Hello!

I'm trying to find some "basic" information about
sympletic geometry.  I can barely find a definition of it
that I can understand.  Why does it only apply to even
dimensions?  How does it relate to physics?  Please 
explain from a background of in-depth understanding rather
than just searching the web--Thanks!!

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Clarification of Question by whack-ga on 20 Jan 2003 17:35 PST

Oops!  I meant "symplectic" !  See how much help I need??

Thanks--

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Request for Question Clarification by mathtalk-ga on 20 Jan 2003 21:59 PST

Hi, whack-ga:

It might be expeditious if you provided a bit of information about
your math background.  Symplectic geometry can be explained readily
only by drawing on some concepts from topology and algebra such as
would often be covered in the first year of graduate studies in
mathematics.  Naturally this does not rule out the potential of a
bright undergraduate or exceptional high school student to delve into
this material, but one's pacing in picking up new material is dictated
as much by one's breadth of knowledge (thus knowing how to learn) as
by the intrinsic difficulty of the subject matter.

A compact explanation can be given of why only even dimension
manifolds admit symplectic forms.

Symplectic geometry may be seen as rooted in the classical physical
applications of Hamiltonian mechanics.  Please clarify what balance
should be sought between explaining these links versus recent
developments.

thanks, mathtalk-ga

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Clarification of Question by whack-ga on 20 Jan 2003 22:49 PST

mathtalk-ga and jm7197-ga--

My background is calculus and abstract algebra,
but no formal topology or graduate math.  I'm
trying to understand specific articles dealing
with quantum mechanics that are way beyond my
grasp.  Thanks so much for responding!

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Request for Question Clarification by mathtalk-ga on 21 Jan 2003 12:37 PST

See my comment at bottom.  Let me know if you wish me to post some
basic introductory material (definition, proof of even dimensions) as
an answer.  I'm afraid that applications to quantum physics would not
be practical to develop in this format.

best wishes, mathtalk-ga

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Clarification of Question by whack-ga on 21 Jan 2003 23:35 PST

mathtalk-ga,

You have already provided great information--thanks much!

As far as an answer--I would be happy with a discussion
of a symplectic form that I could explain to a high-school
student.  Some basic introductory material would be great!

Hi, whack-ga:

I think the discussion of symplectic forms with a high-school student
would go like this:

You know that points in a plane can be identified by Cartesian
coordinates:

O = (0,0), P = (x_1,y_1), Q = (x_2,y_2), etc.

Euclidean geometry studies quantities like length and angle that are
"preserved" by "rigid motions" (rotation, reflection, translation). 
It turns out that the measurement of both length and angle can be
algebraically defined in terms of a vector dot-product "*":

Let u = [x_1 y_1] = vector OP
and v = [x_2 y_2] = vector OQ 
both be non-zero.

The dot-product is:
       u*v = (x_1)(x_2) + (y_1)(y_2).

The distance from O to P is:
            ||u|| = sqrt( u*u ).

The cosine of angle between OP and OQ is:
          ( u*v )/( ||u|| ||v|| ).

Note that OP and OQ are perpendicular if and only if u*v = 0.  For
this reason we refer to dot-products being zero as an orthogonality
condition.

Symplectic geometry studies properties of a modified formula:
       u#v = (x_1)(y_2) - (x_2)(y_1).

Note that while the dot-product u*v = v*u is _commutative_ , this
modified formula is _anti-commutative_ , meaning that:

       u#v = - v#u

Although not obvious, there is a geometric interpretation of this
formula.  It gives the "signed area" of the parallelogram generated by
OP and OQ (the parallelogram with corner points O, P, P+Q, and Q).  If
the rotation from OP to OQ is positive (counterclockwise) and less
than 180 degrees, we get a positive area.  If the rotation is negative
(clockwise) and less than 180 degrees, we get a negative area.  Etc.

This is a very simplified example of what symplectic geometry is
about.  Just as the "origin" in the Euclidean plane can be any point
and we can pick coordinate axes through that point in infinitely many
ways, the coordinates for a symplectic form can be those based around
any point.  The technical name for the sort of "space" made up of
these points is a smooth manifold.  Just as a closed surface such as a
sphere or a donut generalizes the simple idea of a flat plane, so too
do smooth manifolds generalize the higher dimensional spaces.

Here's an outline of the general definition of a symplectic form.  One
has an bilinear "form" W defined at each point of a manifold, so that
with respect to local "coordinates" there:

W(u,v) = -W(v,u) [the anticommutative property]

W varies continuously with the base point at which it is defined; that
is, if O and O' are two very nearby points on the manifold, the
versions of W defined at each of them are very nearly equal.

There are two requirements on W( , ) that must be satisfied to make a
symplectic form.  The first called "nondegeneracy", and it means that
for each nonzero u, there exists a nonzero v such that W(u,v) is
nonzero.

