MUBs, nilpotents and idempotents of Clifford algebras

Kea recently wrote about Rowland’s being banned on arXiv. Rowland’s efforts are kind of similar to mine except that he works with the nilpotents (BB = 0) of a Clifford algebra while I work with the idempotents (BB = B). Some years ago, I had a conversation with Hestenes and discussed these things, and he mentioned that there is a close relationship between nilpotents and idempotents. Since then, I guess I’ve learned enough Clifford algebra to write about that relationship and so this post.

Let’s begin by picking up some intuition from the Pauli matrices, the well known matrix representation of the real Clifford algebra generated by three basis vectors with signature +++. As usual, we will call the three basis vectors x, y, z. The “presentation” of the algebra (i.e. the rules that distinguish it from the free algebra on x, y, and z) are that xx = yy = zz = 1, and that x, y, and z anticommute. The Pauli matrices are as follows:

A few idempotent and nilpotent 2×2 matrices are the following:

The above four are of interest because they exhaust the degrees of freedom of the 2×2 matrices. Of course there are an infinite number of idempotent 2×2 matrices and the same for the nilpotent. To get some intuition about the relationship between nilpotents and idempotents, write the above four matrices in terms of the Pauli spin matrices and unity. (The reader might be more familiar with this sort of thing in the Dirac algebra, where one often has cause for writing a 4×4 matrix in terms of the Dirac bilinears. This amounts to writing 4×4 matrices using a basis set of the 16 Dirac bilinears. The Pauli matrices are only 2×2 and so require a basis set of only four elements. We use the unit matrix and the three Pauli spin matrices.)

The two idempotent matrices can be written as (1+z)/2 and (1-z)/2. The two nilpotent matrices can be written as (x+iy)/2 and (x-iy)/2. Not very similar, eh? I mentioned above that these four matrices form a basis for the 2×2 matrices considered as a 4-dimensional complex vector space. Add them together. You get the matrix 1s in all its elements. This is called the “democratic matrix.” Or sometimes the matrix that is proportional and with trace 1, that is, the nxn matrix with all entries 1/n, is called by this name. In the second case, the matrix is idempotent. And for the 2×2 matrices, the democratic matrix is the density matrix for spin in the +x direction (or twice that matrix).

Note that in the Pauli algebra, i = xyz. So we can rewrite the two nilpotent matrices this way:
(x+iy)/2 = (x+xyz y)/2 = (x-xz)/2 = x(1-z)/2,
(x-iy)/2 = (x-xyz y)/2 = (x+xz)/2 = x(1+z)/2.

In other words, the nilpotent matrices are the same as the idempotent matrices multiplied by x. The reader may be intrigued to find that you can commute the x to the right hand side of the primitive idempotents if you change their signs:
x(1-z)/2 = (1+z)x/2,
x(1+z)/2 = (1-z)x/2.
This property is part of the reason the nilpotents can be defined this way. More generally, since (1+z)/2 is idempotent, that is, since [(1+z)/2][(1+z)/2] = [(1+z)/2], we can rewrite these products as follows:
x(1-z)/2 = [(1+z)/2] x [(1-z)/2],
x(1+z)/2 = [(1-z)/2] x [(1+z)/2].
In other words, these nilpotents are obtained by putting the operator x in between the primitive idempotents. You could also put the operator y in there. You could put a ham sandwich in there so long as it annihilated neither (1+z)/2 nor (1-z)/2.

More general Clifford algebras

More generally, the idempotent structure of a Clifford algebra is defined by the structure of its “complete sets of commuting roots of unity.” In this, “root of unity” means elements of the Clifford algebra that square to one. Perhaps it would be better to write “square root of unity,” but one finds uses for nth roots of unity in creating basis sets for other than Clifford algebras. “Commuting” means that the set of roots of unity all commute with each other. “Complete” mean that there are no other square roots of unity in the Clifford that commute with the set, other than trivial products of the roots one includes. In the quantum sense, a complete set of commuting roots of unity define a complete set of quantum operators (and therefore quantum numbers) for the quantum states of the Clifford algebra. They are sort of like a Cartan subgroup in that they are maximal and Abelian.

For the Pauli algebra, a complete set of commuting roots of unity can have only one element, call it A. Then the primitive idempotents generated by that set is given by (1+A)/2, and (1-A)/2. Larger Clifford algebras can have larger numbers of commuting roots of unity. For example, the Dirac algebra (the gamma matrices of QED), has two commuting roots of unity. One might choose and where I assume the -+++ signature. With more commuting roots of unity, the primitive idempotents are defined as products of idempotents generated by the roots. For the case of the Dirac algebra, the four primitive idempotents generated by {A,B} are:
(1+A)(1+B)/4,
(1+A)(1-B)/4,
(1-A)(1+B)/4,
(1-A)(1-B)/4.

