Emergent time and more from wavefunction collapse in general relativity

Abstract

In this paper, we further develop a recently proposed theory of time based on wavefunction collapse in general relativity. It is based on the postulations that quantum states, which violate the momentum and Hamiltonian constraints, represent instances of time, and stochastic fluctuations of the lapse and shift generate the time evolution under which an initial state gradually collapses toward a diffeomorphism-invariant state. Under the wavefunction collapse, the scale factor monotonically increases, thus acting as a clock. The scalar, vector, and tensor gravitons arise as physical excitations, and the arrow of time for their evolution is set by the initial state. In the long-time limit, the tensor gravitons exhibit emergent unitary dynamics. However, the extra modes are strongly damped due to the non-unitary dynamics that suppress the constraint-violating excitations. The vector mode is uniformly suppressed over all length scales, but the decay rate of the scalar is proportional to its wave vector. This makes the latter a viable candidate for dark matter; excitations with large wavelengths survive over long periods, contributing to long-range interactions, while the fast decay of short-wavelength modes renders them undetectable without sufficient temporal resolution. These are demonstrated for the cosmological constant-dominated universe through semi-classical and adiabatic approximations, which are controlled in the limit of large space dimension.

Figures (5)

• Figure 1: In this theory, an initial state $|\Psi_0 \rangle$ does not satisfy the momentum ($\hat{\cal P}^\mu$) and Hamiltonian ($\hat{\cal H}$) constraints. An evolution is generated by the $\hat{\cal H}$ and $\hat{\cal P}^\mu$ with a series of random lapse and shift functions. An ensemble of such random walks induces a gradual collapse of the state toward a diffeomorphism-invariant state. The stochastic evolution causes the state to evolve toward the direction of increasing scale factor due to the slow dynamics in the region of the Hilbert space with the large scale factor.
• Figure 2: The original contour of $\Pi$ runs along the real axis. It can be deformed to the path of the stationary phase (denoted as the solid line) that goes through two saddle-points, $\Pi_\pm$.
• Figure 3: The dashed lines from left to right represent $\bar{\alpha}(l;L)$ plotted as a function of $l$ for $L=20, 40, 60, 80, 100$, respectively. The solid line denotes the evolution of the scale factor as a function of $L$, $\bar{\alpha}(L;L)$. For the plot, we choose $d=3$, $A=1$, $\Lambda=0.01$ and $\alpha_0=0$.
• Figure 4: The complex energies of the excitations for $s=-1$. As time increases, the energies of the tensor graviton, the vector and the scalar shift toward the real axis, along the imaginary axis and at the $45^{\circ}$ in the lower half of the complex plane, respectively.
• Figure 5: The cubic vertex that produces a scalar of momentum $p$ with negative energy alongside two tensor gravitons of momenta $k$ and $k'$ with positive energies.