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#### Algebraic Quantum Field Theory
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#### Physics
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#Contents#
* table of contents
{:toc}

## Idea

What is called _locally covariant perturbative algebraic quantum field theory_ ([Brunetti-Fredenhagen 00](#BrunettiFredenhagen00), [Brunetti-Fredenhagen-Verch 03](#BrunettiFredenhagenVerch03)) is a formulation of [[local quantum field theory|local]] [[perturbative quantum field theory|perturbative]] [[quantum field theory]] [[AQFT on curved spacetimes|on general spacetimes]] (hence on general [[classical field theory|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]] [[algebra of observables|of observables]] instead of [[C*-algebras]]

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

1. 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](#IlinSlavnov78) but then ignored) that standard [[renormalization]] theory (removal of [[ultraviolet divergences]]) in the guise of [[causal perturbation theory]] ([Epstein-Glaser 73](#EpsteinGlaser73)) automatically produces [[causally local nets]] of [[algebras of observables]] (due to "[[adiabatic switching]]" and [[causal locality]] of the [[S-matrix]], see [there](S-matrix#CausalLocality)), 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) [[perturbative QFT|perturbation theory]] by means of [[formal geometry|formal]] [[Feynman diagram]] expansions of ([[Wick rotation|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 in QCD|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|non-perturbative]] quantum field theory. But as a matter of fact not a single relevant example ([[interaction|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".

$\,$

Locally covariant perturbative quantum field theory provides a synthesis of these two opposites.

$\,$

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 field theory|local]] [[Lagrangian field theory|Lagrangian]] [[classical field theory]] on [[gravity]] [[background fields|backgrounds]], hence a [[natural transformation]]
from ([[globally hyperbolic spacetimes|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 structure|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 functions|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 field theory|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 action|kinetic term]] $L_{kin}$ and an [[interaction]] term $L_{int} \coloneqq L - L_{kin}$, where $L_{kin}$ is such that its [[Euler-Lagrange equation|Euler-Lagrange]] [[equation of motion]] is a [[linear operator|linear]] [[hyperbolic differential operator]] (the "[[wave operator]]" for the [[free field]]);

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

1. for the remaining [[interaction]] $L_{int}$ regarded as an element of the Wick algebra, [[induction|inductively]] construct its [[S-matrix]] as a [[distribution]] with values in the Wick algebra by demanding it to be [[causal additivity|causally additive]] (this is the step of [[causal perturbation theory]] proper).  The theorem of ([Epstein-Glaser 73](#EpsteinGlaser73)) 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]]).

1. 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](#EpsteinGlaser73) building on ideas of Stückelberg and Bogoliubov, while the generalization to [[globally hyperbolic spacetimes]] is due to [Brunetti-Fredenhagen 00](#BrunettiFredenhagen00) (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](#IlinSlavnov78)), but ignored or forgotten, and then rediscovered and popularized in ([Brunetti-Fredenhagen 00](#BrunettiFredenhagen00)).

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](#Collini16)) (for [[phi^4 theory]] of the [[scalar field]]) and in ([Hawkins-Rejzner 16](#HawkinsRejzner16)) (for the [[scalar field]] with regular non-local interaction terms).

There are choices to be made in finding this deformation quantization. In ([Collini 16](#Collini16)) these are parameterized as the choices of initial conditions in the [[induction|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](#EpsteinGlaser73), 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](#Scharf95)).
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 vector field|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](Poisson+manifold#BeniniSchenkel16)), 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](quantization+of+Yang-Mills+theory#JaffeWitten), [Douglas 04](quantization+of+Yang-Mills+theory#Douglas04)).

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 [[quantum field theory on curved spacetime|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](#Schenkel14), [Schreiber 14](https://ncatlab.org/schreiber/show/Higher+field+bundles+for+gauge+fields)).

This remaining problem is meant to be solved by passing from plain [[algebras of observables]] to [[homotopical algebra|homotopical algebras]] ([[higher algebra|higher algebras]]), and hence to a formulation of **[[homotopical algebraic quantum field theory]]** (see [Schenkel 17](#Schenkel17)). 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](causal+perturbation+theory#References)) includes the following:

* [Hollands 07, section 3](#Hollands07)

* [[Robert Wald]], _The Formulation of Quantum Field Theory in Curved Spacetime_ ([arXiv:0907.0416](https://arxiv.org/abs/0907.0416))

* [[Robert  Wald]], _The History and Present Status of Quantum Field Theory in Curved Spacetime_ ([arXiv:gr-qc/0608018](https://arxiv.org/abs/gr-qc/0608018))

* {#FredenhagenRejzner12} [[Klaus Fredenhagen]], [[Katarzyna Rejzner]], _Perturbative algebraic quantum field theory_, In _Mathematical Aspects of Quantum Field Theories_, Springer 2016 ([arXiv:1208.1428](https://arxiv.org/abs/1208.1428))

* {#FredenhagenRejzner15} [[Klaus Fredenhagen]], [[Katarzyna Rejzner]], _Perturbative Construction of Models of Algebraic Quantum Field Theory_ ([arXiv:1503.07814](https://arxiv.org/abs/1503.07814))

* [[Klaus Fredenhagen]], _Quantum field theory and gravitation_, talk at closing colloquium of trimester "Le Monde Quantique", IHES April 2015 ([pdf](https://indico.math.cnrs.fr/event/782/material/slides/1.pdf))

* {#Rejzner16} [[Katarzyna Rejzner]], _[[Perturbative Algebraic Quantum Field Theory]]_, Mathematical Physics Studies, Springer 2016 ([web](https://link.springer.com/book/10.1007%2F978-3-319-25901-7))

* [[Urs Schreiber]], _[[geometry of physics -- perturbative quantum field theory]]_, 2017

* {#Duetsch18} [[Michael Dütsch]], _[[From classical field theory to perturbative quantum field theory]]_, 2018

* {#BrunettiFredenhagenTejzner23} [[Romeo Brunetti]], [[Klaus Fredenhagen]], [[Kasia Rejzner]], *Locally covariant approach to effective quantum gravity*, in *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2212.07800](https://arxiv.org/abs/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

* {#EpsteinGlaser73} [[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

* {#Scharf95} [[Günter Scharf]], _[[Finite Quantum Electrodynamics -- The Causal Approach]]_, Berlin: Springer-Verlag, 1995, 2nd edition

* {#Scharf01} [[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

* {#BrunettiFredenhagenVerch03} [[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](https://arxiv.org/abs/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

* {#Radzikowski96} [[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](http://projecteuclid.org/euclid.cmp/1104287114))

The observation that the [[causal perturbation theory]] of [Epstein-Glaser 73](#EpsteinGlaser73) lends itself to a construction of locally covariant perturvative AQFT is due to

* {#IlinSlavnov78} V. A. Il'in and D. S. Slavnov, _Observable algebras in the S-matrix approach_, Theor. Math. Phys. 36 (1978) 32. ([spire](http://inspirehep.net/record/135575), [doi](http://dx.doi.org/10.1007/BF01035870))

* [Brunetti-Fredenhagen 00, last section](#BrunettiFredenhagen00)

* {#BrunettiDuetschFredenhagen09} [[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](http://arxiv.org/abs/0901.2038))

Extension of [[causal perturbation theory]] to [[AQFT on curved spacetimes|curved spacetimes]] is due to

* {#BrunettiFredenhagen00} [[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](https://arxiv.org/abs/math-ph/9903028))

  following

  * {#Stora93} [[Raymond Stora]], _Differential algebras in Lagrangean field theory_, ETH Lectures, January-February 1993. Manuscript

* {#DuetschFredenhagen00} [[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](https://arxiv.org/abs/hep-th/0001129))

* {#HollandsWald01} [[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](https://arxiv.org/abs/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](https://arxiv.org/abs/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](https://arxiv.org/abs/gr-qc/0404074))

Specifically construction of [[renormalization|renormalized]] [[Yang-Mills theory]] on curved spacetimes is due to

* {#Hollands07} [[Stefan Hollands]], _Renormalized Quantum Yang-Mills Fields in Curved Spacetime_, Rev. Math. Phys.20:1033-1172, 2008 ([arXiv:0705.3340](https://arxiv.org/abs/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](https://inspirehep.net/record/303898/))

which was amplified in

* {#DuetschFredenhagen01} [[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](https://arxiv.org/abs/hep-th/0101079))

and generalized to [[quantum field theory on curved spacetimes]] in [Brunetti-Fredenhagen 00](#BrunettiFredenhagen00) and

* {#BrunettiFredenhagen95} [[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](https://arxiv.org/abs/gr-qc/9510056))

* {#HollandsWald01} [[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](https://arxiv.org/abs/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

* {#Collini16} [[Giovanni Collini]], _Fedosov Quantization and Perturbative Quantum Field Theory_ ([arXiv:1603.09626](https://arxiv.org/abs/1603.09626))

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

* {#HawkinsRejzner16} [[Eli Hawkins]], [[Kasia Rejzner]], _The Star Product in Interacting Quantum Field Theory_ ([arXiv:1612.09157](https://arxiv.org/abs/1612.09157))

The [[perturbative quantum field theory|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

* {#FredenhagenRejzner11a} [[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](https://arxiv.org/abs/1101.5112))

* {#FredenhagenRejzner11b} [[Klaus Fredenhagen]], [[Kasia Rejzner]], _Batalin-Vilkovisky formalism in perturbative algebraic quantum field theory_, Commun. Math. Phys. 317(3), 697–725 (2012) ([arXiv:1110.5232](https://arxiv.org/abs/1110.5232))

and surveyed in

* [Rejzner 16, chapter 7](#Rejzner16)

A survey of the required generalization for [[gauge theory]] with non-trivial [[instanton sectors]] via [[homotopical algebraic quantum field theory]] is in

* {#Schenkel14} [[Alexander Schenkel]], _On the problem of gauge theories
in locally covariant QFT_, talk at _[Operator and Geometric Analysis on Quantum Theory](http://www.science.unitn.it/~moretti/convegno/convegno.html)_ Trento, 2014 ([[SchenkelTrento2014.pdf:file]]) (with further emphasis on this point in the companion talk [Schreiber 14](field+bundle#Schreiber14))

* {#Schenkel17} [[Alexander Schenkel]], _Towards Homotopical Algebraic Quantum Field Theory_, talk at _[Foundational and Structural Aspects of Gauge Theories](https://indico.mitp.uni-mainz.de/event/76/overview)_,
Mainz Institute for Theoretical Physics, 29 May – 2 June 2017. ([pdf](http://aschenkel.eu/Mainz17.pdf))

[[!redirects perturbative AQFT]]
[[!redirects pAQFT]]

[[!redirects locally covariant perturbative quantum field theory]]
[[!redirects locally covariant perturbative quantum field theories]]

[[!redirects locally covariant perturbative algebraic quantum field theory]]
[[!redirects locally covariant perturbative algebraic quantum field theories]]

[[!redirects locally covariant perturbative QFT]]
[[!redirects locally covariant perturbative QFTs]]

[[!redirects locally covariant perturbative AQFT]]

