Departmental Bulletin Paper 実数の量子化 ― 量子重力理論へのひとつのアプローチ ―
Quantization of Real Numbers ― An Approach to Quantum Gravity Theory ―


51pp.49 - 59 , 2017-02 , 広島工業大学
Towards a quantum theory of gravity, this paper presents a possible quantization of real numbers. The procedure to obtain a quantum real is as follows: We first prepare two algebras on the base space R, a function algebra H(R) and the Heisenberg algebra A(R) as the operator algebra on H(R). According to the scheme of, so-called, the deformation quantization, we obtain the deformed Heisenberg algebra Aq(R) with the deformation parameter q ∈ and, then, construct the deformed function algebra Hq(R) on which Aq(R) acts . The base space R introduced formally here is regarded as the quantum real numbers. In particular, we focus on the case where the parameter q is the N-th root of unity and obtain the deformed algebras Aq(R(N)); Hq(R(N)). Here, R(N) := R|qN=1 is proposed as the conclusive quantum real numbers. We further investigate the structure of R(N) and show that R(N) is isomorphic to the space Rq(N):= R×G(N) where the extra dimension G(N) is (para-)Grassmann space. Our conclusion is that the quantum real numbers Rq(N) is just the (para-)superspace.

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