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Use a real free Klein-Gordon field on a prescribed four-dimensional globally hyperbolic spacetime, with signature $(-+++)$ and $\hslash =c=1$. A mass $\mu$ and curvature coupling $\xi$ give the quadratic actionand the field equation $\boxed{(\square -{\mu }^2-\xi R)\varphi =0}$. Minimal coupling is $\xi =0$; for a massless scalar in four dimensions, conformal coupling is $\xi =1/6$. Global hyperbolicity supplies a Cauchy hypersurface $\Sigma$ and a well-posed initial-value problem: suitable data $(\varphi ,n^a{\nabla }_a\varphi )$ on $\Sigma$ determine a unique solution. One assumes appropriate support or decay conditions so no unaccounted flux crosses spatial infinity. The geometry is classical here; quantum backreaction is a further problem.

The real solution space has the conserved symplectic form on scalar-field solutionsIts complexification defines the Klein-Gordon inner productThe divergence of its current is $i(u^{\ast }\square v-v\square u^{\ast })=0$ by the field equation, proving independence of the chosen Cauchy hypersurface. This Hermitian form is not positive definite on the whole complex solution space: conjugating a solution reverses its norm.

Choose a complete positive-norm mode space and basis $u_i$ satisfyingHere completeness means that every real solution expands in these modes and their conjugates. Equivalently, one chooses a compatible complex structure on the Klein-Gordon solution space; this is extra data, not fixed by the field equation. Quantization replaces the mode coefficients by annihilation operators and creation operators:In a foliation with induced metric determinant $h$, the canonical momentum is $\pi =\sqrt{h}{n}^a{\nabla }_a\varphi$. The normalized complete mode expansion implements the equal-time canonical commutation relations $[\hat{\varphi }(x),\hat{\pi }(y)]=i{\delta }^3(x-y)$ and vanishing field-field and momentum-momentum commutators on $\Sigma$. Propagation by the field equation gives microcausality: field observables commute at spacelike separation. Thus there is a local quantum field theory before one has selected a particle interpretation.

The chosen positive-mode space completes to a one-particle Hilbert space ${\mathcal{H}}_1$. Its bosonic Fock space is ${⨁}_{n=0}^{\infty }{\mathrm{Sym}}^n{\mathcal{H}}_1$. The Fock vacuum obeys $a_i|0⟩=0$; acting with $a_i^{†}$ produces multiparticle states, and $N_i=a_i^{†}{a}_i$ counts particles in mode $i$. The vacuum two-point function is $W(x,y)={\sum }_i{u}_i(x)u_i(y{)}^{\ast }$. The Hadamard condition restricts physically admissible states by prescribing their local short-distance singularity; it permits renormalization of the stress-energy tensor but does not select one unique vacuum.

The particle ambiguity is that there is generally no distinguished choice of positive-mode space. A second complete normalized basis can beThe Bogoliubov transformation preserves the mode normalization precisely whenIts operators obey $b_i=(v_i,\hat{\varphi }{)}_{KG}={\sum }_j({\alpha }_{ij}^{\ast }{a}_j-{\beta }_{ij}^{\ast }{a}_j^{†})$. Consequently a state with no $a$-particles generally has $b$-particles if $\beta \ne 0$. A unitary change of basis within the same positive-mode space has $\beta =0$ and merely relabels modes; a change mixing the positive and negative subspaces changes the vacuum and particle notion. Arbitrary coordinates called “time” do not resolve this choice. This is the vacuum ambiguity in a nonstationary spacetime, while the field equation and local commutator remain unchanged.

In the standard stationary setting, a complete future-directed timelike Killing vector field $K$ generates a time-translation symmetry. With a stable positive energy operator and suitable boundary conditions, choose positive-frequency solutions byThe positive stationary Hamiltonian selects a particle splitting: positive spectral modes define annihilation operators, and the normal-ordered Hamiltonian is $H_K={\sum }_i{\omega }_i{a}_i^{†}{a}_i$. Its ground state is the vacuum state in a stationary spacetime. The Killing symmetry transports this splitting unchanged, so no positive-negative mixing is induced by stationary evolution. Degeneracies allow unitary basis changes without changing the vacuum. Stationarity fixes the ground-state particle interpretation relative to this time flow; it does not forbid excited or thermal states.

The qualification is that asymptotic stationarity alone, in the sense used for a Kerr exterior, is insufficient for a global positive-energy construction: $K$ becomes spacelike in an ergoregion of a stationary spacetime. The usual unambiguous construction applies to a strictly stationary spacetime with the stated stability and spectral assumptions, or to appropriate stationary asymptotic regions for scattering. Zero modes or instability need separate treatment. Likewise different observer time flows on different regions need not define the same particles.