[[!redirects locally covariant pAQFT]]


perturbative observables -- table

**[[types]] of [[observables]] in [[perturbative quantum field theory]]**:

$$
  \array{
    &&
    \text{local}
    \\
    &&
    & \searrow
    \\
    \text{field}
    &\longrightarrow&
    \text{linear}
    &\longrightarrow&
    \text{microcausal}
    &\longrightarrow&
    \text{polynomial}
    &\longrightarrow&
    \text{general}
    \\
    &&
    & \nearrow
    \\
    &&
    \text{regular}
  }
$$


perturbative quantization of Chern-Simons theory -- references

### Perturbative quantum Chern-Simons theory
 {#ReferencesPerturbativeCSTheory}

Discussion of  [[perturbative quantum field theory|perturbative]] [[quantization of Chern-Simons theory]] (via [[Kontsevich integrals]]/[[knot graph cohomology]] on [[Jacobi diagrams]], regarding [[Feynman amplitudes as differential forms on configuration spaces of points]] and yielding [[universal Vassiliev invariant|universal]][[Vassiliev invariants]]):

* {#BarNatan91} [[Dror Bar-Natan]], _Perturbative aspects of the Chern-Simons topological quantum field theory_, thesis 1991 ([spire:323500](http://inspirehep.net/record/323500), [proquest:303979053](https://search.proquest.com/docview/303979053), [[BarNatanPerturbativeCS91.pdf:file]])

* {#AxelrodSinger} [[Scott Axelrod]], [[Isadore Singer]], _Chern-Simons Perturbation Theory_, in S. Catto, A. Rocha (eds.) Proc. XXthe DGM Conf. World Scientific Singapore, 1992, 3-45; ([arXiv:hep-th/9110056](http://arxiv.org/abs/hep-th/9110056))

* [[Scott Axelrod]], [[Isadore Singer]], _Chern--Simons Perturbation Theory II_, J. Diff. Geom. 39 (1994) 173-213 ([arXiv:hep-th/9304087](http://arxiv.org/abs/hep-th/9304087))

* {#Kontsevich93} [[Maxim Kontsevich]], _Vassiliev's knot invariants_, Advances in Soviet Mathematics, Volume 16, Part 2, 1993 ([pdf](http://pagesperso.ihes.fr/~maxim/TEXTS/VassilievKnot.pdf))

* {#Kontsevich} [[Maxim Kontsevich]], _Feynman diagrams and low-dimensional topology_, in First European Congress of Mathematics, Vol. II (Paris, 1992), volume 120 of Progr. Math., pages 97–121, Birkhäuser, Basel, 1994. ([pdf](https://www.ihes.fr/~maxim/TEXTS/Feynman%20%20diagrams%20and%20low-dimensional%20topology.pdf))

* {#BarNatan95} [[Dror Bar-Natan]], _Perturbative Chern-Simons theory_,  Journal of Knot Theory and Its RamificationsVol. 04, No. 04, pp. 503-547 (1995) ([doi:10.1142/S0218216595000247](https://doi.org/10.1142/S0218216595000247))

* {#AF96} [[Daniel Altschuler]], [[Laurent Freidel]], _Vassiliev knot invariants and Chern-Simons perturbation theory to all orders_, Commun. Math. Phys. **187** (1997) 261-287 $[$[arXiv:q-alg/9603010](https://arxiv.org/abs/q-alg/9603010), [doi:10.1007/s002200050136](https://doi.org/10.1007/s002200050136)$]$

* {#LambrechtsVolic14} [[Pascal Lambrechts]], [[Ismar Volić]], sections 6 and 7 of _Formality of the little N-disks operad_, Memoirs of the American Mathematical Society ; no. 1079, 2014  ([arXiv:0808.0457](https://arxiv.org/abs/0808.0457), [doi:10.1090/memo/1079](http://dx.doi.org/10.1090/memo/1079))

Review:

* [[Robbert Dijkgraaf]], _Perturbative topological field theory_, In: *Trieste 1993, Proceedings, String theory, gauge theory and quantum gravity '93* 189-227 ([spire:399223](http://inspirehep.net/record/399223/), [[DijkgraafPerturbativeCS93.pdf:file]])

* {#ChmutovDuzhinMostovoy11} [[Sergei Chmutov]], [[Sergei Duzhin]], [[Jacob Mostovoy]], Section 4 of:  _Introduction to Vassiliev knot invariants_, Cambridge University Press, 2012 ([arxiv/1103.5628](http://arxiv.org/abs/1103.5628), [doi:10.1017/CBO9781139107846](https://doi.org/10.1017/CBO9781139107846))

See also at _[[correlator as differential form on configuration space of points]]_ and see at _[[graph complex]]_  as a model for the [[spaces of knots]]_.

The "[[Wheels theorem]]", saying that the  [[perturbative quantization of 3d Chern-Simons theory|perturbative Chern-Simons]] [[Wilson loop observable]] of the [[unknot]] is, as a [[universal Vassiliev invariant]], a [[series]] of wheel-shaped [[Jacobi diagrams]] with [[coefficients]] the [[modified Bernoulli numbers]], is due to

* [[Dror Bar-Natan]], Thang T Q Le, [[Dylan Thurston]], _Two applications of elementary knot theory to Lie algebras and Vassiliev invariants_, Geom. Topol. Volume 7, Number 1 (2003), 1-31 ([euclid.gt/1513883092](https://projecteuclid.org/euclid.gt/1513883092))

following

* [[Dror Bar-Natan]], [[Stavros Garoufalidis]], [[Lev Rozansky]], [[Dylan Thurston]], _Wheels, wheeling, and the Kontsevich integral of  the unknot_ ([q-alg/9703025](http://arxiv.org/abs/q-alg/9703025))


perturbative quantum field theory

+-- {: .rightHandSide}
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###Context###
#### Algebraic Quantum Field Theory
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

What is called _perturbative quantum field theory_ (pQFT) is [[quantum field theory]] where the [[interaction]] (between [[field (physics)|fields]]/[[particles]]) is treated as a tiny [[perturbation]] of the "[[free field theory]]" where no [[interaction]] is assumed to take place ("[[perturbation theory]]"). This is meant to be an approximation to the actual _[[non-perturbative quantum field theory]]_. However, the latter remains elusive except for toy examples of low spacetime dimension, vanishing [[interaction]] and/or [[topological field theory|topological invariance]] and most of the "quantum field theory" in the literature is tacitly understood to be perturbative.

{#pQFTAsInfinitesimalThickenedPoint} Hence pQFT studies the _[[infinitesimal neighbourhood]]_ (also called the _[[formal neighbourhood]]_) of [[free quantum field theories|free]] [[classical field theories]] in the parameter space of all quantum field theories, the rest of the space being the realm of [[non-perturbative QFT]]:

Mathematically this means that the resulting [[quantum observables]] in pQFT (typically: [[scattering amplitudes]] or "[[S-matrices]]" encoded by [[Feynman diagrams]]) are [[formal power series]] in the [[coupling constant]]  $g$ which measures the strength of the [[interaction]] (as well as in _[[Planck's constant]]_, which measures the general strength of [[quantum physics|quantum]]). This distinguishes perturbative quantum field theory from [[non-perturbative quantum field theory]], where the algebras of [[quantum observables]] are supposed to be not formal power series algebras, but [[C*-algebras]].

While just describing a tiny (infinitesimal) part of QFT, perturbative QFT is way better understood than full [[non-perturbative QFT]] (which remains ill-understood to the extent that it has been named a *[Millennium Problem](mass+gap#ReferencesMassGapProblem)*), and has the reputation of still being dramatically successful in practice -- though this applies to some field theories (such as [[QED]]) much more than to others (such as [[QCD]]), see [below](#ReferencesForInverseOfCouplingArgument).

The key object of perturbative QFT is the perturbative _[[scattering matrix]]_ which expresses, as a [[formal power series]] in the ratio of the [[coupling constant]] over [[Planck's constant]], the [[probability amplitude]] of [[scattering]] processes, namely of processes where [[free field theory|free fields]] in a certain [[quantum state|state]] come in from the far past, interact and hence scatter off each other, and then go off in some other [[quantum state]] into the far future. The [[scattering cross sections]] thus defined are the quantities which may be directly measured in scattering [[experiments]], such as the [[LHC]] accelerator.

The perturbative [[S-matrix]] turns out to have an expression as a sum over separate [[scattering amplitudes]] for elementary processes labeled by _[[Feynman diagrams]]_, each of which depicts one specific way for fields ([[particles]]) to interact with each other. That the full S-matrix is the sum over all amplitudes for all these possible scattering processes, the _[[Feynman perturbation series]]_, is an incarnation of the informal heuristic of the [[path integral]] and the [[superposition principle]] in [[quantum physics]], which says that the [[probability amplitude]] for a specific outcome is the sum over the probability amplitudes of all the possible processes that can contribute to this outcome.

For all interesting [[interacting field theories]], such as [[quantum electrodynamics]] and [[quantum chromodynamics]], this [[scattering matrix]] [[formal power series]] necessarily has _vanishing_ [[radius of convergence]] ([Dyson 52](#Dyson52)). If it is assumed that the [[formal power series|formal]] [[Feynman perturbation series]] is the [[Taylor series]] of an actual [[smooth function]] given by the actual [[non-perturbative quantum field theory]] that is being approximated, then this means that it is at least an [[asymptotic series]] (by [this example](asymptotic+series#TaylorSeriesOfSmoothFunctionIsAsymptoticSeries)) whose first couple of terms could sum to a good approximation of the actual value to be computed. Indeed, the sum of the first few [[loop orders]] in the [[S-matrix]] for  [[QED]] and [[QCD]] in the [[standard model of particle physics]] turns out to be in agreement with [[experiment]] to good precision.

(There are however known [[non-perturbative effects]] which are not captured in perturbation theory, such as [[confinement]] in [[QCD]], supposed related to [[instantons in QCD]]. In [[resurgence theory]] one tries to identify these from the [[asymptotic series|asymptotic]] nature of the [[Feynman perturbation series]].)

A key step in the construction of perturbative quantum field theory is the _[[renormalization]]_ of the point interactions. This comes about because given

1. a [[local Lagrangian density]] defining the nature of the [[field (physics)|fields]] and their [[interactions]],

1. a [[vacuum state]] (generally: [[Hadamard state]]) that defines the [[free field theory|free]] [[quantum field theory]] to be perturbed about

it turns out that the construction of the perturbative [[S-matrix]] (the [[Feynman perturbation series]]) still involves at each order a finite-dimensional space of choices to be made. Physically, these are the specification of further high energy interactions not seen in the original [[local Lagrangian density]]; mathematically, this is the choice of extending the [[time-ordered product]] of the interaction, which is an [[operator-valued distribution]], to the locus of coinciding interaction points, in the sense of [[extensions of distributions]].

Historically, perturbative quantum field theory as originally conceived informally by [[Schwinger-Tomonaga-Feynman-Dyson]] in the 1940s, had been notorious for the mysterious conceptual nature of its mathematical principles ("divergences"). The mathematically rigorous formulation of [[renormalization]] ("removal of [[UV-divergences]]") in perturbative quantum field theory on [[Minkowski spacetime]] was established by [Epstein-Glaser 73](#EpsteinGlaser73), based on [Bogoliubov-Shirkov 59](#BogoliubovShirkov59) and [Stückelberg 51](#Stueckelberg51)), now known as **[[causal perturbation theory]]**; laid out in the seminal Erice summer school proceeding ([Velo-Wightman 76](#VeloWightman76)).

