Indecomposable representations with invariant inner product

Consequences of the existence of an invariant (necessarily indefinite) non-degenerate inner product for an indecomposable representation π of a groupG on a space \(\mathfrak{H}\) are studied. If π has an irreducible subrepresentation π₁ on a subspace \(\mathfrak{H}_1 \) , it is shown that there exists an invariant subspace \(\mathfrak{H}_2 \) of \(\mathfrak{H}\) containing \(\mathfrak{H}_1 \) and satisfying the following conditions: (1) the representation π ₁ ^# =π mod \(\mathfrak{H}_2 \) on \(\mathfrak{H}\) mod \(\mathfrak{H}_2 \) is conjugate to the representation (π₁, \(\mathfrak{H}_1 \) ), (2) \(\mathfrak{H}_1 \) is a null space for the inner product, and (3) the induced inner product on \(\mathfrak{H}_2 \) mod \(\mathfrak{H}_1 \) is non-degenerate and invariant for the representation $$\pi _2 = (\pi _2 |_{\mathfrak{H}_2 } )\bmod \mathfrak{H}_1 ,$$ a special example being the Gupta-Bleuler triplet for the one-particle space of the free classical electromagnetic field with \(\mathfrak{H}_1 \) =space of longitudinal photons and \(\mathfrak{H}_2 \) =the space defined by the subsidiary condition.     

Unbounded derivations and invariant trace states

Let \(\mathfrak{M}\) be a von Neumann algebra with cyclic trace vector ?. Let δ(A)=i[H, A] be a spatial derivation of \(\mathfrak{M}\) implemented by an operatorH such thatH?=0 andH is essentially self-adjoint onD(δ)?. It follows that: $$e^{itH} \mathfrak{M}e^{ - itH} = \mathfrak{M},t \in \mathbb{R}.$$      

Normal and locally normal states

It is shown that if \(\mathfrak{A}\) is an irreducibleC* algebra on a Hilbert space ? andN is the set of normal states of \(\mathfrak{A}\) then the weak and uniform topologies onN coincide and are identical to the weak*- \(\mathfrak{A}\) topology for each \(\mathfrak{A} \supset \mathfrak{L}\mathfrak{C}\) (?). It is further shown that all weak* topologies coincide with the uniform topology on the set of normal states with finite energy or with finite conditional entropy. A number of continuity properties of the spectra of density matrices, the mean energy, and the conditional entropy are also derived. The extension of these results to locally normal states is indicated and it is established that locally normal factor states are characterized by a doubly uniform clustering property.     

Independence of local algebras in Quantum Field Theory

It is shown that localC*-algebras \(\mathfrak{A}\) (O ₁) and \(\mathfrak{A}\) (O ₂) associated with spacelike separated regionsO ₁ andO ₂ in the Minkowski space are independent. The proof is accomplished by a theorem concerning the structure of theC*-algebra generated by \(\mathfrak{A}\) (O ₁) and \(\mathfrak{A}\) (O ₂).     

A Garding domain for representations of some Hilbert Lie groups

Given a Banach representation of a Hilbert Lie group, the Lie algebra \(\mathfrak{G}\) of which is the closure of the union of an increasing sequence of finite dimensional subalgebras, we construct a Gårding domain on which we differentiate the group representation to a representation of a dense subalgebra of \(\mathfrak{G}\) .     

Three-loop relation of quark\overline {MS} and pole masses

We calculate, exactly, the next-to-leading correction to the relation between the \(\overline {MS} \) quark mass, \(\bar m\) , and the scheme-independent pole mass,M, and obtain $$\begin{gathered} \frac{M}{{\bar m(M)}} \approx 1 + \frac{4}{3}\frac{{\bar \alpha _s (M)}}{\pi } + \left[ {16.11 - 1.04\sum\limits_{i = 1}^{N_F - 1} {(1 - M_i /M)} } \right] \hfill \\ \cdot \left( {\frac{{\bar \alpha _s (M)}}{\pi }} \right)^2 + 0(\bar \alpha _s^3 (M)), \hfill \\ \end{gathered} $$ as an accurate approximation forN _F?1 light quarks of massesM _i <M. Combining this new result with known three-loop results for \(\overline {MS} \) coupling constant and mass renormalization, we relate the pole mass to the \(\overline {MS} \) mass, \(\bar m\) (μ), renormalized at arbitrary μ. The dominant next-to-leading correction comes from the finite part of on-shell two-loop mass renormalization, evaluated using integration by parts and checked by gauge invariance and infrared finiteness. Numerical results are given for charm and bottom \(\overline {MS} \) masses at μ=1 GeV. The next-to-leading corrections are comparable to the leading corrections.     

