Division Algebras, Lattices, Physics, Windmill Tilting by Geoffrey Dixon (PDF)

Description

Complex numbers, quaternions, octonions, integral domains, laminated lattices, the Leech lattice, with applications to theoretical physics, including the Standard Model of quarks and leptons with spinors and gauge groups.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐A somewhat disorganised approach to some interesting ideas. I’d recommend getting Conway’s books on Octonions and Sphere Packings there is not much in this book that you won’t find in Conway’s books. I’d hoped for a fresh perspective on some of these issues but too much of the book is taken up with the author’s rants about academia and numerology.

⭐If you’ve heard about quaternions or octonions and their speculated uses in the description of physical law, if you have basic college-level training in math and algebra, if you’ve ever wondered about finding nature’s symmetries from the Standard Model in physics from beautifully simple algebraic constructs, or if you’re even familiar with these topics since the 1970’s and would like to see personal view on history paired with a compact presentation of 30 years of research and vision, then this your book. I’ve read Dixon’s “Division Algebras: Octonions Quaternions Complex Numbers and the Algebraic Design of Physics” (1994) a little while ago, and left off both inspired and confused. This new book, “Division Algebras, Lattices, Physics, and Windmill Tilting”, is not only much cheaper, it is also much more compact and approachable in the mathematical presentation. Dixon’s seminal proposal from the 1980’s and 1990’s has since become a benchmark for octonionic models in physical mathematics when aiming at modeling fundamental particle interactions. It is great to see his T = RxCxHxO spinor introduced in this accessible, concise, and easily quotable format. Next to this, I enjoyed the fascinating personal history on how Dixon’s ideas unfolded in academia, his “Multiverse Critique”, the discussion on octonion products and lattices, and a section on his more recent work on ternary systems. You may enjoy Dixon’s brisk humor and cheek, such as: “Michael Atiyah, a deity of mathematical physics at Oxford, recently gave a talk at Princeton extolling the virtues of applying all the division algebras in physics (he suspected that the octonions would be linked to gravity, which is wrong, but still …).” There’s an overall tone throughout the book that is positively inspiring towards trying out things, where researchers shouldn’t be scolded for their mere area of interest as long as their methods are sound: “If no one else on the planet was working in this area, would I feel compelled by the promise of its ideas to carry on alone, despite the potential stigma of being labeled a crackpot?” In my view, the book is an octonion-in-physics must-have.

⭐A fun read for anyone who loves mathematical physics. Down-to-earth calculations regarding fascinating, beautiful mathematical structures, interspersed with personal anecdotes and speculative comments. The informality and life of a blog, but the rigor and care of an advanced textbook — great combination!

⭐Upon first scan of Geoffrey M. Dixon’s “Division Algebra, Lattices, Physics, Windmill Tilting,” my eyes settled upon section 9, “Ternary Stuff”. Drawn to “C_3: Ternary Version of C,” in which he develops a 3D Complex space with ternary product, I recalled Asimov’s “The Gods Themselves,” where aliens mate in threes! Most importantly, Dixon uses “mathematical resonance,” a notion which reflects his belief that “mathematics and physics, at their most profound levels, are unified: they are one science” [p.8]. This resonance seems to guide his pursuit of math-physics, i.e., he pursues only research into what he finds mathematically resonant! (An admirable MOtivation, IMO.)Chapter after chapter, Dixon reveals amazing results that flow from this resonance and his struggle to be true to it. In his “Multiverse Critique” [pp 44-49] he shows that all the talk about multiverses doesn’t mean just any multiverse is possible, but only those which would conform to the math-physics of mathematical reasonance. (Science fiction writers take note!)I was powerfully struck by his simple parenthetical comment [p 121]: “…QM (quantum mechanics) and GR (general relativity) should not be unified, they should be derived.” IMHOpinion, current mathematical physics could benefit immensely from such a philosophical bent.Not only is this a book for the mathematically inclined, it is for anyone who wishes to be true to his/her inner motivation to “Do it your way!” I can’t help but feel that any reader (including the so-inclined graduate students) will find wonderful direction and example from Dixon’s long journey and his compelling findings.– John Shuster, 1-12-2012

⭐No he podido leerlo en mi Kindle! Ese es el mensaje que aparece cuando lo intento…

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