The correct definition of the [[adiabatic switching|adiabatic limit]] ("removal of IR divergencies") was understood in [Il'in-Slavnov 78](#IlinSlavnov78) and eventually developed by [Dütsch-Fredenhagen 01](#DuetschFredenhagen01), [Brunetti-DütschFredenhagen 09](#BrunettiDuetschFredenhagen09), this is now called **[[perturbative algebraic quantum field theory]]**.
The rigorous derivation of the previously informal [[Feynman rules]] and their [[dimensional regularization]] for computation of [[scattering amplitudes]] was achieved in [Keller 10 (IV.12)](#Keller10), [Dütsch-Fredenhagen-Keller-Rejzner 14](#DuetschFredenhagenKellerRejzner14).
Quantization of [[gauge theories]] ([[Yang-Mills theory]]) in [[causal perturbation theory]]/[[perturbative AQFT]] was then discussed (for trivial [[principal bundles]] and restricted to [[gauge invariant observables]]) in the spirit of [[BRST-complex]]/[[BV-formalism]] in ([Fredenhagen-Rejzner 11b](#FredenhagenRejzner11b)).
The generalization of all these constructions from [[Minkowski spacetime]] to perturbative quantum fields on more general [[spacetimes]] (i.e. for more general [[gravity|gravitational]] [[background fields]] such as appearing in [[cosmology]] or [[black hole]] physics) was made possible due to the identification of the proper generalization of [[vacuum states]] and their [[Feynman propagators]] to [[Hadamard states]] on [[globally hyperbolic spacetimes]] in [Radzikowski 96](#Radzikowski96). The resulting rigorous perturbative [[QFT on curved spacetimes]] was developed in a long series of articles by [[Stefan Hollands|Hollands]], [[Robert Wald|Wald]], [[Romeo Brunetti|Brunetti]], [[Klaus Fredenhagen|Fredenhagen]] and others, now called _[[locally covariant perturbative AQFT]]_.

While this establishes a rigorous construction of perturbative quantum field theory on general gravitational backgrounds, the construction principles had remained somewhat ad-hoc: The [[axioms]] for the perturbative [[S-matrix]] (equivalently for the [[time-ordered products]] or [[retarded products]] of field operators) were well motivated by comparison with the [[Dyson series]] in [[quantum mechanics]], by the heuristics of the [[path integral]] and not the least by their excellent confirmation by [[experiment]], but had not been derived from first principles of [[quantization]].  Then in [Dütsch Fredenhagen 01](#DuetschFredenhagen01) it was observed that the [[Wick algebras]] of [[quantum observables]] in [[free quantum field theory]] are equivalently the [[Moyal deformation quantization]] of the canonical [[Poisson bracket]] (the _[[Peierls bracket]]_ or _[[causal propagator]]_) on the [[covariant phase space]] of the free field theory (or rather of a choice of [[Hadamard state]] for it) and [Collini 16](#Collini16) showed that under suitable conditions the perturbative [[interacting observable algebra]] is the [[Fedosov deformation quantization]] of [[covariant phase space]] of the interacting theory. A general argument to this extent was given in [Hawkins-Rejzner 16](#HawkinsRejzner16).

This suggests that the construction of the full [[non-perturbative quantum field theory]] ought to be given by a [[strict deformation quantization]] of the [[covariant phase space]]. But presently no example of such for non-trivial interaction in [[spacetime]] [[dimension]] $\geq 4$ is known.
In particular the [[phenomenology|phenomenologically]] interesting case of a complete construction of interacting field theories on 4-dimensional spacetimes is presently unknown. For the case of [[Yang-Mills theory]] this open problem to go beyond perturbative quantum field theory is one of the "Millennium Problems" (see at _[[quantization of Yang-Mills theory]]_). For the case of [[quantum gravity]] this is possibly the $10^4$-year problem that the field is facing. But observe that as a perturbative ([[effective quantum field theory|effective]]") quantum field theory, [[quantum gravity]] does fit into the framework of  perturbative QFT, is mathematically well-defined and makes predictions, see the references [there](quantum%20gravity#ReferencesAsAnEffectiveFieldTheory).

## Details

A comprehensive introduction is at _[[geometry of physics -- perturbative quantum field theory]]_.

## Properties

* [[main theorem of perturbative renormalization theory]]

## Related concepts

* [[renormalization]], [[radiative correction]]

* [[causal perturbation theory]], [[perturbative AQFT]]

[[!include products in pQFT -- table]]

* [[non-perturbative field theory]]

## References

### General

The original informal conception of perturbative QFT is due to [[Schwinger-Tomonaga-Feynman-Dyson]]:

* {#Dyson49} [[Freeman Dyson]], _The raditation theories of Tomonaga, Schwinger and Feynman_, Phys. Rev. 75, 486, 1949 ([pdf](http://web.ihep.su/dbserv/compas/src/dyson49b/eng.pdf))

The rigorous formulation of renormalized perturbative quantum field theory in terms of [[causal perturbation theory]] was first accomplished in

* {#EpsteinGlaser73} [[Henri Epstein]], [[Vladimir Glaser]], _[[The Role of locality in perturbation theory]]_, Annales Poincaré Phys. Theor. A 19 (1973) 211 ([Numdam](http://www.numdam.org/item?id=AIHPA_1973__19_3_211_0 ))

with precursors in

* {#Stueckelberg49} [[Ernst Stückelberg]], D. Rivier, Helv. Phys. Acta, 22 (1949) 215.

* {#Stueckelberg51} [[Ernst Stückelberg]], J. Green, Helv. Phys. Acta, 24 (1951) 153.

* [[Ernst Stückelberg]], A. Peterman, , _La normalisation des constants dans la theorie des quanta_, Helv. Phys. Acta 26, 499 (1953);

* {#BogoliubovShirkov59} [[Nikolay Bogoliubov]], [[Dmitry Shirkov]], _Introduction to the Theory of Quantized Fields_, New York (1959)

Traditional review includes:

* [[Matthew Schwartz]], *Quantum Field Theory and the Standard Model*, Cambridge University Press (2014) [[ISBN:9781107034730](https://www.cambridge.org/de/universitypress/subjects/physics/theoretical-physics-and-mathematical-physics/quantum-field-theory-and-standard-model?format=HB&isbn=9781107034730), [doi:10.1017/9781139540940](https://doi.org/10.1017/9781139540940), [pdf](https://indico.ihep.ac.cn/event/11134/attachments/3783/4308/QFT-Schwartz.pdf)]

A seminal compilation of the resulting rigorous understanding of [[renormalization]] is

* {#VeloWightman76} G. Velo and [[Arthur Wightman]] (eds.) _Renormalization Theory_ Proceedings of the 1975 Erice summer school, NATO ASI Series C 23, D. Reidel, Dordrecht, 1976

Concrete computations in rigorous [[causal perturbation theory]] have been spelled out for [[quantum electrodynamics]] in

* {#Scharf95} [[Günter Scharf]], _[[Finite Quantum Electrodynamics -- The Causal Approach]]_, Berlin: Springer-Verlag, 1995, 2nd edition

and for [[Yang-Mills theory]], [[quantum chromodynamics]] and [[perturbative quantum gravity]] in

* {#Scharf01} [[Günter Scharf]], _[[Quantum Gauge Theories -- A True Ghost Story]]_, Wiley 2001

The treatment of the IR-divergencies by organizing the perturbative [[quantum observables]] into a [[local net of observables]] was first suggested in

* {#IlinSlavnov78} V. A. Il'in and D. S. Slavnov, _Observable algebras in the S-matrix approach_, Theor. Math. Phys. 36 (1978) 32 ([spire](http://inspirehep.net/record/135575), [doi](http://dx.doi.org/10.1007/BF01035870))

and then developed to _[[perturbative algebraic quantum field theory]]_ in

* {#DuetschFredenhagen01} [[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](https://arxiv.org/abs/hep-th/0001129))

* {#BrunettiDuetschFredenhagen09} [[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](http://arxiv.org/abs/0901.2038))

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]]) in the spirit of [[BRST-complex]]/[[BV-formalism]] in

* {#FredenhagenRejzner11a} [[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](https://arxiv.org/abs/1101.5112))

* {#FredenhagenRejzner11b} [[Klaus Fredenhagen]], [[Kasia Rejzner]], _Batalin-Vilkovisky formalism in perturbative algebraic quantum field theory_, Commun. Math. Phys. 317(3), 697–725 (2012) ([arXiv:1110.5232](https://arxiv.org/abs/1110.5232))

and surveyed in:

* {#Rejzner16} [[Kasia Rejzner]], section 7 of _[[Perturbative Algebraic Quantum Field Theory]]_, Springer 2016

* [[Urs Schreiber]], _[[geometry of physics -- perturbative quantum field theory]]_, 2017

* {#Duetsch18} [[Michael Dütsch]], _[[From classical field theory to perturbative quantum field theory]]_, 2018

The generalization of all these constructions to quantum fields on general [[globally hyperbolic spacetimes]] (perturbative [[AQFT on curved spacetimes]]) was made possible by the results on [[Hadamard states]] and [[Feynman propagators]] in

* {#Radzikowski96} [[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](http://projecteuclid.org/euclid.cmp/1104287114))

and then developed in a long series of articles by [[Stefan Hollands]], [[Robert Wald]], [[Romeo Brunetti]], [[Klaus Fredenhagen]] and others. For this see the references at _[[AQFT on curved spacetimes]]_.

The observation that perturbative quantum field theory is equivalently the [[formal deformation quantization]] of the defining [[local Lagrangian density]] is for [[free field theory]] due to

* {#DuetschFredenhagen01} [[Michael Dütsch]], [[Klaus Fredenhagen]], _Perturbative algebraic quantum field theory and deformation quantization_, Proceedings of the Conference on Mathematical Physics in Mathematics and Physics, Siena June 20-25 (2000) ([arXiv:hep-th/0101079](http://xxx.uni-augsburg.de/abs/hep-th/0101079))

* {#HirschfeldHenselder02} A. C. Hirshfeld, P. Henselder, _Star Products and Perturbative Quantum Field Theory_, Annals Phys. 298 (2002) 382-393 ([arXiv:hep-th/0208194](https://arxiv.org/abs/hep-th/0208194))

and for interacting field theories ([[causal perturbation theory]]/[[perturbative AQFT]]) due

* {#Collini16} [[Giovanni Collini]], _Fedosov Quantization and Perturbative Quantum Field Theory_ ([arXiv:1603.09626](https://arxiv.org/abs/1603.09626))

* {#HawkinsRejzner16} [[Eli Hawkins]], [[Kasia Rejzner]], _The Star Product in Interacting Quantum Field Theory_ ([arXiv:1612.09157](https://arxiv.org/abs/1612.09157))

For more see the references at _[[perturbative algebraic quantum field theory]]_.

