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Split Octonions,

Split CliffordAlgebras,

Witt Index, SymplecticSpaces,

and

Spin(4,3), Cl(4,3):

 
 The split octonions O" have signature (----++++)  of a symmetric quadratic form - X0^2 - X1^2 - X2^2 - X3^2 + Y0^2 + Y1^2 + Y2^2 + Y3^2  instead of (++++++++)  of a symmetric quadratic form + X0^2 + X1^2 + X2^2 + X3^2 + Y0^2 + Y1^2 + Y2^2 + Y3^2 == + X0^2 + X1^2 + X2^2 + X3^2 + X4^2 + X5^2 + X6^2 + X7^2  as the octonions O.  To build the split octonions and the octonions, use the Cayley-Dickson process: Start with an algebra A, with conjugation * .  Define A+ to be the vector space A + A with a new product defined by (x,y)(z,w) = ( x z - w* y , w x + y z* )  and  define A- to be the vector space A + Awith a new product defined by (x,y)(z,w) = ( x z + w* y , w x + y z* )   Let R denote the real numbers, C denote the complex numbers, H denote quaternions, and O denote octonions.  Then:  C = R+ H = C+ O = H+ and  R- is something different, the Lorentz numbers L  C- is not really new and different, but is the 2x2 matrix algebra over R, which can be denoted M2(R) This is because a complex number a + bi can be represented as a 2x2 real matrix   a -b b a   with complex conjugation defined by the matrix operation   1 0 0 -1   H- is something different, the split octonions O" .  Notice that because L, M2(R), and O" have non-trivial "lightcones", they are NOT division algebras, although they ARE normed algebras.  The only division algebras over the reals are R, C, H, O and the only normed algebras over the reals are R, C, L, H, M2(R), O, and O".  That means that all normed algebras over the reals are subalgebras of either the octonions O or the split octonions O".   Now, look at the 7-dimensional imaginary spaces Im(O) and Im(O")  Neither O nor O" are associative, but you can find 3 of the 7 imaginaries that, with the real axis, make an associative subalgebra, either quaternions H, in the case of O, or M2(R), in the case of O".  Call such a set of 3 an Associative Triple. Call the other 4 imaginaries a Coassociative Square.  FOR O THE SIGNATURE OF THE ASSOCIATIVE TRIPLE IS DIFFERENT FROM THE COASSOCIATIVE SQUARE; FOR O" THE SIGNATURE OF THE ASSOCIATIVE TRIPLE IS THE SAME AS THE COASSOCIATIVE SQUARE.  
The quadratic form of the Split Octonions is the Symmetric Quadratic Form  - X0^2 - X1^2 - X2^2 - X3^2 + Y0^2 + Y1^2 + Y2^2 + Y3^2  of the real Clifford Algebra Cl(4,4).  If p = q, the real Clifford Algebra Clf(p,q) is called a Split Clifford Algebra.  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.  Let, for example, p be less than q and q = p + k.  Then the vector space R^(p,q) = R^(p,p+k) of the real Clifford Algebra Cl(p,p+k) can be decomposed into two parts:  a positive definite part R^k  and  a split part R^(p,p).  In turn, the split part R^(p,p) can be decomposed into two null subspaces N^p and N'^p each of dimension p.  The resulting decomposition of R^(p,p+k) into the direct sum N^p + N'^p + R^k is called a Witt Decomposition.  Why are the spaces N^p and N'^p called null subspaces?  You can choose a basis { X1, ... , Xp } for N^p and a basis { Y1, ... , Yp } be for N'^p such that the plane spanned by { Xi, Yi } is a Hyperbolic plane for all i from 1 to p.  Therefore, the Witt Decomposition decomposes the vector space R^(p,q) = R^(p,p+k)into p copies of the Hyperbolic plane plus a k-dimensional definite space (in the example, positive definite).  The number of copies of the Hyperbolic plane is the Witt Index.  Some examples of the Witt Index in physics are:  For Example, begin with R^2 of U(1) = Spin(2) of Cl(0,2) with quadratic form + Y1^2 + Y2^2 then add one Hyperbolic Plane to get the quadratic form - X0^2 + Y0^2 + Y1^2 + Y2^2 of Spin(1,3) of Cl(1,3) of Witt Index 1 of the Lorentz group of Minkowski space, and then add a second Hyperbolic Plane (and renumber the basis) to get the quadratic form - X0^2 - X1^2 + Y0^2 + Y1^2 + Y2^2 + Y^3^2 of Spin(2,4) of Cl(2,4) of Witt Index 2 of the Conformal Group and Conformal Space.   For Another Example, begin with R^24 (the 24-dimensional Euclidean space of the Leech lattice)of Spin(24) of Cl(0,24) with quadratic form + Y1^2 + ... + Y24^2 then add one Hyperbolic Plane to get the quadratic form - X0^2 + Y0^2 + Y1^2 + ... + Y24^2 of Spin(1,25) of Cl(1,25) of Witt Index 1 of the 26-dimensional space of the Lorentz Leech lattice that can be used to construct a representation space for the largest finite simple group, the Monster group. The 26-dimensional space is also used in superstring theory.    

