Fernando Marchesano

* [webpage](https://www.ift.uam-csic.es/en/one-member/marchesa)

## Selected writings

On [[intersecting D-brane models]]

* {#Marchesano03} [[Fernando Marchesano]], _Intersecting D-brane Models_ ([arXiv:hep-th/0307252](https://arxiv.org/abs/hep-th/0307252))

On the [[Higgs mechanism]] in [[intersecting D-brane models]]:

* {#IbanezMarchesanoRabadan01} [[Luis Ibáñez]], [[Fernando Marchesano]], R. Rabadan, section 7 of _Getting just the Standard Model at Intersecting Branes_, JHEP 0111:002, 2001 ([arXiv:hep-th/0105155](https://arxiv.org/abs/hep-th/0105155))

* {#CremadesIbanezMarchesano02} D. Cremades, [[Luis Ibáñez]], [[Fernando Marchesano]], _Intersecting brane models of particle physics and the Higgs mechanism_, JHEP, 0207, 022 2002 ([arXiv:hep-th/0203160](https://arxiv.org/abs/hep-th/0203160))

On [[Yukawa couplings]]:

* {#CremadesIbanezMarchesano03} D. Cremades, [[Luis Ibáñez]], [[Fernando Marchesano]], _Yukawa couplings in intersecting D-brane models_, JHEP 0307 (2003) 038 ([arXiv:hep-th/0302105](https://arxiv.org/abs/hep-th/0302105))

On [[MSSM]] model building:

* {#MarchesanoShiu04} [[Fernando Marchesano]], [[Gary Shiu]], _Building MSSM Flux Vacua_, JHEP0411:041, 2004 ([arXiv:hep-th/0409132](https://arxiv.org/abs/hep-th/0409132))

On [[string phenomenology]] with [[orbifold]]/[[orientifold]] [[intersecting D-brane models]] and [[Gepner models]]:

* [[Fernando Marchesano]], [[Bert Schellekens]], [[Timo Weigand]], *D-brane and F-theory Model Building*, in: *[[Handbook of Quantum Gravity]]*, Springer (2023) [[arXiv:2212.07443](https://arxiv.org/abs/2212.07443)]

Survey of [[string phenomenology]]:

* [[Fernando Marchesano]], [[Gary Shiu]], [[Timo Weigand]], *The Standard Model from String Theory: What Have We Learned?* [[arXiv:2401.01939](https://arxiv.org/abs/2401.01939)]

## Related entries

* [[intersecting D-brane model]]

* [[RR-field tadpole cancellation]]

* [[Yukawa coupling]]

* [[Fukaya category]]

category: people


Fernando Muro

__Fernando Muro__ is an algebraic topologist from Seville (Sevilla, Andalusia, Spain), interested in [[homotopy theory]], [[triangulated categories]], [[higher categories]]...

* [homepage](http://personal.us.es/fmuro)

## Selected writings

Generalizing the [[model structure on chain complexes]] to [[bicomplexes]]:

* {#MuroRoitzheim19} [[Fernando Muro]], [[Constanze Roitzheim]], *Homotopy Theory of Bicomplexes*, Journal of Pure and Applied Algebra **223** 5 (2019) 1913-1939 [[arXiv:1802.07610](https://arxiv.org/abs/1802.07610), [doi:10.1016/j.jpaa.2018.08.007](https://doi.org/10.1016/j.jpaa.2018.08.007)]

[[!redirects F. Muro]]


Fernando Pastawski

* [Institute page](https://www.physik.fu-berlin.de/en/einrichtungen/ag/ag-eisert/people/pastawski/index.html)

## Selected writings

Introducing the [[HaPPY code]] (a [[quantum error correcting code]] whose generating [[tensor network]] exhibits [[holographic entanglement entropy]]):

* {#PYHP15} [[Fernando Pastawski]], [[Beni Yoshida]], [[Daniel Harlow]], [[John Preskill]], _Holographic quantum error-correcting codes: Toy models for the bulk/boundary correspondence_, JHEP 06 (2015) 149 ([arXiv:1503.06237](https://arxiv.org/abs/1503.06237))

More on [[quantum error correcting codes]] via [[holographic tensor networks]]:

* [[Fernando Pastawski]], [[John Preskill]], *Code properties from holographic geometries*, Phys. Rev. X 7, 021022 (2017) ([arXiv:1612.00017](https://arxiv.org/abs/1612.00017))

Introducing [[Majorana dimer codes]] ([[quantum error correcting codes]] induced by [[tensor networks]] exhibiting [[holographic entanglement entropy]]):

* [[Alexander Jahn]], [[Marek Gluza]], [[Fernando Pastawski]], [[Jens Eisert]], *Majorana dimers and holographic quantum error-correcting code*, Phys. Rev. Research 1, 033079 (2019) ([arXiv:1905.03268](https://arxiv.org/abs/1905.03268))

based on:

* [[Alexander Jahn]], [[Marek Gluza]], [[Fernando Pastawski]], [[Jens Eisert]], *Holography and criticality in matchgate tensor networks*, Science Advances 5, eaaw0092 (2019) ([arXiv:1711.03109](https://arxiv.org/abs/1711.03109), [doi:10.1126/sciadv.aaw0092](https://advances.sciencemag.org/content/5/8/eaaw0092))

category: people


Fernando Quevedo

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

## Selected writings

On [[bottom-up model building]] for [[intersecting D-brane models]]:

* {#AldazabalIbanezQuevedoUranga00} G. Aldazabal, [[Luis Ibáñez]], [[Fernando Quevedo]], [[Angel Uranga]], _D-Branes at Singularities: A Bottom-Up Approach to the String Embedding of the Standard Model_, JHEP 0008:002 2000 ([arXiv:hep-th/0005067v2](https://arxiv.org/abs/hep-th/0005067))

On [[moduli stabilization]]:

* [[Vijay Balasubramanian]], Per Berglund, [[Joseph Conlon]], [[Fernando Quevedo]], _Systematics of Moduli Stabilisation in Calabi-Yau Flux Compactifications_,  JHEP 0503:007,2005 ([arXiv:hep-th/0502058](http://arxiv.org/abs/hep-th/0502058))

On [[supergravity]] coupled to the (non-supersymmetric) [[standard model of particle physics]]:

* [[Cliff P. Burgess]], [[Fernando Quevedo]], *Who's Afraid of the Supersymmetric Dark? The Standard Model vs Low-Energy Supergravity* ([arXiv:2110.13275](https://arxiv.org/abs/2110.13275))

On [[moduli stabilization]] in [[string theory]]:

* [[Liam McAllister]], [[Fernando Quevedo]], *Moduli Stabilization in String Theory*, in: *[[Handbook of Quantum Gravity]]* [[arXiv:2310.20559](https://arxiv.org/abs/2310.20559)]

## Related  entries

* [[moduli stabilization]]

category: people


Fernando Sancho de Salas

* [Institute page](http://mat.usal.es/en/personal.htm)

## related $n$Lab entries

* [[finite topological space]]

category: people


Fernando Zalamea

__Fernando Zalamea Traba__ is a mathematician at Universidad Nacional de Colombia at Bogotá with main interests in [[categorical logic]], non-classical logic, [[history]] and [[philosophy]] of mathematics. He is also writing essays in cultural studies.

### On the difference between modern and contemporary mathematics

Just as [[Albert Lautman]] identified 5 features characteristic of advanced (post mid-19th century) mathematics (1-5 in this list), Zalamea identifies an additional set of 5 characteristics (6-10 in this list) which emerge only in the mid 20th century and (in addition to 1-5, which are conserved) define contemporary mathematics:

1. the _complex hierarchisation_ of various theories, irreducible to systems of _intermediate_ deduction;

2. the _richness_ of the models, irreducible to linguistic manipulation;

3. the _unity_ of structural methods and of conceptual polarites, beyond the effective multiplicity of models;

4. the _dynamics_ of the creative activity, in a permanent back-and-forth between freedom and saturation, open to the Platonic division and the Platonic dialectic;

5. the _mathematically demonstrable relation_ between what is multiple on a given level and what is singular on another, through a sophisticated lattice of mixed ascents and descents.

6. the structural _impurity_ of arithmetic (Weil's conjectures, Langlands program, the theorems of Deligne, Faltings and Wiles etc.)

7. the systematic _geometrization_ of all environments of mathematics (sheaves, homologies, cobordisms, geometrical logic etc.)

8. the _schematization_ and the liberation from set theoretical, algebraic, and topological restrictions (groupoids, categories, schemas, topoi, motifs etc..)

9. the _fluxion_ and deformation of the usual boundaries of mathematical structures (nonlinearity, noncommutativity, nonelemantarity, quantization etc)

10. the _reflexivity_ of theories and models onto themselves (classification theory, fixed-point theorems, monstrous models, elementary/nonelementary classes etc..)

### On reintegrating our dispersed culture

from

* [Peirce and Latin American "razonabilidad": forerunners of Transmodernity](http://lnx.journalofpragmatism.eu/wp-content/uploads/2009/11/09-zalamea.pdf), 2009

> A programmatic construction of a sheaf of partial cultural gluings could then be articulated around three main pieces of information: (I) Methodological forces: topological and transformational thinking, universal relatives, logic of sheaves, residuation theory, pendulum weaving, etc. (II) Cultural realizations: critical theory, metaphoric sedimentation, contaminating strata, mediating hierarchies, etc. (III) Projective goals: description of  reflective residues (gluing local and global), dense fabrics (joining multipolar threads), plastic generic forms (allowing continuity and dislocating dualisms), etc. It is our contention that (i) Peirce's system and many mathematical tools, both modern (Riemann, Galois) and contemporary (Grothendieck), provide all the necessary theoretical background to support (I); (ii) Germany's critical dialectical tradition and Latin America's TRANS essayists give good examples of how to deal with (II); and (iii) the very "end" of Postmodernism as such, with its reformulations within Modernism and Transmodernism, show the imperative of the integrated relativity, plasticity and contamination sought in (III). Precise labours on these problems will take years, but, with many non-standard tools at hand, pragmaticism and "razonabilidad" can lead the way.

###References

* [web](http://www.docentes.unal.edu.co/fzalameat)

* Fernando Zalamea, _Filosofía sintética de las matemáticas contemporáneas_, (Spanish) Obra Selecta. Editorial Universidad Nacional de Colombia, Bogotá, 2009. 231 pp. [MR2599170](http://www.ams.org/mathscinet-getitem?mr=2599170), ISBN: 978-958-719-206-3, [pdf](<http://files.acervopeirceano.webnode.es/200000065-18c1b19bb9/Zalamea-Fil-Sint-Mat-Cont.pdf>). Transl. into English by Zachary Luke Fraser: _Synthetic philosophy of contemporary mathematics_, Sep. 2011. [bookpage](http://www.urbanomic.com/pub_syntheticmath.php).
* G. Maddalena, F. Zalamea, _A new analytic/synthetic/horotic paradigm. From mathematical gesture to synthetic/horotic reasoning._, European journal of pragmatism and American philosphy 2009 [pdf](http://lnx.journalofpragmatism.eu/wp-content/uploads/2012/12/18_maddalena_zalamea.pdf)
* Fernando Zalamea Traba, _Ariada y Penélope. Redes y mixturas en el mundo contemporáneo_, 2004
* Fernando Zalamea Traba, _America – una trama integral: Transversalidad, bordes y abismos en la cultura americana, siglos XIX y XX_, Biblioteca abierta. Estudios interdisciplinarios 2009
* F. Zalamea, Versus Laboratory Seminar 24: _Sheaf Logic & Philosophical Synthesis_, [mp3 audio](https://archive.org/download/VersusLaboratorySeminar24SheafLogicPhilosophicalSynthesisWith/Sem24-FernandoZalamea_02.mp3)
* [[Giuseppe Longo]], _[Synthetic Philosophy of Mathematics and Natural Sciences: Conceptual Analyses from a Grothendieckian Perspective. (Reflections on "Synthetic Philosophy of Contemporary Mathematics" by Fernando Zalamea).](http://www.di.ens.fr/users/longo/files/PhilosophyAndCognition/Review-Zalamea-Grothendieck.pdf)_
* Fernando Zalamea (editor), _Rondas en Sais. Ensayos sobre matemáticas y cultura contemporánea._ (Essays on mathematics and contemporary culture, by Moreno, Javier; de Lorenzo, Javier; Villaveces, Andrés; Pérez, Jesús Hernando; Restrepo, Gabriel; Cruz Morales, John Alexánder; Vargas, Francisco; Oostra, Arnold; Ferreirós, José; Zalamea, Fernando; Martín, Alejandro) Universidad Nacional de Colombia, Facultad de Ciencias Humanas 2012 [pdf](https://repositorio.unal.edu.co/bitstream/handle/unal/78631/9789587618693.pdf)

