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Vigier Centenary 2021Journal of Physics: Conference Series 2197 (2022) 012001

IOP Publishingdoi:10.1088/1742-6596/2197/1/012001

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A Review of Majorana fermions and the laws of form

Louis H Kauffman

Department of Mathematics, Statistics and Computer Science 851 South Morgan Street University of Illinois at Chicago Chicago, Illinois 60607-7045 [email protected]

Abstract. This review article is an exploration of G. Spencer-Brown’s well-known work on the Laws of Form and its relationship to physical science, focusing on correspondence with Majorana fermions.

1. Introduction

The purpose of this paper is to explore the interrelationship of sign, space, physics and the concept of distinction, using G. Spencer-Brown’s work “Laws of Form” [Spencer-Brown (1969)] as a pivot and a guide to the evolution of concepts.

The Spencer-Brown mark is a sign that can represent any sign, and so begins in both universal and particular modes. The mark is seen to make a distinction in the space in which it is written, and so can be seen, through this distinction, to refer to itself. By starting with the idea of distinction we find, in the mark, the first sign and the beginning of all possible signs. The mark stands for an observer inseparable from that which is observed. We will explore an interpretation of the mark as a “particle” that can interact with itself in two ways. These are the ways that a form can interact with itself. A form can, when placed adjacent to itself, condense to a single copy of itself.

A form, taking itself as its own argument, can vanish.

Regarding the mark as a representative elementary particle P we have that P can interact with itself to produce either itself or annihilate itself. We can even write

PP = P + 1

To indicate the two possibilities by using the + sign as a form of logical “or” and using 1 as an indication of the state of no particle at all.

=

=



2

Particles that are their own anti-particles are called Majorana Fermions [Kauffman, L. H. (2016a)]. A particle that is a Majorana Fermion and also can interact with itself to produce itself is called a Fibonacci Majorana Fermion. Majorana conjectured the existence of particles that are their own anti-particles. It remains a puzzle to this day to know if such particles exist in Nature. One of the current sources of speculation about such particles occurs because of the structure of Fermion algebra, as we shall see. From the point of view of physics, Fermions are particles like electrons, and they are described by an algebra called the Fermion algebra. Associated with an electron (say) are two mathematical operators

U and U †. You can think of U as associated with the particle and U †

as associated with the anti-particle. The Pauli Exclusion Principle states that two particles cannot occur in the very same state. This leads to the equations

U 2 = 0

U †2 = 0

The remaining equation in the Fermion algebra is

UU † +U †U = 1

You can think of this as saying that it is possible to have an electron and an anti-electron emerge from the vacuum. Here 1 denotes a vacuum and the equation says that pairs of particle and antiparticle can

emerge from the vacuum. The order of the operators is important and so we write both UU † and U †U

We will see that this algebra arises naturally when we think of an elementary process associated with oscillation between marked and unmarked states. Such oscillation is fundamental at the place where physics, observation, thought and understanding occur together. The mathematics of the place where the Fermion algebra emerges is not yet numerical. We use algebra to describe this structure, but the place where it emerges really has only one distinction. There is no counting yet. We are only looking at an oscillation between 0 and 1. Nevertheless, we are very close to counting and to the appearance of both negative and positive numbers when we arrive at oscillation. We show how our temporal analysis shifts when we allow counting and negative as well as positive numbers. Then another species of algebra naturally arises in relation to oscillation between 1 and -1. Here we find the beginnings of Clifford algebra where there are elements a and b with a^2 = 1 and b^2 = 1 and ab + ba = 0. As you can see, this Clifford algebra is a flip from the Fermion algebra with 0 and 1interchanging places. Thus, you might think of the Clifford algebra as a different culture from the Fermion algebra! But remarkably there is a translation between them that is mediated by i, the square root of minus one! The formulas for this mediation are shown below.

U = (a + ib) / 2

U † = (a - ib) / 2

aa = bb = 1,ab + ba = 0

Then the equations U 2 = 0 , U †2 = 0 and UU † +U †U = 1 are a consequence of the Clifford algebra equations: aa = bb = 1, ab + ba = 0. Conversely, if we have U and U* satisfying the Fermion equations then we can write



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a = (U +U †) / 2

b = (U -U †) / 2i

where i is a square root of minus one that commutes with all other algebraic elements. The Clifford algebra equations follow from the Fermion operator equations. Clifford algebra generates Fermion operators. Fermion Operators generate Clifford algebra.

One can speculate that the a and b are themselves physical particles and that the U and U † represent

composites of particles of type a and b. And since a† = a and b† = b , it is reasonable to regard a and b as Majorana particles that are their own antiparticles. These are speculations, but we will show how a very natural line of investigation leads to the Clifford algebras and to these structures and indeed to the Dirac equation, so that one can make a case for these algebraic structure to be fundamental at that place where physics, mathematics and indeed the inception of language begins. These structures are evocations of fundamental process in which mathematics, physicality and understanding are emerging together.

If we begin with the idea that the mark represents a distinction that can act on itself to annihilate itself, then we can consider a feedback process with equation

Jt+1 = Jt.

In Section 2, we consider this process and the algebra that can naturally describe it. We show in this section that the Fermion algebra is a description of this oscillatory process. This is

a different motivation for the Fermion algebra than the usual quantum context. It is a natural evolution of algebra in the description of discrete oscillation. The only mathematics that is used in this section is the two valued mathematics of the marked and unmarked states, written in the usual notation of 0 and 1. We point out that the recursion itself can be seen as a temporal unfolding of the apparently paradoxical equation

J = J .

Section 3 shows how re-entry and oscillation leads to Fermion algebra and the emergence of the square root of minus one in the context of Clifford algebra. Here we allow the mathematics to expand to include -1 as well as +1. Once this is available, the relationship between Fermion algebra and Clifford algebra (as we have outlined it above) comes forward and the paradox to be resolved is

i = -1/i,

the analogue in ordinary arithmetic of the truth value paradox in logic. In Section 4, we take a wider view and outline the historical emergence of algebra, pointing out that

temporal and non-commutative interpretations came late in the sequence of events, and that the reader will be able to see that our temporal interpretations of Fermion and Clifford algebras are a jump from the historical progression and yet related to it in important ways. It is also important to realize that the square root of minus one first appeared as a commutative entity, but in our temporal jump it first appears in a Clifford algebraic and non-commutative context.

Section 5 gives another point of view about the temporal interpretation of the square root of minus one, showing that if we regard e = (-1,1) as an oscillation from -1 to 1 and e* = (1,-1) = - e as an oscillation from 1 to -1, then one can define a new algebra element {e} so that {e}{e} = ee* = -1 and consistently so that the new algebra is associative. This is the formal idea that the oscillation combines with itself just shifted from itself by one time step, and that is the source of the minus one. In examining this algebraic structure and requiring associativity, we show that there is necessarily an element h = {1}



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so that heh = e*. This shows that the method we have used in associating an algebra to an oscillation

is justified on abstract grounds. This construction is to Clifford algebras what the Cayley-Dickson process is to its family of algebras.