If one has N commuting roots of unity, one ends up with 2^N primitive idempotents.

MUBs and Nilpotents

A subject we’ve been discussing a lot recently are Mutually Unbiased Bases or MUBs. Two bases are “mutually unbiased” if the transition probabilities between the states of one basis and the states of the other are all equal. For the case of the Pauli algebra, two bases are mutually unbiased if they are specified with respect to perpendicular directions. Thus the largest possible number of mutually unbiased bases for the Pauli algebra is 3; say spin in the x, y, and z directions.

From the intuition hint at the beginning of the post, we suspect that one obtains the nilpotents of a Clifford algebra by taking the primitive idempotents, and multiplying them (on the left say) by a root of unity chosen from a mutually unbiased basis. Does this work? Yes it does, and I leave it as a cute exercise for the reader. Given two complete sets of commuting roots of unity, say B_n and C_n, the product c = (1 +- C_1)(1 +- C_2)…(1 +- C_N)/2^N is a primitive idempotent, and the product of this with B_1 is a nilpotent.

Nilpotents and Idempotents in Elementary Particles

The form of the nilpotents of the Pauli algebra, (x+iy)/2 and (x-iy)/2 should remind the reader of the raising and lowering operators of the Pauli algebra. The raising operator takes spin down and converts it to spin up, and it annihilates spin up. The lowering operator converts spin up to spin down and annihilates spin down. Now we could see which operator is which by writing them out in matrix form, but matrix representations are so arbitrary that this is a bit disgusting to a mathematician.

Instead, let us first note that one can convert the usual quantum states into density matrix form and that these forms are precisely the primitive idempotents. Spin up is (1+z)/2, while spin down is (1-z)/2. These two states annihilate each other, that is, they multiply to give zero. So if we write the nilpotents in idempotent form, it is clear what they do to spin up and spin down. Clearly, (x+iy)/2 = [(1+z)/2] x [(1-z)/2] is the raising operator. And (x-iy)/2 is the lowering operator.

Furthermore, we see that the choice of “x” is arbitrary. We could just as easily have chosen y as the method of converting the z-basis primitive idempotents to uh, primitive nilpotents. Our raising and lowering operators would still raise and lower, but y would give different phases to the raised and lowered states.

In QFT, one usually assumes a quantum vacuum, that is, a quantum state upon which the creation and annihilation operators act so as to create and eliminate particles. Since spin-1/2 is a fermion, one cannot create two particles with the same quantum states, so one requires that the creation operators (and therefore the annihilation operators, which create antiparticles) be nilpotent. This is a good reason for using the nilpotents of a Clifford algebra to define the quantum states (rather than the idempotents which are more natural from a density matrix / quantum mechanical point of view).

And in fact that is exactly what Rowland does. One otherwise needs Grassman numbers so as to make the creation and annihilation operators nilpotent.

My work uses idempotents instead because I work in the density matrix formalism. For me, the creation and annihilation operators are just mathematical conveniences. They have arbitrary phases. In reality, elementary particles cannot be created or destroyed without any other consequence. To create a positron from nothing violates conservation of charge; physically, one must also create some negatively charged object at the same time. So for me, the creation and annihilation operators are just mathematical conveniences.

Density matrix formalism

If one rewrites the density matrix formalism back down into the state vector formalism of creation and annihilation operators, the primitive idempotents become counting or number operators. To see this, consider the effect of a raising operator followed by a lowering operator:

[ x(1-z)/2 ] [ x(1+z)/2 ]
= [ (1+z)/2 x ] [ x(1+z)/2 ]
= [ (1+z)/2] [ xx ] [ (1+z)/2 ]
= [ (1+z)/2] [ (1+z)/2 ]
= [ (1+z)/2]

That is, a raising operator followed by a lowering operator is just the density matrix for spin up. In the usual vacuum formalism, such an operator is a counting operator; it counts the number of particles in the up state. The eigenvalues are 0 if there are none and 1 if there is one. Thus, in a way, density matrix formalism amounts to restricting one’s quantum field theory to the number operators.

P.S. The topic of Rowland’s work came up in correspondence having to do with Painleve coordinates and the gauge gravity theory of Lasenby, Gull and Doran discussed by Hestenes, and the huge number of hits I was getting on this blog in relation to that. I’m going to write a post critical of general relativity as compared to gauge gravity one of these days. Basically, gauge gravity gives identical results as general relativity outside of event horizons but is built on a flat underlying metric. So it gets all the nice experimental results, but avoids closed timelike circuits and other topological nonsense. Rowland’s work is similar to mine in that he begins with the Clifford / geometric algebra and demands that all symmetries arise from that. He also talks a lot about geometric or Berry phase.