category: people, philosophy

[[!redirects Fernando Zalamea Traba]]


Fethi Kadhi

__Fethi Kadhi__ is a mathematician at the University of Manouba, Tunis, Tunisia.

Ecole Nationale des Sciences de l'informatique

* [web](https://sites.google.com/a/ensi-uma.tn/fethi-kadhi/home)

* Category Theory: [Reedy Diagrams](https://arxiv.org/pdf/1609.09623.pdf) [Model categories](https://arxiv.org/pdf/1610.06535.pdf)

* Optimization theory
[RAIRO 2002](https://www.researchgate.net/publication/245276868_Generalized_Characterization_of_the_Convex_Envelope_of_a_Function)
[JOTA 2001](http://faculty.uoh.edu.sa/f.kadhi/cv2010.files/first%20paper.pdf)

* Stochastic Processes
[SP](https://sites.google.com/a/ensi-uma.tn/fethi-kadhi/stochastic-processes)


Feynman category

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## Contents
* table of contents
{:toc}

## Idea
_Feynman categories_  ([Kaufmann-Ward 13](#KaufmannWard13)) are monoidal categories with extra structures and properties that provide categorical way to corepresent operad-like objects.

They are a [[biequivalence|biequivalent]] approach to [[coloured operads]] and patterns ([Kaufmann-Ward 13](#KaufmannWard13),[Caviglia 15](#Caviglia15), [Batanin-Kock-Weber 15](#BataninKockWeber15)).

## Definition

Let $C$ be a [[groupoid]], $C^\otimes$ the [[free construction|free]] [[symmetric monoidal category]] on $C$ and $M$ a [[symmetric monoidal category]].

([Kaufman-Ward 2013, Defn 2.1](#KaufmanWard13)) A symmetric [[strong monoidal functor]] $\tau: C^\otimes\to M$ is a __Feynman category__ if the following are satisfied

* _isomorphisms condition_: $\tau^\otimes$ induces an equivalence of symmetric monoidal categories $C^\otimes \cong M_{iso}$

* _hereditary condition_: $\tau$ and $\tau^\otimes$ induce an equivalence of symmetric monoidal categories  $(C\downarrow M)^\otimes_{iso}\cong(M\downarrow M)_{iso}$

* _size condition_: For any $\ast \in C$, $(M \downarrow \ast)$ is [[essentially small]].

(Getzler 2009) A symmetric strong monoidal functor $\tau: C^\otimes\to M$ is a __regular pattern__ if the following are satisfied

* $\tau$ is [[essentially surjective functor|essentially surjective]]
* the induced functor of presheaves $\tau^{\hat{}} : M^{\hat{}}\to (C^\otimes)^{\hat{}}$ is strong monoidal for the [[Day convolution]] product

The latter condition on [[comma categories]] ensures the existence of certain (pointed) [[Kan extension]]s.

The condition of a pattern or that of being hereditary guarantees that there is a left adjoint free functor for the forgetful functor of restricting a monoidal functor from $M$ to $C$.
This leads to a monadacity theorem for strong monoidal functors from $M$ to some cocomplete $D$, namely that they are equivalent to algebras over the monad of the adjunction.

Furthermore any morphisms $f$ in the 2-category of Feynman categories defines a pull--back $f^*$ on strong monoidal functors and a left adjoint push-forward  $f_!$ computed by a left Kan extension. The theorem is that the Kan extension is strong monoidal.

## Properties

### Equivalence with coloured operads
 {#EquivalenceWithColouredOperads}
The relationship to patterns and groupoid colored operads
is given in [Kaufmann-Ward 13](#KaufmannWard13) and was further upgraded to an [[equivalence of 2-categories]] between the [[2-category]] of Feynman categories and that of [[coloured operads]] ([Caviglia 15](#Caviglia15), [Batanin-Kock-Weber 15](#BataninKockWeber15)).
This allows to import results from Feynman categories such as W-constructions to the other theories.

## References

Related items include [[operad]], [[Feynman transform]].

