Nonlinear transformations in Hilbert space

Vitalism, from a physicist's standpoint, suggests the introduction of nonlinear transformations in Hilbert space. Two such transformations are introduced and studied in some detail. They are hard to detect by conventional experiments, although they may be very important for living organisms. They can, however, give rise to nonlocal effects, and thus provide a possible physical basis for some parapsychological phenomena, in particular precognition.A recent history in which these two views are contrasted has been written by Smith.     

Embedding quantum logics in Hilbert space

We show that an orthomodular lattice is embeddable in a Hilbert space if and only if states of a certain kind exist. A physical motivation for the existence of such states is given and a connection is provided between the quantum logic, algebraic, and operational approaches to quantum mechanics.     

Affine connection structure of the charged symplectic 2-form

It is shown that the charged symplectic form in Hamiltonian dynamics of classical charged particles in electromagnetic fields defines a generalized affine connection on an affine frame bundle associated with spacetime. Conversely, a generalized affine connection can be used to construct a symplectic 2-form if the associated linear connection is torsion-free and the antisymmetric part of theR ^4* translational connection is locally derivable from a potential. Hamiltonian dynamics for classical charged particles in combined gravitational and electromagnetic fields can therefore be reformulated as aP(4)=O(1, 3)R ^4* geometric theory with phase space the affine cotangent bundleAT ^* M of spacetime. The sourcefree Maxwell equations are reformulated as a pair of geometrical conditions on the ^4* curvature that are exactly analogous to the source-free Einstein equations.     

Nonadiabatic geometric phase in quaternionic Hilbert space

We develop the theory of the nonadiabatic geometric phase, in both the Abelian and non-Abelian cases, in quaternionic Hilbert space.     

Bath-induced correlations in an infinite-dimensional Hilbert space

Quantum correlations between two free spinless dissipative distinguishable particles (interacting with a thermal bath) are studied analytically using the quantum master equation and tools of quantum information. Bath-induced coherence and correlations in an infinite-dimensional Hilbert space are shown. We show that for temperature T> 0 the time-evolution of the reduced density matrix cannot be written as the direct product of two independent particles. We have found a time-scale that characterizes the time when the bath-induced coherence is maximum before being wiped out by dissipation (purity, relative entropy, spatial dispersion, and mirror correlations are studied). The Wigner function associated to the Wannier lattice (where the dissipative quantum walks move) is studied as an indirect measure of the induced correlations among particles. We have supported the quantum character of the correlations by analyzing the geometric quantum discord.     

Hilbert space factorization and partial measurements

A quantum system whose state vector belongs to a finite-dimensional Hilbert space is considered. If this space has a dimension that is a composite number, one can factor the space into a tensor product of sub-spaces. An observable that acts only in one of these subspaces is called a partial measurement. Some of the properties and the interpretation of such partial measurements are discussed.     

Hilbert space representations of Lie algebras

We describe a new approach to the general theory of unitary representations of Lie groups which makes use of the Gelfand-Segal construction directly on the universal enveloping algebra of any Lie algebra. The crucial observation is that Nelson's theory of analytic vectors allows the characterisation of certain states on the universal enveloping algebra such that the corresponding representations of the universal enveloping algebra are the infinitesimal part of unitary representations of the associated simply connected Lie group. In the first section of the paper we show that with the aid of Choquet's theory of representing measures one can derive a simple new approach to integral decomposition theory along these lines.In the second section of the paper we use these methods to study the irreducible unitary representations of general semi-simple Lie groups. We give a simple proof that theK-finite vectors studied by Harish-Chandra [5] are all analytic vectors. We also give new proofs of some of Godement's results [2] characterising spherical functions of height one, at least for unitary representations. Compared with [2] our method has the possible advantage of obtaining the characterisations by infinitesimal methods instead of using an indirect argument involving functions on the group. We point out that while being purely algebraic in nature, this approach makes almost no use of the deep and difficult theorems of Harish-Chandra concerning the universal enveloping algebra [5].Our work is done in very much the same spirit as that of Power's recent paper [8]. The main difference is that by concentrating on a more special class of positive states we are able to carry the analysis very much further without difficulty.     

