# E. C. G. Sudarshan's research while affiliated with University of Texas at Austin and other places

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## Publications (401)

We elaborate on the notion of generalized tomograms, both in the classical
and quantum domains. We construct a scheme of star-products of thick
tomographic symbols and obtain in explicit form the kernels of classical and
quantum generalized tomograms. Some of the new tomograms may have interesting
applications in quantum optical tomography.

We construct a relativistic Hamiltonian model for Higgs resonance. Resonance trajectories are studied as a function of the parameters of the model. The possibility of a narrow width Higgs resonance in 1 TeV region cannot be ruled out from general considerations. Our model may serve as a phenomenological guide in the search for Higgs resonance.

Some non-linear generalizations of classical Radon tomography were recently
introduced by M. Asorey et al [Phys. Rev. A 77, 042115 (2008), where the
straight lines of the standard Radon map are replaced by quadratic curves
(ellipses, hyperbolas, circles) or quadratic surfaces (ellipsoids,
hyperboloids, spheres). We consider here the quantum version...

The notion of f-oscillators generalizing q-oscillators is discussed. For the classical and quantum cases, an interpretation
of the f-oscillator is provided as corresponding to a special nonlinearity of vibration for which the frequency of the oscillation
depends on the energy. The f-coherent states generalizing the q-coherent states are constructed...

We construct a non-Markovian dynamical map that accounts for systems correlated to the environment. We refer to it as a canonical dynamical map, which forms an evolution family. The relationship between inverse maps and correlations with the environment is established. The mathematical properties of complete positivity is related to classical corre...

After a general description of the tomographic picture for classical systems,
a tomographic description of free classical scalar fields is proposed both in a
finite cavity and the continuum. The tomographic description is constructed in
analogy with the classical tomographic picture of an ensemble of harmonic
oscillators. The tomograms of a number...

Metastable states are associated with discrete complex eigenvalues of the Hamiltonian in the analytically continued spaces of quantum mechanics formulated in dual spaces. The necessary modifications when two resonances approach each other and coincide are discussed and illustrated in some exactly soluble models. The Hamiltonian no longer can be dia...

Two classical models for particles with internal structure and which describe Regge trajectories are developed. The remarkable geometric and other properties of the two internal spaces are highlighted. It is shown that the conditions of positive time-like four-velocity and energy momentum for the classical system imply strong and physically reasona...

It is shown that for quantum systems the vector field associated with the equations of motion may admit alternative Hamiltonian descriptions, both in the Schrödinger and Heisenberg picture. We illustrate these ambiguities in terms of simple examples.

We analyze the dynamics of coupled classical and quantum systems. The main idea is to treat both systems as true quantum ones and impose a family of superselection rules which imply that the corresponding algebra of observables of one subsystem is commutative and hence may be treated as a classical one. Equivalently, one may impose a special symmet...

The tomographic picture of quantum mechanics has brought the description of quantum states closer to that of classical probability and statistics. On the other hand, the geometrical formulation of quantum mechanics introduces a metric tensor and a symplectic tensor (Hermitian tensor) on the space of pure states. By putting these two aspects togethe...

We scrutinize the effects of non-ideal data acquisition on the tomograms of quantum states. The presence of a weight function, schematizing the effects of a finite window or equivalently noise, only affects the state reconstruction procedure by a normalization constant. The results are extended to a discrete mesh and show that quantum tomography is...

Schrödinger's equation is formulated in terms of a vector space and the time evolution is given by a unitary transformation. This assumes the time invariance of the norm. When we have states corresponding to unstable particles, the possibility of introducing complex energies is resorted to, but this is not either time translation or time inversion...

The relations between dynamical maps and quantum states of bipartite systems are analyzed from the perspective of quantum conditional probability. In particular, we explore new interesting relations between completely positive maps, which correspond to quantum channels, and states of bipartite systems which correspond to correlations between the in...