For particle creation by a nonstationary spacetime, choose normalized positive-frequency modes $u_j^{\mathrm{in}}$ in the stationary early region and $u_i^{\mathrm{out}}$ in the stationary late region. Propagate both sets by the Klein-Gordon equation through the intermediate time-dependent geometry. Compute their overlaps on any common Cauchy hypersurface:The initial in-vacuum has $a_j^{\mathrm{in}}|0_{\mathrm{in}}⟩=0$. Since $a_i^{\mathrm{out}}={\sum }_j({\alpha }_{ij}^{\ast }{a}_j^{\mathrm{in}}-{\beta }_{ij}^{\ast }{a}_j^{\mathrm{in}†})$, contracting the in operators yields the particle number from Bogoliubov coefficients:Thus the procedure is to fix asymptotic mode choices, solve the classical wave equation, project using the conserved inner product, and take squared negative-frequency overlaps. Particle energy comes from the time-dependent background, so particle creation does not contradict the linear deterministic field equation. For an initial state diagonal in in-mode occupation numbers $n_j$, with no anomalous correlations, the more general expectation is ${\sum }_j[|{\alpha }_{ij}{|}^2{n}_j+|{\beta }_{ij}{|}^2(n_j+1)]$.

For continuous spectra, sums become integrals and normalized wave packets avoid meaningless squares of delta functions. A finite total count requires the appropriate convergence of the double sum or integral. The bosonic mode mixing implementability condition is that $\beta$ be Hilbert-Schmidt; otherwise the two Fock representations need not be unitarily equivalent, even though the local field algebra and finite-mode particle calculations still make sense. This distinction between deterministic field evolution and a chosen representation is essential to quantum field theory in curved spacetime.

Fix a classical globally hyperbolic spacetime with metric signature $(-+++)$; in this solution take $\hslash =1$. A free real Klein-Gordon field can be specified bywhere $\mu$ is its mass, $\xi$ its curvature coupling and $\mathcal{R}$ the scalar curvature. Global hyperbolicity ensures a well-posed initial-value problem on a Cauchy hypersurface $\Sigma$ and the existence of retarded and advanced propagators. Thus compactly supported field and normal-derivative data determine a classical solution. This fixes the dynamics, but not a Fock vacuum.

For real solutions with suitable support or falloff, the symplectic form on scalar-field solutions iswith $n$ the future unit normal. The field equation makes the current conserved, so this symplectic form is independent of $\Sigma$ when boundary flux vanishes. Quantize the initial data by the canonical commutation relation: with $\pi =n^a{\nabla }_a\varphi$ and the delta function defined relative to $d\Sigma$, $[\hat{\varphi }(x),\hat{\pi }(y)]=i{\delta }_{\Sigma }(x,y)$ and the two equal-field commutators vanish. Equivalently, construct the field algebra using the causal propagator. A state on that algebra is additional input.

To construct a particle representation, complexify the classical solution space. Its conserved Klein-Gordon inner product isIt is indefinite on all complex solutions. Choose a complete positive-norm subspace and an orthonormal mode basis $u_i$ with $(u_i,u_j)={\delta }_{ij}$, $(u_i^{\ast },u_j^{\ast })=-{\delta }_{ij}$ and $(u_i,u_j^{\ast })=0$. Such a choice is encoded by a compatible complex structure on the Klein-Gordon solution space. Its positive subspace gives the one-particle Hilbert space, and the associated bosonic Fock space contains symmetrized many-particle states. The field expansion isThe creation operator $a_i^{†}$ adds a particle in mode $i$, the annihilation operator removes one, and the number operator is $N_i=a_i^{†}{a}_i$. For continuous mode labels the sums and Kronecker deltas become integrals and delta functions, or one can work with normalized wave packets.

The ambiguity is precisely that the field equation and global hyperbolicity do not select that positive subspace. A different normalized basis may mix $u_j$ and $u_j^{\ast }$ by a Bogoliubov transformation, and then its annihilation operators mix $a_j$ and $a_j^{†}$. Its Fock vacuum and number operators differ. The Hadamard condition constrains physically acceptable short-distance singularities and allows local renormalization, but it still leaves many states. Hence there is generally no observer-independent particle count on an arbitrary dynamical geometry.

In a stable strictly stationary spacetime, a globally future timelike Killing vector field $K$ gives a preferred time translation. Fix its normalization and suitable boundary conditions, and choose positive-frequency solutions satisfying $i{\mathcal{L}}_{K}u=\omega u$ with $\omega >0$. The corresponding positive spectral subspace gives the preferred vacuum state in a stationary spacetime. Unitary changes of basis within it leave the vacuum and the particle notion unchanged. This construction assumes a well-defined positive stationary generator; stationarity by itself is insufficient if $K$ becomes spacelike, as in a Kerr ergoregion, or if unstable or zero modes obstruct the ground-state construction. It selects a preferred ground state under the stated assumptions, not a unique state among all thermal and excited states.

