THE PRINCIPLES OF NEWTONIAN AND QUANTUM MECHANICS The Need for Planck’s Constant, h

The aim of this book is to expose the mathematical machinery underlying Newtonian
mechanics and two of its refinements, semi-classical and non-relativistic
quantum mechanics. A recurring theme is that these three Sciences are all
obtained from a single mathematical object, the Hamiltonian flow, viewed as
an abstract group. To study that group, we need symplectic geometry and
analysis, with an emphasis on two fundamental topics:
Symplectic rigidity (popularly known as the “principle of the symplectic
camel”). This principle, whose discovery goes back to the work of M. Gromov
in the middle of the 1980’s, says that no matter how much we try to deform a
phase-space ball with radius r by Hamiltonian flows, the area of the projection
of that ball on a position-momentum plane will never become inferior to 7rr2.
This is a surprising result, which shows that there is, contrarily to every belief,
a “classical uncertainty principle”. While that principle does not contradict
Liouville’s theorem on the conservation of phase space volume, it indicates that
the behavior of Hamiltonian flows is much less “chaotic” than was believed.
Mathematically, the principle of the symplectic camel shows that there is a
symplectic invariant (called Gromov’s width or symplectic capacity), which is
much “finer” than ordinary volume. Symplectic rigidity will allow us to define
a semi-classical quantization scheme by a purely topological argument, and will
allow us to give a very simple definition of the Maslov index without invoking
the WKB method.
The metaplectic representation of the symplectic group. That representation
allows one to associate in a canonical way to every symplectic matrix
exactly two unitary operators (only differing by their signs) acting on the
square integrable functions on configuration space. The group Mp(n) of all
these operators is called the metaplectic group, and enjoys very special properties;
the most important from the point of view of physics since it allows
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the explicit resolution of all Schrodinger’s equations associated to quadratic
Hamiltonians. We will in fact partially extend this metaplectic representation
in order to include even non-quadratic Hamiltonians, leading to a precis and
mathematically justifiable form of Feynman’s path integral.
An important issue that is addressed in this book is that of quantum mechanics
in phase space. While it is true that the primary perception we, human
beings, have of our world privileges positions, and their evolution with time,
this does not mean that we have to use only, mathematics in configuration
space. As Basil Hiley puts it “…since thoughts are not located in space-time,
mathematics is not necessarily about material things in space-time”. Hiley is
right: it is precisely the liberating power — I am tempted to say the grace —
of mathematics that allows us to break the chains that tie us to one particular
view of our environment. It is unavoidable that some physicists will feel uncomfortable
with the fact that I am highlighting one unconventional approach
to quantum mechanics, namely the approach initiated by David Bohm in 1952,
and later further developed by Basil Hiley and Bohm himself. To them I want
to say that since this is not a book on the epistemology or ontology of quantum
mechanics (or, of physics, in general), I had no etats d’dme when I used the
Bohmian approach: it is just that this way of seeing quantum mechanics is the
easiest way to relate classical and quantum mechanics. It allows us to speak
about “particles” even in the quantum regime which is definitely an economy of
language… and of thought! The Bohmian approach has moreover immediately
been well-accepted in mathematical circles: magna est Veritas et praevalebit…
While writing this book, I constantly had in mind two categories of readers:
my colleagues – mathematicians, and my dear friends – physicists. The
first will, hopefully, learn some physics here (but presumably, not the way it
is taught in usual physics books). The physicists will get some insight in the
beautiful unity of the mathematical structure, symplectic geometry, which is
the most natural for expressing both classical and quantum mechanics. They
will also get a taste of some sophisticated new mathematics (the symplectic
camel, discussed above, and the Leray index, which is the “mother” of all
Maslov indices). This book is therefore, in a sense, a tentative to reconcile
what Poincare called, in his book Science and Hypothesis, the “two neighboring
powers”: Mathematics and Physics. While Mathematics and Physics
formed during centuries a single branch of the “tree of knowledge” (both were
parts of “natural philosophy”), physicists and mathematicians started going
different ways during the last century (one of the most recent culprits being
the Bourbaki school). For instance, David Hilbert is reported to have said that
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“Physics is too difficult to leave to physicists”, while Albert Einstein characterized
Hilbert’s physics (in a letter to Hermann Weyl) as “infantile”. To be
fair, we must add that Einstein’s theory was really based on physical principles,
while Hilbert’s travail in physics was an exercise in pure mathematics
(we all know that even today many mathematical texts, which claim to be of
physical interest, are too often just pure mathematics dressed up in a phony
physical language).
