The mathematical structure of quantum real numbers TeX Export

@article{citeulike:4498657, 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^{*}\$ 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 \rho \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.}, archivePrefix = {arXiv}, author = {Corbett, John V.}, citeulike-article-id = {4498657}, citeulike-linkout-0 = {http://arxiv.org/abs/0905.0944}, citeulike-linkout-1 = {http://arxiv.org/pdf/0905.0944}, eprint = {0905.0944}, keywords = {algebras}, month = {May}, posted-at = {2009-05-23 20:41:34}, priority = {1}, title = {The mathematical structure of quantum real numbers}, url = {http://arxiv.org/abs/0905.0944}, year = {2009} }

TY - JOUR ID - citeulike:4498657 L3 - citeulike-article-id:4498657 N2 - 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 $O^{*}$ 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 \rho \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. TI - The mathematical structure of quantum real numbers KW - algebras AU - Corbett, John PY - 2009/May/07/ UR - http://arxiv.org/abs/0905.0944 ER -