Chris Tickle, in August 2001, asked me about Clifford algebras: "... What would be the first thing you would tell me, what would be the first example you would show me,and what would be the first application you would show me? ...". Here is substantially what I said in reply: The first thing that I would tell you would be that Clifford algebras (working over the real numbers and using Euclidean signature) have the structure of the binomial triangle, so that the Clifford algebra Cl(n) has structure like: n Total Dimension 0 1 2^0 = 1= 1x1 1 1 1 2^1 = 2= 1+1 2 1 2 1 2^2 = 4= 2x2 3 1 3 3 1 2^3 = 8= 4+4 4 1 4 6 4 1 2^4 = 16= 4x4 5 1 5 10 10 5 1 2^5 = 32=16+16 6 1 6 15 20 15 6 1 2^6 = 64= 8x8 7 1 7 21 35 35 21 7 1 2^7 = 128=64+64 8 1 8 28 56 70 56 28 8 1 2^8 = 256=16x16 9 1 9 36 84 126 126 84 36 9 1 2^9 = 512=256+256 ... etc ... ------------------ The first example that I would show you would be the simplest: Cl(0) is 1-dimensional and looks like the real numbers. To me this is important because it means that very big Cl(n) algebras arise naturally from the single element 0. It lets me, in my physics model,
build a math model for the whole universe from 0,
and I think that it is fun to construct everything from nothing.
------------------
The first "application" would be that
Cl(2) is 4-dimensional (it is the quaternions),
but
its graded structure is 1 2 1
The first 1 (called grade 0) is the scalars
(here I am working with real numbers as scalars).
The 2 (called grade 1) is a 2-dimensional vector space
(which I take to be the complex plane).
This illustrates that the grade-1 elements of Clifford algebras
form vector spaces, and can be called vectors.
The second 1 (called grade 2) is the
Lie algebra of the rotation group of the plane,
the 1-dimensional U(1) circle rotation group.
This illustrates that the grade-2 elements of Clifford algebras
form the Lie algebra of rotations in the space of vectors,
and the grade-2 elemenents can be called bivectors.
(As a remark, note that I skipped Cl1), with structure 1+1,
which is the complex numbers. In a real lecture I might mention
that I skipped Cl(1) to get to Cl(2).)
=================================================================
If I could go further, I would say:
Cl(3) has structure 1 3 3 1,
with 3-dim vectors and 3-dim bivectors,
representing the 3-dim Lie algebra of rotations in 3-dim space;
Cl(4) has structure 1 4 6 4 1,
with 4-dim vectors and 6-dim bivectors,
representing the 6-dim Lie algebra of rotations in 4-dim space
(if you go to Minkowski signature, the 6-dim Lie algebra
gives you 3-dim spatial rotations plus 3-dim Lorentz boosts); and
If you go up to large n (large with respect to 8),
the structure of Cl(n) can be understood
by using a factorization theorem, the Periodicity Theorem:
Cl(n) = Cl(8) x Cl(n-8) (where x denotes tensor product).
Therefore, ALL Clifford algebras can be understood in terms
of Cl(8) and the Cl(k) for k less than 8,
so,
in a sense,
Cl(8) is the fundamental building block of ALL big Clifford algebras,
because
they can be embedded in a Cl(8n) = Cl(8) x ...(n times)... x Cl(8)
which is just a tensor product of a lot of Cl(8) algebras.
That is why Cl(8) is the basic building block of my physics model.
Here are more details about why I like and use Clifford Algebras:
Clifford algebras have a natural Bit-Representation related to Information Theory.
Clifford algebras Cl(N) for very large N may describe Simplex Physics at the Highest Energies, Cl(N) breaks up into the Periodicity-8 tensor product
of N/8 factors, each of which is Cl(8). Cl(8) is 16x16 = 256-dimensional:
The reduction means that you can understand very large things as nested versions of Cl(8), which by itself is a 256-dim thing that is small enough for the human mind to comprehend.
In Quantum Consciousness, states of mind correspond to elements of very large Clifford algebras, such as Cl(10^18), and consciousness processes resemble Quantum Games.
Human divination systems are manifestations of a fundamental way of perceiving phenomena of Life, the Universe, and Everything. The most fundamental (and oldest) such system is IFA, also known as VoDou, which, like the Clifford Algebra Cl(8), is based on 16x16 = 256 elements. Other systems, such as I Ching, which is based on 8x8=64 elements, seem to me to correspond to subsets of IFA - VoDou.
I learned about Clifford Algebras from studying under David Finkelstein at Georgia Tech since around 1981. Both David Finkelstein and I use the the Periodicity-8 tensor product Cl(N) = Cl(8) x ...N/8 times... x Cl(8)) in our physics models. With respect to his model, David Finkelstein says "... An assembly with Clifford statistics we call a squad. The Spinorial Chessboard shows how the dynamics, a large squad of chronons, can spontaneously break down into a Maxwellian assembly of squads of 8 chronons each. Squads of 8 are special this way. ...". Also, see, for example, the paper hep-th/0009086, Clifford Algebra as Quantum Language, by James Baugh, David Ritz Finkelstein, Andrei Galiautdinov, and Heinrich Saller, and the paper by D. Finkelstein and E. Rodriguez, The quantum pentacle, International Journal of Theoretical Physics 23, 887-894 (1984).
I use the conventions of Ian Porteous with respect to signature of Clifford Algebras.
Some very useful links are:
I also discuss Clifford Algebras
CLIFFORD ALGEBRAS are the ALGEBRAS of
Lie algebras of rotations will be defined by Clifford algebra commutators of pairs of vectors.
Spinors will appear as rows or columns in Clifford matrix algebras.
Real Clifford algebras have 8-fold Periodicity.
Clifford Algebras can be used to construct the Division Algebras.
2x2 Matrices of Clifford Numbers give Moebius Transformations.
