The Peculiar Math That Could Underlie the Laws of Nature

New findings are fueling an old suspicion that fundamental particles and forces spring from strange eight-part numbers called “octonions.”

Cohl Furey, a mathematical physicist at the University of
Cambridge, is finding links between the Standard Model of
particle physics and the octonions, numbers whose
multiplication rules are encoded in a triangular diagram called
the Fano plane. Susannah Ireland for Quanta Magazine

by Natalie Wolchover

Senior Writer/Editor

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July 20, 2018

Original link:  https://www.quantamagazine.org/the-octonion-math-that-could-underpin-physics-20180720/

DJS note:  for an explanation of octonion numbers see http://amedleyofpotpourri.blogspot.com/2018/07/octonion.html

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In 2014, a graduate student at the University of Waterloo, Canada, named Cohl Furey rented a car and drove six hours south to Pennsylvania State University, eager to talk to a physics professor there named Murat Günaydin. Furey had figured out how to build on a finding of Günaydin’s from 40 years earlier — a largely forgotten result that supported a powerful suspicion about fundamental physics and its relationship to pure math.

The suspicion, harbored by many physicists and mathematicians over the decades but rarely actively pursued, is that the peculiar panoply of forces and particles that comprise reality spring logically from the properties of eight-dimensional numbers called “octonions.”

As numbers go, the familiar real numbers — those found on the number line, like 1, π and -83.777 — just get things started. Real numbers can be paired up in a particular way to form “complex numbers,” first studied in 16th-century Italy, that behave like coordinates on a 2-D plane. Adding, subtracting, multiplying and dividing is like translating and rotating positions around the plane. Complex numbers, suitably paired, form 4-D “quaternions,” discovered in 1843 by the Irish mathematician William Rowan Hamilton, who on the spot ecstatically chiseled the formula into Dublin’s Broome Bridge. John Graves, a lawyer friend of Hamilton’s, subsequently showed that pairs of quaternions make octonions: numbers that define coordinates in an abstract 8-D space.

John Graves, the Irish lawyer and
mathematician who discovered the octonions in
1843. MacTutor History of Mathematics

There the game stops. Proof surfaced in 1898 that the reals, complex numbers, quaternions and octonions are the only kinds of numbers that can be added, subtracted, multiplied and divided. The first three of these “division algebras” would soon lay the mathematical foundation for 20th-century physics, with real numbers appearing ubiquitously, complex numbers providing the math of quantum mechanics, and quaternions underlying Albert Einstein’s special theory of relativity. This has led many researchers to wonder about the last and least-understood division algebra. Might the octonions hold secrets of the universe?

“Octonions are to physics what the Sirens were to Ulysses,” Pierre Ramond, a particle physicist and string theorist at the University of Florida, said in an email.

Günaydin, the Penn State professor, was a graduate student at Yale in 1973 when he and his advisor Feza Gürsey found a surprising link between the octonions and the strong force, which binds quarks together inside atomic nuclei. An initial flurry of interest in the finding didn’t last. Everyone at the time was puzzling over the Standard Model of particle physics — the set of equations describing the known elementary particles and their interactions via the strong, weak and electromagnetic forces (all the fundamental forces except gravity). But rather than seek mathematical answers to the Standard Model’s mysteries, most physicists placed their hopes in high-energy particle colliders and other experiments, expecting additional particles to show up and lead the way beyond the Standard Model to a deeper description of reality. They “imagined that the next bit of progress will come from some new pieces being dropped onto the table, [rather than] from thinking harder about the pieces we already have,” said Latham Boyle, a theoretical physicist at the Perimeter Institute of Theoretical Physics in Waterloo, Canada.

Decades on, no particles beyond those of the Standard Model have been found. Meanwhile, the strange beauty of the octonions has continued to attract the occasional independent-minded researcher, including Furey, the Canadian grad student who visited Günaydin four years ago. Looking like an interplanetary traveler, with choppy silver bangs that taper to a point between piercing blue eyes, Furey scrawled esoteric symbols on a blackboard, trying to explain to Günaydin that she had extended his and Gürsey’s work by constructing an octonionic model of both the strong and electromagnetic forces.

“Communicating the details to him turned out to be a bit more of a challenge than I had anticipated, as I struggled to get a word in edgewise,” Furey recalled. Günaydin had continued to study the octonions since the ’70s by way of their deep connections to string theory, M-theory and supergravity — related theories that attempt to unify gravity with the other fundamental forces. But his octonionic pursuits had always been outside the mainstream. He advised Furey to find another research project for her Ph.D., since the octonions might close doors for her, as he felt they had for him.

