FLRW Fields and Mode Quantization

A covariant scalar on FLRW reduces to decoupled canonical oscillators only after the time coordinate and field rescaling are fixed. The Wronskian, not a choice of “positive frequency,” enforces the equal-time commutator.

Required background. Covariant scalar fields fixes $P=\Box+m^2+\xi R$, and canonical quantization on curved backgrounds fixes the symplectic form. Helpful background. Review adiabatic states and time-dependent particle creation.

From the covariant equation to canonical modes

For

$$ds^2=a^2(\eta)(d\eta^2-d\mathbf x^2),$$

the site d’Alembertian gives

$$\Box\phi=a^{-2}\left(\phi''+2\mathcal H\phi'-\nabla^2\phi\right), \qquad R_{\rm site}=-6\frac{a''}{a^3}.$$

Set $\chi=a\phi$. Fourier modes of $P\phi=0$ obey

$$v_k''+\left[k^2+a^2m^2-(1+6\xi)\frac{a''}{a}\right]v_k=0.$$

This is the convention check: $m=0$ and $\xi=-1/6$ remove the FLRW potential. Expand

$$\phi(\eta,\mathbf x)=\frac1{a(\eta)} \int\frac{d^3k}{(2\pi)^{3/2}} \left[a_{\mathbf k}v_k(\eta)e^{i\mathbf k\cdot\mathbf x} +a_{\mathbf k}^{\dagger}v_k^*(\eta)e^{-i\mathbf k\cdot\mathbf x}\right].$$

The canonical momentum is $\pi=a^2\phi'$. With

$$[a_{\mathbf k},a_{\mathbf q}^{\dagger}]=\delta^3(\mathbf k-\mathbf q), \qquad v_kv_k^{*\prime}-v_k'v_k^*=i,$$

one obtains $[\phi(\eta,\mathbf x),\pi(\eta,\mathbf y)]=i\delta^3(\mathbf x-\mathbf y)$. The mode equation conserves this Wronskian exactly. Parker’s oscillator construction makes the distinction between canonical normalization and time-dependent particles explicit Parker 1969, §§II–III, pp. 1059–1065.

First application: quantize a massive coupled scalar

Choose initial data at $\eta_0$,

$$v_k(\eta_0)=\frac1{\sqrt{2W_k}},\qquad v_k'(\eta_0)=\left(-iW_k-\frac{W_k'}{2W_k}\right)v_k(\eta_0),$$

with real positive $W_k$ and a specified ultraviolet expansion. Direct substitution gives the Wronskian $i$. Evolving the complex mode or two real fundamental solutions preserves it; numerical drift in

$$\epsilon_W(k,\eta)=\left|v_kv_k^{*\prime}-v_k'v_k^*-i\right|$$

is therefore a solver error, not particle production. The Gaussian two-point function follows from $v_k$ and any declared occupation/pairing covariance.

Symplectic basis changes and state data

Any second normalized solution may be written

$$\widetilde v_k=A_kv_k+B_kv_k^*, \qquad \lvert A_k\rvert^2-\lvert B_k\rvert^2=1.$$

This is a change of complex structure on the same real solution space. If the operators are transformed inversely, the field $\phi$ and its commutator are unchanged. If instead the original annihilation operators are retained while $\widetilde v_k$ is used as the positive-frequency mode, one has selected a different Gaussian state. A nonzero $B_k$ can therefore describe either new coordinates on one state or new state data, depending on what is held fixed.

For numerical work, evolve a real fundamental pair $f_k,g_k$ with initial matrix equal to the identity. Its determinant is the conserved symplectic area. A complex normalized mode is then a particular linear combination of this pair. This representation makes three checks independent: the differential-equation residual tests time evolution, the determinant tests canonical normalization, and the covariance eigenvalues test state positivity. Passing one does not imply the others.

The equal-time correlator also provides a dimensional check. Since $\phi=\chi/a$, the vacuum kernel contains $\lvert v_k\rvert^2/a^2$, whereas the canonical momentum correlator contains the properly differentiated combination $a^2\phi'=a(v_k'-\mathcal Hv_k)$ mode by mode. Omitting the $-\mathcal Hv_k$ term changes the canonical covariance and generally spoils the uncertainty relation even when the mode Wronskian itself remains exact.

For a real field, the $\mathbf k$ and $-\mathbf k$ sectors are conjugate rather than independent oscillators. Enforcing that reality condition and the stated Fourier measure prevents an otherwise common factor-of-two error in spectra, number densities, and stress mode sums.

The structure map places canonical modes before every state and observable choice. Inspect the branch where the same normalized modes support inequivalent Gaussian states.

Mode evolution plus the Wronskian fixes the canonical field algebra; positive frequency and particle number require additional state and basis data. Schematic; not to scale.

Domain and failure conditions

The chapter’s canonical domain table compares mode normalization with later observables. This derivation assumes a spatially flat four-dimensional FLRW patch, a smooth positive $a$, and the rescaled canonical variable $\chi=a\phi$.

Adversarial test. Change to cosmic time and a rescaled variable $q=a^{3/2}\phi$. Its oscillator frequency and Wronskian look different. Transform the canonical momentum and symplectic form as well: the field commutator and two-point function agree. An intermediate occupation obtained by diagonalizing either instantaneous Hamiltonian need not agree; only an asymptotic or detector-defined claim with controlled translation survives.

The failure map marks a variable-dependent frequency as insufficient evidence for production.

Canonical transformations preserve field observables when momentum and normalization are transformed; unqualified instantaneous particles do not share that invariance. Schematic; not to scale.

References

• Parker, L., “Quantized Fields and Particle Creation in Expanding Universes. I,” Physical Review 183, 1057–1068 (1969), doi:10.1103/PhysRev.183.1057.