To construct SymplecticSpace

 begin with the Split real Clifford Algebra Cl(p,p) based on a Symmetric Inner Product IPsym(X,Y) = IPsym(Y,X) =  - X1 Y1 - ... - Xp Yp  + X(p+1) Y(p+1)  + ... + X(2p) Y(2p)  which gives the Cl(p,p) quadratic form QF(X,X) =  - X1^2 - ... - Xp^2  + X(p+1)^2  + ... + X(2p)^2  then consider the Skew-Symmetric Inner Product IPskew(X,Y) = - IPskew(Y,X) =   X1 Y2 - X2 Y1 + ... + X(2p-1) Y(2p) - X(2p) Y(2p-1)  Notice that the space spanned by the p odd-numbered basis elements { X1, X3, ... , X(2p-1) } is a p-dimensional null space because IPskew(X,Y) restricted to that space is zero, and the space spanned by the p even-numbered basis elements { X2, X4, ... , X(2p) } is an orthogonal p-dimensional null space because IPskew(X,Y) restricted to that space is zero.  The p pairs of basis elements { X1, X2 } , ... , { X(2p-1), X(2p) } each form a 2-real-dimensional Symplectic Plane somewhat analogous to a 2-real-dimensional Hyperbolic Plane in a Witt Decomposition of the vector space of a Split Clifford Algebra Cl(p,p).  Any Symplectic Space is just the product of a number of Symplectic Planes. Therefore, any Symplectic Space has even dimension.  

How do you represent SymplecticSpaces?

One way is to notice that each Symplectic Plane has a Skew-Symmetric Inner Product  IPskew(X,Y) = - IPskew(Y,X) =  X1 Y2 - X2 Y1  that produces a skew Quadratic Form  QFskew(X,X) =  X1 X2 - X2 X1  that is the commutator of the basis elements { X1, X2 }  Therefore, if you want a non-trivial represention of the Symplectic Plane in terms of a real Division Algebra and you also want to have an associative representation so that you can make a matrix representation of any order, then you cannot use the commutative Division Algebras (the Real numbers R or the Complex Numbers C) and you cannot use the non-associative Octonions O, so you must represent the Symplectic Plane in terms of the associative non-commutative Quaternions Q.  Since the Quaternions Q are 4-real-dimensional, they are not a 1-1 representation of a 2-real-dimensional Symplectic Plane, so that the Quaternions represent a pair of Symplectic Planes, and the fundamental Quaternionic Symplectic Space is 4-real-dimensional.   The Quaternionic representation of the Skew Inner Product is the n-quaternionic dimensional (4n-real-dimensional) Quaternionic Skew-Hermitian IPskewH(X,Y) =  X1* j Y1 + ... + Xn* j Yn  where * denotes quaternionic conjugation and j is the element j of the quaternionic basis {1,i,j,k} that defines the standard isomorphism   Q = C + Cj = C^2  relating the Quaternions Q to 2-complex-dimensional C^2.   Generalized rotations in the n-quaternionic-dimensional space produce the Cn series of Sp(n) Lie groups, which have the fibration Sp(n+1) / Sp(n)xS3 = QPn, Quaternionic Projective n-space. Note that the 3-sphere S3 = Sp(1) = SU(2) = Spin(3).    Another way to represent Symplectic Spaces is to work directly with the real Symplectic Planes, starting with the Skew-Symmetric Inner Product, usually  with the basis reordered from the ordering used above, so that IPskew(X,Y) = - IPskew(Y,X) =   X1 Y(p+1) - X2(p+1) Y1 + ... + Xp Y(2p) - X(2p) Yp which can be written in terms of the basis { X1, ... , Xp; X(p+1), ... , X(2p) } and the /\ product of exterior algebra as  X1 /\ Y(p+1) + ... + Xp /\ Y(2p)  Since the p-dimensional exterior algebra has 2^p elements, with graded structure   0  1    2               k               p-2    p-1  p  1  p  p(p-1)/2 ... p!/(k!(p-k)!) ... p(p-1)/2   p   1   such that the top line is the grade and the bottom line is the dimensionality of the space of elements of that grade,  and since, if we relable the basis { X1, ... , Xp; X(p+1), ... , X(2p) } as { U1, ... , Up; V1, ... , Vp }   each set { U1, ... , Up } and { V1, ... , Vp } spans a p-dimensional null space of the Symplectic Space.   Every p-dimensional null space of the Symplectic Space is a null space of maximal dimension. It is transverse to at least one of the 2^p null spaces.  

Two subspaces of a vector space are defined to be transverse ifftheir sum is the whole vector space.

Two p-dimensional spaces of R^2p are transverse iff theirintersection is zero.

