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.     

Representations of canonical anticommutation relations and implementability of canonical transformations

It is proved that irreducible representations of CAR are determined by the groups of implementable automorphisms of the correspondingC*-algebra. This is done by a study of implementable canonical transformations. Some results in the same directions for factor representations are given.     

A Markov chain approach to renormalization group transformations

We aim at an explicit characterization of the renormalized Hamiltonian after decimation transformation of a one-dimensional Ising-type Hamiltonian with a nearest-neighbor interaction and a magnetic field term. To facilitate a deeper understanding of the decimation effect, we translate the renormalization flow on the Ising Hamiltonian into a flow on the associated Markov chains through the Markov–Gibbs equivalence. Two different methods are used to verify the well-known conjecture that the eigenvalues of the linearization of this renormalization transformation about the fixed point bear important information about all six of the critical exponents. This illustrates the universality property of the renormalization group map in this case.     

A state space force identification method based on Markov parameters precise computation and regularization technique

A time domain force identification approach for linear system is proposed. This approach can found a highly precise force identification model within the scope of general computer precision while it does not cost much computing time. Although the force identification model is accurate, the force identification process, like other inverse methods, is still ill-posed due to the inversion process and the white noise in measured structural responses. The singular value decomposition is used to reveal the intrinsically matter of the ill-posedness of force identification problem and a regularization technique is utilized to deal with this issue. Finally, the proposed method with the aid of regularization technique is successfully applied to identify the input forces in two numerical simulations.     

Covariant coordinate transformations on noncommutative space

We show how to define gauge-covariant coordinate transformations on a noncommuting space. The construction uses the Seiberg-Witten equation and generalizes similar results for commuting coordinates.     

Markov state models and molecular alchemy

In recent years, Markov state models (MSMs) have attracted a considerable amount of attention with regard to modelling conformation changes and associated function of biomolecular systems. They have been used successfully, e.g. for peptides including time-resolved spectroscopic experiments, protein function and protein folding , DNA and RNA, and ligand-receptor interaction in drug design and more complicated multivalent scenarios. In this article, a novel reweighting scheme is introduced that allows to construct an MSM for certain molecular system out of an MSM for a similar system. This permits studying how molecular properties on long timescales differ between similar molecular systems without performing full molecular dynamics simulations for each system under consideration. The performance of the reweighting scheme is illustrated for simple test cases, including one where the main wells of the respective energy landscapes are located differently and an alchemical transformation of butane to pentane where the dimension of the state space is changed.     

Quantum state transformations and the Schubert calculus

Recent developments in mathematics have provided powerful tools for comparing the eigenvalues of matrices related to each other via a moment map. In this paper, we survey some of the more concrete aspects of the approach with a particular focus on applications to quantum information theory. After discussing the connection between Horn’s Problem and Nielsen’s Theorem, we move on to characterizing the eigenvalues of the partial trace of a matrix.     

Structural-phase transformations in low-stability condensed state systems

The conditions of formation of low-stability condensed state systems, their behavior and structure are investigated. The objects under study are alloys and compounds undergoing structural-phase transitions of the second type or close to it. The low-stability (pre-transitional) state is treated here as the state of a system near its structural-phase transformations, in which its structure and properties exhibit anomalies. An attempt is made to interpret the system from the physical standpoint relying on a new insight into its state, in which the traditionally accepted phase-transition point is represented by a range of values of the parameter controlling the transition. The material state within this range of values is structurally weakly stable in terms of slight variations in the controlling parameter. It is shown that the thermodynamics of structural-phase transformations of the material in this transient state is significantly affected by the interaction of structure defects.     

The geometry of state space

The geometry of the state space of a finite-dimensional quantum mechanical system, with particular reference to four dimensions, is studied. Many novel features, not evident in the two-dimensional space of a single spin, are found. Although the state space is a convex set, it is not a ball, and its boundary contains mixed states in addition to the pure states, which form a low-dimensional submanifold. The appropriate language to describe the role of the observer is that of flag manifolds.     

Ribaucour transformations for isometric immersions of pseudo-Riemannian space forms

In this paper, we give the explicit construction for isometric immersions of pseudo-Riemannian space forms via loop group actions. As a consequence, we obtain a Ribaucour transformation and its permutability theorem for isometric immersions of pseudo-Riemannian space forms, which is a generalization of the permutability theorem for surfaces in R³$R^3$, and a family of new isometric immersions of the pseudo-Riemannian space forms from a given one via a purely algebraic algorithm.     

