Hermitian Conjugate Research Articles

Green Functions of the Stationary Schrödinger Equation and Tomographic Representation of Quantum Mechanics

Invertible maps of operators of quantum observables onto functions of c-number arguments and their associative products are reviewed. In particular, the symplectic tomography map is discussed and an expression connecting an arbitrary operator and its tomographic symbol is written down. This formula is applied to obtain explicit expressions for tomographic symbols, which are symplectic tomograms of Green functions of stationary and nonstationary Schrodinger equations written for the case of harmonic oscillator. The connection between the so-called classical propagator Π(X,μ,ν,t,X′,μ′,ν′,0) and the tomographic symbol of the evolution operator of nonstationary Schrodinger equation is found. The spin tomography is presented as a map of operators acting in spinor space onto functions of c-arguments. As an example, the spin located in a magnetic field is considered and the tomographic symbol of resolventa is obtained. Tomographic symbols of hermitian conjugate operators are shown to be complex conjugate functions.

Nonlinear supersymmetry in quantum mechanics: algebraic properties and differential representation

We study the nonlinear (polynomial, N-fold,…) supersymmetry algebra in one-dimensional QM. Its structure is determined by the type of conjugation operation (Hermitian conjugation or transposition) and described with the help of the Hamiltonian projection on the zero-mode subspace of supercharges. We show that the SUSY algebra with transposition symmetry is always polynomial in the Hamiltonian if supercharges represent differential operators of finite order. The appearance of the extended SUSY with several (complex or real) supercharges is analyzed in details and it is established that no more than two independent supercharges may generate a nonlinear superalgebra which can be appropriately specified as N=2 SUSY. In this case we find a nontrivial hidden symmetry operator and rephrase it as a nonlinear function of the Hamiltonian on the physical state space. The full N=2 nonlinear SUSY algebra includes “central charges” both polynomial and nonpolynomial (due to a symmetry operator) in the Hamiltonian.

Noncommutative phase and unitarization of GLp,q(2)

In this article, imposing Hermitian conjugate relations on the two-parameter deformed quantum group GLp,q(2) is studied. This results in a noncommutative phase associated with the unitarization of the quantum group. After the achievement of the quantum group Up,q(2) with pq real via a noncommutative phase, the representation of the algebra is built by means of the action of the operators constituting the Up,q(2) matrix on states.

Characteristic polynomials of complex random matrix models

We calculate the expectation value of an arbitrary product of characteristic polynomials of complex random matrices and their Hermitian conjugates. Using the technique of orthogonal polynomials in the complex plane our result can be written in terms of a determinant containing these polynomials and their kernel. It generalizes the known expression for Hermitian matrices and it also provides a generalization of the Christoffel formula to the complex plane. The derivation we present holds for complex matrix models with a general weight function at finite- N, where N is the size of the matrix. We give some explicit examples at finite- N for specific weight functions. The characteristic polynomials in the large- N limit at weak and strong non-hermiticity follow easily and they are universal in the weak limit. We also comment on the issue of the BMN large- N limit.

Metrics on the real quantum plane

Using the frame formalism we determine some possible metrics and metric-compatible connections on the noncommutative differential geometry of the real quantum plane. By definition, a metric maps the tensor product of two 1-forms into a “function” on the quantum plane. It is symmetric in a modified sense, namely in the definition of symmetry one has to replace the permutator map with a deformed map σ fulfilling some suitable conditions. Correspondingly, also the definition of the Hermitian conjugate of the tensor product of two 1-forms is modified (but reduces to the standard one if σ coincides with the permutator). The metric is real with respect to such modified * -structure.

NONRELATIVISTIC SPIN-½ PARTICLE IN AN ARBITRARY NON-ABELIAN MAGNETIC FIELD IN TWO SPATIAL DIMENSIONS

The (group and spin space) matrix Hamiltonian describing the dynamics of a non-relativistic spin-½ particle moving in a static, but spatially dependent, non-Abelian magnetic field in two spatial dimensions is shown to take the form of an anticommutator of a nilpotent operator and its Hermitian conjugate. Consequently, the (group space) matrix Hamiltonians for the two different spin projections form partners of a supersymmetric quantum mechanical system. The resulting supersymmetry algebra is exploited to explicitly construct the exact zero energy ground state wave function(s) for the system. The remaining eigenstates and eigenvalues of the two partner Hamiltonians form positive energy degenerate pairs.

Clifford algebraic spinor and the Dirac wave equations

The hyperbolic complex (HC) space is congruent with Minkowski space time.HC is a special kind of non-Euclidean space with continuous odd-points. The Clifford algebraic spinor and the Dirac wave equation can be introduced in the hyperbolic complex space. The Clifford algebraic spinor contains eight independent elements and the Dirac wave equations 64 coefficients. For Dirac particles 4×8 and for antiparticles 4×8 variables which are Hermitian conjugate to each other (on four dimensional space-time).

