The Arrow of Time as a Sectoral Projection

The Arrow of Time as a Sectoral Projection A compact statement of how a directed time can arise from a globally symmetric state, with explicit placement in the existing literature and one constraint that follows from the calculation. The question The microscopic laws of physics are, to excellent approximation, symmetric under time reversal. Nothing in Maxwell's equations, the Schrodinger equation, or general relativity distinguishes past from future. Yet heat flows one way, memories are of the past, and entropy increases. The standard resolution says the asymmetry lies in the state rather than the laws: the universe began in a very special low entropy condition, and everything since is the approach to equilibrium. This works, but it moves the question rather than answering it. Why was the initial state so special? Penrose has quantified how special, and the number is extreme. The mechanism Suppose instead that the global state is pure, symmetric under time reversal, and entangled across some division of its degrees of freedom. An observer has access to one part only, and their description is obtained by tracing out the rest. Three things follow. The global entropy is zero and stays zero, because a pure state has no entropy and unitary evolution preserves that. The entropy of the accessible part is not zero, and it grows as the two parts become more entangled. And each part has its own time parameter along which its own entropy increases, while the global state distinguishes no direction at all. What an observer measures as entropy production is the growth of entanglement with what they cannot reach. This idea is not new, and the paper says so Priority does not lie with this work, and the paper devotes a section to the antecedents rather than a closing footnote. Sakharov proposed twin universes with opposite arrows of time in 1967. Boyle, Finn and Turok have developed a CPT symmetric cosmology and published directly on the arrow of time within it. The thermofield double construction gives the sharpest formal version, a global pure state whose restriction to either half is exactly thermal. And the Unruh effect is the most directly calculable instance of all: the Minkowski vacuum is pure, symmetric and entangled, but restricted to an accelerating observer it is thermal with a definite temperature. No cosmology is needed and the result is standard quantum field theory. The paper states the mechanism compactly and places it. That is exposition, not discovery, and it is labelled as such throughout. The one result that is not restatement If the two parts are coupled, the coherence between them decays. Treating one part as an open quantum system with the other as its environment, the form of that decay is fixed by theorem rather than chosen, and the resulting order parameter turns out to be fully determined by the relaxation history. It contains no free saturation constant. Its asymptotic value follows from an integral over the relaxation coefficient, and for any coefficient that does not decay unreasonably fast, that integral diverges and the system reaches complete decoherence. This constrains a class of constructions. Models that postulate a small frozen residual coherence and attach observable consequences to it are inconsistent with their own dynamics, unless the relaxation is tuned to stay weak across the entire cosmic history, which conflicts with the efficient early relaxation such models typically also require. Complete decoherence is in any case the physically required outcome. It is precisely the statement that the two parts are classically separated, which is what the absence of observable interference between them demands. What the mechanism does not do The paper lists this explicitly, because the account is sometimes credited with more than it delivers. It fixes the magnitude of nothing. No density, no ratio, no rate follows from the structure. It does not eliminate the need for an initial condition but substitutes one, since the global state must be entangled in the appropriate way. It does not explain dark matter or the baryon asymmetry, and attempts to have the same structure supply those fail for independent reasons. And it makes no distinctive prediction: every observable consequence is shared with standard cosmology plus standard decoherence. The mechanism reorganizes an explanation. It does not generate a test. Who might find this useful Anyone building a model in which two sectors, two branches, or a system and its environment are related by a symmetry, and in which a coherence parameter appears. The constraint in the paper applies to any such construction: that parameter is not yours to choose. Michael Lehmann 79730 Murg/Germany mi.lehmann@gmx.de