algebroids BRST complex…
This is one chapter in a series on Mathematical Quantum Field Theory. The previous chapter is 8. Phase space. The next chapter is 10. Gauge symmetries. 9. Propagators In this chapter we discuss the following topics: Background Fourier analysis and Plane wave modes Microlocal analysis and UV-Divergences Cauchy principal values Propagators for the free scalar…
The following is one chapter of a series on Mathematical Quantum Field Theory. The previous chapter is 7. Observables. The next chapter is 9. Propagators. 8. Phase space It might seem that with the construction of the local observables (def. 7.36) on the on-shell space of field histories (prop. 5.10) the field theory defined by…
The following is one chapter in a series on Mathematical Quantum Field Theory. The previous chapter is 6. Symmetries. The next chapter is 8. Phase space. 7. Observables In this chapter we discuss these topics: General observables Polynomial off-shell observables and Distributions Polynomial on-shell observables and Distributional solutions to PDEs Local observables and Transgression Infinitesimal…
The following is one chapter in a series on Mathematical Quantum Field Theory. The previous chapter is 5. Lagrangians. The next chapter is 7. Observables. 6. Symmetries We have introduced the concept of Lagrangian field theories ##(E,\mathbf{L})## in terms of a field bundle ##E## equipped with a Lagrangian density ##\mathbf{L}## on its jet bundle…
This is one chapter in a series on Mathematical Quantum Field Theory. The previous chapter is 4. Field Variations. The next chapter is 6. Symmetries. 5. Lagrangians Given any type of fields (def. 3.1), those field histories that are to be regarded as “physically realizable” (if we think of the field theory as a…
This is chapter 4 of a series on Mathematical Quantum Field Theory. The previous chapter is 3. Fields. The next chapter is 5. Lagrangians. 4. Field variations Given a field bundle as in def. 3.1 above, then we know what type of quantities the corresponding field histories assign to a given spacetime point (a…
The following is one chapter in a series on Mathematical Quantum Field Theory The previous chapter is 2. Spacetime. The next chapter is 4. Field variations. 3. Fields A field history on a given spacetime ##\Sigma## (a history of spatial field configurations, see remark 3.2 below) is a quantity assigned to each point of…
The following is one chapter in a series on Mathematical Quantum Field Theory. The previous chapter is 1. Geometry. The next chapter is 3. Fields 2. Spacetime Relativistic field theory takes place in spacetime. The concept of spacetime makes sense for every dimension ##p+1## with ##p \in \mathbb{N}##. The observable universe has macroscopic dimensions…
This is the first chapter in a series on Mathematical Quantum Field Theory. The next chapter is 2. Spacetime. 1. Geometry The geometry of physics is differential geometry. This is the flavor of geometry which is modeled on Cartesian spaces ##\mathbb{R}^n## with smooth functions between them. Here we briefly review the basics of differential geometry…
These notes mean to give an expository but rigorous introduction to the basic concepts of relativistic perturbative quantum field theories, specifically those that arise as the perturbative quantization of a Lagrangian field theory — such as quantum electrodynamics, quantum chromodynamics, and perturbative quantum gravity appearing in the standard model of particle physics. Table of Contents…
This is the beginning of a series that gives an introduction to perturbative quantum field theory (pQFT) on Lorentzian spacetime backgrounds in its rigorous formulation as locally covariant perturbative algebraic quantum field theory. This includes the theories of quantum electrodynamics (QED) and electroweak dynamics, quantum chromodynamics (QCD), and perturbative quantum gravity (pQG) — hence the…
A notorious open problem: What is the non-perturbative theory formerly known as Strings? We still have no fundamental formulation of “M-theory” – the hypothetical theory of which 11-dimensional supergravity and the five string theories are all special limiting cases. Work on formulating the fundamental principles underlying M-theory has noticeably waned. […]. If history is…
This here is my personal story. For an alternative introduction see my talk: Higher Structures in Mathematics and Physics. Initially, I discovered higher category theory and higher homotopy theory for myself in my Ph.D. work, in the course of analyzing the supersymmetric quantum mechanics of the superstring on loop space. Driven, as I am,…
The Connes-Lott-Chamseddine-Barrett model is the observation that the standard model of particle physics — as a classical action functional, but including its coupling to gravity and subsuming a fair bit of fine detail — may succinctly be encoded in terms of operator algebraic data called a “spectral triple”. This involves some non-commutative algebra, and…
