[Axiom-developer] (no subject)

[root]

Camm Maguire writes:

>Greetings!  If/when I can clear the deck with pending GCL issues, I'd
>really like to dive in to the guts of axiom regarding issues like
>this.  Please accept my apologies for being so non-helpful at the
>present time.
>
>I think Bertfried brings up an interesting distinction between the
>needs of cutting edge theoretical mathematicians, and more
>'practical' sorts like certain types of physicists and most engineers
>who really need an effective compendium of what is known about a
>subject and a tool to help verify its application.  While the former
>is interesting to me, I'm really mostly in the latter camp, which I
>think is also a much larger group in general.  There is no reason why
>the needs of both cannot be met as far as I can see.
>
>>From this latter perspective, the key groups in physics are not that
>many.  SO(2) which most would refer to as U(1), while extremely
>simple, has profound implications for electricity and magnetism as well
>as quantum mechanics, arguably setting the stage for the general
>gauge-invariant pattern of the interaction of force with matter.
>SO(3) and its quantum-mechanically allowed double cover SU(2), governs
>the rotational symmetry of our three dimensional space, as well as
>providing a separation of fundamental particles into Bose and Fermi
>statistical camps.  SO(3,1) and its double cover SL(2,C) governs
>relativity and the division of antimatter from matter.  These, IMHO,
>are the truly well understood groups with nevertheless far reaching
>implications.  SU(3) describing the symmetry of the strong
>force/quantum chromodynamics is basically understood, but I think the
>implications of asymptotic confinement are still being digested
>somewhat.  Higher up in the Lie Group chain, SU(5) governs one of the
>simpler schemes for a Grand Unified (field) Theory (GUT), while the
>exceptional groups (e.g. E8) pertain to strings.  All of these are
>still quite speculative, IMHO, in their applications to the real
>world.  
>
>Tacking on the group of translations onto SL(2,C) gives the poincare
>group, the classifications of the irreducible representations of which
>was one of Wigner's most famous achievements.  Another interesting
>item is the connection between the generators of SU(2) and the
>Heisenberg group containing the essential modifications of kinematics
>from the classical to the quantum worlds.
>
>To these I'd also add the 'point' groups chemists use to classify the
>spectra of molecules based on their symmetry.  Quite powerful
>conclusions can be drawn from symmetry alone.
>
>Take care,
>
>root <address@hidden> writes:
>
>> Camm,
>> 
>> re: quantum theory and groups. Bertfried is working on Clifford and
>> Hopf algebras. One of the concerns seems to be the choice of an
>> efficient data structure. Since I (as yet) know nothing about the
>> subject I can't give good advice. However, I'm looking around at
>> some books to help me learn.
>> 
>> Axiom has some group theory (LyndonWords, LieAlgebra, Clifford Algebra, 
>> etc) already. They are not in the paper version of the book but are
>> in the electronic version.
>> 
>> I'm hoping to extend Axiom with domains which are useful in physics.
>> >From my reading it appears that SU(3), SO(2), etc are of interest.
>

The correct attack on these kinds of problems in Axiom is to first
figure out the category hierarchy. Within algebra you can find a
nice structure of:

 fields
   rings
     groups
       monoids

etc. In one textbook I saw this hierarchy diagrammed but have never
been able to find it again.  Does there exist a book which shows the
Venn diagram or containment hierarchy for the kind of groups you
mention?  Implicit in the discussion above is that such a thing exists
but I haven't ever found it written down. If we could write down how
these groups are contained within each other (and what specialized
names they go by, such as Poincare groups) we'd be well on our way to
having a good, general purpose way of constructing and representing
them. I'm sure this is all well understood but I've never seen it
written down. Does anyone know of diagrams of this kind?

Ideally you'd be able to declare variables of type SU(3).

Tim