The axiomatics is proposed in

* {#KaufmannWard13} Ralph M. Kaufmann, Benjamin C. Ward, _Feynman categories_, Astérisque 387 (2017), vii+161pp ([arxiv:1312.1269](https://arxiv.org/abs/1312.1269))

A more recent survey is in

* {#Kaufman17} Ralph M. Kaufmann, _Lectures on Feynman categories_ ([arxiv:1702.06843](https://arxiv.org/abs/1702.06843))

The [[biequivalence]] of Feynman catgeories with [[coloured operads]] is proven in

* {#Caviglia15} [[Giovanni Caviglia]], Section A.1 of: _The Dwyer-Kan model structure for enriched coloured PROPs_ ([arXiv:1510.01289](https://arxiv.org/abs/1510.01289))

* {#BataninKockWeber15} [[Michael Batanin]], [[Joachim Kock]], [[Mark Weber]], _Regular patterns, substitudes, Feynman categories and operads_, Theory Appl. Categ. 33 (2018), 148--192 ([arXiv:1510.08934](https://arxiv.org/abs/1510.08934))

A representation-theoretical viewpoint is given in

* Ralph M. Kaufmann, _Feynman categories and Representation Theory_, ([arXiv:1911.10169](https://arxiv.org/abs/1911.10169))

A useful generalization is exhibited in

* Ralph M. Kaufmann, Jason Lucas, _Decorated Feynman categories_, [arxiv/1602.00823](https://arxiv.org/abs/1602.00823)

Certain bialgebras and Hopf algebras appear by universal constructions in the setting of Feynman categories:

*  Imma Gálvez-Carrillo, Ralph M. Kaufmann, Andrew Tonks, _Three Hopf algebras and their common simplicial and categorical background_, [arxiv:1607.00196](https://arxiv.org/abs/1607.00196)

> We consider three a priori totally different setups for Hopf algebras from number theory, mathematical physics and algebraic topology. These are the Hopf algebras of Goncharov for multiple zeta values, that of Connes--Kreimer for renormalization, and a Hopf algebra constructed by Baues to study double loop spaces. We show that these examples can be successively unified by considering simplicial objects, cooperads with multiplication and Feynman categories at the ultimate level. These considerations open the door to new constructions and reinterpretation of known constructions in a large common framework.

Role of left [[Kan extension]]s of specific kind in operadic theory, including in the setup of Feynman categories is investigated in

*  Mark Weber, _Algebraic Kan extensions along morphisms of internal algebra classifiers_, [arxiv/1511.04911](https://arxiv.org/abs/1511.04911)

Getzler's axiomatics of regular patterns is similar in spirit to Feynman categories.

* [[Ezra Getzler]], _Operads revisited_, in: Algebra, arithmetic, and geometry: in honor of Yu. I. Manin. Vol. I, vol. 269 of Progr. Math., pp. 675–698 (2009) [math/0701767](https://arxiv.org/abs/math/0701767)

A connection of Feynman categories to (a generalization of) profunctors and to rewriting systems within a proposal to categorification of the [[cyclic operad]]s are exhibited in

* [[Pierre-Louis Curien]], Jovana Obradović, _Categorified cyclic operads_, [arxiv/1706.06788](https://arxiv.org/abs/1706.06788)

Interesting pair of functors (not an adjoint pair!) between operadic categories and Feynman categories is among the topics studied in

* [[Michael Batanin]], [[Martin Markl]], _Koszul duality for operadic categories_, [arxiv.org/abs/1812.02935](https://arxiv.org/abs/1812.02935)

[[!redirects Feynman categories]]


Feynman diagram

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#### Algebraic Qunantum Field Theory
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=--
#### Measure and probability theory
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#Contents#
* table of contents
{:toc}

## Idea

### General

For $\mu_S$ a [[Gaussian probability measure]] and $\exp(-I)\mu_S$ a perturbation with $I$ [[polynomial ring|polynomial]] at least of degree 3, there is a combinatorial expression for the [[moments]]/[[expectation values]] of $\exp(-I)\mu_S$ as a sum over certain [[graphs]] whose $k$-ary vertices are labeled by the monomials of degree $k$ in $I$. (This is such that for $I = 0$ it reduces to [[Wick's lemma]].)

These graphs are called _Feynman diagrams_.

### In perturbative quantum field theory

Feynman graphs play a central role in [[perturbative quantum field theory]], where $\exp(I)\mu_S$ plays the role of an [[action functional]] on a space of [[field (physics)|fields]], $\mu_S$ is the exponentiated [[kinetic action]] and hence the [[measure]] for [[free fields]], while $\exp(I)$ is the [[interaction]] part of the [[action functional]]: the order-$k$ monomials in $I$ encode an interaction of $k$ [[field (physics)|fields]]. In this context the corresponding Feynman diagrams are traditionally thought of as depicting interaction processes of quanta of these fields, with propagation along the edges and interaction at the vertices. But this interpretation has its limits, which is partly reflected in speaking of "[[virtual particles]]". A precise interpretation is given by _[[worldline formalism]]_.

## Details

### For finitely many degrees of freedoms
 {#ForFinitelyManyDegreesOfFreedom}

We discuss Feynman diagrams for a single real [[scalar field]] on a [[discrete space]] of $k\in \mathbb{N}$ points. This contains in it already all the aspects of real Feynman diagrams in [[perturbative quantum field theory]]  and in this context everything is easily well-defined. The generalization to more field components is immediate and simply obtained by thinking of all "$\phi$" in the following as taking values in some appropriate [[representation]] space and all products of $\phi$s as given by suitable [[intertwiners]] and [[inner products]], otherwise the form of the formulas remains the same. Similarly, in generalization to continuous space (non-finitely many degrees of freedom) all the diagrammatics remains the same, the only issue now is to make sense (namely via [[renormalization]]) of the numerical value that is assigned to any one Feynman diagram.

So a [[field (physics)|field]] $\phi$ here is a map $[k]\to \mathbb{R}$ from the $k$-element [[finite set]] to the [[real numbers]], and hence the space of all field configurations is $\mathbb{R}^k$.

Fix then a $k \times k$ real-valued [[matrix]] $A \coloneqq (A_{x y}) \in Mat_{k\times k}(\mathbb{R})$ of non-vanishing [[determinant]] $det A \neq 0$.

