Positive Frequency and Particle Number

A particle is not specified by the Klein–Gordon equation alone. It is specified only after the real solution space has been split into positive- and negative-frequency parts, or equivalently after a compatible complex structure has been chosen. A stationary time translation or an asymptotic stationary region can sometimes make that choice natural; on a generic time-dependent spacetime it does not. The resulting number operator is therefore a property of a field representation and a mode split, not a local scalar attached to the quantum state.

Required background. Partial Differential Equations and Green Functions supplies the well-posed solution space and conserved flux used below. Geodesics and Jacobi Fields provides normalized observer tangents and proper-time evolution; no geodesic-deviation result is needed here.

Helpful background. Flat-Spacetime Quantum Field Theory supplies Fock-space creation and annihilation operators. Causality and Global Structure provides the distinction between a local observer and a global or asymptotic time translation. Quantum Fields in Curved Spacetime defines the prescribed-background setting and its state dependence.

The symplectic solution space

Let $S$ be the real vector space of smooth real solutions of

$$P\phi=(\Box-m^2-\xi R)\phi=0$$

with suitably compact Cauchy data on a globally hyperbolic spacetime. If $\Sigma$ is a Cauchy surface with future-directed unit normal $n^a$, the conserved symplectic form is

$$\sigma(\phi_1,\phi_2) =\int_\Sigma d\Sigma\,n^a \bigl(\phi_1\nabla_a\phi_2-\phi_2\nabla_a\phi_1\bigr).$$

Conservation makes this expression independent of the chosen Cauchy surface. This is the bounded PDE input: global hyperbolicity supplies evolution from Cauchy data, and the field equation makes the flux closed. It does not select which real solutions should be called positive frequency.

For complex solutions define the Klein–Gordon product

$$(u,v)_{\mathrm{KG}} =i\int_\Sigma d\Sigma\,n^a \bigl(\bar u\nabla_a v-v\nabla_a\bar u\bigr).$$

With the site signature $(-,+,+,+)$, a normalized Minkowski mode proportional to $e^{-i\omega t}$ has positive norm. Complex conjugation reverses the norm, so simply complexifying $S$ has not yet produced a one-particle Hilbert space.

A complex structure selects positive frequency

A compatible complex structure acts on a real symplectic completion of the solution space, with the compact-data solutions as its test subspace. This completion matters: a positive-frequency split can be spatially nonlocal, so $J$ need not preserve compact support. On its invariant domain, $J$ satisfies

$$\begin{gathered} J^2=-1,\qquad \sigma(J\phi_1,J\phi_2)=\sigma(\phi_1,\phi_2),\\ \mu_J(\phi_1,\phi_2)=\sigma(\phi_1,J\phi_2), \qquad \mu_J(\phi,\phi)\gt0\quad(\phi\ne0). \end{gathered}$$

The positivity condition is diagonal; $\mu_J(\phi_1,\phi_2)$ need not be positive for distinct arguments. After complexification, $J$ has $+i$ and $-i$ eigenspaces,

$$S_{\mathbb C}=S_J^+\oplus S_J^-, \qquad S_J^- = \overline{S_J^+}.$$

One harmonic oscillator makes the construction tangible. For initial data $(q,p)$, frequency $\omega>0$, and $\sigma((q,p),(q',p'))=qp'-pq'$, choose

$$J(q,p)=(-p/\omega,\omega q), \qquad \sigma((q,p),J(q,p))=\omega q^2+p^2/\omega.$$

Then $J^2=-1$ and the last expression is positive. The positive-frequency data $(1,-i\omega)$ satisfy $J(1,-i\omega)=i(1,-i\omega)$ and evolve as $e^{-i\omega t}$. Thus the abstract choice of $J$ really selects which oscillation counts as an annihilation mode.

We call $S_J^+$ positive frequency and complete it in its positive Klein–Gordon norm to obtain the one-particle Hilbert space $\mathcal H_J$. Thus $J$, the positive-frequency subspace, the pure quasifree vacuum, and its Fock representation are different descriptions of the same extra choice. Hollands and Wald emphasize that the algebra and its states can be defined without first selecting such a preferred representation; that is crucial because a generic curved spacetime provides no preferred vacuum criterion analogous to Poincaré invariance Hollands and Wald 2015, expanded arXiv version, §2.1, pp. 9–20, PDF.

