Theo M. Nieuwenhuizen
CCPP, NYU &
ITP, Univ. of Amsterdam
Armen Allahverdyan, Yerevan
Roger Balian, Saclay
Europhysics Letters 2003 arXiv 2004,Vaxjo 2005, 2006
Beyond the Quantum 2007 arXiv 2008
Opus Magnum, in progress
155 pages, Physics Reports
Foundations of Probability and Physics -6
Linneus University, Vaxjo, 16-6, 2011

Statistical interpretation of QM
Problems & paradoxes + the big questions
The model: system S + apparatus A spin½ A = M + B = magnet + bath
Selection of collapse basis & fate of Schrodinger cats
Registration of the Q-measurement & classical measurement
Post measurement & the Born rule
The Q measurement problem elucidated
Summary

Density matrix describes ensemble of identically prepared systems
Pure state is limiting case; same meaning (no special role)
Ensemble can be real: (many particles: bundle at LHC one trapped ion in photon field, repeated excitation) or virtual: (as in classical statistical physics)
QM = tool for calculating averages from density matrix ρ
=> QM = about what we can measure, not about what is epistomology  ontology
Quantum measurement theory describes ensemble of measurements on ensemble of identically prepared systems

Einstein, Bohr, de Broglie, von Neumann, Wigner,
Bohm, Bell, Balian, van Kampen ...
Collapse = non-unitary
Small part of apparatus described by QM  why not measurement process?
Preferred basis paradox: on which basis will reduction take place?
When does collapse happen ? How long does it take?
What happens to Schrödinger cats?
The biggest of them all:
The quantum measurement problem:
QM = statistical theory, but in experiments we see individual outcomes
Solvable within QM ?? => probably not …or??

R1: simulate as much as possible real experiments;
R2: ensure unbiased, robust, permanent registration by macroscopic pointer
R3: to amplify quantum signal: involve an apparatus initially in a metastable state and evolving towards one of the stable states under the influence of S; the transition of A, instead of occurring spontaneously, is triggered by S;
R4: include a bath where the free energy may be dumped;
R5: be solvable so as to provide a complete scenario of the joint evolution of S + A and to exhibit the characteristic times;
R6: conserve the tested observable;
R7: lead to a final state devoid of “Schrodinger cats”;
R8: satisfy Born’s rule for the registered results;
R9: produce, for ideal measurements or preparations, the required diagonal correlations between the tested system and the indication of the pointer,
R10: be sufficiently flexible to allow discussing imperfect measurements.

System S: s = spin½. For ideal measurement:
Apparatus = Magnet + Bath (A = M + B).
M = N spins½ σ (n) . Pointer = magnetization
B = phonon bath: harmonic oscillators

• J
2
• J
4 term: Curie-Weiss magnet term: allows first order phase transition
• Extreme case of first order transition: J
2
= 0, J
4
= J

Standard harmonic oscillator bath: x,y,z components of the spins of M couple to their own harmonic oscillators
Bath characterized by correlator K(tt’)
Weak MB coupling: γ << 1 allows to work at lowest order in γ

Tested system S : arbitrary density matrix uncorrelated with A
Apparatus A starts in mixed state ,
(large system cannot be prepared in pure state)
Magnet M: N spins ½, starts as paramagnet ( mixed state )
Bath: Gibbs state ( mixed state ) β = 1/T
Apparatus starts in mixed state => system may end up in mixed state
No unitarity paradox

What selects collapse basis? The interaction Hamiltonian !
Liouville-von Neumann eqn
The 4 sectors decouple ideality of the measurement
Motion of the spin a=z: Diagonal terms of r(t) conserved ->= r ii
(t) = r ii
(0 ): Born probabilities a=x,y: Off-diagonal terms evolve => disappearence of Schrodinger cats

Short times: spin-spin couplings in magnet are irrelevant x,y components of spin n of M moves in the magnetic field g of spin S
Larmor procession damping by zero-point flucts of bath
Schrodinger cat: dephasing decoherence as in NMR comes in later
Recurrences eliminated by: decoherence by bath N B >> 1 or even by a spread in g’s
Creation of multi-particle correlations: weak, dying out
Schrodinger cat terms die quickly in cascade of weak multi-spin correlations