The relation of the construction via [[causal perturbation theory]] to the [[Feynman perturbation series]] in terms of [[Feynman diagrams]] was understood in

* {#GarciaBondiaLazzarini00} [[Jose Gracia-Bondia]], S. Lazzarini, _Connes-Kreimer-Epstein-Glaser Renormalization_ ([arXiv:hep-th/0006106](https://arxiv.org/abs/hep-th/0006106))

* {#Keller10} [[Kai Keller]], chapter IV of _Dimensional Regularization in Position Space and a Forest Formula for Regularized Epstein-Glaser Renormalization_, PhD thesis ([arXxiv:1006.2148](https://arxiv.org/abs/1006.2148))

* {#DuetschFredenhagenKellerRejzner14} [[Michael Dütsch]], [[Klaus Fredenhagen]], [[Kai Keller]], [[Katarzyna Rejzner]], _Dimensional Regularization in Position Space, and a Forest Formula for Epstein-Glaser Renormalization_, J. Math. Phy.
55(12), 122303 (2014) ([arXiv:1311.5424](https://arxiv.org/abs/1311.5424))

Non-rigorous but widely used textbooks:

* [[Michael Peskin]], [[Daniel Schroeder]], _An Introduction to Quantum Field Theory_, 1995 ([spire:407703](https://inspirehep.net/literature/407703), [ISBN 9780201503975](https://www.routledge.com/An-Introduction-To-Quantum-Field-Theory/Peskin-Schroeder/p/book/9780201503975))

(...)

### Non-convergence of the perturbation series
 {#ReferencesNonConvergenceOfThePerturbationSeries}

The argument that the perturbation series of realistic pQFTs necessarily [[divergent series|diverges]], in fact has vanishing [[radius of convergence]] (is at best an [[asymptotic series]]) goes back to

* {#Dyson52} [[Freeman Dyson]], _Divergence of perturbation theory in quantum electrodynamics_, Phys. Rev. 85, 631, 1952 ([spire](http://inspirehep.net/record/29799?ln=en))

and is made more precise in

* {#Lipatov77} [[Lev Lipatov]], _Divergence of the Perturbation Theory Series and the Quasiclassical Theory_, Sov.Phys.JETP 45 (1977) 216–223 ([pdf](http://jetp.ac.ru/cgi-bin/dn/e_045_02_0216.pdf))

recalled for instance in

* {#Suslov05} [[Igor Suslov]], section 1 of _Divergent perturbation series_, Zh.Eksp.Teor.Fiz. 127 (2005) 1350; J.Exp.Theor.Phys. 100 (2005) 1188 ([arXiv:hep-ph/0510142](https://arxiv.org/abs/hep-ph/0510142))

* Justin Bond, last section of _Perturbative QFT is Asymptotic; is Divergent; is Problematic in Principle_ ([pdf](https://mcgreevy.physics.ucsd.edu/s13/final-papers/2013S-215C-Bond-Justin.pdf))

* {#FloryHellingSluka12} [[Mario Flory]], [[Robert C. Helling]], Constantin Sluka, *How I Learned to Stop Worrying and Love QFT* [[arXiv:1201.2714](https://arxiv.org/abs/1201.2714)]

* {#Strocchi13} [[Franco Strocchi]], §2.2 of: _An Introduction to Non-Perturbative Foundations of Quantum Field Theory_, Oxford University Press (2013) [[doi:10.1093/acprof:oso/9780199671571.001.0001](https://doi.org/10.1093/acprof:oso/9780199671571.001.0001)]

* {#HollandsWald14} [[Stefan Hollands]], [[Robert Wald]], Section 4.1 of: _Quantum fields in curved spacetime_, Physics Reports **574** (2015) 1-35 [[arXiv:1401.2026](https://arxiv.org/abs/1401.2026), [doi:10.1016/j.physrep.2015.02.001](https://doi.org/10.1016/j.physrep.2015.02.001)]

* Marco Serone, from 2:46 on in _A look at $\phi^4_2$ using perturbation theory_ ([recording](https://www.youtube.com/watch?v=J4nxvY1rOhI))

{#ReferencesForInverseOfCouplingArgument} The argument that the perturbation series should be trustworthy for number of terms smaller than the inverse of the [[coupling constant]] is recalled in [Flory, Helling & Sluka 2012, p. 8 & eq. (34) & Sec. 2.5](#FloryHellingSluka12).

Exposition also in:

* [[Jakob Schwichtenberg]], *[divergence-perturbation-series-qft](http://jakobschwichtenberg.com/divergence-perturbation-series-qft)*

For the example of [[phi^4 theory|$\phi^4$-theory]] this non-convergence of the perturbation series is discussed in

* {#Helling} [[Robert C. Helling]], p. 4 of _Solving classical field equations_ ([pdf](http://homepages.physik.uni-muenchen.de/~helling/classical_fields.pdf), [[HellingClassicalQFT.pdf:file]])

* {#BakulevShirkov10} Alexander P. Bakulev, [[Dmitry Shirkov]], section 1.1 of _Inevitability and Importance of Non-Perturbative Elements in Quantum Field Theory_, Proceedings of the 6th Mathematical Physics Meeting, Sept. 14--23, 2010, Belgrade, Serbia (ISBN 978-86-82441-30-4), pp. 27--54 ([arXiv:1102.2380](https://arxiv.org/abs/1102.2380))

See also

* Carl M. Bender, Carlo Heissenberg, _Convergent and Divergent Series in Physics_ ([arXiv:1703.05164](https://arxiv.org/abs/1703.05164))

And see at _[[perturbation theory]] -- [On divergence/convergence](perturbation+theory#ReferencesDivergenceConvergence)_

{#ReferencesBreakdownAtLargeNumberOfInsertions} Discussion of further issues, even when [[resummation]] is thought to apply, arising for [[n-point functions]] at large $n$ (large number of external particles in a scattering process):

failure of unitarity (for [[phi^n interaction|$\phi^n$-theory]]):

* Sebastian Schenk, *The Breakdown of Resummed Perturbation Theory at High Energies* ([arXiv:2109.00549](https://arxiv.org/abs/2109.00549))

failure of [[locality]] (for [[perturbative quantum gravity]] and [[perturbative string theory]]):

* [[Sudip Ghosh]], [[Suvrat Raju]], *Loss of locality in gravitational correlators with a large number of insertions*, Phys. Rev. D 96, 066033 (2017) ([arXiv:1706.07424](https://arxiv.org/abs/1706.07424))

* [[Sudip Ghosh]], [[Suvrat Raju]], *The Breakdown of String Perturbation Theory for Many External Particles*, Phys. Rev. Lett. 118, 131602 (2017) ([arXiv:1611.08003](https://arxiv.org/abs/1611.08003))

### L-infinity algebra structure

Further identification of [[L-infinity algebra]]-[[structure]] in the [[Feynman amplitudes]]/[[S-matrix]] of [[Lagrangian field theory|Lagrangian]] [[perturbative quantum field theory]]:

* {#Froeb18} [[Markus Fröb]], _Anomalies in time-ordered products and applications to the BV-BRST formulation of quantum gauge theories_ ([arXiv:1803.10235](https://arxiv.org/abs/1803.10235))

* {#Arvanitakis19} [[Alex Arvanitakis]], _The $L_\infty$-algebra of the S-matrix_ ([arXiv:1903.05643](https://arxiv.org/abs/1903.05643))

[[!redirects perturbative quantum field theories]]

[[!redirects perturbative field theory]]
[[!redirects perturbative field theorys]]

[[!redirects perturbative QFT]]
[[!redirects perturbative QFTs]]

[[!redirects pQFT]]
[[!redirects pQFTs]]


perturbative string theory vacuum

+-- {: .rightHandSide}
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###Context###
#### Vacua
+-- {: .hide}
[[!include vacua -- contents]]
=--
#### String theory
+-- {: .hide}
[[!include string theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[perturbative quantum field theory]] a _[[vacuum state]]_ is the information needed to turn a product of field [[observables]] such as $\mathbf{\Phi}^a(x) \mathbf{\Phi}^b(y)$ into a [[function]] (or rather: [[generalized function]]/[[distribution]]) of the insertion points $x$ any $y$, namely the [[n-point function]] (here [[2-point function]], also called the _[[Hadamard propagator]]_)

$$
  \langle \mathbf{\Phi}^a(x) \mathbf{\Phi}^b(y)\rangle
$$

which may be regarded as the [[probability amplitude]] for a quantum in state $b$ at spacetime point $y$ to turn into a quantum in state $a$ at spacetime point $x$, _in the given state_ that the fields are in, which is defined thereby (see at _[[state in AQFT]]_).

In the [[worldline formalism]] of field theories these _[[propagators]]_ arise from a 1-dimensional [[field theory]] on the "[[worldline]]" of ([[virtual particle|virtual]]) [[particles]] running from $y$ to $x$.

Now by the very definition of [[perturbative string theory]], these particles are replaced by [[strings]] whose dynamics is now encoded in a 2d field theory on the [[worldsheet]] of strings, specifically a [[2d superconformal field theory]] ([[2d SCFT]]) of [[central charge]] 15. Hence now it is the _[[2d SCFT]]_ which defines the _[[vacuum state]]_ that the perturbative string theory is in.

This is then called a _[[perturbative string theory vacuum]]_.

If this [[2d SCFT]] arises from [[quantization]] of a [[sigma-model]], then this is called a _geometric background_, otherwise it is a purely algebraically defined _[[non-geometric string vacuum]]_.

In practice full [[2d SCFTs]] are hard to construct, and often one considers them by [[perturbative quantum field theories|perturbation theory]] of a "[[sigma-model]]" which is defined by a [[spacetime]] manifold equipped with extra [[field (physics)|fields]] (e.g. the [[B-field]] etc.). It turns out that to low order these [[background field]] configurations that define [[sigma-model]] [[2d SCFTs]] are given by [[solutions]] to [[equations of motion]] of [[supergravity]] theories (e.g. [[type II supergravity]] for [[type II string theory]], etc.)

Therefore often such [[supergravity]] solutions equipped with some extra data that makes them consistent CFT backgrounds at higher order are referred to as _vacua_ for string theory. But this is in general a coarse approximation. The full vacua are the full [[2d SCFTs]] that define the [[worldsheet]] theory of the string.

The collection of all string vacua, possibly subject to some assumptions, has come to be called the _[[landscape of string theory vacua]]_.

## Related concepts

[[!include states and observables -- content]]

## References

See the references at

* [[string phenomenology]]

* [[landscape of string theory vacua]]

On (in-)stability of non-[[supersymmetry|supersymmetric]] [[AdS spacetime|AdS]] vacua in [[string theory]]:

* [[Iosif Bena]], [[Krzysztof Pilch]], [[Nicholas Warner]], _Brane-Jet Instabilities_,  J. High Energ. Phys. 2020, 91 (2020) ([arXiv:2003.02851](https://arxiv.org/abs/2003.02851))

[[!redirects perturbative string theory vacua]]

[[!redirects string theory vacuum]]
[[!redirects string theory vacua]]

[[!redirects string vacuum]]
[[!redirects string vacua]]

[[!redirects perturbative superstring vacuum]]
[[!redirects perturbative superstring vacua]]


perverse sheaf

#Contents#
* table of contents
{:toc}

## Idea

To a space (typically with singularities) of a certain kind (there are variants) one associates a category whose objects are called **perverse sheaves**. They are neither perverse nor [[sheaves]].

They are related to some sheaf categories and notably generalize [[intersection cohomology]]. Perversity there is a parameter involved in the grading of intersection cohomology groups. Another feature similar to sheaves is that they are somehow determined by the local data; there is a famous gluing due to [[Sasha Beilinson]]. In one of the approaches (see [MacPherson's notes](#MacPherson)) there are even modified [[generalized (Eilenberg-Steenrod) cohomology|Steenrod-Eilenberg axioms]] stated for intersection cohomology.

##Definitions

In this set of examples, $X$ is a complex [[stratified space]] (i.e. a stratification $S$ of $X$ is a collection of locally complex submanifolds which satisfy some conditions).

