Re: [Axiom-developer] Re: Clifford algebras in AXIOM

[Camm Maguire]

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.
> 
> Tim
> 
> 
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> 
> 

-- 
Camm Maguire                                            address@hidden
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