261 research outputs found
A Topos Foundation for Theories of Physics: II. Daseinisation and the Liberation of Quantum Theory
This paper is the second in a series whose goal is to develop a fundamentally
new way of constructing theories of physics. The motivation comes from a desire
to address certain deep issues that arise when contemplating quantum theories
of space and time. Our basic contention is that constructing a theory of
physics is equivalent to finding a representation in a topos of a certain
formal language that is attached to the system. Classical physics arises when
the topos is the category of sets. Other types of theory employ a different
topos. In this paper, we study in depth the topos representation of the
propositional language, PL(S), for the case of quantum theory. In doing so, we
make a direct link with, and clarify, the earlier work on applying topos theory
to quantum physics. The key step is a process we term `daseinisation' by which
a projection operator is mapped to a sub-object of the spectral presheaf--the
topos quantum analogue of a classical state space. In the second part of the
paper we change gear with the introduction of the more sophisticated local
language L(S). From this point forward, throughout the rest of the series of
papers, our attention will be devoted almost entirely to this language. In the
present paper, we use L(S) to study `truth objects' in the topos. These are
objects in the topos that play the role of states: a necessary development as
the spectral presheaf has no global elements, and hence there are no
microstates in the sense of classical physics. Truth objects therefore play a
crucial role in our formalism.Comment: 34 pages, no figure
Physical Logic
In R.D. Sorkin's framework for logic in physics a clear separation is made
between the collection of unasserted propositions about the physical world and
the affirmation or denial of these propositions by the physical world. The
unasserted propositions form a Boolean algebra because they correspond to
subsets of an underlying set of spacetime histories. Physical rules of
inference, apply not to the propositions in themselves but to the affirmation
and denial of these propositions by the actual world. This physical logic may
or may not respect the propositions' underlying Boolean structure. We prove
that this logic is Boolean if and only if the following three axioms hold: (i)
The world is affirmed, (ii) Modus Ponens and (iii) If a proposition is denied
then its negation, or complement, is affirmed. When a physical system is
governed by a dynamical law in the form of a quantum measure with the rule that
events of zero measure are denied, the axioms (i) - (iii) prove to be too rigid
and need to be modified. One promising scheme for quantum mechanics as quantum
measure theory corresponds to replacing axiom (iii) with axiom (iv) Nature is
as fine grained as the dynamics allows.Comment: 14 pages, v2 published version with a change in the title and other
minor change
Collapse models with non-white noises II: particle-density coupled noises
We continue the analysis of models of spontaneous wave function collapse with
stochastic dynamics driven by non-white Gaussian noise. We specialize to a
model in which a classical "noise" field, with specified autocorrelator, is
coupled to a local nonrelativistic particle density. We derive general results
in this model for the rates of density matrix diagonalization and of state
vector reduction, and show that (in the absence of decoherence) both processes
are governed by essentially the same rate parameters. As an alternative route
to our reduction results, we also derive the Fokker-Planck equations that
correspond to the initial stochastic Schr\"odinger equation. For specific
models of the noise autocorrelator, including ones motivated by the structure
of thermal Green's functions, we discuss the qualitative and qantitative
dependence on model parameters, with particular emphasis on possible
cosmological sources of the noise field.Comment: Latex, 43 pages; versions 2&3 have minor editorial revision
Contextual logic for quantum systems
In this work we build a quantum logic that allows us to refer to physical
magnitudes pertaining to different contexts from a fixed one without the
contradictions with quantum mechanics expressed in no-go theorems. This logic
arises from considering a sheaf over a topological space associated to the
Boolean sublattices of the ortholattice of closed subspaces of the Hilbert
space of the physical system. Differently to standard quantum logics, the
contextual logic maintains a distributive lattice structure and a good
definition of implication as a residue of the conjunction.Comment: 16 pages, no figure
The Equivalence Postulate of Quantum Mechanics
The Equivalence Principle (EP), stating that all physical systems are
connected by a coordinate transformation to the free one with vanishing energy,
univocally leads to the Quantum Stationary HJ Equation (QSHJE). Trajectories
depend on the Planck length through hidden variables which arise as initial
conditions. The formulation has manifest p-q duality, a consequence of the
involutive nature of the Legendre transform and of its recently observed
relation with second-order linear differential equations. This reflects in an
intrinsic psi^D-psi duality between linearly independent solutions of the
Schroedinger equation. Unlike Bohm's theory, there is a non-trivial action even
for bound states. No use of any axiomatic interpretation of the wave-function
is made. Tunnelling is a direct consequence of the quantum potential which
differs from the usual one and plays the role of particle's self-energy. The
QSHJE is defined only if the ratio psi^D/psi is a local self-homeomorphism of
the extended real line. This is an important feature as the L^2 condition,
which in the usual formulation is a consequence of the axiomatic interpretation
of the wave-function, directly follows as a basic theorem which only uses the
geometrical gluing conditions of psi^D/psi at q=\pm\infty as implied by the EP.