For example, the null spaces spanned by { U1, ... , Up } and { V1, ... , Vp } are transverse to each other.   Unlike p-dimensional Euclidean space, in which any k-dimensional space can be rotated into any other k-dimensional space, although any nonzero vector in Symplectic Space can be symplectically rotated into any other nonzero vector, it is NOT true that any nonzero Symplectic Plane can be symplectically rotated into any other nonzero Symplectic Plane. This approach seems more abstract to me than the Quaternion approach, but this abstract approach is very useful in constructing such things as Poisson brackets of Hamiltonians based on phase spaces, or cotangent bundles to configuration manifolds.  Since Symplectic Spaces are always of even dimension 2p, similar physics applications in odd dimensions (2p-1) can be constructed as contact structures, based on the set of contact elements, or (p-1)-dimensional subspaces of the p-dimensional tangent spaces of the p-dimensional configuration manifold. The space of contact elements has dimensionality p + -1 = 2p - 1 because p coordinates are needed to specify the point of contact and p-1 coordinates to specify which subspace.    Such physics application are described in Mathematical Methods of Classical Mechanics, 2nd. ed., by V. I. Arnold, Springer-Verlag 1989; and Classical Mathematical Physics, Dynamical Systems and Field Theories, 3rd. ed., by Walter Thirring and translated by Evans M. Harrell II, Springer 1997.        
 The Lie group of rotations of 7-dimensional real Euclidean space is SO(7), while the rotation group in 7-dimensional real space with signature (---++++) is SO(4,3).  SO(4,3) is to SO(7) as the split octonions O" are to the octonions O,  Now look at the automorphism groups of O and of O".  The automorphism group of the octonions O is the 14-dimensional exceptional Lie group G2  G2 is a subgroup of SO(Im(O)) = SO(7) and  the (also 14-dimensional) automorphism group of the split octonions O" can be denoted by G"2.  G"2 is a subgroup of SO(Im(O")) = SO(4,3)  In particular, G2 and G"2 are the subgroups of SO(7) and SO(4,3) that preserve Associative Triples.  
  You can also look at SO(3,4) from the point of view of the Clifford algebra Cl(4,3) of the real 7 dimensional vector space of signature 4,3. It gives the covering group Spin(4,3) of SO(4,3).  Signature can make a difference with respect to Clifford algebras. As matrix algebras:  Cl(0,8) = Cl(8,0) = Cl(4,4) = M16(R), the 16x16 real matrices,  but  Cl(7,0) = M8(C), the 8x8 complex matrices, Cl(0,7) = M8(R) + M8(R), the sum of two sets of 8x8 real matricesand Cl(3,4) = M8(C), the 8x8 complex matrices, Cl(4,3) = M8(R) + M8(R), the sum of two sets of 8x8 real matrices.  However, (see page 146 of the book Clifford Algebras and the Classical Groups, by Ian Porteous, Cambridge 1995), when you go from the Clifford algebra Cl(p,q) to the Spin group Spin(p,q) that is defined by that Clifford algebra, you find that Spin(p,q) = Spin(q,p) , so that Spin(3,4) = Spin(4,3) and it makes sense to talk about Spin(7) = Spin(7,0) = Spin(0,7)  Be careful to realize that, although Cl(0,7) = Cl(4,3) as matrix algebras, the difference in their underlying vector space signatures means that Spin(7) is not the same as Spin(3,4).   Now look at 7-spheres.  For octonions O, define S7 = { x in O : |x| = 1 }  For split octonions O", define S7+ = { x in O" : |x| = +1 } S7- = { x in O" : |x| = -1 } and S7+- as the union of S7+ and S7-  Just as there is a fibration Spin(7) / G2 = S7 there is a fibration Spin(4,3) / G"2 = S7+   Just as there is a fibration Spin(8) / Spin(7) = S7 there is a fibration Spin(4,4) / Spin(4,3) = S7+-   Just as there is a fibration Spin(9) / Spin(7) = S15 there is a fibration Spin(4,5) / Spin(4,3) = M15  where M15 = { x in R(8,8) : |x| = -1 or |x| = -1 } where R8,8 is real 16-dim space with signature (8,8).   
 References:  Spinors and Calibrations, by F. Reese Harvey, Academic Press 1990; e-mail conversations with Onar Aam; Clifford Algebras and the Classical Groups, by Ian Porteous, Cambridge 1995; e-mail conversation with Scott Chang; Representation of Compact Lie Groups, by Theodor Broecker and Tammo tom Dieck, Springer-Verlag 1985; Mathematical Methods of Classical Mechanics, 2nd. ed., by V. I. Arnold, Springer-Verlag 1989; Classical Mathematical Physics, Dynamical Systems and Field Theories, 3rd. ed., by Walter Thirring and translated by Evans M. Harrell II, Springer 1997.    Notation here is not quite conventional, because of ASCII limitations.  I use the signature convention of Porteous, rather than that of Harvey.  
   

 

 

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