patfla mentions the ambiguity in mathematics notation of the lettter "Q".
It is used for the rational numbers, and also for the Quaternions.
However,
some people use the letter "H" (the "H" coming from "Hamilton") for the Quaternions to avoid that ambiguity,
and
others (including me) often use "Q" for the Quaternions hoping that context will make clear what is meant.
Also, some people don't like to use "O" for Octonions ( because "O" is also used for "orthogonal" as in the orthogonal group O(p,q) ) and so instead they use "Ca" (the "Ca" coming from Cayley).
patfla also says "... In SO(4,1) the 4 refers to dimensions with a positive norm – the 1 to a single dimension with a negative norm.
One can only assume that the 4 are spatial dimensions and the 5th, t, is time. So how come t = time has a negative norm? ...".
SO(p,q) and Clifford algebra Cl(p,q) notations are also not unanimously followed in the math community. Some people use (p,q) to denote signature with p + dimensions and q - dimensions, and some use it the other way around. Since both notations are found not only in papers but also in textbooks, you need to figure out which convention is used in whatever you are reading.
As to whether SO(3,1) or SO(1,3) should represent transformations of physical spacetime, you can look at the Clifford algebras Cl(3,1) or Cl(1,3) from whence they come (their Lie algebras are the bivector parts of those Clifford algebras, using the Lie bracket product).
Here I will use +++- and ---+ to be clear instead of (p,q) type notation.
(see for example F. Reese Harvey's book "Spinors and Calibrations" (Academic 1990))
Cl(+++-) = M(2,Q) = 2x2 matrices of Quaternions
Cl(---+) = M(4,R) = 4x4 matrices of Real numbers
So, the question of +++_ versus ---+ signature of physical spacetime becomes: do you want Quaternionic or Real structure?
John Baez has a web page about that question at
http://math.ucr.edu/home/baez/symplectic.html
Here are a few quotes from that web page:
"... spin-1/2 particles in nonrelativistic quantum mechanics are naturally quaternionic if we take time reversal into account! ...
[quoting Toby Bartels] "... we want an operator T: H -> H
...[where]... H ... is a 2-dimensional complex vector space ... the Hilbert space of a non-relativistic spin-1/2 particle ...
to describe the effect of time reversal on our spin-1/2 particle ...
an antiunitary operator with these properties does exist, and is unique up to phase.
It satisfies T2 = -1
Now what have we got? A quaternionic structure on H.
[in terms of Quaternion basis elements {1,i,j,k}] T = j.
... so H becomes a 1-dimensional quaternionic Hilbert space!
... So: spin-1/2 particles in nonrelativistic quantum mechanics are naturally quaternionic if we take time reversal into account!
... f the spin is an integer, T2 = 1 is a real structure, making the Hilbert space the complexification of a real Hilbert space. ..." [end of quote of Toby Bartels]
Even better, it turns out that the same stuff applies to representations of the Poincare group: the reps corresponding to fermions are quaternionic, while the reps corresponding to bosons are real - and the operator j turns out to be nothing other than the CPT operator! ...".
See also
John Baez's week156 at
http://math.ucr.edu/home/baez/week156.html
where he says in a footnote "... Squark found in Volume 1 of Weinberg's "Quantum Field Theory" that the CPT operator on the Hilbert space of a spin-j representation of the Poincare group is an antiunitary operator with (CPT)^2 = -1^2 j. So indeed we do have (CPT)^2 = 1 in the bosonic case, making these representations real, and (CPT)^2 = -1 in the fermionic case, making these representations quaternionic. ...".
Note that all the above is consistent with the general approaches of
Geoffrey Dixon (T = CxQxO acting as a spinor space, with fermionic Quaternions acting like spinors, and generalizing the above to include Octonions)
and
Garrett Lisi using
248-dim E8 = 120-dim bosonic adjoint SO(16) + 128-dim fermionic half-spinor SO(16)
where, on the Lie Group level,
E8 / SO(16) = 128-dim fermionic half-spinor SO(16) = (OxO)P2 = the octo-octonionic projective plane known as Rosenfeld's elliptic projective plane. Rosenfeld is at Penn State and has a web page at
http://www.math.psu.edu/katok_s/BR/init.html
patfla also quotes Geoffrey Dixon as describing T = CxQxO as "... the complexification of the quaternionization of the octonions ..."
and then asks:
"... So what is that?
You seem to be shoe-horning 8 dimensions first into 4 and then into 2 ...".
It is not so much shoe-horning O into Q into C as it is
starting with 8-dim O
then expanding to QxO by letting each element of O be 4-dim Q to go to 4x8 = 32-dim
and
finally expanding again to CxQxO by letting each element of QxO be 2-dim complex to go to 2x32 = 64-dim.
Geoffrey Dixon then (as described in his book) uses two copies of T = CxQxO as his basic 64+645 = 128-dim spinor space
which seems to correspond to Garrett Lisi's E8 / SO(16) = (OxO)P2 = 128-dim spinor-type fermion space.
What makes me think that E8 physics is realistic and probably true is that so many different points of view (octonion, Clifford algebra, Lie algebra, symmetric space geometry, ...) all seem to fit with it consistently and to describe what we observe in the physics of gravity and the standard model.
Tony Smith
PS - I should also note the ambiguity of notation using T for time and T for CxQxO.
All in all, I think that you have to pay close attention to context when reading math/physics literature.
