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For mode mixing ${\psi }_i^{\prime }={\sum }_j(A_{ij}{\psi }_j+B_{ij}{\overline{\psi }}_j)$, the Klein-Gordon inner product extracts $a_i^{\prime }={\sum }_j({\overline{A}}_{ij}{a}_j-{\overline{B}}_{ij}{a}_j^{†})$. The canonical commutation relation is preserved exactly when $A{A}^{†}-B{B}^{†}=I$ and $A{B}^T=B{A}^T$. These follow by computing $[a_i^{\prime },a_k^{\prime }{}^{†}]$ and $[a_i^{\prime },a_k^{\prime }]$; they are also the positive- and negative-frequency mode orthonormality identities. In infinitely many modes, convergence and the existence of a common bosonic Fock space representation require additional analysis; the expected total particle number is finite only if $B$ is a Hilbert-Schmidt operator.

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.

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 $[a(\mathbf{p}),a^{†}(\mathbf{q})]=(2\pi {)}^{3}2E_p{\delta }^3(\mathbf{p}-\mathbf{q})$.

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.

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.

Use units $\hslash =c=1$ and the Minkowski metric $g=\mathrm{diag}(1,-1,-1,-1)$. The canonical momentum of the real scalar field is $\pi =\partial \mathcal{L}/\partial \dot{\varphi }=\dot{\varphi }$. Canonical quantization replaces the classical fields by operator-valued distributions and imposes the equal-time canonical commutation relationsThese relations are imposed on an initial slice, with smeared fields understood. In the Heisenberg picture, simultaneous unitary time evolution of both operators preserves their commutators. Since the momentum is $\dot{\varphi }$, the requested equal-time commutator is $[\varphi (t,\mathbf{x}),\dot{\varphi }(t,\mathbf{y})]=i{\delta }^{(3)}(\mathbf{x}-\mathbf{y})$.

The Legendre transform in mechanics, applied pointwise, gives the canonical Hamiltonian density of a real scalar field. ThusTo obtain its dynamics rather than assume them, use the Heisenberg equation of motion $\dot{O}=i[H,O]$. The momentum-square term gives $[H,\varphi ]=-i\pi$. The field-square term gives $[H,\pi ]=i{m}^2\varphi$, while differentiating the Dirac delta function in the gradient term and integrating by parts gives $[H,\pi ]=-i{\nabla }^2\varphi +i{m}^2\varphi$. ConsequentlyBoundary terms vanish for suitably decaying smeared fields. This is the Klein-Gordon equation as an operator identity.

A spatial Fourier transform reduces the Klein-Gordon equation to ${\ddot{\varphi }}_{\mathbf{p}}+E_p^2{\varphi }_{\mathbf{p}}=0$, where $E_p=\sqrt{{\mathbf{p}}^2+m^2}$. Its two frequency branches and the Hermitian adjoint condition giveThe invariant-normalized scalar mode extraction can be written particularly simply asSubstituting the mode expansion of a free field proves these inversion formulas: the opposite-frequency coefficient is proportional to $E_p-E_q$ and vanishes after spatial Fourier inversion. Alternatively they are the corresponding Klein-Gordon inner products. Taking their commutators using the equal-time fields gives $(E_p+E_q)(2\pi {)}^3{\delta }^{(3)}(\mathbf{p}-\mathbf{q})$ in the mixed case. In the two equal-type cases the delta function enforces $\mathbf{q}=-\mathbf{p}$, multiplying $E_p-E_q=0$. ThereforeConversely these mode commutators reproduce $[\varphi ,\dot{\varphi }]=i{\delta }^{(3)}$: the two frequency contributions each supply half the spatial Dirac delta function.

For the covariant oscillator Hamiltonian of a real scalar field, insert the mode expansion of a free field into $H$. Spatial integration sets the two momenta equal in the mixed terms and opposite in the two-creator or two-annihilator terms. The latter have coefficient $-E_p^2+{\mathbf{p}}^2+m^2=0$. The mixed terms giveThe constant $E_0$ is the divergent zero-point energy; a finite box and a momentum cutoff make its meaning explicit. Normal ordering sets the free vacuum energy to zero and leaves all commutators unchanged. Directly applying the mode commutators to the mixed expression yieldsChoose the Fock vacuum by $a(p)|0⟩=0$. Each creation operator adds a boson of energy $E_p$ and four-momentum $p$, while each annihilation operator removes one. Products of commuting creation operators give bosonic statistics from commuting creation operators and unrestricted occupation numbers. The real scalar field has one spin-zero species, identical to its antiparticle. Its one-particle states obey $⟨q|p⟩=(2\pi {)}^{3}2E_p{\delta }^{(3)}(\mathbf{p}-\mathbf{q})$, precisely the relativistic normalization of a one-particle state.

