Intertwining technique for the one-dimensional stationary Dirac equation

The technique of differential intertwining operators (or Darboux transformation operators) is systematically applied to the one-dimensional Dirac equation. The following aspects are investigated: factorization of a polynomial of Dirac Hamiltonians, quadratic supersymmetry, closed extension of transformation operators, chains of transformations, and finally particular cases of pseudoscalar and scalar potentials. The method is widely illustrated by numerous examples.     

Bosonization in (1 + 1) dimensions: An elementary analysis

The expression of a free, massless, Fermi field in terms of scalar field components in a Lorentz basis is obtained. The infrared “disease” of the scalar field is seen to play a positive role in enforcing the fermionic selection rules. Particular attention is paid to the Poincaré transformation properties, both of the Fermi construct and of the Bose constituents. The way that two charge operators are contained in the scalar theory, and the necessary enlargement of the (indefinite metric) space on which the scalar field acts, are discussed in detail.     

压缩算符在自由标量场理论中的推广

引入了一种在量子场论中构造压缩算符的办法:考虑两个具有不同质量的同一标量场的自由哈密顿量,通过博戈留波夫变换,导出广义压缩算符,该算符把一个基态映射到另一个。该算符作用的有效性分别在量子场论的狄拉克表象和薛定谔泛函表象中得到了验证。我们相信,在任意实标量场理论中,只要存在两组以线性变换联系起来的生成湮灭算符,压缩算符就被类似的方法找到。     

Generalization of Wigner's Unitary-Antiunitary Theorem for Indefinite Inner Product Spaces

We present a generalization of Wigner's unitary-antiunitary theorem for pairs of ray transformations. As a particular case, we get a new Wigner-type theorem for non-Hermitian indefinite inner product spaces.     

Casimir effect for a scalar field via Krein quantization

In this work, we present a rather simple method to study the Casimir effect on a spherical shell for a massless scalar field with Dirichlet boundary condition by applying the indefinite metric field (Krein) quantization technique. In this technique, the field operators are constructed from both negative and positive norm states. Having understood that negative norm states are un-physical, they are only used as a mathematical tool for renormalizing the theory and then one can get rid of them by imposing some proper physical conditions.     

On the quantization of a kinetic energy with variable inertia

It is shown that fundamental properties of kinetic energy determine its operator in wave mechanics almost completely: The part containing differential operators is unequivocally given by the Laplacian calculated for the metric which is defined by the classical kinetic energy. The only remaining ambiguity is an additive scalar function proportional to ?². Invariance properties with respect to infinitesimal transformations may reduce the number of coordinates on which this function depends. In certain cases it must be constant.     

A QCD sum rule study of the light scalar mesons

We examine the interpretation of the light scalar meson nonet as bound states of the scalar diquark and the scalar antidiquark using the QCD sum rule approach. Our results are obtained by means of the operator product expansion (OPE) including operators up to dimension 8. They show no evidence of the coupling of the tetraquark states to the light scalar meson nonet.     

Vertex Operators in 4D Quantum Gravity Formulated as CFT

We study vertex operators in 4D conformal field theory derived from quantized gravity, whose dynamics is governed by the Wess-Zumino action by Riegert and the Weyl action. Conformal symmetry is equal to diffeomorphism symmetry in the ultraviolet limit, which mixes positive-metric and negative-metric modes of the gravitational field and thus these modes cannot be treated separately in physical operators. In this paper, we construct gravitational vertex operators such as the Ricci scalar, defined as space-time volume integrals of them are invariant under conformal transformations. Short distance singularities of these operator products are computed and it is shown that their coefficients have physically correct signs. Furthermore, we show that conformal algebra holds even in the system perturbed by the cosmological constant vertex operator as in the case of the Liouville theory shown by Curtright and Thorn.     

Some Unitary Operators Derived Using the Quantum State and Its Application

Some unitary operators are derived using quantum state by depending on the technique of integration within an ordered product of operators, for example parity operator, displacement operator, squeezed operator, etc. The characteristics of these operators are analyzed. Their unitary transformations play an essential role in some transformations. As applications, the dynamic problems of the double momentum coupling harmonic oscillators are solved exactly.     

An algebraic approach to the non-symmetric Macdonald polynomial

In terms of the raising and lowering operators, we algebraically construct the non-symmetric Macdonald polynomials which are simultaneous eigenfunctions of the commuting Cherednik operators. We also calculate Cherednik's scalar product of them.     

Unitary equivalence of different bipartite entangled states under unitary transformations

The unitary equivalence of different bipartite entangled states with continuous variables under unitary transformations are investigated. With the help of the technique of integration within an ordered product of operators, the corresponding unitary operators are also derived. These results may deepen people's understanding to the various bipartite entangled states, and enrich the representations and transformations theory in quantum mechanics.     

p Dissipative Operator

In this paper the authors prove that the generalized positive p selfadjoint (GPpS) operators in Banach space satisfy the generalized Schwarz inequality, solve the maximal dissipative extension representation of pp dissipative operators in Banach space by using the inequality and introducing the generalized indefinite inner product (GIIP) space, and apply the result to a certain type of Schrödinger operator.     

Galilean and dynamical invariance of entanglement in particle scattering

Particle systems admit a variety of tensor product structures (TPSs) depending on the algebra of observables chosen for analysis. Global symmetry transformations and dynamical transformations may be resolved into local unitary operators with respect to certain TPSs and not with respect to others. Symmetry-invariant and dynamical-invariant TPSs are defined and various notions of entanglement are considered for scattering states.     