Section 6 shows how the Einstein’s special relativity interfaces with Clifford algebras at the pattern of the Pythagorean Theorem and square zero elements of the algebra. Then Section 7 shows how this is related to the Dirac operator, the Dirac equation and the nilpotent approach to quantum mechanics of Peter Rowlands. Section 9 discusses braiding and Majorana Fermions. We first show how the representation of a Majorana Fermion as the mark of distinction leads directly to a braid group representation. We then compare this with the Ivanov representation that comes from the Clifford algebras that we have described in this paper. Section 9 is an epilogue and a discussion of ideas in the paper. 2. Discrimination and fermions

In this paper we point out that we find non-commutative algebra and indeed Clifford algebra and the related Fermion Algebra right at the beginning of a temporal analysis of elementary processes. We regard this as illuminating, and we will now tell this story in detail. The operator algebra for Fermions arises very early in our story. We begin with the concept of a first distinction.

The concept of a first distinction leads to an oscillation and need for the concept of time. A first distinction can only become a distinction if the contents of the distinction are seen to differ in value. But this difference is a further distinction beyond the first distinction. If we are to stay at the level of only first distinction, then the first distinction may at once disappear. Then it can reappear in a timeless oscillation. It is this process that is a first thing that arises from no-thing.

How shall this primordial vibration be indicated? Let us use 0 for void or absence and 1 for presence.

…01010101010101010101010101010101010101…

That is, we use 0 and 1 in the sense of Boolean arithmetic:

0^0 = 0 0^1 = 1^0 = 0

1^1 = 1 and

0 + 0 = 0 0 + 1 = 1 + 0 = 1

1 + 1 = 1.

(Since a^b is, for 0 and 1, the usual multiplication in arithmetic, we will notate it without the ^ symbol from now on.) The reader should recall the interpretations of Boolean arithmetic. We can take 0 to denote an empty universe and 1 to denote the whole universe. We take ^ to denote intersection and + to denote union. Then the rules for ^ are statements such as the intersection of emptiness with emptiness is empty. The intersection of the whole universe with itself is the whole universe. The rules for + are statements like the union of emptiness with emptiness is empty. The union of emptiness with the universe is the

... ...



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universe. The union of the universe with the universe is the universe. Boolean arithmetic also contains the complementation operation ‘with 0’ = 1 and 1’ = 0. Primordial vibration is an oscillation between nothing, 0, and something, 1, where that something is all that there is in the primitive emergence. At this stage, while 1 may emerge, 1 is in fact not different from nothing, and 0 while it stands for nothing, is not nothing. Each is an attempt to become something, either by standing for nothing or by attempting to distinguish a something. Each attempt, in the first place, is not locked into existence. Each attempt falls back into the inchoate nothingness and loses its existence, only to re-emerge again and again. The oscillation is not separate from its own observation. The oscillation is a movement of attention that each time joins with the that of which is attended and disappears only to reappear once again. Would you know it was a second time? Of course not. That would be an extra distinction. The world appears as its own observer, but being indistinct from itself, disappears. What happens next, that we have the possibility to distinguish the oscillation itself?

How do we leave the state of temporality and come to a discussion of a distinction apparently separate from us and yet in relation to us? Let us attempt to describe that process. The Series …010101010101… and the Fermion Algebra

Return to the series alternating between 0 and 1 with Boolean values.

…01010101010101…

Apply a temporal analysis to the series. We take as given two basic forms of observation/participation with the sequence, denoted by

p = (0,1) and q = (1,0).

In the first case the participation is in rhythm from 0 = > 1 and in the second case in rhythm from 1= > 0. We denote these by p and q respectively. In q we have the sense of form dissolving to void. In p we have the sense of form arising from void.

Here is the Boolean sense of form. We can regard 1 as a first form, dominant over void. 0 is a notational representative for void. In examining these waveforms, it is natural to have an operator that shifts from one wave form to its complementary waveform. To this end we introduce h such that

(a,b) h = h (b,a) and h h = 1. In this way we have

h (a,b) h = (b,a).

h is a new operator whose square is the identity: h h = 1.

(a,b) denotes a process that shifts from a to b. h (a,b) h denotes a process the shifts from b to a.

With h h =1, it follows that (a,b) h = h (b,a).

Shifting h across (a,b) converts it into (b,a). In this way we see that while (a,b) and (b,a) appear to

be spatial forms in the sense that they interact with themselves and each other at their individual coordinate places, (a,b) h has a built in temporality when we let it interact with another form or with

itself. For example,



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(a,b) h (a,b) h = (a,b)(b,a) = (ab,ba) = (ab,ab) = ab[1,1] = ab

While (a,b)(a,b) = (aa,bb) = (a,b) when a and b are either 0 or 1. Take p = (1,0) and q = (0,1) as the two views of the sequence. Note that

pq = (1,0)(0,1) = (0,0) = 0.

p + q = (1,0) + (0,1) = (1,1) = 1.

We have

and

as the two basic temporal elements for the algebra of the 0-1 oscillation.

Note that

,

.

Hence and , making p and q time shifts of each other.

Then

,

and

.

Furthermore

,

.

Hence

.

Thus, we have shown that

and .

These are the fundamental equations for the creation and annihilation operators for a Fermionic particle in quantum mechanics [Kauffman 2016a]. More than one Fermi particle cannot occupy the same space. This is the moral of the equations where the square of an operator is 0. Fermi particles can (irrespective

U = ph U † = qh

ph = (1,0)h =h(0,1) =hq

qh =hp

hph = q hph = q

U 2 = phph = pq = 0

(U †)2 = qhqh = qp = 0

UU † = phqh = pphh = pp = p

U †U = qhph = qq = q

UU † +U †U = p + q = 1

U 2 = (U †)2 = 0 UU † +U †U = 1



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of order) be created in pairs from the vacuum. This is the moral of the second equation where pairs

or can appear from 1 (the whole, the vacuum). Fermions embody special distinctions and it is remarkable that we find them emerging from the

process algebra for a single distinction. In the next section we shall see how Fermions emerge from the Clifford algebra associated with the numerical plus/minus sequence.

3. From Distinctions to Clifford Algebra

Here we consider the idea of a distinction and regard the idea and its instantiation in language as fundamental for the development of mathematics and physics. It should be clear that distinction is fundamental to mathematics in that one must perform acts of distinction to produce, construct mathematics and communicate it. In set theory the entire structure is based on the act of collection, and the most elementary such act is the formation of the empty set { }, collecting nothing. Here we give a quick précis of the symbolic system of G. Spencer-Brown in his book Laws of Form [Spencer-Brown (1969)].

The Spencer-Brown mark is a sign that can represent any sign, and so begins in both universal and particular modes. The mark is seen to make a distinction in the space in which it is written, and so can be seen, through this distinction, to refer to itself. By starting with the idea of distinction we find, in the mark, the first sign and the beginning of all possible signs. The mark stands for an observer inseparable from what is observed.

Let there be a distinction, with inside denoted I, and outside denoted O.

Regard the mark as an operator that takes inside to outside and outside to inside.

Then

. Note that it follows that

For any state X (inside or outside) we have

Introduce the unmarked state by letting the inside be unmarked.

Then we can replace I by nothing , and so

Therefore, the value of the outside is identified with the mark, and we have the equation

.

UU † U †U

I =O

O = I

I =O = I

O = I =O

X = X

=O

O =

=



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The value of the outside is identified with the result of crossing from the unmarked inside.