For the Feynman propagator, put $z=x-y$ and contract the two fields in the Fock vacuum. Only $a{a}^{†}$ survives, giving the Wightman functionThe stipulated time-ordered product convention is $i{\Delta }_F(z)=\theta (z^0)W(z)+\theta (-z^0)W(-z)$. In a contour integral over $p^0$, the Feynman i-epsilon prescription places the positive-energy pole just below the real axis and the negative-energy pole just above it. For $z^0>0$ close below, clockwise, obtaining $-i{e}^{-i{E}_p{z}^0}/(2E_p)$; for $z^0<0$ close above, obtaining $-i{e}^{i{E}_p{z}^0}/(2E_p)$. Changing $\mathbf{p}\mapsto -\mathbf{p}$ in the second contribution therefore provesActing with $\square +m^2$ multiplies the integrand by $-(p^2-m^2)$. The remaining factor tends to $-1$ as a distribution, so Fourier inversion givesThe sign also follows from the derivative jump of a free scalar time-ordered two-point function: the first time derivative of $i{\Delta }_F$ jumps by $⟨[\dot{\varphi },\varphi ]⟩=-i{\delta }^{(3)}$, so that of ${\Delta }_F$ jumps by $-{\delta }^{(3)}$.

In a globally static region, choose the time coordinate along the future timelike Killing vector field $K={\partial }_t$. A positive-frequency solution of the Klein-Gordon equation has $i{\mathcal{L}}_K{p}_i={\omega }_i{p}_i$, with ${\omega }_i>0$. Assume appropriate boundary conditions and a Cauchy surface so that the conserved Klein-Gordon inner product isFor a real scalar, take the negative-frequency modes to be $n_i=p_i^{\ast }$. Choose a complete normalized basis withThe negative norm of the conjugate modes is crucial; this is not a positive-definite inner product on all classical solutions.

Promote the real Klein-Gordon field to the operatorThese canonical commutation relations implement the canonical field and conjugate-momentum commutator. The static vacuum state in a stationary spacetime satisfies $a_i|0⟩=0$, and the Hamiltonian is a sum of oscillators, ${\sum }_i{\omega }_i(a_i^{†}{a}_i+1/2)$ before zero-point regularization. Continuous labels replace sums by correctly normalized integrals. For a complex charged scalar there are separate particle and antiparticle oscillators, with the same mode-mixing argument in each sector.

Choose independently normalized past and future bases, and use the matrix indices of the question. Complex conjugation of the future positive-frequency expansion immediately givesConservation of the Klein-Gordon inner product gives $A^{†}A-B^{†}B=I$ and $A^{†}{B}^{\ast }-B^{†}{A}^{\ast }=0$ in this column-index convention. These ensure preservation of the canonical commutation relations.

Let $a_j$ annihilate past modes and $b_i$ annihilate future modes. Extracting the future coefficient from the field gives the Bogoliubov transformationThe minus sign follows from $(n_j^{-},n_k^{-})=-{\delta }_{jk}$. In the in-vacuum, only $⟨0_{\mathrm{in}}|a_j{a}_k^{†}|0_{\mathrm{in}}⟩={\delta }_{jk}$ contributes to the expectation of the future number operator. ThereforeThis is particle number from Bogoliubov coefficients: the intervening time-dependent geometry makes a past positive-frequency mode contain a future negative-frequency component. In infinite volume, localized wave packets avoid delta-function normalization artifacts; a common Fock-space implementation needs the usual square-summability condition on the creation-mixing coefficients.

