Quantum fields in curved spacetime

TL;DR

This work presents a mathematically precise, algebraic formulation of quantum fields in curved spacetime (QFTCS), stressing the distributional nature of fields, locality, covariance, and Hadamard-state structure as the backbone for physically meaningful predictions. It develops the free theory via an algebra of observables and Hadamard states, then extends to nonlinear observables and perturbative interactions, detailing the construction of time-ordered products and renormalization ambiguities in curved backgrounds. The review also discusses quintessential applications (Unruh, Hawking, de Sitter, inflation) and surveys gauge fields through BRST methods, before outlining open questions toward nonperturbative formulations and quantum gravity. Overall, it clarifies how to define composite operators, control singularities, and perform perturbative QFTCS in a generally covariant setting, with implications for black hole thermodynamics and early-universe cosmology.

Abstract

We review the theory of quantum fields propagating in an arbitrary, classical, globally hyperbolic spacetime. Our review emphasizes the conceptual issues arising in the formulation of the theory and presents known results in a mathematically precise way. Particular attention is paid to the distributional nature of quantum fields, to their local and covariant character, and to microlocal spectrum conditions satisfied by physically reasonable states. We review the Unruh and Hawking effects for free fields, as well as the behavior of free fields in deSitter spacetime and FLRW spacetimes with an exponential phase of expansion. We review how nonlinear observables of a free field, such as the stress-energy tensor, are defined, as well as time-ordered-products. The "renormalization ambiguities" involved in the definition of time-ordered products are fully characterized. Interacting fields are then perturbatively constructed. Our main focus is on the theory of a scalar field, but a brief discussion of gauge fields is included. We conclude with a brief discussion of a possible approach towards a nonperturbative formulation of quantum field theory in curved spacetime and some remarks on the formulation of quantum gravity.

Figures (5)

• Figure 1: Conformal diagram and values of the point-pair invariant $Z=Z(x,y)$ as $y$ is varied and $x$ is kept fixed. For the sake of easier visualization, we are giving the diagram in the case of $d=2$ dimensional deSitter spacetime, where the left and right vertical boundaries are to be identified. For $d>2$ dimensions, the diagram would basically consist of only the shaded "left half", with the vertical boundary lines corresponding to the north- and south pole of the $S^{d-1}$ Cauchy surface.
• Figure 2: Conformal diagram for deSitter spacetime, and the static chart. Again, we are drawing the case $d=2$. The case $d>2$ would correspond to the shaded square having the bifurcation surface $\cong S^{d-2}$ in the middle. The vertical boudaries of the shaded square correspond to the north- and south pole of the Cauchy surface $S^{d-1}$.
• Figure 3: Conformal diagram of extended Schwarzschild spacetime ('eternal black hole').
• Figure 4: Conformal diagram of collapsing star spacetime.
• Figure 5: Shown here is the wave-front set of the time-ordered products \ref{['msc']} and its relationship with embedded Feynman graphs ${\mathscr G}$ in $\mathscr{M}$. Through each line $e$ flows a 'momentum' $p_e$ indicated by $\rightarrow$, which is a parallel transported, cotangent null vector. At each vertex $x_i$ the corresponding vector $k_i \in T^*_{x_i} \mathscr{M}$ in the wave front set is characterized by the 'momentum conservation rule' $k_i = \sum_{\rm in} \textcolor{blue}{p_e} - \sum_{\rm out} \textcolor{blue}{p_e}$ counting the momenta associated with the incoming vs. outgoing edges $e$ with opposite sign.

Theorems & Definitions (2)

• Theorem 1
• Theorem 2