nLab perturbative algebraic quantum field theory

Idea

What is called locally covariant perturbative algebraic quantum field theory (Brunetti-Fredenhagen 00, Brunetti-Fredenhagen-Verch 03) is a formulation of local perturbative quantum field theory on general spacetimes (hence on general classical background field configurations of the field theory of gravity) which is both mathematically rigorous as well as closely connected to the traditional non-rigorous discussions, hence to phenomenology and experiment.

The approach uses the Haag-Kastler axioms for algebraic quantum field theory in a variant that

1. demands only formal power series star-algebras of observables instead of C*-algebras

   (this reflects the perturbation theory in Planck's constant and the coupling constant);

2. evaluates local nets of observables not just on causal subspaces of Minkowski spacetime but on more general globally hyperbolic Lorentzian manifolds

(this reflects the extension to general curved spacetime backgrounds).

What connects this to the traditional discussion of QFT is the observation (in essence already due to Il’in-Slavnov 78 but then ignored) that standard renormalization theory (removal of ultraviolet divergences) in the guise of causal perturbation theory (Epstein-Glaser 73) automatically produces causally local nets of algebras of observables (due to “adiabatic switching” and causal locality of the S-matrix, see there), in fact that considering these local nets instead of their would-be colimiting global algebra serves to define the perturbation theory even in the presence of infrared divergences. (Specifically: This takes care of the “algebraic adiabatic limit”, which defines the quantum observables. It does not yet in itself define the “weak adiabatic limit” for the would-be vacuum quantum state.)

To put this into perspective, notice that 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 this 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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Locally covariant perturbative quantum field theory provides a synthesis of these two opposites.

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Quantum field theory is not only extremely rich in its phenomenology but also its formulation involves a plethora of mathematical tools, such as Lorentzian geometry, spin geometry, analysis (especially Fréchet manifold theory for jet bundles and spaces of sections) variational calculus, symplectic geometry, quantization (specifically formal deformation quantization for perturbative quantum field theory) functional analysis (specifically microlocal analysis for causal perturbation theory) and operator algebra. While lcpAQFT is mathematically rigorous, hence unambiguous and precise, it remains a long story from its first principles to the computation of scattering amplitudes. To ease the overview, we now indicate the global structure of the topic first.

The input datum is a local Lagrangian classical field theory on gravity backgrounds, hence a natural transformation from (globally hyperbolic) Lorentzian manifolds to field bundles over these spacetimes equipped with a local Lagrangian density on their jet bundle.

This induces for each Lorentzian spacetime the corresponding covariant phase space, a presymplectic smooth space.

Passage to the corresponding perturbative quantum field theory means to choose a formal deformation quantization of a subalgebra of the algebra of smooth functions on the covariant phase space (the “observables”). This formal deformation is a formal power series star-algebra whose formal parameter is identified with Planck's constant and it is this formal power series nature which makes this just a perturbative quantization (while a non-perturbative quantization would be required to yield a C*-algebra of quantum observables, or something similar).

Finally what makes this locally covariant is that the resulting assignment of formal power series algebras of observables to spacetimes is required to still be a functor satisfying causal locality (a local net). It is in this sense that the output of the procedure is an algebraic quantum field theory in the sense of the Haag-Kastler axioms, only that instead of C*-algebras it assigns formal power series star-algebras.

In summary, constructing a locally covariant perturbative quantum field theory means to

• prescribe a classical field theory encoded by a functor

  $L \;\colon\;\left\{ \text{spacetimes} \right\} \longrightarrow \left\{ \array{ \text{field bundles equipped with} \\ \text{local Lagrangian densities} }\right\}$

  and consider the induced covariant phase space

  $\omega \;\colon\; \left\{ \text{spacetimes} \right\} \longrightarrow \{ \text{pre-symplectic smooth spaces} \}$

• choose a formal deformation quantization of a subalgebra of the smooth functions on the covariant phase space

  $A_{obs} \;\colon\; \left\{ \text{spacetimes} \right\} \longrightarrow \left\{ \text{formal power series star algebra} \right\}$

  such that this satisfies causal locality.