For gravitational collapse, the initial vacuum can be defined before the hole forms, while late outgoing modes are defined using the asymptotic stationary time at future null infinity. The complete future basis also includes modes falling through the event horizon; exterior outgoing modes alone are not a complete basis of all propagated data. Tracing an outgoing mode backwards near a nonextremal horizon gives the exponential ray relation $v=v_H-C{e}^{-\kappa u}$, where $u$ is late retarded time and $\kappa$ is the surface gravity. An outgoing factor $e^{-i\omega u}$ becomes proportional to $(v_H-v{)}^{i\omega /\kappa }$ in the early advanced coordinate, and contains both frequency signs when Fourier decomposed there. Their squared Bogoliubov-coefficient ratio is $e^{-2\pi \omega /\kappa }$; the canonical normalization then gives a Bose factor $(e^{2\pi \omega /\kappa }-1{)}^{-1}$, multiplied by the greybody factor, the transmission probability through the exterior potential. Thus Hawking radiation has temperature $T_H=\kappa /(2\pi )$ in units $\hslash =k_B=c=1$, with greybody factor modifications to the spectrum. This is a free-field calculation on a fixed or slowly evolving background; eventual backreaction is an additional problem, and the nonextremal exponential map must not be blindly used when $\kappa =0$.

Use units $\hslash =c=1$ and the Minkowski metric $g_{\mu \nu }=\mathrm{diag}(1,-1,-1,-1)$, so $p\cdot x=E_{p}t-\mathbf{p}\cdot \mathbf{x}$. The canonical momentum of the real scalar field is $\pi =\dot{\varphi }$, and the Hamiltonian operator isCanonical quantization of a real scalar field promotes $\varphi$ and $\pi$ to operator-valued distributions satisfying the equal-time canonical commutation relationProducts and operator commutators can first be interpreted after spatial smearing. An additive vacuum constant in $H$ does not affect the following equations. In the Heisenberg picture, $\dot{O}=i[H,O]$. Applying the canonical commutation relation to the quadratic Hamiltonian operator, and integrating the gradient term by parts, givesThus the Klein-Gordon equation is an operator equation. The spatial Fourier transform reduces it to ${\ddot{\varphi }}_{\mathbf{p}}+E_p^2{\varphi }_{\mathbf{p}}=0$, with $E_p=\sqrt{{\mathbf{p}}^2+m^2}$. Its two frequency branches, together with the Hermitian adjoint condition ${\varphi }^{†}=\varphi$, give the mode expansion of a free fieldThe measure $d\mu (p)$ is Lorentz invariant; its normalization fixes that of the creation and annihilation operators rather than imposing any extra physical assumption.

For invariant-normalized scalar mode extraction, take the Klein-Gordon inner product of $f_p$ with the field. The positive-frequency contribution has coefficient $E_p+E_q$ and spatial integral $(2\pi {)}^3{\delta }^3(\mathbf{p}-\mathbf{q})$. The negative-frequency contribution has coefficient $E_p-E_q$ and spatial integral $(2\pi {)}^3{\delta }^3(\mathbf{p}+\mathbf{q})$, so it vanishes because $E_{-p}=E_p$. ConsequentlyTaking the Hermitian adjoint, with the order of the scalar factors immaterial, yieldsEquivalently $a(\mathbf{p})=\int d^{3}x{f}_p^{\ast }(E_p\varphi +i\pi )$ and $a^{†}(\mathbf{q})=\int d^{3}y{f}_q(E_q\varphi -i\pi )$. At a common time their operator commutator therefore isThe same calculation gives $[a(\mathbf{p}),a(\mathbf{q})]=[a^{†}(\mathbf{p}),a^{†}(\mathbf{q})]=0$: the surviving factor is an energy difference supported at opposite spatial momenta. These are the bosonic canonical commutation relations in relativistic normalization of a one-particle state.

Choose the Fock vacuum by $a(\mathbf{p})|0⟩=0$ and $⟨0|0⟩=1$. Only the annihilator-creator product contributes to the Wightman function, givingUsing time ordering and the reverse product for $y^0>x^0$, the Feynman propagator, with the convention $D_F=i{\Delta }_F$ in this calculation, isIn particular the denominator in the spatial measure is $2E_p$ in both terms.

