Subject: Re: An answer to a John Baez puzzle (I hope :
)

From: baez@math.ucr.edu (john baez)

Date:
1997/04/06

Message-Id:
<5i905g$2jm@charity.ucr.edu>

Newsgroups:
sci.physics.research

[More Headers]

In article
<E8899E.51E@world.std.com>,

Doug B Sweetser
<sweetser@alum.mit.edu> wrote:

>John Baez
wrote:

>>What sort of vector space has

>>both
a nondegenerate metric AND a symplectic structure?
Such

>>a thing would let you study both bosons and
fermions, in a

>>unified way. It would obviously be very
important.

Ah yes, that old puzzle. I include the answer at
the end of

this post. Anyone who wants to solve it on their own,
don't read

the end of this post!

>I can solve a closely
related puzzle:

>

>Q*: What sort of field has both a
nondegenerate metric AND a

> symplectic
structure?

>

>A*: The field of quaternions has both
properties.

In my puzzle I was requiring that the metric
g(v,w) and

symplectic structure omega(v,w) be real-valued, since
that's

what comes up naturally in the study of fermions and
bosons,

respectively. Your metric and symplectic structure are
quaternionic.

Perhaps that's not too much of a stretch: after all, if
you like

quaternions, why not accept quaternion-valued metrics
and

symplectic structures?

However, if I'm not mistaken,
you can also find a natural

*real-valued* metric and symplectic
structure on the quaternions.

The reason is that the group of
automorphisms of the quaternions

is SU(2), which is a subgroup
of U(2), and U(2) is the intersection

of O(4) (the group of linear
transformations of R^4 preserving the

standard metric) and
Symp(4) (the group of linear transformations of

R^4 preserving
the standard symplectic structure.) There should

thus be a *real-
valued* metric and a *real-valued* symplectic

structure on the
quaternions, both preserved by all automorphisms

of the
quaternions.

If the above groups are not familiar, take a look
at

http://math.ucr.edu/home/baez/week90.html

and
the links to earlier "Weeks".

I don't know how
the metric and symplectic structure I'm hinting

at are related to
the ones your propose, and I'm too lazy to work

it out. My metric
might be the real part of your metric, but my

symplectic
structure can't be the real part of your symplectic

structure, since
the real part of your symplectic structure is

zero!

>
omega(v, w) = 1/2 (v* w - w* v)

>

> = 1/2 (vt - vx I -
vy J - vz K)(wt + wx I + wy J + wz K)

> - (wt - wx I - wy J
- wz K)(vt + vx I + vy J + vz K)

>

> = ((vt wx - vx wt) -
(vy wz - vz wy)) I

>

> ((vt wy - vy wt) - (vz wx - vx
wz)) J

>

> ((vt wz - vz wt) - (vx wy - vy wx))
K

>A question remains: did I stretch the rules too much to
solve

>the puzzle?

Well, you sure didn't solve *my*
puzzle, but you raised some

interesting issues, which is just as
good. In particular, you

might be interested in thinking about the
relation between the

group Symp(2n) --- the group of linear
transformations preserving

a symplectic structure on R^{2n} ---
and the group Sp(n) --- the

group of n x n quaternionic matrices
preserving lengths. I say

a wee bit of what's known about this in
the issues of "This Week's

Finds" you can get to
starting with the one listed above. One

basic fact is that they are
"real forms" of the same complex Lie

group.

Warning: in the literature you will find a bewildering
variety

of notations for what I'm calling Symp(2n) and Sp(n)
above. In

particular, though they are different groups, both are
often called

the "symplectic group". It took me about
a decade to figure out

what the heck was going on with these
groups, thanks in part to

this notational confusion.

Okay,
now for my puzzle:

Q: What sort of vector space has both a
nondegenerate metric

AND a symplectic structure?

A: A
complex Hilbert space.

This is the beginning of a long, long
story. For more than you

want to know about it, try the book
"Introduction to Algebraic and

Constructive Quantum Field
Theory". For a lot less, try

http://math.ucr.edu/home/baez/harmonic.html

This
explains the relation between:

1) rotations and why there are
spin-1/2 particles

and

2) symplectic transformations
and why the zero-point energy of

the quantum harmonic
oscillator is 1/2

Note: in that web-page I use the notation
"Sp(2n)" for what

I call "Symp(2n)" in the
above post. I warned you!

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