We can alternatively define $X$ as an [[algebraic variety]] (where a stratification $S$ of $X$ is a collection of subvarieties which satisfy some conditions).

1. $Shv(X)$ := sheaves of complex vector spaces on X (assigns to every connected open set a vector space)

1. Look at (bounded) chain complexes in $Shv(X)$ (up to homotopy) =: $K^b(X)$

1. formally invert quasi-isomorphisms in $K^b(X)$, making this a [[derived category]] =: $D^b(X)$

1. $D_s^b(X)$ :=  subcategory of $D^b(X)$, all sheaves are constructible;

* where $F \in D_s^b(X)$ if for all strata $S$, $i: \hookrightarrow X$;
* $H^k(i^*F)$ is a finite rank local system on $S$ for all $k$, where $k = \text{dim}_{\mathbb{C}}(S)$.

1. $P_S(X)$ := the category of perverse sheaves, an abelian subcategory of the (non-abelian) derived category of sheaves on $X$, $D_s^b(X)$.

In other words, a [[perverse sheaf]] is an object $C$ of the bounded derived category of sheaves with constructible cohomology on a space (X), satisfying some conditions.

(...)

## Terminology
 {#Terminology}

The terminology "perverse sheaf" has an unfortunate connotation to it, at least in some languages (such as German, where it sounds no better than "idiotic sheaf" or the like).

Accordingly, for instance [[Grothendieck]] wrote in _[[Récoltes et semailles]]_:

> What an idea to give such a name to a mathematical thing! Or to any other thing or living being, except in sternness towards a person—for it is evident that of all the 'things' in the universe, we humans are the only ones to whom this term could ever apply.

The origin of the terminology is recalled by one of its inventors on [MO here](http://mathoverflow.net/a/44149/381):

> When MacPherson and I first started thinking about intersection homology, we realized that there was a number that measured the "badness" of a cycle with respect to a stratum. This number had the property that when you (transversally) intersected two cycles, their "badness" would add. The best situation occurs for cocycles, in which case that number was zero, and the intersection of two cocycles was again a cocycle. The worst situation was for ordinary homology, in which case that number could be as large as the codimension of the stratum. In that case, the intersection of two cycles could even fail to be a cycle. After a while it became clear that we needed a name for this number and we tried "degeneracy", "gap", etc., but nothing seemed to fit. It seemed that the bad cycles were being "obstinate", but "obstinateness" did not sound reasonable. Finally we said, "let's just call it the perversity, and we'll find a better word later". We tried again later, with no success. (We did not realize that in some languages the word is obscene.) When we first went to talk with Dennis Sullivan and John Morgan about these ideas, we were calling the resulting groups "perverse homology", but Sullivan suggested the alternative, "intersection homology", which seemed fine with us. This was 1974-75. Later, when it was discovered that, for any perversity, there is an abelian category of sheaves, whose simple objects are the intersection cohomology sheaves (with that perversity) of closures of strata, Deligne coined the term "faisceaux pervers".

## References

* [[Alexander Beilinson]], [[Joseph Bernstein]], [[Pierre Deligne]], *Faisceaux pervers*, Astérisque **100** (1982) [[ISBN:978-2-85629-878-7](https://smf.emath.fr/publications/faisceaux-pervers), [pdf](https://publications.ias.edu/sites/default/files/Faisceaux%20pervers.pdf), [MR86g:32015](http://www.ams.org/mathscinet-getitem?mr=751966)]

* {#MacPherson} [[Robert MacPherson]], _Intersection Homology and Perverse Sheaves_, [pdf](http://faculty.tcu.edu/gfriedman/notes/ih.pdf)

* Reinhardt Kiehl, Rainer Weissauer, _Weil conjectures, perverse sheaves and l'adic Fourier transform_, Ergebnisse Der Mathematik Und Ihrer Grenzgebiete __42__, Springer 2001.

* Mark Andrea de Cataldo, Luca Migliorini, _What is a perverse sheaf ?_, [arxiv/1004.2983](http://arxiv.org/abs/1004.2983), Notices of Amer. Math. Soc. May 2010, [pdf](http://www.ams.org/notices/201005/rtx100500632p.pdf)

* Ryan Reich, _Notes on Beilinson's "How to glue perverse sheaves"_, [arxiv/1002.1686](http://arxiv.org/abs/1002.1686)

* Geordie Williamson, _An illustrated guide to perverse sheaves_, ([pdf](http://people.mpim-bonn.mpg.de/geordie/perverse_course/lectures.pdf))

An article aiming at the categorification of perverse sheaves is

* [[Mikhail Kapranov]], Vadim Schechtman, _Perverse Schobers_, [arxiv/1411.2772](http://arxiv.org/abs/1411.2772)

category: sheaf theory, algebraic geometry
[[!redirects perverse sheaves]]


Petar Pavesic

**Petar Pavešić** is a Slovenian topologist, with a position in Ljubljana.

Web pages: [1](http://www.fmf.uni-lj.si/si/imenik/3321/), [2](http://www.fmf.uni-lj.si/~pavesic), [research and articles](http://www.fmf.uni-lj.si/~pavesic/index_files/Page317.htm)

* _Diagram cohomologies using categorical fibrations_, J. pure appl. algebra __112__ (1996) 73-90, [pdf](http://www.fmf.uni-lj.si/~pavesic/RESEARCH/Diagram%20cohomologies%20using%20categorical%20fibrations.pdf)

category:people

[[!redirects Petar Pavešić]]


Peter Aczel

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    "caption": "(picture [by S. Awodey](https://groups.google.com/g/HomotopyTypeTheory/c/8Xi1PWFV7z4/m/P-0bM603DgAJ))"
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Peter Aczel (1941–2023)

* [webpage](http://www.cs.man.ac.uk/~petera/)

* [Wikipedia entry](https://en.wikipedia.org/wiki/Peter_Aczel)

> In [2012 at IAS](https://www.ias.edu/idea-tags/homotopy-type-theory), Peter Aczel initiated a working group on informal rigorous reasoning in [[type theory]], whose activities expanded into the writing of the [[HoTT book]]. Without Peter’s initiative there would be no book, nor the ensuing success of [[homotopy type theory]]. [[A Bauer, Aug 2023](https://groups.google.com/g/HomotopyTypeTheory/c/8Xi1PWFV7z4/m/OjGU9A2BDwAJ)]

## Selected writings

Introducing the [[structure identity principle]] in [[homotopy type theory]]:

* [[Peter Aczel]], *On Voevodsky’s Univalence Axiom*, talk at Third European Set Theory Conference (2011) [[pdf](http://www.cs.man.ac.uk/~petera/Recent-Slides/Edinburgh-2011-slides_pap.pdf), [[Aczel-Univalence.pdf:file]]]

On [[constructive set theory]]:

* {#AczelRathjen01} [[Peter Aczel]], [[Michael Rathjen]], _Notes on Constructive Set theory_, 2001 ([pdf](https://events.math.unipd.it/3wftop/pdf/AczelRathjen.pdf), [[AczelRathjenCST.pdf:file]])

* [[Peter Aczel]], *Constructive Set Theory*, 2008 ([pdf](http://www.cs.man.ac.uk/~petera/mathlogaps-slides.pdf))

## Related entries

* [[taboo]]

* [[homotopy type theory]]

category: people


Peter Arndt

* [website](http://www.dpmms.cam.ac.uk/site2002/People/arndt_p.html)

## selected writings

* [[Peter Arndt]], _Abstract motivic homotopy theory_, thesis 2017 ([web](https://repositorium.ub.uni-osnabrueck.de/handle/urn:nbn:de:gbv:700-2017021015476?mode=full),  [pdf](https://repositorium.ub.uni-osnabrueck.de/bitstream/urn:nbn:de:gbv:700-2017021015476/6/thesis_arndt.pdf), [[ArndtAbstractMotivic.pdf:file]])

   also: lecture at _[Geometry in Modal HoTT](https://felix-cherubini.de/abstracts.html)_, 2019 ([recording I](https://www.youtube.com/watch?v=f0wpcNs8hQo))

* {#Arndt17} [[Peter Arndt]], _Abstract motivic homotopy theory_, thesis 2017 ([web](https://repositorium.ub.uni-osnabrueck.de/handle/urn:nbn:de:gbv:700-2017021015476?mode=full),  [pdf](https://repositorium.ub.uni-osnabrueck.de/bitstream/urn:nbn:de:gbv:700-2017021015476/6/thesis_arndt.pdf), [[ArndtAbstractMotivic.pdf:file]], [recording II](https://www.youtube.com/watch?v=sTl8637a2Zo))

  (on [[motivic homotopy theory]])

## related $n$Lab entries

* [[motivic homotopy theory]]

* [[Snaith's theorem]]

category: people


Peter Bantay

_Peter Bantay_ is a Hungarian [[mathematical physics|mathematical physicist]] working at Rolland Eötvös University, Budapest. His interests include [[orbifold]] models, [[vertex operator algebra]]s, [[modular invariance]], [[mapping class group]]s, characters of RCFT, generfalizations of [[Moonshine]] VOA etc.

* Modular differential equations for characters of RCFT, [arxiv/1004.2579](http://front.math.ucdavis.edu/1004.2579)

category: people


Peter Borwein

* [Wikipedia entry](https://en.wikipedia.org/wiki/Peter_Borwein)

## Selected writings

Efficient representations of [[pi]] and related [[irrational numbers]] in [[exact real computer arithmetic]]  (the [[Bailey-Borwein-Plouffe algorithm]]):

* [[David Bailey]], [[Peter Borwein]], [[Simon Plouffe]]: *On the Rapid Computation of Various Polylogarithmic Constants*, Math. Comp. **66** (1997) 903-913 [[doi:1997-66-218/S0025-5718-97-00856-9](https://www.ams.org/journals/mcom/1997-66-218/S0025-5718-97-00856-9)]

category: people


Peter Bouwknegt

* [website](http://tpsrv.anu.edu.au/Members/bouwknegt)

## Selected writings

On [[W-algebras]]:

* [[Peter Bouwknegt]], Jim McCarthy, [[Krzysztof Pilch]], _The $W_3$ algebra: modules, semi-infinite cohomology, and BV algebras_, Lec. Notes in Phys. __42__, Springer 1996 ([doi:10.1007/978-3-540-68719-1](https://link.springer.com/book/10.1007/978-3-540-68719-1))

On [[topological T-duality]] in relation to [[T-folds]]:

* [[Peter Bouwknegt]], [[Ashwin S. Pande]], *Topological T-duality and T-folds*, Advances in Theoretical and Mathematical Physics **13** 5 (2009) [[arXiv:0810.4374](https://arxiv.org/abs/0810.4374), [doi:10.4310/ATMP.2009.v13.n5.a6](https://dx.doi.org/10.4310/ATMP.2009.v13.n5.a6)]

category: people


Peter Bubenik

* [Personal page](https://academic.csuohio.edu/bubenik_p/)

* [Research group page](https://people.clas.ufl.edu/peterbubenik/researchgroup/)

## Selected writings

On [[category theory|category theoretic]] aspects of [[persistence objects]] (as used in [[topological data analysis]]:

* {#BubenikScott14} [[Peter Bubenik]], [[Jonathan Scott]] _Categorification of Persistent Homology_, Discrete & Computational Geometry **51** (2014) 600-627 [[doi:10.1007/s00454-014-9573-x](https://doi.org/10.1007/s00454-014-9573-x)]

category: people


Peter C Marks

[mail](mailto:peter.c.marks@gmail.com)


Peter D. Jarvis

* [Wikipedia entry](https://en.wikipedia.org/wiki/Peter_D._Jarvis)

* [InSpire page](https://inspirehep.net/authors/1004461)

* [MathGenealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=114526)

## Selected writings

On antisymmetric rank-2 [[field (physics)|fields]] (2-form field, [[Kalb-Ramond fields]]):

* [[Peter D. Jarvis]], [[Jean Thierry-Mieg]], *Antisymmetric tensor fields: actions, symmetries and first order Duffin-Kemmer-Petiau formulations* [[arXiv:2311.01675](https://arxiv.org/abs/2311.01675)]

* [[Jean Thierry-Mieg]], [[Peter D. Jarvis]], *Conformal invariance of antisymmetric tensor field theories in any even dimension* [[arXiv:2311.01701](https://arxiv.org/abs/2311.01701)]

category: people

[[!redirects Peter Jarvis]]


Peter Donovan

* [Mathematics Genealogy](http://www.genealogy.math.ndsu.nodak.edu/id.php?id=142914)

## Related entries

* [[twisted K-theory]]

category: people


Peter Draxl

* [Mathematics Genealogy page](https://genealogy.math.ndsu.nodak.edu/id.php?id=57046)

## Selected writings

On [[skew fields]]:

* [[Peter Draxl]], _Skew fields_, London Math. Soc. Lecture notes __81__, Cambridge University Press 1983 ([doi:10.1017/CBO9780511661907](https://doi.org/10.1017/CBO9780511661907), ISBN:9780511661907)

category: people


Peter Dybjer

* [webpage](http://www.cse.chalmers.se/~peterd/)

## Selected writings

Introducing the notion of [[inductive types]] and [[inductive families]] in [[type theory]]:

* [[Peter Dybjer]], *Inductive sets and families in Martin-Löf's type theory and their set-theoretic semantics*, Logical frameworks (1991) 280-306 [[doi:10.1017/CBO9780511569807.012](https://doi.org/10.1017/CBO9780511569807.012), [pdf](http://www.cse.chalmers.se/~peterd/papers/Setsem_Inductive.pdf)]

* [[Peter Dybjer]], *Inductive families*, Formal Aspects of Computing **6** (1994) 440–465 [[doi:10.1007/BF01211308](https://doi.org/10.1007/BF01211308), [doi:10.1007/BF01211308](https://doi.org/10.1007/BF01211308), [pdf](http://www.cse.chalmers.se/~peterd/papers/Inductive_Families.pdf)]

On [[categorical models of dependent types]]:

* [[Peter Dybjer]], *Internal type theory*, Types for proofs and programs (Torino, 1995), Lecture Notes in Comput. Sci. **1158** Springer (1996) 120-134 [[doi:10.1007/3-540-61780-9_66](https://doi.org/10.1007/3-540-61780-9_66), [pdf](http://www.cse.chalmers.se/~peterd/papers/InternalTT.pdf), [MR1474535](http://www.ams.org/mathscinet-getitem?mr=1474535)]

Generalization of [[inductive types]] to [[inductive-recursive types]]:

* [[Peter Dybjer]], *A general formulation of simultaneous inductive-recursive definitions in type theory*, The Journal of Symbolic Logic **65** 2 (2000) 525-549 [[doi:10.2307/2586554](https://doi.org/10.2307/2586554), [pdf](http://www.cse.chalmers.se/~peterd/papers/Inductive_Recursive.pdf)]

* [[Peter Dybjer]], [[Anton Setzer]], *Indexed induction-recursion*, in *Proof Theory in Computer Science* PTCS 2001. Lecture Notes in Computer Science**2183** Springer (2001) [[doi:10.1007/3-540-45504-3_7](https://doi.org/10.1007/3-540-45504-3_7), [pdf](http://www.cse.chalmers.se/~peterd/papers/InductionRecursionInitialAlgebras.pdf)]

For moree see:

* [[Peter Dybjer]], *[Inductive and inductive-recursive definitions in type theory](http://www.cse.chalmers.se/~peterd/papers/inductive.html)* (link list)

On [[internal type theory]]:

* [[Peter Dybjer]], *Internal type theory*, in:  *Types for Proofs and Programs*, Lecture Notes in Computer Science **1158** (1996) 120-134 [[doi:10.1007/3-540-61780-9_66](https://doi.org/10.1007/3-540-61780-9_66), [pdf](http://www.cse.chalmers.se/~peterd/papers/InternalTT.pdf)]

On the [[categorical semantics]] of [[W-types]] by [[initial algebras over an endofunctor]]:

* [[Peter Dybjer]], *Representing inductively defined sets by wellorderings in Martin-Löf's type theory*, Theoretical Computer Science **176** 1–2 (1997) 329-335 [doi:10.1016/S0304-3975(96)00145-4]

On the [[categorical model of dependent types]] by [[locally cartesian closed categories]]:

* {#ClairambaultDybjer} [[Pierre Clairambault]], [[Peter Dybjer]], *The Biequivalence of Locally Cartesian Closed Categories and Martin-Löf Type Theories*, in *Typed lambda calculi and applications*, Lecture Notes in Comput. Sci. **6690** Springer (2011) [[arXiv:1112.3456](http://arxiv.org/abs/1112.3456), [doi:10.1007/978-3-642-21691-6_10](https://doi.org/10.1007/978-3-642-21691-6_10)]

category: people


Peter Eccles

[[!redirects P. Eccles]]
Peter Eccles is a British mathematician based in Manchester.

* [home page](http://www.maths.manchester.ac.uk/~peter/)

## Related entries

* [[Barratt-Eccles operad]]

category:people


Peter Engel

## Selected writings

On [[lattice (discrete subgroup)|lattice theory]]:

* [[Peter Engel]], [[Louis Michel]], [[Marjorie Senechal]], *Lattice Geometry*, IHES/P/04/45 (2004) $[$[pdf](https://cds.cern.ch/record/859509/files/cer-002542451.pdf), [cds:859509](https://cds.cern.ch/record/859509)$]$

On [[crystallography]] with [[crystallographic groups]]:

* [[Peter Engel]], *Geometric Crystallography -- An Axiomatic Introduction to Crystallography*, D. Reidel Publishing (1986) $[$[doi:10.1007/978-94-009-4760-3](https://doi.org/10.1007/978-94-009-4760-3)$]$

category: people


Peter F. Stadler

* [institute page](http://www.bioinf.uni-leipzig.de/~studla/)

* [Wikipedia entry](https://de.wikipedia.org/wiki/Peter_F._Stadler)

## Selected writings

On [[Cayley graph spectra]]:

* {#RockmoreKostelecHordijkStadler02} [[Dan Rockmore]], [[Peter Kostelec]], [[Wim Hordijk]], [[Peter F. Stadler]], *Fast Fourier Transform for Fitness Landscapes*, Applied and Computational Harmonic Analysis Volume 12, Issue 1, January 2002, Pages 57-76 ([doi:10.1006/acha.2001.0346](https://doi.org/10.1006/acha.2001.0346))

category: people

[[!redirects Peter Stadler]]


Peter Franek

* [webpage](http://www.cs.cas.cz/~franek/)

## Selected writings

On [[persistent Cohomotopy]] and its improvement over [[well groups]]:

* [[Peter Franek]], [[Marek Krčál]], _On Computability and Triviality of Well Groups_, Discrete Comput Geom (2016) 56: 126 ([arXiv:1501.03641](https://arxiv.org/abs/1501.03641), [doi:10.1007/s00454-016-9794-2](https://doi.org/10.1007/s00454-016-9794-2))

* [[Peter Franek]], [[Marek Krčál]], _Persistence of Zero Sets_, Homology, Homotopy and Applications, Volume 19 (2017) Number 2 ([arXiv:1507.04310](https://arxiv.org/abs/1507.04310), [doi:10.4310/HHA.2017.v19.n2.a16](http://dx.doi.org/10.4310/HHA.2017.v19.n2.a16))

* [[Peter Franek]], [[Marek Krčál]], _Cohomotopy groups capture robust Properties of Zero Sets via Homotopy Theory_, talk at [ACAT meeting 2015](https://www2.ist.ac.at/acat) ([pfd slides](https://www2.ist.ac.at/fileadmin/user_upload/events_pages/acat/ACAT2015_Marek_Krcal.pdf))

* [[Peter Franek]], [[Marek Krčál]], [[Hubert Wagner]], _Solving equations and optimization problems with uncertainty_, J Appl. and Comput. Topology (2018) 1: 297 ([arxiv:1607.06344](https://arxiv.org/abs/1607.06344), [doi:10.1007/s41468-017-0009-6](https://doi.org/10.1007/s41468-017-0009-6))

Discussion of [[homotopy groups]] in [[computational topology]]:

* [[Marek Filakovsky]], [[Peter Franek]], [[Uli Wagner]], Stephan Zhechev, _Computing simplicial representatives of homotopy group elements_, J Appl. and Comput. Topology (2018) 2: 177. ([arXiv:1706.00380](https://arxiv.org/abs/1706.00380))

category: people


Peter Freund

* [Wikipedia entry](https://en.wikipedia.org/wiki/Peter_Freund)

## Selected writings

On [[gravitational instantons]]:

* {#EguchiFreund67} [[Tohru Eguchi]], [[Peter Freund]], _Quantum Gravity and World Topology_, Phys. Rev. Lett. 37, 1251 (1967) ([doi:10.1103/PhysRevLett.37.1251](https://doi.org/10.1103/PhysRevLett.37.1251))

* {#Freund78} [[Peter Freund]], _World topology and gauged internal symmetries_, Proc. 19th Int. Conf. High Energy Physics, Tokyo 1978 ([spire:137780](http://inspirehep.net/record/137780/), [pdf](https://cds.cern.ch/record/870701/files/c78-08-23-p617.pdf))

Introducing [[Freund-Rubin compactifications]]:

* {#FreundRubin80} [[Peter Freund]], M. A. Rubin, _Dynamics Of Dimensional Reduction_, Phys. Lett. B 97, 233 (1980). (doi:10.1016/0370-2693(80)90590-0)

On [[Kaluza-Klein theory]]:

* [[Thomas Appelquist]], A. Chodos, [[Peter Freund]], *Modern Kaluza-Klein Theories*, Frontiers in Physics, Addison-Wesley (1987) [[spire:255996](https://inspirehep.net/literature/255996)]

On [[p-adic string theory]]:

* [[Peter Freund]], Mark Olson, _Non-archimedean strings_, Physics Letters B 199,2 (1987) (arXiv:10.1016/0370-2693(87)91356-6)

* {#FreundWitten87} [[Peter Freund]], [[Edward Witten]], _Adelic string amplitudes_, Phys. Lett. B 199, 191 (1987). ([web record](http://adsabs.harvard.edu/abs/1987PhLB..199..191F))

* Lee Brekke,  [[Peter Freund]],  Mark Olson, [[Edward Witten]], _Non archimedean  string dynamics_, Nucl. Phys. B302, 3 (1988) (arXiv:10.1016/0550-3213(88)90207-6)

## Related entries

* [[Freund-Rubin compactification]]

