## A flea on Schrödinger’s Cat

### New Directions in the Foundations of Physics 2014

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**ħ**→0 is the same as m→∞^{2}/2m
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

**And this shows that**__exponential splitting of the lowest pair of energy levels.__**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 Everett
interpretation is

**So the**__pure unitary time evolution is compatible with wavefunction collapse.__**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?

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