Forbidden Zones in Hilbert Space


Oh, how mysterious are the ways of infinite-dimensional Hilbert space, that have brought this man to repentence, before our very eyes.
—Chairman (myself), Council of Economic Advisors, Christmas entertainment skit, Harvard Graduate Economics Club, 1979

[jch] [[ddff-ltd] | Global Broadband Strategies] | POSTED: 12.30.04 @15:58
Forbidden Zones in Hilbert Space (search for Cherry 2000)

A. The problem: Hilbert space is big

The interpretation problem stems from the vastness of the Hilbert space, which, by the principle of superposition, admits arbitrary linear combinations of any states as a possible quantum state. This law, thoroughly tested in the microscopic domain, bears consequences that defy classical intuition: It appears to imply that the familiar classical states should be an exceedingly rare exception. And, naively, one may guess that superposition principle should always apply literally: Everything is ultimately made out of quantum “stuff”. Therefore, there is no a priori reason for macroscopic objects to have definite position or momentum. As Einstein noted localization with respect to macrocoordinates is not just independent, but incompatible with quantum theory. How can one then establish correspondence between the quantum and the familiar classical reality?

1. Copenhagen Interpretation

Bohr’s solution was to draw a border between the quantum and the classical and to keep certain objects – especially measuring devices and observers – on the classical side (Bohr, 1928; 1949). The principle of superposition was suspended “by decree” in the classical domain. The exact location of this border was difficult to pinpoint, but measurements “brought to a close” quantum events. Indeed, in Bohr’s view the classical domain was more fundamental: Its laws were self-contained (they could be confirmed from within) and established the framework necessary to define the quantum.

The first breach in the quantum-classical border appeared early: In the famous Bohr – Einstein double-slit debate, quantum Heisenberg uncertainty was invoked by Bohr at the macroscopic level to preserve wave-particle duality. Indeed, as the ultimate components of classical objects are quantum, Bohr emphasized that the boundary must be moveable, so that even the human nervous system could be regarded as quantum provided that suitable classical devices to detect its quantum features were available. In the words of John ArchibaldWheeler (1978; 1983) who has elucidated Bohr’s position and decisively contributed to the revival of interest in these matters, “No [quantum] phenomenon is a phenomenon until it is a recorded (observed) phenomenon”.

This is a pithy summary of a point of view – known as the Copenhagen Interpretation (CI) – that has kept many a physicist out of despair. On the other hand, as long as a compelling reason for the quantum-classical border could not be found, the CI Universe would be governed by two sets of laws, with poorly defined domains of jurisdiction. This fact has kept many a student, not to mention their teachers, in despair (Mermin 1990a; b; 1994).

2. Many Worlds Interpretation

The approach proposed by Hugh Everett (1957a, b) and elucidated by Wheeler (1957), Bryce DeWitt (1970) and others (see DeWitt and Graham, 1973; Zeh, 1970; 1973; Geroch, 1984; Deutsch, 1985, 1997, 2001) was to enlarge the quantum domain. Everything is now represented by a unitarily evolving state vector, a gigantic superposition splitting to accommodate all the alternatives consistent with the initial conditions. This is the essence of the Many Worlds Interpretation (MWI). It does not suffer from the dual nature of CI. However, it also does not explain the emergence of classical reality.

The dffculty many have in accepting MWI stems from its violation of the intuitively obvious “conservation law” – that there is just one Universe, the one we perceive. But even after this question is dealt with, many a convert from CI (which claims allegiance of a majority of physicists) to MWI (which has steadily gained popularity; see Tegmark and Wheeler, 2001, for an assessment) eventually realizes that the original MWI does not address the “preferred basis question” posed by Einstein (see Wheeler, 1983; Stein, 1984; Bell 1981, 1987; Kent, 1990; for critical assessments of MWI). And as long as it is unclear what singles out preferred states, perception of a unique outcome of a measurement and, hence, of a single Universe cannot be explained either.

In essence, Many Worlds Interpretation does not address but only postpones the key question. The quantum - classical boundary is pushed all the way towards the observer, right against the border between the material Universe and the “consciousness”, leaving it at a very uncomfortable place to do physics. MWI is incomplete: It does not explain what is effectively classical and why. Nevertheless, it was a crucial conceptual breakthrough: