A theory of time based on wavefunction collapse

TL;DR

This work proposes that time is not fundamental but emerges from a continual collapse of a gauge-non-invariant state toward a gauge-invariant one, realized via a stochastic projection on the temporal-diffeomorphism orbit and an enlarged Hilbert space. By applying this collapse-based time to a FRW minisuperspace model with scale factor $\alpha$ and a scalar field $\phi$, the authors derive a time parameter $T$ related to $T^2 \sim 1/\langle \hat H^2 \rangle$ and show that, at large $T$, the dynamics become linear and unitary through an effective Hamiltonian $\hat H_{\rm eff}(T)$, while-time direction is dynamically guided by the configuration-dependent mass. The cosmological analysis reveals distinct epochs—pre-radiation, radiation, matter, and dark-energy domination—each governed by explicit $T$-dependent evolutions and crossovers at $\alpha_A$, $\alpha_B$, and $\alpha_C$, with a final transition in the dark-energy era where the evolution becomes diffusive after a critical scale $\alpha^* = \tfrac{1}{2}\log(\Lambda_1/\Lambda_0)$. These results provide a concrete link between quantum measurement, gauge constraints, and cosmological dynamics, and suggest potential observational signatures via fluctuations in $\langle \hat H^2 \rangle$ across cosmic time, while outlining steps toward extending the framework to full general relativity.

Abstract

We propose that moments of time arise through the failed emergence of the temporal diffeomorphism as gauge symmetry, and that the passage of time is a continual process of an instantaneous state collapsing toward a gauge-invariant state. Unitarity and directedness of the resulting time evolution are demonstrated for a minisuperspace model of cosmology.

Figures (4)

• Figure 1: The continual collapse of a gauge non-invariant initial state toward a gauge invariant state as a time evolution.
• Figure 2: $|\chi_{q ,\pm}(z)|^2$ normalized by its peak value at $z_{q}^*$ with increasing $q=0,2,4,6,8$ from left to right curves. The scale factor at which the wavefunction is peaked increases with $q$ because a larger momentum of $\phi$ gives rise to a larger momentum for $\alpha$.
• Figure 3: (a) The amplitude of the wavefunction in the dark-energy-dominated era with $\Lambda_1=1$, $\Lambda_0=e^{-60}$ and $\Phi_+(0,q) = \delta(q)$, $\Phi_-(0,q) = 0$. Each curve represents $|\Psi(\alpha,0,T)|^2$ as a function of $\alpha$ for $T=e^{10},e^{15},e^{20},e^{25},e^{30},e^{31},e^{32},e^{33}$ from left to right. The curve in the thick (blue) line is at $T^*=e^{30}$, which marks the crossover from the $\Lambda_1$-dominated to the $\Lambda_0$-dominated evolution. At the crossover time, the wavefunction is peaked around $\alpha^* = 30$. For $T<T^*$, the peak of the wavefunction moves to larger values of $\alpha$ with increasing $T$. Beyond $T^*$, the wavefunction gets broader in $T$ with the peak position fixed at $\alpha^*$. (b) The logarithm of the amplitude of the gauge invariant wavefunction $\Psi_{E=0,q}(\alpha,\phi)$ in Eq. \ref{['eq:PsiEqn3']} for the same choice of parameters as in (a).
• Figure 4: (a) The conditional probability for $x$ at time $t$ for the gauge invariant wavefunction $\Psi(x,t) =$$e^{ -\frac{\alpha^2}{2} (x-t)^2 + i k_0 (x-t) }$ with $\alpha=2$ and $k_0=1$. It describes a wave-packet well localized in $x$ propagating with speed $1$. (b) The conditional probability for $x$ at an instant defined in basis $| {\mathcal{T}} \rangle_+$ with $\gamma=2$. The state of $x$ at time ${\mathcal{T}}$ is delocalized in $x<{\mathcal{T}}$ because a moment of time in this basis includes the far past of the original basis.