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^[1] in a seminal 1967 paper, and based on earlier work by Peter G. Bergmann^[2]. using the so-called canonical quantization techniques for constrained Hamiltonian systems invented by P. A. M. Dirac^[3]. 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. A black hole concept drawing by NASA. Physics (from the Greek, φυσικός (physikos), natural, and φύσις (physis), nature) is the science of the natural world dealing with the fundamental constituents of the universe, the forces they exert on one another, and the results produced by these forces. ... Hamiltonian mechanics is a re-formulation of classical mechanics that was invented in 1833 by William Rowan Hamilton. ... The factual accuracy of this article is disputed. ... General relativity (GR) is the geometrical theory of gravitation published by Albert Einstein in 1915. ... Dr. Bryce S. DeWitt (January 8, 1923—September 23, 2004) was a theoretical physicist best known for his role in formulating the fundamental Wheeler_deWitt equation. ... Paul Adrien Maurice Dirac, (August 8, 1902 - October 20, 1984) was a British theoretical physicist and a founder of the field of quantum physics. ... Gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally. ... In the physics of gauge theories, gauge fixing (also called choosing a gauge) denotes the act of removing redundant field variables. ... In theoretical physics, the Hartle-Hawking state, named after James Hartle and Stephen Hawking, is a hypothetical vector in the Hilbert space of a theory of quantum gravity that describes the wave function of the Universe. ... In theoretical physics, Regge calculus is a simplified form of general relativity, introduced by the Italian theoretician Tullio Regge in the early 1960s. ... In theoretical physics, the Wheeler-deWitt equation is an equation that a wave function of the Universe should satisfy in a theory of quantum gravity. ... Loop quantum gravity (LQG), also known as loop gravity and quantum geometry, is a proposed quantum theory of spacetime which attempts to reconcile the seemingly incompatible theories of quantum mechanics and general relativity. ...

The quantization is based on decomposing the metric tensor as follows, In mathematics, in Riemannian geometry, the metric tensor is a tensor of rank 2 that is used to measure distance and angle in a space. ...

where the summation over repeated indices is implied, the index 0 denotes time τ = x⁰, Greek indices run over all values 0,...,3 and Latin indices run over spatial values 1,...3. The function N is called the lapse function and the functions β_k are called the shift functions. The spatial indices are raised and lowered using the spatial metric γ_ij and its inverse γ^ij: γ_ijγ^jk = δ_i^k and βⁱ = γ^ijβ_j, γ = detγ_ij, where δ is the Kronecker delta. Under this decomposition the Einstein-Hilbert Lagrangian becomes, up to total derivatives, For other topics related to Einstein see Einstein (disambig) In mathematics, especially in applications of linear algebra to physics, the Einstein notation or Einstein summation convention is a notational convention useful when dealing with coordinate equations or formulas. ... In mathematics, the Kronecker delta or Kroneckers delta, named after Leopold Kronecker (1823-1891), is a function of two variables, usually integers, which is 1 if they are equal, and 0 otherwise. ... In general relativity, Einsteins field equations can be derived from an action principle starting from the Einstein-Hilbert action: where g is the (pseudo)Riemannian metric, R is the Ricci scalar, n is the number of spacetime dimensions and k is a constant which depends on the units chosen... In mathematics, a total derivative is a combination of partial derivatives. ...

where ⁽³⁾R is the spatial scalar curvature computed with respect to the Riemannian metric γ_ijand K_ij is the extrinsic curvature, In Riemannian geometry, the scalar curvature (or Ricci scalar) is the simplest way of describing the curvature of a Riemannian manifold. ... In Riemannian geometry, a Riemannian manifold is a real differentiable manifold in which each tangent space is equipped with an inner product in a manner which varies smoothly from point to point. ...

where donates covariant differentiation with respect to the metric γ_ij. DeWitt writes that the Lagrangian "has the classic form 'kinetic energy minus potential energy,' with the extrinsic curvature playing the role of kinetic energy and the negative of the intrinsic curvature that of potential energy." While this form of the Lagrangian is manifestly invariant under redefinition of the spatial coordinates, it makes general covariance opaque. To meet Wikipedias quality standards, this article or section may require cleanup. ...

Since the lapse function and shift functions may be eliminated by a gauge transformation, they do not represent physical degrees of freedom. This is indicated in moving to the Hamiltonian formalism by the fact that their conjugate momenta, respectively π and πⁱ, vanish identically (on shell and off shell). These are called primary constraints by Dirac. A popular choice of gauge, called synchronous gauge, is N = 1 and β_i = 0, although they can, in principle, be chosen to be any function of the coordinates. In this case, the Hamiltonian takes the form Gauge theories are a class of physical theories based on the idea that symmetry transformations can be performed locally as well as globally. ... In physics, particularly in classical field theory, configurations of a physical system that satisfy classical equations of motion are called on shell, and those that do not are called off shell. ...

where

and π^ij is the momentum conjugate to γ_ij. Einstein's equations may be recovered by taking Poisson brackets with the Hamiltonian. Additional on-shell constraints, called secondary constaints by Dirac, arise from the consistency of the Poisson bracket algebra. These are and . This is the theory is being quantized in approaches to canonical quantum gravity. In mathematics and classical mechanics, the Poisson bracket is an important operator in Hamiltonian mechanics, playing a central role in the definition of the time-evolution of a dynamical system in the Hamiltonian formulation. ...

References

1. ^  B. S. DeWitt (1967). Quantum theory of gravity. I. The canonical theory. Phys. Rev. 160: 1113–48.
2. ^  see, e.g. P. G. Bergmann, Helv. Phys. Acta Suppl. 4, 79 (1956) and references.
3. ^  P. A. M. Dirac (1950). Generalized Hamiltonian dynamics. Can. J. Math. 2: 129–48. P. A. M. Dirac (1964). Lectures on quantum mechanics, New York: Yeshiva University.