* [[p-adic string theory]]

* [[7d supergravity]]

category: people


Peter Freyd

* [website](https://www.math.upenn.edu/people/peter-j-freyd)

* [Wikipedia](http://en.wikipedia.org/wiki/Peter_J._Freyd)

## Selected writings

On the relation between [[toposes]] and [[abelian categories]] and introducing the unifying notion of *[[AT-categories]]*:

* {#Freyd99} [[Peter Freyd]] in reply to [[Vaughan Pratt]], *Abelian-topos (AT) categories*, mailing list comments (1999) [[catlist:1999/atcat](http://www.mta.ca/~cat-dist/catlist/1999/atcat), [[Freyd_AT-categories.txt:file]]]

On [[abelian categories]]:

* {#Freyd} [[Peter Freyd]], *Abelian Categories*, Harper and Row (1964), Reprints in Theory and Applications of Categories **3** (2003) 23-164 [[tac:tr3](http://www.tac.mta.ca/tac/reprints/articles/3/tr3abs.html)]

On [[category theory]]:

* {#Freyd65} [[Peter Freyd]], _The theories of functors and models_, in: Proceedings of _Symposium on the Theory of Models_, North Holland, 1965 ([doi:10.1016/C2013-0-11897-1](https://doi.org/10.1016/C2013-0-11897-1))

* [[Peter Freyd]], [[Andre Scedrov]], *[[Categories, Allegories]]*, Mathematical Library Vol 39, North-Holland (1990) ([ISBN 978-0-444-70368-2](https://www.elsevier.com/books/categories-allegories/freyd/978-0-444-70368-2))

On [[continuous functors]] (and introducing the notion of [[orthogonal factorization systems]]):

* [[Peter Freyd]], [[Max Kelly]], *Categories of continuous functors*, J. Pure. Appl. Algebra **2** (1972) 169-191 [doi:10.1016/0022-4049(72)90001-1]

On [[bireflective subcategories]] with [[ambidextrous adjoints]]:

* {#FOPTST99} [[Peter Freyd]], [[Peter O’Hearn]],  [[A. John Power]], [[M. Takeyama]], [[Ross Street]], [[Robert D. Tennent]], *Bireflectivity*, Theoretical Computer Science **228** 1–2 (1999) 49-76 [doi:10.1016/S0304-3975(98)00354-5]

On [[real numbers|real]] [[analysis]]:

* [[Peter Freyd]], *Algebraic real analysis*, Theory and Applications of Categories, Vol. 20, 2008, No. 10, pp 215-306 ([tac:20-10](http://www.tac.mta.ca/tac/volumes/20/10/20-10abs.html))

## Related entries

* [[Freyd-Mitchell embedding theorem]]

* [[adjoint functor theorem]]

* [[AT category]]

* [[paracategory]]

* [[allegory]]

category: people

[[!redirects Freyd]]
[[!redirects P Freyd]]
[[!redirects P. Freyd]]
[[!redirects PJ Freyd]]
[[!redirects P J Freyd]]
[[!redirects P. J. Freyd]]
[[!redirects Peter Freyd]]
[[!redirects Peter J Freyd]]
[[!redirects Peter J. Freyd]]


Peter Gilkey

* [webpage](http://pages.uoregon.edu/gilkey/)

## Selected writings

On [[topology]]/[[homotopy theory]] and [[differential geometry]] in [[gauge theory]] and [[gravity]] (such as in [[fiber bundles in physics]]):

* [[Tohru Eguchi]], [[Peter Gilkey]], [[Andrew Hanson]], _Gravitation, gauge theories and differential geometry_, Physics Reports Volume 66, Issue 6, December 1980, Pages 213-393 (doi:10.1016/0370-1573(80)90130-1)

category: people


Peter Goddard

* [webpage](https://www.ias.edu/scholars/goddard)

## Selected writings

Proving the [[Goddard-Thorn no-ghost theorem]]:

* {#GoddardThorn72} [[Peter Goddard]], [[Charles Thorn]], _Compatibility of the dual Pomeron with unitarity and the absence of ghosts in the dual resonance model_, Phys. Lett., B 40, No. 2 (1972), 235-238 ([spire:74899](http://inspirehep.net/record/74899/))

Early discussion of [[quantization]] of the [[string]] [[sigma-model]] in [[light-cone gauge]]:

* [[Peter Goddard]], [[Jeffrey Goldstone]], [[Claudio Rebbi]], [[Charles B. Thorn]], *Quantum dynamics of a massless relativistic string*, Nuclear Physics B **56** 1 (1973) 109-135 [doi:10.1016/0550-3213(73)90223-X]

On [[electric-magnetic duality]] ([[S-duality]]):

* {#GNO77} [[Peter Goddard]], [[Jean Nuyts]], [[David Olive]], *Gauge Theories And Magnetic Charge*, Nucl. Phys. B **125** (1977) 1-28 [doi:10.1016/0550-3213(77)90221-8, [spire:111435](https://inspirehep.net/literature/111435)]

On the [[Wess-Zumino-Witten model]] [[2d CFT]] (via [[Kac-Moody algebra]] and [[Virasoro algebra]]):

* [[Peter Goddard]], [[David Olive]], *Kac-Moody and Virasoro algebras in relation to quantum physics*,  International Journal of Modern Physics A **01** 02 (1986) 303-414 [[doi:10.1142/S0217751X86000149](https://doi.org/10.1142/S0217751X86000149), [spire:18583](https://inspirehep.net/literature/18583)]

## Related entries

* [[Goddard-Thorn no-ghost theorem]]

category: people


Peter H. H. Fantham

* [Mathematics Genealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=15868)

## Selected writings

On [[strict (2,1)-categories]] in the guise of [[Grpd]]-[[enriched categories]], motivated from the [[homotopy 2-category]] of [[topological spaces]]:

* [[Peter H. H. Fantham]], [[Eric J. Moore]], *Groupoid Enriched Categories and Homotopy Theory*, Canadian Journal of Mathematics **35** 3 (1983) 385-416 ([doi:10.4153/CJM-1983-022-8](https://doi.org/10.4153/CJM-1983-022-8))

category: people

[[!redirects Peter Fantham]]

[[!redirects P.H.H. Fantham]]


Peter Haine

* [webpage](http://math.mit.edu/~phaine/)

## Selected writings

On [[ambidextrous adjunctions]]:

* [[Peter Haine]], *Ambidexterity* (2018) [[pdf](https://math.berkeley.edu/~phaine/files/Ambidexterity_4.pdf), [[Haine-Ambidexterity.pdf:file]]]

On [[differential geometry]] with emphasis on [[differential forms]] and their [[de Rham cohomology]]:

* {#GuilleminHaine19} [[Victor Guillemin]], [[Peter Haine]], _Differential Forms_, World Scientific (2019) [[doi:10.1142/11058](https://doi.org/10.1142/11058), draft:[pdf](https://math.mit.edu/classes/18.952/2018SP/files/18.952_book.pdf)]

On [[pyknotic sets]]:

* {#BarHai19} [[Clark Barwick]], [[Peter Haine]], _Pyknotic objects, I.  Basic notions_, ([arXiv:1904.09966](https://arxiv.org/abs/1904.09966))

On [[Whitehead-generalized cohomology theory|Whitehead-generalized]] [[differential cohomology]] via [[sheaves of spectra]]:

* {#AmabelDebrayHaine21} [[Araminta Amabel]], [[Arun Debray]], [[Peter J. Haine]] (eds.), _Differential Cohomology: Categories, Characteristic Classes, and Connections_. Based on [Fall 2019 talks at MIT's Juvitop seminar](https://math.mit.edu/juvitop/pastseminars/2019_Fall.html) by: A. Amabel, D. Chua, A. Debray, S. Devalapurkar, D. Freed, P. Haine, M. Hopkins, G. Parker, C. Reid, and A. Zhang. ([arXiv:2109.12250](https://arxiv.org/abs/2109.12250))

On [[poset-stratified spaces]]:

* [[Peter J. Haine]], _On the homotopy theory of stratified spaces_ ([arXiv:1811.01119](https://arxiv.org/abs/1811.01119))

## Related $n$Lab entries

* [[pyknotic set]]

category: people

[[!redirects Peter J. Haine]]


Peter Heinig

* [MO user page](https://mathoverflow.net/users/108556/peter-heinig)

category: people


Peter Higgs

* [Wikipedia entry](http://en.wikipedia.org/wiki/Peter_Higgs)

## Related entries

* [[Higgs boson]]

category: people


Peter Hilton

* [Wikipedia entry](http://en.wikipedia.org/wiki/Peter_Hilton)

* [Biography (History of Mathematics)](http://www-history.mcs.st-andrews.ac.uk/Biographies/Hilton.html)

## Obituary

The following Obituary appeared in the paper in Binghamton, NY, where he had lived for some years.

>PETER J. HILTON
>of Binghamton

>Peter Hilton, 87, Distinguished Professor of Mathematics at Binghamton University, died on Saturday, November 6, 2010. He is survived by his wife Margaret, sons and daughter-in-law, Nicholas Hilton, Binghamton and Timothy and Catherine Hilton, Seattle, Wa., two grandsons, one great granddaughter and brother and sister-in-law,  Dr. Sydney and Mary Hilton, North Wales, U.K.

>Peter was born in London, and educated at Oxford University. During World War II, at age 18, he was recruited from Oxford, because of his mathematical ability and knowledge of German, to work at Bletchley Park, the secret British facility dedicated to breaking German codes.  This project was led by Alan Turing, the celebrated mathematician and founder of computer science, with whom the young Peter Hilton worked closely. Initially, Peter worked on breaking the Enigma code, and, later, on the more refined Fish code. Once the British Official Secrets Act was lifted in the 1980's, his lectures about the years at Bletchley Park were highly popular at venues all over the world. He gave several such lectures at Binghamton University.

>After the War Peter obtained his doctorate from Oxford.  Peter went on to hold academic positions at Cambridge and Manchester Universities, and a Chair at the University of Birmingham.   In 1962, he moved to the United States where he was Professor of Mathematics, first at Cornell, then at the University of Washington and the Battelle Institute. He held the Louis D. Beaumont Chair at Case Western Reserve University for a number of years, ending in 1982 when he became Distinguished Professor at Binghamton University, retiring in 1995.

>Peter Hilton was one of the most influential mathematicians of his generation. He made major contributions to algebraic topology and homological algebra. His influence on these subjects has been profound. In his later years he was also a significant figure in Mathematics Education, especially in Continental Europe. He published hundreds of research articles and many books on mathematics and mathematics education, and he lectured at conferences into his mid-eighties.