Furey posing for a portrait on the grounds of Trinity Hall,
Cambridge, where she often works on a yoga mat.

Susannah Ireland for Quanta Magazine

But Furey didn’t — couldn’t — give up. Driven by a profound intuition that the octonions and other division algebras underlie nature’s laws, she told a colleague that if she didn’t find work in academia she planned to take her accordion to New Orleans and busk on the streets to support her physics habit. Instead, Furey landed a postdoc at the University of Cambridge in the United Kingdom. She has since produced a number of results connecting the octonions to the Standard Model that experts are calling intriguing, curious, elegant and novel. “She has taken significant steps toward solving some really deep physical puzzles,” said Shadi Tahvildar-Zadeh, a mathematical physicist at Rutgers University who recently visited Furey in Cambridge after watching an online series of lecture videos she made about her work.

Furey has yet to construct a simple octonionic model of all Standard Model particles and forces in one go, and she hasn’t touched on gravity. She stresses that the mathematical possibilities are many, and experts say it’s too soon to tell which way of amalgamating the octonions and other division algebras (if any) will lead to success.

“She has found some intriguing links,” said Michael Duff, a pioneering string theorist and professor at Imperial College London who has studied octonions’ role in string theory. “It’s certainly worth pursuing, in my view. Whether it will ultimately be the way the Standard Model is described, it’s hard to say. If it were, it would qualify for all the superlatives — revolutionary, and so on.”

Peculiar Numbers

I met Furey in June, in the porter’s lodge through which one enters Trinity Hall on the bank of the River Cam. Petite, muscular, and wearing a sleeveless black T-shirt (that revealed bruises from mixed martial arts), rolled-up jeans, socks with cartoon aliens on them and Vegetarian Shoes–brand sneakers, in person she was more Vancouverite than the otherworldly figure in her lecture videos. We ambled around the college lawns, ducking through medieval doorways in and out of the hot sun. On a different day I might have seen her doing physics on a purple yoga mat on the grass.

Furey, who is 39, said she was first drawn to physics at a specific moment in high school, in British Columbia. Her teacher told the class that only four fundamental forces underlie all the world’s complexity — and, furthermore, that physicists since the 1970s had been trying to unify all of them within a single theoretical structure. “That was just the most beautiful thing I ever heard,” she told me, steely-eyed. She had a similar feeling a few years later, as an undergraduate at Simon Fraser University in Vancouver, upon learning about the four division algebras. One such number system, or infinitely many, would seem reasonable. “But four?” she recalls thinking. “How peculiar."

Video: Cohl Furey explains what octonions are and what
they might have to do with particle physics.

Susannah Ireland for Quanta Magazine

After breaks from school spent ski-bumming, bartending abroad and intensely training as a mixed martial artist, Furey later met the division algebras again in an advanced geometry course and learned just how peculiar they become in four strokes. When you double the dimensions with each step as you go from real numbers to complex numbers to quaternions to octonions, she explained, “in every step you lose a property.” Real numbers can be ordered from smallest to largest, for instance, “whereas in the complex plane there’s no such concept.” Next, quaternions lose commutativity; for them, a × b doesn’t equal b × a. This makes sense, since multiplying higher-dimensional numbers involves rotation, and when you switch the order of rotations in more than two dimensions you end up in a different place. Much more bizarrely, the octonions are nonassociative, meaning (a × b) × c doesn’t equal a × (b × c). “Nonassociative things are strongly disliked by mathematicians,” said John Baez, a mathematical physicist at the University of California, Riverside, and a leading expert on the octonions. “Because while it’s very easy to imagine noncommutative situations — putting on shoes then socks is different from socks then shoes — it’s very difficult to think of a nonassociative situation.” If, instead of putting on socks then shoes, you first put your socks into your shoes, technically you should still then be able to put your feet into both and get the same result. “The parentheses feel artificial.”

The octonions’ seemingly unphysical nonassociativity has crippled many physicists’ efforts to exploit them, but Baez explained that their peculiar math has also always been their chief allure. Nature, with its four forces batting around a few dozen particles and anti-particles, is itself peculiar. The Standard Model is “quirky and idiosyncratic,” he said.