These two days of discussions on physics with special attention to my scientific quests gives me great gratification. So many of my friends who have distinguished themselves as physicists have put their minds to present such excellent expositions. The contributions that I have made over the last half century, when handled by these experts, show the...

We propose to substitute Newton’s constant G
N for another constant G
2, as if the gravitational force would fall off with the 1/r law, instead of the 1/r
2; so we describe a system of natural units with G
2, c and ℏ. We adjust the value of G
2 so that the fundamental length L = L
Pl is still the Planck’s length and so G
N = L × G
2. We argue for t...

Quantum channels can be mathematically represented as completely positive trace-preserving maps that act on a density matrix. A general quantum channel can be written as a convex sum of `extremal' channels. We show that for an $N$-level system, the extremal channel can be characterized in terms of $N^2$-$N$ real parameters coupled with rotations. W...

In a recent letter one of us pointed out how differences in preparation procedures for quantum experiments can lead to non-trivial differences in the results of the experiment. The difference arise from the initial correlations between the system and environment. Therefore, any quantum experiment that is prone to the influences from the environment...

The quasidistributions corresponding to the diagonal representation of
quantum states are discussed within the framework of operator-symbol
construction. The tomographic-probability distribution describing the quantum
state in the probability representation of quantum mechanics is reviewed. The
connection of the diagonal and probability representat...

The quasidistributions corresponding to the diagonal representation of quantum states are discussed within the framework of operator-symbol construction. The tomographic-probability distribution describing the quantum state in the probability representation of quantum mechanics is reviewed. The connection between the diagonal and probability repres...

We analyze the structure of the subset of states generated by unital completely positive quantum maps, A witness that certifies that a state does not belong to the subset generated by a given map is constructed. We analyse the representations of positive maps and their relation to quantum Perron-Frobenius theory.

We expand the set of initial states of a system and its environment that are known to guarantee completely positive reduced dynamics for the system when the combined state evolves unitarily. We characterize the correlations in the initial state in terms of its quantum discord [1]. We prove that initial states that have only classical correlations l...

We construct a non-Markovian canonical dynamical map that accounts for systems correlated with the environment. The physical meaning of not completely positive maps is studied to obtain a theory of non-Markovian quantum dynamics. The relationship between inverse maps and correlations with the environment is established. A generalized non-Markovian...

We introduce several possible generalizations of tomography for quadratic surfaces. We analyze different types of elliptic, hyperbolic and hybrid tomograms. In all cases it is possible to consistently define the inverse tomographic map. We find two different ways of introducing tomographic sections. The first method operates by deformations of the...

Using general construction of star-product the q-deformed Wigner–Weyl–Moyal quantization procedure is elaborated. The q-deformed Groenewold kernel determining the product of quantum observables is given in explicit form for small nonlinearities corresponding to nonlinear vibrations of classical and quantum q-oscillators. The deformation of Groenewo...

A Markov approximation in open quantum dynamics can give unphysical results when a map acts on a state that is not in its domain. This is examined here in a simple example, an open quantum dynamics for one qubit in a system of two interacting qubits, for which the map domains have been described quite completely. A time interval is split into two p...

I review my scientific research career for the last 50 years, with emphasis on the
issue of 'Poincaré recurrences': I stress some ideas of mine which became so popular that they
have been taken up (recurred) by others, sometimes forgetting the original source.

We discuss the geometry of states of quantum systems in an n-dimensional Hilbert space in terms of an explicit parameterization of all such systems. The geometry of the space of pure as well as mixed states for n-state systems is discussed. The parameterization is particularly useful since it allows for the simple construction and isolation of vari...

We investigate the spin-statistics connection in arbitrary dimensions for hermitian spinor or tensor quantum fields with a
rotationally invariant bilinear Lagrangian density. We use essentially the same simple method as for space dimension D=3. We find the usual connection (tensors as bosons and spinors as fermions) for D=8n+3,8n+4,8n+5, but only b...

