Concepts of coarse graining in quantum mechanics

单洁,kimik3,X. Ma, Independent Researcher, Institute for First-Space Ontology and Emergent Physics Email: maexiang@163.com; 15552786932@139.com; maexiang2026@Outlook.com

Quantum mechanics is not a universal theory. This paper establishes a rigorous tripartite partition of mechanical applicability based on the coarse-graining scale L_c within the Cosmic String-Field and Coarse-Graining Meta-Theory (CSF-CMT). Layer I (L_c \sim l_{\text{string}}): meta-field wave dynamics governing the first-space initial state \Psi_I^{(0)}, where no particle concept, Hilbert space, or probability interpretation exists. Layer II (l_{\text{string}} \ll L_c \ll L_{\text{macro}}): standard quantum mechanics as an emergent effective theory, where particles arise as coarse-grained excitations and the Heisenberg uncertainty relation manifests as an information-cutoff effect. Layer III (L_c \to \infty): classical mechanics and deterministic geometry. The atomic horizon r_{AH} is identified as the physical boundary between Layer I and Layer II. This partition resolves the century-old quantum-classical boundary problem by demonstrating it to be a natural phase transition in coarse-graining scale, not a mysterious wave-function collapse.

F. A. Kaempffer

P. T. Landsberg

The concepts microscopic reversibility, detailed balancing, equal probabilities, equilibrium, $H$ theorem are often used in (coarse grained) quantum statistics. It is pointed out that each has a (fine grained) quantum-mechanical counterpart. These concepts are formulated, and their interrelations discussed, at an abstract mathematical level. The results are then interpreted by taking first the fine grained and then the coarse grained point of view. A type of transition matrix arises which does not seem to have been investigated so far, and some of its properties are discussed.

Carlo Heissenberg, Augusto Sagnotti

Mohammad Reza Pahlavani

James B. Hartle

These are the author's lectures at the 1992 Les Houches Summer School, "Gravitation and Quantizations". They develop a generalized sum-over-histories quantum mechanics for quantum cosmology that does not require either a preferred notion of time or a definition of measurement. The "post-Everett" quantum mechanics of closed systems is reviewed. Generalized quantum theories are defined by three elements (1) the set of fine-grained histories of the closed system which are its most refined possible description, (2) the allowed coarse grainings which are partitions of the fine-grained histories into classes, and (3) a decoherence functional which measures interference between coarse grained histories. Probabilities are assigned to sets of alternative coarse-grained histories that decohere as a consequence of the closed system's dynamics and initial condition. Generalized sum-over histories quantum theories are constructed for non-relativistic quantum mechanics, abelian gauge theories, a single relativistic world line, and for general relativity. For relativity the fine-grained histories are four-metrics and matter fields. Coarse grainings are four-dimensional diffeomorphism invariant partitions of these. The decoherence function is expressed in sum-over-histories form. The quantum mechanics of spacetime is thus expressed in fully spacetime form. The coarse-grainings are most general notion of alternative for quantum theory expressible in spacetime terms. Hamiltonian quantum mechanics of matter fields with its notion of unitarily evolving state on a spacelike surface is recovered as an approximation to this generalized quantum mechanics appropriate for those initial conditions and coarse-grainings such that spacetime geometry