In the Standard Model, elementary particles are manifestations of three “symmetry groups” — essentially, ways of interchanging subsets of the particles that leave the equations unchanged. These three symmetry groups, SU(3), SU(2) and U(1), correspond to the strong, weak and electromagnetic forces, respectively, and they “act” on six types of quarks, two types of leptons, plus their anti-particles, with each type of particle coming in three copies, or “generations,” that are identical except for their masses. (The fourth fundamental force, gravity, is described separately, and incompatibly, by Einstein’s general theory of relativity, which casts it as curves in the geometry of space-time.)

Sets of particles manifest the symmetries of the Standard Model in the same way that four corners of a square must exist in order to realize a symmetry of 90-degree rotations. The question is, why this symmetry group — SU(3) × SU(2) × U(1)? And why this particular particle representation, with the observed particles’ funny assortment of charges, curious handedness and three-generation redundancy? The conventional attitude toward such questions has been to treat the Standard Model as a broken piece of some more complete theoretical structure. But a competing tendency is to try to use the octonions and “get the weirdness from the laws of logic somehow,” Baez said.

Furey began seriously pursuing this possibility in grad school, when she learned that quaternions capture the way particles translate and rotate in 4-D space-time. She wondered about particles’ internal properties, like their charge. “I realized that the eight degrees of freedom of the octonions could correspond to one generation of particles: one neutrino, one electron, three up quarks and three down quarks,” she said — a bit of numerology that had raised eyebrows before. The coincidences have since proliferated. “If this research project were a murder mystery,” she said, “I would say that we are still in the process of collecting clues.”

The Dixon Algebra

To reconstruct particle physics, Furey uses the product of the four division algebras, R⊗C⊗H⊗O (R for reals, C for complex numbers, H for quaternions and O for octonions) — sometimes called the Dixon algebra, after Geoffrey Dixon, a physicist who first took this tack in the 1970s and ’80s before failing to get a faculty job and leaving the field. (Dixon forwarded me a passage from his memoirs: “What I had was an out-of-control intuition that these algebras were key to understanding particle physics, and I was willing to follow this intuition off a cliff if need be. Some might say I did.”)

Whereas Dixon and others proceeded by mixing the division algebras with extra mathematical machinery, Furey restricts herself; in her scheme, the algebras “act on themselves.” Combined as R⊗C⊗H⊗O, the four number systems form a 64-dimensional abstract space. Within this space, in Furey’s model, particles are mathematical “ideals”: elements of a subspace that, when multiplied by other elements, stay in that subspace, allowing particles to stay particles even as they move, rotate, interact and transform. The idea is that these mathematical ideals are the particles of nature, and they manifest the symmetries of R⊗C⊗H⊗O.

As Dixon knew, the algebra splits cleanly into two parts: C⊗H and C⊗O, the products of complex numbers with quaternions and octonions, respectively (real numbers are trivial). In Furey’s model, the symmetries associated with how particles move and rotate in space-time, together known as the Lorentz group, arise from the quaternionic C⊗H part of the algebra.  The symmetry group SU(3) × SU(2) × U(1), associated with particles’ internal properties and mutual interactions via the strong, weak and electromagnetic forces, come from the octonionic part, C⊗O.

Günaydin and Gürsey, in their early work, already found SU(3) inside the octonions. Consider the base set of octonions, 1, e₁, e₂, e₃, e₄, e₅, e₆ and e₇, which are unit distances in eight different orthogonal directions: They respect a group of symmetries called G2, which happens to be one of the rare “exceptional groups” that can’t be mathematically classified into other existing symmetry-group families. The octonions’ intimate connection to all the exceptional groups and other special mathematical objects has compounded the belief in their importance, convincing the eminent Fields medalist and Abel Prize–winning mathematician Michael Atiyah, for example, that the final theory of nature must be octonionic. “The real theory which we would like to get to,” he said in 2010, “should include gravity with all these theories in such a way that gravity is seen to be a consequence of the octonions and the exceptional groups.” He added, “It will be hard because we know the octonions are hard, but when you’ve found it, it should be a beautiful theory, and it should be unique.”

Holding e₇ constant while transforming the other unit octonions reduces their symmetries to the group SU(3). Günaydin and Gürsey used this fact to build an octonionic model of the strong force acting on a single generation of quarks.