The positive and not completely positive maps of density matrices, which are contractive maps, are discussed as elements of a semigroup. A new kind of positive map (the purification map), which is nonlinear map, is introduced. The density matrices are considered as vectors, linear maps among matrices are represented by superoperators given in the f...

We study the effects of preparation of input states in a quantum tomography experiment. We show that maps arising from a quantum process tomography experiment (called process maps) differ from the well know dynamical maps. The difference between the two is due to the preparation procedure that is necessary for any quantum experiment. We study two p...

Simple examples are constructed that show the entanglement of two qubits being both increased and decreased by interactions on just one of them. One of the two qubits interacts with a third qubit, a control, that is never entangled or correlated with either of the two entangled qubits and is never entangled, but becomes correlated, with the system...

Phenomenological treatments of unstable states in quantum theory have been known for six decades and have been extended to more complex phenomena. But the twin requirement of causality ruling out a physical state with complex energy and the apparent decay of unstable states necessitates generalizing quantum mechanics beyond the standard Dirac formu...

The stochastic evolution of a quantum system can be expressed by a dynamical map that acts as a superoperator on a density matrix. If all eigenvalues of this map are positive, the map is said to be completely positive. If the dynamical map comes from the reduced unitary evolution of a bipartite system, the map depends on the correlations, and can h...

We make the connection between initial quantum correlations and not completely positive maps. Though this has been suggested in literature for some time now, our arguments are supported by explicit calculations. In the process we will discuss our work in relation with quantum process tomography. We are especially interested in recent experiments th...

From the time dependence of states of one of them, the dynamics of two interacting qubits is determined to be one of two possibilities that differ only by a change of signs of parameters in the Hamiltonian. The only exception is a simple particular case where several parameters in the Hamiltonian are zero and one of the remaining nonzero parameters...

Simple examples are presented of Lorentz transformations that entangle the
spins and momenta of two particles with positive mass and spin 1/2. They apply
to indistinguishable particles, produce maximal entanglement from finite
Lorentz transformations of states for finite momenta, and describe entanglement
of spins produced together with entanglemen...

The method of constructing the tomographic probability distributions describing quantum states in parallel with density operators is presented. Known examples of Husimi-Kano quasi-distribution and photon number tomography are reconsidered in the new setting. New tomographic schemes based on coherent states and nonlinear coherent states of deformed...

The example of nonpositive trace-class Hermitian operator for which Robertson-Schroedinger uncertainty relation is fulfilled is presented. The partial scaling criterion of separability of multimode continuous variable system is discussed in the context of using nonpositive maps of density matrices. Comment: 11 pages, to be submitted to Physics Lett...

The relation between completely positive maps and compound states is investigated in terms of the notion of quantum conditional probability.

Quantum stochastic processes are characterized in terms of completely nonnegative quantum dynamical maps which form a convex set. The canonical form of such a map is in terms of R

For systems described by finite matrices, an affine form is developed for the maps that describe evolution of density matrices for a quantum system that interacts with another. This is established directly from the Heisenberg picture. It separates elements that depend only on the dynamics from those that depend on the state of the two systems. Whil...

Lorentz transformations of spin density matrices for a particle with positive mass and spin 1/2 are described by maps of the kind used in open quantum dynamics. They show how the Lorentz transformations of the spin depend on the momentum. Since the spin and momentum generally are entangled, the maps generally are not completely positive and act in...

The mechanism of describing quantum states by standard probabilities (tomograms) instead of wave function or density matrix is elucidated. Quantum tomography is formulated in an abstract Hilbert space framework, by means of decompositions of the identity in the Hilbert space of hermitian linear operators, with trace formula as scalar product of ope...

The partial scaling transform of the density matrix for multiqubit states is introduced to detect entanglement of quantum states. The transform contains partial transposition as a special case. The scaling transform corresponds to partial time scaling of subsystem (or partial Planck's constant scaling) which was used to formulate recently separabil...

