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Geometric Algebra: The “Royal Road” To Quantum Gravity

Abstract

Euclid is said to have chided Ptolemy I that there is no “royal road” to geometry. Fortunately, there is a “royal road” to quantum gravity – namely, geometric algebra. In geometric algebra, geometrical elements, such as points, vectors, bivectors, ..., up to n-volumes, are represented by matrices. The kind of matrix – dyreal, real, complex, quaternionic, or dyquaternionic – depends only on the metric signature s (the number of spatial dimensions minus the number of temporal dimensions). The rank of the matrix depends only on n, the total number of dimensions, spatial plus temporal. Geometric algebras are periodic in s, but recursive in n. The recursion is generated from the basis vectors of either the Euclidean plane or the Minkowskian plane. The product of these two planes is 4-dimensional space-time if the resulting dimensions are not curled up, but it generates the Standard Model of physics if the resulting dimensions are curled up. Recursive generation leads to an expanding space-time lattice with the Standard Model at each node of the lattice. The rate of the recursion sets the cosmological constant. The other roads of Lee Smolin’s Three Roads to Quantum Gravity, namely M-Theory and Loop Quantum Gravity, can each be expressed as a geometric algebra. By subsuming the alternatives, geometric algebra is indeed the “royal road” to quantum gravity.

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