. This equation can be read on the left as “cross from the inside” and it can be read on the right as “name of the outside”. Once the inside is unmarked, then the mark itself can be seen to represent a first distinction. The language of the mark is self-referential.

says that either mark names the other. The mark stands for itself, for the distinction that the mark makes (with the help of the mathematical observer). We now have a Primary Arithmetic:

We have one sign and two laws or rules about that sign:

The Law of Crossing: .

The Law of Calling: .

At this stage in the development of the sign, these laws are statements about naming and about the crossing of the boundary of an initial distinction. The initial distinction can be the distinction made by the sign itself. From here one can develop this arithmetic, its algebra, relations with logic, generalizations that include ordinary arithmetic, rational and real numbers, multiple valued logics, combinatorial structures and much more. If one takes the idea of distinction seriously then one can see any mathematical structure from this point of view. We add that the self-referential nature of the mark leads naturally to the consideration of the equation

where self-reference or re-entry occurs at the algebraic level. Since the mark makes a distinction between its inside and its outside, this equation suggests that J

must itself have a sign that indicates a form that re-enters its own indicational space. J must have a sign as shown below.

Now one has an explicit oscillation in the sense that if , then . And so J will oscillate between the marked and the unmarked states. One can point out that time is a best solution to paradox, and one can point out that historical seemingly paradoxical equations in arithmetic such as

can be put in the form . If we take this last equation for seriously in the light of , then we see that i gives rise to an oscillation between plus and minus one. This elementary process to algebra that enfolds the square root of minus one in a non-commutative framework.

To see this first note that we may represent the oscillation by an ordered pair

(-1,+1) or (+1,-1)

=

=

=

=

J = J

J = J = J = =

i2 = -1 i = -1/ i J = J



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(you can start at +1 and oscillate or you can start at -1 and oscillate). (-1,+1): -1,+1,-1,+1,-1,+1, -1,+1,-1,+1,-1,+1, -1,+1,-1,+1,-1,+1, -1,+1,-1,+1,-1,+1,… (+1,-11): +1,-1,+1,-1,+1, -1,+1,-1,+1,-1,+1, -1,+1,-1,+1,-1,+1, -1,+1,-1,+1,-1,+1,-1,…

We can relate these by adding to our algebra a symbol so that (a,b) = (b,a). We will assume

that = 1 and so we have that (a,b) = (b,a) . If we let e = (-1,+1) then e* = –e = (+1,-1) then

e = -e . If we let (a,b)(c,d) = (ac, bd) (a natural way to combine oscillations) then ee = 1 and so we

have an algebra generated by e and with e = - e. But now look at

L = e .

LL = e e = e( - e) = -ee = -1.

We have constructed L as a square root of negative one by making an arithmetic of oscillations. The element is a time shifting operator and we have seen L = e as an especially time sensitive

oscillation between plus and minus one. This is a new dynamic way to think about the square root of minus one. This way puts the square root of minus one in the context of the non-commutative Clifford algebra generated by and e with the relations

ee = 1, = 1 and e + e = 0.

By thinking in this way, we have jumped across the entire history of the square root of minus one and landed on it in a non-commutative algebra.

As we will see in this paper, it is this Clifford algebra that is so closely connected with quantum physics and the Dirac equation. It is also the case that in making this development that we have circumvented the idea that i should be commuting with all the algebra elements that it interacts with. Indeed historically this was so, and we will adopt the attitude that it is natural to have both commuting i and non-commuting L = e . With this understood, there is a magic trick that can now be performed.

Let I = ie, J = i and K = e.

Remember that ii = -1 and I commutes with everyone. Then

II = ieie = iiee = -1, JJ= i i =ii = -1,

KK = e e = -ee = -1,

IJK = iei e = iiee = -1.

Thus, we have shown that II = JJ = KK = IJK = -1 and so I, J and K generate the quaternions. Hamilton’s quaternions arise naturally from the oscillatory process associated with the arithmetical paradox i = -1/i.

We will make use of the Clifford algebra and this viewpoint. Fermions and Standard Arithmetic

Returning to Section 2 and the Fermion algebra, corresponds to the Pauli exclusion principle.

Two identical Fermions cannot occupy the same place.

h h hh h h h hh

h h h

hh h

h h

h

h h h h

h

h h

h h h hh hh h

UU = 0 =U †U †



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The equation can be interpreted as saying that the combination of particle and antiparticle can emerge from void where now void is the multiplicative void of 1. This corresponds to the unmarked state at the Boolean level. If we allow standard arithmetic, then we can consider subtraction and we find that

e = p – q = (0,1) – (1,0) = (-1,1)

can be interpreted as oscillation between -1 and +1. This is an oscillation between absence and presence, but now in the language of standard arithmetic. We have that –e = (1, -1), the complementary oscillation, and

e (p - q) p q q - p = - e.h h h h h h h h= = - =

Thus, we have an algebra of e and h with ee = 1= h h and eh = -h e.

This is the Clifford algebra as we have discussed it above. If we write t = eh then

e e e(- e) = 1.tt h h= = -

Thus, along with the emergence of the Clifford algebra, we have a construction of the square root of

minus one that is essentially temporal. The alternation of -1 and +1, made temporally sensitive,

interacts with itself to produce negative unity.

In the next section we discuss the historical context of algebra, and to see the nature of the jumps we have made in the last two sections to obtain complex numbers, quaternions and Clifford algebras. 4. The emergence of algebra

We now examine the emergence of the square root of negative unity and the subsequent development of algebras up to the present day. Once standard arithmetic was in the world there were also the rational numbers of the form p/q where p and q are integers and q is not zero. And along with the rationals, real numbers began to emerge that

were not rational such as 2 and p . These last two numbers were regarded as necessary since they

represented respectively, the length of the diagonal of a square of side one and one half the circumference of a circle of radius one. Eventually a theory of real numbers emerged that was capacious enough to apparently correspond to the points on an infinite line, containing the integers as a discrete and evenly spaced subset. Very natural problems occurred even before Euclid such as: Given that x+y = p and xy = q where p and q are given real numbers, find x and y. 2p can be the perimeter of a rectangular field of sides x and y, q can be the area of this field. To solve the equation, suppose that

x =(p + u)/2 and

y = (p – u)/2

so that

x+ y = p.

Then

UU † +U †U = 1



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(p + u)(p - u)/4 = xy = q and so

p2 - u2 = 4q

We rewrite and deduce that

u = ± p2- 4q .

This means the Perimeter/Area problem cannot always be solved. It can happen that p2 - 4q < 0 .

For example, we could have p = 1 and q = 1 and then that p2 - 4q = -3 and u = ± -3 . But what is

-3 ? Indeed, what is -1 ?

Letting i = -1 , we have indicated the historical invention of the square root of negative unity as a

new form of number, commuting with all the previously created numbers, and acting to provide solutions to quadratic equations such as the above. Cardano and Tartaglia in the 1500’s were able to use these imaginary numbers to find the real roots of certain cubic equations. It took a long time for understanding and interpretation of these new numbers to emerge. All that time, the context held that i was a new form

of number with i2 = -1, and that one could consider the collection of complex numbers of the form a+ ib where a and b are real numbers. These can be formally multiplied and added as shown below.

a + ib + c + id = (a + c) + i(b + d),

(a + ib)(c + id) = (ac –bd) + i(bc + ad).

