Different representations of R-function and parameterized quantization scheme

Three different representations of R-function distribution are deduced for a given density operator of light field in terms of Q-function, Wigner function and positive P-representation, respectively. Furthermore, the τ-parameterized quantization scheme is proposed, where the P-representation, Weyl correspondence, and Q-function are as its special cases. As applications, the photon-count distribution from R-function and the nonclassical depth of photon-added coherent state are examined.     

The Essence of Operators’ Weyl Ordering Scheme and New Operator Identities for Converting Q−P Ordering to Weyl Ordering

We find new operator formulas for converting Q?P and P?Q ordering to Weyl ordering, where Q and P are the coordinate and momentum operator. In this way we reveal the essence of operators’ Weyl ordering scheme, e.g., Weyl ordered operator polynomial ${_{:}^{:}}\;Q^{m}P^{n}\;{_{:}^{:}}$ , $$\begin{aligned} {_{:}^{:}}\;Q^{m}P^{n}\;{_{:}^{:}} =&\sum_{l=0}^{\min (m,n)} \biggl( \frac{-i\hbar }{2} \biggr) ^{l}l!\binom{m}{l}\binom{n}{l}Q^{m-l}P^{n-l} \\ =& \biggl( \frac{\hbar }{2} \biggr) ^{ ( m+n ) /2}i^{n}H_{m,n} \biggl( \frac{\sqrt{2}Q}{\sqrt{\hbar }},\frac{-i\sqrt{2}P}{\sqrt{\hbar }} \biggr) \bigg|_{Q_{\mathrm{before}}P} \end{aligned}$$ where ${}_{:}^{:}$ ${}_{:}^{:}$ denotes the Weyl ordering symbol, and H _m,n is the two-variable Hermite polynomial. This helps us to know the Weyl ordering more intuitively.     

Quantum and Classical Thermal Correlations in the XXZ Spin-1/2 Chain

We compute the quantum dissonance Q (non-entangled quantum correlation), entanglement E, quantum discord D (total quantum correlation) and classical correlation C in the eight-qubit XXZ spin-1/2 chain at finite temperatures. We find that not only D but also Q and C can clearly detect the critical points associated to quantum phase transitions for this model at finite temperatures. Moreover, Q can detect the special points of the system where the entanglement just appears or completely vanishes. Finally, we obtain two simple dominance relations: C≥E and D≥E+Q. Except these there are no other simple ordering relations in this model.     

Baxter Q-operators of the XXZ chain and R-matrix factorization

We construct Baxter operators as generalized transfer matrices being traces of products of generic R matrices. The latter are shown to factorize into simpler operators allowing for explicit expressions in terms of functions of a Weyl pair of basic operators. These explicit expressions are the basis for explicit expression for Baxter Q-operators and for investigating their properties.     

Mathematics of noncanonical quantum theory

The Weyl relatione ^?isQ e ^itP e ^isQ =e ^it(P+sI) is generalized so as to hold for noncanonical couples (P,Q) implying the commutation relationi[P,Q]=C whereC is arbitrary bounded self-adjoint. It is shown that if 0 is not in the closure of the numerical range ofC then bothP andQ are spectrally continuous and neither bounded from below nor above. The dynamical equations in noncanonical theory are established. It is shown thatH (which is no longer given by correspondence) cannot be bounded from below (above) ifC≦0 (C≧0),C≠0.     

Newton-Leibniz integration for ket-bra operators in quantum mechanics (IV)—Integrations within Weyl ordered product of operators and their applications

We show that the technique of integration within normal ordering of operators [Hong-yi Fan, Hai-liang Lu, Yue Fan, Ann. Phys. 321 (2006) 480-494] applied to tackling Newton-Leibniz integration over ket-bra projection operators, can be generalized to the technique of integration within Weyl ordered product (IWWOP) of operators. The Weyl ordering symbol is introduced to find the Wigner operator’s Weyl ordering form Δ(p,q) =  δ(p − P)δ(q − Q) , and to find operators’ Weyl ordered expansion formula. A remarkable property is that Weyl ordering of operators is covariant under similarity transformation, so it has many applications in quantum statistics and signal analysis. Thus the invention of the IWWOP technique promotes the progress of Dirac’s symbolic method.     