To verify the four-dimensional Fourier transform representation, let $\tau =x^0-y^0$ and integrate the energy variable first:The Feynman i-epsilon prescription puts the positive-energy pole at $E_p-i0$ and the negative-energy pole at $-E_p+i0$. For $\tau >0$, the contour integral closes in the lower half-plane with clockwise orientation. The residue theorem multiplies the residue $i{e}^{-i{E}_p\tau }/(2E_p)$ by $-i$, giving $e^{-i{E}_p\tau }/(2E_p)$. For $\tau <0$, it closes counterclockwise above: multiplying the residue $-i{e}^{i{E}_p\tau }/(2E_p)$ by $i$ gives $e^{i{E}_p\tau }/(2E_p)$. Thus $I(\tau )=e^{-i{E}_p|\tau |}/(2E_p)$. After inserting the spatial exponential, reversing $\mathbf{p}$ in the negative-time term gives exactly the time ordering expression above. HenceThis equality is understood as a distribution limit; the same prescription gives $(\square +m^2)D_F(x-y)=-i{\delta }^4(x-y)$.

Use a free real scalar field obeying a linear Klein-Gordon equation, for example $(\square -{\mu }^2-\xi R)\varphi =0$. This linearity is the assumption that permits mode evolution by a Bogoliubov transformation. Work with a complete set of normalized modes, or normalizable wave packets when the spectrum is continuous. The stationary asymptotic regions supply time translations relative to which positive-frequency solutions and their complex conjugates can be distinguished.

For solutions define the conserved Klein-Gordon inner productHere $\Sigma$ is a Cauchy surface with future-directed normal. Its conservation follows by applying the wave equation to the divergence of its current, provided boundary flux is included or vanishes. Let $u_j$ be a positive-frequency basis in the past and $v_s$ one in the future, each propagated as a solution through the intermediate geometry. Normalize them by $(u_i,u_j)={\delta }_{ij}$, $(u_i^{\ast },u_j^{\ast })=-{\delta }_{ij}$ and $(u_i,u_j^{\ast })=0$, with the corresponding identities for the future basis. The two field expansions areThe canonical commutation relations give $[a_i,a_j^{†}]={\delta }_{ij}$ and $[b_s,b_t^{†}]={\delta }_{st}$. The in-vacuum satisfies $a_j|0_{-}⟩=0$, while the out-vacuum satisfies $b_s|0_{+}⟩=0$. They are generally different vacua.

Fix the coefficient convention by expanding future modes in past modes:The minus sign is due to the negative norm of the conjugate modes. Taking the Klein-Gordon inner product with the field, which is antilinear in its first entry, givesBoth frequency sectors are needed: if $\beta \ne 0$, a future annihilation operator contains a past creation operator. This is the mechanism of particle number from Bogoliubov coefficients, not a classical nonzero mean field.

Computing $[b_s,b_t^{†}]$ and $[b_s,b_t]$ explicitly gives the canonical identities for a bosonic Bogoliubov transformation:Completeness also gives the inverse $a_j={\sum }_s({\alpha }_{sj}{b}_s+{\beta }_{sj}^{\ast }{b}_s^{†})$. Thus the evolved past vacuum, viewed in the future Fock space, obeysFor a finite-mode treatment with invertible $\alpha$, solve this condition by a multimode squeezed vacuum:The canonical identities make $Z$ symmetric. Commuting $b_s$ through the exponential gives $b_s|\Psi ⟩={\sum }_t{Z}_{st}{b}_t^{†}|\Psi ⟩$, which directly verifies the vacuum condition. Its normalization is $|C|=\det (I-Z{Z}^{†}{)}^{1/4}$. A general initial state is obtained by applying its initial creation-operator polynomial and expressing those operators in the future basis. In infinitely many modes, a common unitary Fock representation requires the creation-mixing coefficients to be Hilbert-Schmidt; finite wave-packet counts avoid treating delta-normalized modes as individual normalizable states. These are the bosonic mode mixing implementability conditions, rather than an assumption of arbitrary finite total creation over an infinite time interval.

For the requested number operator of one future mode, set $N_s=b_s^{†}{b}_s$. Expectations below are in the evolved past vacuum $|\Psi ⟩$, equivalently in $|0_{-}⟩$ using the Heisenberg-mode operators above. Acting on the past vacuum gives $b_s|0_{-}⟩=-{\sum }_j{\beta }_{sj}^{\ast }{a}_j^{†}|0_{-}⟩$, whose squared norm isThe mean particle number vanishes exactly when that mode has no negative-frequency mixing.