On the energy-dependence of the mass-operator and the „effective-interaction“

We consider a microscopic system \(\mathfrak{S}\) coupled to a bath \(\mathfrak{B}\) and establish a non-Markoffian master equation for the reduced statistical operator of \(\mathfrak{S}\) , valid in theBorn approximation. Discussing in detail theBorn approximation we find as a general condition for its validity that a certain “strength function” should not degenerate to one or more extremly sharp and high lines.     

Spectral signs of the internal heavy-atom effect in aliphatic carbonyl compounds

The CNDO/S method has been applied to the internal effect of Si on the electronic spectrum of the acetone molecule; there is a considerable bathochromic shift and an increase in the \(S_0 \to S_{n\pi ^ * } \) intensity for theα-silyl ketones, while theβ-silyl ketons give only an increase in the intensity of \(S_0 \to S_{n\pi ^ * } \) absorption relative to acetone. The heavy atom substantially alters \(f_{S_0 \to T_{n\sigma ^* } } \) and \(\tau _{T_{n\sigma ^* } }^0 \) but has little effect on \(f_{S_0 \to T_{n\pi ^* } } \) and \(\tau _{T_{n\pi ^* } }^0 \) .     

Singular $$ \mathcal{R}$$ -Matrices and Drinfeld's Comultiplication

We compute the $\mathcal{R}$ -matrix which intertwines two dimensional evaluation representations with Drinfeld comultiplication for ${\text{U}}_q \left( {\widehat{{\text{sl}}}_{\text{2}} } \right)$ . This $\mathcal{R}$ -matrix contains terms proportional to the δ-function. We construct the algebra $A\left( \mathcal{R} \right)$ generated by the elements of the matrices L^±(z) with relations determined by $\mathcal{R}$ . In the category of highest-weight representations, there is a Hopf algebra isomorphism between $A\left( \mathcal{R} \right)$ and an extension $\overline {\text{U}} _q \left( {\widehat{{\text{sl}}}_{\text{2}} } \right)$ of Drinfeld's algebra.     

Functional Properties of Hörmander’s Space of Distributions Having a Specified Wavefront Set

The space \({\mathcal{D}_\Gamma^\prime}\) of distributions having their wavefront sets in a closed cone \({\Gamma}\) has become important in physics because of its role in the formulation of quantum field theory in curved spacetime. In this paper, the topological and bornological properties of \({\mathcal{D}_\Gamma^\prime}\) and its dual \({\mathcal{E}_\Lambda^\prime}\) are investigated. It is found that \({\mathcal{D}_\Gamma^\prime}\) is a nuclear, semi-reflexive and semi-Montel complete normal space of distributions. Its strong dual \({\mathcal{E}_\Lambda^\prime}\) is a nuclear, barrelled and (ultra)bornological normal space of distributions which, however, is not even sequentially complete. Concrete rules are given to determine whether a distribution belongs to \({\mathcal{D}_\Gamma^\prime}\) , whether a sequence converges in \({\mathcal{D}_\Gamma^\prime}\) and whether a set of distributions is bounded in \({\mathcal{D}_\Gamma^\prime}\) .     

Back to the Amitsur–Levitzki Theorem: a Super Version for the Orthosymplectic Lie Superalgebra $${\mathfrak{o}}{\mathfrak{s}}{\mathfrak{p}}\left( {1,2n} \right)$$

We prove an Amitsur–Levitzki type theorem for the Lie superalgebras $\mathfrak{o}\mathfrak{s}\mathfrak{p}\left( {1,2n} \right)$ ) inspired by Kostant's cohomological interpretation of the classical theorem. We show that the Lie superalgebras $\mathfrak{g}\mathfrak{l}\left( {p,q} \right)$ cannot satisfy an Amitsur–Levitzki type super identity if pq≠0 and conjecture that neither can any other classical simple Lie superalgebra with the exception of $\mathfrak{o}\mathfrak{s}\mathfrak{p}\left( {1,2n} \right)$ .     

Comparison of strange antibaryon and strange meson production inK + p interactions at 32 GeV/c

New experimental results are presented on inclusive production properties of \(\bar \Sigma ^{ * + } \) (1385) and \(\bar \Sigma ^{ * + } \) (1385) inK ⁺ p interactions at 32 GeV/c. The analysis is based on significantly larger statistics than previously available. A comparison is also made of invariantx-distributions ofK ⁰/ \(\bar K^0 \) , \(\bar \Lambda \) and \(\bar \Xi ^ + \) and of \(\bar \Sigma ^{ * \pm } \) (1385) andK ^*+(892). These spectra exhibit regularities expected from the quark-recombination picture when it is assumed that the strange mesons and antibaryons are produced off the strange \(\bar s\) -valence-quark in the incidentK ⁺ meson. Transverse momentum distributions are also presented forK ^*+(892) and \(\bar \Sigma ^{ * \pm } \) (1385) and found to be very similar. The results on strange antibaryon average multiplicities disagree strongly with a recent version of the additive quark model.     

