Geometry Of Hilbert Space Research Articles

Statistical geometry in quantum mechanics

A statistical model M is a family of probability distributions, characterised by a set of continuous parameters known as the parameter space. This possesses natural geometrical properties induced by the embedding of the family of probability distributions into the Hilbert space H. By consideration of the square-root density function we can regard M as a submanifold of the unit sphere in H. Therefore, H embodies the `state space' of the probability distributions, and the geometry of M can be described in terms of the embedding of in H. The geometry in question is characterised by a natural Riemannian metric (the Fisher-Rao metric), thus allowing us to formulate the principles of classical statistical inference in a natural geometric setting. In particular, we focus attention on the variance lower bounds for statistical estimation, and establish generalisations of the classical Cramer-Rao and Bhattacharyya inequalities. The statistical model M is then specialised to the case of a submanifold of the state space of a quantum mechanical system. This is pursued by introducing a compatible complex structure on the underlying real Hilbert space, which allows the operations of ordinary quantum mechanics to be reinterpreted in the language of real Hilbert space geometry. The application of generalised variance bounds in the case of quantum statistical estimation leads to a set of higher order corrections to the Heisenberg uncertainty relations for canonically conjugate observables.

On the Identifiability of Minimal VARMA Representations

By studying the geometry of relevant Hilbert spaces, we give a characterization of the identifiable standard representations of multivariate ARMA models in terms of the autocovariance function.

Complete interpolating sequences for Paley-Wiener spaces and Muckenhoupt's ($A_p$) condition

We describe the complete interpolating sequences for the Paley-Wiener spaces L^p_\pi (1 < p < \infty) in terms of Muckenhoupt's ( A_p ) condition. For p=2 , this description coincides with those given by Pavlov [9], Nikol'skii [8], and Minkin [7] of the unconditional bases of complex exponentials in L^2 (– \pi , \pi) . While the techniques of these authors are linked to the Hilbert space geometry of L^2_\pi , our method of proof is based on turning the problem into one about boundedness of the Hilbert transform in certain weighted L^p spaces of functions and sequences.

Geometry of Quantum Statistical Inference.

An efficient geometric formulation of the problem of parameter estimation is developed, based on Hilbert space geometry. This theory, which allows for a transparent transition between classical and quantum statistical inference, is then applied to the analysis of exponential families of distributions (of relevance to statistical mechanics) and quantum mechanical evolutions. The extension to quantum theory is achieved by the introduction of a complex structure on the given real Hilbert space. We find a set of higher order corrections to the parameter estimation variance lower bound, which are potentially important in quantum mechanics, where these corrections appear as modifications to Heisenberg uncertainty relations for the determination of the parameter.

Comment on 'Velocity autocorrelation function in fluctuating hydrodynamics'

For original work see ibid., vol.4, p.L303, 1992. Tsekov and Radoev recently studied the velocity autocorrelation function in a hydrodynamic model using the generalized Langevin equation formalism originally due to Mori and Zwanzig. The more recent recurrence relations formulation of the generalized Langevin equation provides a simplification. It also lends a perspective on the model of Tsekov and Radoev and points out the model's limitations. A refinement of the model is possible by consideration of the Hilbert space geometry as developed by the recurrence relations formulation.

Geometry of projective Hilbert space.

Geodesic equations determined by the metric in a projective Hilbert space openP are found. It is shown that connections in the tangent bundle over openP consistent with these geodesics can be introduced. A connection with zero torsion is proposed. Quantum-system dynamics follows the geodesics of another connection which is found for the case of the constant Hamiltonian. A set of geometrical quantities is defined, conserved by a quantum evolution.

The L problem of moments in two dimensions

The classical solution to the L problem of moments due to N. I. Akhiezer and M. G. Krein is partially generalized to the two-dimensional case. A parallel between the apparently quite different one-dimensional and two-dimensional L problems of moments is established by means of the theories of the phase shift and of the principal function. This point of view brings into the field the geometry of Hilbert spaces and the analysis of the pairs of not necessarily commuting self-adjoint operators acting on them.