For standard applications this $A$ is a discretized version of the [[Laplacian]] and then the expression

$$
  S_{kin} = E_{kin} \coloneqq
  \tfrac{1}{2}\sum_{x,y = 1}^k \phi_x A_{x y} \phi_y
$$

is the [[kinetic energy]] and [[kinetic action]] of the field configuration $\phi$. More concretely, think of the set of $k$ space points as being a discretization of a circle and write $(\phi_p)$ for the corresponding ([[discrete Fourier transform|discrete]]) [[Fourier transform]] of $(\phi_x)$. Then the kinetic term of the [[free field theory|free]] [[scalar field]] on this space is given by $A$ which is the [[diagonal matrix]] in the $p$-basis with

$$
  A_{p,p} = p^2 - m^2
  \,,
$$

where $m$ is a constant called the [[mass]] of the [[field (physics)|field]] $\phi$.

A [[sum]] over all values of $\phi$ is the finite (and hence well-defined) analog of a [[path integral]]. The [[Gaussian integral]] of $A$ is called the _[[partition function]]_:

$$
  \begin{aligned}
  Z_0
  & \coloneqq
  \int  \exp(- S_{kin}(\phi)) \,D\phi
  \\
  & \coloneqq
  \int_{\mathbb{R}^k}
  \,
  \exp(- \tfrac{1}{2}\sum_{x,y = 1}^k \phi_{x} A_{x,y} \phi_y  )
  \;
  d\phi_{1} \cdots d\phi_{k}
  \\
  & =
  (2\pi)^{k/2} (det A)^{-1/2}
  \end{aligned}
$$

An _[[n-point function]]_ $\langle \phi_{x_1} \phi_{x_2} \cdots \phi_{x_n}\rangle$ is an  $n$th [[moment]] of this Gaussian distribution:

$$
  \begin{aligned}
  \langle \phi_{x_1} \phi_{x_2} \cdots \phi_{x_n}\rangle
  & \coloneqq
  \frac{1}{Z_0}
  \int
    \left(
    \exp(-S_{kin}(\phi))
    \phi_{x_1}\phi_{x_2}\cdots \phi_{x_n}
    \right)
    D \phi
  \\
  & \coloneqq
  \frac{1}{Z_0}
  \int_{\mathbb{R}^k}
  \left(
  \exp(-\tfrac{1}{2} \phi_x A_{x,y}\phi_y)
  \,
  \phi_{x_1}\phi_{x_2} \cdots\phi_{x_n}
  \right)
  d\phi_1 \cdots d\phi_k
  \end{aligned}
$$

In order to compute these conveniently, pass to the [[generating function]] obtained by adding a [[source]] [[variable]] $J = (J_x)$:

$$
  \begin{aligned}
  Z(J)
  & \coloneqq
  \int( \exp(- S_{kin}(\phi) + J \cdot\phi) ) D\phi
  \\
  & \coloneqq
  \int_{\mathbb{R}^k}
    \exp(- \tfrac{1}{2}\sum_{x = 1}^k \phi_{x} A_{x,y} \phi_y
    + \sum_{x = 1}^k J_x \phi_x
   )
  \;
  d\phi_{1} \cdots d\phi_{k}
  \\
    & =
    Z_0 \exp(\tfrac{1}{2} \sum_{x = 1}^k J_x A^{-1}_{x,y} J_y)
  \end{aligned}
  \,,
$$

where $A^{-1} = (A^{-1}_{x,y})$ is the [[inverse matrix]] of $A$, which appears by computing the new integral here again as a [[Gaussian integral]] after [[completing the square]] in the exponent.

In applications to field theory this $A^{-1}$ is called the _[[Feynman propagator]]_. In the standard example where $A$ is the kinetic term of the [[free field theory|free field]] of [[mass]] $m$ then in Fourier-transformed components $A^{-1}$ is diagonal with components

$$
  A^{-1}_{p,p} = \frac{1}{p^2 - m^2}
  \,.
$$

This means, incidentally. that in the non-finite case of interest in physics, $A$ is not actually naively invertible after all, as $p^2 = m^2$ on the mass [[shell]]. In this case one has to replace the naive $A^{-1}$ with its [[zeta function regularization|zeta-function regularized version]].

The way this works is insightful even when the naive $A^{-1}$ does exist. Notice that using "[[Schwinger parameterization]]" the [[propagator]] is equivalently rewritten as a [[Mellin transform]] [[integral]]:

$$
  A^{-1}_{p,p} = \int_0^\infty \exp(- \tau A_{p,p}) d\tau
  \,.
$$

Again in the example of the standard [[scalar field]] kinetic term expressed in Fourier diagonalization $A_{p,p}= p^2 - m^2$ then with $X \colon [0,\tau] \to \mathbb{R}$ a parameterization of the straight line with slope $p$, the exponent is equivalently

$$
  \begin{aligned}
    \tau A_{p,p}
    & =
    \tau (p^2 - m^2)
    \\
    & =
    \int_0^\tau (\dot X^2 - m^2)
  \end{aligned}
  \,.
$$

This now happens to be the standard [[action functional]] ([[Polyakov action]]) for a [[sigma model]] describing the propagation of a particle along its [[worldline]].
This means that the propagator of the [[scalar field]] may be thought of as coming from the [[path integral]] of a scalar particle along its [[worldline]]. This perspective is called the "[[worldline formalism]]", it is a formalization of [[second quantization]], expressing the dynamics of [[field (physics)|fields]] in terms of that of their particle "quanta" running along [[worldlines]] of the form of the corresponding Feynman diagrams (to which we finally come in a moment).

