Essential self-adjointness: implications for determinism and the classical–quantum correspondence

Abstract

It is argued that seemingly "merely technical" issues about the existence and uniqueness of self-adjoint extensions of symmetric operators in quantum mechanics have interesting implications for foundations problems in classical and quantum physics. For example, pursuing these technical issues reveals a sense in which quantum mechanics can cure some of the forms of indeterminism that crop up in classical mechanics; and at the same time it reveals the possibility of a form of indeterminism in quantum mechanics that is quite distinct from the indeterminism of state vector collapse. More generally, the examples considered indicate that the classical-quantum correspondence is more intricate and delicate than is generally appreciated. The aim of the article is to give a series of examples that reveal why the technical issues about self-adjointness are relevant to the philosophy of science and that help to make the issues accessible to philosophers of science.

Randomness? What Randomness?

“…the set of all 1-random sequences, which makes it bizarre that (barring a few exceptions related to Chaitin's number Ω) not a single one of these 1-random sequences can actually be proved to be 1-random, cf. Appendix B. Theorem 4.1 has further amazing consequences: Corollary 4.2 With respect to µ ∞ , almost every infinite outcome sequence x of a fair coin flip is Borel normal, 34 incomputable, 35 and contains any finite string infinitely often. 36 This follows because any 1-random sequence has these properties with certainty, see Calude (2010), §6.4.…”

“…(A. 35), showing that the Born probability µ a for single outcomes induces the Bernoulli process probability µ ∞ a on the space σ(a) N of infinite outcome sequences. As mentioned before, I specialize to fair quantum coin flips producing 50-50 Bernoulli processes, of which there are examples galore: think of measuring the third Pauli matrix σ z = diag(1, −1) in a state like ψ = (1, 1)/ √ 2.…”

“…This means that any group x1 · · · xn of digits in x occurs with relative frequency equal to 2 −n 35. This means that there is no computable function f : N → N taking values in {0, 1} that produces x, where computability is meant in the old sense of recursion theory, or, equivalently, in the sense of Turing machines or indeed modern computers.…”

Can the World Beshown to be Indeterministic after all?

“…However, Earman (2009) points to a number of examples where the Hamiltonian is not essentially self-adjoint. Although it is certainly a question that philosophers of physics ought to tackle systematically, I will not here delve into whether the Hamiltonian is essentially self-adjoint in all physically possible situations.…”

Homotopy and Path Integrals in the Time-dependent Aharonov-Bohm Effect

“…Bearing in mind the relation between non-trivial homotopy (i.e., the fundamental homotopy group has more than one element) and the absence of essential self adjointness [4,5,13] it is not surprising that quantum mechanics has a greater richness than classical mechanics (see also [19]). Other examples come to mind.…”