Lucy Reading-Ikkanda/Quanta Magazine

Furey has gone further. In her most recent published paper, which appeared in May in The European Physical Journal C, she consolidated several findings to construct the full Standard Model symmetry group, SU(3) × SU(2) × U(1), for a single generation of particles, with the math producing the correct array of electric charges and other attributes for an electron, neutrino, three up quarks, three down quarks and their anti-particles. The math also suggests a reason why electric charge is quantized in discrete units — essentially, because whole numbers are.

However, in that model’s way of arranging particles, it’s unclear how to naturally extend the model to cover the full three particle generations that exist in nature. But in another new paper that’s now circulating among experts and under review by Physical Letters B, Furey uses C⊗O to construct the Standard Model’s two unbroken symmetries, SU(3) and U(1). (In nature, SU(2) × U(1) is broken down into U(1) by the Higgs mechanism, a process that imbues particles with mass.) In this case, the symmetries act on all three particle generations and also allow for the existence of particles called sterile neutrinos — candidates for dark matter that physicists are actively searching for now. “The three-generation model only has SU(3) × U(1), so it’s more rudimentary,” Furey told me, pen poised at a whiteboard. “The question is, is there an obvious way to go from the one-generation picture to the three-generation picture? I think there is.”

This is the main question she’s after now. The mathematical physicists Michel Dubois-Violette, Ivan Todorov and Svetla Drenska are also trying to model the three particle generations using a structure that incorporates octonions called the exceptional Jordan algebra. After years of working solo, Furey is beginning to collaborate with researchers who take different approaches, but she prefers to stick with the product of the four division algebras, R⊗C⊗H⊗O, acting on itself. It’s complicated enough and provides flexibility in the many ways it can be chopped up. Furey’s goal is to find the model that, in hindsight, feels inevitable and that includes mass, the Higgs mechanism, gravity and space-time.

Already, there’s a sense of space-time in the math. She finds that all multiplicative chains of elements of R⊗C⊗H⊗O can be generated by 10 matrices called “generators.” Nine of the generators act like spatial dimensions, and the 10th, which has the opposite sign, behaves like time. String theory also predicts 10 space-time dimensions — and the octonions are involved there as well. Whether or how Furey’s work connects to string theory remains to be puzzled out.

So does her future. She’s looking for a faculty job now, but failing that, there’s always the ski slopes or the accordion. “Accordions are the octonions of the music world,” she said — “tragically misunderstood.” She added, “Even if I pursued that, I would always be working on this project.”

The Final Theory

Furey mostly demurred on my more philosophical questions about the relationship between physics and math, such as whether, deep down, they are one and the same. But she is taken with the mystery of why the property of division is so key. She also has a hunch, reflecting a common allergy to infinity, that R⊗C⊗H⊗O is actually an approximation that will be replaced, in the final theory, with another, related mathematical system that does not involve the infinite continuum of real numbers.

That’s just intuition talking. But with the Standard Model passing tests to staggering perfection, and no enlightening new particles materializing at the Large Hadron Collider in Europe, a new feeling is in the air, both unsettling and exciting, ushering a return to whiteboards and blackboards. There’s the burgeoning sense that “maybe we have not yet finished the process of fitting the current pieces together,” said Boyle, of the Perimeter Institute. He rates this possibility “more promising than many people realize,” and said it “deserves more attention than it is currently getting, so I am very glad that some people like Cohl are seriously pursuing it.”

Boyle hasn’t himself written about the Standard Model’s possible relationship to the octonions. But like so many others, he admits to hearing their siren song. “I share the hope,” he said, “and even the suspicion, that the octonions may end up playing a role, somehow, in fundamental physics, since they are very beautiful.”

Octonion

From Wikipedia, the free encyclopedia

 

In mathematics, the octonions are a normed division algebra over the real numbers, usually represented by the capital letter O, using boldface O or blackboard bold ${\displaystyle \mathbb {O} }$. Octonions have eight dimensions; twice the number of dimensions of the quaternions, of which they are an extension. Along with the real numbers R, complex numbers C, and quaternions H, octonions complete the set of numbers capable of being added, subtracted, multiplied or divided; as such, they are believed by some researchers to have fundamental importance in physical theory^. They are noncommutative and nonassociative, but satisfy a weaker form of associativity; namely, they are alternative.

Octonions are not as well known as the quaternions and complex numbers, which are much more widely studied and used. Despite this, they have some interesting properties and are related to a number of exceptional structures in mathematics, among them the exceptional Lie groups. Additionally, octonions have applications in fields such as string theory, special relativity, and quantum logic. Applying the Cayley-Dickson construction to the octonions produces the sedenions.

