When the Source Is a Field, Not a Particle: Gravitational Decoherence for Quantum Field Theory

Paper L: The Second-Quantized Diosi-Penrose Formalism from the Quantum-Geometric Duality Series

A Formula Built for the Wrong Objects

The Diosi-Penrose rate

$\Gamma_{\mathrm{DP}} = \frac{G M^2}{\hbar\, d}$

is the workhorse of every prediction in this series. It is also a first-quantized formula. Strip the notation away and what it describes is a single rigid mass $M$ occupying two positions separated by $d$. The Universe does not actually contain such objects. It contains quantum fields, whose excitations are photons, phonons, atoms, and primordial curvature perturbations. A photon Fock-state superposition $(\ket{n}+\ket{m})/\sqrt{2}$ does not have a position to displace and a mass to substitute into the numerator.

The Diosi master equation knows about this. Its coupling is not to position but to the mass density operator $\hat{\mu}(\mathbf{x})$. The natural relativistic generalization is

$\hat{\mu}(\mathbf{x}) \;\longrightarrow\; \frac{\hat{T}^{00}(\mathbf{x})}{c^2}\,.$

Energy density, not just rest mass, sources the gravitational field. Once this substitution is taken seriously, the master equation becomes well-defined for any superposition of field states whose stress-energy expectations differ between branches.

The Master Formula

The decoherence rate for a superposition $(\ket{\Psi_1}+\ket{\Psi_2})/\sqrt{2}$ with stress-energy expectation values $\langle\hat{T}^{00}\rangle_{1,2}$ is

$\Gamma = \frac{G}{\hbar c^4}\int d^3x\,d^3y\;\frac{\Delta T^{00}(\mathbf{x})\,\Delta T^{00}(\mathbf{y})}{|\mathbf{x}-\mathbf{y}|}\,,$

with $\Delta T^{00} = \langle\hat{T}^{00}\rangle_1 - \langle\hat{T}^{00}\rangle_2$. This is the gravitational self-energy of the energy-density difference, divided by $\hbar$. It is the same physics as $GM^2/(\hbar d)$: two branches carry distinguishable gravitational fields, the overlap decays, and the self-energy sets the rate.

The pointer basis is worth pausing on. The Diosi kernel couples to $\hat{T}^{00}(\mathbf{x})$, so eigenstates of the energy density are immune to decoherence. For point particles, energy density is concentrated at $\hat{\mathbf{X}}$, so the pointer basis is position. For fields, it is energy-momentum eigenstates or quasi-classical configurations --- not field-amplitude eigenstates. The same mechanism that picks out positions for billiard balls picks out classical waveforms for fields.

Fock States and the Consistency Check

For a single-mode Fock superposition $(\ket{n}+\ket{m})/\sqrt{2}$ in a cubic box of side $L$, the energy density difference is spatially uniform, and the only non-trivial step is a geometric integral over the box:

$\int_{\mathrm{cube}}\!\!\int_{\mathrm{cube}} \frac{d^3x\,d^3y}{|\mathbf{x}-\mathbf{y}|} = C_{\mathrm{cube}}\,L^5\,, \qquad C_{\mathrm{cube}} \approx 1.192\,.$

(Chandrasekhar evaluated this in 1969 for self-gravitating cubes; numerical agreement is good to many digits.) The result is exact:

$\Gamma_{\mathrm{Fock}} = \frac{G\,(n-m)^2\,\hbar\,\omega^2\,C_{\mathrm{cube}}}{c^4\,L}\,.$

The rate scales as $(n-m)^2$, as $\omega^2$, and as $1/L$. The numbers are sobering. An optical photon in a meter-scale cavity decoheres at $\sim 10^{-48}$ s$^{-1}$, a decoherence time some forty orders of magnitude longer than the age of the Universe. Microwave photons and GHz phonons are worse. Gravitational decoherence does not threaten any photonics experiment ever conceived.

The consistency check is what justifies the whole construction. Take a single massive particle: $n=1$, $m=0$, $\omega = mc^2/\hbar$, $L \sim d$. Plug in:

$\Gamma = \frac{G\,\hbar\,(mc^2/\hbar)^2\,C_{\mathrm{cube}}}{c^4\,d} = \frac{G\,m^2\,C_{\mathrm{cube}}}{\hbar\,d}\,.$

The Diosi-Penrose rate is recovered exactly, with $C_{\mathrm{cube}} \to 1$ in the localized-wavepacket limit. The QFT extension is not a new theory. It is the same theory done correctly for fields.

Where It Matters: Massive Bosons and Cosmology

Coherent and squeezed states obey the same family of formulas, with $n \to |\alpha|^2$ for coherent superpositions and a factor of $\sinh^4(r)$ for squeezed-vs-vacuum superpositions. Two cases stand out.

For a Bose-Einstein condensate of $N$ atoms of mass $m$ in a trap of size $L$, a Schrodinger-cat state with $N$ versus $0$ atoms decoheres at $\Gamma_{\mathrm{BEC}} = G N^2 m^2 C_{\mathrm{cube}}/(\hbar L)$. The $N^2$ scaling is the many-body enhancement that has been the target of cold-atom interferometry for two decades. The Fock-state derivation reproduces it from first principles without invoking any new physics.

