Twisted geometries: A geometric parametrisation of SU(2) phase space (1001.2748v3)

Abstract: A cornerstone of the loop quantum gravity program is the fact that the phase space of general relativity on a fixed graph can be described by a product of SU(2) cotangent bundles per edge. In this paper we show how to parametrize this phase space in terms of quantities describing the intrinsic and extrinsic geometry of the triangulation dual to the graph. These are defined by the assignment to each triangle of its area, the two unit normals as seen from the two polyhedra sharing it, and an additional angle related to the extrinsic curvature. These quantities do not define a Regge geometry, since they include extrinsic data, but a looser notion of discrete geometry which is twisted in the sense that it is locally well-defined, but the local patches lack a consistent gluing among each other. We give the Poisson brackets among the new variables, and exhibit a symplectomorphism which maps them into the Poisson brackets of loop gravity. The new parametrization has the advantage of a simple description of the gauge-invariant reduced phase space, which is given by a product of phase spaces associated to edges and vertices, and it also provides an abelianisation of the SU(2) connection. The results are relevant for the construction of coherent states, and as a byproduct, contribute to clarify the connection between loop gravity and its subset corresponding to Regge geometries.

• The paper introduces twisted geometries as a novel parametrization of the SU(2) phase space in loop quantum gravity, bridging intrinsic and extrinsic curvature.
• It develops a Poisson structure and symplectomorphism that map discrete geometric variables onto conventional holonomy-flux variables.
• The approach abelianizes the gauge-invariant phase space and facilitates coherent state construction, advancing semiclassical analysis in quantum geometry.

Insights into the Parametrization of SU(2) Phase Space in Loop Quantum Gravity

This paper presents the novel concept of "twisted geometries" as a parametric tool for describing the SU(2) phase space in the context of loop quantum gravity (LQG). The authors, Laurent Freidel and Simone Speziale, introduce a geometric framework that provides an alternative representation of the phase space associated with LQG on a fixed graph. Rather than relying solely on holonomy-flux variables, this work conceptualizes the phase space using discrete geometry elements that bridge the gap between intrinsic and extrinsic curvature representations.

Core Contributions and Theoretical Framework

The primary contribution of this research is the introduction of twisted geometries, characterized by a set of variables associated with each triangle in a triangulation: the area, two unit normals as viewed from the adjacent polyhedra, and an additional angle related to the extrinsic curvature. Importantly, the paper reveals a symplectomorphism mapping these twisted geometries into the conventional holonomy-flux phase space of LQG.

The symplectomorphism stems from the treatment of the holonomy-flux variables as elements of a cotangent bundle to SU(2), thereby enabling a geometric interpretation that distinguishes between the intrinsic and extrinsic geometry. The presence of an angle per edge, denoted ξ, is highlighted as a critical factor for augmenting the holonomy-flux variables with extrinsic curvature information.

Detailed Analysis of the Poisson Structure

The authors meticulously construct a Poisson structure for the twisted geometries, ensuring closure of the algebra by exploiting the Hopf map on SU(2). This leads to the identification of the twisted geometry space as a presymplectic manifold, allowing for the reduction into a symplectic phase space through quotient by the kernel of the symplectic form.

Abelianization and Implications for Coherent States

A salient outcome of this parametrization is its ability to abelianize the gauge-invariant phase space of LQG, providing an edge-based decomposition into conjugate variables, and a reduction of n-valent vertex data into 2(n-3) shape parameters. This factorization yields simpler routes towards quantization and is particularly poised to inform the construction of coherent states. Such progress is potentially transformative in understanding the semiclassical limit of quantum geometry.

Prospective Developments and Impact on Quantum Gravity Research

The authors suggest that this framework's utility extends beyond theoretical refinements within LQG. By facilitating more effective coherent state construction, twisted geometries may offer new avenues for addressing the semiclassical limit problem, providing a pathway for deriving classical spacetime geometries from quantum states. Future research should aim to explore the implications of twisted geometries on spin foam models and their associated graviton propagator computations.

Further, the relationship between the discrete variables associated with twisted geometries and the standard decomposition of SU(2) gauge fields supports a reevaluation of the role of fluxes as intrinsic-extrinsic mixed entities. This perspective may be crucial for rigorously establishing the connection between LQG and continous classical geometries beyond the Planck scale.

Conclusion

Freidel and Speziale's work on twisted geometries adds a significant layer to our understanding of loop quantum gravity's algebraic and geometric underpinnings. By reconciling discrete geometric interpretations with classical curvature properties, this paper not only enriches the established frameworks but also sets the stage for innovative explorations within quantum gravity landscapes. Subsequent research should continue to probe the quantization of these geometries and their role in advancing quantum-to-classical correspondence theories.