Bogoliubov Coefficients and Pair Creation

Bogoliubov coefficients compare two complete positive-frequency choices. The $\alpha$ coefficients measure overlap with the old positive-frequency modes, while $\beta$ measures overlap with their complex conjugates. This immediately determines transformed occupation numbers, but it does not by itself establish a globally unitary map between the two Fock spaces. That last statement is an infinite-dimensional Hilbert–Schmidt question.

Required background. Partial Differential Equations and Green Functions supplies the common Klein–Gordon solution space and its conserved inner product. Geodesics and Jacobi Fields provides the observer and asymptotic trajectories used to interpret the two time evolutions; no Jacobi-field estimate enters the algebra.

Helpful background. Flat-Spacetime Quantum Field Theory supplies canonical commutators and Fock vacua. Causality and Global Structure provides the asymptotic-region language. Quantum Fields in Curved Spacetime defines the fixed-background and state conventions.

Two normalized mode bases

Let $\{u_i\}$ and $\{v_A\}$ be complete complex mode bases for the same real Klein–Gordon equation on the same globally hyperbolic spacetime. To keep the algebra visible, take discrete labels; for continuous spectra every sum is replaced by the declared spectral integral and every Kronecker delta by a Dirac delta. With the $(-,+,+,+)$ convention,

$$(u_i,u_j)_{\mathrm{KG}}=\delta_{ij},\quad (\bar u_i,\bar u_j)_{\mathrm{KG}}=-\delta_{ij},\quad (u_i,\bar u_j)_{\mathrm{KG}}=0,$$

and the same relations hold for the $v$ basis. Expand

$$v_A=\sum_i\left(\alpha_{Ai}u_i+\beta_{Ai}\bar u_i\right).$$

Taking inner products gives the projection formulas

$$\alpha_{Ai}=(u_i,v_A)_{\mathrm{KG}}, \qquad \beta_{Ai}=-(\bar u_i,v_A)_{\mathrm{KG}}.$$

Thus $\beta$ is precisely the negative-frequency part of a new positive-frequency mode relative to the old split. The definition presupposes the same classical solution space, a conserved product, complete normalized bases, and specified boundary conditions. A coefficient without these data is not yet a particle-production prediction.

Canonical identities from inner-product preservation

Substitute the expansion into $(v_A,v_B)_{\mathrm{KG}}=\delta_{AB}$. Positive- and negative-norm terms enter with opposite signs, yielding

$$\sum_i\left(\alpha_{Ai}\bar\alpha_{Bi} -\beta_{Ai}\bar\beta_{Bi}\right)=\delta_{AB}.$$

Likewise, $(v_A,\bar v_B)_{\mathrm{KG}}=0$ gives

$$\sum_i\left(\alpha_{Ai}\beta_{Bi} -\beta_{Ai}\alpha_{Bi}\right)=0.$$

In matrix notation these are $\alpha\alpha^\dagger-\beta\beta^\dagger=1$ and $\alpha\beta^T-\beta\alpha^T=0$. They are not optional normalizations: they are exactly what makes the transformed field obey the canonical commutation relations. The inverse completeness identities must also hold when the bases are genuinely complete.

Writing the field in either basis,

$$\widehat\Phi =\sum_i\left(u_i a_i+\bar u_i a_i^\dagger\right) =\sum_A\left(v_A b_A+\bar v_A b_A^\dagger\right),$$

and projecting with $v_A$ gives

$$b_A=\sum_i\left(\bar\alpha_{Ai}a_i-\bar\beta_{Ai}a_i^\dagger\right).$$

Direct substitution verifies $[b_A,b_B^\dagger]=\delta_{AB}$ and $[b_A,b_B]=0$. This calculation fixes the conjugations and signs; changing the convention for which basis is expanded reverses the displayed transformation, so coefficients should never be copied without its defining equation.

Birrell and Davies provide a chapter-level synthesis of this mode algebra and its curved-spacetime applications; their Chapter 3 is used for orientation here, not as a theorem locator Birrell and Davies 1982, Chapter 3.

Occupations and pair correlations

Let $|0_u\rangle$ obey $a_i|0_u\rangle=0$. Then

$$\langle0_u|b_A^\dagger b_B|0_u\rangle =\sum_i\beta_{Ai}\bar\beta_{Bi}, \qquad \langle0_u|N_B|0_u\rangle =\sum_A\sum_i|\beta_{Ai}|^2,$$

whenever the sums exist. The anomalous correlator $\langle0_u|b_A b_B|0_u\rangle$ is also generally nonzero and is fixed by $\alpha$ and $\beta$. On a spatially homogeneous background, translation symmetry commonly restricts the mixing to momentum pairs $\mathbf k$ and $-\mathbf k$; the old vacuum is then a squeezed state in the new basis. This is the precise sense in which pair creation emerges from mode mixing, as developed for expanding universes by Parker Parker 1969, complete article, pp. 1057–1068.

