quantum geometry
This is a contributed topic on spacetime
quantization
and loop quantum gravity
often descibed as quantum geometry.
In 4 dimensions, one of the attractive programs
of spacetime quantization is “quantum geometry”, often represented as “loop quantum gravity” .
Loop quantum gravity starts with a Hamiltonian
formulation of the first order formalism, with constraints, written in analogy to the (3+1)-dimensional case that take the form:
and
where the indices
and
are the spatial indices on a surface of constant time,
is the
gauge-covariant derivative for the connection
, and the
are the spatial components of the curvature two-form.
Ponzano-Regge and Turaev-Viro models are examples of “spin foam” models that is, they are models based on simplicial complexes
with faces, edges, and vertices labeled by group representations
and intertwiners.
spin foam
models are based on a fixed triangulation of spacetime, with edge lengths serving as the basic gravitational variables. An alternative scheme is “dynamical triangulation”, in which edge lengths are fixed and the path integral is represented as a sum over triangulations.
Dynamical triangulation is a useful alternative to spin foams
that has been shown to provide a useful method in two-dimensional
gravity.
Several quantum observables
whose expectation values generally give topological information about the nature of quantized spacetime have been already considered but- with very few exceptions- the results in this area has remain largely mathematical in nature; thus, surprisingly little is understood about the physics of such observables, although some are most likely to be related to length and perhaps volumes, whereas other observables are connected to scattering amplitudes for quantum paricles.
“Perhaps the most important lesson of (2+1)-dimensional quantum gravity is that general relativity can, in fact, be quantized.” (download here a concise online review)
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As of this snapshot date, this entry was owned by bci1.