A flea on Schrödinger’s Cat
Continuing the conference talks presentations, in “Asymptotic Theory Reduction and the Measurement Problem”, Klaas Landsman introduced a fresh start on the measurement problem.
Landsman started by defining low-level and high-level theories:
L = lower-level theory = fundamental theory (physics) = reducing theory (philosophy)
H = higher-level theory = phenomenological theory (physics) = reduced theory (philosophy)
Here are some examples:
L = Quantum Physics (ħ→0) H = Classical Physics
L = Statistical Mechanics (N→∞) H = Thermodynamics
L = Molecular Dynamics H = Hydrodynamics
L = Wave Optics (wavelength→0) H = Geometric Optics
Then there are a couple of observations:
- H is deﬁned and understood by itself
- H has ‘novel’ feature(s) not present in L (classical physics has counterfactual definiteness, thermodynamics allows irreversibility, etc)
Also a quick observation is that technically ħ→0 is apparently impossible because ħ is a constant. However ħ→0 is actually shorthand for changing a dimensionless combination. For example ħ2/2m→0 is the same as m→∞
Now back to the measurement problem. Here there is no consensus among physicists: some claim it is not a problem, some that it is a pseudo-problem, some that it is a very serious problem. Basically the problem is that quantum mechanics fails to predict that measurements have outcomes:
-theoretically, Schrödinger’s Cat states of L yield mixed limit states of H
-experimentally, outcomes are sharp, hence pure states in H
Regardless of physicist’s consensus (or lack of it), this can be stated in a mathematical precise way:
H is ħ→0 limit of L, but limit L→H induces the wrong classical states.
The proposed solution: asymptotic reduction similar with spontaneous symmetry breaking. Here is how it works in a completely soluble example:
Start with a symmetric double well potential in classical physics which has reflection symmetry. A test classical particle at rest can reside either at the left or right potential well bottom, but not in the middle (between the two minima) because it is an unstable equilibrium point. Hence a symmetric invariant state (a mixed state) in unphysical.
Now solve the same problem in quantum mechanics and observe that the ground state (the lowest energy state) in symmetric! Then we can take the limit ħ→0 and obtain two sharp localization peaks.
But then here comes a flea!
It can be shown analytically in several problems (double well potential, quantum Ising model, quantum Curie-Weisz model), that a tiny perturbation induces an exponential splitting of the lowest pair of energy levels. And this shows that the ground state of perturbed Hamiltonian shifts to a localized state and the density matrix not only decoheres but gets single peak!
In turn this implies that asymptotic emergence does not exist because tiny perturbations have exponentially large effects! In short, reduction is real, emergence is not.
The “Old Measurement Problem” solutions also involve reduction. The “Old Measurement Problem” demands to show that a pure state evolves to a mixt state. The proposed solutions were:
-Classical description of apparatus (Borh) for ħ→0 or N→∞
-Superselection sector (Hepp, Emch, Wigner) for N→∞
-Decoherence (Joos, Zeh, Zurek) for t→∞
Then selection of one term in mixture would completely solve the measurement problem.
The proposed solution shows how pure unitary time evolution is compatible with wavefunction collapse. So the
interpretation is NOT the only game in
town when it comes to unitary evolution explanation of the collapse.
There are several open problems related to this approach:
- Where does the “flea” come from?
- Are the perturbations deterministic or stochastic?
- Is the mechanism dynamically viable?
- Is the mechanism experimentally testable?
- Is the mechanism universal?