## Selected writings

On the [[Whitehead product]]:

* {#HiltonWhitehead53} [[Peter Hilton]], [[J. H. C. Whitehead]], _Note on the Whitehead Product_, Annals of Mathematics Second Series, Vol. 58, No. 3 (Nov., 1953), pp. 429-442  ([jstor:1969746](https://www.jstor.org/stable/1969746))

* {#Hilton55} [[Peter Hilton]], _On the homotopy groups of unions of spheres_, J. London Math. Soc., 1955, 30, 154–172 ([[Hilton55.pdf:file]])

On [[homotopy theory]]:

* {#Hilton53} [[Peter J. Hilton]], *An introduction to homotopy theory*, Cambridge University Press 1953 ([doi:10.1017/CBO9780511666278](https://doi.org/10.1017/CBO9780511666278))

On [[internalization]], [[H-spaces]], [[monoid objects]], [[group objects]] in [[algebraic topology]]/[[homotopy theory]] and introducing the [[Eckmann-Hilton argument]]:

* [[Beno Eckmann]], [[Peter Hilton]], Theorem 1.12: *Structure maps in group theory*, Fundamenta Mathematicae 50 (1961), 207-221
([doi:10.4064/fm-50-2-207-221](https://www.impan.pl/en/publishing-house/journals-and-series/fundamenta-mathematicae/all/50/2/94854/structure-maps-in-group-theory))

* [[Beno Eckmann]], [[Peter Hilton]], _Group-like structures in general categories I multiplications and comultiplications_, Math. Ann. 145, 227–255 (1962) ([doi:10.1007/BF01451367](https://doi.org/10.1007/BF01451367))

* [[Beno Eckmann]], [[Peter Hilton]], _Group-like structures in general categories III primitive categories_,  Math. Ann. **150** 165–187 (1963) ([doi:10.1007/BF01470843](https://doi.org/10.1007/BF01470843))

On [[category theory]] and [[homology]]:

* {#Hilton69} [[Peter Hilton]] (ed.) _Category Theory, Homology Theory and Their Applications_,

  vol 1: Lecture Notes in Mathematics **86**, Springer (1969) [[doi:10.1007/BFb0079380](https://doi.org/10.1007/BFb0079380)]

  vol 2: Lecture Notes in Mathematics **92**, Springer (1969) [[doi:10.1007/BFb0080761](https://doi.org/10.1007/BFb0080761)]

  vol 3: Lecture Notes in Mathematics **99**, Springer (1969) [[doi:10.1007/BFb0081959](https://doi.org/10.1007/BFb0081959)]

On [[generalized cohomology]], the [[Chern-Dold character]] and [[topological K-theory]]:

* {#Hilton71} [[Peter Hilton]], _General cohomology theory and K-theory_, London Mathematical Society Lecture Note Series 1, Cambridge University Press (1971) ([doi:10.1017/CBO9780511662577](https://doi.org/10.1017/CBO9780511662577))

On [[homological algebra]]:

* {#HiltonStammbach71} [[Peter Hilton]],  U. Stammbach, _A course in homological algebra_, Springer-Verlag, New York, 1971, Graduate Texts in Mathematics, Vol. 4.

On [[nilpotent groups]], [[nilpotent modules]] and [[nilpotent topological spaces]]:

* [[Peter Hilton]], _Nilpotency in group theory and topology_, Publicacions de la Secció de Matemàtiques Vol. 26, No. 3 (1982), pp. 47-78 ([jstor:43741908](https://www.jstor.org/stable/43741908))

On [[nilpotent topological space|nilpotent]] [[semi-simplicial sets]]:

* [[Peter Hilton]], _On a generalization of nilpotency to semi-simplicial complexes_ ([pdf](http://plms.oxfordjournals.org/content/s3-10/1/604.full.pdf))

On [[diagram chasing lemmas]]:

* Temple Fay, [[Keith Hardie]], [[Peter Hilton]], _The two-square lemma_, Publicacions Matemàtiques, Vol 33 (1989) ([pdf](http://dmle.cindoc.csic.es/pdf/PUBLICACIONSMATEMATIQUES_1989_33_01_10.pdf))

On the [[Mayer-Vietoris sequence]]:

* {#EckmannHilton} [[Beno Eckmann]], [[Peter Hilton]], _Unions and intersections in homotopy theory_, Comment. Math. Helv. 3 (1964),2 93-307, [doi](http://dx.doi.org/10.1007/BF02566918)

On the early history of [[algebraic topology]] and [[homotopy theory]] (starting with *[[Analysis Situs]]*):

* [[Peter Hilton]], *Subjective History of Homology and Homotopy Theory*, Mathematics Magazine **61** 5 (1988) 282-291 $[$[doi:10.2307/2689545](https://doi.org/10.2307/2689545)$]$

See also:

* A link to Peter Hilton's Alan Turing lecture 2006 is [here](http://www.colossus-computer.com/hilton.htm).

## Related entries

* [[Eckmann-Hilton argument]]

* [[Whitehead product]]

* [[homotopy theory]]

* [[semi-simplicial set]]

* [[homological algebra]]

* [[diagram chasing lemmas]]

* [[injective object]]

* [[category theory]]

category: people

[[!redirects Hilton]]
[[!redirects Peter J. Hilton]]
[[!redirects P. J. Hilton]]
[[!redirects Peter John Hilton]]


Peter Hochs

* [old webpage](http://www.analysis.uni-hannover.de/~hochs/)

* [current webpage](http://www.maths.adelaide.edu.au/people/peter.hochs)

category: people


Peter I. Booth

* [institute page](https://www.mun.ca/math/our-people/faculty/professores-emeriti/peter-booth/)

* [Maths genealogy entry](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=67740)

## Selected writings

Discussion of [[exponential law for spaces|exponential objects]] in [[slice categories]] of [[compactly generated topological spaces]] (towards [[locally cartesian closed category|local cartesian closure]], see at *[[parameterized homotopy theory]]*):

* {#Booth70} [[Peter I. Booth]], *The Exponential Law of Maps I*, Proceedings of the London Mathematical Society **s3-20** 1 (1970) 179-192 $[$[doi:10.1112/plms/s3-20.1.179](https://doi.org/10.1112/plms/s3-20.1.179)$]$

* {#Booth71} [[Peter I. Booth]], *The exponential law of maps. II*, Mathematische Zeitschrift **121** (1971) 311–319 $[$[doi:10.1007/BF01109977](https://doi.org/10.1007/BF01109977)$]$

* {#BoothBrown78a} [[Peter I. Booth]], [[Ronnie Brown]], *Spaces of partial maps, fibred mapping spaces and the compact-open topology*, General Topology and its Applications **8**  2 (1978) 181-195 $[$doi:10.1016/0016-660X(78)90049-1$]$

* {#BoothBrown78b} [[Peter I. Booth]], [[Ronnie Brown]], *On the application of fibred mapping spaces to exponential laws for bundles, ex-spaces and other categories of maps*, General Topology and its Applications **8** 2 (1978) 165-179 $[$doi:10.1016/0016-660X(78)90048-X$]$

On [[adjoint strings]] of [[functors]]:

* {#Booth72} [[Peter I. Booth]], *Sequences of adjoint functors*, Archiv der Mathematik **23** (1972)  489-493 [[doi:10.1007/BF01304920](https://doi.org/10.1007/BF01304920)]

category: people

[[!redirects Peter Booth]]


Peter J. Huber

* [Wikipedia entry](https://en.wikipedia.org/wiki/Peter_J._Huber) ([de](https://de.wikipedia.org/wiki/Peter_J._Huber))

* [MathGenealogy page](https://www.mathgenealogy.org/id.php?id=18363)

* Andreas Buja, Hans R. Künsch: *A Conversation with Peter Huber*, Statistical Science **23** 1 (2008) 120-135 [[arXiv:0808.0777](https://arxiv.org/abs/0808.0777), [doi:10.1214/07-STS251](https://doi.org/10.1214/07-STS251)]

## Selected writings

On abstract [[homotopy theory]] and using early discussion of [[adjoint functors]] and their induced [[comonad|co]][[monads]] and [[canonical resolutions]]:

* {#Huber61} [[Peter J. Huber]], *Homotopy theory in general categories*, Mathematische Annalen **144** (1961) 361–385 [[doi:10.1007/BF01396534](https://doi.org/10.1007/BF01396534)]

## Related entries

* [[statistics]]

category: people

[[!redirects Peter Huber]]


Peter Johnstone

__Peter Tennant Johnstone__ is a Professor of Mathematics at the University of Cambridge. He works on [[topos theory]].

* [university page](https://www.dpmms.cam.ac.uk/person/ptj1000) ([archived](https://web.archive.org/web/20160527024824/https://www.dpmms.cam.ac.uk/people/ptj1000/))

* Wikipedia entry

* [MathematicsGenealogy entry](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=24514)

## Selected writings

On [[topos theory]]:

*  _[[Topos Theory 1977|Topos Theory]]_ (1977)

*  _[[Elephant|Sketches of an elephant: a topos theory compendium]]_, Oxford University Press (2002)

On [[indexed categories]]:

* [[P. T. Johnstone]], [[R. Paré]], [[R. D. Rosebrugh]], [[D. Schumacher]], [[R. J. Wood]], [[G. C. Wraith]], _Indexed Categories and Their Applications_, Lecture Notes in Mathematics **661** (1978) [[doi:10.1007/bfb0061362](https://doi.org/10.1007/bfb0061362)]

On [[Stone spaces]] and [[Stone duality]]:

*  _[[Stone Spaces]]_ (1982)

On [[commutativity of limits with colimits]]:

* [[Marie Bjerrum]], [[Peter Johnstone]], [[Tom Leinster]], [[William F. Sawin]], *Notes on commutation of limits and colimits*, Theory and Applications of Categories **30** (2015), 527-532 [[arXiv:1409.7860](http://arxiv.org/abs/1409.7860), [tac:3015](http://www.tac.mta.ca/tac/volumes/30/15/30-15abs.html)]

category: people

[[!redirects Johnstone]]
[[!redirects P. T. Johnstone]]
[[!redirects P. Johnstone]]
[[!redirects Peter Tennant Johnstone]]
[[!redirects Peter T. Johnstone]]


Peter Kloeden

* [webpage](https://www.uni-frankfurt.de/47832616/Peter-Kloeden)

category: people


Peter Kostelec

* [Mathematics Genealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=89575)

## Selected writings

On [[Cayley graph spectra]]:

* {#RockmoreKostelecHordijkStadler02} [[Dan Rockmore]], [[Peter Kostelec]], [[Wim Hordijk]], [[Peter F. Stadler]], *Fast Fourier Transform for Fitness Landscapes*, Applied and Computational Harmonic Analysis Volume 12, Issue 1, January 2002, Pages 57-76 ([doi:10.1006/acha.2001.0346](https://doi.org/10.1006/acha.2001.0346))

category: people


Peter Kristel

* [personal page](https://peterkristel.com/)

* [institute page](https://math-inf.uni-greifswald.de/institut/ueber-uns/mitarbeitende/kristel/)

## Selected writings

On [[2-vector bundles]] for [[2-vector spaces]] regarded ([here](2-vector+space#ReferencesAsAlgebrasWithBimodules)) as algebras with bimodules between them:

* [[Peter Kristel]], [[Matthias Ludewig]], [[Konrad Waldorf]], *The insidious bicategory of algebra bundles* [[arXiv:2204.03900](https://arxiv.org/abs/2204.03900)]

On [[2-representations]] of the [[string 2-group]] on [[2-vector spaces]] and the construction of the [[stringor bundle]]:

* [[Peter Kristel]], [[Matthias Ludewig]], [[Konrad Waldorf]], *A representation of the string 2-group*, [[arXiv:2206.09797](https://arxiv.org/abs/2206.09797)]

category: people


Peter Kronheimer

* [webpage](http://www.math.harvard.edu/~kronheim/)

## related $n$Lab pages

* [[ALE space]], [[ADE classification]]

category: people


Peter Landin

* [Wikipedia entry](https://en.wikipedia.org/wiki/Peter_Landin)

category: people