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| {{Beyond the Standard Model|expanded=[[Quantum gravity]]}}
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| In [[physics]], '''canonical quantum gravity''' is an attempt to quantize the canonical formulation of general relativity (or '''canonical gravity'''). It is a [[Hamiltonian mechanics|Hamiltonian]] formulation of [[Albert Einstein|Einstein]]'s [[general relativity|general theory of relativity]]. The basic theory was outlined by [[Bryce DeWitt]]{{ref|dewitt}} in a seminal 1967 paper, and based on earlier work by [[Peter G. Bergmann]]{{ref|bergmann}} using the so-called canonical quantization techniques for constrained Hamiltonian systems invented by [[Paul Dirac]].{{ref|dirac}} Dirac's approach allows the quantization of systems that include [[gauge symmetry|gauge symmetries]] using Hamiltonian techniques in a fixed [[gauge fixing|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]].
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| The quantization is based on decomposing the [[metric tensor]] as follows,
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| : <math>g_{\mu\nu}dx^\mu \, dx^\nu =(-\,N^2+\beta_k\beta^k)dt^2+2\beta_k \, dx^k \, dt+\gamma_{ij} \, dx^i \, dx^j</math>
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| where the summation over repeated indices is [[Einstein summation convention|implied]], the index 0 denotes time <math>\tau=x^0</math>, Greek indices run over all values 0, . . ., ,3 and Latin indices run over spatial values 1, . . ., 3. The function <math>N</math> is called the '''lapse function''' and the functions <math>\beta_k</math> are called the '''shift functions.''' The spatial indices are raised and lowered using the spatial metric <math>\gamma_{ij}</math> and its inverse <math>\gamma^{ij}</math>: <math>\gamma_{ij}\gamma^{jk}=\delta_i{}^k</math> and <math>\beta^i=\gamma^{ij}\beta_j</math>, <math>\gamma=\det\gamma_{ij}</math>, where <math>\delta</math> is the [[Kronecker delta]]. Under this decomposition the [[Einstein–Hilbert action|Einstein–Hilbert Lagrangian]] becomes, up to [[total derivative]]s,
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| : <math>L=\int d^3x\,N\gamma^{1/2}(K_{ij}K^{ij}-K^2+{}^{(3)}R)</math>
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| where <math>{}^{(3)}R</math> is the spatial [[scalar curvature]] computed with respect to the [[Riemannian manifold|Riemannian metric]] <math>\gamma_{ij}</math> and <math>K_{ij}</math> is the [[second fundamental form|extrinsic curvature]],
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| :<math>K_{ij}= -\frac{1}{2}(\mathcal{L}_{n}\gamma)_{ij} =\frac{1}{2}N^{-1}\left(\nabla_j\beta_i+\nabla_i\beta_j-\frac{\partial\gamma_{ij}}{\partial t}\right),</math>
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| where <math>\mathcal{L}</math> denotes Lie-differentiation, <math>n</math> is the unit normal to surfaces of constant <math>t</math> and <math>\nabla_i</math> denotes [[covariant derivative|covariant differentiation]] with respect to the metric <math>\gamma_{ij}</math>. Note that <math> \gamma_{\mu\nu} = g_{\mu\nu} + n_{\mu}n_{\nu} </math>. 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.
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| 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 <math>\pi</math> and <math>\pi^i</math>, vanish identically ([[on shell and off shell]]). These are called ''primary constraints'' by Dirac. A popular choice of gauge, called [[synchronous gauge]], is <math>N=1</math> and <math>\beta_i=0</math>, although they can, in principle, be chosen to be any function of the coordinates. In this case, the Hamiltonian takes the form
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| :<math>H=\int d^3x\mathcal{H},</math>
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| where
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| :<math>\mathcal{H}=\frac{1}{2}\gamma^{-1/2}(\gamma_{ik}\gamma_{jl}+\gamma_{il}\gamma_{jk}-\gamma_{ij}\gamma_{kl})\pi^{ij}\pi^{kl}-\gamma^{1/2}{}^{(3)}R</math>
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| and <math>\pi^{ij}</math> is the momentum conjugate to <math>\gamma_{ij}</math>. Einstein's equations may be recovered by taking [[Poisson bracket]]s with the Hamiltonian. Additional on-shell constraints, called ''secondary constraints'' by Dirac, arise from the consistency of the Poisson bracket algebra. These are <math>\mathcal{H}=0</math> and <math>\nabla_j\pi^{ij}=0</math>. This is the theory which is being quantized in approaches to canonical quantum gravity.