A few words about the technical knowledge required for an optimal understanding
of the text. The mathematical tools that are needed are introduced
in due time, and are rather elementary (undergraduate linear algebra and calculus
suffice, together with some knowledge of the rudiments of the theory of
differential forms). This makes the book easily accessible to a rather large and
diversified scientific audience, especially since I tried as much as possible to
write a “self-contained” text (a few technical Appendices have been added for
the reader’s convenience). A word to my colleagues – mathematicians: this
book can be read without any particular prior knowledge of physics, but it is
perhaps somewhat unrealistic to claim that it is an introduction “from scratch”
to the subject. Since I have tried to be intelligible by both mathematicians and
physicists, I have made every effort to use rigorous, but simple mathematics.
I have, however, made every effort to avoid Bourbachian rigor mortis.
This book is structured as follows:
Chapter 1 is devoted to a review of the basic principles of Newtonian and
quantum mechanics, with a particular emphasis on its Bohmian formulation,
and the “quantum motion” of particles, which is in a sense simpler than the
classical motion (there are no “caustics” in quantum mechanics: the latter
only appear at the semi-classical level, when one imposes classical motion to
the wave functions).
Chapter 2 presents modern Newtonian mechanics from the symplectic point
of view, with a particular emphasis on the Poincare-Cartan form. The latter
arises in a natural way if one makes a certain physical hypothesis, which we call,
following Souriau, the “Maxwell principle”, on the form of the fundamental
force fields governing the evolution of classical particles. The Maxwell principle
allows showing, using the properties of the Poincare-Cartan invariant, that
Newton’s second law is equivalent to Hamilton’s equations of motion for these
force fields.
In Chapter 3, we study thoroughly the symplectic group. The symplectic
group being the backbone of the mathematical structure underlying Newtonian
mechanics in its Hamiltonian formulation, it deserves as such a thorough study
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in its own right. We then propose a semi-classical quantization scheme based
on the principle of symplectic rigidity. That scheme leads in a very natural
way to the Keller-Maslov condition for quantization of Lagrangian manifolds,
and is the easiest way to motivate the introduction of the Maslov index in
semi-classical mechanics.
In Chapter 4, we study the so fundamental notion of action, which is most
easily apprehended by using the Poincare-Cartan invariant introduced in Chapter
2. An important related notion is that of generating function (also called
Hamilton’s “two point characteristic functions”). We then introduce the notion
of Lagrangian manifold, and show how it leads to an intrinsic definition
of the phase of classical completely integrable systems, and of all quantum
systems.
Chapter 5 is devoted to a geometrical theory of semi-classical mechanics in
phase space, and will probably be of interest to theoretical physicists, quantum
chemists and mathematicians. This Chapter is mathematically the most
advanced, and can be skipped in a first reading. We begin by showing how
the Bohmian approach to quantum mechanics allows one to interpret the wave
function as a half-density in phase space. In the general case, wave forms are
(up to a phase factor) the square roots of de Rham forms defined on the graph
of a Lagrangian manifold. The general definition of a wave form requires the
properties of Leray’s cohomological index (introduced by Jean Leray in 1978);
it is a generalization of the Maslov index, which it contains as a “byproduct”.
We finally define the “shadows” of our wave forms on configuration space:
these shadows are just the usual semi-classical wave functions familiar from
Maslov theory.
Chapter 6 is devoted to a rather comprehensive study of the metaplectic
group Mp(n). We show that to every element of Mp( we can associate an integer
modulo 4, its Maslov index, which is closely related to the Leray index. This
allows us to eliminate in a simple and elegant way the phase ambiguities, which
have been plaguing the theory of the metaplectic group from the beginning.
We then define, and give a self-contained treatment, of the inhomogeneous
metaplectic group IMp(n), which extends the metaplectic representation to
affine symplectic transformations. We also discuss, in a rather sketchy form,
the difficult question of the extension of the metaplectic group to arbitrary
(non-linear) symplectic transformations, and Groenewold-Van Hove’s famous
theorem.
The central theme of Chapter 7 is that although quantum mechanic cannot
be derived from Newtonian mechanics, it nevertheless emerges from it via the
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theory of the metaplectic group, provided that one makes a physical assumption
justifying the need for Planck’s constant h. This “metaplectic quantization”
procedure is not new; it has been known for decades in mathematical
circles for quadratic Hamiltonians. In the general case, there is, however an
obstruction for carrying out this quantization, because of Groenewold-Van
Hove’s theorem. This theorem does however not mean that we cannot extend
the metaplectic group to non-quadratic Hamiltonians. This is done by using
the Lie-Trotter formula for classical flows, and leads to a general metaplectic
representation, from which Feynman’s path integral “pops out” in a much
more precise form than in the usual treatments.
The titles of a few Sections and Subsections are followed by a star * which
indicates that the involved mathematics is of a perhaps more sophisticated
nature than in the rest of the book. These (sub)sections can be skipped in a
first reading.
This work has been partially supported by a grant of the Swedish Royal
Academy of Science.
Maurice de Gosson, Blekinge Institute of Technology, Karlskrona,
March 2001