Geometry of the Unification of Quantum Mechanics and Relativity of a Single Particle

The paper summarizes, generalizes and reveals the physical content of a recently proposed framework that unifies the standard formalisms of special relativity and quantum mechanics. The framework is based on Hilbert spaces H of functions of four space-time variables x,t, furnished with an additional indefinite inner product invariant under Poincaré transformations. The indefinite metric is responsible for breaking the symmetry between space and time variables and for selecting a family of Hilbert subspaces that are preserved under Galileo transformations. Within these subspaces the usual quantum mechanics with Shrödinger evolution and t as the evolution parameter is derived. Simultaneously, the Minkowski space-time is embedded into H as a set of point-localized states, Poincaré transformations obtain unique extensions to H and the embedding commutes with Poincaré transformations. Furthermore, the framework accommodates arbitrary pseudo-Riemannian space-times furnished with the action of the diffeomorphism group.     

Antiperiodic spin-1/2 XXZ quantum chains by separation of variables: Complete spectrum and form factors

In this paper we consider the spin-1/2 highest weight representations for the 6-vertex Yang–Baxter algebra on a finite lattice and analyze the integrable quantum models associated to the antiperiodic transfer matrix. For these models, which in the homogeneous limit reproduces the XXZ spin-1/2 quantum chains with antiperiodic boundary conditions, we obtain in the framework of Sklyanin?s quantum separation of variables (SOV) the following results: I) The complete characterization of the transfer matrix spectrum (eigenvalues/eigenstates) and the proof of its simplicity. II) The reconstruction of all local operators in terms of Sklyanin?s quantum separate variables. III) One determinant formula for the scalar products of separates states, the elements of the matrix in the scalar product are sums over the SOV spectrum of the product of the coefficients of the states. IV) The form factors of the local spin operators on the transfer matrix eigenstates by one determinant formulae given by simple modifications of the scalar product formulae.     

Correspondence between quantum-optical transform and classical-optical transform explored by developing Dirac’s symbolic method

By virtue of the new technique of performing integration over Dirac’s ket–bra operators, we explore quantum optical version of classical optical transformations such as optical Fresnel transform, Hankel transform, fractional Fourier transform, Wigner transform, wavelet transform and Fresnel–Hadmard combinatorial transform etc. In this way one may gain benefit for developing classical optics theory from the research in quantum optics, or vice-versa. We cannot only find some new quantum mechanical unitary operators which correspond to the known optical transformations, deriving a new theorem for calculating quantum tomogram of density operators, but also can reveal some new classical optical transformations. For examples, we find the generalized Fresnel operator (GFO) to correspond to the generalized Fresnel transform (GFT) in classical optics. We derive GFO’s normal product form and its canonical coherent state representation and find that GFO is the loyal representation of symplectic group multiplication rule. We show that GFT is just the transformation matrix element of GFO in the coordinate representation such that two successive GFTs is still a GFT. The ABCD rule of the Gaussian beam propagation is directly demonstrated in the context of quantum optics. Especially, the introduction of quantum mechanical entangled state representations opens up a new area in finding new classical optical transformations. The complex wavelet transform and the condition of mother wavelet are studied in the context of quantum optics too. Throughout our discussions, the coherent state, the entangled state representation of the two-mode squeezing operators and the technique of integration within an ordered product (IWOP) of operators are fully used. All these have confirmed Dirac’s assertion: “...for a quantum dynamic system that has a classical analogue, unitary transformation in the quantum theory is the analogue of contact transformation in the classical theory”.     

Optical Operator Method Studied via Fresnel Operator Decomposition and Coherent State Representation

We find that the mapping from classical optical transformations to the optical operator method can be realized by using the coherent state representation and the technique of integration within an ordered product of operators. The optical Fresnel operator derived in (Commun. Theor. Phys. (Beijing, China) 38 (2002) 147) can unify those frequently used optical operators. Various decompositions of Fresnel operator into the exponential canonical operators are obtained.     

General Formalism for Mapping of Two-Mode Classical Canonical Transformations to Quantum Unitary operators

Two types of canonical transformations in two-mode classical phase space are mapped into the quantum mechanical Hilbert space to produce some new normally ordered unitary operators. These operators are evaluated in the coordinate (momentum) representations using the "integration within ordered product technique, and the mapping is maniferrtly apparent in the derivation. New generalixed coherent states are constructed in terms of these operators, and the uncertainty relations for these states are analysed.     

The Newton-Wigner and Wightman localization of the photon

A quantum theory of the photon is developed in a natural manner. Newton-Wigner and Wightman demonstrated that the photon could not be strictly localized according to natural criteria. These investigations involved the identification of an elementary system with a uirrep of the Poincare group. We identify a particle with the localized measurement of the states satisfying the uirrep. In the case of zero mass and unit spin, the photon is identified with those components of the state that can be localized. A c-number four-vector potential and Lorentz condition are derived from the relativistic wave equation. The Wightman localization is demonstrated for the three independent space components of the vector potential, and the photon is identified with these components. A position operator and probability density follow immediately from the localization. A consequence of the subjective definition of a photon is that the transformations of the vector potential are unitary, and hence the unitary scalar product can be obtained for the four-vector potential. A Hilbert space is defined for the three space components of the vector potential. A position operator and probability density are derived from the scalar product, which compare directly with those obtained from the localization and the non-relativistic theory. As the longitudinal and scalar polarizations do not contribute to the measured transition probability, they are considered virtual. Lastly, a conserved four-vector current is derived from the scalar product. The possibility of observing a strict localization of the photon in the laboratory is suggested.