Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 311 / 4

For complex solutions of the Klein-Gordon equation, define the currentIts divergence isbecause $\square \beta =m^2\beta$ and $\square \overline{\alpha }=m^2\overline{\alpha }$ for real $m^2$. The Klein-Gordon inner product is the flux ${\int }_{\Sigma }{n}_a{j}^{a}d\Sigma$. Applying Stokes theorem to a slab between two Cauchy hypersurfaces gives equality of the fluxes, with the future normal on each surface and opposite boundary orientations.

For noncompact surfaces this requires the integrals to converge and no net flux through spatial infinity, for example spacelike compact support or suitable decay. Global hyperbolicity alone does not ensure convergence for arbitrary solutions. Under these conditions,Complex conjugation also givesThus it is a Hermitian form, antilinear in its first argument and linear in its second. It is indefinite on the full complex solution space; positivity will arise only on a chosen positive-frequency solution subspace.

In Minkowski spacetime with signature $(-+++)$, $p\cdot x=-{\omega }_{\mathbf{p}}t+\mathbf{p}\cdot \mathbf{x}$, where ${\omega }_{\mathbf{p}}=p^0=\sqrt{|\mathbf{p}{|}^2+m^2}>0$. Hence $i{\partial }_t{\psi }_{\mathbf{p}}={\omega }_{\mathbf{p}}{\psi }_{\mathbf{p}}$: these are positive-frequency solutions relative to the inertial time translation.

On a constant-time Cauchy hypersurface, the Klein-Gordon inner product and the spatial Fourier transform identity giveAlso $({\overline{\psi }}_{\mathbf{p}},{\overline{\psi }}_{\mathbf{q}})=-{\delta }^3(\mathbf{p}-\mathbf{q})$ and $({\psi }_{\mathbf{p}},{\overline{\psi }}_{\mathbf{q}})=0$: the latter is proportional to $({\omega }_{\mathbf{p}}-{\omega }_{\mathbf{q}}){\delta }^3(\mathbf{p}+\mathbf{q})$, which vanishes. This is the Klein-Gordon plane-wave normalization.

For completeness, spatial Fourier transformation of the field equation gives $\ddot{\hat{\Phi }}(t,\mathbf{p})+{\omega }_{\mathbf{p}}^2\hat{\Phi }(t,\mathbf{p})=0$. Its solutions have the two branches $e^{\mp i{\omega }_{\mathbf{p}}t}$. Selecting the positive-frequency branch leaves precisely the Fourier superpositions of ${\psi }_{\mathbf{p}}$. These plane waves form a generalized basis, normalized with the Dirac delta distribution, rather than individually normalizable vectors. For a normalizable packet,

The Klein-Gordon inner product satisfies $(\overline{\alpha },\overline{\beta })=-(\beta ,\alpha )$, so $({\overline{\psi }}_i,{\overline{\psi }}_j)=-{\delta }_{ij}$. Orthogonality of positive and negative modes in the field expansion therefore gives $({\psi }_i,\Phi )=a_i$ and $({\overline{\psi }}_i,\Phi )=-a_i^{†}$.

Extracting the primed coefficient in the same way, and using antilinearity in the first entry,Consequently the Bogoliubov transformation of the creation and annihilation operators isThe minus sign comes from the negative norm of the conjugate modes, and the complex conjugates come from the first argument of the Hermitian pairing.

Use the canonical commutation relation for the creation and annihilation operators and the transformation derived above. Directly,Requiring these to equal ${\delta }_{ik}$ and zero, respectively, and conjugating the equations yields the canonical identities for a bosonic Bogoliubov transformation:The commutator of the primed creation operators then vanishes by taking adjoints. These are exactly the corresponding positive-mode norm and positive-negative orthogonality conditions for the Bogoliubov transformation. They are sufficient for these algebraic commutators whenever the sums are well defined. Infinitely many modes introduce further questions of convergence and whether the two choices admit a common bosonic Fock space representation; the matrix identities alone do not settle unitary implementability.

Choose the unprimed modes as the inertial positive-frequency modes in the far past, and the primed modes as those in the far future. In the in-vacuum and out-vacuum construction, $|0_{\mathrm{in}}⟩$ is annihilated by every $a_j$. The number of late-time quanta in mode $i$ is measured by $N_i^{\prime }=a_i^{\prime }{}^{†}{a}_i^{\prime }$. Expanding using the Bogoliubov transformation, the only surviving vacuum contraction is $⟨0_{\mathrm{in}}|a_j{a}_k^{†}|0_{\mathrm{in}}⟩={\delta }_{jk}$. Thus the particle number from Bogoliubov coefficients isIf $B\ne 0$, at least one row has positive norm and late-time observers detect particles. If $B=0$, each $a_i^{\prime }$ is a linear combination of past annihilation operators and annihilates the same state, so all these occupation numbers vanish. The intervening nonstationary geometry can mix positive and negative frequencies, realizing vacuum ambiguity in a nonstationary spacetime even though the asymptotic inertial choices are unambiguous.

For infinitely many modes, the expressions are understood with suitable convergence or wave-packet regularization. The total expected number is ${\sum }_{ij}|B_{ij}{|}^2$, finite precisely when $B$ is a Hilbert-Schmidt operator in the specified orthonormal mode spaces. A divergent sum is not a finite occupation number in a common vacuum representation; it signals the extra infinite-dimensional issue rather than changing the finite-mode calculation.