Finite W-Superalgebras and Truncated Super Yangians

Let ${Y_{m|n}^{\ell}}$ be the super Yangian of general linear Lie superalgebra for ${\mathfrak{gl}_{m|n}}$ . Let ${e \in \mathfrak{gl}_{m\ell|n\ell}}$ be a “rectangular” nilpotent element and ${\mathcal{W}_e}$ be the finite W-superalgebra associated to e. We show that ${Y_{m|n}^{\ell}}$ is isomorphic to ${\mathcal{W}_e}$ .     

Separation of variables and quantum numbers for Weyl neutrino fields on curved space-time

We construct, via separation of variables, a first-order differential operator whose commutator with the Weyl neutrino operator is proportional to it in the class \(\mathfrak{D}\) of Petrov type D vacuum and electrovac backgrounds with cosmological constant. Further, we characterize tensorially all such operators on an arbitrary background space-time. These results when combined explain the existence of a two-index Killing spinor in the class \(\mathfrak{D}\) .     

Instability of the chiral invariant vacuum due to gluon exchange

It has been shown by Finger, Horn and Mandula in the Tamm-Dancoff approximation that Coulomb exchange induces vacuum instability for α _s bigger than some critical value. We show here in all generality that the critical coupling is lower using the Bogolioubov-Valatin variational method. For Coulomb exchange we find \(\tfrac{4}{3}\alpha _s^{crit} = 1\) instead of \(\tfrac{4}{3}\alpha _s^{crit} = \tfrac{3}{2}\) , and adding transverse gluon exchange with the Breit interaction, \(\tfrac{4}{3}\alpha _s^{crit} = \tfrac{1}{3}\) . It is remarkable that these values of α _s ^crit are not far from the range of perturbative QCD.     

Production of π0 mesons and charged hadrons in\bar v neon and ν neon charged current interactions

The average multiplicities of charged hadrons and of π⁺, π^? and π⁰ mesons, produced in \(\bar v\) Ne and νNe charged current interactions in the forward and backward hemispheres of theW ^±-nucleon center of mass system, are studied with data from BEBC. The dependence of the multiplicities on the hadronic mass (W) and on the laboratory rapidity (y _Lab) and the energy fraction (z) of the pion is also investigated. Special care is taken to determine the π⁰ multiplicity accurately. The ratio of average π multiplicities \(\frac{{2\left\langle {n_{\pi ^O } } \right\rangle }}{{[\left\langle {n_{\pi ^ + } } \right\rangle + \left\langle {n_{\pi ^ - } } \right\rangle ]}}\) is consistent with 1. In the backward hemisphere \(\left\langle {n_{\pi ^O } } \right\rangle \) is positively correlated with the charged multiplicity. This correlation, as well as differences in multiplicities between \(\mathop v\limits^{( - )} \) and \(\mathop v\limits^{( - )} \) , \(\mathop v\limits^{( - )} \) scattering, is attributed to reinteractions inside the neon nucleus of the hadrons produced in the initial \(\mathop v\limits^{( - )} \) interaction.     

Characteristics of π−,\bar p,and antinuclei frompp collisions

An investigation of inclusivepp→π^?+? in terms of the covariant Boltzmann factor (BF) including the chemical potential μ indicates a) that the temperatureT increases less rapidly than expected from Stefan's law, b) that a scaling property holds for the fibreball velocity of π^? secondaries, leading to a multiplicity law like ～E _cm ^1/2 at high energy, and c) that μ_π is related to the quark mass: μ_π=2m _q ?m _π the quark massm _q determined by \(T_{\pi ^ - } \) at \(\bar pp\) threshold beingm _q =3T_π?330 MeV. Because ofthreshold effects \(T_{\bar p}< T_{\pi ^ - } \) , whereas \({{\mu _p } \mathord{\left/ {\vphantom {{\mu _p } {\mu _{\pi ^ - } }}} \right. \kern-0em} {\mu _{\pi ^ - } }} \simeq {3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-0em} 2}\) as expected from the quark contents of \(\bar p\) and π. The antinuclei \(\bar d\) and \({{\bar t} \mathord{\left/ {\vphantom {{\bar t} {\overline {He^3 } }}} \right. \kern-0em} {\overline {He^3 } }}\) observed inpp events are formed by coalescence of \(\bar p\) and \(\bar n\) produced in thepp collision. Semi-empirical formulae are proposed to estimate multiplicities of π^?, \(\bar p\) and antinuclei.