The headline case is inflation.

The Inflationary Application

Quantum scalar perturbations during inflation begin in the Bunch-Davies vacuum and, after horizon crossing, are squeezed by the de Sitter parametric amplifier. The squeezing parameter grows linearly with the number of e-folds elapsed: $r_k \approx N_k$. The mean occupation number grows exponentially, $\bar{n}_k \approx e^{2N_k}/4$. By the time a CMB-scale mode has been outside the horizon for fifty e-folds, it contains something like $10^{40}$ inflatons in a single Fourier mode.

The standard story stops here: enormous occupation numbers, the argument goes, make the perturbations "effectively classical." This conflates classicality of expectation values with classicality of the quantum state, and it does not work. A highly squeezed state is still a pure state, and the off-diagonal density-matrix elements do not disappear by themselves. A decoherence mechanism is required, and inflationary cosmology has been searching for a convincing one for thirty years.

The QFT-extended master equation supplies it. Apply the rate formula to a single squeezed mode in a Hubble volume, with $\omega_k \approx H_{\mathrm{inf}}$ and $L \sim c/H_{\mathrm{inf}}$. The energy fluctuation is $\Delta E_k \sim \hbar H_{\mathrm{inf}} \bar{n}_k$, and the rate becomes

$\frac{\Gamma_k}{H_{\mathrm{inf}}} \;\sim\; \varepsilon_{\mathrm{grav}} \cdot \frac{e^{4 N_k}}{16}\,, \qquad \varepsilon_{\mathrm{grav}} \equiv \left(\frac{\hbar H_{\mathrm{inf}}}{E_P}\right)^{\!2}\,.$

The dimensionless parameter $\varepsilon_{\mathrm{grav}}$ is tiny --- $\sim 10^{-12}$ for GUT-scale inflation --- but the squeezing factor $e^{4N_k}$ grows fast. Setting $\Gamma_k = H_{\mathrm{inf}}$ and solving:

$N_{\mathrm{dec}} \approx \frac{1}{4} \ln\!\left(\frac{16}{\varepsilon_{\mathrm{grav}}}\right)\,.$

For GUT-scale inflation, $N_{\mathrm{dec}} \approx 7.7$. Eight e-folds after a mode crosses the horizon, its quantum coherence is gone. Since observable CMB modes spend fifty to sixty e-folds outside the horizon during inflation alone, classicalization occurs vastly earlier than recombination, vastly earlier than re-entry, for every mode we can see.

The logarithmic dependence on $\varepsilon_{\mathrm{grav}}$ is striking. Across twelve orders of magnitude in the inflationary energy scale, $N_{\mathrm{dec}}$ varies by less than a factor of two. The mechanism is robust against ignorance about which inflationary model the Universe ran.

What Is Established, What Is Estimated

The Fock-state rate is exact. The consistency reduction to $GM^2/(\hbar d)$ is exact. These are theorems given the master equation.

The inflationary number is a motivated estimate. Three approximations control its $O(1)$ coefficient. The Diosi kernel $1/|\mathbf{x}-\mathbf{y}|$ is the flat-space Newtonian propagator, not the de Sitter Green function; on super-Hubble scales these differ. The stress-energy $T^{00}$ is gauge-dependent, while the gauge-invariant comoving curvature perturbation $\zeta$ should ultimately replace it in a covariant formulation. The Markov approximation enters at intermediate $N_k$. None of these affects the parametric scaling $\varepsilon_{\mathrm{grav}} e^{4N}$; all of them can shift the prefactor by an $O(1)$ amount.

There is also a striking concordance to note. Kiefer, Polarski, and Starobinsky derived classicalization from environmental decoherence --- tracing out short-wavelength modes to decohere long-wavelength ones --- and found the same parametric scaling, $\Gamma/H \sim \varepsilon_{\mathrm{grav}}$. The mechanism here is different: self-gravity of a single mode, no environment required. That two independent channels produce the same parametric answer suggests the underlying physics is the Hamiltonian constraint of general relativity, the same constraint that drove the $G^1$ result of Paper K.

What the Universe Looks Like Under This Picture

Primordial perturbations are quantum-mechanically generated, classically observed, and the bridge between is built by self-gravity. The power spectrum is unaffected --- decoherence kills off-diagonal density matrix elements, not diagonal ones --- which explains why the standard inflationary predictions work despite ignoring decoherence entirely. Bell-inequality violations between modes are wiped out long before any observation could detect them. The CMB is statistically classical because gravity made it so.

It also closes a small loop. Paper A introduced gravitational decoherence as the answer to "why are big things classical?" Paper K derived it from the Wheeler-DeWitt constraint. Paper M gave its tabletop signature. Paper L says the same mechanism, applied to fields, classicalizes the largest structures we can observe, using the same parametric coupling $\varepsilon_{\mathrm{grav}}$. One formula, point particles to primordial perturbations.

The next question is whether the de Sitter kernel and a gauge-invariant treatment of $\zeta$ confirm the $O(1)$ coefficient or shift it. The parametric answer is locked in. The numerical answer is the work that comes after.

This is Paper L of the Quantum-Geometric Duality series, extending the Diosi master equation from point masses to quantum fields and applying the resulting formalism to the classicalization of inflationary perturbations.