The octonions were discovered in 1843 by John T. Graves, inspired by his friend W. R. Hamilton's discovery of quaternions. Graves called his discovery octaves, and mentioned them in a letter to Hamilton dated 16 December 1843, but his first publication of his result in (Graves 1845) was slightly later than Arthur Cayley's article on them. The octonions were discovered independently by Cayley^[2] and are sometimes referred to as Cayley numbers or the Cayley algebra. Hamilton described the early history of Graves' discovery.^[3]

Definition

The octonions can be thought of as octets (or 8-tuples) of real numbers. Every octonion is a real linear combination of the unit octonions:

${\displaystyle \{e_{0},e_{1},e_{2},e_{3},e_{4},e_{5},e_{6},e_{7}\},}$

where e₀ is the scalar or real element; it may be identified with the real number 1. That is, every octonion x can be written in the form

${\displaystyle x=x_{0}e_{0}+x_{1}e_{1}+x_{2}e_{2}+x_{3}e_{3}+x_{4}e_{4}+x_{5}e_{5}+x_{6}e_{6}+x_{7}e_{7},\,}$

with real coefficients {x_i}.

Addition and subtraction of octonions is done by adding and subtracting corresponding terms and hence their coefficients, like quaternions. Multiplication is more complex. Multiplication is distributive over addition, so the product of two octonions can be calculated by summing the products of all the terms, again like quaternions. The product of each pair of terms can be given by multiplication of the coefficients and a multiplication table of the unit octonions.

Most off-diagonal elements of the table are antisymmetric, making it almost a skew-symmetric matrix except for the elements on the main diagonal, as well as the row and column for which e₀ is an operand.

The table can be summarized as follows:^[5]
${\displaystyle e_{i}e_{j}={\begin{cases}e_{j},&{\text{if }}i=0\\e_{i},&{\text{if }}j=0\\-\delta _{ij}e_{0}+\varepsilon _{ijk}e_{k},&{\text{otherwise}}\end{cases}}}$

where ${\displaystyle \delta _{ij}}$ is the Kronecker delta and ${\displaystyle \varepsilon _{ijk}}$ is a completely antisymmetric tensor with value +1 when ijk = 123, 145, 176, 246, 257, 347, 365.

The above definition though is not unique, but is only one of 480 possible definitions for octonion multiplication with e₀ = 1. The others can be obtained by permuting and changing the signs of the non-scalar basis elements. The 480 different algebras are isomorphic, and there is rarely a need to consider which particular multiplication rule is used. Each of these 480 definitions is invariant up to signs under some 7-cycle of the points (1234567), and for each 7-cycle there are four definitions, differing by signs and reversal of order. A common choice is to use the definition invariant under the 7-cycle (1234567) with e₁e₂ = e₄ as it is particularly easy to remember the multiplication.

A variation of this sometimes used is to label the elements of the basis by the elements ∞, 0, 1, 2, ..., 6, of the projective line over the finite field of order 7. The multiplication is then given by e_∞ = 1 and e₁e₂ = e₄, and all expressions obtained from this by adding a constant (mod 7) to all subscripts: in other words using the 7 triples (124) (235) (346) (450) (561) (602) (013). These are the nonzero codewords of the quadratic residue code of length 7 over the Galois field of two elements, GF(2). There is a symmetry of order 7 given by adding a constant mod 7 to all subscripts, and also a symmetry of order 3 given by multiplying all subscripts by one of the quadratic residues 1, 2, 4 mod 7.^[6][7]

The multiplication table for a geometric algebra of signature (−−−−) can be given in terms of the following 7 quaternionic triples (omitting the identity element): (I,j,k), (i,J,k), (i,j,K), (I,J,K), (∗I,i,m), (∗J,j,m), (∗K,k,m) in which the lowercase items are vectors (mathematics and physics) and the uppercase ones are bivectors and ∗ = mijk (which is in fact the Hodge star operator). If the ∗ is forced to be equal to the identity then the multiplication ceases to be associative, but the ∗ may be removed from the multiplication table resulting in an octonion multiplication table.

Note that in keeping ∗ = mijk associative and thus not reducing the 4-dimensional geometric algebra to an octonion one, the whole multiplication table can be derived from the equation for ∗. Consider the gamma matrices. The formula defining the fifth gamma matrix shows that it is the ∗ of a four-dimensional geometric algebra of the gamma matrices.

