Unimodular quantum cosmology in the connection representation: A minimal model

1. Problem Statement

The paper addresses the "problem of time" in canonical quantum gravity, where the Hamiltonian constraint typically annihilates physical states, preventing standard time evolution. The author investigates Unimodular Gravity (UG) as a conservative modification of General Relativity (GR) where the spacetime four-volume element is fixed. In UG, the Hamiltonian is no longer a constraint but generates time evolution, leading to a Schrödinger-type equation.

The specific problem tackled is the quantization of UG in the connection representation (using Ashtekar-Barbero variables) for a minimal cosmological model (homogeneous, isotropic, spatially flat, and matter-free). The goal is to isolate the specific quantum effects arising from the unimodular condition and determine how they alter the structure of canonical quantum cosmology, particularly regarding the cosmological constant ($\Lambda$) and the behavior of the wave function at zero volume.

2. Methodology

A. Classical Hamiltonian Analysis

• Variables: The theory uses the Ashtekar-Barbero connection ($A^i_a$) and densitized triad ($E^a_i$).
• Action: The starting point is the Holst action with a scalar field $\Lambda$ acting as the unimodular cosmological constant and a fixed scalar density $\alpha$.
• Symmetry Reduction: The model assumes a flat FLRW metric. The variables are reduced to a single pair of canonical variables: $c(\tau)$ (related to the connection) and $p(\tau)$ (related to the scale factor $a(\tau)$).
• Constraint Handling: The analysis identifies primary constraints ($p_N \approx 0, p_\Lambda \approx 0$) and secondary constraints (Hamiltonian constraint $\Phi \approx 0$ and unimodular constraint $\Theta \approx 0$).
  • Unlike standard GR, these constraints are second-class.
  • The author adopts the reduced phase space approach, explicitly solving the second-class constraints to eliminate dependent variables ($N$ and $\Lambda$) in favor of independent variables ($c$ and $p$).
• Reduced Hamiltonian: The resulting reduced Hamiltonian is $H = -3\mathcal{E} (k\beta^2)^{-1} |p|^{-1} c^2$, where $\mathcal{E}$ is a constant related to the fixed volume density. Crucially, $H$ is not a constraint; it generates time evolution.

B. Canonical Quantization

• Representation: Quantization is performed in the $p$-representation (where $\hat{p}$ is multiplicative and $\hat{c}$ is a derivative operator).
• Schrödinger Equation: The dynamics are governed by $i\hbar \partial_\tau \Psi = \hat{H} \Psi$.
• Operator Ordering: A symmetric factor ordering is chosen for the Hamiltonian operator $\hat{H} = -3\mathcal{E}(k\beta^2)^{-1} |\hat{p}|^{-1/2} \hat{c}^2 |\hat{p}|^{-1/2}$ to ensure hermiticity.
• Eigenvalue Problem: The stationary states satisfy $\hat{H}\Psi = \hbar\omega\Psi$, leading to a differential equation solvable in terms of Airy functions.

C. Semiclassical Analysis

• The WKB approximation is applied to the oscillatory sector ($\Lambda > 0$) to construct wave packets.
• The coherence of these wave packets is analyzed to understand the relationship between the spread of the cosmological constant and the geometry.

3. Key Contributions and Results

A. Incompatibility of Negative Cosmological Constant

• Result: The paper demonstrates that a negative cosmological constant ($\Lambda < 0$) is inconsistent within this minimal quantum model.
• Reasoning:
  • For $\Lambda < 0$, the solutions to the Schrödinger equation involve Airy functions that grow exponentially at large $|p|$.
  • To ensure the wave function is normalizable (suppressing exponential growth), one coefficient must be zero.
  • However, requiring the regularity of the operator $\hat{c}$ and the self-adjointness of the Hamiltonian at $p=0$ imposes a specific ratio between coefficients.
  • These two requirements are mutually exclusive for $\Lambda < 0$, forcing the wave function to vanish trivially ($\Psi = 0$).
• Implication: This suggests that in the original unimodular gravity formulation with this specific quantization, a negative $\Lambda$ is forbidden, mirroring the classical Friedmann equation impossibility for $\Lambda < 0$ in this context.

B. Vanishing Wave Function at Zero Volume (DeWitt Boundary)

• Result: For a positive cosmological constant ($\Lambda > 0$), the eigenstates are oscillatory and vanish at $p=0$ (zero spatial volume).
• Mechanism: This is not an imposed boundary condition (like the DeWitt boundary condition in standard Loop Quantum Cosmology) but a kinematic consequence of the unimodular constraint.
  • The unimodular constraint implies the lapse function $N \propto |p|^{-3/2}$, which diverges at $p=0$.
  • To solve the eigenvalue equation with this singular operator, the wave function must behave as $|p|^{1/2}$ near zero, forcing $\Psi(p=0) = 0$.
• Significance: This provides a natural mechanism for the wave function to vanish at the singularity without invoking quantum bounce dynamics.

C. Semiclassical Fluctuations and the Cosmological Constant Problem

• Result: The paper derives a relation for the relative fluctuation ($r$) of the spatial volume, lapse function, and the Hamiltonian eigenvalue (which corresponds to $\Lambda$).
  $$r \sim \frac{\hbar k}{\Lambda V_4}$$
  where $V_4$ is the spacetime four-volume.
• Interpretation:
  • In standard GR, $\Lambda$ is a fixed parameter. In UG, physical states are superpositions of different $\Lambda$ eigenvalues.
  • The coherence of a wave packet (semiclassicality) depends on the value of $\Lambda$ itself.
  • Applying this to the observable Universe (identifying $V_4$ with the Hubble volume and $\Lambda$ with the observed value), the relative fluctuation is extremely small ($r \sim 10^{-120}$).
• Clarification: The author emphasizes this does not solve the cosmological constant problem (it does not explain why $\Lambda$ is small). Rather, it shows that given the observed small $\Lambda$, the universe is naturally in a highly coherent semiclassical state. The smallness of $\Lambda$ ensures the suppression of the wave packet's spread.

4. Significance

1. Resolution of Time: The work successfully demonstrates how Unimodular Gravity resolves the problem of time in quantum cosmology by promoting the Hamiltonian to a generator of evolution, resulting in a standard Schrödinger equation.
2. Operator Regularity: It highlights a critical constraint in quantum cosmology: the requirement for operators to be well-defined (regular) and self-adjoint can strictly forbid certain classical solutions (like negative $\Lambda$) that might otherwise be considered.
3. Origin of Boundary Conditions: It provides a derivation for the vanishing of the wave function at zero volume based on the unimodular condition itself, rather than as an ad-hoc assumption.
4. New Perspective on Semiclassicality: It reframes the concept of semiclassicality in quantum gravity. Instead of just geometric fluctuations around a background, semiclassicality in UG is linked to the superposition of different cosmological constants. The observed universe's coherence is directly tied to the smallness of the cosmological constant.
5. Limitations and Future Work: The author notes that these results are specific to the "minimal" model (no matter, flat space). Future work is needed to determine if these features (exclusion of negative $\Lambda$, vanishing at zero volume) persist when matter fields or spatial curvature are included, or if they are artifacts of the symmetry reduction.

In summary, Yamashita's paper provides a rigorous connection-representation quantization of unimodular gravity, revealing that the unimodular condition fundamentally alters the quantum structure of cosmology, leading to a natural exclusion of negative $\Lambda$ and a kinematic resolution of the singularity, while offering a novel consistency check for the semiclassical nature of our universe.