We put forward a quantum model of cosmology that is exactly periodic but avoids the Boltzmann Brain problem. If the universe is described by a quantum state evolving unitarily in a finite-dimensional Hilbert space, its evolution will be recurrent: given enough time, the state will return arbitrarily close to its initial state. There is a worry that such a scenario cannot be phenomenologically acceptable, because the state will spend most of its time in a high-entropy equilibrium macrostate, with rare fluctuations downward in entropy, and the vast majority of observers will be minimal fluctuations away from equilibrium, or ``Boltzmann Brains." Here we show that this is not necessarily true. If the differences in energy eigenvalues are commensurable, the evolution is not simple recurrent, but exactly periodic. Moreover, if the state starts at minimum thermodynamic entropy, its evolution can feature a distinguished entropy excursion that is much more pronounced than one would expect from the conventional expression $P(ΔS) \propto \exp(-ΔS)$. This excursion could represent our Big Bang, with relatively few Boltzmann fluctuations occurring in the subsequent equilibrium phase before a Big Crunch occurs and the cycle begins again. We speculate on the spacetime interpretation of this kind of quantum universe.
We put forward a quantum model of cosmology that is exactly periodic but avoids the Boltzmann Brain problem. If the universe is described by a quantum state evolving unitarily in a finite-dimensional Hilbert space, its evolution will be recurrent: given enough time, the state will return arbitrarily close to its initial state. There is a worry that such a scenario cannot be phenomenologically acceptable, because the state will spend most of its time in a high-entropy equilibrium macrostate, with rare fluctuations downward in entropy, and the vast majority of observers will be minimal fluctuations away from equilibrium, or ``Boltzmann Brains." Here we show that this is not necessarily true. If the differences in energy eigenvalues are commensurable, the evolution is not simple recurrent, but exactly periodic. Moreover, if the state starts at minimum thermodynamic entropy, its evolution can feature a distinguished entropy excursion that is much more pronounced than one would expect from the conventional expression P(ΔS)∝exp(−ΔS). This excursion could represent our Big Bang, with relatively few Boltzmann fluctuations occurring in the subsequent equilibrium phase before a Big Crunch occurs and the cycle begins again. We speculate on the spacetime interpretation of this kind of quantum universe.
There are two kinds of quantum fluctuations relevant to cosmology that we focus on in this article: those that form the seeds for structure formation in the early universe and those giving rise to Boltzmann brains in the late universe. First, structure formation requires slight inhomogeneities in the density of matter in the early universe, which are then amplified by the effect of gravity, leading to clumping of matter into stars and galaxies. According to inflation theory, quantum fluctuations form the seeds of these inhomogeneities. However, these quantum fluctuations are described by a quantum state that is homogeneous and isotropic, and this raises a problem, connected to the foundations of quantum theory, as the unitary evolution alone cannot break the symmetry of the quantum state. Second, Boltzmann brains are random agglomerates of particles that, by extreme coincidence, form functioning brains. Unlikely as these coincidences are, they seem to be predicted to occur in a quantum universe as vacuum fluctuations if the universe continues to exist for an infinite (or just very long) time; in fact, they are predicted to occur over and over, even forming the majority of all brains in the history of the universe. We provide a brief introduction to the Bohmian version of quantum theory and explain why in this version Boltzmann brains, an undesirable kind of fluctuation, do not occur (or at least not often), while inhomogeneous seeds for structure formation, a desirable kind of fluctuation, do.
We consider a closed macroscopic quantum system in a pure state ψ evolving unitarily with Hamiltonian H in a finite-dimensional Hilbert space H. Following von Neumann, we assume a decomposition of H into mutually orthogonal subspaces Hν, where each ν represents a macro state, and Hν contains the pure states compatible with ν. We define the quantum Boltzmann entropy for a macro state ν as SqB(ν)=kBlogdimHν, where kB is the Boltzmann constant. Goldstein et al. [http://arxiv.org/abs/1903.11870] have suggested, as a version of the second law of thermodynamics, that SqB predominantly increases with time for realistic macroscopic systems. Here, we discuss precise versions of this statement and confirm them for several simple models of H and Hν.
Abstract The ontological status of time remains unresolved in fundamental physics. In quantum mechanics, time appears as an external classical parameter, whereas in general relativity it is dynamical and intertwined with spacetime geometry. Canonical approaches to quantum gravity, most notably the Wheeler–DeWitt formulation, eliminate time altogether, leading to the well-known “problem of time.” These tensions motivate frameworks in which time is not fundamental but emergent. In this work, we propose a discrete, update-based framework in which the universe evolves through a sequence of global quantum states indexed by an abstract update parameter rather than by a physical time coordinate. Evolution is formulated as a completely positive, trace-preserving (CPTP) quantum channel acting on the universal density operator, ensuring exact conservation of probability and global quantum information at every update. The framework is deliberately conservative: it preserves the formal structure of quantum mechanics and quantum field theory, enforces vacuum stability, and supports global conservation laws without introducing new dynamical postulates. Within this setting, time emerges relationally at the level of subsystems. As subsystems become increasingly entangled with their environments, entropy gradients arise that define an effective arrow of time, closely aligned with the thermodynamic arrow. The resulting temporal flow is relative, statistical, and observer-dependent, while the global evolution remains information-preserving and, in the closed-system limit, reversible. We explore the implications of this framework for cosmology and black hole physics. The Big Bang is reinterpreted as an initial quantum state rather than a singular beginning of time, avoiding temporal singularities. Black hole evaporation is shown to be compatible with unitary evolution when the full quantum state is considered, naturally reproducing the expected Page curve and resolving the apparent information paradox. Overall, the framework provides a coherent, minimal setting in which time, irreversibility, and cosmological history emerge from quantum correlations rather than from a fundamental temporal background.
