Quantum Fermion Fields in FLRW Spacetimes

• Quantum fermion fields in FLRW metrics are described by Dirac and Majorana formulations on curved spacetimes, employing mode expansions and Hamiltonian formalism.
• The framework applies adiabatic regularization up to fourth order to achieve UV-finite stress-energy tensors, crucial for consistent backreaction analyses.
• Applications include analyzing particle production, vacuum polarization, and modifications from quantum gravitational effects in early-universe scenarios.

A quantum fermion field in a Friedmann–Lemaître–Robertson–Walker (FLRW) metric is the paper of fermionic degrees of freedom (primarily Dirac and Majorana fields) propagating and interacting on cosmological backgrounds described by the FLRW line element. This problem is central to understanding quantum matter in the early universe, the quantum structure of spacetime, and the emergence of macroscopic phenomena (such as particle production, vacuum polarization, and backreaction) due to quantum fields in a dynamical geometry. The technical treatments span canonical quantization, adiabatic regularization, mode expansions, backreaction mechanisms, and implications for cosmological observables.

1. Canonical and Quantum Field Theoretic Formalisms

The quantum treatment of fermion fields in FLRW metrics is built upon the Dirac equation in curved spacetime,

$i \gamma^\mu D_\mu \psi - m \psi = 0,$

where $\gamma^\mu$ are spacetime-dependent gamma matrices constructed from tetrads, and $D_\mu$ includes the spin connection. In spatially flat FLRW,

$ds^2 = dt^2 - a^2(t) (dx^2 + dy^2 + dz^2),$

the Dirac field is typically expanded in terms of spinor harmonics or plane wave modes. For closed FLRW spacetime, expansion in spinor harmonics on $S^3$ is employed, yielding mode-by-mode decoupling with time-dependent frequencies proportional to inverse scale factor and mode indices.

In the Hamiltonian formalism, this leads to mode-specific Hamiltonians,

$$H_{n,p} = N_0 \left[ \nu(x^0) (x \bar{x} + y \bar{y}) + m (y x + \bar{x} \bar{y}) \right],$$

where $\nu(x^0) = a^{-1}(x^0)(n + 3/2)$ and $(n,p)$ label the harmonics (Tavakoli et al., 10 May 2025). The quantum evolution for each mode is governed by a Schrödinger equation with time-dependent parameters reflecting both the expanding geometry and, in quantum cosmological models, the dressing by background quantum corrections.

2. Ultraviolet Regularization and Stress-Energy Renormalization

Expectation values of composite operators, notably the stress-energy tensor $T_{\mu\nu}$, require rigorous ultraviolet regularization. The adiabatic regularization approach, successfully extended to fermions (Rio et al., 2014), constructs mode functions as series in metric derivatives: $$h_k(t) \sim \sqrt{\frac{\omega - m}{\omega}} \exp\left[ -i \int \omega(t') dt' \right] \left[1 + \text{adiabatic corrections}\right],$$ with subtraction terms up to fourth adiabatic order guaranteeing UV-finiteness: $$T_{00}^{\text{ren}} = \frac{1}{2\pi^2 a^3} \int dk\, k^2 \left[P_k - P_k^{(0)} - P_k^{(2)} - P_k^{(4)} \right].$$ This framework is directly applicable to cosmological particle production, vacuum polarization, and backreaction calculations in radiation-dominated or inflationary scenarios.

At late times in a radiation-dominated universe, massive fermionic modes yield a stress-energy tensor with energy density scaling as $1/a^3$ and negligible pressure, thus behaving as classical cold matter. Massless fermion modes exhibit conformal behavior, with rapid redshifting and vanishing late-time energy densities (Rio et al., 2014).

3. Propagators and Mode Expansions

The momentum-space approach to Dirac fields in FLRW backgrounds allows for explicit construction of propagators, central to perturbative QFT calculations (Cotaescu, 2018). The field expansion

$$\psi(t, x) = \int d^3p \sum_\sigma \left[ U_{p,\sigma}(x) a(p, \sigma) + V_{p,\sigma}(x) b^\dagger(p, \sigma) \right]$$

employs fundamental spinors with time-dependent coefficients satisfying charge-conjugation and normalization.

A new integral representation for the Feynman propagator is formulated,

$$S_F(t_c, t_c', x) = \frac{1}{8\pi^4 (a(t_c)a(t_c'))^{3/2}} \int d^3p\, e^{ip\cdot x} \int_{-\infty}^{\infty} ds\, \mathcal{W}_s(t_c) \frac{\gamma^0 s - \gamma^i p^i}{s^2 - p^2 + i\epsilon} \mathcal{W}_s^\ast(t_c'),$$

with the auxiliary integration variable $s$ absorbing step function structure required for causal propagation.

For massless (neutrino) fields, the conformal invariance enables analytical propagator construction in any FLRW geometry—flat, open, or closed—via rescaling of the solution in Minkowski spacetime by the conformal factor $a^{-3/2}(t)$ (Cotaescu, 2018).