The latter formula is obtained by assuming that multiplication distributes over addition and that all numbers commute with one another. The context always held multiplication is commutative.

It was difficult for mathematicians to find a way to understand what these new complex numbers could mean. Around 1675, the Oxford mathematician John Wallis had a beautiful idea that would place i at ninety degrees to the real in e and one unit above it in distance. We illustrate his idea in Figure 1. There you see a right triangle with a horizontal hypotenuse and the point at the right angle a distance h above the horizontal line. The hypotenuse is divided into segments x and y on either side of the lower end of the perpendicular to the base from the right angle. It follows from similar triangles that x/h = h/y.

Hence h2 = xy. This is classical geometry. Wallis, a proponent of the relatively new negative numbers

on the real line, said let x be on one side of zero and y on its other side. Let x and y have size 1 and so

also h. But then the equation h2 = xy becomes h2 = (-1)(+1) = -1. So, h is a square root of minus 1!

This insight was not picked up by the contemporaries of John Wallis. It was not found again until 1800 when a surveyor Argand and the mathematician Carl Friedrich Gauss both noted that i could be interpreted as the unit point on an axis perpendicular to the real line and that any complex number a + ib could be seen as a point in the Cartesian plane. Of course the Cartesian plane of points (a,b) as a context for geometry had been found by that time. With this articulation of Gauss and Argand, it became clear that multiplication by i corresponded to rotation by ninety degrees counterclockwise about the origin in the plane. With i perched at ninety degrees already, one more rotation took it to -1 and the geometric interpretation of I was born. See again, Figure 1. The rise of complex analysis was rapid after that and entered into many mathematical and physical applications.



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In all this time, i was regarded as a number that commuted with all the other numbers that it encountered. Sir William Rowan Hamilton changed all that. He looked at the beautiful way that complex numbers encapsulate all rotations of the plane and wanted to generalize the complex numbers to an algebraic structure on three-dimensional space that would accomplish rotations in three dimensions. Hamilton succeeded after a 15-year search. He had to realize that he needed a non-commutative system and that it should partake of four dimensions.

Figure 1. Geometric Interpretations of -1

Hamilton discovered that the three spatial directions could be labeled by unit lengths I, J and K with

II = JJ = KK = IJK = -1. Given that his system is associative, these equations tell the whole story. For example, from IJK = -1 and II = -1 we get I(IJK) = -I whence –JK = - I whence JK = I. And similarly IJ= K, JK = I, KI = J with JI = - IJ, KJ = - JK, IK = - KI. Hamilton’s quaternions do not commute with one another! The general quaternion has the form a + bI + cJ + dK where a,b,c and d are real numbers. This is in direct analogy to writing a complex number such as a + bI and indeed there are many copies of the complex numbers inside the quaternions. Now we have square roots of unity that are thoroughly mixed with non-commutativity. The story of how rotations of three-dimensional space can be represented by quaternions is very beautiful but we shall not repeat it here [Kauffman, L. H. (2012)]. Hamilton found the quaternions in 1843. They were the first example of a non-commutative algebra discovered in mathematics. Hamilton was enthralled by the quaternions and spent much of the rest of his life working on them. He had an intuition that their fourth dimension should be time, and as we know that is correct in relation to special relativity. Indeed, Lorentz transformations can be described via quaternions, and electromagnetism can be written in these terms. But that was not all. In the very year that Hamilton discovered quaternions, his colleague Graves found an 8-dimensional algebraic system that followed the same pattern, but was not associative. This was rediscovered by Arthur Cayley a few years later and become known as the Octonions or the Cayley Numbers. Eventually Leonard Dickson (in 1919) found an algebraic recursive scheme that generates all these algebras and more. This Cayley-Dickson Process can be summarized as described below.



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One starts with a given algebra A with a “conjugation” operation x à x* on it that is of order two. Thus x** = x for any x. One then defines a new algebra A^ consisting in ordered pairs of elements of A so that (a,b) in A^ consists in a and b given elements of A. Conjugation is extended to A^ by (a,b)* = (a*, -b) and addition is done so that

(a,b) + (c,d) = (a + c, b + d).

Multiplication is given by the formula

(a,b)(c,d) = (ac – db*, cb + a*b).

This is the Cayley-Dickson Process. Order of multiplication and the use of the conjugations is important.

To see how this works, start with A = R, the real numbers and r* = r for all real numbers r. Then R^ consists in pairs (a,b) and (a,b)* = (a, - b). The multiplication is (a,b)(c,d) = (ac – db*, cb + a*b) = (ac – bd, bc + ab) and note that the elements of R commute with one another. It is easy to see that R^ is just the complex numbers when we identify i = (0,1). Then i(c,d) = (0,1)(c,d) = (-d,c) and this can be compared with i(c + id) = ic – d = -d + ic as we write complex numbers in the customary way. The next departure gives the quaternions as R^^, and from the quaternions the Cayley-Dickson process produces the octonions. The algebras lose properties as the process is repeated. We start with the commutative, ordered, real numbers. Then we get the unordered complex numbers. Then we get the non-commutative quaternions, and then the non-associative octonions. All of these algebras, so far, are division rings in the sense that non-zero elements have inverses. After the octonions the process continues, but the algebras are not division rings. See the book [Chatelin, F. (2012)] by Francois Chatelin for more information about this infinity of algebras that emerge along with the complex numbers and the square root of minus one through the Cayley-Dickson Process.

Returning to complex numbers, the prescription a+ ib = (a,b) identifies them as points in a plane, and once this identification is taken seriously then it is easy to see that multiplication by i is a rotation by ninety degrees since indeed that is the geometry of i(a,b) = (-b,a). With the advent of non-commutative algebra associated with vectors, the familiar operations of dot product and vector cross product emerged along with their well-known interpretations in relation to projections, length and area. There was more to come. Grassmann (1844) had a remarkable and beautiful idea that produced more algebras and goes to the root of relationships with length, area and volume. His work was not fully appreciated until the 20th century.

Grassmann’s idea was this. Let a and b be vectors. We want an algebraic product a^b of a and b to represent the area of a parallelogram spanned by a and b. We then see that we must demand that aâ = 0 for any a, since the area of the parallelogram for a vector with itself is zero. We also need the distributive rules

(a + b)^c = a^c + b^c

and

c^{a + b) = câ + c^b.

and rules for scalars k such as

(ka)^b = a^(kb).

These rules model how the area will decompose when the vector is a sum. But now we see that since (a + b) will form a zero area with itself,

0 = (a+b)^(a+b) = aâ + a^b + bâ + b^b = + a^b + bâ



14

Thus 0 = a^b + bâ and we conclude that

a^b + bâ = 0 The algebra is non-commutative and we must deal with signed areas.

With this beginning Grassman put an algebra structure on n- dimensional space with vector basis {e1, e2, …, en}. The basic relations are

ekêk = 0

and

eiêj = - ejêi

whenever i is not equal to j.

This part is very easy for us to say because we are used to the concept of vector space. Such was not the case when Grassmann made his discovery.

