Extending $\mathcal{P}\mathcal{T}$ Symmetry from Heisenberg Algebra to E2 Algebra

The E2 algebra has three elements, J, u, and v, which satisfy the commutation relations [u,J]=iv, [v,J]=−iu, [u,v]=0. We can construct the Hamiltonian H=J ²+gu, where g is a real parameter, from these elements. This Hamiltonian is Hermitian and consequently it has real eigenvalues. However, we can also construct the PT\mathcal{P}\mathcal{T}-symmetric and non-Hermitian Hamiltonian H=J ²+igu, where again g is real. As in the case of PT\mathcal{P}\mathcal{T}-symmetric Hamiltonians constructed from the elements x and p of the Heisenberg algebra, there are two regions in parameter space for this PT\mathcal{P}\mathcal{T}-symmetric Hamiltonian, a region of unbroken PT\mathcal{P}\mathcal{T} symmetry in which all the eigenvalues are real and a region of broken PT\mathcal{P}\mathcal{T} symmetry in which some of the eigenvalues are complex. The two regions are separated by a critical value of g.     

$\ensuremath{\mathcal{P}}$$\ensuremath{\mathcal{T}}$ Symmetry vs. Hermiticity

The relation between the P\ensuremath{\mathcal{P}} T\ensuremath{\mathcal{T}} symmetry and Hermiticity is discussed. In the finite-dimensional linear space, any Hermitian matrix is a special case of P\ensuremath{\mathcal{P}} T\ensuremath{\mathcal{T}}-symmetric matrices. Explicit results in 2×2 are shown. The early belief that the P\ensuremath{\mathcal{P}} T\ensuremath{\mathcal{T}}-symmetric quantum mechanics is a generalization of the conventional Hermitian quantum mechanics is confirmed.     

Inclusion relations for the spaces L(ℝ), \mathcal{H}\mathcal{K} (ℝ) ∩ \mathcal{B}\mathcal{V} (ℝ), and L 2(ℝ)

The inclusion relations for the spaces $ \mathcal{H}\mathcal{K} $ (I), L(I), $ \mathcal{H}\mathcal{K} $ (I) ∩ $ \mathcal{B}\mathcal{V} $ (I), and L ²(I) are found. On unbounded intervals, functions in $ \mathcal{H}\mathcal{K} $ (I) ∩ $ \mathcal{B}\mathcal{V} $ (I) need not be Lebesgue integrable.     

Why is \mathcal{CPT} Fundamental?

Lüders and Pauli proved the theorem based on Lagrangian quantum field theory almost half a century ago. Jost gave a more general proof based on “axiomatic” field theory nearly as long ago. The axiomatic point of view has two advantages over the Lagrangian one. First, the axiomatic point of view makes clear why is fundamental—because it is intimately related to Lorentz invariance. Secondly, the axiomatic proof gives a simple way to calculate the transform of any relativistic field without calculating , and separately and then multiplying them. The purpose of this pedagogical paper is to “deaxiomatize” the theorem by explaining it in a few simple steps. We use theorems of distribution theory and of several complex variables without proof to make the exposition elementary.     

Hadronic η and $ \eta{^\prime}$ decays

The hadronic decays η, ↦3π and ↦ηππ are investigated within the framework of U(3) chiral effective field theory in combination with a relativistic coupled-channels approach. Final state interactions are included by deriving s- and p-wave interaction kernels for meson-meson scattering from the chiral effective Lagrangian and iterating them in a Bethe-Salpeter equation. Very good overall agreement with currently available data on decay widths and spectral shapes is achieved.     