4. Quantum Cosmological Backreaction and Rainbow Metrics

Quantum corrections to the background geometry—particularly in loop quantum cosmology (LQC)—result in "dressed" metrics, where geometric operators (e.g., volume or scale factor) in the quantum gravitational sector acquire expectation values (Tavakoli et al., 10 May 2025). Fermionic mode evolution incorporates these corrections: $$i\hbar \partial_t \psi_{np} = [ \lambda_n \ell^{-2} \langle H_0^{-1/2} V^{2/3}(T) H_0^{-1/2} \rangle (x x^\dagger + y y^\dagger) + m \ell^{-3} \langle H_0^{-1/2} V(T) H_0^{-1/2} \rangle (y x + x^\dagger y^\dagger) ] \psi_{np}.$$ Born-Oppenheimer approximations further allow for the systematic inclusion of fermionic backreaction, with each mode inducing a shift in the global Hamiltonian constraint proportional to its occupation number and energy eigenvalue.

In the Planck regime, the sign of the fermionic mode occupation governs the modification of the minisuperspace potential and the timing and character of quantum cosmological bounces. Excited pair states lower the bounce density and introduce time-asymmetry, while vacuum occupation raises the barrier and delay the bounce. At large volumes, massive fermions yield a constant energy density component, acting as an emergent cosmological constant.

5. Torsionful Gravity and Vacuum Effects

Extensions involving torsionful backgrounds (Einstein–Cartan theory with square torsion term) introduce additional interaction channels for quantum fermion fields (Capolupo et al., 8 Oct 2025). The coupling between classical Dirac field sources and quantized fermion fields in the FLRW metric modifies both axial currents and energy-momentum tensors via quantum vacuum contributions.

Diagonalization of the non-free Hamiltonian via a Bogoliubov transformation creates vacuum states $|0_c(t_0)\rangle$ characterized by nonzero condensates: the spatial components of axial current and energy density scale as $C^{-4}(t)$ (with $C(t)$ the scale factor), significant during inflation or early cosmic epochs. Such quantum contributions alter the evolution iteratively through backreaction, with higher-order corrections speculated to impact the dark sector.

6. Fermion Mixing, Inequivalent Representations, and Phenomenological Implications

Quantum field theory of fermion mixing in curved backgrounds (notably FLRW) generalizes the Pontecorvo oscillation framework to include curvature-dependent Bogoliubov coefficients and unitarily inequivalent Hilbert spaces (Capolupo et al., 2023). The flavor vacuum, constructed via time-dependent Bogoliubov transformations of the mass vacuum, forms a condensate of particle-antiparticle pairs and is unitarily inequivalent to the mass vacuum even across different Cauchy surfaces.

Transition probabilities between flavors involve expectation values of charge operators in the flavor Fock space: $$P_{a \to b}(T, T_0) = \langle \psi_{a}(T_0) | :Q_b(T): | \psi_{a}(T_0) \rangle - \langle 0_F(T_0)| :Q_b(T): |0_F(T_0)\rangle.$$ Oscillation formulae incorporate curvature-dependent amplitude and phase modifications, with implications for cosmological neutrino oscillations, structure formation, and possible contributions to cosmological fluids.

7. Quantum Simulators and Nonperturbative Methodologies

Recent advancements utilize quantum-spin systems to emulate quantum field theories of Majorana fermions in curved spacetimes, including FLRW (Kinoshita et al., 10 Oct 2024). Via explicit "dictionaries" connecting spin model parameters (exchange couplings, magnetic fields) to spacetime metric functions ($\alpha$, $\gamma$, etc.), the transverse-field Ising model with time-dependent transverse field simulates expanding-universe particle production:

$$H = -\frac{1}{2\epsilon} \sum_j [ \sigma_j^x \sigma_{j+1}^x + (1 - \epsilon m a(\eta)) \sigma_j^z ],$$

where $a(\eta)$ is the (conformal time-dependent) scale factor. Measurements of multi-point spin correlators and entanglement Hamiltonians probe phenomena such as particle production and analogues of the Unruh effect, demonstrating experimental accessibility.

Summary Table: Major Quantum Effects for Fermion Fields in FLRW Metric

Aspect	Formalism/Result	Context/Significance
Stress-energy renorm.	Adiabatic regularization, mode sums (Rio et al., 2014)	UV-finite, essential for backreaction
Propagator structure	Integral representations, mode expansion (Cotaescu, 2018)	Analytical solutions, QFT calculations
Quantum geometry	LQC-dressed metrics, rainbow metrics (Tavakoli et al., 10 May 2025)	Fermion backreaction, cosmological const.
Torsion effects	Bogoliubov vacuum, axial current (Capolupo et al., 8 Oct 2025)	Early-universe, inflation, dark sector
Fermion mixing	Bogoliubov transformation, flavor vacuum (Capolupo et al., 2023)	Neutrino oscillations, cosmic fluids
Quantum simulation	Spin–Majorana dictionary (Kinoshita et al., 10 Oct 2024)	Experimental probes, nonperturbative phenomena

This topic integrates canonical quantization, semiclassical and quantum cosmology, backreaction mechanisms, propagator analysis, and new experimental approaches, with significant consequences for quantum structure, early universe physics, and the interpretation of cosmological observations. The regularization and dynamical dressing of quantum fields in FLRW backgrounds are essential for resolving singularities, producing realistic stress-energies, and determining the evolution of quantum matter in the universe.