The Dirac Operator is the natural differential operator for spinors.
References and Background Material for the Clifford Paths for the D4-D5-E6-E7-E8 VoDou Physics Model:
Now, we must construct the Clifford algebras. BEGIN with a real vector space V(p,q) of dimension n
and signature (p,q), where n = p+q,
p is the dimension of the negative definite subspace of V,
and q is the dimension of the positive definite subspace.
There is no real consensus as to the sign convention here,
so I chose this one, following Ian Porteous.
Ian Porteous, in his book Clifford Algebras and the Classical Groups (Cambridge 1995), says at pages 123-124:
"... It is shown that, for any finite-dimensional real quadratic space X, there is a real associative algebra, A say, with unit element 1, containing isomorphic copies of R and X as linear subspaces such that, for all x in X, x^2 = -x^(2).If the algebra A is also generated as a ring by the copies of R and X or, equivalently, as a real algebra by {1} and X, then A is said to be a (real) Clifford algebra for X (Clifford's term ... was geometric algebra).
It is shown that such an .algebra can be chosen so that there is also on A an algebra anti-involution A -> A ; a -> a' such that, for all x in X , x' = -x . ...
To simplify notations in the above definitions, R and X have been identified with their copies in A. More strictly, there are linear injections a : R -> A and i : X -> A such that, for all x in X, (i(x))^2 = -a(x^(2)), the unit element in A being a(1).
The minus sign in the formula x^2 = -x^(2) can be a bit of a nuisance at times. One could get rid of it at the outset, simply be replacing the quadratic space X by its negative. However, it turns up anyway in applications, and so we keep it in.
...the existence of a Clifford algebra for an arbitrary n-dimensional quadratic space X is implied by the existence of a Clifford algebra for the neutral non-degenerate space R(n,n). Such an algebra is constructed below in Corollary 15.18. ... [The Corollary 15.18 says (at page 131) that for each n, the endomorphism algebra R(2^n) (of 2^n x 2^n matrices) is a universal Clifford algebra for the neutral non-degenerate space R(n,n). That is, Cl(n,n) = R(2^n). Porteous's proof is by induction. The basis is that R is a universal Clifford algebra for R(0,0) ... Porteous says (at page 167) that, given a universal real Clifford algebra A for a finite-dimensional non-degenerate real quadratic space X, the real algebra A(2) of 2x2 matrices with entries in A is a universal real Clifford algebra for the real quadratic space X (+) R(1,1), whose elements are represented in A(2) by matrices called vectors in A(2) of the form, for x in X and u,v in R
x v
u -x
].. An alternative construction of a Clifford algebra for a real quadratic space X depends on the prior construction of the (infinite-dimensional) tensor algebra of X, regarded as a linear space. The Clifford algebra is then defined as a quotient algebra of the tensor algebra, this definition being that adopted by Chevalley (1954). ...".Porteous defines x^(2) on on page 22, saying: "... a^(2) ... being called the square or quadratic norm of a ...".
In other words, according to the - sign in a^2 = -a^(2) on page 123, the Clifford product square has the opposite sign of the vector space scalar product square.
Therefore, in Porteous's convention, the Clifford algebra Cl(p,q) has
For some people, that seems so counterintuitive that they "... get rid of it at the outset simply by replacing the quadratic space X by its negative ...", as Porteous says on page 124. However, Porteous and I are happy with that, and leave it alone.
It should be emphasized that there are other people who do replace X by its negative, so that there is a lot of diversity in signature and sign conventions in the literature.
If the vector space has a mixed signature,
in which neither p nor q is zero,
the minimum of p and q is the Witt Index.
If p = q, the Clifford Algebra is called
a Split Clifford Algebra.
The scalars of V(p,q) have dimension 1.
Since scalars have no vector components,
call the scalars "elements of grade 0".
The vectors of V(p,q) have dimension n.
Call the vectors "elements of grade 1".
INTRODUCE the exterior /\ product. It is a "creator" because a /\ X "adds" the "dimension" of a to the "dimensions" of X.
The exterior /\ product of 2 vectors in V(p,q) defines a 2-dimensional area. Since it is the product of 2 vectors, it is a bivector. The exterior /\ product is alternating, or antisymmetric, so that a /\ b = -b /\ a for all a,b in V(p,q), and the bivectors of V(p,q) have dimension n(n-1)/2. Call the bivectors "elements of grade 2". The exterior /\ product of k vectors in V(p,q) defines a k-dimensional hyper-volume. Since it is the product of k vectors, it is a k-vector. The k-vectors have dimension n!/k!(n-k)!. Call the k-vectors "elements of grade k". The exterior product of n vectors in V(p,q) defines the n-dimensional volume element. Since it is the product of n vectors, it is an n-vector. It is also the pseudo-scalar, the volume, and the determinant of the n vectors. The psuedoscalars, like the scalars, have dimension 1. Call the pseudoscalars "elements of grade n". Since they represent n-dimensional volumes, they are related to n-dimensional hypercubes.
Since the exterior /\ product is alternating,
or antisymmetric,
and a /\ a = - a /\ a = 0 for all a in V(p,q),
the exterior product of more than n vectors vanishes,
and there are no elements of grade higher than n.
Now we can combine the
scalar, vector, bivector, ... , k-vector, ..., pseudoscalar,
spaces into one big combined space whose total dimension is
the sum of the dimensions of each of the component spaces.
Call this combined space the
graded algebra /\V(p,q) of the vector space V(p,q)
and
call the subspace of elements of grade k the
grade k subspace /\kV(p,q) of /\V(p,q).