Back to the computation of the $n$-point function. By construction, it is now equally expressed by [[partial derivatives]] of the [[generating function]] with respect to the [[source]] [[variable]] $J$ and evaluated at $J = 0$, and this in turn is a combinatorial expression just in products of the [[propagator]]:

$$
  \begin{aligned}
    \langle \phi_{x_1} \cdots \phi_{x_n}  \rangle
    &=
    \frac{1}{Z_0}
    \left(
    \frac{\partial}{\partial J_{x_1}}
    \cdots
    \frac{\partial}{\partial J_{x_n}}
    Z(J)
    \right)_{\vert J = 0}
    \\
    & =
   \left(
   \frac{\partial}{\partial J_{x_1}}
   \cdots
   \frac{\partial}{\partial J_{x_n}}
   \exp(\tfrac{1}{2} \sum_{x = 1}^k J_{x} A^{-1}_{x y} J_y)
   \right)_{\vert J = 0}
   \\
   & =
   \underset{pairings}{\sum}  A^{-1}_{x_{j_1} x_{j_2}} \cdot A^{-1}_{x_{j_3} x_{j_4}} \cdots A^{-1}_{x_{j_{n-1}}x_{j_n}}
  \end{aligned}
  \,.
$$

Here the last [[equality]] -- known as [[Wick's theorem]] -- comes from simple inspection: take the derivatives inside the [[exponential series]] and observe that then the only summands non-vanishing at $J = 0$ appears for even $n$ and are those where all derivatives hit the monomial $\left(\tfrac{1}{2} \sum_{x = 1}^k J_{x} A^{-1}_{x y} J_y\right)^{n/2}$.

Thinking of $A^{-1}_{x y}$ here as labeling an [[edge]] (a "[[worldline]]") from [[vertex]] $x$ to vertex $y$ This is the source of all Feynman digrammatics.

Now consider a [[polynomial]] $V(\phi)$ of degree $\geq 3$.  In applications to field theory this represents the [[potential energy]] or (self)[[interaction]] of the field configuration. The difference of the [[kinetic energy]] and the [[potential energy]] is called the (here: "Wick rotated"/"Euclidean") [[action]]

$$
  \begin{aligned}
    S & =  S_{kin} + S_{int}
    \\
    & = \tfrac{1}{2} \sum_{x,y = 1}^k \phi_x A_{x y} \phi_y - g V(\phi)
  \end{aligned}
  \,.
$$

The prefactor $g$ is called the _[[coupling constant]]_.

Putting everything together, the integral over the full [[action]] may be expressed as a [[power series]] in the [[coupling constant]] $g$ of the [[moments]] with respect to the kinetic action of the powers of the interaction term:

$$
  \begin{aligned}
  Z(g)
  & =
  \frac{1}{Z_0}\int \exp(- S(\phi)) \, D\phi
  \\
  & \coloneqq
  \frac{1}{Z_0}
  \int \left(\exp(-\tfrac{1}{2} \sum_{x,y = 1}^k \phi_x A_{x y} \phi_y + g V(\phi)) \right)
  \, d\phi_1 \cdots d\phi_k
  \\
  & =
  \langle \exp(g V(\phi))  \rangle
  \\
  & =
  1 + g \langle V(\phi)\rangle + \frac{g^2}{2} \langle V(\phi) ^2\rangle + \cdots
  \end{aligned}
$$

By [[Wick's theorem]] stated above, each $\langle V(\phi)^\ell\rangle$ is equivalently expressed as a sum over products of components of the [[propagator]] $A^{-1}_{x y}$. Thinking of each such propagator term as an [[edge]] produces a diagram, this is the corresponding Feynman diagram.

For instance, for a cubic point interaction

$$
  V(\phi) = \sum_x \phi_x^3
$$

then

$$
  \begin{aligned}
    \langle V(\phi)^2 \rangle
    &\coloneqq
    \langle (\sum_{x= 1}^k \phi_x^3)^2 \rangle
    \\
    & = \sum_{x_1, x_2 = 1}^k
    \langle \phi_{x_1}\phi_{x_1} \phi_{x_1}\phi_{x_2}\phi_{x_2}\phi_{x_2}\rangle
    \\
    & =
    prefactor
    \;
    \sum_{x_1,x_2 = 1}^k
    \underset{dumbbell\, diagram}{\underbrace{A^{-1}_{x_1 x_1} A^{-1}_{x_1 x_2} A^{-1}_{x_2 x_2}}}
    +
    prefactor
    \;
    \sum_{x_1, x_2 = 1}^k
    \underset{theta\, diagram}{\underbrace{A^{-1}_{x_1 x_2} A^{-1}_{x_1 x_2} A^{-1}_{x_1 x_2}}}
  \end{aligned}
$$

Here the first summand corresponds to the "dumbbell" Feynman diagram of the form

and the second summand corresponds to the "theta" Feynman diagram of the form

.

### In perturbative quantum field theory
 {#DetailsForPerturbativeQuantumFieldTheory}

For details see at

* [[S-matrix]] the section _[Feynman perturbation series](S-matrix#FeynmanDiagrams)_

$\,$

[[!include Wick algebra -- table]]

$\,$

[[!include Feynman diagrams in causal perturbation theory -- summary]]

## Related concepts

* [[vacuum diagram]]

* [[perturbative quantum field theory]]

* [[S-matrix]], [[perturbation series]]

* [[causal perturbation theory]]

* [[path integral quantization]]

* [[scattering amplitude]], [[string scattering amplitude]]

* [[Feynman propagator]]

* [[worldline formalism]]

* [[Schwinger-Dyson equation]]

* [[tadpole]]

* [[loop order]]

* [[radiative correction]]

* [[graph complex]], [[formality of the little n-disk operad]]

## References

### General

Traditional review:

* [[Radovan Dermisek]], _Path integral for interacting field_ ([pdf](http://www.physics.indiana.edu/~dermisek/QFT_09/qft-I-5-4p.pdf))

* [[Stefan Weinzierl]], *Introduction to Feynman Integrals* [[arXiv:1005.1855](https://arxiv.org/abs/1005.1855)]

* [[Stefan Weinzierl]], *Feynman Integrals*, UNITEXT for Physics, Springer (2022) [[arXiv:2201.03593](https://arxiv.org/abs/2201.03593), [doi:10.1007/978-3-030-99558-4](https://doi.org/10.1007/978-3-030-99558-4)]  (816 pages)

