Over at viXra log, Philip Gibbs had a nice post on quantum mechanics and non-locality. In traditional quantum mechanics, it is often assumed one is constructing projective spaces over the complex field. However, as John Baez has noted at the n-category cafe, one can always formulate quantum mechanics over the quaternions and octonions as well. In order for octonionic quantum mechanics to be properly formulated, the Jordan formulation must be used in order to define projective spaces. Even then, one is limited to constructing a projective plane in the best case, due to algebraic topological constraints.
Back in my undergrad days, I was interested in studying quantum mechanics over arbitrary division algebras, which inevitably leads to the study of Jordan algebras as normed spaces over the reals. In the octonionic case, first studied by Jordan, Wigner and von Neumann back in the 1920's, one can have an algebra of 3x3 Hermitian operators in the maximal case. This case yields the exceptional Jordan algebra, with its corresponding projective space OP^2, the Cayley-Moufang plane. Even in this somewhat pathological case, it is possible to construct a 27-dimensional normed vector space over the reals. This is done by defining an inner product on the exceptional Jordan algebra, (X,Y)=tr(XoY), which induces a positive definite form, the norm, (X,X)=tr(X^2)=|X|^2. This norm also works for any nxn Jordan algebras over R,C,H. In all these cases, the length of a Hermitian operator is zero if and only if it's the zero vector (zero matrix). This means, in particular there are no rank one operators with zero length, and hence our projective spaces as manifolds, are easily described with the number of charts given by the degree of the Jordan algebra. In quantum mechanics this means we can normalize our rank one operators and the norm squared acquires a nice probabilistic interpretation.
When one attempts to give a similar normed space construction for Jordan algebras over the split composition algebras, it turns out the story isn't so nice. The first property that goes out the window is positive definiteness. So in quantum mechanics over split composition algebras there are a bunch of rank one projectors that have zero length. To this, one may say, "so what?" Well, for one, one can't assign a probabilistic interpretation to pure states described by these vectors. Now one may reply, "so just mod these out and define your projective space accordingly" Sure, we can try to do this but what if the physics actually requires the use of these pathological rank one operators?
Quantum mechanics over split composition algebras has already found use in M-theory compactifications, especially in describing extremal black hole charge vectors. In M-theory on T^5 and T^6, the charge vector spaces are actually Jordan algebras over the split octonions. In the black hole context, rank one operators describe 1/2 BPS states with zero entropy. This can be seen by noting rank one operators are those with zero determinant. So what does the (semi)norm mean in this context? I'm not really sure yet. In a literal sense, it gives the distance squared of an operator from the zero matrix. If one borrows some terminology from D-brane constructions, perhaps the norm can be interpreted as giving a type of tension, proportional to some theoretical mass. This would give an interpretation to the non-trivial charge vectors with zero norm: they describe some type of "massless" 1/2 BPS black holes. The other non-zero norm, rank one charge vectors describe "massive" 1/2 BPS black holes. The spectral decomposition of a full rank 3x3 Hermitian operator in the charge space then says that a 1/8 BPS black hole can be viewed as a bound state of elementary massive 1/2 BPS black holes, in some sense.
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2 comments:
The split quaternions are isomorphic to the algebra of 2x2 matrices over reals with the determinant as the norm. No wonder they come up when looking at qubits.
If you can do quantum maechanics using 2x2 matrices, can you do it with NxN matrices? Would that just be a Yangs-Mills theory?
I suppose you just have to second quantise to overcome the bad behavior of the norm.
Yes, people already do Yang-Mills theory with NxN matrices over the reals, complex numbers and quaternions. In those cases, you get O(N), U(N) and Sp(N) symmetry respectively. Such groups arise as Chan-Paton gauge symmetry groups in string theory, where Yang-Mills theory is seen as a special case of D-brane dynamics.
One could conceivably do Yang-Mills theory with split composition algebras. In the split octonion case, for 2x2 and 3x3 matrices one recovers SO(5,4) and F4(4) gauge symmetry. If we take some hints from quantum information theory and twistor strings, we can consider larger symmetry groups that merely preserve the determinant (hence preserve collinearity in projective space). For qubits and qutrits, such groups are known as SLOCC groups. For 2x2 and 3x3 matrices over the split octonions one recovers SO(5,5) and E6(6) symmetry. For M-theory on T^5 and T^6, corresponding to D=6 and D=5, N=8 supergravity, SO(5,5) and E6(6) act as T-duality and U-duality groups, respectively. These groups preserve the fraction of supersymmetry of BPS states. So for example, 1/2 BPS extremal black holes cannot be transformed into 1/4 BPS black holes via the action of such groups.
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