Gary,
Thanks for the interest in my essay, and for the questions, which give me the opportunity to give some details, for you and others who may be interested.
> I am a believer in quaternions as the basis for Physics so I am able generally to follow your arguments.
Yes, quaternions appear more than we normally see in the usual formulations of physics. The Clifford algebras admit matrix representations in terms of real, complex, or quaternionic numbers (see this wiki page).
The Clifford algebra of a real n-dimensional vector space endowed with a metric (or scalar product or inner product) whose diagonal form is (+1,...,+1,-1,...,-1), where +1 occurs r times and -1 occurs s times, and r+s=n, is denoted by Clr,s. In all cases the representations are 2qx2q matrices, with elements from R, C, or H. The number q is determined so that the total real dimension of the matrix algebra is 2qx2q x dim K = 2n, where K = R, C, or H.
In particular, the Clifford algebra of the three dimensional Euclidean space is isomorphic with the 2x2 complex matrices, but also with the complex quaternion algebra, that is, quaternions a+bi+cj+dk, where a,b,c,d are complex (the complex imaginary unit i' is different than the quaternion i). For the Minkowski metric (+1,-1,-1,-1), the Clifford algebra Cl1,3 is represented by the 2x2 matrix algebra of quaternions. Dirac matrices are actually complex, because he worked with the complexified of Cl1,3, but there is a change of basis which puts them in quaternionic form, where each quaternion element of the matrix is represented by a 2x2 complex matrix.
> The notion that all the information of the universe could be encoded at each point of the universe is profound. It is more than I can wrap my mind around.
An analytic function is one that can be expressed as a power series, like
[math]f(x) = f(a) + frac{f'(a)}{1!}(x-a) + frac{f''(a)}{2!}(x-a)^2 + frac{f'''(a)}{3!}(x-a)^3 +... [/math]
You see that if you know f(a) and all of its derivatives f'(a), f''(a) etc, you can find f(x) for any x. So all information about the function f is contained at a, in f(a), f'(a), f''(a) etc. It is also possible for a function of more variables, like f(x,y,z,t), or for a field, to be analytic like this, in which case you will need all partial derivatives at that point.
In physics, fields satisfy systems of partial differential equations, whose coefficients are usually constant. They have analytic solutions, if the initial conditions are analytic too. If they are not, then the solutions are not analytic. The holomorphic functions I mention are also analytic, but they are always like this, once they satisfy the Cauchy-Riemann equations. So this is a big difference, since there are no non-analytic solutions for the Cauchy-Riemann equations.
> Equation 1 simply looks like multiplication of a pair of vectors to me. That is the opposite of the sum of the dot product and the cross product.
It is a multiplication of a pair of vectors, made of their scalar product a·b and their exterior product a∧b. But by associativity, this can be extended to all multivectors of the Clifford algebra. The exterior product is not a vector, and the scalar product is, of course, a number. But they combine together into an algebra, which is the Clifford algebra. In the matrix representation, the scalar is represented by a number multiplied with the identity matrix. So all these are represented as matrices of the same dimension.
> Regarding Equation 2 (Dirac Operator), is the function f a vector? If so, then is d/dx in the k direction and d/dy in the j direction?
Yes. Here I am talking about a 2-dimensional space V whose metric is diagonalized to (-1,-1). Its Clifford algebra is Cl0,2, and its matrix representation is just 1x1 matrices of quaternions (see this wiki page). The basis of the vector space V is given by two vectors, which can be chosen to be the quaternions j and k. So a vector (x,y) can be written as xj+yk. The exterior product j∧k is just the third quaternion, i. This i acts on the basis j,k by rotating it with 90 degrees. So you are right, d/dx is in the k direction and d/dy in the j direction. And if we take f(x,y) to have values in V, f(x,y)=fx(x,y)j + fy(x,y) k, and equation (2) is just the Cauchy-Riemann equations. But xj+yk is not a complex number, so how you identify f(x,y) with a complex function f(z), where z=x+iy? We notice that xj+yk=(x+iy)j. This is because the vector space, even if it is complex vector space, it doesn't have preferred real and imaginary parts. Real and imaginary parts make sense for complex numbers, but this is a complex vector space, so you have to choose which direction is its imaginary one, similar to how you choose in a 1 dimensional real vector space which vector corresponds to +1 and which to -1.
We can also take f(x,y) of the form f(x,y)=f0 + f1 i, and equation (2) gives the Cauchy-Riemann equations in this case too.
> You lose me when you get to the discussion of the standard model. Allow me to ask a question though ... can that vast menagerie of particles actually be fundamental? And how can something be fundamental if it is not stable? Surely the thing to which these particles decay must be fundamental instead?
Only the leptons (electron, muon, taon and their neutrinos), quarks, exchange bosons and the Higgs boson are considered to be fundamental, and I will explain why. A fundamental particle can decay, for example a u quark can decay into a d quark (which is lighter) and a W+ boson, which in turn can decay into a positron and a neutrino. But the u quark is still fundamental, because it is not actually made of a d quark and a W+ boson. This notion of "fundamental" came from the idea of using symmetries to classify particles. Wigner and Bargmann did this, considering the spacetime flat (Minkowski), and taking as internal degrees of freedom only the spin, but this can be extended to charges, colors, weak isospin etc.
So here is the idea of Wigner and Bargmann. They knew from quantum mechanics that the wavefunction of a particle has to have its squared amplitude integrable to 1 over the entire space. Its evolution should preserve this property. But also any rotation or translation or Lorentz boost of a wavefunction should change it into a wavefunction which has the same property. So they wanted to represent the spacetime symmetries, given by the PoincarГ© transformation, by unitary or antiunitary transformations of the wavefunction, so that the squared amplitude is preserved under these transformations. They gave a theorem which states that transformations representing the PoincarГ© group transform the wavefunctions by unitary or antiunitary transformations only if they are spinors, of various spins, 0, 1/2, 1, 3/2 etc. So they classified the particles by using the representations of the PoincarГ© group like this. In fact, these are representations not of the PoincarГ© group (translations and Lorentz transformations), but of the group made of translations and the spin group SL(2,C), which is a double cover of the (proper orthochronous) Lorentz group. To include time and space inversions we have to add a conjugate representation of the spin group, and this is how we get the Dirac spinors. The spinors they obtained are spinor fields, so they depend on (x,y,z,t). But composite particles depend on (x,y,z,t) more than once, and are reducible. Only the irreducible representations correspond to elementary particles, not made of other particles.
The hundreds of mesons and barions are not elementary, being made of quarks and antiquarks.
Should these many elementary particles be considered fundamental? I would say that the quantum fields are "more fundamental".
In the model I proposed and which you mention, all leptons and quarks of a generation are just parts of a field with values in that Clifford algebra. When you represent this in the matrix form I shown at page 8 in the essay, you see there the electron, neutrino, and up and down quarks of all their colors, and their antiparticles. Their various discrete charges are there, and also the exchange bosons, arising from the symmetries of this algebra. And the space where the Higgs lives is there too. I have to do more work to fit three families of leptons and quarks, and also gravity.
Best regards,
Cristi