CHARGED SPIN-1/2 PARTICLE IN AN ARBITRARY MAGNETIC FIELD IN TWO SPATIAL DIMENSIONS: A SUPERSYMMETRIC QUANTUM MECHANICAL SYSTEM

It is shown that the 2×2 matrix Hamiltonian describing the dynamics of a charged spin-1/2 particle with g-factor 2 moving in an arbitrary, spatially dependent, magnetic field in two spatial dimensions can be written as the anticommutator of a nilpotent operator and its Hermitian conjugate. Consequently, the Hamiltonians for the two different spin projections form partners of a supersymmetric quantum mechanical system. The resulting supersymmetry algebra can then be exploited to explicitly construct the exact zero energy ground state wave function for the system. Modulo this ground state, the remainder of the eigenstates and eigenvalues of the two partner Hamiltonians form positive energy degenerate pairs. We also construct the spatially asymptotic form of the magnetic field which produces a finite magnetic flux and associated zero energy normalizable ground state wave function.

Use of the reciprocal basis in neutral meson mixing

In the presence of CP violation, the effective Hamiltonian matrix describing a neutral meson anti-meson system does not commute with its hermitian conjugate. As a result, this matrix cannot be diagonalized by a unitary transformation and one needs to introduce a reciprocal basis. Although known, this fact is seldom discussed and almost never used. Here, we use this concept to highlight a parametrization of the Hamiltonian matrix in terms of physical observables, and we show that using it reduces a number of long and tedious derivations into simple matrix multiplications. These results have a straightforward application for propagation in matter. We also comment on the (mathematical) relation with neutrino oscillations.

Extended Supersymmetry in Four-Dimensional Euclidean Space

Since the generators of the two SU(2) groups which comprise SO(4) are not Hermitian conjugates of each other, the simplest supersymmetry algebra in four-dimensional Euclidean space more closely resembles the N=2 than the N=1 supersymmetry algebra in four-dimensional Minkowski space. An extended supersymmetry algebra in four-dimensional Euclidean space is considered in this paper; its structure resembles that of N=4 supersymmetry in four-dimensional Minkowski space. The relationship of this algebra to the algebra found by dimensionally reducing the N=1 supersymmetry algebra in ten-dimensional Euclidean space to four-dimensional Euclidean space is examined. The dimensional reduction of N=1 super Yang–Mills theory in ten-dimensional Minkowski space to four-dimensional Euclidean space is also considered.

Optimization of entanglement witnesses

An entanglement witness (EW) is an operator that allows to detect entangled states. We give necessary and sufficient conditions for such operators to be optimal, i.e. to detect entangled states in an optimal way. We show how to optimize general EW, and then we particularize our results to the non-decomposable ones; the latter are those that can detect positive partial transpose entangled states (PPTES). We also present a method to systematically construct and optimize this last class of operators based on the existence of ``edge'' PPTES, i.e. states that violate the range separability criterion [Phys. Lett. A{\bf 232}, 333 (1997)] in an extreme manner. This method also permits the systematic construction of non-decomposable positive maps (PM). Our results lead to a novel sufficient condition for entanglement in terms of non-decomposable EW and PM. Finally, we illustrate our results by constructing optimal EW acting on $H=\C^2\otimes \C^4$. The corresponding PM constitute the first examples of PM with minimal ``qubit'' domain, or - equivalently - minimal hermitian conjugate codomain.

Some consequences of exchangeability in random-matrix theory.

Properties of infinite sequences of exchangeable random variables result directly in explicit expressions for calculating asymptotic densities of eigenvalues rho(infinity)(lambda) of any ensemble of random matrices H whose distribution depends only on tr(H+H), where H+ is the Hermitian conjugate of H. For real symmetric matrices and for Hermitian matrices, the densities rho(infinity)(lambda) are constructed by summing up Wigner semicircles with varying radii and weights as confirmed by Monte Carlo simulations. Extensions to more general matrix ensembles are also considered.

A FERMIONIC HODGE STAR OPERATOR

A fermionic analogue of the Hodge star operation is shown to have an explicit operator representation in models with fermions, in space–times of any dimension. This operator realizes a conjugation (pairing) not used explicitly in field theory, and induces a metric in the space of wave function(al)s just as in exterior calculus. If made real (hermitian), this induced metric turns out to be identical to the standard one constructed using hermitian conjugation; the utility of the induced complex bilinear form remains unclear.

Single-shot half k-space high-resolution gradient-recalled EPI for fMRI at 3 Tesla.

Half k-space gradient-recalled echo-planar imaging (GR-EPI) is discussed in detail. T2* decay during full k-space GR-EPI gives rise to unequal weighting of the lines of k-space, loss of signal intensity at the center of k-space, and a point-spread function that limits resolution. In addition, the long readout time for high-resolution full k-space acquisition gives rise to severe susceptibility effects. These problems are substantially reduced by acquiring only half of k-space and filling the empty side by Hermitian conjugate formation. Details of the pulse sequence and image reconstruction are presented. The point-spread function is 3(1/2) times narrower for half than full k-space acquisition. Experiments as well as theoretical considerations were carried out in a context of fMRI using a whole-brain local gradient and an RF coil at 3 Tesla. Using a bandwidth of +/-83 kHz, well-resolved single-shot images of the human brain, as well as good quality fMRI data sets were obtained with a matrix of 192 x 192 over 16 x 16 cm2 FOV using half k-space techniques. The combination of high spatial resolution using the methods presented in this article and the high temporal resolution of EPI opens opportunities for research into fMRI contrast mechanisms. Increase of percent signal change as the resolution increases is attributed to reduction of partial volume effects of activated voxels. Histograms of fMRI pixel responses are progressively weighted to higher percent signal changes as the resolution increases. The conclusion has been reached that half k-space GR-EPI is generally superior to full k-space GR-EPI and should be used even for low-resolution (64 x 64) EPI.

