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Zeitschrift für Naturforschung A

On Classical Representations of Convex Descriptions

Slawomir Bugajski

Published Mar 1, 1993• 3 citations
Publisher

Abstract

Abstract It is demonstrated that if V* is not a vector lattice, where V is a base norm Banach space, then there is no commutative observable providing a classical representation for V. This observation generalizes a similar result of Busch and Lahti, obtained for V - the trace class of operators on a separable complex Hilbert space.
DOI
10.1515/zna-1993-0305

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In the quantum probability theory the 6-field of random events is replaoed by the lattice of orthogonal projectors in a separable infinite dimensional Hilbert space H.A countably additive function from this lattices to the unit interval constitutes a state, the non-commutative analogue of a probability measure.The Theorem of Gleason [4] asserts that every state is of the form n--tr jtT, where jt runs over all projectors and T is a probability operator on H, i.e. a positive linear operator of unit trace.Conversely, every probability operator determines a state by the Gleason formula.From now onwards let S stand for the set of all states, i.e. all probability operators on H.We shall dienote by the space of-? ell nuclear linear operators acting in H with the norm jjïj|.j= tr{TT*) 1 ^2.Of coarse S is e closed and convex subset Of TyIn quantum theory to every physical quantity or observable th^re corresponds e sslf»adjoint not necessarily bounded linear operator on Hs By 0 ws shall denote the set of all obt ;ry£bles-Given he-0 f tac pi'cb^ivility distribution of A at the etato T is defined for ail Eoi'oi subsets E of the real
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Abstract We show that any decoherence functional D can be represented by a spanning vector-valued measure on a complex Hilbert space. Moreover, this representation is unique up to an isomorphism when the system is finite. We consider the natural map U from the history Hilbert space K to the standard Hilbert space H of the usual quantum formulation. We show that U is an isomorphism from K onto a closed subspace of H and that U is an isomorphism from K onto H if and only if the representation is spanning. We then apply this work to show that a quantum measure has a Hilbert space representation if and only if it is strongly positive. We also discuss classical decoherence functionals, operator-valued measures and quantum operator measures.
In principle, the lattice of elementary propositions of a generic quantum system admits a representation in real, complex or quaternionic Hilbert spaces as established by Solèr’s theorem (1995) closing a long standing problem that can be traced back to von Neumann’s mathematical formulation of quantum mechanics. However up to now there are no examples of quantum systems described in Hilbert spaces whose scalar field is different from the set of complex numbers. We show that elementary relativistic systems cannot be described by irreducible strongly-continuous unitary representations of SL(2, C) on real or quaternionic Hilbert spaces as a consequence of some peculiarity of the generators related with the theory of polar decomposition of operators. Indeed such a ”naive” attempt leads necessarily to an equivalent formulation on a complex Hilbert space. Although this conclusion seems to give a definitive answer to the real/quaternionic-quantum-mechanics issue, it lacks consistency since it does not derive from more general physical hypotheses as the complex one does. Trying a more solid approach, in both situations we end up with three possibilities: an equivalent description in terms of a Wigner unitary representation in a real, complex or quaternionic Hilbert space. At this point the ”naive” result turns out to be a definitely important technical lemma, for it forbids the two extreme possibilities. \nIn conclusion, the real/quaternionic theory is actually complex. This improved approach is based upon the concept of von Neumann algebra of observables. Unfortunately, while \nthere exists a thorough literature about these algebras on real and complex Hilbert spaces, an analysis on the notion of von Neumann algebra over a quaternionic Hilbert space is \ncompletely absent to our knowledge. There are several issues in trying to define such a mathematical object, first of all the inability to construct linear combination of operators \nwith quaternionic coeffients. Restricting ourselves to unital real *-algebras of operators we are able to prove the von Neumann Double Commutant Theorem also on quaternionc \nHilbert spaces. Clearly, this property turns out to be crucial.
Starting from a thorough analysis of the conjugate H‾\overline{H}H of a complex Hilbert space HHH, including its significant importance regarding a representation of the tensor product of two complex Hilbert spaces and its impact to the theorem of Fréchet-Riesz over to a revisit of applications of nuclear and absolutely ppp-summing operators in algebraic quantum field theory (AQFT) in the sense of Araki, Haag and Kastler (p=2p=2p=2) and more recently in the framework of general probabilistic spaces (p=1p=1p=1), we will outline that Banach operator ideals in the sense of Pietsch, or equivalently tensor products of Banach spaces in the sense of Grothendieck are even lurking in the foundations and philosophy of quantum physics and quantum information theory. In particular, we concentrate on their importance in AQFT (Theorem 5.27). In doing so, we revisit the role of trace-class operators in quantum theory and construct the enveloping $\tup{C}^\adj$-algebra, corresponding to an arbitrarily given normed operator ideal (Proposition 5.3 and Theorem 5.5). Applications are presented, including a purely linear algebraic description of the quantum teleportation process, thereby showing a link to quantum information theory, also due to the emergence of the Hadamard-Walsh transform and the controlled NOT gate (Example 4.18). All Hilbert spaces discussed in this paper may be nonseparable (and hence infinite-dimensional).