The Wave Function is a philosophy anthology about the wave function of
quantum mechanics. The wave function
specifies the state of the quantum mechanical system in a way similar to how
the ideal gas law specifies the state of a dilute gas. You can make more or less of that, if you
wish. But if you’re a philosopher, you’ll
make more.

This book is not really about the wave function, what it
does, and how to care for it, it is a discussion of David Albert’s thesis
exposited in the 1996 paper, “Elementary Quantum Mechanics.” In this paper he looked the wave function as
a real thing, and said that if it is real, then the universe must exist in
3N-dimensions, where N is the number of particles in the universe. This is because the wave function a system of
particles is a collection of positions for those particles.

I’ll discuss each chapter in turn. You might think the description is a little short
for some of them, but the review has gotten pretty long.

1. David Albert, “Wave Function Realism”

In this essay, David Albert of the philosophy department of
Columbia University discusses his idea on how to view the wave function
realistically. Realist, in the
philosophical sense, of the wave function is a real thing, and so its nature
can be used to tell us something about the nature of the rest of the world.

Since this is the view that most of the remainder of the
essays discuss, and all address, it’s a good thing to go into detail about this
here. If the wave function is a real,
physical object, it is a kind of field.
In physics, the word field refers to an object that can be represented
as a function, which can be scalar, vector, or tensor-valued, that has
different values at different points in space.
The velocity field of a stream, for example, is a vector field that
tells you how fast and in what direction the water in that stream is moving at
that point. In a steady state, even
though the water is different at every instant, the current is the same at every
point. Its domain is the physical, three
dimensional space that composes the stream (technically, it could be all
space), and its range is the three dimensional velocity vectors that the water
can travel at (magnitudes and directions.

What Albert noticed is that the domain of the field is, in
all of physics, the 3D space that we live in or a subset of it. All of physics, that is, except in quantum
mechanics, where the domain of the wave function is the possible positions of
each of the particles that the wave function describes (is that true?*), and so
instead of being a 3-dimensional space, it is a 3N-dimensional space with N
being the number of particles. Albert’s
leap was to say that since quantum mechanics is the foundational theory of the
world, this 3N-dimensional space is the REAL world whereas our usual
3-dimensional space is an apparition based on the relationships between large
number of particles.

The reason why we don’t see the 3N world is basically a
brain-in-a-vat type of problem.

2. Valia Allori, “Primitive Ontology and the Structure of
Physical Theories”

Valia Allori, a philosopher at Northern Illinois University,
tries to understand all this in a very philosophical way. She invents sub-categories within categories
that you’d never heard of. In this case,
she starts talking about the “primitive ontology” of a theory. This is all, if I recall, along the same
program as Albert.

An ontology in the philosophy of science is the collection
of thing in the world on which the theory can function, whether they be atoms
or charges or point particles. A
primitive ontology is the minimum ontology for the theory to function. This varies from theory to theory, and it has
a set of “primitive variables” which create the minimum parameterization that
allows you to translate the objects of the primitive ontology into mathematics.

Allori analyzes three and a half theories with this system:
Bohmian mechanics, the Ghirardi-Rimini-Weber (2x versions), and the many-worlds
interpretation. These three
interpretations keeps coming up, and not very many more, so I wonder if most of
the philosophy of quantum mechanics is a detailed response to John Bell,
especially the collection The Speakable and Unspeakable in Quantum Mechanics –
since those were, really the three that he detailed in that book.

3. Steven French , “Whither Wave Function Realism”

Steven French, a philosopher at the University of Leeds,
wonders whether the wave function is the right thing for the realist
philosopher of science to consider as part of the ontology of the theory. He feels that overestimating the importance
of the wave function in using quantum mechanics to tell us about the world
underdetermines the theory and leaves us with a rather vague idea about what
really exists.

4. Sheldon Goldstein and Nino Zanghi, “Reality and the Role
of the Wave Function in Quantum Theory”

Sheldon Goldstein, a mathematician at Rutgers, and Nino
Zanghi, a physicist at the University of Genoa,
wonder just what it is that a wave function can be, and there are
several things that look at. First of
all, there can be no such thing as the wave function in the world. It is just a computational tool. Next, it could be an epistemic representation
of our subjective knowledge of the system.
That is, it isn’t physical but it has something to do with the state of
something physical – basically, the state of our brains. Or it can be some fact or object in the world
– a thing in the world. That it, the
wave function could be nothing, it could be epistemic, or it could be real.

