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A test space is the set of outcome-sets associated with a collection of experiments. This notion provides a simple mathematical framework
for the study of probabilistic theories—notably, quantum mechanics—in which one is faced with incommensurable random quantities.
In the case of quantum mechanics, the relevant test space, the set of orthonormal bases of a Hilbert space, carries significant
topological structure. This paper inaugurates a general study of topological test spaces. Among other things, we show that
any topological test space with a compact space of outcomes is of finite rank. We also generalize results of Meyer and Clifton-Kent
by showing that, under very weak assumptions, any second-countable topological test space contains a dense semi-classical
test space.
I wish to dedicate this paper to the memory of Frank J. Hague III. 相似文献
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Let V be a braided vector space, i.e., a vector space together with a solution \({\hat{R}\in {{End}}(V\otimes V)}\) of the Yang–Baxter equation. Denote \({T(V):=\bigoplus_k V^{\otimes k}}\) . We associate to \({\hat{R}}\) a one-parameter family of solutions \({T(\hat{R})\in {\rm End}(T(V)\otimes T(V))}\) of the Yang–Baxter equation on the tensor space T (V). Main ingredients of the solution are braid analogues of the binomial coefficients and of the Pochhammer symbols. The association \({\hat{R}\rightsquigarrow T(\hat{R})}\) is functorial with respect to V. 相似文献
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Oliver Rudolph 《Communications in Mathematical Physics》2000,214(2):449-467
The basic mathematical framework for super Hilbert spaces over a Gra?mann algebra with a Gra?mann number-valued inner product
is formulated. Super Hilbert spaces over infinitely generated Gra?mann algebras arise in the functional Schr?dinger representation
of spinor quantum field theory in a natural way.
Received: 8 November 1999 / Accepted: 25 April 2000 相似文献
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N. S. Kronfli 《International Journal of Theoretical Physics》1969,2(4):345-349
The asymptotic condition is formulated for a system whose theory is more general than quantum mechanics. Its logic forms an orthocomplemented weakly modular -lattice. The set of states , consisting of all the probability measures on , is endowed with the most suitable metric physically, called here the natural one. In this space it is proved that the asymptotic condition implies the existence of two convex automorphisms
+- of which we call the wave-automorphisms. From these theS-automorphism
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+ is defined and corresponds to the scattering operator in conventional quantum theory. 相似文献
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Stanley Gudder 《Foundations of Physics》1999,29(3):409-422
Various axiomatic models for unsharp quantum measurements are investigated. These include effect spaces (E-spaces), effect test spaces (E-test spaces), effect algebras, and test groups. It is shown that a test group G is the universal group of an E-test space if and only if G is strongly atomistic. It follows that if G is strongly atomistic, then G is an interpolation group. We then demonstrate that if G is an interpolation group, then G is the universal group of an E-space. Finally, it is shown that an E-space is isomorphic to an E-test space if and only if it is strongly atomistic. 相似文献
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Every quantum state can be represented as a probability distribution over the outcomes of an informationally complete measurement.
But not all probability distributions correspond to quantum states. Quantum state space may thus be thought of as a restricted
subset of all potentially available probabilities. A recent publication (Fuchs and Schack, , 2009) advocates such a representation using symmetric informationally complete (SIC) measurements. Building upon this work we
study how this subset—quantum-state space—might be characterized. Our leading characteristic is that the inner products of
the probabilities are bounded, a simple condition with nontrivial consequences. To get quantum-state space something more
detailed about the extreme points is needed. No definitive characterization is reached, but we see several new interesting
features over those in Fuchs and Schack (, 2009), and all in conformity with quantum theory. 相似文献
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