While this is straightforward as a prescription, finding any formal deformation quantization on infinite-dimensional phase spaces is highly non-trivial. (Notice that essentially all the existing literature on formal deformation quantization considers it on finite-dimensional manifolds.) But there is an algorithm that yields an effective solution together with a parameterization of the available choices, called causal perturbation theory. This algorithm proceeds as follows:

1. Choose a decomposition of the Lagrangian density $L$ into a kinetic term $L_{kin}$ and an interaction term $L_{int} \coloneqq L - L_{kin}$, where $L_{kin}$ is such that its Euler-Lagrange equation of motion is a linear hyperbolic differential operator (the “wave operator” for the free field);

2. construct its Wick algebra which is the vector space of microcausal functionals on phase space equipped with the Moyal star product induced by the causal Green function of that hyperbolic differential operator;

3. for the remaining interaction $L_{int}$ regarded as an element of the Wick algebra, inductively construct its S-matrix as a distribution with values in the Wick algebra by demanding it to be causally additive (this is the step of causal perturbation theory proper). The theorem of (Epstein-Glaser 73) states that this has a solution, and that the ambiguity in the solution is at each step of the induction parametrized by a finite set of real numbers (given, at stage $n$, by extending the n-point function distributions to the diagonal).

4. Finally the desired algebra of observables on some spacetime $X$ is the Wick algebra on microcausal functions on any slightly larger spacetime $\tilde X$ deformed by conjugation with the S-matrix evaluated (being a distribution) on any bump function on $\tilde X$ that is unity on $X \subset \tilde X$.

For Minkowski spacetime this algorithm is due to Epstein-Glaser 73 building on ideas of Stückelberg and Bogoliubov, while the generalization to globally hyperbolic spacetimes is due to Brunetti-Fredenhagen 00 (at least for scalar field theory). That this does in fact yield a local net (causal locality) was first observed in (Il’in-Slavnov 78), but ignored or forgotten, and then rediscovered and popularized in (Brunetti-Fredenhagen 00).

When comparing with the bulk of the literature, the reader should beware that most texts do not discuss this algorithm as a means for formal deformation quantization. Instead, this algorithm of causal perturbation theory was extracted from experience and justified itself by coinciding with other established recipes of quantum field theory to the extent that these are well-defined, and hence ultimately by producing scattering amplitudes in QED, QCD and the standard model of particle physics which, when truncated to low order in the formal deformation parameter $\hbar$, turn out to coincide to high precision with experiment.

That this algorithm in fact constructs the formal deformation quantization of the covariant phase space of the given local Lagrangian density $L$ was understood only very recently in (Collini 16) (for phi^4 theory of the scalar field) and in (Hawkins-Rejzner 16) (for the scalar field with regular non-local interaction terms).

There are choices to be made in finding this deformation quantization. In (Collini 16) these are parameterized as the choices of initial conditions in the inductive construction of Fedosov's deformation quantization and shown to be equivalent to the choices of causal splitting of distributions in the above algorithm of causal perturbation theory due to Epstein-Glaser 73, which in turn equivalently are the choices known as renormalization coefficients in traditional perspectives to perturbative quantum field theory (which in a mathematically rigorous approach might better be called “normalization” coefficients as in Scharf 95, section 4.3). The main theorem of perturbative renormalization states that the space of these choices is a torsor over a group, called the Stückelberg-Petermann renormalization group.

Finally notice that there is also a choice involved in restricting attention to the microcausal functionals inside the algebra of all smooth functions on phase space. For deformation quantization on finite-dimensional manifolds it is understood that the full algebra of smooth functions is to be quantized, but for the infinite-dimensional phase space of field theory the full algebra of functions does not even support the Poisson bracket which is to be deformed. The subalgebra of microcausal functionals just happens to be technically under control, closed under the Poisson bracket, and large enough to contain the point-interaction terms that are relevant in quantum field theories observed in nature (QED, QCD, perturbative quantum gravity). It seems to be unclear and in fact un-studied what a lift of the whole program to larger subalgebras would mean or entail.

