Dr. A. Garrett Lisi

Fractured Atlas
Deferential Geometry

To a theoretical physicist, the world is a giant jigsaw puzzle of theories joined together by experimental facts and mathematical relationships. Currently, our two best theories are General Relativity (Einstein's elegant, geometric theory of gravity) and Quantum Field Theory (describing the zoo of subatomic particles and their quantum interactions). But our puzzle is incomplete, as these two pieces do not fit together easily.

The purpose of Deferential Geometry is to assemble the known pieces by "rotating" them into mathematical representations matched with the others. (The same physics theory can often be expressed using different mathematical structure.) A study of GR and QFT reveals the theoretical rotations necessary to join them in a beautiful new structure encompassing all particles and forces of the standard model in a single geometric field. After a similar rotation of quantum mechanics to match the mathematics of thermodynamics, a complete picture emerges in which the quantum vibrations of this field are driven by a universal background of action - just as molecules of water are driven to motion when placed on a hot stove.

Led by well-established physical theory and this new understanding, we embark on an exciting exploration into the heart of physics.

Guided by an ideal of mathematical beauty and respect for existing theory, three foundational ideas are proposed, leading towards a unified picture of reality.

1) The successful GUT program is extended by including the Higgs multiplet and gravitational fields in a unified bosonic connection. The Higgs, with vev related to the cosmological constant, multiplies the GR frame and combines with the spin connection, as in the MacDowell-Mansouri approach.

2) Consideration of Fadeev-Popov ghosts for this bosonic connection suggests their role as physical fermions. The resulting BRST extended connection -- explicitly represented as a large matrix of 1-forms and Grassmann numbers -- economically encapsulates all gravitational and standard model fields. The curvature, F, of this unified geometric connection correctly reproduces known couplings and charges, with the emergent group structure of the Pati-Salam SU(4)XSU(2)XSU(2) GUT. A restricted BF action describes the dynamics.

3) Completing the picture, a heuristic derivation of quantum mechanics is proposed. Beginning with basic information theory, the path integral formulation of QM is obtained from a single physical principle: the existence of a universal reservoir of action.

The exploration of these steps to unification is open source physics -- facilitated by a new research and educational tool, a personal wiki notebook:

physicswiki.org
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