Choose an orthonormal basis $\{u_j\}$ of $\mathcal H_J$,

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

where sums become integrals and Kronecker deltas become Dirac deltas for continuous labels. In the corresponding Fock representation,

$$\widehat\Phi(x)=\sum_j\bigl(u_j(x)a_j+\bar u_j(x)a_j^\dagger\bigr), \qquad N_J=\sum_j a_j^\dagger a_j.$$

The vacuum $|0_J\rangle$ obeys $a_j|0_J\rangle=0$. A unitary change of basis within $S_J^+$ changes the labels of the modes but not the total positive-frequency subspace or total number. A transformation that mixes $S_J^+$ with $S_J^-$ changes the complex structure and hence changes the particle interpretation. The infinite sum defining $N_J$ is an unbounded, global operator and requires its natural domain; it is not a locally covariant field.

When geometry makes the split natural

Suppose a region admits a complete timelike Killing field $K^a$ and the Klein–Gordon dynamics has a positive self-adjoint generator there. Modes can then be chosen as spectral modes of the Killing flow,

$$i\mathcal L_K u_{\omega\lambda}=\omega u_{\omega\lambda}, \qquad \omega>0,$$

where $\lambda$ denotes the remaining quantum numbers. This realizes the convention $e^{-i\omega t}$. A static ground-state condition can then distinguish a useful $J_K$. Likewise, if the geometry becomes stationary in the remote past or future, separate in- and out-splits may be defined by their asymptotic boundary conditions.

These qualifications matter. A Killing field that becomes spacelike does not define positive energy everywhere; a merely coordinate time is not a physical selector; and two inequivalent asymptotic regions can select different splits. Fulling’s static-coordinate construction on a portion of flat spacetime is the classic demonstration that a different time evolution can lead to a different Fock particle interpretation even without curvature Fulling 1973, complete article, pp. 2850–2862. The inertial/Rindler comparison develops the same lesson operationally: the mode decomposition changes while invariant detector probabilities must agree when calculated consistently Crispino, Higuchi, and Matsas 2008, arXiv version, §§III.A.1–5, pp. 19–25, PDF. Their formulas use the opposite metric signature; the products and mode phases here have been translated to $(-,+,+,+)$.

Number is relative, not nonexistent

Saying that particle number is relative does not mean it is meaningless. Once the geometry, boundary conditions, normalized modes, representation, and measurement question are fixed, $N_J$ is a precise observable in that representation. In an asymptotic scattering problem it can encode an out-particle yield; in a stationary cavity it can describe quanta of the cavity Hamiltonian. Birrell and Davies give a useful chapter-level synthesis of these constructions, but their Chapter 3 is not being used here as a theorem locator Birrell and Davies 1982, Chapter 3.

What fails is the inference that every curved spacetime or every local experiment possesses one basis-independent total particle number. Local correlation functions can be compared without choosing $J$, and a detector response additionally depends on its trajectory and coupling protocol. The next page makes the change of split quantitative through Bogoliubov coefficients.

Check your understanding

Let $u$ be a normalized positive-frequency mode and define a second normalized candidate by

$$v=\cosh r\,u+\sinh r\,\bar u, \qquad r\in\mathbb R.$$

Check its Klein–Gordon norm and compute the $v$-particle occupation in the $u$-vacuum. Which input, rather than the state itself, has changed?

Solution

Because $(u,u)_{\mathrm{KG}}=1$, $(\bar u,\bar u)_{\mathrm{KG}}=-1$, and the cross terms vanish, $(v,v)_{\mathrm{KG}}=\cosh^2r-\sinh^2r=1$. The associated annihilator is $b=\cosh r\,a-\sinh r\,a^\dagger$. Hence $b|0_u\rangle=-\sinh r\,a^\dagger|0_u\rangle$ and $\langle0_u|b^\dagger b|0_u\rangle=\sinh^2r$. This calculation also checks the zero-squeezing limit: at $r=0$, the two modes and both number operators coincide. For nonzero $r$, the state remained the same vector; what changed was the positive-frequency subspace, equivalently the complex structure and its number operator. The nonzero answer therefore does not diagnose a change in the field state by itself.

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.