Hidden diffeos in the Hamiltonian formulation of a background independent field theory

TL;DR

The paper analyzes a background-independent HM modification of the Husain-Kuchař model using geometric Hamiltonian methods to uncover hidden diffeomorphism symmetry. By formulating the theory on $M=\mathbb{R}\times\Sigma$ and performing a careful Legendre transform, the authors derive primary and secondary constraints and compute the Hamiltonian vector fields on the constraint surfaces. In the $\mathfrak{su}(2)$ case, they obtain explicit expressions for the Hamiltonian flows, showing that 3D diffeomorphisms are generated by first-class combinations of constraints alongside internal $SU(2)$ gauge transformations with parameters $\Lambda=a-\xi\lrcorner A$, plus an extra $N$-dependent symmetry. The work clarifies the symmetry structure of this diff-invariant model and illustrates how geometric (GNH) methods reveal diffeomorphism generators that Dirac’s traditional approach can miss, with implications for broader background-independent quantization programs.

Abstract

We analyze from a geometric perspective the Hamiltonian formulation of a recent modification of the Husain-Kuchař model where, while preserving the connection as a dynamical variable, the other field is restricted to be the exterior covariant derivative of a Lie algebra-valued function. We prove that 3-dimensional diffeomorphisms can be accommodated among the local gauge transformations of the model in addition to the internal gauge symmetries.

Theorems & Definitions (23)

• Definition 1
• Definition 2
• Lemma 3
• Remark 4
• Lemma 5
• Remark 6
• Proposition 7
• proof
• Proposition 8
• proof