Using pure entangled Schmidt states, we show that m-positivity of a map is bounded by the ranks of its negative Kraus matrices. We also give an algebraic condition for a map to be m-positive. We interpret these results in the context of positive maps as entanglement witnesses, and find that only 1-positive maps are needed for testing entanglement.

The arguments that are often put forward to justify the singular importance given to completely positive maps over more generic maps in describing open quantum evolution are studied. We find that these do not appear convincing on closer examination. Positive as well as not positive maps are good candidates for describing open quantum evolution.

The evolution of two qubits coupled by a general nonlocal interaction is studied in two distinct regimes. In the first regime the purity of the individual qubits is interchanged through the entanglement shared by the two. We illustrate how this can be a mechanism for decoherence. In the second regime, the interaction entangles two initially pure qu...

We define extension maps as maps that extend a system (through adding ancillary systems) without changing the state in the original system. We show, using extension maps, why a completely positive operation on an initially entangled system results in a non positive mapping of a subsystem. We also show that any trace preserving map, either positive...

We consider a geometrization, i.e., we identify geometrical structures, for the space of density states of a quantum system. We also provide few comments on a possible application of this geometrization for composite systems.

The problem of constructing a necessary and sufficient condition for establishing the separability of continuous variable systems is revisited. Simon [Phys. Rev. Lett. 84 (2000) 2726] pointed out that such a criterion may be constructed by drawing a parallel between the Peres' partial transpose criterion for finite-dimensional systems and partial t...

Spinors play a role in classical physics, since the fact that the rotation group is not simply connected has consequences for macroscopic bodies. Here we relate the orientations of a rigid body to two-dimensional complex spinors by combining O. Rodrigues parameterization of rotations with Cartans 2:1 identification of spinors with complex isotropic...

We reply to the critique by Pucchini and Vucetich of our construction of a non-relativistic proof of the spin-statistics connection using SU(2) invariance and a Weiss-Schwinger action principle.

Linear maps of matrices describing evolution of density matrices for a quantum system initially entangled with another are identified and found to be not always completely positive. They can even map a positive matrix to a matrix that is not positive, unless we restrict the domain on which the map acts. Nevertheless, their form is similar to that o...

The positive and not completely positive maps of density matrices are discussed. Probability representation of spin states (spin tomography) is reviewed and U(N)-tomogram of spin states is presented. Unitary U(∞)-group tomogram of photon state in Fock basis is constructed. Notion of tomographic purity of spin states is introduced. An entanglement c...

The claim that there is an inconsistency of quantum-classical dynamics [1] is investigated. We point out that a consistent formulation of quantum and classical dynamics which can be used to describe quantum measurement processes is already available in the literature [2]. An example in which a quantum system is interacting with a classical system i...

Entangled and separable states of a bipartite (multipartite) system are
studied in the tomographic representation of quantum states. Properties of
tomograms (joint probability distributions) corresponding to entangled states
are discussed. The connection with star-product quantization is presented.
U(N)-tomography and spin tomography as well as the...

The density matrix of composite spin system is discussed in relation to the adjoint representation of unitary group U(n). The entanglement structure is introduced as an additional ingredient to the description of the linear space carrying the adjoint representation. Positive maps of density operator are related to random matrices. The tomographic p...

SummaryA unified statistical description for both Integrable and non-Integrable systems in terms of the dynamics of correlations is presented by Prigogine. A generalized Quantum Field theory based on analytic continuation is proposed by Sudarshan and Boya. A Time operator of the unstable Friedrichs model and the associated eigenvectors are discusse...

We investigate some examples of quantum Zeno dynamics, when a system
undergoes very frequent (projective) measurements that ascertain whether it
is within a given spatial region. In agreement with previously obtained results,
the evolution is found to be unitary and the generator of the Zeno dynamics
is the Hamiltonian with hard-wall (Dirichlet) bo...