Our graded algebra /\V(n) has
the structure of the Yang Hui triangle
(also called the Meru-Prastera triangle
or the Pascal triangle). Here, it is shown up to n=8:
n Total
Dimension
0 1 2^0 = 1= 1x1
1 1 1 2^1 = 2= 1+1
2 1 2 1 2^2 = 4= 2x2
3 1 3 3 1 2^3 = 8= 4+4
4 1 4 6 4 1 2^4 = 16= 4x4
5 1 5 10 10 5 1 2^5 = 32=16+16
6 1 6 15 20 15 6 1 2^6 = 64= 8x8
7 1 7 21 35 35 21 7 1 2^7 = 128=64+64
8 1 8 28 56 70 56 28 8 1 2^8 = 256=16x16
THE GRADED ALGEBRA /\V(p,q) THAT WE HAVE CONSTRUCTED HAS:
total dimension 2^n
a natural 1-1 mapping, called the Hodge dual *,
from /\kV(p,q) onto /\(n-k)V(p,q)
since both subspace have dimension n!/k!(n-k)!
and
an exterior /\ product such that,
for all a in V(p,q) and X in /\kV(p,q),
a /\ X is in /\(k+1)V(p,q)
Our algebra /\V(p,q) lets us build up multivector spaces, and interchange some of them by using the Hodge dual *.
For V(p,q) = R(0,3), we can even define a cross-product x
a x b = * ( a /\ b )
using * and /\ because /\R(0,3) is graded
1 3 3 1
so that * takes 3-dim /\2R(0,3) into 3-dim /\1R(0,3)
We can use the cross-product x to define ROTATIONS in R(0,3),
but for V(p,q) NOT of the form 1 3 3 1
WE CAN NOT YET DEFINE ROTATIONS.
TO DEFINE ROTATIONS, WE NEED an "ANNIHILATOR" to
correspond to our "CREATOR" exterior /\ product.
Since we have a bilinear metric form (a,b) for all a,b in V(p,q), we can use it to define an "annihilator" interior |_ product a |_ b = (a,b) for all a,b in V(p,q). We can extend the interior product for a in V(p,q) to a |_ X for X in the graded algebra /\V(p,q) by requiring that for all u in /\V(p,q): ( u , a |_ X ) = ( a /\ u , X )
The INTERIOR |_ product is an "annihilator" because x |_ Y "annihilates" the "dimension" of x from the "dimensions" of Y. Now, we would like to combine the exterior product and the interior product into one single product that should describe our geometry: THE CLIFFORD GEOMETRIC PRODUCT a.X for all a in V(p,q) and X in /\V(p,q) is defined by a.X = a /\ X - a |_ X The Clifford product extends naturally to define X.Y for all X,Y in /\V(p,q) (See the references for details about the extension.) So we will now use the Clifford product . for elements of our graded algebra, and from now on will denote our graded algebra with Clifford product by Cl(p,q). The Clifford product of a vector by a multivector is in some sense an extension of the set-theoretic XOR from sets and subsets to vector spaces and subspaces. The antisymmetric a /\ X part is the subspace spanned by either a or X (or their linear combinations), and the symmetric a |_ X part that is subtracted out is the subspace common to both a and X. What the Clifford product has, that XOR does not have, is the concept of sign, or orientation, a concept that is sort of like the set-theorectic concept of ordered sets.
Now we can use the Clifford . [,] COMMUTATOR operating on
BIVECTORS a /\ b and c /\ d
to define the LIE ALGEBRA
of rotations in the vector space V(p,q).
The Clifford . [,] COMMUTATOR
[ a /\ b , c /\ d ] = a /\ b . c /\ d - c /\ d . a /\ b =
= a /\ b /\ c /\ d - c /\ d /\ a /\ b +
+ a /\ b |_ c /\ d - c /\ d |_ a /\ b =
= (b |_ c) a /\ d - (d |_ a) c /\ b
takes bivectors into bivectors and closes to form the
LIE ALGEBRA of rotations
in the vector space V(p,q).
For further details, see reference books and papers cited therein.
Note that if we were restricted to the exterior /\ [,] COMMUTATOR
we could not define the Lie algebra of rotations in vector spaces
of dimension greater than 2, because
[ a /\ b , c /\ d ] = a /\ b /\ c /\ d - c /\ d /\ a /\ b = 0
always vanishes, and vanishing commutators cannot define
nonabelian Lie algebras.
As to generalizations of the 3-dimensional cross product x,
there is only one: the 7-dimensional octonionic cross product x.
However, we would have to use some peculiarly octonionic structures:
the "associative" 3-form PHI in R(0,7) such that
(a _| PHI) /\ (b _| PHI) /\ PHI = 6 (a,b) VOL
where _| is the adjoint of |_ and VOL is the 7-dim volume element;
and
the Lie algebra G2 of the automorphism group of octonions,
G2 being a 14-dimensional Lie algebra that defines
a 21-14 = 7-dimensional subspace in the bivector space of
/\R(0,7) , which is graded 1 7 21 35 35 21 7 1
(G2 can be defined as the 49-dim 7x7 matrices factored by
the 35-dim /\3R(0,7), i.e., as maps of R(0,7) preserving
a general trilinear form).
Therefore, I will not go further into the
octonionic cross product here.
For details, see the references and citations therein.
WHAT DOES Cl(p,q) LOOK LIKE? Cl(p,q) is a matrix algebra for even n = p+q, or the sum of two matrix algebras for odd n = p+q. Depending upon signature, the matrix algebras may be real R, complex C, or quaternionic Q (or H). Here are the matrix algebras for Cl(p,q) for all p,q up to 8:
(The horizontal axis is q (positive definite) ,
the vertical axis is p (negative definite))
(Quaternions here are denoted H for Hamilton.)
In some sense, this table shows all the types of real Clifford algebras,
because there is
an 8-fold Periodicity Theorem for real Clifford Algebras.
Michael Gibbs has written a Constructive Proof of Clifford Periodicity 8.
