Abstract
It is argued that the Schrödinger equation does not yield a correct description of the quantum-mechanical time evolution of states of isolated physical systems featuring events. A general statistical law replacing unitary Schrödinger evolution of states is then formulated within the so-called ETH-Approach to Quantum Mechanics. This law eliminates the infamous “measurement problem.” Our general concepts and results are illustrated by an analysis of simple models describing a very heavy atom coupled to the quantized radiation field. In the limit where the speed of light tends to infinity these models can be treated quite explicitly.
Highlights
In Search of a New Law of Nature “... their attempts to see in the very inadequacy of the conventional interpretation of quantum theory a deep physical principle have often led physicists to adopt obscurantist, mystical, positivist, psychical, and other irrational worldviews.” (David Deutsch [1])In this paper we attempt to add a missing piece that has long been searched for to the puzzle of Quantum Mechanics, namely an appropriate notion of states1 and a general statistical law governing the time evolution of states of isolated physical systems featuring events
In the weak-coupling regime, the Schrödinger-picture time evolution of states of the atom is well approximated by unitary evolution—the one we are used to from text books on elementary Quantum Mechanics—we find that, in the models considered here, the law of evolution of states of the atom in the E T H -Approach smoothly interpolates between unitary deterministic Schrödinger evolution, appropriate for closed systems, and classical Markovian evolution of the probabilities μ(·), appropriate for isolated open systems of matter very strongly coupled to the radiation field
Our main goal in this paper has been to illustrate the E T H -Approach to Quantum Mechanics, which many readers may find rather abstract, with a discussion of simple, concrete models, which are, sophisticated enough to exhibit some of the main subtleties and virtues of the E T H -Approach
Summary
In Search of a New Law of Nature “... their attempts to see in the very inadequacy of the conventional interpretation of quantum theory a deep physical principle have often led physicists to adopt obscurantist, mystical, positivist, psychical, and other irrational worldviews.” (David Deutsch [1]). The E T H -Approach to QM yields the following picture of the dynamics of states in Quantum Mechanics: The evolution of states of an isolated open system S featuring actual events, in the sense of Definition 6 stated above, is determined by a (continuous-time) stochastic branching process, whose state space is referred to as the non-commutative spectrum, ZS, of S (see [11]). Suffice it to say that an actuality setting in at time t, described by a partition of unity πξ | ξ ∈ X ⊂ E≥t , corresponds to measuring a physical quantitiy X ∈ OS (see Eq (1)) iff the projections πξ | ξ ∈ X can be well approximated (in the norm on the linear space B(HS) given by the scalar product induced by the state ωt —see [11,13]) by spectral projections of the self-adjoint operator X (t ) ∈ E≥t representing Xat some time t t D[(n0,)n ] := A[n,n ] ⊗ 1|hS , E[(n0,)n ] := A[n,n ] ⊗ B(hS), n < n , E≥(0n) := A≥n ⊗ B(hS)
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