The second moment needs the anomalous contractionThe canonical commutation relations give $N_s^2=b_s^{†2}{b}_s^2+N_s$. Moreover,Its vacuum and two-particle pieces are orthogonal. Applying the commutation relations twice showsThe two pairings each contribute $n_s^2$ to the squared norm. Hence the full answer isThus $\mathrm{Var}(N_s)=n_s(n_s+1)+|c_s{|}^2$. These are the out-mode number fluctuations in the in-vacuum. Omitting $c_s$ would assume a property not given in the general question. A thermal single mode has $c_s=0$, giving $⟨N_s^2⟩=2n_s^2+n_s$. A single-mode squeezed vacuum state instead has $|c_s{|}^2=n_s(n_s+1)$, giving $⟨N_s^2⟩=3n_s^2+2n_s$.

If the counted observable is the total number over a finite set of future modes, its moments follow by retaining cross-mode correlations as well. Define $C_{st}=⟨b_s^{†}{b}_t⟩$ and $M_{st}=⟨b_s{b}_t⟩$. The same vacuum pairings give $⟨N_s{N}_t⟩=C_{ss}{C}_{tt}+|C_{st}{|}^2+|M_{st}{|}^2+{\delta }_{st}{C}_{ss}$. Thus $⟨N_{\mathrm{tot}}⟩=\mathrm{tr}C$ and $⟨N_{\mathrm{tot}}^2⟩=(\mathrm{tr}C{)}^2+\mathrm{tr}(C^2)+\mathrm{tr}(M^{†}M)+\mathrm{tr}C$, with $C_{st}={\sum }_j{\beta }_{sj}{\beta }_{tj}^{\ast }$ and $M_{st}=-{\sum }_j{\alpha }_{sj}^{\ast }{\beta }_{tj}^{\ast }$. Independent thermal counts cannot be assumed for a general correlated squeezed state.

For Hawking radiation, choose the incoming vacuum at past null infinity of a collapse spacetime and propagate late outgoing modes backwards. Near a nonextremal forming event horizon, the logarithmic tortoise-coordinate divergence leads to the Hawking exponential ray mapwhere $u$ is late retarded time, $U$ is an early regular affine null coordinate and $\kappa >0$ is the final surface gravity. A late positive-frequency mode $e^{-i\omega u}$ therefore becomes proportional to $(U_H-U{)}^{i\omega /\kappa }$ on rays escaping just before horizon formation. This is not positive-frequency with respect to $U$: it has both frequency signs and hence nonzero $\beta$.

The thermal factor can be seen directly. With $x=U_H-U>0$, $a=\omega /\kappa$ and a regulator $\epsilon >0$, the relevant Fourier integrals areAs $\epsilon \downarrow 0$, the arguments of the two complex factors approach $\pm \pi /2$. Their squared-modulus ratio for the suppressed and enhanced frequency branches is $e^{-2\pi a}$. Thus the thermal ratio of Hawking Bogoliubov coefficients, combined with the canonical normalization, yields the bosonic occupationin units $\hslash =c=k_B=1$. Scattering through the exterior potential multiplies the asymptotic occupation by the appropriate greybody factor.

A complete future mode description includes modes crossing the horizon as well as modes reaching future null infinity. The global incoming vacuum develops correlations between these sectors; restricting to the exterior gives approximately thermal outgoing occupation rather than a claim that the complete pure state became a thermal mixed state. This is the particle-production picture of a collapsing black hole, not an incoming thermal bath on an eternal geometry. Including backreaction turns the emitted positive energy into black-hole mass loss. The renormalized quantum stress tensor is not subject to the pointwise classical positivity used in the black-hole area theorem, so this does not contradict its classical proof.

Hawking radiation is a prediction of quantum field theory in curved spacetime, not of classical collapse alone. Consider a free massless Klein-Gordon field in the collapse vacuum for Hawking radiation: initially no incoming particles from past null infinity, with the short-distance state regular for freely falling observers through the forming event horizon. At late times use the approximately fixed exterior Schwarzschild metric; neglect backreaction during the observation interval. First set $G=c=\hslash =k_B=1$.

