The Spin Chronicles (Part 14): The Universe and Clifford group actions

 I can't refrain myself from starting this post with a quote from the Introduction to the paper "Conformal Mappings, Hyperanalyticity and Field Dynamics" by V.V. Kassandrov (1993): Here it goes:

"1. Introduction

It seems impossible to imagine that the primary Void, which contains the divine diversity of forms, determines the dynamics of the World. However, we have to believe and try to understand it, else we are destined to stay on the primitive level of ‘geographical’ phenomenology.

Maybe it is wise to give up, for a while, all the principles and laws established by our great predecessors, even such ‘fundamental’ laws as Lorentz invariance, gauge symmetry or the minimal action principle. We should not flatter ourselves with their apparent beauty – they are neither basic nor primary, since they themselves are to be explained and grounded. It is also valid for most remarkable empirical facts, such as the invariance of light velocity, the quantization of electric charge, the universality of the inverse square law.

We may regard these facts as axioms and then proceed as far as possible in constructing a fruitful physical theory. This is just the way Einstein created his special theory of relativity (STR).

It seems more attractive to find a new interpretation of well-known physical phenomena, where the existence of the correspondent correlation becomes self-evident (as the equality of gravitational and inert masses after a geometric treatment of gravitation in general theory of relativity (GTR)). There still exists another way of thought, the most thankless one but promising. This is the way bequeathed by Pythagoras, Hamilton, Eddington and other Grands, the way chosen by Einstein in the last half of his life. Here we suppose the existence of exceptional relational structures of a purely mathematical nature and intend to discover them and formulate a unique abstract principle on their basis. The characteristic differential equations, algebro-geometrical properties and other relations should then follow from this principle only. Here it is necessary to bring in conformity all the mathematical categories and solutions with the

quantities describing real objects. If and only if the complete correspondence is achieved, we should consider the Universe as a ‘reflection’, a ‘realization’ of this abstract Principle! 

 

The Universe as a ‘reflection’, a ‘realization’ of this abstract Principle! 

One has to understand that such ideology contradicts the widespread treatment of mathematics as a merely subjective formalism. On the contrary, one has to assume that there do exist completely objective laws in its very structure (see [1]), which have to be discovered and identified with physical laws. In this article we try to show that the afore-stated approach is not so fabulous as it seems at the first sight. Naturally, at once it is impossible to establish that very Principle whose structure carries the main part of physical phenomenology.

Nevertheless, even within the frame work of Hamilton’s and his followers’ well-known ideas, it becomes possible to reveal the wonderful correlations between the properties of an exceptional mathematical structure – the algebra of quaternion-like type – and the physics and geometry of space-time.[...]"

Число, время, свет || Владимир Кассандров || Программа Гордона №313 (04.11.2003)

This is exactly the philosophy I am trying to follow in this series. In The Spin Chronicles (Part 13) we have introduced the group G:

G = {g∈Cl(V): ∆(u) = 1},

where ∆(u) = B_ν(u,u) = (p₀)² - p².

In fact we do not even need ∆(u) in this definition - we need only the Clifford conjugation ν itself, since the element ν(u)u automatically belongs to the center of Cl(V), and we have, as can be verified from the form of the product, and the form of  ν(u) (Part 11):

uv = (p₀q₀ + p·q,  p₀q + q₀p ₊ i p⨯q),

ν(u) = (p₀,-p).

Therefore

ν(u)u = uν(u) =  ((p₀)² - p²)1.

Exercise 1. Verify the last formula.

Once we have a group, we analyze its natural actions. By definition we say that a group G acts on a set S if there is a map G ⨯ S ⟶S, denoted (g,s) ⟼ g·s, such that, for all s we have e·s = s, where e is the identity of the group, and for  all g,h in G and all s in S we have g·(h·s) = (gh)·s. In our case for the construction of natural actions we have at our disposal the automorphism π, and two anti-automorphisms ν, and τ. The following four actions are natural:

1). g·u = g u τ(g),

2). g·u = g u ν(g),

3). g·u = π(g) u τ(g),

4). g·u  = π(g) u ν(g).

Exercise 2. Verify that all four cases are indeed group actions.

We will consider them one by one.

Case 1. We recall the form of τ:

τ(u) = (p₀,p)*,

where "*" denotes the complex conjugation. Every element of Cl(V) can be decomposed as

u = (u+τ(u))/2 + (u-τ(u))/2.

We can write it as u = u_R + u_I. Then  τ(u_R) = u_R,  τ(u_I) = - u_I. In other words we split Cl(V) into a direct sum of two eigensubspaces of τ. It can be easily checked that these eigenspaces are invariant under the action 1).

Exercise 3. Verify the last statement.

In order to analyze the group action we will consider its one-parameter subgroup of the form g(t)=exp(t X), where X is in Cl(V), t is real parameter. To have ν(g(t))g(t)=1, since

ν(exp( t X)) = exp(t ν(X)),

we must have ν(X)=-X. Therefore X must be of the form (0,v+iω), where v and ω are real vectors.

Exercise 4. Prove the last statement.

We will continue our analysis in the next post.