Modern physics faces a constellation of foundational puzzles: the Big Bang singularity, time's arrow, dark matter and dark energy, the cosmological constant problem, fine-tuning, quantum measurement, and quantum gravity unification. This Perspective proposes a unified resolution grounded in complexity-driven selection from diverse quantum paths. Prior to the Big Bang, reality comprises a pre-temporal quantum substrate (lacking classical time coordinate but governed by unitary quantum evolution), evolving reversibly across multiple trajectories—including opposing time arrows. The Big Bang marks an emergence threshold (λ) where decoherence locks spacetime parameters, yielding our classical universe as a "survivor" of local emergence rather than a closed totality. Dark matter and dark energy represent unevolved quantum paths—residuals exerting gravitational influence without full manifestation. This framework reframes cosmology as the study of layered emergence, dissolving "why" questions into "when/how" selection dynamics, with testable predictions in CMB perturbations, quantum decoherence experiments, and gravitational wave observations. By shifting from reductionism to emergentism, it suggests a universal pattern: threshold-gated emergence from diverse substrates, applicable from the cosmos to consciousness.
We consider the temporal evolution of an isolated system of particles from a non-equilibrium state of entropy S = S′ to the equilibrium state of maximum entropy S = S max, S′ ≤ S max. The application of usual density matrix theory to the temporal development of S leads us to the unexpected result P = dS/dt = 0. The entropy does not change in time. This result is valid in case of the equilibrium state. However it is wrong for a non-equilibrium state, which could not irreversibly change from S′ to S max by entropy production P > 0. This contradicts the second law of thermodynamics. The paradoxial result can be traced back to the Liouville theorem. To overcome this difficulty, we introduce the t–P space. The time t and the entropy production P are observables in quantum thermodynamics. They are defined by the eigenvalue equations t|t〉 = t|t〉 and P|P〉 = P|P〉. The operators t and P do not commute. We have a non-vanishing commutator, [t,P] = ik, and consequently a t–P uncertainty relation, ΔtΔP ≥ k/2. The Boltzmann constant k is characteristic of quantum thermodynamics and represents the atomic entropy unit. — The time evolution operator U(τ) = exp[−(i/k)τP] shifts the time t by τ, U(τ)|t〉 = |t + τ〉. The generator of τ is the operator P of entropy production. Then the eigenvector |P〉 is also an eigenket of U(t), and it follows with progressing time τ ≥ 0 that τP ≥ 0 and hence P = dS/dt ≥ 0. This is the second law of thermodynamics, which is correctly reproduced in quantum thermodynamics. — Exchanging −τ for +τ replaces the equations of forward time +τ by the equations of backward time −τ. Independent of the direction of the arrow of time, the entropy of the system always increases or remains constant in the limit of the equilibrium state, dS ≥ 0. This is characteristic of irreversibility, for the system does not return to its initial state on time reversal. — The t–P uncertainty relation allows to discuss the temporal evolution of the expectation value 〈P〉 of entropy production and the lifetime Δt of the transient states, the system passes through on approaching the equilibrium state. In case of a non-equilibrium state 〈P〉, ΔP, and Δt are finite. On approaching the equilibrium state, 〈P〉 and ΔP decrease, consequently Δt increases. Finally, in the case of equilibrium (eq), we have 〈P〉eq = 0, ΔP eq = 0, and Δt eq → ∞. The lifetime of the equilibrium state is unlimited, because the entropy production 〈P〉 and the uncertainty of entropy production ΔP exactly vanish. The temporal evolution of the system can be described by means of a phenomenological equation.
An intricate quantum statistical effect guides us to a deterministic, non-causal quantum universe with given fixed initial and final state density matrix. A concept is developed on how and where something like macroscopic physics can emerge. The concept does not allow to incorporate philosophically indispensable free will decisions. If the quantum world and its conjugate evolve independently one can replace both fixed final states by a matching common one. This allows for external manipulations done in the quantum world and its conjugate which do not otherwise alter the basic structure. In a big bang / big crunch universe the expanding part can be attributed to the quantum world and the contracting part to the conjugate one. The obtained bi-linear picture has a number of beautiful and exciting consequences.