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

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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