We examine a number of recent proofs of the spin-statistics theorem. All, of course, get the target result of Bose-Einstein
statistics for identical integral spin particles and Fermi-Dirac statistics for identical half-integral spin particles. It
is pointed out that these proofs, distinguished by their purported simple and intuitive kinematic chara...

Clocks are dynamical systems, notwithstanding our usual formalism in which time is treated as an external parameter. A dynamical variable which may be identified with time is explicitly constructed for a variety of simple dynamical systems. This variable is canonically conjugate to the Hamiltonian. The complications brought about by the semibounded...

We show that the manifold of density matrices can be derived from CP^{N^2-1} by the action of SU(N). We give some preliminary observations on the structure of this manifold.

A recent paper by Peshkin [1] has drawn attention again to the problem of understanding the spin statistics connection in non-relativistic quantum mechanics. Allen and Mondragon [2] has pointed out correctly some of the flaws in Peshkin's arguments which are based on the single valuedness under rotation of the wave functions of systems of identical...

I review theories and problems of inconsistencies in the description of higher spin wave equations.

We review the procedure of purification of quantum states which provides recovery of interference and entanglement. The phase space distributions and quantum tomograms are discussed. The superposition principle and the composition law of density matrices are studied.

Forms of dynamics of open finite level systems is formulated. We give a presentation of stochastic dynamics of such systems in terms of maps. Completely positive maps are classified and parametrized. If the system is coupled to a companion system, the contraction of the unitary evolution of the combined system leads to a completely positive map of...

I present an overview of the paradigm of quantum computing that is emerging as a result of recent advances in a variety of fields, including fundamentals of quantum mechanics, information theory, quantum optics and atomic physics. The essentials of a practical quantum computer are discussed and a few algorithms that may be implemented on such a com...

Here we apply our SU(N) and U(N) parameterizations to the question of entanglement in the two qubit and qubit/qutrit system. In particular, the group operations which entangle a two qubit pure state will be given, as well as the corresponding manifold that the operations parameterize. We also give the volume of this manifold, as well as the hypothe...

Here we apply our SU(N) and U(N) parameterizations to the question of entanglement in the two-qubit system. In particular, the group operations which entangle a two-qubit pure state will be given, as well as the corresponding manifold that the operations parameterize. We also give the volume of this manifold, as well as the hypothesized volume for...

The basic formulae governing the fluctuations of counts registered by photoelectric detectors in an optical field are derived. The treatment, which has its origin in Purcell's explanation of the Hanbury Brown-Twiss effect, is shown to apply to any quasi-monochromatic light, whether stationary or not, and whether of thermal origin or not. The repres...

In a previous paper [1] an Euler angle parametrization for SU (4) was given. Here we present the derivation of a generalized Euler angle parametrization for SU (N). The formula for the calculation of the Haar measure for SU (N) as well as its relation to Marinov's volume formula for SU (N) [2, 3] will also be derived. As an example of this parametr...

In a previous paper [J. Phys. A, Math. Gen. 35, 10467–10501 (2002; Zbl 1047.22012)] an Euler angle parameterization for SU(N) was given. Here we present a generalized Euler angle parameterization for U(N). The formula for the calculation of the volume for U(N), ℂP N as well as other SU(N) and U(N) cosets, normalized to this parameterization, will a...

We present a systematic calculation of the volumes of compact manifolds which appear in physics: spheres, projective spaces, group manifolds and generalized flag manifolds. In each case we state what we believe is the most natural scale or normalization of the manifold, that is, the generalization of the unit radius condition for spheres. For this...

We show how a set of POVMs, expressed as a set of $\mu$ linear maps, can be performed with a unitary transformation followed by a von-Neumann measurement with an ancillary system of no more than $\mu N^2$ dimensions. This result shows that all generalized linear transformations and measurements on density matrices can be performed by unitary transf...