Optimal entropic uncertainty relation in two-dimensional Hilbert space

The exact lower bound on the sum of the information entropies is obtained for arbitrary pairs of observables in two-dimensional Hilbert space. The result coincides with that given by Garrett and Gull for the particular case of real transformation matrices and state vectors. A weaker analytical bound is also obtained.     

Minimal supports in quantum logics and Hilbert space

It is shown that if a fully atomic, complete orthomodular lattice satisfies a minimal support condition (m.s.c.), then it satisfies Piron's axioms, and is thereby shown to be the projection lattice of a generalized Hilbert space. It is shown, conversely, that m.s.c. holds in Hilbert space subspace lattices. The physical justification for m.s.c. is provided in the context of a property lattice (A, ) for a realistic entity (A, ) in the sense of Foulis-Piron-Randall. This context provides a clear focus on key issues in the debate over the appropriateness of requiring quantum logics to be represented over Hilbert spaces.     

有限维希尔伯特空间中的非线性相干态

在有限维希尔伯特空间中构造了非线性相干态,并研究了该量子态的压缩效应和反聚束效应,给出了压缩效应和反聚束效应与有限维希尔伯特空间维数关系的数值计算结果.     

Relativistic resonances as non-orthogonal states in Hilbert space

We analyze the energy-momentum properties of relativistic short-lived particles with the result that they are characterized by two 4-vectors: in addition to the familiar energy-momentum vector (timelike) there is an energy-momentum `spread vector' (spacelike). The wave functions in space and time for unstable particles are constructed. For the relativistic properties of unstable states we refer to Wigner's method of Poincaré group representations that are induced by representations of the space-time translation and rotation groups. If stable particles, unstable particles and resonances are treated as elementary objects that are not fundamentally different one has to take into account that they will not generally be orthogonal to each other in their state space. The scalar product between a stable and an unstable state with otherwise identical properties is calculated in a particular Lorentz frame. The spin of an unstable particle is not infinitely sharp but has a `spin spread' giving rise to `spin neighbors'. This opens the possibility of a non-zero scalar product between states with unequal spin. - A first practical application of non-orthogonal states is seen in diffraction dissociation reactions whose large cross-sections are attributed to interference of states that are `partially identical'. Received: 28 June 2002 / Revised version: 13 November 2002 / Published online: 14 March 2003 RID="a" ID="a" e-mail: walter.blum@cern.ch RID="b" ID="b" e-mail: hns@mppmu.mpg.de     

Preserving some numerical correspondencesbetween Hilbert space effects

In this paper we introduce some natural numerical quantities which measure differentkinds of correspondences between Hilbert space effects. We show that every objective map of the space ?(H) of all effects on the Hilbert space H which preserves any of these quantities is implemented by a unitary or antiunitary operator on H.     

Tensor products of Hilbert space effect algebras

A definition of a tensor product in the category of Hilbert space effect algebras is introduced such that the tensor product reflects as much as possible of the physically important properties of the components. It is shown that in the complex case, there are two candidates to the tensor product, which are not equivalent. The situation is similar to the tensor product in the category of projection lattices, but also the two dimensional case is included. The case of tensor product of two classical systems is also investigated.     

Measures on Hilbert space and quantum states

A representation of quantum mechanics of one particle by means of integration in Hilbert space H is embedded into a multi-particle theory on boson Fock space F₊. We prove that the grand canonical statistical operator on F₊ can be represented by means of a gaussian measure on the one-particle Hilbert space H.     

Plane frame functions and pure states in Hilbert space

An elementary proof of Gleason's theorem for pure states in a Hilbert space is given. This theorem follows from a general result concerning plane frame functions in real inner product spaces.