Re: [Fis] Group Theory, Quantum Mechanics, and Music

Dear Michael:

Your posting (included below) brings many questions to mind.

Most perplexing is the question of what you are seeking to communicate.

I find no mention of the subject of my post, chemical structure, and your post.

As this is a multi disciplinary list with a wide range of readership
it is desirable to explicitly state the propositions that are of
interest to you.

While I could imagine numerous possible propositions that could be
implied from your communication, as I understand the post, it
provides a concise review of standard material on mechanics. (The
review includes a wide range of mathematical species (real line,
Hilbert spaces, operators, lie algebras, unitary group, vector
fields, etc, but does not mention labelled bipartite graphs related
to chemical structures.) Then, it introduce some recent concepts on
possible relations between mechanics and music.

I am unaware of any specific process or logic that starts from
mechanical principles and generates chemical structures. Quantum
mechanics is based on the notion that if one starts from a chemical
structure, one can, in some cases, generate a mechanical view of the
molecule. Is it possible that you are presupposing that chemical
principles are extensible in the same sense as Peano's postulates
generate the nature numbers? Or, the extensibility of other common
mathematical structures? If not, what is the connection between the
post and my post?

As an aside, I am curious about the choice of words that are invoked
in your writings.

Is it possible to distinguish between "definitio quid res" and
"definitio quid nominis" in your usage? (I view this distinction as
critical in distinguishing between natural science and mathematical
or model theoretic abstraction.)

Alternatively, from a pure conceptual perspective, the question could be asked:
If the concept of music is the same as the concept of a group, what
is the generative role of an individual's CNS in creating the
emotional response to a particular song / piece / performance? Or,
what distinguishes two individual's response to the same music?

As another alternative way of stating this question:
While it is conceivable that a correspondence within one mind may
generate a perfect matching between a conceptualization of a group
and a conceptualization of a "music", is it possible that two
genetically distant minds would generate a different group for the
same "music"? As possible sources of the distinctive interpretations
one could invoke differential hearing ranges, differential hearing
sensitivity, differential training in music, differential training in
mathematics, differential moods, and so forth.

It goes without saying that mechanical thinking is extraordinarily
useful for solving mechanical problems. Our challenge, it seems to
me, is distinguishing mechanical from non mechanical problems. In
earlier posts I referred to this distinction as the difference
between Shannon information and human communication. Speciation of
relations?

Cheers

Jerry LR Chandler

>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 Mon Jun 9 23:29:30 2003