Spin-measurement formally described

From the point of view of the thermal interpretation a spin-measurement in the Schrödinger picture looks like this:

<.> is the state of the universe at time t, which is monotonic and linear on the algebra E of all variables.

ES is the algebra of system variables. For a single spin this is the algebra of complex 2x2 matrices [A11,A12;A21,A22]. J: ES —> E is a unitary representation which specifies exactly which of the many spins in the universe is being represented. The subsystem is described in the Schrödinger picture: At an arbitrary time t the system S is in the state ρt, which is determined by

<J(A)>t = trace ρt A

for all A in ES. If the relationship

ρt = ψtψt*

holds at time t, one says one has prepared the subsystem in the pure state ψt.

At time t, S is in a pure state |s> (s=1,2) if ρt = |s><s| holds, so that

<J(A)>t = Ass

for all A in ES. This will be in EIG(s) for certain prepared times, but not normally in general. At certain other times t, S is instead in a pure superposition ψt, so that ρt = ψtψt* and then

<J(A)>t = ψt*t

for all A in ES. At unprepared times the system is, in general, in a mixed state.

z is the measured macroscopic pointer variable which measures s. The reaction time of the detector (until equilibrium has been reached) is R; the subsequent dead time (until a further reliable measurement is possible) is T. The system prepared at time t is measured insofar as the thermodynamics equilibrium value

st := <z>t+R

is read off up to an accuracy ε.

For a sensible measurement device it is assumed that (up to the measurement accuracy ε)

st = s

for all t in EIG(s), as long as two successive measurements are separated by time at least R+T. This can be tested in a calibration phase.

The unitary dynamics of the universe is given by

<f>t :=<U(t)*>fU(t)0,


U(t)*U(t) = U(t)U(t)* = 1.

This is all one knows a priori. Obviously one can - in contrast to in Wigner's idealised analysis - not in general follow how the measurement value must appear in a prepared system. Instead of this it must be explained by an analysis using the methods of statistical mechanics. For a suitably modelled interaction this provides the required probability structure and the Born rule.

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