A (p,q)-Deformation of Quantum Mechanics**The project supported by Natural Science Foundation of Jiangxi Province of China

On a one-dimensional lattice with a geometric sequence spacing, the Hermitian conjugation of a (p,q)-derivative operator is discussed by means of (p,q)-integration. Then a (p,q)-deformation of both the Heisenberg algebra for the canonical coordinates and the Heisenberg–Weyl algebra for the harmonic oscillator is presented. It is shown that although in the algebraic aspect the (p,q)-deformation discussed here is identical with q-deformation given by Truong, the (p,q)-deformed Schrödinger picture is in fact different from the q-deformed one.

Meson-baryon-baryon vertex function and the Ward-Takahashi identity.

Ohta proposed a solution for the well-known difficulty of satisfying the Ward-Takahashi identity for a photo-meson-baryon-baryon amplitude (\ensuremath{\gamma}MBB) when a dressed meson-baryon-baryon (MBB) vertex function is present. He obtained a form for the \ensuremath{\gamma}MBB amplitude which contained, in addition to the usual pole terms, longitudinal seagull terms which were determined entirely by the MBB vertex function. He arrived at his result by using a Lagrangian which yields the MBB vertex function at tree level. We show that such a Lagrangian can be neither Hermitian nor charge conjugation invariant. We have been able to reproduce Ohta's result for the \ensuremath{\gamma}MBB amplitude using the Ward-Takahashi identity and no other assumption, dynamical or otherwise, and the most general form for the MBB and \ensuremath{\gamma}MBB vertices. However, contrary to Ohta's finding, we find that the seagull terms are not robust. The seagull terms extracted from the \ensuremath{\gamma}MBB vertex occur unchanged in tree graphs, such as in an exchange current amplitude. But the seagull terms which appear in a loop graph, as in the calculation of an electromagnetic form factor, are, in general, different. The whole procedure says nothing about the transverse part of the (\ensuremath{\gamma}MBB) vertex and its contributions to the amplitudes in question. \textcopyright{} 1996 The American Physical Society.

The Calculation of Off-Shell Interaction Revised

Various methods for calculating the off-shell interaction is discussed. The Hermitian conjugation property of is restored. But different methods give rise to a little different results and raise an uncertainty of the off-shell interaction. The origin of the non-Galilean invariant term is still unknown. The reason of a too small coupling constant is clarified.

On the normalization and hermiticity of amplitudes in 4D heterotic superstrings

We consider how to normalize the scattering amplitudes of 4D heterotic superstrings in a Minkowski background. We fix the normalization of the vacuum amplitude (the string partition function) at each genus, and of every vertex operator describing a physical external string state in a way consistent with unitarity of the S-matrix. We also provide an explicit expression for the map relating the vertex operator of an incoming physical state to the vertex operator describing the same physical state, but outgoing. This map is related to hermitian conjugation and to the hermiticity properties of the scattering amplitudes.

Smeared propagators for lattice hadron spectroscopy

We propose to replace ordinary propagators in lattice operator correlations entering the determination of hadron masses with space-time smeared propagators. These are defined as the inverse of the quadratic operator in the fermion action times its hermitian conjugate. We obtain a cleaner determination of hadron masses, comparable to the effect of using smeared operators.

On the analytic properties of chiral solitons in the presence of the ω-meson

A thorough study is performed of the analytical properties of the fermion determinant for the case that the time components of (axial) vector fields do not vanish. For this purpose the non--Hermitian Euclidean Dirac Hamiltonian is generalized to the whole complex plane. The Laurent series are proven to reduce to Taylor series for the corresponding eigenvalues and --functions as long as field configurations are assumed for which level crossings do not occur. The condition that no level crossings appears determines the radius convergence. However, the need for regularization prohibits the derivation of an analytic energy functional because real and imaginary parts of the eigenvalues are treated differently. Consistency conditions for a Minkowski energy functional are extracted from global gauge invariance and the current field identity for the baryon current. Various treatments of the Nambu--Jona--Lasinio soliton are examined with respect to these conditions. Motivated by the studies of the Laurent series for the energy functional the Euclidean action is expanded in terms of the $\omega$--field. It is argued that for this expansion the proper--time regularization scheme has to be imposed on the operator level rather than on an expression in terms of the one--particle eigenenergies. The latter treatment is plagued by the inexact assumption that the Euclidean Dirac Hamiltonian and its Hermitian conjugate can be diagonalized simultaneously. It is then evident that approaches relying on counting powers of the $\omega$--field in the one--particle eigenenergies are