8-fold Periodicity is related to homotopy periodicity:
Cl(p,q+8) = Cl(p,q) (X) R(16)
(where (X) is the tensor product and R(16) is the 16x16 real matrix algebra).
It is also true that Cl(p+8,q) = Cl(p,q) (X) R(16),
Cl(p+1,q) = Cl(q+1,p) and Cl(p,q) = Cl(p+4,q-4).
For Cl(p,q,R) Clifford algebras with real scalar field R, p negative signatures, q positive signatures) the Cl(p,q,R) Clifford algebras are matrix algebras (or sums of two of them) with either real R, complex C, or quaternion Q coefficients, appearing as an 8-periodic square array:
q 0 1 2 3 4 5 6 7
p
0 R C Q 2Q Q(2) C(4) R(8) 2R(8)
1 2R R(2) C(2) Q(2) 2Q(2) Q(4) C(8) R(16)
2 R(2) 2R(2) R(4) C(4) Q(4) 2Q(4) Q(8) C(16)
3 C(2) R(4) 2R(4) R(8) C(8) Q(8) 2Q(8) Q(16)
4 Q(2) C(4) R(8) 2R(8) R(16) C(16) Q(16) 2Q(16)
5 2Q(2) Q(4) C(8) R(16) 2R(16) R(32) C(32) Q(32)
6 Q(4) 2Q(4) Q(8) C(16) R(32) 2R(32) R(64) C(64)
7 C(8) Q(8) 2Q(8) Q(16) C(32) R(64) 2R(64) R(128)
You can have periodicity 8 with matrix coefficients as either R, C, or Q. For example, consider the following table made from diagonals of the above 8x8 table repeated (the * refers to a Clifford algebra that has a tensor product of R(16), and ** refers to two such tensorings):
Periodicity diagonals of Cl(p,q,R) real Clifford algebras with matrix entries of types R, C, and Q (and another Q - see note below):
k 0 1 2 3
p,q
0,k R C Q 2Q
1,k+1 R(2) C(2) Q(2) 2Q(2)
2,k+2 R(4) C(4) Q(4) 2Q(4)
3,k+3 R(8) C(8) Q(8) 2Q(8)
4,k+4 R(16) C(16) Q(16) 2Q(16)
5,k+5 R(32) C(32) Q(32) *2Q(2)
6,k+6 R(64) C(64) *Q(4) *2Q(4)
7,k+7 R(128) *C(8) *Q(8) *2Q(8)
8,k+8 **R **C **Q **2Q
( Note that Cl(1,3) of Minkowski space has quaternionic structure Q(2), which is nicely consistent with quaternion spinors as physical fermions. )
( Note also that I have put in a column for k = 3 that seems to look like a column for pairs of quaternions. I put that column in because I like octonions, and although Cl(0,3) = 2Q is not the octonions, it is closely related to them and a pair-of-quaternion column is as close as I can get to an octonion column. )
There are two ways of thinking of a Clifford algebra as being Real, Complex, ... etc. One is, as described immediately above, the nature of the coefficients in the matrix representation of the Clifford algebra.
The other is based on the fact that conventionally a Clifford algebra Cl(n,F) is defined over an n-dimensional vector space with scalar field F.
What happens if we go to complex
vector spaces over the complex numbers C?
Over C, signature is irrelevant
(you can Wick rotate from one signature to another).
Here are the matrix algebas for complex n-dimensional
Clifford algebras ClC(n) for n up to 16:
Notice that:
the complex Clifford algebras of even dimension 2p
come from the real Clifford algebras of split signature (p,p);
the complex Clifford algebras of odd dimension 2p+1
come from the real Clifford algebras of signature (p+1,p);
and
the complex Clifford algbra ClC(4) = M4(C) (shown in blue)
of complex 4x4 matrices is the complexification of
the 4-dimensional Dirac gamma matrices, and
that it is the complexification of 4-dim Dirac gammas
over (real) 4-dim spacetime of ANY signature.
Complex Clifford Periodicity is of period 2:
ClC(N + 2) = ClC(N) (X) ClC(2) = ClC(N) (X) C(2)
where C(2) is the 2x2 complex matrix algebra.
Now that we have constructed the Clifford algebras,
Sometimes I use the term Spinor, or Spin(p,q), when I really should use the term Pin, or Pin(p,q). A physical significance of the difference is that Spinors and Spin(n) are related to the even subalgebra of the Clifford algebra Cl(p,q) ( where Cle(p,q) = Cl(p,q-1) and Cle(p,0) = Cl(0,p-1) ) ) and so do not contain some reflection-related characteristics (such as parity reversal, etc.), while such things are contained in Pin and Pin(p,q) because they are related to the full Clifford algebra Cl(p,q) including its odd part. A paper by Marcus Berg, Cecile DeWitt-Morette, Shangjr Gwo, and Eric Kramer, math-ph/0012006, discusses Pin and Spin.
Spinors of Cl(p,q) come from Cle(p,q).
while
Pinors of Cl(p,q) come from Cl(p,q) itself.
My version of spinors of positive definite Euclidean spaces,
and their dimensions:
1 S1 = R 16 S9 = O2
2 S2 = C 32 S10 = (CxO)2
4 S3 = H 64 S11 = (HxO)2
8 S4 = H+H 128 S12 = (HxO)2+(HxO)2
8 S5 = H2 128 S13 = (H2xO)2
8 S6 = C4 128 S14 = (C4xO)2
8 S7 = O 128 S15 = (OxO)2
16 S8 = O+O 256 S16 = (OxO)2+(OxO)2
Compare the terminology used by John Baez in his
paper, math.RA/0105155, about Octonions
(also mentioned in his week 168),
for spinors of positive definite Euclidean Spaces,
and their dimensions:
1 S1 = R 16 S9 = O2
2 S2 = C 32 S10 = (CxO)2
4 S3 = H 64 S11 = (HxO)2
4 S+/- 4 = H 64 S+/- 12 = (HxO)2
8 S5 = H2 128 S13 = (H2xO)2
8 S6 = C4 128 S14 = (C4xO)2
8 S7 = O 128 S15 = (OxO)2
8 S+/- 8 = O 128 S+/- 16 = (OxO)2
If D5 is taken to be Spin(1,9), as in the D4-D5-E6-E7-E8 VoDou Physics model, then spinors are R16+R16 of Cl(1,8) = Cle(1,9).