It is familiar that Einstein gravity may be formulated in terms of a vielbein field together with a “spin connection”, subject to the constraint that the torsion vanishes. There is a little industry trying to suggestively rewrite Einstein’s field equations in this “first-order formalism” and speculating about how this might shed light on the…
In 2002, Pierre Deligne proved a remarkable theorem on what mathematically is called Tannakian reconstruction of tensor categories. Here I give an informal explanation what this theorem says and why it has profound relevance for theoretical particle physics: Deligne’s theorem on tensor categories combined with Wigner’s classification of fundamental particles implies a strong motivation…
At the end of this page, we will have come full circle back to the first article in the series 20 years ago — The M-Theory Conjecture. We now construct prequantum field theories — in the sense discussed at Higher Prequantum Geometry I, II, III, IV, V — of “WZW-type”, using the methods from Examples of Prequantum…
After having constructed gauge fields and higher gauge fields in the previous article by systems of ##L_\infty##-algebroid-valued differential forms on simplex bundles, we now use the same method to construct prequantum field theories — in the sense discussed at Higher Prequantum Geometry I, II, III, IV, V — of higher Chern-Simons type with such higher gauge fields. We begin with…
After having recalled ordinary gauge fields from a dg-algebraic perspective in the previous article, here I discuss how in these terms we easily get the concept of higher (nonabelian) gauge fields, i.e. of gauge fields whose “vector potential” is not just a 1-form but involves differential forms of higher degree. These will be…
After having motivated the need for prequantum field theory and having laid out its principles (i. extremal action, ii. global action, iii. covariant phase space, iv. local observables), it is time to look at examples. The key classes of examples — which are considerably larger than one may think — are 1) field theories of Chern-Simons type…
This article discusses how the previous considerations naturally follow the concepts of local observables of local field theories and of the Poisson bracket on them; as well as that of conserved currents and the variational Noether theorem relating them to symmetries. At the same time, all these concepts are promoted to prequantum local field theory….
The Euler-Lagrange ##p##-gerbes discussed in the previous article are singled out as being exactly the right coherent refinement of locally defined local Lagrangians that may be integrated over a ##(p+1)##-dimensional spacetime/worldvolume to produce a function, the action functional. In a corresponding manner, there are further refinements of locally defined Lagrangians by differential cocycles that are adapted to…
The previous article ended with the concept of classical locally variational field theories, of which a class of examples are field theories of higher WZW type. To recall, in the diagrammatics that we established, such theories were concisely expressed like so: Here we discuss how asking such theories to have a well-defined global action…
The previous article motivated the importance of considering “pre-quantum field theory” in-between classical and quantum field theory. This article here reviews modern classical field theory from a perspective that will be useful for this purpose. Most field theories of relevance in theory and in nature are local Lagrangian field theories (and those that do…
Before proceeding with a discussion of the super p-brane sigma models, whose emergence from the superpoint I discussed in the previous article, we need to speak a bit about the fundamentals of local Lagrangian field theory in general, of which these sigma-models are examples. The geometry that underlies the physics of Hamilton and Lagrange’s…
In the previous article we saw that the generalization of super-Lie algebras to homotopy super-Lie n-algebras (super L-infinity algebras) has been found, decades back, to be at the heart of supergravity and M-theory — somewhat secretly so, in the dual guise of “FDA”s. Lie algebras are infinitesimal symmetries. Super Lie algebras are infinitesimal supersymmetries. Lie…
The previous article in this series claimed that the mathematics of the 21st century that had fallen into the 1970s in the form of string theory is the same mathematics that Grothendieck had dreamed about in his pursuit of stacks around that same time, and which meanwhile has come to full existence: higher geometry…
While the world didn’t end, after all, 15 years back at the turn of the millennium, in hindsight it is curious that, almost unnoticed, something grand did come to a halt around that time. Or almost. The 90s had seen a firework of structural insight into the mathematical nature of string theory, an unprecedented…