Cayley–Dickson construction

A more systematic way of defining the octonions is via the Cayley–Dickson construction. Just as quaternions can be defined as pairs of complex numbers, the octonions can be defined as pairs of quaternions. Addition is defined pairwise. The product of two pairs of quaternions (a, b) and (c, d) is defined by

${\displaystyle (a,b)(c,d)=(ac-d^{*}b,da+bc^{*}),}$

where ${\displaystyle z^{*}}$ denotes the conjugate of the quaternion z. This definition is equivalent to the one given above when the eight unit octonions are identified with the pairs

(1,0), (i,0), (j,0), (k,0), (0,1), (0,i), (0,j), (0,k)

Fano plane mnemonic

A mnemonic for the products of the unit octonions.^[8]

 

A 3D mnemonic visualization showing the 7 triads as hyperplanes through the Real (${\displaystyle e_{0}}$) vertex of the octonion example given above.^[8]

A convenient mnemonic for remembering the products of unit octonions is given by the diagram at the right, which represents the multiplication table of Cayley and Graves.^[4][9] This diagram with seven points and seven lines (the circle through 1, 2, and 3 is considered a line) is called the Fano plane. The lines are oriented. The seven points correspond to the seven standard basis elements of Im(O) (see definition below). Each pair of distinct points lies on a unique line and each line runs through exactly three points.

Let (a, b, c) be an ordered triple of points lying on a given line with the order specified by the direction of the arrow. Then multiplication is given by

ab = c and ba = −c

together with cyclic permutations. These rules together with

• 1 is the multiplicative identity,
• e_i² = −1 for each point in the diagram

completely defines the multiplicative structure of the octonions. Each of the seven lines generates a subalgebra of O isomorphic to the quaternions H.

Conjugate, norm, and inverse

The conjugate of an octonion

${\displaystyle x=x_{0}\,e_{0}+x_{1}\,e_{1}+x_{2}\,e_{2}+x_{3}\,e_{3}+x_{4}\,e_{4}+x_{5}\,e_{5}+x_{6}\,e_{6}+x_{7}\,e_{7}}$

is given by

${\displaystyle x^{*}=x_{0}\,e_{0}-x_{1}\,e_{1}-x_{2}\,e_{2}-x_{3}\,e_{3}-x_{4}\,e_{4}-x_{5}\,e_{5}-x_{6}\,e_{6}-x_{7}\,e_{7}.}$

Conjugation is an involution of O and satisfies (xy)^∗ = y^∗ x^∗ (note the change in order).

The real part of x is given by

${\displaystyle {\frac {x+x^{*}}{2}}=x_{0}\,e_{0}}$

and the imaginary part by

${\displaystyle {\frac {x-x^{*}}{2}}=x_{1}\,e_{1}+x_{2}\,e_{2}+x_{3}\,e_{3}+x_{4}\,e_{4}+x_{5}\,e_{5}+x_{6}\,e_{6}+x_{7}\,e_{7}.}$

The set of all purely imaginary octonions span a 7 dimension subspace of O, denoted Im(O).
Conjugation of octonions satisfies the equation

${\displaystyle x^{*}=-{\frac {1}{6}}(x+(e_{1}x)e_{1}+(e_{2}x)e_{2}+(e_{3}x)e_{3}+(e_{4}x)e_{4}+(e_{5}x)e_{5}+(e_{6}x)e_{6}+(e_{7}x)e_{7}).}$

The product of an octonion with its conjugate, x^∗ x = x x^∗, is always a nonnegative real number:

${\displaystyle x^{*}x=x_{0}^{2}+x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}+x_{5}^{2}+x_{6}^{2}+x_{7}^{2}.}$

Using this the norm of an octonion can be defined, as

${\displaystyle \|x\|={\sqrt {x^{*}x}}.}$

This norm agrees with the standard Euclidean norm on R⁸.

The existence of a norm on O implies the existence of inverses for every nonzero element of O. The inverse of x ≠ 0 is given by

${\displaystyle x^{-1}={\frac {x^{*}}{\|x\|^{2}}}.}$

It satisfies x x⁻¹ = x⁻¹ x = 1.

Properties

Octonionic multiplication is neither commutative:

${\displaystyle e_{i}e_{j}=-e_{j}e_{i}\neq e_{j}e_{i}\,}$ if ${\displaystyle i,j}$ are distinct and non-zero,

nor associative:

${\displaystyle (e_{i}e_{j})e_{k}=-e_{i}(e_{j}e_{k})\neq e_{i}(e_{j}e_{k})\,}$ if ${\displaystyle i,j,k}$ are distinct, non-zero or if ${\displaystyle e_{i}e_{j}\neq \pm e_{k}}$.