where Cle(p,q) = Cl(p,q-1) and Cle(p,0) = Cl(0,p-1)
My version of spinors of negative definite Euclidean spaces,
and their dimensions:
1 S1 = R 16 S9 = R16
2 S2 = R+R 32 S10 = R16+R16
2 S3 = R2 32 S11 = R32
4 S4 = C2 64 S12 = C32
8 S5 = H2 128 S13 = H32
16 S6 = H2+H2 256 S14 = H32+H32
16 S7 = H4 256 S15 = H64
16 S8 = C8 256 S16 = C128
Consider x in the grade-1 1-vector space X of Cl(N).
The map x to -x extends from X to
the map u to u* for all u in Cl(N).
For k-vectors u, u* = u if k is even
and u* = -u if k is odd.
The Clifford Group CLG(N) of Cl(N) is the
set of invertible elements s in Cl(N) such that
for all x in X, s x s*^(-1) is in X.
Since each of its elements is composed of
a finite number of hyperplane reflections,
CLG(N) is a group of orthogonal transformations.
The Pinor group Pin(N) of Cl(N) is
the subgroup of the Clifford Group CLG(N)
whose elements are of quadratic norm 1.
Pin(N) is the double cover of O(N).
Pinors are the minimal ideals on
which the Pinor Group Pin(N) acts.
Since the even-grade elements of Cl(N)
form the even Clifford subalgebra Cle(N)
of half the dimension of Cl(N),
where Cle(p,q) = Cl(p,q-1) and Cle(p,0) = Cl(0,p-1),
we can define CLGe(N) as the
subgroup of CLG(N) whose elements are each
the product of an even number of Cl(N) basis elements.
(Note that rotations are defined by
and even number of reflections,
and Clifford group elements are reflections,
so you get coverings of the rotation group SO(N)
from the even Cle(N) and CLGe(N).)
The Spinor Group Spin(N) of Cl(N) is
the subgroup of CLGe(N)
whose elements are of quadratic norm 1.
Spin(N) is the double cover of SO(N).
Spinors are the minimal ideals on
which the Spinor Group Spin(N) acts.
While Pin(N) comes from Cl(N),
Spin(N) comes from Cle(N).
Having said all this (insofar as it is correct, it is
based on Pertti Lounesto's book), I will now give
a slightly different definition of Spinors
that is perhaps simpler and more intuitive, and also,
to quote Ian Porteous, perhaps "slightly dishonest"
in that it does not fully honor
the distinction between Pin(N) and Spin(N):
SPINORS ARE THE MINIMAL IDEALS OF CLIFFORD ALGEBRAS.
Since the Clifford algebras are matrix algebras,
spinors are either rows or columns
in the Clifford matrix algebras.
Row or Column spinors are left or right minimal ideals.
Since a minimal (left) ideal X of an algebra is
an irreducible linear subspace of the algebra such that,
for all x in X and all y in the algebra, yx is in X,
we can find a Primitive Idempotent element P in X
such that X is made up of all elements of the form yP
where y is in the algebra.
P is called Idempotent because PP = P
and P is called Primitive because it is
not the sum of two non-zero idempotents of the algebra.
In matrix form, P is similar to
a matrix with a single non-zero entry on its main diagonal,
and all other entries are zero.
Since odd dimensional Clifford algebras are
the sum of two matrix algebras,
spinors for odd dimensional Clifford algebras
are rows (or columns) of one of the matrix algebras,
the matrix subalgebras of the even-grade elements of
the graded Clifford algebra.
Since even dimensional Clifford algebras are matrix algebras,
spinors for even dimensional Clifford algebras
are rows (or columns) of the matrix algebra.
Since the even-grade elements of the
even dimensional Clifford algebras form a subalgebra
of the full graded Clifford algebra,
made up of two diagonal blocks each of whose rows
(and columns) are half the original size,
the even dimensional spinors are reducible into
two irreducible half-spinors.
For dimensions up to 8, here are the dimensions of
the spinors (with real structure) of the Clifford algebras:
n Total Spinor
Dimension Dimension
0 1 2^0 = 1= 1x1 1
1 1 1 2^1 = 2= 1+1 1
2 1 2 1 2^2 = 4= 2x2 2 = 1+1
3 1 3 3 1 2^3 = 8= 4+4 2
4 1 4 6 4 1 2^4 = 16= 4x4 4 = 2+2
5 1 5 10 10 5 1 2^5 = 32=16+16 4
6 1 6 15 20 15 6 1 2^6 = 64= 8x8 8 = 4+4
7 1 7 21 35 35 21 7 1 2^7 = 128=64+64 8
8 1 8 28 56 70 56 28 8 1 2^8 = 256=16x16 16 = 8+8
To see why the restriction to real structure is important,
consider, for example, Cl(1,3) = M(2,Q) and Cl(3,1) = M(4,R).
Both are 16-dim Clifford algebras of 4-dimensional vector spaces,
although of different signatures -+++ and +---,
but
the full spinor space of real Cl(3,1) is 4x1 = 4-dimensional,
while
the full spinor space of quaternionic Cl(1,3) is 2x4 = 8-dimensional.
As can be seen from the Clifford CheckerBoard Table,
for any N there is always a signature (p,q) with p+q = N
such that Cl(pq) has real structure.
Now, look at the Yang Hui triangle.