See also

* [[Alain Connes]], [[Matilde Marcolli]], section 3 of _[[Noncommutative Geometry, Quantum Fields and Motives]]_

* Wikipedia, _[Feynman diagram](https://en.wikipedia.org/wiki/Feynman_diagram)_

* F. T. Brandt, J. Frenkel, D. G. C. McKeon, *Feynman diagrams in terms of on-shell propagators* [[arXiv:2206.14860](https://arxiv.org/abs/2206.14860)]

* Zhi-Feng Liu, Yan-Qing Ma, *Determining Feynman Integrals with Only Input from Linear Algebra*, Physical Review Letters, Volume 129, Issue 22, 23 November 2022 ([doi:10.1103/PhysRevLett.129.222001](
https://doi.org/10.1103/PhysRevLett.129.222001
), [arXiv:2201.11637](https://arxiv.org/abs/2201.11637))

Discussion via [[D-modules]]:

* Johannes Henn, Elizabeth Pratt, Anna-Laura Sattelberger, Simone Zoia, *$D$-Module Techniques for Solving Differential Equations in the Context of Feynman Integrals* [[arXiv:2303.11105](https://arxiv.org/abs/2303.11105)]

### In causal perturbation theory
 {#ReferencesInCausalPerturbationTheory}

Discussion of Feynman diagrams in the rigorous formulation of [[causal perturbation theory]] and [[perturbative AQFT]] is due to

* {#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))

parts of which also appears as

* {#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))

An exposition of this is in

* {#Brouder10} [[Christian Brouder]], _Multiplication of distributions_, 2010 ([[BrouderProductOfDistributions.pdf:file]])

Relation to the [[Hopf algebra]] structure on Feynman diagrams due to [Kreimer 97](renormalization#Kreimer97) is discussed in

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

Relation to [[periods]] (in the sense [here](period#ReferencesInPerturbativeQuantumFieldTheory)) is discussed in

* {#Rejzner16} [[Kasia Rejzner]], _Renormalization and periods in perturbative Algebraic Quantum Field Theory_ ([arXiv:1603.02748](https://arxiv.org/abs/1603.02748))

See also

* {#HawkinsRejzner16} [[Eli Hawkins]], [[Kasia Rejzner]], section 5.2 of _The Star Product in Interacting Quantum Field Theory_ ([arXiv:1612.09157](https://arxiv.org/abs/1612.09157))

Review is in

* [[Katarzyna Rejzner]], section 6.5.2 of _Perturbative Algebraic Quantum Field Theory_, Mathematical Physics Studies, Springer 2016 ([pdf](https://link.springer.com/book/10.1007%2F978-3-319-25901-7))

### In terms of motivic structures
 {#ReferencesInTermsOfMotivicStructures}

In terms of [[motives]] (see also _[[motives in physics]]_)

* [[Spencer Bloch]], [[Helene Esnault]], [[Dirk Kreimer]], _On motives associated to graph polynomials, ([pdf](http://preprints.ihes.fr/2013/P/P-13-24.pdf))

* [[Spencer Bloch]], Pierre Vanhove, The elliptic dilogarithm for the sunset graph, IHES preprint P-13-24, pdf

### In homological BV-quantization
  {#ReferencesInHomologicalBVQuantization}

A clean derivation of the [[Feynman rules]] for finite-dimensional spaces of fields in terms of [[quantization]] by passing to [[cochain cohomology]] of [[BV-complexes]] (see at _[BV-formalism -- Homological quantization](geometric%20quantization#AsIndexOfSpinCDiracOperator))_ is in

* [[Owen Gwilliam]], [[Theo Johnson-Freyd]], _How to derive Feynman diagrams for finite-dimensional integrals directly from the BV formalism_ (2011) ([arXiv:1202.1554](http://arxiv.org/abs/1202.1554))

with a review in the broader context of [[factorization algebras of observables]] in section 2.3 of

* [[Owen Gwilliam]], _Factorization algebras and free field theories_ ([[GwilliamThesis.pdf:file]])

### As string diagrams (morphisms in monoidal categories)
 {#AsStringDiagrams}

It has been observed that Feynman diagrams, notably in [[gauge theory]],
are in particular [[string diagrams]] (in the sense of [[category theory]],
not here in any sense of [[string theory]]!) in the given
[[category of representations]]: the edges are labelled by
[[particle]] [[species]], hence by [[Wigner classification]]
by [[irreps]], and the vertices are labeled by representation
[[homomorphisms]] ("[[intertwiners]]") which, indeed, label the
[[interaction]] of particles in the Feynman diagram.

A review of this formulation is in

* [[John Baez]], [[Aaron Lauda]], _A Prehistory of n-Categorical Physics_, Deep beauty, 13-128, Cambridge Univ. Press, Cambridge, 2011 ([arXiv:0908.2469](http://arxiv.org/abs/0908.2469))

Moreover, one may think of [[string diagrams]] in [[monoidal categories]] as providing [[categorical semantics]] for [[proof nets]] (see there for more) in multiplicative [[linear logic]]. Under this
identification then Feynman diagrams have a relation to [[proof nets]]. Something like this is discussed in

* [[Richard Blute]], [[Prakash Panangaden]], _Proof nets as formal Feynman diagrams_ ([pdf](http://www.indiana.edu/~iulg/qliqc/phi.pdf))

[[!redirects Feynman diagrams]]

[[!redirects Feynman rule]]
[[!redirects Feynman rules]]

[[!redirects Feynman graph]]
[[!redirects Feynman graphs]]

[[!redirects Feynman integral]]
[[!redirects Feynman integrals]]


Feynman diagrams in causal perturbation theory -- summary

+-- {: .num_example}
###### Example
**([[Feynman amplitudes]] in [[causal perturbation theory]] -- example of [[QED]])**

