903 research outputs found
A topos perspective on the Kochen-Specker theorem: II. Conceptual Aspects, and Classical Analogues:
In a previous paper, we have proposed assigning as the value of a physical
quantity in quantum theory, a certain kind of set (a sieve) of quantities that
are functions of the given quantity. The motivation was in part physical---such
a valuation illuminates the Kochen-Specker theorem; and in part
mathematical---the valuation arises naturally in the topos theory of
presheaves.
This paper discusses the conceptual aspects of this proposal. We also
undertake two other tasks. First, we explain how the proposed valuations could
arise much more generally than just in quantum physics; in particular, they
arise as naturally in classical physics. Second, we give another motivation for
such valuations (that applies equally to classical and quantum physics). This
arises from applying to propositions about the values of physical quantities
some general axioms governing partial truth for any kind of proposition.Comment: Small changes and correction
Perennials and the Group-Theoretical Quantization of a Parametrized Scalar Field on a Curved Background
The perennial formalism is applied to the real, massive Klein-Gordon field on
a globally-hyperbolic background space-time with compact Cauchy hypersurfaces.
The parametrized form of this system is taken over from the accompanying paper.
Two different algebras and of
elementary perennials are constructed. The elements of
correspond to the usual creation and annihilation operators for particle modes
of the quantum field theory, whereas those of are the
smeared fields. Both are shown to have the structure of a Heisenberg algebra,
and the corresponding Heisenberg groups are described. Time evolution is
constructed using transversal surfaces and time shifts in the phase space.
Important roles are played by the transversal surfaces associated with
embeddings of the Cauchy hypersurface in the space-time, and by the time shifts
that are generated by space-time isometries. The automorphisms of the algebras
generated by this particular type of time shift are calculated explicitly.Comment: 31 pages, revte
A topos perspective on the Kochen-Specker theorem: I. Quantum States as Generalized Valuations
The Kochen-Specker theorem asserts the impossibility of assigning values to
quantum quantities in a way that preserves functional relations between them.
We construct a new type of valuation which is defined on all operators, and
which respects an appropriate version of the functional composition principle.
The truth-values assigned to propositions are (i) contextual; and (ii)
multi-valued, where the space of contexts and the multi-valued logic for each
context come naturally from the topos theory of presheaves.
The first step in our theory is to demonstrate that the Kochen-Specker
theorem is equivalent to the statement that a certain presheaf defined on the
category of self-adjoint operators has no global elements. We then show how the
use of ideas drawn from the theory of presheaves leads to the definition of a
generalized valuation in quantum theory whose values are sieves of operators.
In particular, we show how each quantum state leads to such a generalized
valuation.Comment: Clarification of situation for situation for operators with
continuous spectr
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