Physical systems and their measurement

In the thermal interpretation a physical system is defined by an algebra E_{S} of associated variables, on which is defined a trace tr_{S} with the property

tr_{S} fg = tr_{S} gf.

Typical examples are the algebra of operators on a small Hilbert subspace of the universal Hilbert space (giving traditional quantum systems) or algebras generated by certain integrals of the form

f = ∫ dx^{3} a^{*}(x) c(x,∇x) a(x)

where c(x,∇x) differs essentially from zero only at the (more or less spread out) location of the system (this gives classical subsystems at the hydrodynamic level).

A physical system S at time t has the objective values <f>_{t} for all f in E_{S}, where <.> is the universal ensemble.

These objective properties can be completely described by the reduced density matrix ρ_{S}(t) in E_{S}, which is completely determined by

<f>_{t} := tr_{S}ρ_{S}(t)f

and the state of the universe.

It is, incidentally, extremely unlikely that the state ρ_{S}(t) is pure, unless one restricts the algebra to a matrix algebra of low dimension (for example 8-dimensional for a system of three spin variables) and prepares the state carefully.

In order to justify the description 'objective', we must explain how they are measured. The value <f>_{t} of a variable f of the physical system S is observable in principle, as long as it does not vary too quickly in time or space, and as long as the system is long-lived enough to perform the measurement. To measure <f>_{t} one needs a macroscopic device (that is, a many particle system described by statistical mechanics) coupled to the system in such a way that

1. the dynamics induced by the coupling do not destroy the value <f> (otherwise one measures something, but not what one hoped for), and

2. the system possesses a macroscopic pointer variable x, and according to the theory the coupling can be shown to imply that after a sufficient time a state of equilibrium is reached, in which (for the simplest case) <x> = K<f> holds for a known constant K. Then one can calculate <f> from the observed value of <x>: <f> = <x>/K. In this case one says that one has measured <f>.

This is a precise definition of the measurement process, which, in contrast to all previous interpretations of quantum mechanics, allows one to analyse the measurement process on the basis of the fundamental model alone, in particular without the use of probabilities, wave-function collapse, or the like.

In general there are certain problems with ensuring that 1. or 2. hold on the basis of the many particle dynamics (which is the only allowed basis for the procedure in 2.) and as a result one can in general only determine <f> approximately. When one reckons more carefully, one can see that one can in fact determine a single <f> with adequate precision. But for a pair of complementary variables (position and momentum, or spin in orthogonal directions) simultaneous measurability is fundamentally restricted by an uncertainty relation; see Section 3 in EEEQ.

Arnold Neumaier (Arnold.Neumaier@univie.ac.at) A theoretical physics FAQ