Short description: Time evolution of quantum states and the conceptual role of time in quantum theory

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The quantum time evolution of a system describes how its wavefunction or quantum state changes over time according to the laws of quantum mechanics. In standard formulations, time is treated as an external parameter that governs the evolution of physical states.

A central conceptual issue related to time in quantum theory is the 'problem of time, which highlights a conflict between quantum mechanics and general relativity. Quantum mechanics treats time as universal and absolute, whereas relativity describes time as dependent on spacetime structure.^[1]^[2]

Time evolution in quantum mechanics

In non-relativistic quantum mechanics, the time evolution of a state $| ψ (t) ⟩$ is governed by the Schrödinger equation:

$i ℏ \frac{\partial}{\partial t} ψ (t) = H ψ (t),$

where $H$ is the Hamiltonian operator representing the total energy of the system. The formal solution is given by the unitary time-evolution operator

$| ψ (t) ⟩ = e^{- i H t / ℏ} | ψ (0) ⟩ .$

This evolution preserves probability and provides the standard description of dynamics in quantum theory. In both classical mechanics and standard quantum mechanics, time plays a special role as an external parameter, and observables are defined at specific instants of time.^[3]^: 759

Time in general relativity

In general relativity, time is no longer a unique external background parameter, but part of the dynamical structure of spacetime. The field equations are formulated in terms of spacetime geometry rather than evolution with respect to a fixed time variable. These different roles of time in quantum theory and relativity are fundamentally incompatible.^[3]

Impact on quantum gravity

Quantum gravity attempts to reconcile or unify quantum mechanics and general relativity, the current theory of gravity.^[4] The problem of time is central to these efforts. It remains unclear how time is related to quantum probability, whether time is fundamental or emergent, and whether it is exact or approximate in a more complete theory.^[5]

The frozen formalism problem

One of the most discussed aspects of the problem of time is the frozen formalism problem. In ordinary quantum mechanics, the wavefunction evolves in time according to the Schrödinger equation. In canonical quantum gravity, however, the corresponding equation takes the form of the Wheeler–DeWitt equation:

$\hat{H} (x) | ψ ⟩ = 0,$

where the Hamiltonian becomes a constraint. This suggests that the wavefunction of the universe is static or “frozen,” even though time-dependent behavior clearly appears at smaller scales in the physical world.^[3]^: 762

Wavepacket displacement

A motional wavepacket is initially displaced to the minimum of the potential energy surface, after which it begins to encircle the conical intersection, denoted CI. b, Initial wavepacket density in 2D (left), and integrated over Q1 (right). c, After sufficient time evolution, the two components of the wavepacket destructively interfere due to geometric phase, giving a nodal line along Q2 = 0 (dotted line). d, Motional wavepacket density at the maximum interference time T . e, If the geometric phase were neglected, the two wavepacket components would interfere constructively. f, Density at t = T with geometric phase neglected. Contours in b, d, and f correspond to the potential energy surface E−. g, The Jahn-Teller Hamiltonian HJT is engineered in an ion-trap quantum simulator with a single 171Yb+ ion. The ion (white sphere) is confined in a Paul trap and HJT is realised using two simultaneous laser-induced interactions (purple and pink, corresponding to colour-coded terms in HJT)." "The effects of geometric phase on dynamics around a conical intersection can be directly observed from the motional probability density, fig. 1a–d."

"As the initial wave-packet, we choose the ground state of the non-interacting vibrational Hamiltonian, H0 = ω(a†1a1 +a† 2a2), displaced to the potential-energy minimum at Q1 = −κ/ω, Q2 = 0 (fig. 1a–b). During the time evolution, the wavepacket splits into two components evolving in opposite directions around the conical intersection. The two components overlap at Q1 > 0, causing destructive interference at the nodal line Q2 = 0, where their equal and opposite geometric phases lead to a vanishing density (fig. 1c–d).

By contrast, if geometric phase were disregarded, the two wavepacket fragments would interfere constructively, reaching maximum amplitude at Q2 = 0 (fig. 1e–f).

Proposed solutions

Several approaches have been proposed to address the problem of time. Work by Don Page and William Wootters suggests that time may emerge through entanglement between an evolving subsystem and an internal clock subsystem within a larger timeless universe.^[6]^[7]^[8]

In 2013, Ekaterina Moreva and collaborators performed an experimental test of the Page–Wootters idea, showing for photons that time can appear for internal observers while remaining absent for external observers, in agreement with the timeless structure of the Wheeler–DeWitt equation.^[9]^[10]^[11]

Another proposal is the thermal time hypothesis, developed by Carlo Rovelli and Alain Connes, in which time is associated with the thermodynamic or statistical state of a system rather than being fundamental.^[12]^[13]