Complex Structures and One-Particle Spaces

A symplectic phase space does not yet say which classical oscillations should count as annihilation modes. A compatible complex structure makes that choice geometrically. Its compatibility preserves the canonical form, while its positivity turns the real solution space into a one-particle Hilbert space. The construction is powerful precisely because it exposes which ingredient is chosen rather than dictated by the field equation.

Required background. Flat-Spacetime Quantum Field Theory supplies positive-frequency modes and oscillator quantization, while Variational and Symplectic Geometry defines the real symplectic solution space on which the complex structure acts.

Helpful background. Partial Differential Equations and Green Functions provides the causal solution map behind the test-function quotient, and Causality and Global Structure establishes the globally hyperbolic setting in which Cauchy data and spacelike-compact solutions agree.

For immediate use, the causal propagator sends a compact source $f$ to the homogeneous spacelike-compact solution $Ef$; sources differing by $Ph$ represent the same solution. The symplectic space below is this quotient written in solution language.

Compatible complex structures

Let $(\mathcal V,\sigma)$ be the real symplectic space of scalar solutions, with all degeneracies already quotiented out. A compatible positive complex structure is a real-linear map $J:\mathcal V\to\mathcal V$ satisfying

$$J^2=-\mathbf1, \qquad \sigma(Jv,Jw)=\sigma(v,w), \qquad \mu_J(v,w):=\sigma(v,Jw), \qquad \mu_J(v,v)>0\quad(v\ne0).$$

The second condition says that $J$ is symplectic. Together with $J^2=-1$, it makes $\mu_J$ symmetric: $\sigma(w,Jv)=\sigma(v,Jw)$. The third condition is the nontrivial positivity requirement. Merely finding a real operator whose square is $-1$ is not enough.

For infinite-dimensional fields, topology matters as well. The displayed definition applies on a real phase space preserved by $J$; use $\mu_J(v,v)^{1/2}$ as the pre-Hilbert norm and complete when necessary. Do not assume that $J$ preserves the original space of spacelike-compact solutions. For example, on an ultrastatic background with positive spatial operator $A$, the ground-state choice acts on Cauchy data as $J(q,p)=(-A^{-1/2}p,A^{1/2}q)$. Fractional powers of $A$ are generally nonlocal and need not preserve compact support. One instead works in an appropriate completion, or uses the map $K$ below directly on compact-data solutions. Continuity and domains must be checked for the actual construction. The standard one-particle construction is summarized in Wald 1994, §§3.1–3.3, pp. 23–52.

The one-particle inner product

Declare multiplication by $i$ on the real vector space to be $J$. With the convention that Hilbert products are antilinear in the first entry, define

$$\langle Kv,Kw\rangle_{\mathcal H} =\frac12\bigl(\mu_J(v,w)+i\sigma(v,w)\bigr).$$

Here $K$ denotes the real-linear inclusion into the completion $\mathcal H$. The factor $1/2$ is a normalization, not decoration. It gives

$$2\operatorname{Im}\langle Kv,Kw\rangle_{\mathcal H}=\sigma(v,w),$$

so the Fock field $\Phi(v)=a(Kv)+a^*(Kv)$ will obey $[\Phi(v),\Phi(w)]=i\sigma(v,w)\mathbf1$.

The formula is genuinely Hermitian. Compatibility implies

$$\langle KJv,Kw\rangle =\frac12\bigl(\sigma(v,w)-i\mu_J(v,w)\bigr) =-i\langle Kv,Kw\rangle,$$

as required when the first argument is antilinear. Positivity of $\mu_J$ gives $\langle Kv,Kv\rangle>0$ for nonzero $v$. These calculations answer the defining question: $J^2=-1$, symplectic compatibility, positive $\sigma(\cdot,J\cdot)$, and suitable continuity are the conditions that turn a classical phase space into a one-particle quantum space.

More generally, one may start with a real-linear map $K:\mathcal V\to\mathcal H$ whose complex span is dense and which satisfies $2\operatorname{Im}\langle Kv,Kw\rangle=\sigma(v,w)$. This formulation includes one-particle structures not presented by a complete $J$ on the original topology. A positive complex structure is the clean route to a pure quasifree representation, not the definition of every mixed state.

Positive frequency as an eigenspace

Complexify the real solution space. The projectors

$$P_+=\frac12(\mathbf1-iJ), \qquad P_-=\frac12(\mathbf1+iJ)$$

select the $+i$ and $-i$ eigenspaces of $J$. With the Fourier convention $e^{-i\omega t}$, the $+i$ eigenspace is the positive-frequency subspace in the oscillator example below. Complex conjugation exchanges the two subspaces, so a real solution is recovered by combining its positive- and negative-frequency parts.