Cat suppression factor time
• Reduction is ongoing (not once-and-for-all), like the repeated Stosszahlansatz in the Boltzmann equation
• Through resonant excitation of bath modes
( is energy needed to flip spin of M)

Solve Q-dynamics of diagonal elements to second order in the coupling to bath
In sector with s z
=1: analogy to classical measurement of classical Ising spin s z
=1
Magnetization m involves only eigenvalues , as in classical statistical physics
Measure a spin with a “classical” apparatus of magnet and a bath
Dynamics
Free energy F=U-TS: minima are stable states of free energy m

Free energy landscape: classical Curie-Weiss model
At g=0: m
High T: paramagnet is stable m
Low T: paramagnetic initial state; stable ferromagnetic states can act as measuring apparatus

Perform measurement: turn on coupling field h=g s z m
Bath is needed to dump the free energy m
Procedure:
Turn off coupling g after minimum is reached
Magnet goes to ferromagnetic minimum at g=0. Stays there a time exp(N)
(Result is stable and may be read off, or not)

•
• Finally: sharp peak at m=+m
F in sector s z
=+1 m=−m
F in sector s z
=−1

Magnet ends up in up/down ferromagnetic state
Sign of magnetization maximally correlated with sign spin S
Probabilities satisfy Born rule
No Schrodinger cat terms
Physical disappearance: sums of many oscillating terms vanish despite of mathematical survival: individual terms have fixed amplitude

The macroscopic magnet itself is well understood.
Initial paramagnet relaxes to stable  or  ferromagnet
Ensemble theory describes breaking of ergodicity
Individual  and  ferromaget is stable, Poincare time is very large
In Q measurement this transition is triggered by coupling to the spin S
Long life time of FM state ensures uniqueness of outcome for m= ± m
F and for the measured spin s z
= ± 1 that is fully correlated with it
No survival of Schrodinger cat terms, that could spoil this view.
One may select individual outcomes with magnetization 
(possible because ferromagnetic  state longlived)
This unambiguous subensemble is a pure ensemble of spins  of S

Summary: the measurement problem elucidated
• Collapse basis determined by interaction Hamiltonian
• Measurement in two steps: cats die & registration of the result very fast is slower cascade of small multi-particle correlations ongoing reduction
• Registration : some classical features (  Bohr) macroscopic pointer: irreversible dynamics, entropy increase
• Born rule results from the dynamics
• Observation of pointer is irrelevant for outcomes (  Wigner)
(results may be read off, or processed automatically)
• Statistical interpretation of QM
• Individual outcomes due to irreversible phase transition in magnet long lived pointer indication, no cat terms

How to teach QM
• Copenhagen interpretation
• Statistical interpretation: minimalistic, simple

Statistical interpretation: QM describes ensembles, not single systems
Q-measurement theory = about ensembles of measurements
• Solution gives probabilities for outcomes of experiments:
• system in collapsed state + apparatus in pointer state
• Density matrix describes my knowledge about the ensemble of identically prepared systems
• Should be updated after measurement

Statistical interpretation of QM
Einstein again wrote on it even in 1955
Kemble : in his book 1937
Ballentine: Rev Mod Phys 1975 van Kampen: Am J Phys 1988
Balian: Am J Phys 1989
Statistical interpretation: a density matrix (mixed or pure) describes an ensemble of systems
Stern-Gerlach expt: ensemble of particles in upper beam described by |up>
Q-measurement theory describes the statistics of outcomes of an ensemble of measurements on an ensemble of systems

It cannot and should not describe individual experiments
(otherwise than in a probablistic sense)

• Why do we see individual outcomes in individual measurements?
• Perhaps because there is an underlying reality in which the events occur (“world”, “nature”, “life”)?
• Should there exist a theory for that?
Hidden-variables theory, sub-quantum mechanics

• Copenhagen: each system has its own wavefunction, that has the most complete info in a statistical sense.
There is an underlying wavefunction of system + apparatus
• Ghirardi-Rimini-Weber GRW: collapse is a physical process, modification of QM needed to achieve the collapses.
• Multi-universe: at each measurement one of the branches is chosen, other ones in other universes.
• Wigner’s friend: the observer “finishes” the measurement
• Decoherence picture: pointer is a mode of the bath
( The air measures my height )

as an identification at the macroscopic level
In practice: describes statistics of pointer states determine relative frequencies (von Mises)
Post-measurement state of tested system is a mixed state,