In [[perturbative quantum field theory]], **[[Feynman diagrams]]** are labeled [[multigraphs]] that encode [[product of distributions|products of]] [[Feynman propagators]], called _[[Feynman amplitudes]]_ ([this prop.](S-matrix#FeynmanPerturbationSeriesAwayFromCoincidingPoints)) which in turn contribute to [[probability amplitudes]] for physical [[scattering]] processes -- [[scattering amplitudes]]:

The [[Feynman amplitudes]] are the summands in the [[Feynman perturbation series]]-expansion of the _[[scattering matrix]]_

$$
  \mathcal{S}
  \left(
    S_{int}
  \right)
  =
  \underset{k \in \mathbb{N}}{\sum}
  \frac{1}{k!}
  \frac{1}{(i \hbar)^k}
  T( \underset{k \, \text{factors}}{\underbrace{S_{int}, \cdots , S_{int}}} )
$$

of a given [[interaction]] [[Lagrangian density]] $L_{int}$.

The Feynman amplitudes are the summands in an expansion of the _[[time-ordered products]]_ $T(\cdots)$ of the [[interaction]] with itself, which, away from coincident vertices, is given by the [[star product]] of the [[Feynman propagator]] $\Delta_F$ ([this prop.](S-matrix#TimeOrderedProductAwayFromDiagonal)), via the [[exponential]] contraction

$$
  T(S_{int}, S_{int})
  \;=\;
  prod \circ \exp
  \left(
     \hbar
      \int \Delta_{F}^{a b}(x,y)
        \frac{\delta}{\delta \mathbf{\Phi}^a(x)}
          \otimes
        \frac{\delta}{\delta \mathbf{\Phi}(y)}
    \right)
    ( S_{int} \otimes S_{int} )
   \,.
$$

Each [[edge]] in a [[Feynman diagram]] corresponds to a factor of a [[Feynman propagator]]  in $T( \underset{k \, \text{factors}}{\underbrace{S_{int} \cdots S_{int}}} )$, being a [[distribution of two variables]]; and each [[vertex]] corresponds to a factor of the [[interaction]] [[Lagrangian density]] at $x_i$.

For example [[quantum electrodynamics]] in [[Gaussian-averaged Lorenz gauge]] involves (via [this example](S-matrix#FieldSpeciesQED)):

1. the [[Dirac field]] modelling the [[electron]], with [[Feynman propagator]] called the _[[electron propagator]]_ ([this def.](Feynman+propagator#FeynmanPropagatorForDiracOperatorOnMinkowskiSpacetime)), here to be denoted

   $$
     \Delta \phantom{AAAA} \text{electron propagator}
   $$

1. the [[electromagnetic field]] modelling the [[photon]], with [[Feynman propagator]] called the _[[photon propagator]]_ ([this prop.](A+first+idea+of+quantum+field+theory#PhotonPropagatorInGaussianAveragedLorenzGauge)), here to be denoted

   $$
     G \phantom{AAAA} \text{photon propagator}
   $$

1. the [[electron-photon interaction]]

   $$
     L_{int}
     \;=\;
     \underset{
       \text{interaction}
     }{
     \underbrace{
       i g
       (\gamma^\mu)^\alpha{}_\beta
     }
     }
     \,
     \underset{
       { \text{incoming} \atop \text{electron}  }
       \atop
       \text{field}
     }{\underbrace{\overline{\psi_\alpha}}}
     \;
     \underset{
       {
         \,
         \atop
         \text{photon}
       }
       \atop
       \text{field}
     }{\underbrace{a_\mu}}
     \;
     \underset{
       {\text{outgoing} \atop \text{electron} }
       \atop
       \text{field}
     }{\underbrace{\psi^\beta}}
   $$

The [[Feynman diagram]] for the [[electron-photon interaction]] alone is

where the solid lines correspond to the [[electron]], and the wiggly line to the [[photon]]. The corresponding [[product of distributions]] is (written in [[generalized function]]-notation)

$$
  \underset{
    \text{loop order}
  }{
  \underbrace{
    \hbar^{3/2-1}
  }
  }
  \underset{
    \text{electron-photon}
    \atop
    \text{interaction}
  }{
  \underbrace{
    i g (\gamma^\mu)^\alpha{}_\beta
  }
  }
  \,.
  \,
  \underset{
    {\text{incoming} \atop \text{electron}}
    \atop
    \text{propagator}
  }{
  \underbrace{
    \overline{\Delta(-,x)}_{-, \alpha}
  }
  }
  \underset{
    {
      \,
      \atop
      \text{photon}
    }
    \atop
    \text{propagator}
  }{
  \underbrace{
     G(x,-)_{\mu,-}
  }
  }
  \underset{
    { \text{outgoing} \atop \text{electron} }
    \atop
    \text{propagator}
  }{
  \underbrace{
    \Delta(x,-)^{\beta, -}
  }
  }
$$

Hence a typical Feynman diagram in the [[QED]] [[Feynman perturbation series]] induced by this [[electron-photon interaction]] looks as follows:

where on the bottom the corresponding [[Feynman amplitude]] [[product of distributions]] is shown;
now notationally suppressing the contraction of the internal indices and all prefactors.

For instance the two solid [[edges]] between the [[vertices]] $x_2$ and $x_3$ correspond to the two factors of $\Delta(x_2,x_2)$:

This way each sub-graph encodes its corresponding subset of factors in the [[Feynman amplitude]]:

> graphics grabbed from  [Brouder 10](Feynman+diagram#Brouder10)

A priori this [[product of distributions]] is defined away from coincident vertices: $x_i \neq x_j$. The definition at coincident vertices $x_i = x_j$ requires a choice of _[[extension of distributions]]_ to the [[diagonal]] locus. This choice is the [[renormalization|("re-")normalization]] of the [[Feynman amplitude]].

=--