On a stationary spacetime a complete timelike Killing flow can provide a distinguished spectral splitting when the corresponding generator has a suitable positive-energy theory. On a generic time-dependent spacetime, the differential equation supplies no preferred $J$. Choosing initial data, an instantaneous Hamiltonian, an adiabatic prescription, or asymptotic conditions can each produce one, but these are extra structures. Kay and Wald emphasize the role and limits of symmetry-selected quasifree structures in Kay and Wald 1991, §§2–3, pp. 60–100.

Oscillator normalization and the Fourier sign

For one oscillator of frequency $\omega>0$, write a phase-space vector as $(q,p)$ and set

$$\sigma((q,p),(q',p'))=qp'-pq', \qquad J(q,p)=\left(-\frac p\omega,\omega q\right).$$

Then $J^2=-1$, $J$ preserves $\sigma$, and

$$\mu_J((q,p),(q,p))=\omega q^2+\frac{p^2}{\omega}>0.$$

Hamiltonian evolution is

$$T_t=\cos(\omega t)\mathbf1-\sin(\omega t)J.$$

If $Jv=iv$, this becomes $T_tv=e^{-i\omega t}v$, exactly matching the site’s Fourier convention. Changing the sign of $J$ would exchange positive and negative frequency and would make $\mu_J$ negative with the present symplectic orientation. The three compatibility checks therefore also fix the convention-sensitive sign.

The normalized complex amplitude

$$z=\sqrt{\frac\omega2}\,q+\frac{i}{\sqrt{2\omega}}p$$

satisfies $\lvert z\rvert^2=\mu_J(v,v)/2=\langle Kv,Kv\rangle$. It is the one-mode coordinate represented by a creation operator acting on the Fock vacuum.

Comparing two choices

Let $J_1$ and $J_2$ be two positive compatible complex structures. They define the same abstract CCR algebra because $\sigma$ has not changed, but generally define different notions of annihilation operator, vacuum, and particle number. Relative to $J_1$, the change of splitting has a complex-linear part $\alpha$ and an antilinear part $\beta$. Every finite-dimensional change is unitarily implementable, but a field requires a global summability test.

For a diagonal tower of oscillator modes, take real squeezing parameters $r_n$ so that

$$\alpha_n=\cosh r_n, \qquad \beta_n=\sinh r_n, \qquad \lvert\alpha_n\rvert^2-\lvert\beta_n\rvert^2=1.$$

As an abstract example, set $r_n=n^{-1/2}$. Every individual $\beta_n$ is finite and tends to zero. Nevertheless $\sum_n\lvert\beta_n\rvert^2$ diverges because $\sinh^2(n^{-1/2})\sim1/n$. Thus modewise mild mixing does not establish unitary equivalence of the full Fock representations. This oscillator tower does not assert that a particular spacetime realizes these coefficients; an actual geometry requires solving its mode equations. The precise Hilbert–Schmidt theorem is developed in Bogoliubov Implementability.

This example also prevents a common category error: inequivalent representations do not mean different canonical algebras. The algebra is fixed by $\sigma$; the state selects its GNS representation; $J$ is one efficient way to select a pure quasifree state.

Exercise: test a proposed complex structure

On the oscillator phase space above, consider $J_c(q,p)=(-cp,q/c)$ for a positive constant $c$. Does this define a compatible positive complex structure, and which value reproduces the frequency-$\omega$ choice?

Solution

Yes. Applying the map twice gives $J_c^2(q,p)=(-q,-p)$, and direct substitution shows $\sigma(J_cv,J_cw)=\sigma(v,w)$. Moreover, $\sigma(v,J_cv)=q^2/c+cp^2$, which is strictly positive for nonzero $v$ precisely when $c>0$. These are three independent tests: the square produces complex multiplication, symplectic compatibility preserves the canonical bracket, and positivity supplies the Hilbert norm. Comparing $(-cp,q/c)$ with $(-p/\omega,\omega q)$ gives $c=1/\omega$. Thus positivity alone permits a family, while the oscillator dynamics selects the member compatible with its frequency and the chosen positive-frequency convention.

References

• Kay, Bernard S., and Robert M. Wald. “Theorems on the Uniqueness and Thermal Properties of Stationary, Nonsingular, Quasifree States on Spacetimes with a Bifurcate Killing Horizon.” Physics Reports 207, no. 2 (1991): 49–136. DOI 10.1016/0370-1573(91)90015-E.
• Wald, Robert M. Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics. Chicago Lectures in Physics. University of Chicago Press, 1994. Publisher record.