The left-side border line is all 1's
since there is only 1 dimension of scalars.
The next line is
1
2
3
4
5
6
7
8
the dimension of the vector space V(p,q) of the Clifford algebra Cl(p,q).
The next line is
1
3
6
10
15
21
28
the dimension of the bivector subspace of the Clifford algebra.
The bivector subspace closes under the commutator
[a,b] = a.b - b.a operation, which defines the Lie algebra
of the Lie group Spin(p,q) of the Clifford algebra Cl(p,q).
Spin(p,q) is the simply connected (except for n=0,1,2)
2-1 covering group of the rotation group SO(p,q) of
the vector space V(p,q) underlying Cl(p,q).
EVERY REPRESENTATION OF Spin(p,q) CAN BE CONSTRUCTED FROM
the scalar graded subspace of Cl(p,q),
the vector graded subspace of Cl(p,q), and
the spinors (or two half-spinors, for even p+q)
BY USING THE OPERATIONS OF EXTERIOR /\ PRODUCT,
TENSOR PRODUCT, OR SUM or DIFFERENCE.
John Baez has a nice introduction to the
Euclidean Clifford algebras, in which he states:
"What's a Clifford algebra?
Well, there are various flavors.
But one of the nicest --- let's call it C_n ---
is just the associative algebra over the real numbers
generated by n anticommuting square roots of -1.
That is, we start with n fellows called
e_1, ... , e_n
and form all formal products of them,
including the empty product, which we call 1
[that gives a total of 2^n products,
which are considered to be independent and
graded by the number of distinct factors.
They are the algebra basis].
Then we form all real linear combinations of these products,
and then we impose the relations
e_i^2 = -1
e_i e_j = - e_j e_i.
What are these algebras like?
Well, C_0 is just the real numbers,
since none of these e_i's are thrown into the stew.
C_1 has one square root of minus 1,
so it is just the complex numbers.
C_2 has two square roots of minus 1, e_1 and e_2, with
e_1 e_2 = - e_2 e_1
Thus C_2 is just the quaternions, with e_1, e_2, and e_1e_2
corresponding to Hamilton's i, j, and k.
How about the C_n for larger values of n?
Well, here is a little table up to n = 8:
C_0 R
C_1 C
C_2 H
C_3 H + H
C_4 H(2)
C_5 C(4)
C_6 R(8)
C_7 R(8) + R(8)
C_8 R(16)
What do these entries mean?
Well, R(n) means the n x n matrices with real entries.
Similarly, C(n) means the n x n complex matrices,
and H(n) means the n x n quaternionic matrices.
All these become algebras with the usual matrix addition
and matrix multiplication.
Finally, if A is an algebra,
A + A means the algebra consisting of pairs of guys in A,
with the obvious rules for addition and multiplication:
(a, a') + (b, b') = (a + b, a' + b')
(a, a') (b, b') = (ab, a'b')
...
How about n larger than 8?
Well, here a remarkable fact comes into play.
Clifford algebras display a certain sort of "period 8" phenomenon.
Namely,
C_{n+8} consists of 16 x 16 matrices with entries in C_n ! "
John Baez goes beyond Euclidean signature, and discusses spinors,
in his Week 93.
The "period 8" periodicity is used in the D4-D5-E6-E7-E8 VoDou physics model
at energies above the Planck energy.
The isomorphism of the quaternions H with
the Clifford algebra Cl(2) is due to the
existence of the 3-dimensional vector cross-product
that allows the bivector ij to be identified with the vector k,
and the fact that the spinor space of Cl(2) is sqrt(2^2) =
Aside from dimensions 1 and 3,
a vector cross-product only exists in dimension 7.
The non-isomorphism of the octonions O with
the Clifford algebra Cl(3) is due to the
nonassociativity of the 7-dimensional vector cross-product.
To represent the octonions,
you have to start with the 7 vector generators of Cl(7).
They can be written as 8x8 matrices (spinors of Cl(7)),
and represent left-multiplication by imaginary octonions.
To get to the full octonion multiplication table,
you have to define a new multiplication that
includes the 21 Cl(7) bivector commutators of the 7 Cl(7) vectors.
That then gives you 7+21 = 28 matrices that are real 8x8.
The 28 are then the Spin(8) Lie algebra bivectors of Cl(8).
(see Gunaydin and Gursey, J. Math. Phys 14 (1973) 1651-1667)
Clifford Algebras and Division Algebras:
The EMPTY set can be denoted by a vector space of dimension -1,
so that Cl(-1) corresponds to the VOID.
Cl(0) has dimension 2^0 = 1 and corresponds to the real numbers
and to time.
The even subalgebra is EMPTY.
Cl(0) is the 1x1 real matrix algebra.
Cl(1) has dimension 2^1 = 2 = 1 + 1
= 1 + i
and corresponds to the complex numbers
and to 2-dim space-time.
The even subalgebra is 1 Cl(0).
Cl(1) is the 1x1 complex matrix algebra.
Cl(2) has dimension 2^2 = 4 = 1 + 2 + 1
= 1 + {j,k} + i
and corresponds to the quaternions
and to 4-dim space-time.
The even subalgebra is 1 + i Cl(1).
Cl(2) is the 1x1 quaternion matrix algebra.
Cl(3) has dimension 2^3 = 8 = 1 + 3 + 3 + 1
= 1 + {I,J,K} + {i,j,k} + E
and corresponds to the octonions
and to 8-dim space-time.
The even subalgebra is 1 + {i,j,k} Cl(2).
Cl(3) is the direct sum
of two 1x1 quaternion matrix algebras.