The nondegeneracy condition implies that the underlying manifold will
have to be even dimensional, which is why we used the plane as an
example above, rather than say three-dimensional space.

The second condition required of a symplectic form is called being
"closed".  This is more technical and would require a deeper
development of notation to explain properly.  It has an analog in
simple vector geometry called Jacobi's identity, concerning a triple
sum of cross-products:

Ax(BxC) + Bx(CxA) + Cx(AxB) = 0 [the Jacobi identity]

For a non-technical (but not high school level either) discussion of
these and other aspects of your subject, see this 1992 paper by Gotay
and Isenberg:

[The Symplectization of Science]
http://www.math.hawaii.edu/~gotay/Symplectization.pdf

regards, mathtalk-ga

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Clarification of Answer by mathtalk-ga on 09 Feb 2003 13:34 PST

Thank YOU for taking time to rate my answer, and for posting such an
interesting question.  Your humorous phrasing made me chuckle more
than once!

-- mathtalk-ga

Thanks for all the time you spent with comments, also--The
paper that you refer to has lots of great information, too!

whack-ga

These answers might lead you to more questions. Without knowing your
level of math knowledge, I can only offer two answers (simple answer)
or text-book definition).

simple answer:  Symplectic Geometry is a non-Euclidian geometry. The
geometry most learn in junior high school and high school is Euclidean
Geometry. You know, "parallel lines never meet".  Other types of
geometry such as spherical geometry states that "parallel lines always
meet".


text-book definition:

A symplectic form on a smooth manifold M is a smooth closed 2-form "w"
on "M" which is nondegenerate such that at every point "m", the
alternating bilinear form  on the tangent space  is nondegenerate.

A symplectic form on a vector space V over  is a function f(x,y)
(defined for all  and taking values in ) which satisfies 
http://mathworld.wolfram.com/s3img3143.gif  and 
http://mathworld.wolfram.com/s3img3149.gif.


f is called non-degenerate if  for all y implies that x = 0.
Symplectic forms can exist on M (or V) only if M (or V) is
even-dimensional. An example of a symplectic form over a vector space
is the complex Hilbert space with inner product  given by

 



FYI, if interested, here are other forms of geometry.

Absolute Geometry, Affine Geometry, Cartesian Coordinates,
Combinatorial Geometry, Computational Geometry, Coordinate Geometry,
Differential Geometry, Discrete Geometry, Enumerative Geometry,
Finsler Geometry, Inversive Geometry, Kawaguchi Geometry, Minkowski
Geometry, Nil Geometry, Non-Euclidean Geometry, Ordered Geometry,
Plane Geometry, Projective Geometry, Sol Geometry, Solid Geometry,
Spherical Geometry, Stochastic Geometry, Thurston's Geometrization
Conjecture

Hi, whack-ga:

I'd love to help, but as Euclid is reported to have told King Ptolemy
I of Egypt, "There is no royal road to geometry."

[Euclid]
http://fclass.vaniercollege.qc.ca/web/mathematics/people/euclid.htm

Between where you are and a familiarity with applications of
symplectic geometry in quantum physics there is a road, but it will
not be possible to cover it for you in the space of a Google Answers
textbox.

I could explain in detail why odd dimensional manifolds cannot support
a symplectic form, but you might be able to follow for yourself the
proof (modelled along the lines of a Gram-Schmidt calculation) of the
first theorem here:

[Lectures on Symplectic Geometry by Ana Cannas da Silva]
http://books.pdox.net/Math/Lectures%20on%20Symplectic%20Geometry.pdf

which shows that as a corollary to a representation theorem for
skew-symmetric bilinear forms.

That on-line book (222 pages, takes a while to download even on my
cable modem) seems to me an excellent resource, certainly for someone
who wishes to recapitulate the classical Hamiltonian mechanics from a
modern perspective.

Other than providing a detailed explanation of the definition (which
you may already feel comfortable with) and proof of the even
dimensional part of your question, I doubt that I can help.  There are
some on-line courses and other educational programs that you should
consider if you wanted to make the investment in learning this
material.  This site in particular, about an on-line course taught at
the University of Houston, seems aimed at engineers with a background
similar to yours:

[CSDC: Applications of Exterior Differential Forms]
http://www.uh.edu/~rkiehn/ed3/ed3homep.htm

The author provides an email address which you might use to contact
him about the availability of the class, although his lecture notes
are available on-line for independent study.  He also has this PDF
file on a related subject:

[CSDC: Hamiltonian Extremal and Symplectic Physics]
http://www22.pair.com/csdc/ed3/ed3fre12.htm

For researchers with a more advanced background, there is this
semester long symposium-style program starting in March 2003:

[IPAM: Symplectic Geometry and Physics]
http://www.ipam.ucla.edu/programs/sgp2003/

regards, mathtalk-ga