The Philosophy of Quantum Mechanics

The trouble with the philosophy of quantum mechanics, I'm told, is that as soon as you've found a position, you lose your momentum.' Three very different books by philosophers on quantum mechanics have recently appeared in print. Peter Gibbin's Particles and Paradoxes is a splendid introduction to current problems in the field, suitable as an introductory text. Subtitled 'The Limits of Quantum Logic', it concludes with a lengthy essay on what quantum logic can and can't do. Michael Redhead's book, Incompleteness, Nonlocality and Realism: A Prolegomenon to the Philosophy of Quantum Mechanics, covers much the same ground at an advanced level. Here, too, the aim is to acquaint the reader with the important new work that has emerged over the past twenty years or so, following the seminal results of Bell and of Kochen and Specker in the sixties. The book reflects the considerable contribution by Redhead himself to the debate on the foundations of quantum mechanics, and I would regard it as the definitive study of what we have learnt about interpretative positions that no longer appear viable. The Metaphysics of Quantum Theory by Henry Krips has more ambitious aspirations, and claims to provide a new realist interpretation of quantum mechanics that resolves the measurement problem and other puzzles and paradoxes of the theory. I must confess to a certain feeling of awe for anyone who can juggle seventy one 'principles' (usefully collected for the reader at the end of the book-count theml) but, mindful of the uncertainty principle that applies here, I am rather less sanguine about the merits of the interpretation. It is not hard to see why quantum mechanics is conceptually puzzling. The theory incorporates an algorithm (formulated in terms of the geometry of Hilbert space) for assigning probabilities to events, or propositions, or ranges of values of physical magnitudes (the dynamical variables or 'observables' of the theory). This algorithm resists the representation of the probabilities generated by the statistical states (quantum states) as measures over what might be termed 'property states': assignments of values to the physical magnitudes, or 'lists' of properties, or assignments of truth values to the propositions of the system. Formally, such a property state might be represented by an ultrafilter in an

Complementation in Krein Spaces

A generalization of the concept of orthogonal complement is introduced in complete and decomposable complex vector spaces with scalar product. Complementation is a construction in the geometry of Hilbert space which was applied to the invariant subspace theory of contractive transformations in Hilbert space by James Rovnyak and the author [6]. The concept was later formalized by the author [3]. Continuous and contractive transformations in Krein spaces appear in the estimation theory of Riemann mapping functions [4]. It is therefore of interest to know whether a generalization of complementation theory applies in Krein spaces. Such a generalization is now obtained. The results are also of interest in the invariant subspace theory of continuous and contractive transformations in Krein spaces [5]. The vector spaces considered are taken over the complex numbers. A scalar product for a vector space )1 is a complex-valued function (a, b) of a and b in )1 which is linear, symmetric, and nondegenerate. Linearity means that the identity (aa + 3b, c) = a(a, c) + 3(b, c) holds for all elements a, b, and c of )1 when a and , are complex numbers. Symmetry means that the identity (b, a) = (a, b) holds for all elements a and b of )1. Nondegeneracy means that an element a of )1 is zero if the scalar product (a, b) is zero for every element b of )1. Every element b of )1 determines a linear functional bon )1 which is defined by b-a = (a, b) for every element a of )1. The weak topology of )1 is the weakest topology with respect to which bis a continuous linear functional on )1 for every element b of )1. The weak topology of )1 is a locally convex topology having the property that every continuous linear functional on )1 is of the form bfor an element b of )1. The element b is then unique. The antispace of a vector space with scalar product is the same vector space considered with the negative of the given scalar product. A fundamental example of a vector space with scalar product is a Hilbert space. A Krein space is a vector space with scalar product which is the orthogonal sum of a Hilbert space and the antispace of a Hilbert space. Received by the editors October 10, 1986 and, in revised form, February 23, 1987. The results of the paper were presented to the Department of Mathematics, Indiana and Purdue University in Indianapolis, on March 27, 1987, as the Ernest J. Johnson Colloquium. 1980 Mathematics Subject Classification (1985 Revision). Primary 46D05. Research supported by the National Science Foundation. ?1988 American Mathematical Society 0002-9947/88 $1.00 + $.25 per page

Maximal angles of purely resistive systems

The geometry of Hilbert space suggests the natural concept of an angle associated with every performance of any Hilbert input-output system. A physical interpretation of this angle is suggested. In particular, it is shown (theorem 1) that the purely resistive n-ports are "sectorial operators." Every resistive n-port has a maximal angle which cannot be excelled by any forced regime.

Mathematical model of the spectral decomposition of periodic and non-periodic geophysical stationary random signals

A model is discussed for the spectral decomposition of stationary periodic and non-periodic functions, suitable for studying various geophysical signals. The properties of the individual components of the random spectrum of the periodic stationary random function was shown in the geometry of Hilbert space of random quantities, and these were used to construct the periodic continuation of the random stationary function, given in a finite interval.

Stationary and Related Stochastic Processes: Sample Function Properties and Their Applications.

Book on stationary and related stochastic processes covering sample function properties and applications, Hilbert space geometry, etc

Stochastic Processes As Curves in Hilbert Space

Regular complex-valued random processes $x(t)$ with finite moments of second order are studied by methods of Hilbert space geometry. A representation formula (4) is given for the process $x(t)$ in terms of “past and present innovations”. The number N is called the complete spectral multiplicity of the process $x(t)$ and is the smallest number for which such a representation exists. It is shown that the multiplicity of $x(t)$ is uniquely determined by the corresponding correlation function and that one can always find a harmonizing process $x(t)$ which has the multiplicity prescribed in advance.