The main point of most of these papers is to analyze and
criticize Albert’s wave function realism, so it is the last that is
interesting. If the wave function is
real, there are two possibilities: it could be nomological or material, or at
least partially one or the other will a little subjectiveness or nothingness
thrown in. If it is nomological, it is
a fact about the world, like Gauss’ Law.
If it is material, it is a real thing, like a changed pith ball. But again, they give themselves a little
wiggle room by allowing the wave function to be either quasi-nomological or
quasi-material. It might be factish or
thinglike.

Just like the Allori paper, Goldstein and Zanghi analyze a
group of different interpretations of quantum mechanics to determine what role
the wave function plays in each according to this categorization. If you’re interested enough in which is what
and what is which, you’re probably interested enough to read the book, so I’ll
save myself some time and not make out a table.

5. Peter Lewis, “Dimension and Illusion”

Peter Lewis, a Dartmouth philosopher that was at the
University of Miami when The Wave Function was published, gives a pragmatic
analysis of Albert’s thesis. And it’s no
surprise what a pragmatist will think about a 3N-dimensional world.

6. Tim Maudlin, “The Nature of the Quantum State”

Time Maudlin, New York University Philosopher, provides the
most direct assault on Albert in this book.
That is, he goes after the main method of analysis – producing an
ontology from the mathematics – in order to show that 3N-dimensional space isn’t
necessary. He does this both by careful
analysis of Alberts 1996 paper and with an analogy to Fourier’s Analytical
Theory of Heat, which provided a metaphysical cover for the caloric fluid model
of heat.

That particular induction was natural. The equations in the theory of heat flow are
the same as those as for current flow in liquids. So, if you don’t have any idea about
statistical mechanics, it’s the most natural thing in the world to see heat as a
current of some sort of fluid instead of just energy transfer.

And of course, that didn’t work.

Maudlin’s conclusion is justified: looking at the
mathematical form that a theory has to take does not require you to take
implications of the mathematics to be real – to be in the ontology of the
theory, as the philosophers put it. Not
only is not necessary, it’s not even a good reason.

7. Bradley Monton, “Against 3N-Dimensional Space”

Bradley Monton, who worked at the University of Colorado at
Boulder at the time but now philosophizes at Wuhan University, sets the tone
for this one with his first section “Quantum Mechanics is False.” Why does he say that? Because he feels that General
Relativity is the more fundamental theory of the two, mostly because quantum
mechanics synchronize their watches. This
may seem trivial, but it’s a major problem in using string theory to construct
a theory of gravity.

His main argument against the 3N-dimensional space and in
favor of 3-dimensional space as being the fundamental dimensionality of the
world is that 3-dimensional space more accurately reflects what physicists think
about the world and how they carry out experiments. And, Monton argues, unless 3N-dimensional
space can make itself useful, then there’s no good reason to take it as
fundamental.

8. Alyssa Ney, “Ontological Reduction and the Wave Function
Ontology”

Alyssa Ney, a philosopher at the University of California at
Davis, gives an account of “ontological reduction,” how one set of things can
be reduced to another set of things. In
this case, she gives an account of how our 3-dimensional experience can reduce
to the 3N-dimensional space of the wave function. You can think of this in analogy to
scientific reductionism where biology can be reduced to chemistry, for example,
for a certain idea about what biology is.
Chemistry never gives you the full picture of biology, but we have faith
that between chemistry and physics, everything about living things can be
explained in some reasonable way – although not predicted.

9. Jill North, “Structure of the Quantum World”

Jill North, now of Rutgers, once of Cornell, discusses how
Albert’s program is supported by the dynamics of the world. If the wave function changes in
3N-dimensions, then a 3N-universe is the best explanation of it. I didn’t see it before, but I see it now:
North’s view of the wave function is of the universal variety, and the
universal wave function is the most physical assumption of the many-world’s
hypothesis.

10. David Wallace, “A Prolegomenon to the Ontology of the
Everett Interpretation”

David Wallace moved from Oxford to the University of
Southern California to do his philosophizing.
Here, he talks a lot about the many-worlds interpretation.

* In the case of
identical particles, the wave function gives the probability amplitude of
finding *a* particle there. It doesn’t
tell you which one.