Narrow Results

We extend the mathematical theory of quantum hypothesis testing to the general W*-algebraic setting and explore its relation with recent developments in non-equilibrium quantum statistical mechanics. In particular, we relate the large deviation principle for the full counting statistics of entropy flow to quantum hypothesis testing of the arrow of time.

Previously, we have shown that the Kasner solution

The conventional tacit assumption according to which quantum processes take place in the Newtonian time topology is abandoned. The chronotopological space is based on the interaction proper time neighborhood. It is disconnected, satisfies the separation axioms of the topological space , and is developed and used to solve some problems related to quantum field theory. In chronotopology the time evolution operator, , breaks down into the alternative = either . The unitary, U, and the reduction, R, dynamics are implemented by means of two time evolution operators, and , derived from a quantized version of Gel'fand's theory of the generalized random and infinitely divisible fields. As an application the solution of the paradox of Schrödinger's cat is presented.

This paper proves, via an analytical approach, that 170 (out of 256) Boolean CA rules in a one-dimensional cellular automata (CA) are time-reversible in a generalized sense. The dynamics on each attractor of a time-reversible rule N is exactly mirrored, in both space and time, by its bilateral twin ruleN^†. In particular, all 69 period-1 rules, 17 (out of 25) period-2 rules, and 84 (out of 112) Bernoulli rules are time-reversible.

The remaining 86 CA rules are time-irreversible in the sense that N and N^† mirror their dynamics only in space, but not in time. In this case, each attractor of N defines a unique arrow of time.

A simple "time-reversal test" is given for testing whether an attractor of a CA rule is time-reversible or time-irreversible. For a time-reversible attractor of a CA rule N the past can be uniquely recovered from the future of N^†, and vice versa. This remarkable property provides 170 concrete examples of CA time machines where time travel can be routinely achieved by merely hopping from one attractor to its bilateral twin attractor, and vice versa. Moreover, the time-reversal property of some local rules can be programmed to mimic the matter–antimatter "annihilation" or "pair-production" phenomenon from high-energy physics, as well as to mimic the "contraction" or "expansion" scenarios associated with the Big Bang from cosmology.

Unlike the conventional laws of physics, which are based on a unique universe, most CA rules have multiple universes (i.e. attractors), each blessed with its own laws. Moreover, some CA rules are endowed with both time-reversible attractors and time-irreversible attractors.

Using an analytical approach, the time-τ return map of each Bernoulli στ-shift attractor of all 112 Bernoulli rules are shown to obey an ultra-compact formula in closed form, namely,.

or its inverse map.

These maps completely characterize the time-asymptotic (steady state) behavior of the nonlinear dynamics on the attractors. In-depth analysis of all but 18 global equivalence classes of CA rules have been derived, along with their basins of attraction, which characterize their transient regimes.

Above all, this paper provides a rigorous nonlinear dynamics foundation for a paradigm shift from an empirical-based approach à la Wolfram to an attractor-based analytical theory of cellular automata.

If our universe underwent inflation, its entropy during the inflationary phase would have been substantially lower than it is today. Because a low-entropy state is less likely to be chosen randomly than a high-entropy one, inflation is unlikely to arise through randomly-chosen initial conditions. To resolve this puzzle, we examine the notion of a natural state for the universe, and argue that it is a nearly-empty space–time. If empty space has a small vacuum energy, however, inflation can begin spontaneously in this background. This scenario explains why a universe like ours is likely to have begun via a period of inflation, and also provides an origin for the cosmological arrow of time.

We point out that time’s arrow is naturally induced by quantum mechanical evolution, whenever the systems have a very large number $\mathcal{𝒩}$ of nondegenerate states and a Hamiltonian bounded from below. When $\mathcal{𝒩}$ is finite, the arrow is imperfect, since evolution can resurrect past states. In the limit $\mathcal{𝒩}\to \infty$ the arrow is fixed by the “tooth of time”: the decay of excited states induced by spontaneous emission to the ground state, mediated by interactions and a large number of decay products which carry energy and information to infinity. This applies to individual isolated atoms, and does not require a coupling to a separate large heath bath.

Using an elementary example based on two simple harmonic oscillators, we show how a relational time may be defined that leads to an approximate Schrödinger dynamics for subsystems, with corrections leading to an intrinsic decoherence in the energy eigenstates of the subsystem.

Construction of a model of Quantum Gravity, which will be some day in concordance with experiments, is one of the most fascinating tasks we have in modern theoretical physics. There are a plethora of common problems, which must be solved to find a viable candidate for Quantum Gravity. We introduce the concept of the nonlinear graviton and we end with possible experimental evidence for our approach.

Quantum gravity is considered on the base of five-dimensional Dirac equation with complex coordinates. We have used a multi-world Everett-like approach, which allowed us to introduce an arrow of time into dynamics of the theory. Effective metrics arises as coordinate representation of the quantum metric operator, which belongs to the Clifford algebra. We present the results obtained in the study of isotropic and homogenous cosmology in the theory. They include explanation of the matter - antimatter asymmetry in the physical world.

The received view on the problem of the direction of time holds it that time has no intrinsic dynamical properties, and that its apparent asymmetry, to be understood in purely topological terms, is dependent on the directional properties of physical processes. In this paper we shall challenge both claims, in the light of an algebraic representation of time. First, we will show how to give a precise formulation to the intuitive idea that time possesses an intrinsic dynamics; this formulation relies on the fact that the algebraic properties of time can equivalently be understood in dynamical terms. Second, we shall argue that the directional properties displayed by the processes occurring in time depend on the directional properties of time, rather than the converse.