The associative Cl(3) product can be deformed
into the 1x1 octonion nonassociative product:
To get the Cl(3) multiplication table,
note that:
The 1 scalar can be represented by
1 0
0 1
The 3 vectors can be represented by
i 0 j 0 k 0
0 -i 0 -j 0 -k
The 3 bivectors can be represented by
i 0 j 0 k 0
0 i 0 j 0 k
The 1 pseudo-scalar can be represented by
-1 0
0 1
This gives the multiplication table:
1 i j k I J K E
1 1 i j k I J K E
i i -1 k -j E k -j -I
j j -k -1 i -k E i -J
k k j -i -1 j -i E -K
I I E k -j -1 K -J I
J J -k E i -K -1 I J
K K j -i E J -I -1 K
E E -I -J -K I J K 1
To get an octonion multiplication table,
start with an orthonormal basis of 8 octonions, and
pick a scalar real axis 1
and pick (2 sign choices) a pseudoscalar axis E or -E.
Then you have 6 basis elements to designate as i or -i,
which is 6 element choices and 2 sign choices.
Then you have 5 basis elements to designate as j or -j,
which is 5 element choices and 2 sign choices.
Then the underlying quaternionic product fixes ij as k or -k,
which is 2 sign choices.
Then deform the Clifford product by changing EE from 1 to -1,
and IJ from K to -k, JK from I to -i, KI from J to -j,
and deform the cross-terms correspondingly.
A fundamental reason for the deformation is that
the graded structure of the Clifford algebra gives it
the underlying exterior product antisymmetry rule that
A/\B = (-1)^pq B/\A for p-vector A and q-vector B,
while
for Octonions,
you want for all unequal imaginary octonions to have
the antisymmetry rule AB = - BA
and for equal ones to have AA = -1.
There are more details about how the octonion product works,
including how it is related to the Clifford Group and XOR,
in Chapter 2 of From Sets to Quarks.
How many inequivalent octonion multiplication tables are there?
You had 6 i-element choices, 5 j-element choices, and 4 sign choices,
for a total of 6 x 5 x 2^4 = 30 x 16 = 480 octonion products.
Here is an example of an octonion multiplication table:
1 i j k I J K E
1 1 i j k I J K E
i i -1 k -j -E -K J I
j j -k -1 i K -E -I J
k k j -i -1 -J I -E K
I I E -K J -1 -k j -i
J J K E -I k -1 -i -j
K K -J I E -j i -1 -k
E E -I -J -K i j k -1
Cl(4) has dimension 2^4 = 16 = 1 + 4 + 6 + 4 + 1
1
+ {S,T,U,V}
+ {i,j,k,I,J,K}
+ {W,X,Y,Z}
+ E
and corresponds to the sedenions.
The even subalgebra is 1 + {i,j,k,I,J,K} + E Cl(3).
Cl(4) is the 2x2 quaternion matrix algebra.
HERE ARE SOME INTERESTING CHARACTERISTICS OF THE SEDENIONS:
CHARACTERISTIC 1:
As noted by Onar Aam,
Clifford | G R A D E
Algebra | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
---------+-----+-----+-----+-----+-----+-----+-----+-----+-----+
Cl(0)=R | 1 | | | | | | | | |
| | | | | | | | | |
Cl(1)=C | 1 | i | | | | | | | |
| | | | | | | | | |
Cl(2)=Q | 1 | jk | i | | | | | | |
| | | | | | | | | |
Cl(3) O | 1 | IJK | ijk | E | | | | | |
| | | | | | | | | |
Cl(4) | 1 |STUV | ijk |WXYZ | E | | | | |
| | | IJK | | | | | | |
| | | | | | | | | |
Cl(5) | 1 | 5 | 10 | 10 | 5 | 1 | | | |
| | | | | | | | | |
Cl(6) | 1 | 6 | 15 | 20 | 15 | 6 | 1 | | |
| | | | | | | | | |
Cl(7) | 1 | 7 | 21 | 35 | 35 | 21 | 7 | 1 | |
| | | | | | | | | |
Cl(8) | 1 | 8 | 28 | 56 | 70 | 56 | 28 | 8 | 1 |
| | | | | | | | | |
this table shows:
that the real axis is always the 0-grade scalar;
that the even subalgebra of the Cl(1)=C complex numbers is Cl(0)=R;
that the even subalgebra of the Cl(2)=Q quaternions is Cl(1)=C,
with the complex imaginary i the only grade-2 pseudoscalar;
that the even subalgebra of the octonionic Cl(3) O is Cl(2)=Q,
with the quaternion imaginaries ijk as grade-2 bivectors; and
that the even subalgebra of the Cl(4) sedenions is octonionic Cl(3) O,
BUT the octonion imaginaries ijkIJKE are NOT in the same grade,
as ijkIJK are grade-2 bivectors and E is grade-4 pseudoscalar;
so that above the Cl(3) octonions, such as for the Cl(4) sedenions,
the new Cayley-Dickson algebra product does NOT make a division algebra
because it CANNOT be extended consistently from the octonions
because
ALL Cl(4) imaginary octonions are NOT in the same Cl(4) grade,
and
the new Cl(4) imaginaries, 4 grade-1 STUV and 4 grade-3 WXYZ,
have incompatible graded structure with the
Cl(4) octonion basis, 1 grade-0 1, 6 grade-2 ijkIJK, and 1 grade-4 E
(in other words, from Cl(4) on up,
the structure of the even subalgebra is incompatible
with the structure of the odd part of the Clifford algebra).
IN CONTRAST:
all Cl(3) imaginary quaternions ijk are Cl(3) grade-2 bivectors,
and
the new Cl(3) imaginaries, 3 grade-1 IJK and 1 grade-3 E,
have isomorphic graded structure with the
Cl(3) quaternion basis, 1 grade-0 1 and 3 grade-2 ijk; and
the Cl(2) imaginary complex i is the Cl(2) grade-2 bivector,
and
the new Cl(2) imaginaries, 2 grade-1 jk,
have compatible (although not isomorphic) graded structure
with the Cl(2) complex basis, 1 grade-0 1 and 1 grade-2 i.
