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Cambridge University Press, Cambridge, England
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C. Rovelli, Quantum Gravity (Cambridge University Press, Cambridge, England, 2004), p. 480.
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Quantum Gravity
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Rovelli, C.1
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2
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84927757872
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Thomas Thiemann, gr-qc/0110034
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Thomas Thiemann, gr-qc/0110034.
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4
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84927757871
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unpublished
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Alejandro Perez (unpublished).
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Perez, A.1
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7
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33646624346
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C. Rovelli and L. Smolin, Nucl. Phys. B442, 593 (1995); B456, 753(E) (1995).
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Nucl. Phys.
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A. Ashtekar, J. Baez, A. Corichi, and K. Krasnov, Phys. Rev. Lett. 80, 904 (1998).
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Phys. Rev. Lett.
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Ashtekar, A.1
Baez, J.2
Corichi, A.3
Krasnov, K.4
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20
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84927757870
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note
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This remains an open issue in the full theory of LQG. The master constraint approach [20] and the spin foam formulation [21] are current proposals to solve this problem.
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21
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84927757869
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T. Thiemann, gr-qc/0305080
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T. Thiemann, gr-qc/0305080.
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23
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84927757868
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Donald Marolf, gr-qc/0011112
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Donald Marolf, gr-qc/0011112.
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24
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21844520846
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A. Ashtekar, J. Lewandowski, D. Marolf, J. Mourao, and T. Thiemann, J. Math. Phys. (N.Y.) 36, 6456 (1995).
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Ashtekar, A.1
Lewandowski, J.2
Marolf, D.3
Mourao, J.4
Thiemann, T.5
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27
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84927757867
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note
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According to the general analysis of symmetric connections for arbitrary symmetry groups [25] the time component of the connection is a connection on a symmetry-reduced principal fiber bundle. This fiber bundle has as its gauge group the centralizer Zλ := ZG[λ(F)] where G is the gauge group of the theory, F is the isotropy group, and λ is a homomorphism from F to G. For our model the gauge group as well as the isotropy group is SU(2). The homomorphism λ is thus the identity map and the centralizer Zλ only contains the identity. The time component of the connection thus vanishes.
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30
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84927757866
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note
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The factor of 1/√E is added such that Hmatter agrees with the standard form of the matter Hamiltonian used in other formulations. Note that the constraint used here does not have units of energy hence the need for the extra factor.
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31
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84927757865
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note
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Halliwell [31] has considered the path integral for actions that are reparametrization invariant such as the action presented here. Starting with the path integral P = ∫ DADEDB exp[i ∫t′ t″ dt3EȦ + BH] the one form component B is gauge fixed to be constant in time and after including ghost terms to make the path integral independent of the gauge choice the path integral becomes P = ∫ dB(t″ - t′) ∫ DADE exp[i ∫t′t″ dt3EȦ + BH] which with the redefinition T = B(t″ - t′) and t = B(t - t′) takes on the form equivalent to the group averaging one P = ∫ dT ∫ DADE exp[i ∫0T dt̄3EȦ + H] = ∫ dT〈v″|e-iĤT|v′〉.
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33
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84927757864
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note
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Notice that the effect of the cosmological constant term depends on the orientation of the triad which is dynamical for our action.
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35
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17044436179
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Ph.D. thesis, Pennsylvania State University
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Joshua Willis, Ph.D. thesis, Pennsylvania State University, 2004.
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(2004)
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Willis, J.1
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