Zero Hamiltonian and its energies

[Karlisbad]

By the way if we have $$\bold H =0$$ can this H be split into the terms.

$$H^{(0)} (g_{00} , \pi _{00} )+H^{(3)}(g_ {ij} , \pi _ {ij} )=0$$


so we quantizy the term... $$H^{(3)}(g_ {ij} , \pi _ {ij})\Phi = E_{n} \Phi$$


and from this we get the energies..

 

[Careful]

By the way if we have $$\bold H =0$$ can this H be split into the terms.

$$H^{(0)} (g_{00} , \pi _{00} )+H^{(3)}(g_ {ij} , \pi _ {ij} )=0$$


so we quantizy the term... $$H^{(3)}(g_ {ij} , \pi _ {ij})\Phi = E_{n} \Phi$$


and from this we get the energies..

First of all in GR, you have four constraints, second the constraints themselves do only contain the $$g_{ij}, \pi_{ij}$$ . Moreover, these energies as you call them have no physical meaning whatsoever.

 

[Karlisbad]

By the way appart from constraint (i finally learned how to apply Dirac's method to them using Wikipedia ) Does the Einstein Lagrangian has an acceleration term?..i have some troubles to quantize Lagrangians with an acceleration term so:

$$L= A(v)^{2}+BV(x)+C(\dot V)^{2}$$

 

[Careful]

By the way appart from constraint (i finally learned how to apply Dirac's method to them using Wikipedia )

Perhaps you should consult a better reference

Does the Einstein Lagrangian has an acceleration term?..

Yes, the Ricci scalar contains second derivatives with respect to time.

 

[dextercioby]

Usually, second order derivative terms in the HE action can be removed by a part integration. See Dirac (1975) for details.

Daniel.

 

[Careful]

Usually, second order derivative terms in the HE action can be removed by a part integration. See Dirac (1975) for details.

Daniel.

I guess you need to demand in such cases that spacetime is asymptotically static; in either that $$\partial_t g_{\mu \nu} = 0$$ in the infinite past and infinite future (in some a priori chosen coordinate time t).

 

[dextercioby]

Of course, in fact Dirac didn't mention this assumption. He just used it to get a better looking Lagrangian and a faster way to derive the field eqns. Actually, a boundary term occurs which can be set to 0, iff your assumption is validated.

Daniel.

 

[Careful]

Of course, in fact Dirac didn't mention this assumption. He just used it to get a better looking Lagrangian and a faster way to derive the field eqns. Actually, a boundary term occurs which can be set to 0, iff your assumption is validated.

Daniel.

But that would kill off asymtotically de Sitter universes. I guess you do not want that.

 

[Karlisbad]

And to get the Semi-classical Energies..could we make or use Bohr-Sommerfeld formula?..i mean:

$$\oint_{S}d^{4}x \pi _{ab}=\hbar (n+1/2)$$

where S is an hyper-surface, and the "pi's" are the conjugate momenta to the metric g_ab, could we from this expression get the energies?

 

[dextercioby]

But that would kill off asymtotically de Sitter universes.

Which is weird because such solutions are solutions (excuse the tautology) of the field equations one derives by making such an assumption.

Daniel.

 

[Careful]

Which is weird because such solutions are solutions (excuse the tautology) of the field equations one derives by making such an assumption.

Daniel.

I was merely worried about the fact that de Sitter allows for no global timelike Killing field and that therefore any such construction involves a cosmological horizon associated to one worldline I guess (since the metric gets degenerate in such coordinate systems). Therefore, it seems you cannot apply this trick to get out full de Sitter, that is all I meant (working in special coordinate systems can be pretty dangerous, at least that is how I understood your comment about Dirac's trick).