The Grassmann algebra works in any dimensions, but you have to take a second look at it. Go back to the Grassmann algebra generated by, say, unit vectors a, b, c, perpendicular to one another in three space. Suppose we have v = xa + y b + z c and w = ra + sb + tc. Now v^w is supposed to represent an area, but look at what happens when we compute v^w:

v^w = (xa + y b + z c)^( ra + sb + tc) = (xs - yr)a^b + (xt – zr)a^c + (yt – zs)b^c

and you will find out that the area spanned by the two vectors is equal to the square root of the sum of the squares of the coefficients of a^b, a^c and b^c respectively. Thus v^w is a vector with coordinates [(xs-yr), (xt – zr), (yt – zs)] whose length is the area we desire. What are these new coordinate directions a^b, a^c and b^c ? When Grassmann wrote his work, these questions seemed very confusing. Now we have gotten used to them, and we say that if we start with a vector space with basis {a,b,c} and form the Grassmann algebra of it, then we obtain a new vector space with basis

{1, a, b, c, a^b, a^c, b^c, a^b^c}. Thus, we jump from dimension 3 to dimension 8.

The Grassmann algebra is very wise. It knows about areas and volumes. The coefficient of a^b^c in v^w^z for three vectors v,w,z gives the volume of the parallelepiped

spanned by the three vectors. Once the idea to have non-commutative algebras was released by Hamilton, there was a big expansion of new algebras and new possibilities. A close relative of the Grassmann algebra is the Clifford algebra. Clifford, in 1878, combined the algebraic rules of Hamilton and Grassmann to define a new algebraic system, which he himself called geometric algebra. In Clifford algebra we have a vector space with basis {e1, e2, …, en} and the rule is now that ek ek = 1 and ek el = - el ek for k not equal to l. Here I do not use the wedge product, and so distinguish the Clifford algebra from the Grassmann algebra. Suppose that you are given the Clifford algebra generated by a and b and c. Then you can define I = ba, J = cb and K=ac, and find that II = JJ = KK = IJK = -1. Thus, the Clifford algebra on three generators gives us the quaternions. This is perhaps not surprising since if we take I to be a commuting square root of -1, we can define, from the quaternions I, J, K the algebra elements A = iI, B = iJ, C=iK and we have AA = BB = CC = 1 and the A,B,C anti-commute.



15

Thus, this gives a Clifford algebra associated with the quaternions. It should be apparent from this example that Clifford indeed combined the quaternions with the Grassmann algebra and created a high dimensional generalization of the quaternions to boot. Not every element in a Clifford algebra has an inverse. Nevertheless, the Clifford algebras are all associative and this makes them amenable to many applications that would be more difficult in a non-associative context. All of these developments occurred as the notions of linear algebra, vector spaces, linear transformations and matrices came into being. Linear algebra forms the context for studying the representations of associative algebras and many properties of Clifford algebras are understood in that context. It also came to pass that aspects of topology and algebraic topology were deeply related to the structures of these algebras. We are particularly interested in elementary situations that naturally give rise to algebra. We have seen that the development of complex numbers was motivated by the solving of quadratic and cubic equations and this in a commutative context that eventually acquired a geometric interpretation. Then came non-commutativity with Hamilton. Once non-commutativity was articulated a vast development ensued. In the course of all these developments an abstract point of view emerged that is often called “universal algebra”. In this viewpoint, if you want an algebra with certain properties, you can build it by specifying an alphabet and rules for combination of the letters in that alphabet. For example, I can set down S = (a,b | aba = bab, aa=bb= 1). This means that I can write strings of symbols such as ababab and I am allowed to reduce my strings according to the rules aba = bab and aa = bb =1 and it is understood that x1 = 1x = x for any string x. In this example, we have ababab = babbab = ba1ab = baab = bb= 1 in the system S. So, we conclude that the string ababab = 1 in S. You can take S as an exercise and show that it has exactly 6 distinct strings up the equivalence relation generated by these rules.

On the other hand, if we consider G = (a,b | aba = bab ) then one can prove that G is infinite, isomorphic to the three-strand braid group that we will examine in Section 8 of this paper. In universal algebra we construct whatever we like and no longer worry about paradox. It is clear that the notion of an alphabet with strings of symbols representing processes is very close to the basics of language and physics. There is one more aspect of algebra that I would like to mention here.

The famous Russell paradox about the Russell set (the set of all sets that are not members of themselves) can be expressed algebraically. Let Ax denote the statement that “x is a member of A”. Let ~ denote the logical connective “not”. Define the Russell set by the equation Rx = ~xx and note that indeed this says that x is a member of R exactly when x is not a member of x. Now we can see the paradox. For if we place x = R, then we get RR = ~RR and this says that R is a member of R exactly when R is not a member of R. However, viewed equationally this says that RR is a fixed point for negation. In two valued logic one does not have a fixed point for negation. This allows one to see the possibilities for extending either the logic to include values beyond true and false, or to extend the Russell set to be a process that is temporal and so there is no contradiction because it changes to include itself, but in so doing, goes beyond its former self. Algebras that allow processes and the production of fixed points of this kind are part of the logic of combinators and reflexive domains that are used extensively in computer science. Here we see that there is an affinity with the context from which we produced the square root of negative unity via temporal process. In a more general reflexive domain, we could write Gx = F(xx) and then conclude that GG = F(GG), obtaining a fixed point for any F. This is an abstract algebraic analogue of a domain of observables where every observable H has eigenvectors (since a fixed point is an analogue of an eigenvector). See (Kauffman (2005a,2009) ). In this way the idea of a reflexive domain is an extension of universal algebra when a process such as x à F(xx) can acquire a name as in Gx = F(xx) and the entity G is then accepted as an element of the algebra. One can think of a reflexive domain as a language that is always ready to accept new elements in this way. The domain can grow and expand like a physical system or a human language.



16

Another aspect of progress in algebra starts with the notational invention of “bras” and “kets” of P.A.M. Dirac. Dirac used the ket |A> to denote a vector in a complex vector space. The label A can be any label that is convenient for use. Once a label is surrounded with the ket frame it denotes a vector. Similarly, one has the bra <A| indicating a convector. To make this concrete, think of the ket |A> as a column vector. Then <A| is a row vector that is the conjugate transpose of the column vector. Thus <A| |A> denotes the scalar that results from multiplying the row vector and the column vector. Dirac denotes <A| |A> = <A|A>, so we have || = | as an underlying formalism for the bra-ket. Along with the bra-ket there is also the ket-bra |A> <A| and this is a matrix, not a scalar. And note that if P = |A><A|, then PP = |A><A||A><A| = = |A><A|A><A| = <A|A> |A><A| = <A|A> P. Thus, P is a projection mapping when <A|A> = 1. The Dirac notation becomes the beginning point of an algebra that is remarkably good at handling the formalisms needed to deal with quantum mechanics. Dirac’s notational algebra can be regarded as the beginning of Temperley-Lieb connection algebras [Kauffman (2012)] that include topological interactions of connecting arcs in the plane. See Figures 2 and 3. In Figure 2, we illustrate the abstract schematic structure of the Dirac algebra via P = >< and d = <>. Then if we have another projector, noted as Q = ][ and scalar e = [], then we have compositions of these two projectors and we see, as in the figure that there is a constant k so that

PQP = kP and QPQ = kQ.