The exterior metric has $f(r)=1-2M/r$. Introduce the tortoise coordinate and outgoing and ingoing null coordinates,The surface gravity is $\kappa =1/(4M)$. The logarithm in $r_{\ast }$ makes a late escaping ray spend a long exterior coordinate time close to the event horizon. One of the regular outgoing Kruskal–Szekeres coordinates is proportional to $-e^{-\kappa u}$. Smooth propagation backwards through the collapsing body relates this regular coordinate to the incoming coordinate on past null infinity. If $v_H$ labels the last ray able to escape, the late-time Hawking exponential ray map is consequentlyHere the symbol $u(v)$ is the outgoing retarded time evaluated on the ray; the first equation is the leading asymptotic relation, with subleading corrections that vanish at late times. The constant $C$ depends on collapse history, whereas the exponential rate is determined by the final surface gravity. The relation follows from a nondegenerate Killing horizon; a linear ray map would not have the same effect.

An outgoing positive-frequency mode at future null infinity behaves as $p_{\omega }\sim e^{-i\omega u}/\sqrt{4\pi \omega }$, where $\omega >0$ is measured with respect to the asymptotic stationary time. In the near-horizon geometric optics approximation its backward-propagated form on past null infinity isThis is not purely positive frequency in $v$. Its decomposition into incoming positive-frequency modes $f_{{\omega }^{\prime }}\sim e^{-i{\omega }^{\prime }v}/\sqrt{4\pi {\omega }^{\prime }}$ and their complex conjugates is a Bogoliubov transformation:The coefficients follow from the Klein-Gordon inner product, or equivalently from the Fourier transform with the indicated flux normalization. Writing $y=v_H-v$ and $a=\omega /\kappa$, the two Fourier integrals, after a convergence regulator, areThe equality is the defining integral of the Gamma function, continued to these arguments from the right half-plane. Up to phases of unit modulus, $\alpha$ is $\sqrt{{\omega }^{\prime }/\omega }{I}_{+}/(2\pi )$ and $\beta$ is $\sqrt{{\omega }^{\prime }/\omega }{I}_{-}/(2\pi )$. Taking $ϵ\downarrow 0$ on the principal branches, the arguments of $ϵ\pm i{\omega }^{\prime }$ tend to $\pm \pi /2$. Thus $|I_{+}|\propto e^{\pi a/2}$ and $|I_{-}|\propto e^{-\pi a/2}$, giving the thermal ratio of Hawking Bogoliubov coefficientsIn particular, using $|\Gamma (1+ia){|}^2=\pi a/\sinh (\pi a)$,The collapse-dependent phases and $C$ disappear from the ratio. The canonical identities for a bosonic Bogoliubov transformation state that the positive-frequency norm minus the negative-frequency norm is one for a normalized outgoing mode. Combining that identity with the thermal ratio yields the late-time mean occupationThese are the Bose-Einstein distribution and the Hawking temperature. Strictly, monochromatic modes have delta-function normalization, and $\int d{\omega }^{\prime }|{\beta }_{\omega {\omega }^{\prime }}{|}^2$ contains the infinite-duration divergence $\int d{\omega }^{\prime }/{\omega }^{\prime }$. Normalized wave packets with a finite frequency band and a finite retarded-time window remove this artifact. For sufficiently late packets the occupation is the frequency average of the displayed thermal factor, tending to it for a narrow band. Fermionic fields instead give the corresponding denominator $e^{2\pi \omega /\kappa }+1$ because their canonical normalization uses anticommutators.

Four-dimensional propagation adds an important qualification to the word thermal. Separation of the massless Klein-Gordon equation into spherical harmonics gives an exterior radial scattering potentialOnly a fraction ${\Gamma }_{\ell }(\omega )$ of the outgoing mode is transmitted to future null infinity; this is the greybody factor. The scalar particle flux there isThe factor $2\ell +1$ counts the angular modes, and multiplication by $\omega$ gives the spectral energy flux. An observer at infinity therefore receives an outgoing thermal population filtered by curvature scattering, rather than an exact unfiltered blackbody spectrum. In physical units, for mass $M_{\mathrm{phys}}$,There is no incoming thermal bath in the collapse vacuum for Hawking radiation. The outgoing flux has positive Killing energy; its counterpart inside the event horizon can carry negative Killing energy. Including the expectation value of the stress-energy tensor in the gravitational evolution allows the mass to decrease. The fixed-background derivation describes the slowly evolving semiclassical regime and does not determine the endpoint of evaporation. Quantum expectation values need not satisfy the classical null energy condition, so this loss of area is not a contradiction of the classical second law of black-hole mechanics. The essential late-time universality is the combination of a regular initial quantum state and the exponential redshift at the newly formed event horizon.

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.