Quantum Mechanics as Classical Physics

Charles SebensUniversity of Michigan

July 31, 2013

Dirk-André Deckert Michael Hall Howard WisemanUC Davis Griffith University Griffith University

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A Strange Interpretation of QM

Deterministic, no collapse No wave function, no Schrödinger equation Many worlds Worlds do not branch The number of worlds is finite Particles follow Bohmian trajectories Equation of motion of the form

This is a development of the hydrodynamic interpretation, originally proposed by Madelung (1927) and developed by Takabayasi (e.g., 1952) among others.

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OutlineI. A Series of Solutions to the Measurement Problem

I. The Many-worlds Interpretation

II. Bohmian Mechanics

III. Prodigal QM

II. Newtonian Quantum Mechanics

III. A Strength: Probability

IV. A Weakness: Non-quantum States

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Refresher: The Double-slit Experiment

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Solution 1: Everettian QMWhat there is (Ontology):

The Wave Function

What it does (Laws):

Schrödinger Equation

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Solution 2: Bohmian QMOntology:

The Wave Function

Particles

Laws:

Schrödinger Equation

Guidance Equation

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Solution 3: Prodigal QMOntology:

The Wave Function

Particles (in many worlds)

Laws:

Schrödinger Equation

Guidance Equation

This is a significantly altered variant of the proposal in Dorr (2009).

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Two Important Characters: is the density of worlds in configuration

space, scaled so that . By hypothesis, . gives the velocity of the -th particle in the

world where particles are arranged . The evolution of is determined by the velocities of the

particles in the various worlds by a continuity equation:

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Worlds are distributed in accordance with psi-squared:

The Schrödinger equation:

The guidance equation:

The Evolution of From the following facts one can derive the double boxed equation below.

This equation gives a way of calculating the evolution of , and hence , which never references the wave function .

The quantum potential Q.

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Ontology:

Particles (in many worlds)

Law:

Newtonian Force Law

Solution 4: Newtonian QM

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Review of the Alternatives

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The Wave Function in Newtonian QM

The wave function is not part of the fundamental ontology in Newtonian QM. Still, one can introduce an object which conforms to the following conditions:

Worlds are distributed in accordance with psi-squared

The guidance equation is obeyed

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The Quantitative Probability Problem for Everettian QM

Alice makes 10 measurements of the z-spin of the following state:

In a highly idealized scenario, branches… In 252 branches (25%) she will see half up, half down. In 10 branches (1%) she will see nine up, one down (which should

be most likely, happening about 40% of the time).

In general, most branches don’t exhibit Born Rule statistics.

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No Similar Problem for Newtonian QM

Since the wave function is not fundamental but introduced as a means of summarizing information about , it is impossible for the density of worlds in a region of configuration space to deviate from psi-squared. Most worlds are in high-amplitude regions.

In Newtonian QM the uncertainty present is self-locating uncertainty. One needs to determine which world they are in since many are consistent with any particular set of experiences.

Since, in general, many worlds will be consistent with one’s experiences, each should be assigned equal credence (given the state of the universe; see Elga 2004).

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Versus Bohmian Mechanics

In Newtonian QM, each world follows a Bohmian trajectory through configuration space. So, if Bohmian mechanics can reproduce the predictions of textbook QM, Newtonian QM should be able to also.

Some Bohmian trajectories don’t predict quantum statistics. The correct long-run quantum statistics will be observed if the

universe satisfies the quantum equilibrium hypothesis.

Quantum Equilibrium Hypothesis (Teufel 2013, rough version):

“The initial wave function and configuration of the "universe" are such that the empirical distributions of subsystem configurations of (different) subsystems at (different) times with the same conditional wave function are close to the -distribution.”

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Avoiding Anomalous Statistics

In Newtonian QM, it is also possible that one’s own world does not exhibit quantum statistics.

However, it is necessarily true that the majority of worlds in any universe satisfy the quantum equilibrium hypothesis since (by definition of Ψ).

Thus, one should always expect to be in a world that is in quantum equilibrium. So, one should expect to see Born Rule statistics in long-run frequencies of measurements (see Durr et al. 1992).

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How Many Worlds?

We’ve been treating the number of worlds as continuously infinite, described by a density, and a collection of velocity fields, .

Meanwhile, I’ve claimed that the number of worlds is actually finite.

Why finitely many worlds? is easier to interpret. How many? Enough that they can be well-modeled by a

continuum. This is similar to the hydrodynamic limit in fluid mechanics.

If there isn’t really a continuum, is the force law given earlier really a fundamental law? No.

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The Quantization Condition Not all collections of can be described by a wave function. The velocity of a particle is proportional to the gradient of the wave

function’s complex phase. For there to exist a wave function which describes* , the velocity

field must satisfy the following requirement (see Wallstrom 1994; Takabayasi 1952).

Quantization Condition: Integrating particle momenta along any closed loop in configuration space gives a multiple of Planck’s constant.

*That is, a wave function for which and the guidance equation is obeyed by the particle velocities.

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Limitations of Newtonian QM

Yet to be extended to relativistic quantum physics Yet to be extended to multiple particles with spin We don’t yet have the fundamental law(s) The state space is too large in two ways:

States that violate the Quantization Condition States with too few worlds to use the hydrodynamic limit

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Neat Features of Newtonian QM

Wave function is a mere summary of the properties of particles No superpositions No entanglement No collapse No mention of “measurement” in the laws All dynamics arise from Newtonian forces The theory is deterministic Worlds are fundamental, not emergent (so avoids the need to

explain how people and planets arise as structures in the WF) Worlds do not branch (so avoids concerns about personal identity) No qualitative probability problem No quantitative probability problem Immune to Everett-in-denial objection,* not in denial* See Deutsch (1996), Brown & Wallace (2005).

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Neat Features of Newtonian QM

Wave function is a mere summary of the properties of particles No superpositions No entanglement No collapse No mention of “measurement” in the laws All dynamics arise from Newtonian forces The theory is deterministic Worlds are fundamental, not emergent (so avoids the need to

explain how people and planets arise as structures in the WF) Worlds do not branch (so avoids concerns about personal identity) No qualitative probability problem No quantitative probability problem Immune to Everett-in-denial objection, not in denial

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Neat Features of Newtonian QM

Wave function is a mere summary of the properties of particles No superpositions No entanglement No collapse No mention of “measurement” in the laws All dynamics arise from Newtonian forces The theory is deterministic Worlds are fundamental, not emergent (so avoids the need to

explain how people and planets arise as structures in the WF) Worlds do not branch (so avoids concerns about personal identity) No qualitative probability problem No quantitative probability problem Immune to Everett-in-denial objection, not in denial

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References

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End

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Ontological OptionsOption 1: World-particles in Configuration Space

Option 2: World-particles in Configuration Space and 3D Worlds

Option 3: Distinct 3D Worlds

Option 4: Overlapping 3D Worlds

Shown below for two particles in one dimensional space…

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An Unnatural ConstraintTo get a feel for the Quantization Condition, let’s look at the following case:

Consider the , , energy eigenstate of an electron in the Hydrogen atom (treating the force from the nucleus as an external potential).

If the electrons were circling the -axis a little faster or a little slower, they would not be representable by a wave function.

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The Orbital