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Weekly Report

Weekly Report. 091214 Yu- Chiao Chang. A trouble with H-L gravity. Miao Li shows that the constraint system of H-L gravity has some problem, the Poisson brackets of the Hamiltonian density do not form a closed structure, resulting in many new constraints.

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Weekly Report

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  1. Weekly Report 091214 Yu-Chiao Chang

  2. A trouble with H-L gravity • Miao Li shows that the constraint system of H-L gravity has some problem, the Poisson brackets of the Hamiltonian density do not form a closed structure, resulting in many new constraints. • Taking these new constraints into account, it appears that there is no degree of freedom left or the phase space is reduced to one with an odd number of fields.

  3. Continue the last week’s report, we want to compute the Poisson bracket of the Hamiltonian density: • But this is too complicated to compute, so he tried to study a special case: choose ξ to be with η arbitrary.

  4. After a tedious calculation, we obtain the following result expanded with respect to the covariant derivatives of η in different orders. • With the coefficient given by:

  5. And

  6. After some detailed mathematical discussion, Li argued that • Then the effective coefficient is

  7. Because covariant derivatives of η of different order are independent, consistency requires that the coefficients in front of covariant derivatives of η should vanish(Because is a first class constraint and the reason will be given later). • That is

  8. To analyze this condition, we diagonalize the Cotton tensor via vielbein: • This gives 7 independent equation:

  9. The solutions of these constraints can be separated into two classes: (1) together with the property make only one degree of freedom in be physical. Now, the Hamiltonian constraint and three momentum constraints eliminate four conjugate momenta, leaving two components of be physical. Altogether, the phase space is described by three unpaired fields.

  10. (2) This case is rather bad, it indicates that all the conjugate momentaof are unphysical, since three momentum constraints already eliminates three of the six conjugate momenta.

  11. Following, Li uses Dirac’s approach to justified that the Hamiltonian density is a first class constraint not a second. • The Dirac’s approach and the first, second class constraints are new ideas for me. It seems to be the material of the “constraint geometry.” and there is quite a bit issue and discussion on that. • It is worhty taking time to study that to understand Li’s work more deeply.

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