CHARACTERISTIC 2:
Cl(4) is the first Clifford algebra that is NOT made of 1x1 matrices.
CHARACTERISTIC 3: (which is related to triality):
Cl(4) is the first Clifford algebra to have a part (its bivector part)
that does NOT act "consistently" when "operated on"
by the pseudoscalar "volume element" E.
Particularly, although ijkIJK are the 6 bivectors,
ijk are taken by E into IJK while
IJK are taken by E into -i-j-k
This is sometimes described by saying that
ijk are self-dual and IJK are anti-self-dual,
and is related to the fact that
the Cl(4) Lie algebra Spin(4) is NOT irreducible,
since Spin(4) = Spin(3)xSpin(3) = SU(2)xSU(2) = Sp(1)xSp(1) = S3xS3.
What does this have to do with whether or not sedenions
have non-trivial divisors of zero,
and so do not form a division algebra?
If you deform and extend the Cl(4) Clifford product
to a sedenion product, you will extend the ijk and the IJK
so that their extensions overlap, and the bivectors will
be double-covered by a self-dual cover and an anti-self-dual cover,
so that (as in the "overlap" matrix description above)
there are nonzero elements whose sedenion product is zero.
Since the 6 bivectors of Cl(4) are Spin(4),
this is the reducibility of Spin(4) to SU(2) x SU(2),
which is the decomposition of R4 rotations into
rotations in a spatial R3 subspace
and boosts in the corresponding R4 lightcone.
Since the Weyl group of Spin(4) is Z4, the Klein 4-group,
this is the reducibility of Z4 to Z2 x Z2,
which is due to the composite nature of 4,
as opposed to the prime numbers 1, 2, and 3.
Since Spin(4) acts on R4 as the double-cover of the rotation group,
it also acts on the D4 lattice in R4.
Since the D4 lattice is the root vector lattice of Spin(8),
whose vertex figure is a 24-cell,
Spin(4) acts on the 24 = 8+8+8 root vectors of Spin(8).
In particular,
the lightcone boost Z2 of Spin(4) acts on the R4 lightcone
containing the 8+8 = 16 spinor 24-cell root vectors,
with 8 half-spinors on the future lightcone and
8 half-spinors on the past lightcone,
and
the spatial rotation Z2 of Spin(4) rotates the space
of the 7 spatial root vectors of the 8 vector 24-cell root vectors.
The isomorphism between past and future lightcones
gives an isomorphism between the two 8-dim half-spinors,
and
the isomorphism between the two Z2s, or boosts and rotations,
gives an isomorphism between 8-dim vectors and 8-dim half-spinors,
plus an isomorphism between 7-dim space and 7-dim pure half-spinors.
Together, these isomorphisms give 24-cell triality,
which in turn gives triality of the D4 Lie algebra Spin(8).
All 3 of the 8-dim representations of Spin(8) can
be represented by octonions,
and the fourth fundamental representation,
the 28-dim adjoint representation,
can be represented as the group of idempotent-fixing automorphisms
of J3(O), the 3x3 Jordan algebra of octonions.
Since the primitive idempotents are the unit elements
on the diagonal of the 3x3 matrices,
the 28-dim adjoint representation acts
on the off-diagonal triple of octonions in J3(O).
The Clifford Algebra of Spin(8) is Cl(8)
which is the fundamental constituent of Cl(N) for large N,
due to 8-fold Periodicity property Cl(N+8) = Cl(N) (x) Cl(8)
where (x) denotes tensor product.
Surreal octonionic Cl(N), for high N,
may represent sum-over-histories of the Many-Worlds
used to describe Physics, Life, the Universe, and Everything.
Since Cl(k) contains Cl(4) for k at least 4,
by the embedding of Cl(n) as the even subalgebra of Cl(n+1),
the only DIVISION ALGEBRAS based on real Euclidean vector spaces
and their Clifford algebras
and related structures such as Spin(n) Lie Groups
ARE R, C. Q, and O,
and the SEDENIONS do not form a division algebra,
but they do form a ZeroDivisor Algebra.
The octonion product can also be constructed from cross products.
For more about Clifford algebras,
particularly with respect to Cl(0,8),
see Why not SEDENIONS?
and
From Sets to Quarks, particularly Chapter 2.
All Lie algebras can be built from Clifford Algebras.
Also, it is conjectured that
MetaClifford Algebras might be useful.
REFERENCES: Three books that are good places to start reading about the details of Clifford algebras and Spinors are: Clifford Algebras and Spinors, by Pertti Lounesto
(London Mathematical Society Lecture Note Series, No 239)
Spinors and Calibrations, by F. Reese Harvey,
Academic Press (1990).
Clifford Algebras and the Classical Groups, by Ian Porteous,
Cambridge University Press (1995).
Errors can creep into published books and papers in any field.
So that errors don't propagate and become widely-held misconceptions,
it is important to find them and point them out.
The best person I know at doing that is Pertti Lounesto,
who has a web page of counterexamples
that not only points out errors,
but shows why they were probably made,
so that the error-correction process not only corrects,
but also gives deeper insight into the fundamental structures
that led the original authors astray.
It is important that authors not be condemned for making errors,
so long as they are willing to acknowledge them and correct them.
OY! Barry Simon has written YABOGR!
The official title is:
Representations of Finite and Compact Groups (AMS 1996)
What is YABOGR? Read the Book!
What does YABOGR do?
It, together with ideas
of Onar Aam and Ben Goertzel about XOR and set theory,
inspires me to write THIS PAGE ABOUT
SETS, CLIFFORD GROUPS AND ALGEBRAS, AND THE McKAY CORRESPONDENCE.
(any errors you see are due to me,
not to Barry Simon, Onar Aam, or Ben Goertzel)
Tony Smith's Home Page ......