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arXiv:quant-ph/0311110

Geometroneurodynamics

Sisir Roy, Menas Kafatos

Published Nov 17, 2003• 1 citation
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Abstract

Hilbert space structure is assumed as a valid geometric description for neurodynamics, i.e., for applying any kind of quantum formalism in brain dynamics. The orientation selectivity of the neurons is used as a justification to construct a type of statistical distance function which is proportional to the usual distance (or angle) between orientations of the neurons. The equivalence between the statistical distance and the Hilbert-space distance is discussed within this framework. It gives rise to the possibility of reanalysing the issue of measurement and information processing in the brain function

DOI
10.48550/arxiv.quant-ph/0311110

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Abstract. The recent controversy of applicability of quantum formalism to brain dynamics has been critically analyzed. PELLIONISZ and LLINÁS (1982) proposed a functional geometry to understand the internal representation of the events associated to the space-timing of moving objects in the external world. The joint representation of space and time associated to an event as understood by the brain is shown to be different from that understood in modern physics. This indicates that the four-dimensional geometry i.e. Minkowski geometry is not an appropriate description for the internal world. If it is to be the case, the applicability of any kind of quantum field theory in modeling brain function has to be analyzed with great care. Here, the issue of applicability of quantum mechanics to brain function has been discussed in general, from an anatomical perspective and then particular emphasis has been given to the applicability of quantum field theory. 1.
Abstract Our purpose here is to draw attention to an interesting relationship that can be shown to hold between (a) information geometry, and (b) quantum geometry. By ‘information geometry’, we mean the natural geometry associated with families of probability distributions. This is a subject that goes back more than half a century to the pioneering work of Rao (1945), who showed that the Fisher information matrix associated with a parametrised family of probability dis tributions gives rise to a natural Riemannian metric on the parameter space. This then allows one to speak in a precise way of the ‘distance’ between two probability distributions, an idea that turns out to be useful in problems of statistical inference. An extensive literature has thus developed, following this line of enquiry, on applications of differential geometry to statistics (see, e.g., Amari (1985), Barndorff-Nielsen et al. (1986), Murray and Rice (1993)). By ‘quantum geometry’, on the other hand, we mean the geometry of the manifold of states associated with a given quantum mechanical system. In particular, we consider the manifold of ‘pure’ states associated with a given complex Hilbert space. This is the space of ‘rays’ through the origin in the Hilbert space, which has the structure of a complex projective space. It is known that the usual rules of quantum mechanics allow one to construct a natural metric on this space, namely the Fubini-Study metric, the specification of which is equivalent to all the familiar structure associated with ordinary quantum mechanics.
A concept of "statistical distance" is defined between different preparations of the same quantum system, or in other words, between different rays in the same Hilbert space. Statistical distance is determined entirely by the size of statistical fluctuations occurring in measurements designed to distinguish one state from another. It is not related, a priori, to the usual distance (or angle) between rays. One finds, however, that these two kinds of distance are in fact the same, a result which depends on certain peculiarities of quantum mechanics.
Abstract. In this paper we present a survey of the use of differential geometric formalisms to describe Quantum Mechanics. We analyze Schrödinger framework from this perspective and provide a description of the Weyl-Wigner construction. Finally, after reviewing the basics of the geometric formulation of quantum mechanics, we apply the methods presented to the most interesting cases of finite dimensional Hilbert spaces: those of two, three and four level systems (one qubit, one qutrit and two qubit systems). As a more practical application, we discuss the advantages that the geometric formulation of quantum mechanics can provide us with in the study of situations as the functional independence of entanglement witnesses. 1.