(Broken) Gauge symmetries and constraints in Regge calculus

Abstract

We will examine the issue of diffeomorphism symmetry in simplicial models of (quantum) gravity, in particular for Regge calculus. We find that for a solution with curvature there do not exist exact gauge symmetries on the discrete level. Furthermore we derive a canonical formulation that exactly matches the dynamics and hence symmetries of the covariant picture. In this canonical formulation broken symmetries lead to the replacements of constraints by so-called pseudo constraints. These considerations should be taken into account in attempts to connect spin foam models, based on the Regge action, with canonical loop quantum gravity, which aims at implementing proper constraints.We will argue that the long standing problem of finding a consistent constraint algebra for discretized gravity theories is equivalent to the problem of finding an action with exact diffeomorphism symmetries. Finally we will analyze different limits in which the pseudo constraints might turn into proper constraints. This could be helpful to infer alternative discretization schemes in which the symmetries are not broken.

(3 + 1)-dimensional topological phases and self-dual quantum geometries encoded on Heegaard surfaces

“…However such a discrete dynamics for four-dimensional gravity, which is implemented (via the 4-simpex gluing) in a local way, breaks four-dimensional diffeomorphism invariance [100][101][102][103]. To restore this symmetry one would have to consider a continuum limit [19,20,[104][105][106], which can be constructed via an auxiliary coarse graining flow [58].…”

Spin Foam Models with Finite Groups

“…In 3D, the flatness constraints are defined on the plaquettes of the lattice. Constraints act also as generators of gauge transformations (indeed the conditions (77) and (82) impose that physical states have to be gauge invariant), and the flatness constraints can be interpreted as translating the vertices of the dual lattice in space time [39,40,145,146], which can be seen as an action of a diffeomorphism. In 4D, the same interpretation holds only for some exceptional cases corresponding to special lattices that do not lead to space time curvature; see the discussion in [107].…”

“…Such models might arise as fixed points of a Wilsonian renormalization flow and represent the so-called perfect discretizations [33][34][35][36][37][38]. For gravity, this concept is intertwined with the appearance of discrete representation of diffeomorphism symmetry in the lattice models [36][37][38][39][40][41]. Such a discrete notion of diffeomorphism symmetry can arise in the form of vertex translations [42][43][44][45][46].…”

“…Here, one of the main problems is to implement the simplicity constraints [57][58][59][60][61][62][63][64] into the BF-partition function. The status of the symmetries is quite unclear in these models; however, there are indications [40] that in the discrete the reduction of the BF-translation symmetries does not leave a sufficiently large group to be interpretable as diffeomorphisms. Nevertheless, if these models are related to gravity, then there is a possibility that diffeomorphism symmetry as the fundamental symmetry of general relativity arises as a symmetry for large scales.…”

Symplectic and semiclassical aspects of the Schläfli identity

“…See, for example, Barrett and Steele (2003), Livine and Oriti (2003), Dittrich, Freidel and Speziale (2007), and Bahr andDittrich (2009, 2010), where the Schläfli identity plays a crucial role. See also Carfora and Marzuoli (2012) for a modern, comprehensive review of simplicial methods in quantum gravity and other fields, including the role of state sums and their regularizations.…”