The octonions do satisfy a weaker form of associativity: they are alternative. This means that the subalgebra generated by any two elements is associative. Actually, one can show that the subalgebra generated by any two elements of O is isomorphic to R, C, or H, all of which are associative. Because of their non-associativity, octonions do not have matrix representations, unlike quaternions.

The octonions do retain one important property shared by R, C, and H: the norm on O satisfies

${\displaystyle \|xy\|=\|x\|\|y\|}$

This equation means that the octonions form a composition algebra. The higher-dimensional algebras defined by the Cayley–Dickson construction (e.g., the sedenions) all fail to satisfy this property. They all have zero divisors.

Wider number systems exist which have a multiplicative modulus (e.g. 16-dimensional conic sedenions). Their modulus is defined differently from their norm, and they also contain zero divisors.

As shown by Hurwitz, the only normed division algebras over the reals are R, C, H, and O. These four algebras also form the only alternative, finite-dimensional division algebras over the reals (up to isomorphism).

Not being associative, the nonzero elements of O do not form a group. They do, however, form a loop, indeed a Moufang loop.

Commutator and cross product

The commutator of two octonions x and y is given by

${\displaystyle [x,y]=xy-yx.}$

This is antisymmetric and imaginary. If it is considered only as a product on the imaginary subspace Im(O) it defines a product on that space, the seven-dimensional cross product, given by

${\displaystyle x\times y={\frac {1}{2}}(xy-yx).}$

Like the cross product in three dimensions this is a vector orthogonal to x and y with magnitude

${\displaystyle \|x\times y\|=\|x\|\|y\|\sin \theta .}$

But like the octonion product it is not uniquely defined. Instead there are many different cross products, each one dependent on the choice of octonion product.^[10]

Automorphisms

An automorphism, A, of the octonions is an invertible linear transformation of O which satisfies

${\displaystyle A(xy)=A(x)A(y).}$

The set of all automorphisms of O forms a group called G₂.^[11] The group G₂ is a simply connected, compact, real Lie group of dimension 14. This group is the smallest of the exceptional Lie groups and is isomorphic to the subgroup of Spin(7) that preserves any chosen particular vector in its 8-dimensional real spinor representation. The group Spin(7) is in turn a subgroup of the group of isotopies described below.

See also: PSL(2,7) – the automorphism group of the Fano plane.

Isotopies

An isotopy of an algebra is a triple of bijective linear maps a, b, c such that if xy = z then a(x)b(y) = c(z). For a = b = c this is the same as an automorphism. The isotopy group of an algebra is the group of all isotopies, which contains the group of automorphisms as a subgroup.

The isotopy group of the octonions is the group Spin₈(R), with a, b, and c acting as the three 8-dimensional representations.^[12] The subgroup of elements where c fixes the identity is the subgroup Spin₇(R), and the subgroup where a, b, and c all fix the identity is the automorphism group G₂.

Integral octonions

There are several natural ways to choose an integral form of the octonions. The simplest is just to take the octonions whose coordinates are integers. This gives a nonassociative algebra over the integers called the Gravesian octonions. However it is not a maximal order (in the sense of ring theory); there are exactly 7 maximal orders containing it. These 7 maximal orders are all equivalent under automorphisms. The phrase "integral octonions" usually refers to a fixed choice of one of these seven orders.

These maximal orders were constructed by Kirmse (1925), Dickson and Bruck as follows. Label the 8 basis vectors by the points of the projective plane over the field with 7 elements. First form the "Kirmse integers" : these consist of octonions whose coordinates are integers or half integers, and that are half integers (that is, halves of odd integers) on one of the 16 sets

∅ (∞124) (∞235) (∞346) (∞450) (∞561) (∞602) (∞013) (∞0123456) (0356) (1460) (2501) (3612) (4023) (5134) (6245)

of the extended quadratic residue code of length 8 over the field of 2 elements, given by ∅, (∞124) and its images under adding a constant mod 7, and the complements of these 8 sets. Then switch infinity and any one other coordinate; this operation creates a bijection of the Kirmse integers onto a different set, which is a maximal order. There are 7 ways to do this, giving 7 maximal orders, which are all equivalent under cyclic permutations of the 7 coordinates 0123456. (Kirmse incorrectly claimed that the Kirmse integers also form a maximal order, so he thought there were 8 maximal orders rather than 7, but as Coxeter (1946) pointed out they are not closed under multiplication; this mistake occurs in several published papers.)