These are basic properties of projector algebra, and we are deriving them from the way that the Dirac formalism interacts with itself. This can be generalized as in Figure 3 to algebras whose basic elements are collections of non-intersecting arc s that join points on the left and right with themselves and so that the arcs do not cross in the plane. Then one has a big topological/combinatorial generalization of the Dirac kets and bras to algebras that describe this planar combinatorics. It turns out that these Temperley-Lieb connection algebras have many uses in topology and in statistical mechanics. In Figure 4 we illustrate the Temperley-Lieb Category where the number of input strands can be unequal to the number of output strands. The figure contains a hint about how one can use planar combinatorics to find elements P in the Temperley-Lieb algebra with PP = P.

Figure 2. – Abstract Dirac Algebra



17

This advent of algebra that is directly tied to topology and combinatorics is a relatively recent

development and it shows how algebra and geometry, topology, combinatorics are more intricately connected with one another than one might have thought after the initial progress along more abstract lines.

Figure 3 – Temperley-Lieb Connection Algebra

Figure 4 – Temperley-Lieb Category



18

Remark. Having told this story of the development of algebra, it is important to understand it in its historical context. We wish to understand interaction and relationship. Algebra is based on the use of the simple left and right and parenthetical interactions of letters in the Roman alphabet. It is quite extraordinary that strings of adjacent characters in linear ordered arrays seem to be sufficient to carry our theories of what has the appearance of a multidimensional physical world. We begin to understand that much of this success comes from the ability of such strings of characters to encode information. Indeed, in some of the examples we have given here, such as the Temperley-Lieb algebra we have seen that complex topological changes can correspond to the manipulations of such strings of characters. It can then come to pass that problems about such algebras have to be studied by examining the actual topological background to the character strings. We are always ready to accept the possibility to widening contexts to encompass new understanding. If we leave the historical context and begin algebra from the making of a distinction, as indicated in the Introduction and in Sections 2 and 3, then we find that it is natural to consider what can be written in a plane space, rather than on a line. A circle or a simple closed curve makes a distinction in the plane and it is possible to consider complexities of interaction in the plane that may go beyond the direct coding in characters in a line space. Problems in knot theory [Kauffman (2012)] represented as diagrammatic structures in the plane sometimes achieve complete algebraic solutions while many other problems remain open. The interaction of algebra with topological and geometric structures is a source of deep research problems in mathematics. By the same token, it is not obvious that one could or should find that fundamental physical laws can be expressed in simple algebraic mathematical formulas. It is a miracle that Newton’s laws of force and gravitation can be expressed as simply as they can, even in the face of more complex corrections to them such as Einstein’s general relativity. In this paper, we are examining the essential simplicity of the Dirac approach to fermions. We are emphasizing this simplicity by building the Clifford algebraic background for Dirac from nearly nothing but the idea of a distinction. In this way we shed new light on the development of the algebra and the physical understanding. There is much more to be said about all of this and more mathematics to be articulated. We now return to our story. 5. Temporal algebra once again – from associativity

In making a temporal model for the complex numbers we took the concept that e = (-1,1) could stand for an alternation between -1 and 1, and that e* = (1,-1) could be a conjugate oscillation. We wanted to have temporal versions of e and e*, call them {e} and {e*} so that {x}{y} = xy*, that is so that the interaction of {x} and {y} would involve the time delay that causes e to shift to e*. Since e e* = -1, this accomplishes the task to make an element {e} whose square is -1:

{e}{e} = ee* = e(-e) = - (ee) = -1. The purpose of this section is to analyse the algebra behind this scheme and to show that there must be an h such that {x} = x h with h h = 1 under natural hypotheses about x. We will assume that there

is a given algebra structure A and that in this structure there is a (conjugation) operation xà x* with

x** = x for any x, 1*=1, 0*=0

and

(xy)*= x* y* for any x and y in A.

We will assume that our given algebra is associative and that the new algebra is also associative. We further assume that {x}{y} = xy* for all x and y. This is motivated by the discussion in the first paragraph above.



19

Lemma. Given the assumptions above, and that the new algebra obtained from A is also associative, then, assuming that {x}y and x{y} each have the form {w} for some w in A, we have that for invertible elements in the algebra,

{x}{y} = xy*,

{x}y = {xy*},

x{y} = {xy}.

Proof. By associativity, zx*y = ({z}{x})y = {z}({x}y) = {z}{w} = zw*. Thus, X*y = w* and therefore w = w** = (x*y)*= x** y*= xy*. Thus {x}y = {w} = {xy*}. Similarly, xyz* = x({y}{z}) = (x{y}){z} = {p}{z} for some p in A. Thus xyz* = pz* and so xy = p whence x{y} ={p} = {xy}. This completes the proof of the Lemma. Now let h = {1}. Then we have the

Lemma. For any x in A,

x h = {x}

and

h x h = x*.

Proof. Note that h h = {1}{1} = 1 1* = 1. x h = x {1} = {x1} = {x} Thus, h x h = {1}{x} = x*.

This completes the proof of the Lemma.

Remark. We now have that x* = h x h , and note that with this we have

(xy)* = h xy h = h x h h y h = x* y*.

Remark. If we start with A equal to the complex numbers. Then A = C= {a+ib| a and b are real

numbers and ii = -1} and we have a* = a for any real number a, i* = -i and (a+ib)* = a-ib. Thus, conjugation in A is the usual conjugation of complex numbers. Appling the construction given here we have h i h = i* = - i. Thus h i = - i h . The new algebra is generated by 1, i, h , i h with ii = -1, hh = +1, (i h )(i h ) = i h i h = i (-i) = +1. Thus, we arrive once more at our familiar Clifford algebra

but this time starting with the complex numbers as already given. We see from this discussion that it was inevitable that we could choose an element h to accomplish

the conjugation in our oscillation algebras in Sections 2 and 3, and that the construction of the square root of minus one as e h = {e} is an entirely natural mathematical construction. In so doing, we have

underlined the naturality of the temporal interpretation of the square root of minus one. 6. From Pythagoras to Einstein and Dirac

Now we can see a remarkable property of the Clifford Algebra. Suppose that we write a general element of this algebra in the form

U = A + B + C

where A, B and C are real numbers. Now compute the square of this element.

U2 = ( A + B + C)2 =

.

e h e h

e h e h

(eA)2 + (hB)2 + (ehC)2 + ( eh +he )AB + (eeh + ehe )AC + (heh + ehh)BC



20

Just as ( + ) = 0, all the other cross terms vanish. Since 2 = 2 = 1 and ( )2 = -1, we

obtain

U2 = A2 + B2 – C2.

We have proved

Theorem. If U = A + B + C then U2 = A2 + B2 – C2.

Hence U2 = 0 if and only if C2 = A2 + B2.

This Theorem suggests that the Pythagorean Theorem is involved with our algebra for the imaginary

states of the reentering mark. An operator with square zero is called a nilpotent operator.

If U is the Universe then U can act upon itself to annihilate itself. In the algebraic operator mode of speaking, this is in the form of

UU = Nothing

Thus, we have found a Clifford algebraic expression U = A + B + C, where C2 = A2 + B2,

for a fundamental nilpotent, a representative of universe that creates from nothing and takes back to

nothing.

U2 = 0 when C2 = A2 + B2.

