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The usual manifestly unitary quantum theory can be described either in the schrodinger or heisenberg pictures. Both of these pictures are 'non-relativistic', because either the operators or the states evolve in time depending on the picture. The path integral formulation is manifestly relativistic, but in its present form is incomplete in the sense that unitarity is not manifest.

It is possible to interpret canonical quantum mechanics in a relativistic way if we consider the creation operators to be the 'string field theory' limit of quantum gravity.

a

a

where N is the number of particles. This means that in a sense a ~ √N. This property of the creation/annihiliation operators must be preserved by a covariant version of these operators for the following reason. The difference between time and space means that particles should be considered to be 'packets of energy', but there is no notion of a 'packet of momentum'. Momentum is the flux of energy. Covariant theories treat time and space on an 'equal footing', but this is very subtle for the reason mentioned.

For example, in loop quantum gravity, the 'Wheeler-de Witt equation'

H|ψ⟩ = 0

Has de Sitter spacetime as a solution (ψ = e

(p

Where now p

Please feel free to ask any questions, as there are many details to include in one post.

It is possible to interpret canonical quantum mechanics in a relativistic way if we consider the creation operators to be the 'string field theory' limit of quantum gravity.

a

^{+}= x + ip creates a particle of momentum p. This state |p⟩ evolves in time. The corresponding relativistic notion would be a creation operator that creates a 'spacetime history', rather than a state that evolves in time. An example of a spacetime history is exp i(px - Et), which is characterized not by an x and a p, but by E and p. The creation operators satisfya

^{+}a^{-}= Nwhere N is the number of particles. This means that in a sense a ~ √N. This property of the creation/annihiliation operators must be preserved by a covariant version of these operators for the following reason. The difference between time and space means that particles should be considered to be 'packets of energy', but there is no notion of a 'packet of momentum'. Momentum is the flux of energy. Covariant theories treat time and space on an 'equal footing', but this is very subtle for the reason mentioned.

For example, in loop quantum gravity, the 'Wheeler-de Witt equation'

H|ψ⟩ = 0

Has de Sitter spacetime as a solution (ψ = e

^{i(dA^A + A^A^A)}.) Where the term in the exponent is the action for Chern-Simons theory. This equation presumably corresponds to(p

^{2}- m^{2})φ = 0Where now p

^{2}- m^{2}is in some sense a Hamiltonian, and φ is a spacetime history rather than a state that evolves in time.Please feel free to ask any questions, as there are many details to include in one post.

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