Entries by Urs Schreiber

Introduction to Perturbative Quantum Field Theory

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 […]

I am 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.

Super p-Brane Theory Emerging from Super Homotopy Theory

  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 […]

I am 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.

Why Higher Category Theory in Physics?

  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 PhD work, in the course of analyzing the supersymmetric quantum mechanics of the superstring on loop space. Driven, as I am, […]

I am 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.

Spectral Standard Model and String Compactifications

  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 […]

I am 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.

11d Gravity From Just the Torsion Constraint

  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 […]

I am 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.

Why Supersymmetry? Because of Deligne’s theorem.

  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 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 […]

I am 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.

Examples of Prequantum Field Theories IV: Wess-Zumino-Witten-type Theories

    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 […]

I am 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.

Examples of Prequantum Field Theories III: Chern-Simons-type Theories

    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 […]

I am 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.

Examples of Prequantum Field Theories II: Higher Gauge Fields

      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 […]

I am 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.

Examples of Prequantum Field Theories I: Gauge Fields

    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 […]

I am 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.

Higher Prequantum Geometry V: The Local Observables – Lie Theoretically

  This article discusses how from 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. […]

I am 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.

Higher Prequantum Geometry IV: The Covariant Phase Space – Transgressively

    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 integration […]

I am 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.

Higher Prequantum Geometry III: The Global Action Functional – Cohomologically

    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 functional […]

I am 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.

Higher Prequantum Geometry II: The Principle of Extremal Action – Comonadically

    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 are […]

I am 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.

Higher Prequantum Geometry I: The Need for Prequantum Geometry

  Before proceeding with 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 classical mechanics […]

I am 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.

Emergence from the Superpoint

  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 […]

I am 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.

Homotopy Lie-n Algebras in Supergravity

  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 […]

I am 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.

It Was 20 Years Ago Today — the M-theory Conjecture

    While the world didn’t end, after all, 15 years back at the turn of the millenium, 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 unprecendented global […]

I am 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.