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The '''Barrett–Crane model''' is a model in [[quantum gravity]] which was defined using the [[Plebanski action]].<ref name=Barrett1998>{{citation
| author = Barrett, John W.; Louis Crane
| title = Relativistic spin networks and quantum gravity
| journal = J.Math.Phys. 39
| volume = 39
| pages = 3296–3302
| doi = 10.1063/1.532254
| year = 1998
|arxiv = gr-qc/9709028 |bibcode = 1998JMP....39.3296B }}</ref><ref name=Barrett>{{citation
| author = Barrett, John W.; Louis
Crane,
| title = A Lorentzian signature model for quantum general relativity
| journal = Classical and Quantum Gravity
| volume = 17
| doi = 10.1088/0264-9381/17/16/302
|arxiv = gr-qc/9904025 |bibcode = 2000CQGra..17.3101B }}</ref>
 
The <math>B</math> field in the action is supposed to be a <math>so(3, 1)</math>-valued [[2-form]], i.e. taking values in the [[Lie algebra]] of a [[special orthogonal group]]. The term
 
:<math>B^{ij} \wedge B^{kl}</math>
 
in the action has the same symmetries as it does to provide the [[Einstein-Hilbert action]]. But the form of
 
:<math>B^{ij}</math>
 
is not unique and can be posed by the different forms:
*<math>\pm e^i \wedge e^j</math>
*<math>\pm \epsilon^{ijkl} e_k \wedge e_l</math>
where <math>e^i</math> is the [[tetrad]] and <math>\epsilon^{ijkl}</math> is the [[antisymmetric symbol]] of the <math>so(3, 1)</math>-valued 2-form fields.  
 
The Plebanski action can be constrained to produce the [[BF model]] which is a theory of no local [[degrees of freedom (physics and chemistry)|degrees of freedom]]. [[John W. Barrett]] and [[Louis Crane]] modeled the analogous constraint on the summation over [[spin foam]].
 
The Barrett–Crane model on spin foam quantizes the [[Plebanski action]], but its [[functional integration|path integral]] amplitude corresponds to the degenerate <math>B</math> field and not the specific definition 
 
:<math>B^{ij} = e^i \wedge e^j</math>,
 
which formally satisfies the [[Einstein's field equation]] of [[general relativity]]. However, if analysed with the tools of [[loop quantum gravity]] the Barrett–Crane model gives an incorrect long-distance limit [http://arxiv.org/abs/0708.0883], and so the model is not identical to loop quantum gravity.
 
==See also==
* [[EPRL model]]
 
==References==
{{reflist}}
 
{{DEFAULTSORT:Barrett-Crane model}}
[[Category:Loop quantum gravity]]
 
 
{{physics-stub}}

Revision as of 17:10, 30 January 2014

The Barrett–Crane model is a model in quantum gravity which was defined using the Plebanski action.[1][2]

The B field in the action is supposed to be a so(3,1)-valued 2-form, i.e. taking values in the Lie algebra of a special orthogonal group. The term

BijBkl

in the action has the same symmetries as it does to provide the Einstein-Hilbert action. But the form of

Bij

is not unique and can be posed by the different forms:

where ei is the tetrad and ϵijkl is the antisymmetric symbol of the so(3,1)-valued 2-form fields.

The Plebanski action can be constrained to produce the BF model which is a theory of no local degrees of freedom. John W. Barrett and Louis Crane modeled the analogous constraint on the summation over spin foam.

The Barrett–Crane model on spin foam quantizes the Plebanski action, but its path integral amplitude corresponds to the degenerate B field and not the specific definition

Bij=eiej,

which formally satisfies the Einstein's field equation of general relativity. However, if analysed with the tools of loop quantum gravity the Barrett–Crane model gives an incorrect long-distance limit [1], and so the model is not identical to loop quantum gravity.

See also

References

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