Quantum field theory in curved spacetime

Quantum field theory in curved spacetime quantizes matter fields on a prescribed classical spacetime geometry.

Positive-frequency modes in stationary asymptotic past and future regions can differ by a Bogoliubov transformation mixing positive and negative frequency solutions. The conserved Klein-Gordon inner product gives its canonical normalization. The out annihilation operator then contains in creation operators, yielding the displayed number expectation in the initial vacuum. A nonzero mixing coefficient changes the particle content relative to final observers without violating deterministic field evolution.

An observer at constant proper acceleration in the Minkowski vacuum detects a thermal response at this temperature. The associated Rindler coordinates separate the observer's accessible wedge from a horizon. This is an observer-dependent particle/detector response, not an assertion that inertial observers measure a thermal bath.

The Hadamard condition requires the two-point function of a physical free-field state to have the universal local short-distance singularity prescribed by the spacetime geometry and field equation. It permits state-dependent smooth contributions and enables local renormalization of the stress-energy tensor. It does not select a unique Fock vacuum.

A free Klein-Gordon field is a scalar obeying $(\square -m^2-\xi R)\varphi =0$ in the mostly-plus metric signature. Global hyperbolicity provides a well-defined initial-value problem. Quantization additionally needs a state or positive-norm mode splitting; the equation alone does not select particles.

For real Klein-Gordon field solutions with suitable support or boundary conditions, $\Omega ({\varphi }_1,{\varphi }_2)={\int }_{\Sigma }({\varphi }_1{n}^a{\nabla }_a{\varphi }_2-{\varphi }_2{n}^a{\nabla }_a{\varphi }_1)d\Sigma$ is a conserved symplectic form. The field equation makes its current divergence-free. Its complexification gives $(u,v{)}_{KG}=i\Omega (u^{\ast },v)$, the Klein-Gordon inner product. A compatible complex structure on the Klein-Gordon solution space is additional data needed for a particle representation.

On a scalar, the covariant wave operator is $\square \varphi =|g{|}^{-1/2}{\partial }_a(|g{|}^{1/2}{g}^{ab}{\partial }_b\varphi )$. For the metric signature $(-+++)$ it has the flat-space form $-{\partial }_t^2+{\nabla }^2$. This divergence expression is convenient for separation of variables and for integrating the Klein-Gordon equation by parts.

On the real Klein-Gordon field solution space with conserved symplectic form $\Omega$, a compatible real-linear map $J$ obeys $J^2=-I$, preserves $\Omega$, and makes $\mu (f,g)=\Omega (f,Jg)$ positive definite. Its $+i$ eigenspace in the complexified space has positive Klein-Gordon inner product with the convention $(u,v{)}_{KG}=i\Omega (\bar{u},v)$. It defines the one-particle Hilbert space and a free-field Fock vacuum.

A conformally coupled scalar field includes the curvature coupling that makes the massless scalar action invariant under local conformal rescaling in the appropriate dimension. In four-dimensional de Sitter spacetime its mode equation is equivalent to that of a flat-space massless scalar after conformal rescaling.

For complex solutions of the Klein-Gordon equation,The associated current is conserved, so the inner product is independent of the Cauchy hypersurface when boundary flux vanishes.

For a free real scalar field expanded with measure $d^{3}p/[(2\pi {)}^{3}2E_p]$ and $f_p=e^{-ip\cdot x}$, the displayed Klein-Gordon inner product extracts its annihilation operator. Indeed $(f_p,f_q)=(2\pi {)}^{3}2E_p{\delta }^3(\mathbf{p}-\mathbf{q})$ and $(f_p,f_q^{\ast })=0$; spatial Fourier inversion removes the measure. Taking the Hermitian adjoint gives $a^{†}(\mathbf{p})=-i\int d^{3}x{f}_p\overleftrightarrow{{\partial }_t}\varphi$. The equal-time canonical commutation relation then gives $[a(\mathbf{p}),a^{†}(\mathbf{q})]=(2\pi {)}^{3}2E_p{\delta }^3(\mathbf{p}-\mathbf{q})$.

Relative to a timelike Killing field $K$, a positive-frequency solution obeys $i{\mathcal{L}}_{K}u=\omega u$ with $\omega >0$. Such solutions define the one-particle space and annihilation operators of a stationary vacuum.

For $p^0=\sqrt{{\mathbf{p}}^2+m^2}>0$ in Minkowski spacetime, these modes obey $i{\partial }_t{\psi }_{\mathbf{p}}=p^0{\psi }_{\mathbf{p}}$ and have Klein-Gordon inner product $({\psi }_{\mathbf{p}},{\psi }_{\mathbf{q}})={\delta }^3(\mathbf{p}-\mathbf{q})$. Their conjugates have the opposite norm and are orthogonal to the positive-frequency modes. Plane waves are a generalized, delta-normalized basis; normalizable wave packets have positive norm $\int |a(\mathbf{p}){|}^2{d}^{3}p$. Completeness follows by applying the spatial Fourier transform to the Klein-Gordon equation, which gives the two branches $e^{\mp i{p}^{0}t}$ for each momentum.

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 a stable free Klein-Gordon field with a complete timelike Killing flow and a positive self-adjoint one-particle energy generator, its positive spectral subspace selects positive-frequency solutions of positive Klein-Gordon inner product. Their annihilation operators define the ground-state Fock vacuum. Degenerate modes can be mixed by a unitary matrix without changing this splitting or vacuum. Merely being stationary near infinity is insufficient if the Killing field becomes spacelike in an ergoregion of a stationary spacetime; zero modes, instability or additional boundary conditions also require separate treatment. The choice of a ground state does not prevent other states, such as thermal states, from using the same particle operators.

// Target: statistical-physics.bigb

This is the symmetric-particle construction of Fock space.

For one-particle Hilbert space ${\mathcal{H}}_1$, the bosonic Fock space isCreation 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 a Bogoliubov transformation and generally define different particle notions and vacuum states.