# nLab quantum field theory

Contents

This entry provides some broad pointers. For a detailed introduction see geometry of physics – perturbative quantum field theory.

# Contents

## Idea

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.

## Locally covariant perturbative quantum field theory

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.

## Higher categories in quantum field theory

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:

• algebraic quantum field theory: AQFT

• functorial quantum field theory: FQFT.

(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

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.

## References

The early history of the subject is outlined in

Introductions include

A standard textbook written from the perspective of effective field theory is

• Steven WeinbergThe Quantum Theory of Fields, Cambridge University Press (1995)

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

based on the causal perturbation theory of

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

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

• Charles Nash, Differential topology and quantum field theory, Acad. Press 1991.

An indication of the modern foundational picture of quantum mechanics is attempted in

• Hisham Sati, Urs Schreiber, Mathematical foundations of quantum field and perturbative string theory Proceedings of Symposia in Pure Mathematics, AMS (web).