$J^{P}$ Assignments of $\Lambda_c^{+}$ Baryons

The ＂good＂ diquark is employed to study A＋ baryons within a mass loaded flux tube model. The study indicates that all A＋ baryons candidates in the 2008 review by the Particle Data Group （PDG） are well described in the mass loaded flux model. The quantum numbers JP of these A＋ candidates are assigned. If Ac（2765）＋ is an orbitally excited A＋, it is likely the JP=3/2＋one. If ∧c（2765）＋ is an orbitally excited ∑c, there ought to be another JP=3/2＋∧c＋with mass ≈2770 MeV. In the model, there exists no JP=1/2＋∧c＋（≈2700）predicted in existing literature. Ac （2940）＋ is very possible the orbitally excited baryon with     

$\mathcal {}$-Matrix-valued Wave Packet Frames in L2(ℝd,ℂs×r)$L^{2}(\mathbb {R}^{d}, \mathbb {C}^{s\times r})$

We study frame properties of a matrix-valued wave packet system in the matrix-valued function space \(L^{2}(\mathbb {R}^{d}, \mathbb {C}^{s\times r})\), where the lower frame condition is controlled by a bounded linear operator \(\mathcal {K}\) on \(L^{2}(\mathbb {R}^{d}, \mathbb {C}^{s\times r})\) (lower \(\mathcal {K}\)-frame condition, in short). There are many differences between ordinary frames and \(\mathcal {K}\)-frames. The lower \(\mathcal {K}\)-frame condition for matrix-valued wave packet Bessel sequences in \(L^{2}(\mathbb {R}^{d},\mathbb {C}^{s\times r})\) in terms of operators; a trace functional associated with a bounded linear operator on \(L^{2}(\mathbb {R}^{d}, \mathbb {C}^{s\times r})\); and a series associated with a matrix-valued Bessel sequence is presented. It is shown that matrix-valued wave packet frames are stable under small perturbation with respect to wave packet window functions.     

Superconductivity in the doped “pseudo-ladder” compound CaCu$_\mathsf{2}$O$_\mathsf{3}$

An effective anisotropic t-J model for the pseudo-ladder compound CaCu₂O₃ is proposed based on recent experimental studies and band structure calculations. Superconducting pairing mediated by the exchange interaction in the model is investigated as a function of doping away from the antiferromagnetic insulating state. It is shown that strong anisotropy in the electronic spectrum suppresses superconducting temperature in comparison with conventional copper-oxide superconductors with square lattices.     

Three-Body Baryonic B ̄ 0 → Λ c + p ̄ π 0 $\bar {B}^{0}\rightarrow {{\varLambda }}^{+}_{c} \bar {p} \pi ^{0}$ Decay

International Journal of Theoretical Physics - We study the three-body baryonic decay $ \bar {B}^{0}\rightarrow {{\varLambda }}^{+}_{c} \bar {p} \pi ^{0}$ based on the factorization approach. The...     

Sum rule estimate of the subleading non-perturbative contributions to Bs–$\bar{ B}_{s}$ mixing

We use QCD sum rules to compute the matrix elements of the ΔB=2 operators appearing in the heavy-quark expansion of the width difference of the B _s mass eigenstates. The main focus of our analysis is on the subleading operators R ₂ and R ₃, which appear at next-to-leading order in the 1/m _b expansion. The matrix elements of these operators are already essential for precise phenomenology, but their calculation in lattice QCD is lacking and the values given here provide a first estimate of their values. We conclude that the violation of the factorization approximation for these matrix elements due to non-perturbative vacuum condensates is as low as 1–2%.     

Analysis of the vertexes \Xi_{Q}^{*}′QV , \Sigma_{Q}^{*} \Sigma_{Q}^{}V and radiative decays \Xi_{Q}^{*} \rightarrow ′Q \gamma , \Sigma_{Q}^{*} \rightarrow \Sigma_{Q}^{} \gamma

In this article, we study the vertexes $ \Xi_{Q}^{*}$ ′_Q V and $ \Sigma_{Q}^{*}$ $ \Sigma_{Q}^{}$ V with the light-cone QCD sum rules, then assume the vector meson dominance of the intermediate $ \phi$ (1020) , $ \rho$ (770) and $ \omega$ (782) , and calculate the radiative decays $ \Xi_{Q}^{*}$ $ \rightarrow$ ′_Q $ \gamma$ and $ \Sigma_{Q}^{*}$ $ \rightarrow$ $ \Sigma_{Q}^{}$ $ \gamma$ .     