In the last section we found operators and , both of square zero and so that ,

corresponding to the algebra of a Fermion. In the present context there is a good choice for (and we

shall explain below why this is a good choice). It is

and you can easily check that . Note that we have the calculation

.

And so (up to a constant) and also satisfy the Fermion equations, and they have the variability afforded by solutions to the Pythagorean Equation A2 + B2 = C2. Now there is a relationship with physics

that must be mentioned. First of all, there is the remarkable formula

E2 = (pc)2 + (mc2)2

This is the Einstein formula for the energy of a particle with rest mass m that has a momentum p with respect to a given observer [Kauffman, 2016a]. We usually think of the formula E = mc2 for the relationship of mass and energy, but if the mass is moving past at momentum p, then there is a Pythagorean relationship of the total energy E and the energy mc2 of the stationary particle. Thus one has the relativistic formula

e h h e e h e h

e h e h

e h e h

U U † UU † +U †U = 1

U †

U † = eA +hB - ehC

(U †)2 = 0

UU † +U †U = (U +U †)2 = 4(eA+hB)2 = 4(A2 + B2 ) = 4C2

U U †

E = (pc)2+ (mc2 )2



21

making the energy the length of the hypotenuse of a right triangle whose sides are pc and mc2. Here, for the record is the proof of this formula.

We have where is the rest mass of the particle (Einstein’s basic mass, energy

relation). And we have where v is the velocity of the particle and m is the relativistic mass

with

.

The formula then follows by squaring this (Pythagorean) mass equation.

What will happen if we combine our fundamental nilpotent U with this Pythagorean energy formula from special relativity? The remarkable answer is that we arrive at Dirac’s relativistic equation for the electron.

To make the formalism easier, let us take c = 1. We use the convention that the speed of light is equal to 1. Then E2 = p2 + m2 where m is the rest mass of the particle. With this we can write

and obtain a (physical) nilpotent. It follows at once from our Theorem that U2 =

0. So, we have associated mass with , the representative of the on-going time series of the recursion.

And we have associated momentum with the time shift operator, and energy with the time sensitive operator whose square is minus one.

We shall define the dual nilpotent . Here time goes backwards relative to U,

and if U creates a particle, then creates the corresponding anti-particle. See the discussion below. We then find (as above) that

Thus, we have that

and

.

The first equation expresses that a creation of a particle and an antiparticle (in either order) can proceed from pure energy. The second two equations represent the Pauli Exclusion Principle that forbids the existence of identical particles at the same place. There is just no possibility for the production of two identical Fermi particles from pure energy.

Remark. Recall that we have seen this algebra before where we found the Fermion algebra (with E = ½ in the above terms) coming directly from the re-entering mark. Here we took a longer road and used

E = m0c2 m0

p = mv2

m = m0 / 1- v2 / c2

m2 = m0

2 / (1- v2 / c2 )

m2(1- v2 / c2 ) = m0

2

m2(c2 - v2 ) = m0

2c2

(mc2 )2 = (mv)2 c2 +m0

2c4

E2 = p2c2 + m0

2c4

U = me + ph + Eeh

e

U † = me + ph - Eeh

U †

UU † +U †U = (U +U †)2 = 4(me + ph)2 = 4(p2 +m2 ) = 4E2

UU † +U †U = 4E2

U 2 =U †2 = 0



22

arithmetic rather than primary values of marked and unmarked, and we arrived at nilpotents in relation to the Pythagorean relationship E2 = p2 + m2. It is clear that even more thought is needed here about the roots of these relationships in the form and in the physics. From distinctions we find precursors to physics and the depths of this relationship are not yet fully apprehended.

We began this discussion with a tip of the hat to the Pauli Exclusion Principle, and return to it in this way at the end. For the details about how this Fermion algebra is related to the Dirac equation, we recommend that the reader examine [Kauffman 2016a] and the work of Peter Rowlands [Rowlands, 2007] who discovered the nilpotent approach to the Dirac equation.

Dirac devised his equation so that it would correspond to the basic fact that we have

. He needed an operator form of the square root . Dirac reasoned that if

and , then and so he could identify the energy operator

with .

We can back-engineer the nilpotent Dirac equation from the requirement that

should be a solution to the original Dirac equation. See the above references for the details. Now you can see why we took

,

for this corresponds exactly to reversing time in the plane wave and so corresponds creating an anti-particle. The final expression of this Fermionic solution to the Dirac equation is quite remarkable, combining the algebraic form of U and the continuously varying exponential expression in space, time, momentum and energy. At this point we have arrived at relativistic quantum mechanics by starting from Laws of Form.

Remark. Note that we have

and

.

Thus, if

then

and

E2 = p2 +m2

E = p2+ m2 a 2 = b 2 = 1

ab + ba = 0 (a p + bm)2 = p2 +m2 = E2

a p + bm

Uj = (me + ph + ehE)ei( px-Et )


U = me + ph + Eeh


U = A+ iB

U †= A - iB

A = me + ph

B = iEhe



23

Note that

The basic Fermion operators U and U † are expressed directly in terms of the Clifford algebra

generated by A and B. This splitting into A and B corresponds to the two ways to view the energy E, either as E or as the sum of the squares of the momentum and the mass. It has been suggested that since Fermions can be represented in the form U = A + iB, then there may be “particles” that correspond to A and to B. These would be so-called Majorana Fermions that would be their own anti-particles. It remains to be seen if this kind of decomposition will become a reality. There is no question that Clifford algebra has a deep structural contribution to the nature of Fermions.

7. The Dirac operator

In the previous section we discussed the construction of the Dirac operator, but we did not make it explicit. An essential component of quantum theory is the articulation of operators that correspond to

each physical quantity. Thus to the energy E there is an energy operator . To the momentum there is

a momentum operator . To the mass m there is a mass operator .

In the case of the physics the relationship between operators and operands occurs through a

wavefunction such as and we will have that

It is in relation to the wave function that the operators and operands become interchangeable. Dirac’s original equation was then the statement that at the operator level

where

The nilpotent Dirac operator is a variant of this relationship where we write and

we then have that

with the nilpotent so that

A2 = p2 + m2

B2 = E2

AB + BA= 0

B2 = E2 = p2 +m2 = A2

Ep m

y = ei( px-Et )

Ey = Ey

py = py

my = my

E = ap + bm

a2 = b2 = 1

ab + ba = 0.

D = abE + bp - am

Dy = (abE + bp - am)y

U = (abE + bp - am)



24

and

.

Thus is a solution to the Dirac equation. It is a Fermion.

We have written all of this in 1+1 spacetime. The reader can see the full spacetime articulation in our papers (Kauffman 2020, 2021). The simplest precursor to the nilpotent Dirac operator is the laws of form operator

.

with the mark taken to void by the operator D. At this level we are speaking of a universe as it first emerges from one distinction and we are speaking

of the distinction of an observer who finalizes a measurement in quantum mechanics. The final registration of an observation is identical with the creation of a universe in the moment of possibility becoming actuality.

