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On the Invariant and Geometric Structure of the Holomorphic Unified Field Theory

J. W. Moffat, E. J. Thompson

Abstract

We present the invariant structure of a Holomorphic Unified Field Theory in which gravity and gauge interactions arise from a single geometric framework. The theory is formulated on a product principal bundle, with one connection, and curvature equipped with a Hermitian field on a complexification of spacetime. From a single Diff(M) $\times$ G-invariant action, variation yields the Einstein and Yang-Mills equations together with their paired Bianchi identities. A compatibility condition is implemented either definitionally or through an auxiliary penalty functional. It enforces that the antisymmetric part of our Hermitian field is exactly the gauge field's curvature on the real slice.

On the Invariant and Geometric Structure of the Holomorphic Unified Field Theory

Abstract

We present the invariant structure of a Holomorphic Unified Field Theory in which gravity and gauge interactions arise from a single geometric framework. The theory is formulated on a product principal bundle, with one connection, and curvature equipped with a Hermitian field on a complexification of spacetime. From a single Diff(M) G-invariant action, variation yields the Einstein and Yang-Mills equations together with their paired Bianchi identities. A compatibility condition is implemented either definitionally or through an auxiliary penalty functional. It enforces that the antisymmetric part of our Hermitian field is exactly the gauge field's curvature on the real slice.

Paper Structure

This paper contains 7 sections, 2 theorems, 119 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Invariant Geometry of HUFT
  3. Real Slice Geometry
  4. Symmetry Completeness and its Physical Equivalence
  5. Field Geometry and Global Spatial Topology
  6. Quantum Mechanics and Quantum Field Theory from HUFT
  7. Conclusion

Key Result

Theorem 1

For the data above, the following hold on the real slice $B=F$:

Figures (2)

  • Figure 1: The top ribbon is the Spin frame bundle $\pi_{\mathrm{Spin}}:P_{\mathrm{Spin}}\to M$ carrying the tetrad or coframe and spin connection $(e^\alpha,\omega)$; the bottom ribbon is the internal gauge bundle $\pi_G:P_G\to M$ carrying the gauge potential and curvature $(A,F)$. The dashed oval indicates the fiber product over a common base point $x\in M$.
  • Figure 2: The upper ribbon is $M_{\mathbb C}$ with coordinates $z^\mu=x^\mu+i\,y^\mu$, the lower ribbon is the real slice $M$. The projection $\pi:M_{\mathbb C}\!\to M$ sends $(x,y)\mapsto x$. The black dots mark the point $s_0(x)=(x,0)\in M_{\mathbb C}$ (label $s$) and its projection $x\in M$ (label $\pi$), so that $\pi\circ s_0=\mathrm{id}_M$. The label $\mathbb{P}_y$ denotes the rank–$4$ real vector bundle $E_y\to M$ of imaginary directions: for each $x\in M$, the fiber $Y_x=\pi^{-1}(x)\cong\mathbb R^4$ carries coordinates $y^\mu$.

Theorems & Definitions (2)

  • Theorem 1
  • Theorem 2