Wheeler-Dewitt equation and canonical quantum gravity

[ylping]

In the canonical quantum gravity, Hamiltonian equal zero, why? Since the split is space, the time is a constant? or the space-time is Ricci-flat?

In other words, if the space-time is Ricci-flat whether time supersurface, Hamiltonian equal zero?

 

[hellfire]

The hamiltonian is equal to zero in any system which is invariant under reparametrizations. This is always the case for the gravitational field in general relativity. Another example is a relativistic point particle in geodesic motion.

 

[Entropia]

The Wheeler-Dewitt equation and canonical quantum gravity are both important concepts in the field of quantum gravity, which aims to unify the theories of general relativity and quantum mechanics. The Wheeler-Dewitt equation is a mathematical equation that describes the quantum state of the universe, while canonical quantum gravity is a theoretical framework that attempts to quantize the gravitational field.

One of the key features of canonical quantum gravity is that the Hamiltonian, which is a mathematical operator that describes the total energy of a system, is equal to zero. This is because in the canonical formalism, the gravitational field is treated as a constraint rather than a dynamical variable, and therefore does not contribute to the total energy of the system. This is in contrast to other quantum theories where the Hamiltonian is a non-zero operator.

The reason for this is not directly related to the split of space and time, but rather the nature of gravity itself. In general relativity, the gravitational field is described by the curvature of space-time, which is determined by the distribution of matter and energy. In a perfectly flat or Ricci-flat space-time, there is no matter or energy present, and therefore no gravitational field. This means that in a Ricci-flat space-time, the Hamiltonian would indeed be equal to zero.

However, it is important to note that the Wheeler-Dewitt equation and canonical quantum gravity are still theoretical concepts and have not been fully proven or tested. The nature of space-time and gravity is still a subject of ongoing research and debate in the scientific community.