Determination of Quality Factor for Highly Overcoupled EPR Resonators

We reported determination of the loaded quality factor (Q) of highly overcoupled (dielectric, loop-gap, and cavity) resonators used in time-domain electron paramagnetic resonance. We introduced a microwave absorber into resonators and achieved critical-coupling. Due to the deep “Q-dip” of critical-coupling, we can easily determine the loaded Q as low as 10. The loaded Q of resonators with and without the microwave absorber was examined under various overcoupling conditions. We found that the radiation Q (Q _r) can be calculated from the loaded Q of the resonator that contains the microwave absorber. We proposed a simple model that represents the loaded Q of the overcoupled resonator in terms of two parameters, Q ₀ and Q _r. Q ₀ is the effective unloaded Q of the resonator determined for the critically coupled resonator without the microwave absorber and is independent of a degree of coupling. The model can be applied to overcoupling in which the coupling parameter (Q ₀/Q _r) is in the range of 1 to ca. 20.     

Semiclassical Limits of Ore Extensions and a Poisson Generalized Weyl Algebra

We observe [Launois and Lecoutre, Trans. Am. Math. Soc. 368:755–785, 2016, Proposition 4.1] that Poisson polynomial extensions appear as semiclassical limits of a class of Ore extensions. As an application, a Poisson generalized Weyl algebra A₁, considered as a Poisson version of the quantum generalized Weyl algebra, is constructed and its Poisson structures are studied. In particular, a necessary and sufficient condition is obtained, such that A₁ is Poisson simple and established that the Poisson endomorphisms of A₁ are Poisson analogues of the endomorphisms of the quantum generalized Weyl algebra.     

Cohomogeneity-one Einstein–Weyl structures: a local approach

We analyse in a systematic way the (non-) compact n-dimensional Einstein–Weyl spaces equipped with a cohomogeneity-one metric. In that context, with no compactness hypothesis for the manifold on which lives the Einstein–Weyl structure, we prove that, as soon as the (n−1)-dimensional space is a homogeneous reductive Riemannian space with a unimodular group of left-acting isometries G:

• •there exists a Gauduchon gauge such that the Weyl-form is co-closed and its dual is a Killing vector for the metric;
• •in that gauge, a simple constraint on the conformal scalar curvature holds;
• •a non-exact Einstein–Weyl structure may exist only if the (n−1)-dimensional homogeneous space G/H contains a non-trivial subgroup H′ that commutes with the isotropy subgroup H;
• •the extra isometry due to this Killing vector corresponds to the right-action of one of the generators of the algebra of the subgroup H′.

The first two results are well known when the Einstein–Weyl structure lives on a compact manifold, but our analysis gives the first hints on the enlargement of the symmetry due to the Einstein–Weyl constraint.We also prove that the subclass with G compact, a one-dimensional subgroup H′ and the (n−2)-dimensional space G/(H×H′) being an arbitrary compact symmetric Kähler coset space, corresponds to n-dimensional Riemannian locally conformally Kähler metrics. The explicit family of structures of cohomogeneity-one under SU(n/2) being, thanks to our results, invariant under U(1)×SU(n/2), it coincides with the one first studied by Madsen; our analysis allows us to prove most of his conjectures.     

Stochastic processes originating in deterministic microscopic dynamics

We investigate the probability distribution of the scaled trajectory of a test particle moving in an equilibrium fluid according to the laws of classical mechanics, i.e., ifQ(t) is the displacement of the test particle we letQ _A(t) =Q(At)/√A and consider the distribution of the trajectory Q_A(t) in the limit A→∞. The randomness of the motion is due entirely to the randomness of the initial state of the fluid, test particle, or both, and the process is generally non-Markovian. Nevertheless, it can be proven in some cases and we expect it to be true in many more that Q_A (t) looks like Brownian motion in the limit A→∞. Some results for simple model systems are presented.     

Proofs of existence of local potentials for trace-free symmetric 2-forms using dimensionally dependent identities

We exploit four-dimensional tensor identities to give a very simple proof of the existence of a Lanczos potential for a Weyl tensor in four dimensions with any signature, and to show that the potential satisfies a simple linear second-order differential equation, e.g., a wave equation in Lorentz signature. Furthermore, we exploit higher-dimensional tensor identities to obtain the analogous results for (m, m)-forms in 2m dimensions.     

