Quantum field theory in curved spacetime

For complex solutions of the Klein-Gordon equation,

(ϕ_{1}, ϕ_{2})_{K G} = i \int_{Σ} d Σ^{a} (ϕ_{1}^{*} \nabla_{a} ϕ_{2} - ϕ_{2} \nabla_{a} ϕ_{1}^{*}) .

The associated current is conserved, so the inner product is independent of the Cauchy hypersurface when boundary flux vanishes.

Positive-frequency solution

Relative to a timelike Killing field

K

i L_{K} u = ω u

with

ω > 0

. Such solutions define the one-particle space and annihilation operators of a stationary vacuum.

In a strictly stationary spacetime, the timelike Killing flow gives a preferred positive-frequency splitting. The corresponding vacuum is annihilated by every annihilation operator associated with a positive-frequency mode.

For one-particle Hilbert space

H_{1}

F_{s} (H_{1}) = ⨁_{n = 0}^{\infty} Sym^{n} H_{1} .

Creation operators add one symmetrized particle and annihilation operators remove one.

Without a preferred timelike Killing flow there is no canonical positive-frequency splitting. Different choices are related by Bogoliubov transformations and generally define different particle notions and vacuum states.

A Bogoliubov transformation is a linear canonical transformation that mixes positive- and negative-frequency modes, and therefore mixes annihilation and creation operators. A nonzero mixing coefficient makes one vacuum contain particles relative to the other mode decomposition.

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Quantum Field Theory (QFT) in curved spacetime is the framework that combines the principles of quantum mechanics and quantum field theory with general relativity, which describes the gravitational field in terms of curved spacetime rather than a flat background. This approach is essential for understanding physical phenomena in strong gravitational fields, such as near black holes or during the early moments of the universe just after the Big Bang, where both quantum effects and gravitational effects are significant.