User:Tnorsen/Sandbox/Beables

The word beable was introduced by the physicist John Stewart Bell in his article entitled "The theory of local beables" (see Speakable and Unspeakable in Quantum Mechanics, pg. 52). A beable of a physical theory is an object that, according to that theory, is supposed to correspond to an element of physical reality. The word "beable" (be-able) contrasts with the word "observable". While the value of an observable can be produced by a complex interaction of a physical system with a given experimental apparatus (and not be associated to any "intrinsic property" of the physical system), a beable exists objectively, independently of observation. For instance, it can be proven[1] that there exists no physical theory, consistent with the predictions of quantum theory, in which all observables of quantum theory (i.e., all self-adjoint operators on the Hilbert space of quantum states) are beables.

While, in a given theory, an observable does not have to correspond to any beable, the result of the "measurement" of an observable that has actually been carried out in some experiment is physically real (it is represented, say, by the position of a pointer) and must be stored in some beable of the theory. In the words of Bell (Speakable and Unspeakable in Quantum Mechanics, pg. 52):

"This [the theory of local beables] is a pretentious name for a theory which hardly exists otherwise, but which ought to exist. The name is deliberately modelled on 'the algebra of local observables'. The terminology, be-able as against observ-able, is not designed to frighten with metaphysic those dedicated to realphysic. It is chosen rather to help in making explicit some notions already implicit in, and basic to, ordinary quantum theory. For, in the words of Bohr, 'it is decisive to recognize that, however far the phenomena transcend the scope of classical physical explanation, the account of all evidence must be expressed in classical terms'. It is the ambition of the theory of local beables to bring these 'classical terms' into the equations, and not relegate them entirely to the surrounding talk.

The concept of 'observable' lends itself to very precise mathematics when identified with 'self-adjoint operator'. But physically, it is a rather wooly concept. It is not easy to identify precisely which physical processes are to be given status of 'observations' and which are to be relegated to the limbo between one observation and another. So it could be hoped that some increase in precision might be possible by concentration on the beables, which can be described in 'classical terms', because they are there. The beables must include the settings of switches and knobs on experimental equipment, the currents in coils, and the readings of instruments. 'Observables' must be made, somehow, out of beables. The theory of local beables should contain, and give precise physical meaning to, the algebra of local observables."

Local beable

A local beable of a physical theory is a beable that is associated to (or is an element of physical reality of) a given portion of spacetime. For instance, in quantum mechanics, if one is willing to grant beable status to the wave function, then it certainly won't be a local beable, since the wave function is defined in configuration space and is not in general associated with any particular portion of spacetime (in many presentations of quantum field theory it is stressed that, while quantum states are not local, i.e., are not associated to any particular portion of spacetime, observables must be localized). In classical mechanics, the position and momenta of all the particles in a given portion ${\displaystyle R}$ of spacetime are local beables associated to ${\displaystyle R}$. In Maxwell's theory, the eletric and magnectic fields in a given portion ${\displaystyle R}$ of space-time are also local beables associated to ${\displaystyle R}$ (on the other hand, the scalar and vector potential are usually not granted beable status). The beables representing switches and knobs on experimental equipment and the beables representing readings of instruments must be local beables.

Local beables and local causality

In his article "The theory of local beables", Bell introduces the concept of local beable in order to analyse the concept of local causality. His concept of local causality was later more precisely formulated in his article "La Nouvelle Cuisine" (see Speakable and Unspeakable in Quantum Mechanics, pg. 232). In "La Nouvelle Cuisine", Bell stresses his view that it is not possible to seriously analyse the matter of local causality (or, as he puts it, the question "what cannot go faster than light?") unless one is willing to be explicit about the local beables of a theory. In his words (Speakable and Unspeakable in Quantum Mechanics, pg. 234):

"No one is obliged to consider the question 'What cannot go faster than light?'. But if you decide to do so, then the above remarks suggest the following: you must identify in your theory 'local beables'. The beables of the theory are those entities in it which are, at least tentatively, to be taken seriously, as corresponding to something real. The concept of 'reality' is now an embarassing one for many physicists, since the advent of quantum mechanics, and especially of 'complementarity'. But if you are unable to give some special status to things like electric and magnectic fields (in classical eletromagnetism), as compared with the vector and scalar potentials, and British sovereignty, then we cannot begin a serious discussion."

In the previous section, Bell gives examples of things that go faster than light, like British sovereignty ("When the Queen dies in London (may it long be delayed) the Prince of Wales, lecturing on modern architecture in Australia, becomes instantaneously King.") and the scalar potential (with Coulomb gage) in Maxwell's theory. As Bell explains, "Conventions can propagate as fast as may be convenient" and concludes "But then we must distinguish in our theory between what is convention and what is not".