### here - Foundations of Physics

```Quantum Mechanics as
Classical Physics
Charles Sebens
University of Michigan
July 31, 2013
Dirk-André Deckert
Michael Hall
Howard Wiseman
UC Davis
Griffith University
Griffith University
2
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
 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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Outline
I.
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 QM
What there is (Ontology):
The Wave Function
What it does (Laws):
Schrödinger Equation
6
Solution 2: Bohmian QM
Ontology:
The Wave Function
Particles
Laws:
Schrödinger Equation
Guidance Equation
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Solution 3: Prodigal QM
Ontology:
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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The Evolution of
From the following facts one can derive the double boxed equation below.
 Worlds are distributed in accordance with psi-squared:
 The Schrödinger equation:
 The guidance equation:
The quantum
potential Q.
This equation gives a way of calculating the evolution of  , and
hence , which never references the wave function Ψ.
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Solution 4: Newtonian QM
Ontology:
Particles (in many worlds)
Law:
Newtonian Force Law
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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 Ψ 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, 210 = 1,024 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 Ψ(0) and configuration (0) 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 | |2 -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
by the particle velocities.
and the guidance equation is obeyed
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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
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* See Deutsch (1996), Brown & Wallace (2005).
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
22
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
23
References
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End
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Ontological Options
Option 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 Constraint
To get a feel for the Quantization Condition, let’s look at the
following case:
Consider the  = 2,  = 1,  = 1 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
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