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R(2): THE REAL GEOMETRIC ALGEBRA FOR BOTH SPACE-TIME AND QUANTUM BITS

Abstract

Matrix representations of basis vectors for space-times with p spatial dimensions and q temporal dimensions can be generated recursively (as matrices of matrices) from the matrix representations of basis vectors for either the Euclidean plane or the Minkowskian plane. Although these planes have different geometries, they have the same real geometric algebra, generated from sums and products of the basis vectors over the real numbers, which thus serves as the foundation of both quantum mechanics (the Euclidean plane) and relativity (the Minkowskian plane). That algebra is R(2), the algebra of 2x2 matrices of real numbers. The elements of R(2) can be taken to be (a) for the Euclidean plane, the identity matrix I, the space-like basis vectors X and Y, and the time-like bivector YX = T; or (b) for the Minkowskian plane, the identity matrix I, the space-like basis vector X, the time-like basis vector T, and the space-like bivector TX = Y. Geometrically, the so-called complex plane (I, T) is not a plane, since I is a scalar, not a vector. Using the Euclidean plane (X, Y) instead of complex numbers to describe quantum bits (qubits) geometrizes quantum mechanics. Moreover, real geometric algebras distinguish between time and space; complex ones do not. Anti-symmetric real 2x2 matrices are trace-free and time-like. Symmetric trace-free real 2x2 matrices are space-like. The unit time-like real 2x2 matrix T is unique, up to sign, corresponding to particles and anti-particles. The unit space-like real 2x2 matrix S = X cosθ + Y sinθ has an additional degree of freedom, corresponding to quantum phase. This explains why time grows linearly while, asymptotically, space expands exponentially, with a doubling time determined from the measured value of Einstein’s cosmological constant to be about 38 billion years.

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