Quantum field theory in curved spacetime

Definition

Quantum field theory in curved spacetime is quantum field theory on a fixed, classical, curved spacetime that is not affected by the quantum fields. Its basic model, the free real scalar field, is defined as follows. Let $(M, g)$ be a globally hyperbolic, time-oriented Lorentzian manifold of dimension $4$, let $m \ge 0$ and $\xi \in \R$, and let $R$ be the scalar curvature of $g$. Let $P = \square_g + m^2 + \xi R$ be the Klein–Gordon operator of $g$, with $\square_g$ the d'Alembert operator of $g$. Let $E = E^- - E^+$ be the causal propagator, the difference of the advanced and retarded fundamental solutions of $P$. The field algebra is the unital $*$-algebra generated by symbols $\phi(f)$, for $f \in C_c^\infty(M)$ complex-valued, subject to the following relations, for all $f, h \in C_c^\infty(M)$ and $a, b \in \C$.

1. Linearity: $\phi(af + bh) = a\phi(f) + b\phi(h)$.
2. Hermiticity: $\phi(f)^* = \phi(\bar f)$.
3. Field equation: $\phi(Pf) = 0$.
4. Commutation relation: $[\phi(f), \phi(h)] = i E(f, h)\, 1$, where $E(f, h) = \int_M f\, (Eh)\, d\mathrm{vol}_g$.

A state of the theory is a positive normalized linear functional on this algebra.