Spin foam

Beyond the Standard Model
Simulated Large Hadron Collider CMS particle detector data depicting a Higgs boson produced by colliding protons decaying into hadron jets and electrons
Standard Model
Evidence Hierarchy problem Dark matter Dark energy Quintessence Phantom energy Dark radiation Dark photon Cosmological constant problem Strong CP problem Neutrino oscillation
Theories Brans–Dicke theory Cosmic censorship hypothesis Fifth force F-theory Theory of everything Unified field theory Grand Unified Theory Technicolor Kaluza–Klein theory 6D (2,0) superconformal field theory Noncommutative quantum field theory Quantum cosmology Brane cosmology String theory Superstring theory M-theory Mathematical universe hypothesis Mirror matter Randall–Sundrum model N = 4 supersymmetric Yang–Mills theory Twistor string theory Dark fluid Doubly special relativity de Sitter invariant special relativity Causal fermion systems Black hole thermodynamics Unparticle physics Graviphoton Graviscalar Graviton Gravitino Massive gravity Gauge gravitation theory Gauge theory gravity CPT symmetry
Supersymmetry MSSM NMSSM Superstring theory M-theory Supergravity Supersymmetry breaking Extra dimensions Large extra dimensions
Quantum gravity False vacuum String theory Spin foam Quantum foam Quantum geometry Loop quantum gravity Quantum cosmology Loop quantum cosmology Causal dynamical triangulation Causal fermion systems Causal sets Canonical quantum gravity Semiclassical gravity Superfluid vacuum theory
Experiments ANNIE Gran Sasso INO LHC SNO Super-K Tevatron NOvA
v t e

In

The 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 diffeomorphisms of general relativity applies. The resulting path integral represents a sum over all the possible configurations of spin foam.^[how?]

Spin network

A spin network is a two-dimensional 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

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

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,^[2] 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,^[3] 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

${\displaystyle Z:=\sum _{\Gamma }w(\Gamma )\left[\sum _{j_{f},i_{e}}\prod _{f}A_{f}(j_{f})\prod _{e}A_{e}(j_{f},i_{e})\prod _{v}A_{v}(j_{f},i_{e})\right]}$

with:

• a set of 2-complexes ${\displaystyle \Gamma }$ each consisting out of faces ${\displaystyle f}$, edges ${\displaystyle e}$ and vertices ${\displaystyle v}$. Associated to each 2-complex ${\displaystyle \Gamma }$ is a weight ${\displaystyle w(\Gamma )}$
• a set of irreducible representations ${\displaystyle j}$ which label the faces and intertwiners ${\displaystyle i}$ which label the edges.
• a vertex amplitude ${\displaystyle A_{v}(j_{f},i_{e})}$ and an edge amplitude ${\displaystyle A_{e}(j_{f},i_{e})}$
• a face amplitude ${\displaystyle A_{f}(j_{f})}$, for which we almost always have ${\displaystyle A_{f}(j_{f})=\dim(j_{f})}$

See also

• Group field theory
• Loop quantum gravity
• Lorentz invariance in loop quantum gravity
• String-net liquid

References