[Fis] Group Theory, Quantum Mechanics, and Music

Dear FIS colleagues,

Jerry's recent letter raises the issue
of the link between chemical structure
and group theory.

Perhaps, at this stage it is therefore
useful to remind ourselves of the
relation between group theory and
quantum mechanics. I will then
add my new group-theoretic formulation
of symmetries in quantum mechanics,
and show its deep connections to music.

Standardly, two spaces are involved in quantum
mechanics: (1) The configuration space, e.g.,
in standard 1-D problems, this is the real line;
and (2) the infinite dimensional
Hilbert space of functions over
the configuration space.
Schroedinger's equation prescribes the
deterministic evolution of any member of
the Hilbert space over time.
It does so in the following way:
(For ease of exposition, I will use
the term "Hamiltonian operator"
for i/h H, where h is the conventional
h-bar and H is the conventional
Hamiltonian operator.)

A physical situation defines a Hamiltonian,
which is a vector in the Lie algebra
of the unitary group.

The Schroedinger equation for the
physcial situation is a map in which
the Hamiltonian operator
sends any state (member of Hilbert space)
to a velocity vector for that state.
In other words, Schroedinger's equation
defines a vector field on Hilbert space.

Thus, the Hamiltonian, a single vector
in the Lie algebra, prescribes an entire
vector field on Hilbert space.

Now in the standard relation between
a Lie algebra and a Lie group,
the exponential map sends the former
to the latter. Via this map a single vector
in the Lie algebra can be regarded
as generating a 1-parameter subgroup
of the Lie group. In the particular case
of quantum mechanics, the Hamiltonian
(within the Lie algebra) generates a
1-parameter subgroup of the unitary
group.

Now the unitary group is an isometry group
(in fact, rotation group) on Hilbert space.
So the Hamiltonian operator is associated
with a 1-parameter group of rotations of
Hilbert space.

Symmetries of the Hamiltonian flow, commute
with the Hamiltonian operator, and send
flow lines to flow lines. They are also
rotations of Hilbert space. I have argued
that the symmetric action should be represented
by a wreath product in which the
fiber group is the 1-parameter subgroup
generated by the Hamiltonian, confined
to an individual flow line, and the
symmetry action is the control group.
This better represents the process of
constructing experiments, and using
induction on them.

This wreath product group is an example
of what I define as an iso-regular group:
an n-fold wreath product, in which each
level is on one generator, and is represented
as an isometry group.

I then show that the melodic and rhythmic
anticipation hierarchies of music are given
by iso-regular groups. We therefore see
a deep relationship between the structure
of music and the structure of quantum mechanics.
You can read my chapters on music and on
quantum mechanics, on-line at the Springer-Verlag
web-site, which can be accessed via
http://www.rci.rutgers.edu/~mleyton/homepage.htm

best,
Michael Leyton

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Received on Tue Jun 3 14:19:58 2003