Does Cosmology require Hermiticity in Quantum Mechanics?

We explore the consequences of allowing non-Hermitian structures in quantum cosmology by extending the Wheeler DeWitt framework beyond strictly Hermitian dynamics. Using a controlled semiclassical reduction, we show how anti Hermitian contributions propagate into both early universe primordial fluctuations and late-time structure growth as effective damping or gain terms. Confronting this framework with inflationary observables, growth of structure and the observed near flatness of the universe, we derive strong infrared constraints that suppress non Hermiticity across cosmic history. We demonstrate that these bounds are mutually consistent between early and late-time probes and can be partially relaxed in theories beyond General Relativity. Our results establish cosmology as a novel arena for testing foundational aspects of quantum mechanics and suggest that Hermiticity may emerge dynamically along the semiclassical branch describing our universe.

💡 Research Summary

The paper investigates whether the fundamental requirement of Hermiticity in quantum mechanics can be relaxed in the context of quantum cosmology. Starting from the standard Wheeler‑DeWitt (WDW) equation, the authors introduce a non‑Hermitian extension by adding an anti‑Hermitian operator Ĝ to the Hamiltonian constraint, yielding  Ĥ_NH = Ĥ_H + i Ĝ. This term represents a “gain‑loss” generator that, unlike the usual Hermitian part, does not conserve the standard inner‑product current on superspace.

In a minisuperspace truncation (scale factor a and a homogeneous scalar field φ), the modified WDW equation becomes a Klein‑Gordon‑type equation with a complex potential U_R + i U_I. In a WKB treatment the complex phase S = S_R + i S_I produces an exponential factor e^{−S_I/ħ} that acts as a source or sink for the semiclassical wave‑function. Performing a Born‑Oppenheimer split between heavy gravitational variables and light matter/perturbation variables, the anti‑Hermitian piece propagates into an effective non‑unitary Schrödinger equation for the light sector:

 d⟨ψ|ψ⟩/dt = −2ħ ⟨ψ|K(t)|ψ⟩,

where K(t) is directly related to Ĝ. Positive K produces damping, negative K produces amplification.

The authors then translate this abstract quantum‑mechanical effect into observable cosmological consequences. In the early‑universe (inflationary) regime, U_I modifies the primordial power spectrum by introducing a complex mass term for scalar perturbations, potentially altering the amplitude and tilt. Current CMB observations, however, demand that any such modification be extremely small, implying U_I must be highly suppressed during inflation.

In the late‑time universe, the same non‑unitary dynamics manifests as an extra friction (or anti‑friction) term γ(t) in the linear growth equation for matter density contrast δ_m:

 ¨δ_m +