The mathematical structure of quantum real numbers

Abstract

The mathematical structure of the sheaf of Dedekind real numbers $\RsubD(X)$ for a quantum system is discussed. The algebra of physical qualities is represented by an $O^{\ast }$ algebra $\mathcal{M}$ that acts on a Hilbert space that carries an irreducible representation of the symmetry group of the system. $X =\EsubS(\mathcal M)$, the state space for $\mathcal{M}$, has the weak topology generated by the functions $a_Q(\cdot )$, defined for $\hat{A}\in {\mathcal{M}}_{sa}$ and $\forall \hat ρ\in \EsubS(\mathcal M) $, by $a_Q(\hat{\rho })=Tr\hat{A}\hat{\rho }$. For any open subset $W$ of $\EsubS(\mathcal M)$, the function $a_Q{|}_W$ is the numerical value of the quality $\hat{A}$ defined to the extent $W$. The example of the quantum real numbers for a single Galilean relativistic particle is given.

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References in corpus (3)

• Conformal Symmetry and Quantum Relativity
• Quantum Mechanics interpreted in Quantum Real Numbers
• Quantum mechanics as a space-time theory