The word creation is justified most cleanly when the geometry is asymptotically stationary in both the past and future: in-boundary conditions select $u_i$, out-boundary conditions select $v_A$, and an out detector can measure the corresponding excitations. Without such a physical selection, the equations compare two representations but do not privilege either particle vocabulary. Fulling’s flat-spacetime static-coordinate example shows why this qualification is essential Fulling 1973, complete article, pp. 2850–2862.

Modewise finiteness is not unitary implementability

For finitely many oscillators every canonical Bogoliubov transformation is implemented by a unitary squeeze operator. For infinitely many modes, the bosonic Shale condition adds a global requirement: the antilinear map $\beta$ must be Hilbert–Schmidt,

$$\|\beta\|_{\mathrm{HS}}^2 =\sum_{A,i}|\beta_{Ai}|^2<\infty.$$

For bounded symplectic transformations on the common one-particle completion, this condition is necessary and sufficient Shale 1962, Theorem 4.1, pp. 157–158. Only then is there a unitary $U$ on the original Fock space with $b_A=Ua_AU^{-1}$ after matching labels. Equivalently, the old vacuum contains a finite expected total number of new quanta. Hollands and Wald explain the associated criterion for unitary equivalence of pure quasifree representations and why inequivalent representations are normal in field theory Hollands and Wald 2015, expanded arXiv version, §2.1, Eqs. (26)–(27), pp. 17–18, PDF.

The logical distinction is sharp. Set $\beta_{nn}=n^{-1/2}$, $\alpha_{nn}=\sqrt{1+n^{-1}}$, and all off-diagonal entries to zero. Every coefficient is finite and each mode satisfies $|\alpha_{nn}|^2-|\beta_{nn}|^2=1$, yet

$$\|\beta\|_{\mathrm{HS}}^2=\sum_{n=1}^{\infty}\frac1n=\infty.$$

The transformation is therefore not unitarily implementable on the original Fock space. In a continuum, delta normalization and infinite spatial volume require equal care: a finite occupation density is not the same as a finite global Hilbert–Schmidt norm. Finite $\beta$ mode by mode never proves implementability.

Independent conformal check

For a massless conformally coupled scalar in four-dimensional spatially flat FLRW spacetime,

$$ds^2=a(\eta)^2(-d\eta^2+d\mathbf x^2), \qquad m=0,\quad \xi=\frac16,$$

the rescaled field $\chi=a\phi$ obeys the flat wave equation. Its normalized modes retain the phase $e^{-ik\eta}$ through the expansion. If the in and out choices are the conformally related vacua and no boundary spoils conformal invariance, there is no negative-frequency mixing: $\beta_{\mathbf k\mathbf k'}=0$. The canonical identities reduce to $\alpha\alpha^\dagger=1$, and both the out occupation and pair correlator vanish. This checks the formulas while also showing that expansion alone does not logically force particle creation.

The inertial/Rindler detector comparison supplies a complementary warning. Observers may describe the same detector transition with different mode particles while agreeing on the scalar transition probability Crispino, Higuchi, and Matsas 2008, arXiv version, §§III.A.1–5, pp. 19–25, PDF. Their $+---$ formulas have been translated here to the site’s $-+++$ convention. A detector click is therefore not, without further asymptotic structure, a measurement of a unique global $N$.

Check your understanding

Consider the diagonal infinite-mode transformation

$$b_n=\sqrt{1+n^{-1}}\,a_n-n^{-1/2}a_n^\dagger.$$

Check the canonical commutator for each $n$, compute the new occupation of each mode in $|0_a\rangle$, and decide whether a unitary on the $a$-Fock space implements all modes at once.

Solution

For each mode, the commutator is $[b_n,b_n^\dagger]=(1+n^{-1})-n^{-1}=1$, and different modes commute. Substitution in the $a$-vacuum gives $\langle b_n^\dagger b_n\rangle=n^{-1}$, which is finite for every fixed $n$. However, the Hilbert–Schmidt test requires the sum over all modes, and $\sum_n n^{-1}$ diverges. Therefore no single unitary on the original Fock space implements the full transformation. This counterexample proves why a mode-by-mode statement is insufficient.

References

• Birrell, N. D., and P. C. W. Davies, Quantum Fields in Curved Space, Cambridge Monographs on Mathematical Physics, Cambridge University Press (1982), Chapter 3, DOI.
• Crispino, Luis C. B., Atsushi Higuchi, and George E. A. Matsas, “The Unruh Effect and Its Applications,” Reviews of Modern Physics 80 (2008), 787–838, DOI, Open PDF.
• Fulling, Stephen A., “Nonuniqueness of Canonical Field Quantization in Riemannian Space-Time,” Physical Review D 7 (1973), 2850–2862, DOI.
• Hollands, Stefan, and Robert M. Wald, “Quantum Fields in Curved Spacetime,” Physics Reports 574 (2015), 1–35, DOI, Open PDF.
• Parker, Leonard, “Quantized Fields and Particle Creation in Expanding Universes. I,” Physical Review 183 (1969), 1057–1068, DOI.
• Shale, David. “Linear Symmetries of Free Boson Fields.” Transactions of the American Mathematical Society 103, no. 1 (1962): 149–167. DOI.