Role of the Λ(1600) in the ${K}^{-}p \rightarrow {\rm{\Lambda }}{\pi }^{0}{\pi }^{0}$ reaction

Role of the Λ(1600) is studied in the ${K}^{-}p\to {\rm{\Lambda }}{\pi }^{0}{\pi }^{0}$ reaction by using the effective Lagrangian approach near the threshold. We perform a calculation for the total and differential cross sections by considering the contributions from the Λ(1600) and Λ(1670) intermediate resonances decaying into ${\pi }^{0}{{\rm{\Sigma }}}^{* 0}(1385)$ with ${{\rm{\Sigma }}}^{* 0}(1385)$ decaying into ${\pi }^{0}{\rm{\Lambda }}$. Additionally, the non-resonance process from u-channel nucleon pole is also taken into account. With our model parameters, the current experimental data on the total cross sections of the ${K}^{-}p\to {\rm{\Lambda }}{\pi }^{0}{\pi }^{0}$ reaction can be well reproduced. It is shown that we really need the contribution from the Λ(1600) with spin-parity ${J}^{P}=1/{2}^{+}$, and that these measurements can be used to determine some of the properties of the Λ(1600) resonance. Furthermore, we also plot the π⁰Λ invariant mass distributions which could be tested by the future experimental measurements.     

Solution of an analogous Schrödinger equation for \mathcal{P}\mathcal{T} -symmetric sextic potential in two dimensions

We investigate the quasi-exact solutions of an analogous Schrödinger wave equation for two-dimensional non-Hermitian complex Hamiltonian systems within the framework of an extended complex phase space characterized by x = x ₁ + ip ₃, y = x ₂ + ip ₄, p _x = p ₁ + ix ₃, p _y = p ₂ + ix ₄. Explicit expressions for the energy eigenvalues and eigenfunctions for ground and first excited states of a two-dimensional $ \mathcal{P}\mathcal{T} $ -symmetric sextic potential and some of its variants are obtained. The eigenvalue spectra are found to be real within some parametric domains.     

\mathcal{A} -invariance: An Axiomatic Approach to Quantum Relativity

The sort of approach claimed by the title of this article is realizable, at least, within the framework of ADG where we do not assume any “spacetime” supplying the dynamics we employ. The latter classical type of argument can naturally be included herewith along with its concomitant impediments that are emanated therefrom and are essentially “absorbed”, technically speaking, by the proposed mechanism. So our approach, being “manifoldless” (thence, no smoothness, in the standard sense) does not contain any such issue, as before, according to the very definitions, being thus “singularities”-free. As a consequence, the equations that one would be able to formulate within the present set-up will be, by the very essence of the matter, already the quantized ones. Dedicated to Professor Rafael D. Sorkin on the occasion of his 60th birthday with much friendship and recognition of his creative pursuit in theoretical physics.     

Deuteron and antideuteron measurements in Au + Au collisions at $\sqrt{s_{NN}}=200~\mbox{GeV}$ at RHIC-PHENIX

In Run-7 of RHIC operations PHENIX has recorded over 5.4 billion minimum bias events, facilitating a detailed study of deuteron and antideuteron production as a function of centrality and up to transverse momentum of p _T =5 GeV/c. We present transverse momentum p _T and transverse mass m _T spectra, mean transverse momentum 〈p _T 〉, source parameters, particle ratios, and nuclear modification factor R _CP . Deuteron and antideuteron spectra are compared to those of other identified particles and to hydrodynamical predictions.     

$\mathfrak{D}$ -Differentiation in Hilbert Space and the Structure of Quantum Mechanics

An appropriate kind of curved Hilbert space is developed in such a manner that it admits operators of - and -differentiation, which are the analogues of the familiar covariant and D-differentiation available in a manifold. These tools are then employed to shed light on the space-time structure of Quantum Mechanics, from the points of view of the Feynman ‘path integral’ and of canonical quantisation. (The latter contains, as a special case, quantisation in arbitrary curvilinear coordinates when space is flat.) The influence of curvature is emphasised throughout, with an illustration provided by the Aharonov-Bohm effect.     

Vitali $$\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\smile}$}}{l} $$ Anatol’evich Zverev (On his 80th birthday)

Chronicle

Vitali $\overset{\lower0.5em\hbox{$\overset{\lower0.5em\hbox{ Anatol’evich Zverev (On his 80th birthday)