One way to understand why it is not the full space of smooth functions on phase space that should be quantized is the sibling of algebraic deformation quantization known as geometric quantization. This demands that only the Hamiltonian observables be considered, which are those smooth functions $h$ on phase space such that there exists a smooth vector field $v$ with $d h = \omega(v,-)$ (where $\omega$ is the pre-symplectic form). On these at the very least the Poisson bracket is guaranteed to close (for discussion of this in the context of local field theory see Benini-Schenkel 16), in fact geometric pre-quantization is the Lie integration of the resulting Poisson-bracket Lie algebra to the quantomorphism group. Hence the restriction to Hamiltonian observables may be regarded as part of non-perturbative field theory and hence highlights the intrinsic incompleteness of perturbative quantization methods.

At the moment, however, non-perturbative quantization of field theories remains completely non-understood (apart from toy cases such as 2d conformal field theories and certain topological field theories). For the class of Yang-Mills theories it is one of in the list of open “Millenium Problems” of the Clay Mathematics Institute (Jaffe-Witten, Douglas 04).

For more detailed discussion of all of this see at geometry of physics – A first idea of quantum field theory.

Besides lcpQFT being just perturbative (in Planck's constant), 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 meant to be 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.

References

Review

Survey and exposition of locally covariant algebraic quantum field theory beyond causal perturbation theory in Minkowski spacetime (for which see there) includes the following:

• Hollands 07, section 3

• Robert Wald, The Formulation of Quantum Field Theory in Curved Spacetime (arXiv:0907.0416)

• Robert Wald, The History and Present Status of Quantum Field Theory in Curved Spacetime (arXiv:gr-qc/0608018)

• 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)

• Klaus Fredenhagen, Quantum field theory and gravitation, talk at closing colloquium of trimester “Le Monde Quantique”, IHES April 2015 (pdf)

• Katarzyna Rejzner, Perturbative Algebraic Quantum Field Theory, Mathematical Physics Studies, Springer 2016 (web)

• Urs Schreiber, geometry of physics – perturbative quantum field theory, 2017

• Michael Dütsch, From classical field theory to perturbative quantum field theory, 2018

• Romeo Brunetti, Klaus Fredenhagen, Kasia Rejzner, Locally covariant approach to effective quantum gravity, in Handbook of Quantum Gravity, Springer (2023) [arXiv:2212.07800]

Original articles

The method of causal perturbation theory goes back to ideas of Stückelberg and Bogoliubov. The proof that the causal S-matrix required by this method may always be constructed by an inductive process with a finite-dimensional space of choices (“renormalization”) at each stage is due to

• Henri Epstein, Vladimir Glaser, The Role of locality in perturbation theory, Annales Poincaré Phys. Theor. A 19 (1973) 211.

A detailed exposition of this method applied to QED, QCD, Yang-Mills theory and perturbative quantum gravity on Minkowski spacetime is in the textbooks

• Günter Scharf, Finite Quantum Electrodynamics – The Causal Approach, Berlin: Springer-Verlag, 1995, 2nd edition

• Günter Scharf, Quantum Gauge Theories – A True Ghost Story, Wiley 2001

The axioms for the locally covariant formulation of AQFT on curved spacetimes is due to

• Romeo Brunetti, Klaus Fredenhagen, Rainer Verch, The generally covariant locality principle – A new paradigm for local quantum physics, Commun.Math.Phys.237:31-68, 2003 (arXiv:math-ph/0112041)

The construction of Wick algebras of quantum observables of free fields on curved spacetimes via Hadamard distributions relies on the results of

• Marek Radzikowski, Micro-local approach to the Hadamard condition in quantum field theory on curved space-time, Commun. Math. Phys. 179 (1996), 529–553 (Euclid)

The observation that the causal perturbation theory of Epstein-Glaser 73 lends itself to a construction of locally covariant perturvative AQFT is due to