The Kirmse integers and the 7 maximal orders are all isometric to the E₈ lattice rescaled by a factor of 1/√2. In particular there are 240 elements of minimum nonzero norm 1 in each of these orders, forming a Moufang loop of order 240.

The integral octonions have a "division with remainder" property: given integral octonions a and b ≠ 0, we can find q and r with a = qb + r, where the remainder r has norm less than that of b.

In the integral octonions, all left ideals and right ideals are 2-sided ideals, and the only 2-sided ideals are the principal ideals nO where n is a non-negative integer.

The integral octonions have a version of factorization into primes, though it is not straightforward to state because the octonions are not associative so the product of octonions depends on the order in which one does the products. The irreducible integral octonions are exactly those of prime norm, and every integral octonion can be written as a product of irreducible octonions. More precisely an integral octonion of norm mn can be written as a product of integral octonions of norms m and n.

The automorphism group of the integral octonions is the group G₂(F₂) of order 12096, which has a simple subgroup of index 2 isomorphic to the unitary group ²A₂(3²). The isotopy group of the integral octonions is the perfect double cover of the group of rotations of the E₈ lattice.

Milloy & Dunn testimony to EPA on science transparency

This afternoon, I delivered the following remarks on EPA’s proposed science transparency rule at the public hearing. I commented on behalf of myself and Dr. John Dunn. We are trying to open the black box.

Open Black Box

My name is Steve Milloy. I publish JunkScience.com.

I am speaking on behalf of myself and Dr. John Dunn, an emergency physician in Texas.

We are commenting in support the proposed transparency initiative.

Science transparency in EPA regulatory action is long past overdue.

When I first started working on EPA issues in 1990, the main controversy with EPA science was the use of science policy and default assumptions – like the linear no-threshold model of carcinogenesis.

The problem wasn’t necessarily the use of science policy and default assumptions. The problem was, rather, EPA’s failure to disclose the nature of those default assumptions.

In other words: what part was science; what part was guesswork and; what part was politics.

When I first reported on this problem for the Department of Energy in 1994, the Clinton administration tried to censor my report. But they couldn’t and so here we are so many years later finally making progress on this important issue.

More recently, the major problem with EPA science has been what has become known as “secret science.”

Since 1990s, EPA grantees like Harvard’s Doug Dockery and Brigham Young University’s Arden Pope have refused to make available to the public the raw data used in their epidemiologic studies.

And this is true despite the fact that these studies were cited by EPA as the principal scientific bases for major air quality rules like those that were the Obama administration’s war-on-coal.

Worse, prior EPA administrations have actually aided and abetted Dockery and Pope hiding their data from public review. Prior EPA administrations have ignored the requests of the EPA’s own independent science advisors and Congress to release the data.

I can only conclude this is because independent review of the Harvard Six City and American Cancer Society line of studies would prove them to be highly problematic, embarrassing or even fraudulent.

Desperate to defend the indefensible, supporters of Dockery and Pope have wrongly maintained that making the data in question public would violate medial and personal privacy rights.

Nothing could be further from the truth.

For the most part, the data is electronic. Scrubbed files with the key data needed for independent review can easily be made available. No one is interested in any personal and medical data. It has no value.

The state of California has made such data files available for use for years. I know. I have obtained this data (over 2 million death certificates to be precise) and with it enabled research to be published that completely debunks the secret science of Dockery and Pope.

Fear of exposure of their research as faulty, if not fake, is why Dockery and Pope are so scared of producing their data for independent review.

To make these comments current, efforts have been made this month to obtain the Dockery and Pope data. But they continue to keep their data secret.

Given that the Dockery and Pope research and the related PM2.5 research has been funded by taxpayers – to the tune of more than $600 million dollars, and then this research is used to regulate the public, costing untold billions more dollars without providing any public health benefit — the conspiratorial hiding of this secret data is more akin to crime than science.

If EPA wants to regulate, that is fine. But the basis of and reasons for that regulation must be clearly laid out so their can be full and fair debate.

Harvard’s Doug Dockery and Brigham Young’s Arden Pope don’t want independent scientists to check their work for some reason. Dockery and Pope’s supporters may offer whatever excuses they like. But we all know what the reality is – fear of exposure.