An addition rule of impure density operators, which provides a pure state density operator, is formulated. Quantum interference including a visibility property is discussed in the context of the density operator formalism. A measure of entanglement is then introduced as the norm of the matrix equal to the difference between a bipartite density matr...

The most general evolution of the density matrix of a quantum system with a finite-dimensional state space is by stochastic maps which take a density matrix linearly into the set of density matrices. These dynamical stochastic maps form a linear convex set that may be viewed as supermatrices. The property of hermiticity of density matrices renders...

In this paper we give an explicit parametrization for all two qubit density matrices. This is important for calculations involving entanglement and many other types of quantum information processing. To accomplish this we present a generalized Euler angle parametrization for SU(4) and all possible two qubit density matrices. The important group-the...

The generic linear evolution of the density matrix of a system with a finite-dimensional state space is by stochastic maps which take a density matrix linearly into the set of density matrices. These dynamical stochastic maps form a linear convex set that may be viewed as supermatrices. The property of hermiticity of density matrices renders an ass...

It is often stated that quantum mechanics only makes statistical predictions and that a quantum state is described by the various probability distributions associated with it. Can we describe a quantum state completely in terms of probabilities and then use it to describe quantum dynamics? What is the origin of the probability distribution for a ma...

Despite claims that Bell's inequalities are based on the Einstein locality condition, or equivalent, all derivations make an identical mathematical assumption that local hidden-variable theories produce a set of positive-definite probabilities for detecting a particle with a given spin orientation. The standard argument is that because quantum mech...

The generating function is obtained for N photocounts for a gaussian optical field with a lorentzian profile without any restriction to the intervals during which the photodetectors are open. The method may be generalized to arbitrary spectral profiles using the method of Srinivasan and Sukavanam (1971 and 1972).

Quantum Field Theory formulated in terms of hermitian fields automatically leads to a spin-statistic s connection when invariance under rotations is required. In three (or more) dimensions of space this implies Bose statistics for integer spin fields and Fermi statistics for half-integer spin fields. One should recall that spin-H fields in three di...

We obtain the black body radiation formula of Planck by considering independent contributions of multiphoton states

Quantum interference is described in term of density operators only using a formulated composition law for pure-state density operators. In order to retain the relative phases in quantum mechanics, we use fiducial vectors and fiducial projectors. These aspects are illustrated in terms of quantum tomography and density operators. Entanglement is det...

The evolution of a quantum system undergoing very frequent measurements takes place in a proper subspace of the total Hilbert space (quantum Zeno effect). The dynamical properties of this evolution are investigated and several examples are considered.

Superposition principle for spin degrees of freedom is described in terms of density operators only using a formulated composition law of pure-state density operators. Decoherence phenomenon and visibility of the interference pattern are discussed.

The need to retain the relative phases in quantum mechanics implies an addition law parametrized by a phase of two density operators required for the purification of a density matrix. This is shown with quantum tomography and the Wigner function. Entanglement is determined in terms of phase dependent multiplication.

Discovery of the exclusion principle discovery of the electron spin Bose-Einstein statistics identical particle wave functions Fermi-Dirac statistics quantum field theory anticommunication for F-D fields from hole theory to positrons quantization of the K-G equation Pauli's first proof of the SST Fierz's proof of the SST Belinfante's proof of the S...

The relation between the density matrix obeying the von Neumann equation and the wave function obeying the Schrödinger equation
is discussed in connection with the superposition principle of quantum states. The definition of the ray-addition law is given,
and its relation to the addition law of vectors in the Hilbert space of states and the role of...

Complex measure theory is used to widen the scope of the study of stochastic processes and it is shown how, with such an extension, the physical concepts of superposition and diffraction follow automatically. The Dirac-Feynman path integral formalism is seen as a natural development. Several generic Markov processes are studied when extended to com...

