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.

E_{S} is the algebra of system variables. For a single spin this is the algebra of complex 2x2 matrices [A_{11},A_{12};A_{21},A_{22}]. J: E_{S} —> 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 ρ_{t}, which is determined by

<J(A)>_{t} = trace ρ_{t} A

for all A in E_{S}. If the relationship

ρ_{t} = ψ_{t}ψ_{t}^{*}

holds at time t, one says one has prepared the subsystem in the pure state ψ_{t}.

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

<J(A)>_{t} = A_{ss}

for all A in E_{S}. 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 ψ_{t}, so that ρ_{t} = ψ_{t}ψ_{t}^{*} and then

<J(A)>_{t} = ψ_{t}^{*}Aψ_{t}

for all A in E_{S}. 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

s_{t} := <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)

s_{t} = 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},

with

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