This week I have been reviewing Daniele Oriti’s work, reading his Frontiers of Fundamental Physics 14 conference paper – Group field theory: A quantum field theory for the atoms of space and making notes on an earlier paper, Group field theory as the 2nd quantization of Loop Quantum Gravity. I’m quite interested in Oriti’s work as can be seen in the posts:

- Disappearance and Emergence of space and time in quantum gravity
- Quantum cosmology of Loop Quantum Gravity condensates: an example
- Constructing spacetime from the quantum tetrahedron: spacetime as a Bose-Einstein condensate

**Introduction**

We know that there exist a one-to-one correspondence between spin foam models and group field theories, in the sense that for any assignment of a spin foam amplitude for a given cellular complex,

there exist a group field theory, specified by a choice of field and action, that reproduces the same amplitude for the GFT Feynman diagram dual to the given cellular complex. Conversely, any given group field theory is also a definition of a spin foam model in that it specifies uniquely the Feynman amplitudes associated to the cellular complexes appearing in its perturbative expansion. Thus group field theories encode the same information and thus

define the same dynamics of quantum geometry as spin foam models.

That group field theories are a second quantized version of loop quantum gravity is shown to be the result of a straightforward second quantization of spin networks kinematics and dynamics, which allows to map any definition of a canonical

dynamics of spin networks, thus of loop quantum gravity, to a specific group field theory encoding the same content in field-theoretic language. This map is very general and exact, on top of being rather simple. It puts in one-to-one correspondence the Hilbert space of the canonical theory and its associated algebra of quantum observables, including any operator defining the quantum dynamics, with a GFT Fock space of states and algebra of operators and its dynamics, defined in terms of a classical action and quantum equations for its n-point functions.

GFT is often presented as the 2nd quantized version of LQG. This is true in a precise sense: reformulation of LQG as GFT very general correspondence both kinematical and dynamical. Do not need to pass through Spin Foams . The LQG Spinfoam correspondence is obtained via GFT. This reformulation provides powerful new tools to address open issues in LQG, including GFT renormalization and Effective quantum cosmology from GFT condensates.

**Group field theory from the Loop Quantum Gravity perspective:a QFT of spin networks**

Lets look at the second quantization of spin networks states and the correspondence between loop quantum gravity and group field theory. LQG states or spin network states can be understood as many-particle states analogously to those found in particle physics and condensed matter theory.

As an example consider the tetrahedral graph formed by four vertices and six links joining them pairwise

The group elements G_{ij} are assigned to each link of the graph, with G_{ij}=G_{ij}^{-1}. Assume gauge invariance at each vertex i of the graph. The basic point is that any loop quantum gravity state can be seen as a linear combination of states describing disconnected open spin network vertices, of arbitrary number, with additional conditions enforcing gluing conditions and encoding the connectivity of the graph.

**Spin networks in 2nd quantization**

A Fock vacuum is the no-space” (“emptiest”) state |0〉 , this is the LQG vacuum – the natural background independent, diffeo-invariant vacuum state.

The 2nd quantization of LQG kinematics leads to a definition of quantum fields that is very close to the standard non-relativistic one used in condensed matter theory, and that is fully compatible

with the kinematical scalar product of the canonical theory. In turn, this can be seen as coming directly from the definition of the Hilbert space of a single tetrahedron or more generally a quantum polyhedron.

The single field quantum is the spin network vertex or tetrahedron – the so called building block of space.

A generic quantum state is anarbitrary collection of spin network vertices including glued ones or tetrahedra including glued ones.

The natural quanta of space in the 2nd quantized language are open spin network vertices. We know from the canonical theory that they carry area and volume information, and know their pre geometric properties from results in quantum simplicial geometry.

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