The simplest dynamical system is the point particle characterized by its mass. Its state is specified by its position and momentum. The dynamical law is the description of how these quantities change in time. For a free particle, the momentum remains constant while the position increases in the direction of the momentum. The increase is directly pr...

An explicit parameterization is given for the density matrices for $n$-state systems. The geometry of the space of pure and mixed states and the entropy of the $n$-state system is discussed. Geometric phases can arise in only specific subspaces of the space of all density matrices. The possibility of obtaining nontrivial abelian and nonabelian geom...

## Citations

... For reviewing the development history and main research achievements of quantum theory and experiment, please refer to the authoritative collections [2,3], where a large number of research papers have been collected, with no need to repeat here. De Broglie's wave particle duality hypothesis was experimentally verified to be able to describe the quantum behaviour of electrons. ...

... We can assume that the classical Hamiltonian is, once again, the sum of the system and clock Hamiltonians and that the classical state of the system is a phase space distribution in which the states of the system are correlated to the states of the clock. The dynamics of classical states can be represented as (see e.g., [4]): ...

Reference: Classical Evolution without Evolution

... There is also another kind of clock that uses "a measurement" of irreversible eventsbecause of the regularity of radioactive decay of unstable particles, the fraction of surviving particles depends on the time elapsed, so by measuring the quantity of the particles, we measure time (see Sudarshan 2017). ...

... A systematic exploration of the mathematical background for the various quantum tomographic frameworks based on the so called quantizer-dequantizer formalism, has been conducted and the underlying mathematical and physical problems have been described [45], [35], [22], [5], [6]. From these efforts it emerges that an analysis based on the algebraic description of quantum systems, i.e., using the C *algebraic picture of quantum mechanics, would be relevant (see for instance, [36] and [42,Chap. ...

... which is used to construct an orthonormal basis (Pauli matrices in 3 dimensions, Gelfand matrices in 8 dimensions, etc. see e.g. [17,37]). In a classical paper [6] a base of vectors f i was constructed by the Gram-Schmidt orthogonalization, namely consecutive applications of the contracting operator P + defined by ...

... On the one hand, since several studies do not seem to be affected by these issues, the existing approaches continue to be regarded as a useful tool. On the other hand, the systematic formulation of quantum-classical models beyond Ehrenfest dynamics satisfying basic consistency properties remains a subject of current research [15,21,22,26,30,43,52]. ...

... Whereas the generators of evolution in conservative systems can be diagonalized at degenerate eigenvalues, those in dissipative systems can at best be brought into a Jordan canonical form. In scattering theory, the coalescence of eigenvalues manifests itself as double or higher order poles in the complex energy plane [3,4,5,6] that lead to resonances. ...

... There appeared also a number of papers related to the fundamentals of Wigner quantization, or related algebraic quantizations. We mention here in particular the work of Man'ko, Marmo, Zaccaria and Sudarshan [21] , Blasiak, Horzela, Kapus- cik [13, 7, 4], and that of Atakishiyev, Wolf and collaborators [1, 2, 3] in the context of finite oscillator models. More recently, Regniers and Van der Jeugt [32] investigated one-dimensional Hamiltonians with continuous energy spectra as Wigner quantum systems. ...

... These measures are part of a hierarchy of NM [9] and are related to the presence of correlations at intermediate times [10]. Another related definition of NM involves the presence of initial correlations [11][12][13][14][15][16]. Such initial correlations violate the assumption of initially factorised states and hence lead to the breakdown of several well known approaches to the dynamics of the reduced state such as Lindblad dynamics and completely positive trace preserving (CPTP) maps [17]. ...

... In view of quantum physicist Sudarshan ( 1982), 'mind is an interface between the public world described by the physical sciences and the private world of personal experience and individuality.' In this case, the task of psychologist is to understand the latter part, experiences of the being, whereas the quantum physicist would help him to describe mind in terms of 'physic-chemical laws' (Sudarshan 2002(Sudarshan , 2003. ...