• V. A. Il’in and D. S. Slavnov, Observable algebras in the S-matrix approach, Theor. Math. Phys. 36 (1978) 32. (spire, doi)

• Brunetti-Fredenhagen 00, last section

• Romeo Brunetti, Michael Dütsch, Klaus Fredenhagen, Perturbative Algebraic Quantum Field Theory and the Renormalization Groups, Adv. Theor. Math. Physics 13 (2009), 1541-1599 (arXiv:0901.2038)

Extension of causal perturbation theory to curved spacetimes is due to

• 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)

  following

  • Raymond Stora, Differential algebras in Lagrangean field theory, ETH Lectures, January-February 1993. Manuscript

• Michael Dütsch, Klaus Fredenhagen, Algebraic Quantum Field Theory, Perturbation Theory, and the Loop Expansion, Commun.Math.Phys. 219 (2001) 5-30 (arXiv:hep-th/0001129)

• Stefan Hollands, Robert Wald, Local Wick Polynomials and Time Ordered Products of Quantum Fields in Curved Spacetime, Commun. Math. Phys. 223:289-326,2001 (arXiv:gr-qc/0103074)

• Stefan Hollands, Robert Wald, On the Renormalization Group in Curved Spacetime, Commun. Math. Phys. 237 (2003) 123-160 (arXiv:gr-qc/0209029)

• Stefan Hollands, Robert Wald, Conservation of the stress tensor in perturbative interacting quantum field theory in curved spacetimes, Rev.Math.Phys. 17 (2005) 227-312 (arXiv:gr-qc/0404074)

Specifically construction of renormalized Yang-Mills theory on curved spacetimes is due to

• Stefan Hollands, Renormalized Quantum Yang-Mills Fields in Curved Spacetime, Rev. Math. Phys.20:1033-1172, 2008 (arXiv:0705.3340)

The observation that this construction produces formal deformation quantization of the classical field theory was made for the free field and its quantization by Wick algebras is

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

which was amplified in

• Michael Dütsch, Klaus Fredenhagen, Perturbative algebraic field theory, and deformation quantization, in Roberto 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

  (arXiv:hep-th/0101079)

and generalized to quantum field theory on curved spacetimes in Brunetti-Fredenhagen 00 and

• Romeo Brunetti, Klaus Fredenhagen, M. Köhler, The microlocal spectrum condition and Wick polynomials on curved spacetimes, Commun. Math. Phys. 180, 633-652, 1996 (arXiv:gr-qc/9510056)

• Stefan Hollands, Robert Wald, Local Wick Polynomials and Time Ordered Products of Quantum Fields in Curved Spacetime, Commun. Math. Phys. 223:289-326, 2001 (arXiv:gr-qc/0103074)

That more generally the deformation quantization of interacting fields yields the traditional prescription of causal perturbation theory was shown for the scalar field phi^4 theory in

• Giovanni Collini, Fedosov Quantization and Perturbative Quantum Field Theory (arXiv:1603.09626)

and for the interacting scalar field in the toy example of regular non-local interactions in

• Eli Hawkins, Kasia Rejzner, The Star Product in Interacting Quantum Field Theory (arXiv:1612.09157)

The perturbative quantization of gauge theories (Yang-Mills theory) in causal perturbation theory/perturbative AQFT is discussed (for trivial principal bundles and restricted to gauge invariant observables) via BRST-complex/BV-formalism in

• Klaus Fredenhagen, Kasia Rejzner, Batalin-Vilkovisky formalism in the functional approach to classical field theory, Commun. Math. Phys. 314(1), 93–127 (2012) (arXiv:1101.5112)

• Klaus Fredenhagen, Kasia Rejzner, Batalin-Vilkovisky formalism in perturbative algebraic quantum field theory, Commun. Math. Phys. 317(3), 697–725 (2012) (arXiv:1110.5232)

and surveyed in

• Rejzner 16, chapter 7

A survey of the required generalization for gauge theory with non-trivial instanton sectors 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)