functorial quantum field theory
Reshetikhin?Turaev model? / Chern-Simons theory
FQFT and cohomology
physics, mathematical physics, philosophy of physics
theory (physics), model (physics)
experiment, measurement, computable physics
Axiomatizations
Tools
Structural phenomena
Types of quantum field thories
Quantum field theory is the general framework for the description of the fundamental processes in physics as understood today. These are carried by configurations of fields under the generalized rules of quantum mechanics, therefore the name. Notably the standard model of particle physics is a quantum field theory and has been the main motivation for the development of the concept in general.
Historically quantum field theory grew out of attempts to combine classical field theory in the context of special relativity with quantum mechanics. While some aspects of it are understood in exceeding detail, the overall picture of what quantum field theory actually is used to be quite mysterious. There are two main approaches for axiomatizing and formalizing the notion:
algebraic quantum field theory: AQFT – this encodes a quantum field theory as an assignment of operator algebras “of observables” to patches of spacetime;
functorial quantum field theory: FQFT – this encodes a quantum field theory as an assignment of spaces of quantum states to patches of codimension 1, and of maps between spaces of states – the time evolution operator – to cobordisms between such patches.
Both these approaches try to capture the notion of a full quantum field theory. On the other hand, much activity in physics is concerned with perturbative quantum field theory. This is a priori to be thought of as an approximation to a full quantum field theory akin to the approximation of a function by its Taylor series, but not the least because it is often the only available technique, the tools of perturbative quantum field theory are to some extent also taken as a definition of quantum field theory.
The gap for instance between the formal study of the AQFT axioms and physics as done in practice by physicists had to a large extent been due to the fact that AQFT had little to say about perturbative quantum field theory. But recently this has been changing. See perturbative quantum field theory for more.
Recent times have seen major progress in understanding these axiomatizations and connecting them to the structures studied in physics (see the references below), but still the number of interesting phenomena in quantum field theory that physicists handle semi-rigorously and that are waiting for a fully formal understanding is large.
The formulation of quantum field theory has many aspects and perspectives. Two almost complementary threads are the following:
1) perturbation theory by means of formal Feynman diagram expansions of (Wick rotated) path integrals
This is the approach predominant in phenomenology.
It produces the observable numbers which are checked to great precision in experiments starting from the early development of quantum electrodynamics, fully established with the success of quantum chromodynamics and recently culminating in the Higgs field physics seen at the LHC experiment, confirming the standard model of particle physics.
While many of the mathematical intricacies of this approach have found solutions over the decades, most of these rely on global properties of Minkowski spacetime such as translation invariance and existence of an invariant vacuum quantum state, hence on a consistent concept of particles, none of which generalizes robustly to quantum field theory on curved spacetimes of relevance in cosmology, black hole radiation or the instanton vacuum of QCD.
2) algebraic quantum field theory by means of local nets of C*-algebras of observables
This is the approach predominant in mathematical physics.
It produces the structural theorems of quantum field theory, such as the PCT theorem and the spin-statistics theorem and it seamlessly generalizes to QFT on curved spacetimes.
In its insistence on C*-algebras its ambition is to describe the full non-perturbative quantum field theory. But as a matter of fact not a single relevant example (interacting QFT in spacetime dimensions 4 or greater) is known. Indeed, the construction of the motivating example, quantization of Yang-Mills theory, is one of the open “millenium problems”.
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But more recently a synthesis of these two threads has been developed:
3) locally covariant perturbative quantum field theory by means of local nets of formal power series algebras of observables
This rests on the observation (Il’in-Slavnov 78, Brunetti-Fredenhagen 00) that the formulation of renormalization in causal perturbation theory (Epstein-Glaser 73, Scharf 95) produces a local net of formal power series algebras of observables.
This formulation hence allows to construct the usual examples of perturbative quantum field theories in a rigorous fashion, and then to extend them to quantum field theory on curved spacetimes, such as in formulation of perturbative quantum gravity (Brunetti-Fredenhagen-Rejzner 13).
There remains just one problem:
When applied to gauge theory on on curved spacetime the usual axioms on a local net break: either they enforce all characteristic classes of the gauge field (“instanton sectors”) to be trivial, or else the axioms encoding locality are broken (see Schenkel 14, Schreiber 14).
This remaining problem is solved by passing from plain algebras of observables to homotopical algebras (higher algebras), and hence to a formulation of homotopical algebraic quantum field theory (see Schenkel 17). This is still in the making.
(See also higher category theory and physics and (SatiSchreiber)).
Even though quantum field theory has been around for decades and has been very successful both as a phenomenological model in experimental physics as well as a source of deep mathematical structures and theorems, from a mathematical perspective it is still to a large extent mysterious, though recently much progress is being made.