The mark represents a primordial distinction and it also represents a primordial particle. As particle, the mark interacts with itself to annihilate itself (crossing) or to affirm/produce itself (calling). No particle could be more elementary than this. The mark is its own anti-particle, arising and returning to a void that should be compared with the physical vacuum. 8. Braiding Majorana Fermions

In this section we show how Majorana Fermions are closely allied with representations of the Artin Braid group. First we recall the structure of the braid groups. View Figure 5. ,

Figure 5 – Braiding and the Braid Group

Dy =Uy

D(Uy ) =U 2y = 0

Uy

DX = X



25

In this Figure you see depicted the generators of the four-strand braid group, s1,s 2 ,s 3 and their

relations where each generator has an inverse and the identity element in the group, denoted by 1, corresponds to four parallel strands. Each generator corresponds to the weaving of two strands, forming a 180 degree twist.

A braid is a concatenation of these elementary braidings so that it forms a weaving of the four strands as in the example in Figure 5. The generators have two fundamental relations of the form:

s 1s 2s1 =s 2s 1s 2 and s is j =s js i

when |i-j| > 1.

These same types of relation serve to describe the braid group B_n for n strands. Braids are naturally related to particles that move in a plane. Imagine two particles undergoing an exchange of place. The world lines of these particles occur in the spacetime that is a Cartesian product of the plane and a timeline. Thus the spacetime for planar particles in three space and the world line of two particles moving around one another is a braid in that three space. In fact the world lines of a collection of particles moving around in the plane (but not colliding) will be a braid on n strands in three space when there are n particles. So braiding is naturally associated with any particles. We shall give rules for the interaction of Majorana particles that induce algebraic transforms that satisfy the braiding rules. View Figure 6. Compare [Kauffman, Collings (2022)].

Figure 6 – Braiding via a Distinction Operator

In this Figure you see how we can configure an elementary braid as a transform that can be applied

to pairs of Spencer-Brown marks. The crossover transform has the equation T (a,b) = (b ,a) and you

can see from the diagrams that the inverse of T is T '(a,b) = (b,a ) and that when we put this transform

in the context of three strands (by leaving it to be the identity on one strand) then the transforms satisfy all the relations of the braid group. Thus, we have shown that the fact that a Majorana Fermion to



26

annihilates itself, makes rows of these particles, acting on each other by these rules, into a representation of the braid group.

There is another way, via Clifford algebra, that braid group representations for Majorana Fermions occur. View Figure 7. Here we see a row of Majorana Fermions with Clifford algebra labels e1,e2,…,en. With ei ei = 1 and ei ej + ejei = 0 when i and j are unequal. Then let

s i = (1+ ei+1ei ) / 2

for I = 1,2,…,n-1.

One verifies (Kauffman (2016a)) that the s i form a representation of the braid group Bn

. In this

case each s i has order 8 and the representation is finite. Ivanov (Ivanov (2001)) points out that by

defining Tk :Vn ®Vn where Vn

is the vector space span of {e1,e2,…,en} and Tk (ei ) =s keis k

-1 , then

Tk (ek+1) = -ek

Tk (ek ) = ek+1

and otherwise, T_k is the identity, as in Figures 7 and 8. This braid group representation has the same form as our simple LoF representation described in Figure 6. Thus, we see that in this way the Clifford algebra structure and its braiding does involve the self-annihilating property of the Majorana Fermions but indirectly and in a way that asks for deeper understanding.

Figure 7 – Clifford Algebra Braiding

One last point of view on the braiding is shown in Figure 8. There we illustrate that when particles

attached by a band in three-space, the exchange of the particles results in the 360 degree twist in the band. This twist corresponds to the phase change of (-1) that we have seen in these braiding



27

representatons. The braiding representations contain extra information as they assign this phase change to one particle and not the other as shown in Figure 8 and Figure 7.

The twist in the belt is not localized in this way. There are interesting topological questions in relation to this non-locality of the belt in relation to the locality specified by the unitary transformations from the algebra.

Figure 8 – Braiding Majorana Fermions

9. Epilogue

In this paper we have examined the use, development and dynamics of signs in relation to G. Spencer-Brown’s Laws of Form and how algebraic and dynamic structures that come from tis base are related to fundamental physics and with Majorana Fermions. The language of Laws of Form was discovered, according to Spencer-Brown, in making a descent from Boolean algebra in which he found the notation of the mark, the role of the unmarked state and the double-carry of mark as name and mark as transformation. In Boolean algebra and in symbolic logic the negation sign connotes transformation and it does not stand for a value (True or False). In the calculus of indications, viewed from the stance of symbolic logic, the mark is a coalescence of the value True and the sign of negation. This comes about because True is what is not False and the False is unmarked in Laws of Form. But this can not be said without confusion in symbolic logic since there is no inside to the sign ~ of negation.

In Laws of Form we can write and the mark as container (as parenthesis) makes it possible for it to take on its double role of value and operator.

Wittgenstein says (Wittgenstein (1922) - Tractatus [97] 4.0621)

“. . . the sign ‘~’ corresponds to nothing in reality.”

T = F =



28

And he goes on to say (Wittgenstein (1922) - Tractatus 5.511) “How can all-embracing logic which

mirrors the world use such special catches and manipulations? Only because all these are connected

into an infinitely fine network, the great mirror.” For Wittgenstein in the Tractatus, the negation sign is part of the mirror making it possible for thought

to reflect reality through combinations of signs. These remarks of Wittgenstein are part of his early picture theory of the relationship of formalism and the world. In our view, the world and the formalism we use to represent the world are not separate. The observer and the mark are (in the form) identical. This theme of formalism and the world is given a curious twist by an observation that the mark and its laws of calling and crossing can be regarded as the pattern of interactions of the most elementary of possible quantum particles, the Majorana Fermion [Kauffman (2016a)]. A Majorana Fermion is a hypothetical particle that is its own anti-particle. It can interact with itself to either produce itself or to annihilate itself. In the mark we have these two modes of interaction as calling

and crossing . The curious nature of quantum mechanics is seen not in such simple interactions but in the logic of superposition and measurement. Measurement of a quantum state demands the coming into actuality of exactly one of a myriad of possibilities. Thus, we may write

to indicate that the quantum state of a self-interacting Majorana Fermion is a superposition of marked and unmarked states. Upon observation, one or the other (marked or unmarked) will be actual, but before observation, the state is neither marked nor is it unmarked. The equation for this interaction can be written in ordinary algebra as

PP = P + 1 where P stands for the Majorana Particle and 1 stands for the neutral state. Then we recognize a quadratic equation

P2 = P + 1

with solution the Golden Ratio and multifold relationships with the Fibonacci numbers.

Indeed, this is the legacy of the Majorana Fermion as Fibonacci particle, fundamental entity in the most idealistic and yet soon to be practical searches for quantum computing and understanding of particles as well-known as the electron. Each electron appears in the mathematics as an amalgam of two Majorana Fermions. The moving boundary of Sign and Space is changed from the time of Wittgenstein. With negation ~ replaced by the mark <>, the negation does indeed begin to appear in the world. When representation and explanation are insisted upon, then an infinite regress occurs due to the proliferation of signs that must indicate each stage of explanation. When this noise is reduced by the simple recognition of the presence of a distinction, then forms can stand alone and be recognized as being, in form, identical with their creators. Along with the references quoted directly in the text, I have provided a selection of papers that are related to the themes of this essay. There is much to think about in this domain and we have only just begun.

= =

* = +

*

(1+ 5) / 2



29

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