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| All canonical theories of general relativity have to deal with the [[problem of time]]. In short, in general relativity, time is just another coordinate as a result of [[general covariance]]. In quantum field theories, especially in the Hamiltonian formulation, the formulation is split between three dimensions of space, and one dimension of time.
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| ==See also==
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| *[[Loop quantum gravity]] is one of this family of theories.
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| *[[Loop quantum cosmology]] (LQC) is a finite, symmetry reduced model of loop quantum gravity.
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| ==Sources and notes==
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| #{{cite journal |last1=Arnowitt |first1=R. |last2=Deser |first2=S. |last3=Misner |first3=C. W. |title=The Dynamics of General Relativity |year=2008 |journal=[[General Relativity and Gravitation]] |volume=40 |issue=9 |pages=1997–2027 |arxiv=gr-qc/0405109 |doi=10.1007/s10714-008-0661-1|bibcode = 2008GReGr..40.1997A }}
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| #*Originally from {{cite book |last=Witten |first=L. |title=Gravitation: An Introduction to Current Research |year=1962 |pages=227–265|publisher=[[John Wiley & Sons]]}}
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| #{{note|bergmann}} {{cite journal | title = Hamilton–Jacobi and Schrödinger Theory in Theories with First-Class Hamiltonian Constraints | year = 1966 | last1 = Bergmann | first1 = P. | journal = [[Physical Review]] | volume = 144 | issue = 4 | pages = 1078–1080 | doi = 10.1103/PhysRev.144.1078|bibcode = 1966PhRv..144.1078B }}
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| #{{note|dewitt}} {{cite journal | title = Quantum Theory of Gravity. I. The Canonical Theory | year = 1967 | last1 = Dewitt | first1 = B. | journal = [[Physical Review]] | volume = 160 | issue = 5 | pages = 1113–1148 | doi = 10.1103/PhysRev.160.1113 |bibcode = 1967PhRv..160.1113D }}
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| #{{note|dirac}} {{cite journal |last1=Dirac |first1=P. A. M. |title=Generalized Hamiltonian Dynamics |volume=246 |issue=1246 |pages=326–332 |journal=[[Proceedings of the Royal Society of London A]] |year=1958 |doi=10.1098/rspa.1958.0141 | |jstor=100496|bibcode = 1958RSPSA.246..326D }}
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| #{{cite journal |last1=Dirac |first1=P. A. M. |year=1958 |title=The Theory of Gravitation in Hamiltonian Form |journal=[[Proceedings of the Royal Society of London A]] |volume=246 |issue=1246 |pages=333–343 |bibcode=1958RSPSA.246..333D |doi= 10.1098/rspa.1958.0142 |jstor=100497}}
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| #{{cite journal |last1=Dirac |first1=P. A. M. |year=1959 |title=Fixation of Coordinates in the Hamiltonian Theory of Gravitation |journal=[[Physical Review]] |volume=114 |issue=3 |pages=924–930 |doi=10.1103/PhysRev.114.924|bibcode = 1959PhRv..114..924D }}
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| #{{cite book |last = Dirac |first=P. A. M. |year=1964 |title = Lectures on quantum mechanics |publisher=[[Yeshiva University]] |isbn = 0-387-51916-5}}
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| {{Theories of gravitation}}
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| {{quantum gravity}}
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| [[Category:Mathematical methods in general relativity]]
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| [[Category:Quantum gravity]]
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| [[Category:Physics beyond the Standard Model]]
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