If the geometry is suitably stationary in the asymptotic past and future, choose those preferred mode spaces separately, giving the in-vacuum and out-vacuum. Propagate the past modes through the intervening region using the field equation and compare them to the future modes using the conserved Klein-Gordon inner product. Adopt the conventionThe canonical identities for a bosonic Bogoliubov transformation read $\alpha {\alpha }^{†}-\beta {\beta }^{†}=I$ and $\alpha {\beta }^T=\beta {\alpha }^T$. Extracting the future annihilation operator with the same inner product givesIn the in-vacuum, only $⟨a_j{a}_k^{†}⟩={\delta }_{jk}$ contributes to $⟨b_i^{†}{b}_i⟩$. Therefore the particle number from Bogoliubov coefficients isNonzero $\beta$ is the production of future particles from the past vacuum. Summing over future modes gives the total expected particle number when that sum is finite. For infinitely many modes, a Hilbert-Schmidt operator $\beta$ is the condition for unitary implementability between these pure bosonic Fock space representations; finite-volume or wave-packet calculations must respect the relevant measures and convergence. Particle production is determined by the negative-frequency mixing, rather than by identifying a single instantaneous vacuum throughout the time-dependent region.

Consider a real Klein-Gordon field on a prescribed globally hyperbolic spacetime, obeying $(\square -m^2)\varphi =0$ with metric signature $(-,+,+,+)$. A curvature coupling can be included as $m^2\mapsto m^2+\xi R$. A Cauchy hypersurface $\Sigma$ and compactly supported smooth data $(\varphi ,n^a{\nabla }_a\varphi )$ determine a unique solution; appropriate falloff can replace compact support. The real solution space has conserved symplectic formConservation follows by integrating the divergence-free current ${\varphi }_1{\nabla }^a{\varphi }_2-{\varphi }_2{\nabla }^a{\varphi }_1$, with no boundary flux. On complex solutions the conserved Klein-Gordon inner product isIt is indefinite on the full complex solution space.

Choose a complete positive-norm mode subspace, with modes $u_j$ satisfyingEquivalently choose a compatible complex structure on the Klein-Gordon solution space. The mode labels may be continuous, in which case sums and Kronecker symbols become integrals and Dirac delta functions. Construct the one-particle Hilbert space from these modes and its bosonic Fock space. Promote the field to the operator-valued distributionFor a foliation with spatial metric determinant $h$, the conjugate momentum density is $\hat{\pi }=\sqrt{h}{n}^a{\nabla }_a\hat{\varphi }$. Mode completeness gives the equal-time canonical commutation relationsThe Fock vacuum obeys $a_j|0⟩=0$, and $a_j^{†}{a}_j$ counts particles in the chosen mode. For local products and a renormalized stress-energy tensor, physically admissible states are further restricted by the Hadamard condition. The field algebra exists without a preferred Fock vacuum.

A different admissible mode splitting can mix positive and negative norms:Orthonormality imposes the canonical identities for a bosonic Bogoliubov transformationThe same field then hasWithout a preferred notion of positive frequency, a nonstationary spacetime supplies no distinguished mode splitting: particle number and vacuum depend on the choice of modes, although the field equation and field algebra do not. With infinitely many modes the Bogoliubov transformation need not be unitarily implementable; finite total mixing requires a Hilbert-Schmidt operator $\beta$.

For a stable strictly stationary spacetime, a chosen future globally timelike Killing vector field $K$ gives a preferred time translation. Choose modes withWhen the corresponding conserved Killing energy is positive and the spectral problem has suitable boundary conditions and no problematic zero modes, this gives the preferred vacuum state in a stationary spacetime and particles relative to $K$. Positive-frequency mode mixing within that same subspace leaves the Fock vacuum unchanged. Rescaling $K$ by a positive constant changes the frequency units but not their sign.

Stationarity alone, if it only means a Killing field timelike near infinity, is insufficient for a global unique particle interpretation. In the Kerr ergoregion, ${\partial }_t$ is spacelike, so positive frequency relative to $t$ does not automatically select a positive-norm subspace throughout the geometry; superradiance illustrates the difficulty. Additional vacuum and boundary choices remain necessary. The customary stationary answer therefore assumes a suitable timelike stationary flow and a stable positive-energy quantization; it does not assert that every stationary black-hole extension has one globally preferred vacuum.

A strictly stationary spacetime has an everywhere timelike Killing field $K$. Choose a complete orthonormal set of positive-frequency solutions satisfyingwith positive Klein-Gordon inner product. Expandwith the sum replaced by an integral for continuous labels. Equivalently, the coefficients are projections using the Klein-Gordon product: $a_j=(u_j,\hat{\Phi }{)}_{KG}$ and $a_j^{†}=-(u_j^{\ast },\hat{\Phi }{)}_{KG}$.

The stationary vacuum is uniquely selected, up to degeneracies and unitary changes of positive-frequency basis, byfor every $j$. Acting with the $a_j^{†}$ constructs the bosonic Fock space, the symmetric direct sum of all particle-number sectors.

In a nonstationary spacetime no preferred timelike Killing flow exists, so there is no canonical split into positive and negative frequencies. Different splits mix creation and annihilation operators by Bogoliubov transformation and lead to the vacuum ambiguity in a nonstationary spacetime.