View source for Spin foam

{{Short description|Topological structure used in a description of quantum gravity}}
{{Beyond the Standard Model|cTopic=[[Quantum gravity]]}}

In [[physics]], the [[topological]] structure of '''spinfoam''' or '''spin foam'''<ref name="url[gr-qc/0409061] Introduction to Loop Quantum Gravity and Spin Foams">{{cite arXiv |title=[gr-qc/0409061] Introduction to Loop Quantum Gravity and Spin Foams |eprint=gr-qc/0409061 |last1=Perez |first1=Alejandro |year=2004 }}</ref> consists of two-dimensional faces representing a configuration required by [[functional integration]] to obtain a [[Feynman's path integral]] description of [[quantum gravity]]. These structures are employed in [[loop quantum gravity]] as a version of [[quantum foam]].

==In loop quantum gravity==
{{main|Loop quantum gravity}}

The [[Second quantization|covariant formulation]] of loop quantum gravity provides the best formulation of the dynamics of the theory of [[quantum gravity]] – a [[quantum field theory]] where the invariance under [[General covariance|diffeomorphisms]] of [[general relativity]] applies. The resulting path integral represents a sum over all the possible configurations of spin foam.{{how|date=May 2016}}

===Spin network===
{{Main|Spin network}}
A spin network is a two-dimensional [[Graph (discrete mathematics)|graph]], together with labels on its vertices and edges which encode aspects of a spatial geometry. 

A spin network is defined as a diagram like the [[Feynman diagram]] which makes a basis of [[Connection (mathematics)|connection]]s between the elements of a [[differentiable manifold]] for the [[Hilbert space]]s defined over them, and for computations of amplitudes between two different [[hypersurface]]s of the [[manifold]]. Any evolution of the spin network provides a spin foam over a manifold of one dimension higher than the dimensions of the corresponding spin network.{{clarify|date=May 2016}} A spin foam is analogous to [[quantum history]].{{why|date=May 2016}}

===Spacetime===

Spin networks provide a language to describe the [[quantum geometry]] of space. Spin foam does the same job for spacetime.

Spacetime can be defined as a superposition of spin foams, which is a generalized Feynman diagram where instead of a graph, a higher-dimensional complex is used. In [[topology]] this sort of space is called a 2-[[cell complex|complex]]. A spin foam is a particular type of 2-[[cell complex|complex]], with labels for [[vertex (geometry)|vertices]], edges and [[Face (geometry)|faces]]. The boundary of a spin foam is a spin network, just as in the theory of manifolds, where the boundary of an n-manifold is an (n-1)-manifold.

In loop quantum gravity, the present spin foam theory has been inspired by the work of [[Ponzano–Regge model]]. The idea was introduced by Reisenberger and Rovelli in 1997,<ref>{{cite journal |author1=Reisenberger |first=Michael |author2=Rovelli |first2=Carlo |author-link2=Carlo Rovelli |year=1997 |title="Sum over surfaces" form of loop quantum gravity |journal=Physical Review D |volume=56 |issue=6 |pages=3490–3508 |arxiv=gr-qc/9612035 |bibcode=1997PhRvD..56.3490R |doi=10.1103/PhysRevD.56.3490}}</ref> and later developed into the [[Barrett–Crane model]]. The formulation that is used nowadays is commonly called EPRL after the names of the authors of a series of seminal papers,<ref>{{cite journal |author1=Engle |first=Jonathan |author2=Pereira |first2=Roberto |author3=Rovelli |first3=Carlo |author-link3=Carlo Rovelli |author4=Livine |first4=Etera |year=2008 |title=LQG vertex with finite Immirzi parameter |journal=Nuclear Physics B |volume=799 |issue=1–2 |pages=136–149 |arxiv=0711.0146 |bibcode=2008NuPhB.799..136E |doi=10.1016/j.nuclphysb.2008.02.018}}</ref> but the theory has also seen fundamental contributions from the work of many others, such as [[Laurent Freidel]] (FK model) and [[Jerzy Lewandowski]] (KKL model).

==Definition==
The summary partition function for a '''spin foam model''' is 

<math> Z:=\sum_{\Gamma}w(\Gamma)\left[ \sum_{j_f,i_e}\prod_f A_f(j_f) \prod_e A_e(j_f,i_e)\prod_v A_v(j_f,i_e) \right]</math>

with:

* a set of 2-complexes <math>\Gamma</math> each consisting out of faces <math>f</math>, edges <math>e</math> and vertices <math>v</math>. Associated to each 2-complex <math>\Gamma</math> is a weight <math>w(\Gamma)</math>
* a set of irreducible representations <math>j</math> which label the faces and intertwiners <math>i</math> which label the edges.
* a vertex amplitude <math>A_v(j_f,i_e)</math> and an edge amplitude <math>A_e(j_f,i_e)</math>
* a face amplitude <math>A_f(j_f)</math>, for which we almost always have <math>A_f(j_f)=\dim(j_f)</math>

==See also==
* [[Group field theory]]
* [[Loop quantum gravity]]
* [[Lorentz invariance in loop quantum gravity]]
* [[String-net liquid]]

==References==
{{Reflist}}

== External links ==
* {{cite journal |arxiv=gr-qc/9709052 |first=John C. |last=Baez |title=Spin foam models |journal=Classical and Quantum Gravity |year=1998 |volume=15 |issue=7 |pages=1827–1858 |doi=10.1088/0264-9381/15/7/004 |bibcode=1998CQGra..15.1827B |s2cid=6449360 }}
* {{cite journal |arxiv=gr-qc/0301113 |first=Alejandro |last=Perez |title=Spin Foam Models for Quantum Gravity |journal=Classical and Quantum Gravity |date=2003 |volume=20 |issue=6 |pages=R43–R104 |doi=10.1088/0264-9381/20/6/202 |s2cid=13891330 }}
* {{cite arXiv |eprint=1102.3660 |first=Carlo |last=Rovelli |title=Zakopane lectures on loop gravity |date=2011 |class=gr-qc }}

{{Quantum gravity}}

{{DEFAULTSORT:Spin Foam}}
[[Category:Theoretical physics]]
[[Category:Loop quantum gravity]]