As a result, the EP itself implies a dynamical equation that does not require
any further assumption and reproduces both tunnelling and energy quantization.
Several features of the formulation show how the Copenhagen interpretation
hides the underlying nature of QM. Finally, the non-stationary higher
dimensional quantum HJ equation and the relativistic extension are derived.Comment: 1+3+140 pages, LaTeX. Invariance of the wave-function under the
action of SL(2,R) subgroups acting on the reduced action explicitly reveals
that the wave-function describes only equivalence classes of Planck length
deterministic physics. New derivation of the Schwarzian derivative from the
cocycle condition. "Legendre brackets" introduced to further make "Legendre
duality" manifest. Introduction now contains examples and provides a short
pedagogical review. Clarifications, conclusions, ackn. and references adde
A Topos Foundation for Theories of Physics: I. Formal Languages for Physics
This paper is the first in a series whose goal is to develop a fundamentally
new way of constructing theories of physics. The motivation comes from a desire
to address certain deep issues that arise when contemplating quantum theories
of space and time. Our basic contention is that constructing a theory of
physics is equivalent to finding a representation in a topos of a certain
formal language that is attached to the system. Classical physics arises when
the topos is the category of sets. Other types of theory employ a different
topos. In this paper we discuss two different types of language that can be
attached to a system, S. The first is a propositional language, PL(S); the
second is a higher-order, typed language L(S). Both languages provide deductive
systems with an intuitionistic logic. The reason for introducing PL(S) is that,
as shown in paper II of the series, it is the easiest way of understanding, and
expanding on, the earlier work on topos theory and quantum physics. However,
the main thrust of our programme utilises the more powerful language L(S) and
its representation in an appropriate topos.Comment: 36 pages, no figure
Extended Representations of Observables and States for a Noncontextual Reinterpretation of QM
A crucial and problematical feature of quantum mechanics (QM) is
nonobjectivity of properties. The ESR model restores objectivity reinterpreting
quantum probabilities as conditional on detection and embodying the
mathematical formalism of QM into a broader noncontextual (hence local)
framework. We propose here an improved presentation of the ESR model containing
a more complete mathematical representation of the basic entities of the model.
We also extend the model to mixtures showing that the mathematical
representations of proper mixtures does not coincide with the mathematical
representation of mixtures provided by QM, while the representation of improper
mixtures does. This feature of the ESR model entails that some interpretative
problems raising in QM when dealing with mixtures are avoided. From an
empirical point of view the predictions of the ESR model depend on some
parameters which may be such that they are very close to the predictions of QM
in most cases. But the nonstandard representation of proper mixtures allows us
to propose the scheme of an experiment that could check whether the predictions
of QM or the predictions of the ESR model are correct.Comment: 17 pages, standard latex. Extensively revised versio
No extension of quantum theory can have improved predictive power
According to quantum theory, measurements generate random outcomes, in stark
contrast with classical mechanics. This raises the question of whether there
could exist an extension of the theory which removes this indeterminism, as
suspected by Einstein, Podolsky and Rosen (EPR). Although this has been shown
to be impossible, existing results do not imply that the current theory is
maximally informative. Here we ask the more general question of whether any
improved predictions can be achieved by any extension of quantum theory. Under
the assumption that measurements can be chosen freely, we answer this question
in the negative: no extension of quantum theory can give more information about
the outcomes of future measurements than quantum theory itself. Our result has
significance for the foundations of quantum mechanics, as well as applications
to tasks that exploit the inherent randomness in quantum theory, such as
quantum cryptography.Comment: 6 pages plus 7 of supplementary material, 3 figures. Title changed.
Added discussion on Bell's notion of locality. FAQ answered at
http://perimeterinstitute.ca/personal/rcolbeck/FAQ.htm
Quantifying the connectivity of a network: The network correlation function method
Networks are useful for describing systems of interacting objects, where the
nodes represent the objects and the edges represent the interactions between
them. The applications include chemical and metabolic systems, food webs as
well as social networks. Lately, it was found that many of these networks
display some common topological features, such as high clustering, small
average path length (small world networks) and a power-law degree distribution
(scale free networks). The topological features of a network are commonly
related to the network's functionality. However, the topology alone does not
account for the nature of the interactions in the network and their strength.
Here we introduce a method for evaluating the correlations between pairs of
nodes in the network. These correlations depend both on the topology and on the
functionality of the network. A network with high connectivity displays strong
correlations between its interacting nodes and thus features small-world
functionality. We quantify the correlations between all pairs of nodes in the
network, and express them as matrix elements in the correlation matrix. From
this information one can plot the correlation function for the network and to
extract the correlation length. The connectivity of a network is then defined
as the ratio between this correlation length and the average path length of the
network. Using this method we distinguish between a topological small world and
a functional small world, where the latter is characterized by long range
correlations and high connectivity. Clearly, networks which share the same
topology, may have different connectivities, based on the nature and strength
of their interactions. The method is demonstrated on metabolic networks, but
can be readily generalized to other types of networks.Comment: 10 figure
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