Singularities in Weyl gravitational fields

The peculiarities of the scalarS ≡R _ijkl R ^ijkl are exhibited for two axially-symmetric static (Weyl) gravitational fields. By examiningS along curved families of trajectories to the Weyl singularities, examples are found which contradict previous claims by Gautreau and Anderson regarding ‘directional singularities’. Proper circumferences about the Bach and Weyl line-mass singularity are also examined. There is no apparent correlation between the source structure and the behaviour ofS from this analysis.     

Model for nucleon valence structure functions at allx,allp ⊥ and allQ 2 from the correspondence between QCD and DTU

We propose a simple parametrization of the nucleon valence structure functions at allx, allp _⊥ and allQ ². We use the DTU parton model to fix the parametrization at a reference point (Q ₀ ² =3 GeV²) and we mimic the QCD evolution by replacing the dimensioned parameters of the DTU parton model by functions depending onQ ². Excellent agreement is obtained with existing data.     

Scaling violation in the pion structure function

TheQ ²-dependence of the pion structure function extracted from high transverse momentum π⁰ production cross sections is examined. We find, in thex→1 region, that this dependence can be parametrized by a power of (1?x) increasing as ln lnQ ². This is in agreement with the expectations of asymptotic freedom and also seen in the structure function extracted from dilepton production. It is not possible to fit all of theQ ²-dependence with a simple power law term.     

Do we see anomalous dimensions in e+e− inclusive annihilation at Q = 4 GeV?

It is shown in the framework of scale invariant models that a simple ansatz for the anomalous dimensions allows us to determine the experimentally observed variation with Q of the one-charged particle inclusive cross section.Our calculation sheds no light on the behaviour of σ_had with Q since we have worked with the inclusive cross section normalized to the experimentally measured σ_had.     

Type-III and IV interacting Weyl points

3+1-dimensional Weyl fermions in interacting systems are described by effective quasi-relativistic Green’s functions parametrized by a 16-element matrix e _α ^μ in an expansion around the Weyl point. The matrix e _α ^μ can be naturally identified as an effective tetrad field for the fermions. The correspondence between the tetrad field and an effective quasi-relativistic metric g_μν governing the Weyl fermions allows for the possibility to simulate different classes of metric fields emerging in general relativity in interacting Weyl semimetals. According to this correspondence, there can be four types of Weyl fermions, depending on the signs of the components g ⁰⁰ and g ₀₀ of the effective metric. In addition to the conventional type-I fermions with a tilted Weyl cone and type-II fermions with an overtilted Weyl cone for g ⁰⁰ > 0 and, respectively, g ₀₀ > 0 or g ₀₀ < 0, we find additional “type-III” and “type-IV” Weyl fermions with instabilities (complex frequencies) for g ⁰⁰ < 0 and g ₀₀ > 0 or g ₀₀ < 0, respectively. While the type-I and type-II Weyl points allow us to simulate the black hole event horizon at an interface where g ⁰⁰ changes sign, the type-III Weyl point leads to effective spacetimes with closed timelike curves.     

Angular distribution of dileptons in hadronic collisions

The polar and azimuthal angular distributions of the lepton pair produced in hadron-hadron collisions are studied. For large Q_T these are calculated from lowest-order QCD diagrams. For low Q_T we present simple expressions which take into account the effect of primordial quark transverse momentum on the angular distributions.     

Deep-inelastic scattering in κ factorization and the anatomy of the differential gluon structure function of the proton

The differential gluon structure function of the proton, ?(x, Q ²), introduced by Fadin, Kuraev, and Lipatov in 1975 is extensively used in small-x QCD. We report here the first determination of ?(x, Q ²) from experimental data on the small-x proton structure function F _2p (x, Q ²). We give convenient parametrizations for ?(x, Q ²) based partly on the available DGLAP evolution fits (GRV, CTEQ, and MRS) to parton distribution functions and on realistic extrapolations into the soft region. We discuss the impact of soft gluons on various observables. The x dependence of the so-determined ?(x, Q ²) varies strongly with Q ² and does not exhibit simple Regge properties. Nonetheless, the hard-to-soft diffusion is found to give rise to a viable approximation of the proton structure function F _2p (x, Q ²) by the soft and hard Regge components with intercepts Δ_soft=0 and Δ_hard ～ 0.4.