Differential calculus and gauge transformations on a deformed space

We consider a formalism by which gauge theories can be constructed on noncommutative space time structures. The coordinates are supposed to form an algebra, restricted by certain requirements that allow us to realise the algebra in terms of star products. In this formulation it is useful to define derivatives and to extend the algebra of coordinates by these derivatives. The elements of this extended algebra are deformed differential operators. We then show that there is a morphism between these deformed differential operators and the usual higher order differential operators acting on functions of commuting coordinates. In this way we obtain deformed gauge transformations and a deformed version of the algebra of diffeomorphisms. The deformation of these algebras can be clearly seen in the category of Hopf algebras. The comultiplication will be twisted. These twisted algebras can be realised on noncommutative spaces and allow the construction of deformed gauge theories and deformed gravity theory. Dedicated to the 60th birthday of Prof. Obregon.     

Representations of coherent and squeezed states in an extended two-parameter Fock space

Recently an f-deformed Fock space which is spanned by |n〉_λ was introduced. These bases are the eigenstates of a deformed non-Hermitian Hamiltonian. In this contribution, we will use rather new non-orthogonal basis vectors for the construction of coherent and squeezed states, which in special case lead to the earlier known states. For this purpose, we first generalize the previously introduced Fock space spanned by |n〉_λ bases, to a new one, spanned by extended two-parameters bases |n〉λ₁,λ₂. These bases are now the eigenstates of a non-Hermitian Hamiltonian H_λ1,_λ2=a_λ1,_λ2⁺a+(1/2), where a_λ1,_λ2⁺=a⁺+ λ₁a + λ₂ and a are, respectively, the deformed creation and ordinary bosonic annihilation operators. The bases |n〉λ₁,λ₂ are non-orthogonal (squeezed states), but normalizable. Then, we deduce the new representations of coherent and squeezed states in our two-parameter Fock space. Finally, we discuss the quantum statistical properties, as well as the non-classical properties of the obtained states numerically.     

Gauge generators,transformations and identities on a noncommutative space

By abstracting a connection between gauge symmetry and gauge identity on a noncommutative space, we analyse star (deformed) gauge transformations with the usual Leibniz rule as well as undeformed gauge transformations with a twisted Leibniz rule. Explicit structures of the gauge generators in either case are computed. It is shown that, in the former case, the relation mapping the generator with the gauge identity is a star deformation of the commutative space result. In the latter case, on the other hand, this relation gets twisted to yield the desired map.     

The pure state space of quantum mechanics as Hermitian symmetric space

The pure state space of Quantum Mechanics is investigated as Hermitian Symmetric Kähler manifold. The classical principles of quantum mechanics (Quantum Superposition Principle, Heisenberg Uncertainty Principle, Quantum Probability Principle) and Spectral Theory of observables are discussed in this non-linear geometrical context.     

The physical state space of quantum electrodynamics

Starting from the fact that electrically charged particles are massive, we derive a criterion which characterizes the state space of quantum electrodynamics. This criterion clarifies the special role of the electric charge amongst the uncountably many superselection rules in quantum electrodynamics and provides a basis for a general analysis of the infrared problem. Within this framework we establish the existence of asymptotic electromagnetic fields in all charge-sectors, find a general characterization of infra-particles and introduce a notion of asymptotic completeness.     

Non-homogeneous Polygonal Markov Fields in the Plane: Graphical Representations and Geometry of Higher Order Correlations

We consider polygonal Markov fields originally introduced by Arak and Surgailis (Probab. Theory Relat. Fields 80:543–579, 1989). Our attention is focused on fields with nodes of order two, which can be regarded as continuum ensembles of non-intersecting contours in the plane, sharing a number of features with the two-dimensional Ising model. We introduce non-homogeneous version of polygonal fields in anisotropic environment. For these fields we provide a class of new graphical constructions and random dynamics. These include a generalized dynamic representation, generalized and defective disagreement loop dynamics as well as a generalized contour birth and death dynamics. Next, we use these constructions as tools to obtain new exact results on the geometry of higher order correlations of polygonal Markov fields in their consistent regime. Research supported by the Polish Minister of Science and Higher Education grant N N201 385234 (2008-2010).     

Representations of MV-Algebras by Hilbert-Space Effects

It is shown that for every Archimedean MV-effect algebra M (equivalently, every Archimedean MV-algebra) there is an injective MV-algebra morphism into the MV-algebra of all multiplication operators between the zero and identity operator on $\ell_{2}(\mathcal{S}_{0})$ , where $\mathcal{S}_{0}$ is an ordering set of extremal states (state morphisms) on M.