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.     

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.     

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.     

Unfolding the singularities in superspace

A method is described for unfolding the singularities in superspace, \(\mathcal{G} = \mathfrak{M}/\mathfrak{D}\) , the space of Riemannian geometries of a manifoldM. This unfolded superspace is described by the projection $$\mathcal{G}_{F\left( M \right)} = \frac{{\mathfrak{M} \times F\left( M \right)}}{\mathfrak{D}} \to \frac{\mathfrak{M}}{\mathfrak{D}} = \mathcal{G}$$ whereF(M) is the frame bundle ofM. The unfolded space \(\mathcal{G}_{F\left( M \right)}\) is infinite-dimensional manifold without singularities. Moreover, as expected, the unfolding of \(\mathcal{G}_{F\left( M \right)}\) at each geometry [g _o] ∈ \(\mathcal{G}\) is parameterized by the isometry groupIg _o (M) of g₀. Our construction is natural, is generally covariant with respect to all coordinate transformations, and gives the necessary information at each geometry to make \(\mathcal{G}\) a manifold. This construction is a canonical and geometric model of a nonrelativistic construction that unfolds superspace by restricting to those coordinate transformations that fix a frame at a point. These particular unfoldings are tied together by an infinite-dimensional fiber bundleE overM, associated with the frame bundleF(M), with standard fiber \(\mathcal{G}_{F\left( M \right)}\) , and with fiber at a point inM being the particular noncanonical unfolding of \(\mathcal{G}\) based at that point. ThusE is the totality of all the particular unfoldings, and so is a grand unfolding of \(\mathcal{G}\) .     

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).     

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: $$e^{itH} \mathfrak{M}e^{ - itH} = \mathfrak{M},t \in \mathbb{R}.$$      

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 \(\mathfrak{G}\) .     

Studying bi-partite entangled state representations via the integration over ket-bra operators in \mathfrak{Q}-ordering or \mathfrak{P}-ordering

For two particles＇ relative position and total momentum we have introduced the entangled state representation ｜μ〉, and its conjugate state｜ξ〉 In this work, for the first time; we study theln via the integration over ket bra operators in -ordering or -ordering, where Q-ordering means all Qs are to the left, of all Ps and -ordering means all Ps are to the left of all Qs. In this way we newly derive -ordered （or Q-ordered） expansion formulas of the two-mode squeezing operator which can show the squeezing effect on both the two-mode coordinate and momentum eigenstates. This tells that not only the integration over ket bra operators within normally ordered, but also within - ordered （or -ordered） are feasible and useful in developing quantum mechanical representation and transtbrlnation theory.     

Two-dimensional exactly and completely integrable dynamical systems

An investigation of two-dimensional exactly and completely integrable dynamical systems associated with the local part of an arbitrary Lie algebra \(\mathfrak{g}\) whose grading is consistent with an arbitrary integral embedding of 3d-subalgebra in \(\mathfrak{g}\) has been carried out. We have constructed in an explicit form the corresponding systems of nonlinear partial differential equations of the second order and obtained their general solutions in the sense of a Goursat problem. A method for the construction of a wide class of infinite-dimensional Lie algebras of finite growth has been proposed.     

Unbounded derivations ofC*-algebras II

It is demonstrated that a closed symmetric derivation δ of aC?-algebra \(\mathfrak{A}\) generates a strongly continuous one-parameter group of automorphisms of aC?-algebra \(\mathfrak{A}\) if and only if, it satisfies one of the following three conditions

1. (αδ+1)(D(δ))= \(\mathfrak{A}\) , α∈?\{0}.
2. δ possesses a dense set of analytic elements.
3. δ possesses a dense set of geometric elements.

Together with one of the following two conditions

1. ∥(αδ+1)(A)∥≧∥A∥, α∈IR,A∈D(δ).
2. If α∈IR andA∈D(δ) then (αδ+1)(A)≧0 impliesA≧0.

Other characterizations are given in terms of invariant states and the invariance ofD(δ) under the square root operation of positive elements.     

Double hypernuclei and the λ-gL interaction

The binding energy of the double hypernuclei \(\mathop \Lambda \limits^6 \Lambda\) He, \(\mathop {\Lambda \Lambda }\limits^{10}\) Be, and \(\mathop {\Lambda \Lambda }\limits^{11}\) Be and of the corresponding hypernuclei are calculated using the λ+λ+ core model. The parameters of the λ-λ interaction are found and compared to those of the λ-N interaction.     

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.     

Classical {\mathcal{W}}-Algebras and Generalized Drinfeld–Sokolov Hierarchies for Minimal and Short Nilpotents

We derive explicit formulas for λ-brackets of the affine classical \({\mathcal{W}}\) -algebras attached to the minimal and short nilpotent elements of any simple Lie algebra \({\mathfrak{g}}\) . This is used to compute explicitly the first non-trivial PDE of the corresponding integrable generalized Drinfeld–Sokolov hierarchies. It turns out that a reduction of the equation corresponding to a short nilpotent is Svinolupov’s equation attached to a simple Jordan algebra, while a reduction of the equation corresponding to a minimal nilpotent is an integrable Hamiltonian equation on 2h ˇ?3 functions, where h ˇ is the dual Coxeter number of \(\mathfrak{g}\) . In the case when \(\mathfrak{g}\) is \({\mathfrak{sl}_2}\) both these equations coincide with the KdV equation. In the case when \(\mathfrak{g}\) is not of type \({C_n}\) , we associate to the minimal nilpotent element of \(\mathfrak{g}\) yet another generalized Drinfeld–Sokolov hierarchy.     

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.