Bohm on Nilpotency and Quaternions

My interest in physics and its relation to phenomenology, education and sociology stemmed from my meeting Peter Rowlands. But really we work and study in a University to mix with people from whom we learn new things and gain new insights which we wouldn't have otherwise gained. 

Peter's work is based on the mathematics which underpins physical law. It addresses fundamental problems which beset quantum mechanics and relativity theory, and addresses the relationship between classical mechanics and quantum mechanics (which is often seen as a radical paradigm shift - something which has done the social sciences no good, as people have jumped on to the "entanglement" bandwaggon). In Peter's arsenal of mathematical devices, two things stand out: the "nilpotent" - the idea of a mathematical entity which when raised to a power equals zero (it's like the square root of minus 1, but with zero). In Peter's universe, everything grows from nothing.

The other element is Hamilton's Quaternions. These are a 3-dimension complex number, which has the property of anti-commutativity: if i and j are elements of the quaternion, then i * j is not the same as j * i. This anti-commutative behaviour introduces powerful and complex symmetries, which when coupled with the nilpotent, arise from nothing. These ideas have changed me (and I noticed that Peirce was also interested in quaternions).

Now, I have been looking much more closely at physics and quantum mechanics more particularly. David Bohm is a fascinating figure because he connects physical theory with a theory of consciousness and communication. A coherent connection to learning and education isn't far behind, although Bohm didn't quite go there - but that's where I'm interested in going!

But Bohm was ahead of the game. In this passage, he seems to prefigure Peter Rowlands work:

We do not regard terms like 'particle', 'charge', 'mass', 'position', 'momentum', etc as having primary relevance in the algebraic language. Rather, at best, they will have to come out as high-level abstractions. [...] the real meaning of quantum algebra will then be that it is a mathematization of the general language, which enriches the latter and makes possible a more precisely articulated discussion of implicate order than is possible in terms of the general language alone. (p 163)

He then discusses some of the properties of the algebra, and hits on two of the key features which are also important in Rowland's work. The first is the nilpotent:

It is important to emphasise that the 'law of the whole' will not just be a transcription of current quantum theory to a new language. Rather, the entire context of physics (classical and quantum) will have to be assimilated in a different structure, in which space, time, matter, and movement are described in new ways. Such assimilation will then lead on to new avenues to be explored, which cannot even be thought about in terms of current theories. 

First, we recall that we begin with an undefinable total algebra and take out sub-algebras that are suitable for the description of certain contexts of physical research. Now, mathematicians have already worked out certain interesting and potentially relevant features of such sub-algebras.

Thus, consider a given sub-algebra A. Among its terms A(i), there may be some A(n) which are nilpotent, i.e., which have the property that some powers of A(n) (say A(n)^6) are zero. Among these, there is a subset of terms A(p) which are properly nilpotent, i.e. which remain nilpotent when multiplied by any term of the algebra A(i) (so that (A(i)A(p))^s = 0) (p. 169)

This is a slightly different take on the nilpotent from Rowlands: Bohm is saying that invariance in structure depends of the absence of what he calls proper nilpotency ("we should have an algebra that has no properly nilpotent terms" (p170)). Rowlands, by contrasts, sees invariance in terms of conservation, and that this is an aspect of the different dimensions associated with mass (a scalar), charge, space (vector), or time (imaginary scalar).

Bohm also addresses the use of quaternions:

It is significant that by mathematizing the general language in terms of an initially undefined and unspecifiable algebra, we arrive naturally at the sort of algebras used in current quantum theory for 'particles with spin', i.e. products of matrices and quaternions. [....] the quaternions imply invariance under a group of transformations similar to rotations in three-dimensional space. (p/170)

Why does this matter to education? It is because Bohm makes the connection between quantum mechanics and consciousness. His "implicate order" is nothing short of what we perceive in our emotional life minute by minute. Bohm's mechanics and his algebra gives us a technical way of thinking about questions of phenomenology as we reach for apprehension of the implicate order from analysing the explicate order. Education itself, and university particularly, deals ostensibly with issues of the explicate order (because they are ostensible), but it sits on an inner logic about which both Bohm and Rowlands have powerful things to say. That their techniques and approach are similar is not, I think, a coincidence.