Quantum Gravity

Quantum gravity may be viewed as a bipartite matter-geometry system. Evolution of matter-geometry entanglement entropy is an interesting question for issues such as the emergence of QFT on curved spacetime from quantum gravity: emergence would require initial states to evolve to product states. We study this question in a cosmological model. We give numerical evidence that matter-geometry entanglement entropy increases indefinitely (with respect to a relational time) for apparently arbitrary initial states. This result raises the question of whether matter and geometry remain entangled in a large Universe.

An assessment of the particle standard model and an alternative formulation of
the model are presented. An ultraviolet complete particle model is constructed
for the observed particles of the standard model. The quantum field theory
associates infinite derivative entire functions with propagators and vertices, which
make perturbative quantum loops finite and maintain Poincaré invariance and
unitarity of the model. The electroweak model SU(2) X U(1) group is treated as a
broken symmetry group with non-vanishing experimentally determined boson
and fermion masses. A spontaneous symmetry breaking of the vacuum by a
scalar Higgs field is not invoked to restore boson and fermion masses to the
initially massless SU(2) X U(1) Lagrangian of the standard model. The hierarchy
naturalness problem of the Higgs boson mass is resolved and the model
contains only experimentally observed masses and coupling constants. The
renormalization group features of the scalar Higgs field are investigated and the
model is shown to be asymptotically safe and free from the triviality problem. It is
demonstrated that the finite nonlocal quantum field theory satisfies
microcausality. The model can predict a stable vacuum evolution. Experimental
tests to distinguish the standard model from the alternative model are proposed.
The finite quantum field theory can be extended to quantum gravity.

I will describe the relevant representation theory that allows to think of all components of fermions of a single generation of the Standard Model as components of a single Weyl spinor of an orthogonal group whose complexification is SO(14,C). There are then only two real forms that do not lead to fermion doubling. One of these real forms is the split signature orthogonal group SO(7,7). I will describe some exceptional phenomena that occur for the orthogonal groups in 14 dimensions, and then specifically for this real form. The real form SO(7,7) suggests a link to generalised geometry, which I will describe. I will also describe why spinors in general are really differential forms in disguise. 

Tensor models are generalizations of vector and matrix models. They have been introduced in quantum gravity and are also relevant in the SYK model. I will mostly focus on models with a U(N)^d-invariance where d is the number of indices of the complex tensor, and a special case at d=3 with O(N)^3 invariance. The interactions and observables are then labeled by (d-1)-dimensional triangulations of PL pseudo-manifolds. The main result of this talk is the large N limit of observables corresponding to 2-dimensional planar triangulations at d=3. In particular, models using such observables as interactions have a large N limit exactly solvable as it is Gaussian. If time permits, I will also discuss interesting questions in the field: models which are non-Gaussian at large N, the enumeration of triangulations of PL-manifolds, matrix model representation of some tensor models, etc.

The goal of this series is to collect various different formulations of General Relativity, with emphasis on four spacetime dimensions and formulations that use differential forms. The (non-exhaustive) list of formulations to be covered is per this plan:

Lecture 1): Motivations, followed by the usual Einstein-Hilbert to start with, first order Palatini, second order pure affine connection Eddington-Schroedinger.

Lecture 2) Cartan's geometry of soldering. First order Einstein-Cartan tetrad formulation, second order pure spin connection formulation, MacDowell-Mansouri formulation.

Lecture 3) Non-chiral BF-type formulations. Explicit pure spin connection Lagrangian, field redefinitions, BF plus potential for the 2-form field formulation.

Lecture 4) Chiral formulations in four dimensions. Chiral Einstein-Cartan, Plebanski formulation, pure SU(2) connection formulation, SU(2) BF plus potential for the 2-form field. Self-dual gravity. Concluding remarks. 

The goal of this series is to collect various different formulations of General Relativity, with emphasis on four spacetime dimensions and formulations that use differential forms. The (non-exhaustive) list of formulations to be covered is per this plan:

Lecture 1): Motivations, followed by the usual Einstein-Hilbert to start with, first order Palatini, second order pure affine connection Eddington-Schroedinger.

Lecture 2) Cartan's geometry of soldering. First order Einstein-Cartan tetrad formulation, second order pure spin connection formulation, MacDowell-Mansouri formulation.

Lecture 3) Non-chiral BF-type formulations. Explicit pure spin connection Lagrangian, field redefinitions, BF plus potential for the 2-form field formulation.

Lecture 4) Chiral formulations in four dimensions. Chiral Einstein-Cartan, Plebanski formulation, pure SU(2) connection formulation, SU(2) BF plus potential for the 2-form field. Self-dual gravity. Concluding remarks. 

The goal of this series is to collect various different formulations of General Relativity, with emphasis on four spacetime dimensions and formulations that use differential forms. The (non-exhaustive) list of formulations to be covered is per this plan:

Lecture 1): Motivations, followed by the usual Einstein-Hilbert to start with, first order Palatini, second order pure affine connection Eddington-Schroedinger.

Lecture 2) Cartan's geometry of soldering. First order Einstein-Cartan tetrad formulation, second order pure spin connection formulation, MacDowell-Mansouri formulation.

Lecture 3) Non-chiral BF-type formulations. Explicit pure spin connection Lagrangian, field redefinitions, BF plus potential for the 2-form field formulation.

Lecture 4) Chiral formulations in four dimensions. Chiral Einstein-Cartan, Plebanski formulation, pure SU(2) connection formulation, SU(2) BF plus potential for the 2-form field. Self-dual gravity. Concluding remarks. 

The goal of this series is to collect various different formulations of General Relativity, with emphasis on four spacetime dimensions and formulations that use differential forms. The (non-exhaustive) list of formulations to be covered is per this plan:

Lecture 1): Motivations, followed by the usual Einstein-Hilbert to start with, first order Palatini, second order pure affine connection Eddington-Schroedinger.

Lecture 2) Cartan's geometry of soldering. First order Einstein-Cartan tetrad formulation, second order pure spin connection formulation, MacDowell-Mansouri formulation.

Lecture 3) Non-chiral BF-type formulations. Explicit pure spin connection Lagrangian, field redefinitions, BF plus potential for the 2-form field formulation.

Lecture 4) Chiral formulations in four dimensions. Chiral Einstein-Cartan, Plebanski formulation, pure SU(2) connection formulation, SU(2) BF plus potential for the 2-form field. Self-dual gravity. Concluding remarks.