There are essentially two alternative approaches for formalizing quantum field theory and making it accessible to mathematical treatment:
(Other structures which are used to define quantum field theories, such as vertex operator algebras are now more or less understood to be special cases of these two approaches. See there for details.)
Both AQFT and FQFT involve the language of category theory and higher category theory. In fact, a couple of important higher categorical structures were motivated from and first considered in the context of quantum field theory. For instance
John Roberts first conceived the idea of $\infty$-categorical nonabelian cohomology in the context of AQFT.
the description of 2-dimensional CFT and 3-dimensional TFT in terms of modular tensor categories provides a major application of the theory of monoidal categories;
the notion of the (∞,n)-category of cobordisms, which is thought to play a role analogous to, and as fundamental as, that of the sphere spectrum was motivated from FQFT;
In early 1990s A-∞ categories first appeared in works of Kontsevich and Fukaya on the categorical description of twisted sigma models what is then used in the formulation of homological mirror symmetry.
the cobordism hypothesis was formulated by John Baez and Jim Dolan in the context of extended topological FQFT.
various structures involving (∞,1)-operads, such as topological chiral homology and blob homology are motivated by, and find their application in, the algebraic description of quantum field theory;
the description of higher background gauge fields very much motivated the development and study of differential cohomology, which like all notions of cohomology is intrinsically about (∞,1)-topos theory.
There are some indications that such higher categorical structures, such as those appearing in groupoidification, are essential for clarifying some of the mysteries of quantum field theory, such as the path integral. While this is far from being clarified, this is what motivates research in higher categorical structures in QFT.
Ours is the age to figure this out.
The early history of the subject is outlined in
Rudolf Haag, Early papers on quantum field theory (1992-1930) (pdf)
Introductions include
Arnold Neumaier, chapter B of Theoretical Physics FAQ
Gerald B. Folland, Quantum field theory: A tourist guide for mathematicians, Math. Surveys and Monographs 149 (ZMATH)
Richard Borcherds, A. Barnard, Lectures on QFT, (arxiv:math-ph/0204014)
A standard textbook written from the perspective of effective field theory is
An extensive compilation of material on QFT aiming for mathematical precision is
Survey of the mathematically rigorous discussion in locally covariant perturbative quantum field theory is in
Klaus Fredenhagen, Katarzyna Rejzner, Perturbative algebraic quantum field theory, In Mathematical Aspects of Quantum Field Theories, Springer 2016 (arXiv:1208.1428)
Klaus Fredenhagen, Katarzyna Rejzner, Perturbative Construction of Models of Algebraic Quantum Field Theory (arXiv:1503.07814)
based on the causal perturbation theory of
Henri Epstein and Vladimir Glaser, The Role of locality in perturbation theory, Annales Poincaré Phys. Theor. A 19 (1973) 211.
V. A. Il’in and D. S. Slavnov, Observable algebras in the S-matrix approach, Theor. Math. Phys. 36 (1978) 32.
Romeo Brunetti, Klaus Fredenhagen, Microlocal Analysis and Interacting Quantum Field Theories: Renormalization on Physical Backgrounds, Commun. Math. Phys. 208 : 623-661,2000 (math-ph/9903028)
and developed as causal perturbation theory for gauge theory such as quantum electrodynamics in
and for quantum chromodynamics and perturbative quantum gravity in
Discussion of perturbative quantum gravity in this perspective is in
and survey of the generalization to gauge theory via homotopical algebraic quantum field theory is in
Alexander Schenkel, On the problem of gauge theories in locally covariant QFT, talk at Operator and Geometric Analysis on Quantum Theory Trento, 2014 (pdf) (with further emphasis on this point in the companion talk Schreiber 14)
Alexander Schenkel, Towards Homotopical Algebraic Quantum Field Theory, talk at Foundational and Structural Aspects of Gauge Theories, Mainz Institute for Theoretical Physics, 29 May – 2 June 2017. (pdf)
A discussion of aspects of QFT with an eye towards applications in string theory and aimed at mathematicians (though requiring more of a physicist’s mindset than many pure mathematicians will find themselves in) is
Aspects of topology and differential geometry (e.g. connections to index theorems and moduli spaces) are emphasized in
An indication of the modern foundational picture of quantum mechanics is attempted in
See also
Albert Schwarz, Quantum field theory and topology, Grundlehren der Math. Wissen. 307, Springer 1993.
Graeme Segal, Three Roles of Quantum Field Theory , Felix Klein lectures, Bonn (2011)
For further references see FQFT and AQFT.
See also