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Concept
Canonical quantum gravity
Summary
In physics, canonical quantum gravity is an attempt to quantize the canonical formulation of general relativity (or canonical gravity). It is a Hamiltonian formulation of Einstein's general theory of relativity. The basic theory was outlined by Bryce DeWitt in a seminal 1967 paper, and based on earlier work by Peter G. Bergmann using the so-called canonical quantization techniques for constrained Hamiltonian systems invented by Paul Dirac. Dirac's approach allows the quantization of systems that include gauge symmetries using Hamiltonian techniques in a fixed gauge choice. Newer approaches based in part on the work of DeWitt and Dirac include the Hartle–Hawking state, Regge calculus, the Wheeler–DeWitt equation and loop quantum gravity.
Phase space Poisson brackets Hilbert space canonical commutation relation and Schrödinger equation
In the Hamiltonian formulation of ordinary classical mechanics the Poisson bracket is an important concept. A "canonical coordinate system" consists of canonical position and momentum variables that satisfy canonical Poisson-bracket relations,
where the Poisson bracket is given by
for arbitrary phase space functions and . With the use of Poisson brackets, the Hamilton's equations can be rewritten as,
These equations describe a "flow" or orbit in phase space generated by the Hamiltonian . Given any phase space function , we have
In canonical quantization the phase space variables are promoted to quantum operators on a Hilbert space and the Poisson bracket between phase space variables is replaced by the canonical commutation relation:
In the so-called position representation this commutation relation is realized by the choice:
and
The dynamics are described by Schrödinger equation:
where is the operator formed from the Hamiltonian with the replacement and .
Gauge symmetry Hole argument and Diffeomorphism
Canonical classical general relativity is an example of a fully constrained theory.
Official source
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