We present the global existence theorem for semilinear Schrödinger equations satisfying gauge invariance. Combining the local existence results and a priori estimates, we construct global solutions with small initial data.

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This paper studies a class of representations (called quadratic) of the canonical commutation relations over symplectic spaces of arbitrary dimension, which naturally generalizes coherent and symplectic (i.e. quasifree) representations and which has previously been heuristically employed in the special case of finite degrees of freedom in the physics literature. An explicit characterization of canonical quadratic transformations in terms of a ‘standard form’ is given, and it is shown that they can be exponentiated to give representations of the Weyl algebra. Necessary and sufficient conditions are presented for the unitary equivalence of these representations with the Fock representation. Possible applications to quantum optics and quantum field theory are briefly indicated.

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We construct a model for a chemical reaction in the Heisenberg representation. The time evolution exists in the thermodynamic limit, independent of the particle density and the initial state. In mathematical terms, we establish a C^*-dynamical system. Thus we can benefit from general results in algebraic quantum statistical mechanics, showing, for example, that equilibrium states exist. Galilei invariance of our nonrelativistic model is demonstrated by defining it directly on the Galilean space-time manifold, without reference to any coordinate system. PACS 05.30.Fk, 03.70.+k, 11.10.Cd, 11.15.Tk

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Every positive definite Fourier hyperfunction is a Fourier transform of a positive and infra-exponentially tempered measure, which is the generalized Boehner-Schwartz theorem for the Fourier hyperfunctions. To prove this we characterize the distributions with exponential growth via the heat kernel method.

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If U_q(\mathfrak{g}) is a finite-dimensional complex simple Lie algebra, an affinization of a finite-dimensional irreducible representation V of U_q(\mathfrak{g}) is a finite-dimensional irreducible representation \hat{V} of U_q(\hat{\mathfrak{g}}) which contains V with multiplicity one, and is such that all other U_q(\mathfrak{g})-types in \hat{V} have highest weights strictly smaller than that of V. We define a natural partial ordering {\preceq} on the set of affinizations of V. If \mathfrak{g} is of rank 2, we show that there is a unique minimal element with respect to this order and give its U_q(\mathfrak{g})-module structure when \mathfrak{g} is of type A₂ or C₂.

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It is proved that the groupoid of nonsingular partial isomorphisms of a Lebesgue space (X, μ) is weakly contractible in a “strong” sense: we present a contraction path which preserves invariant the subgroupoid of μ-preserving partial isomorphisms as well as the group of nonsingular transformations of X. Moreover, let \mathscr{R} be an ergodic measured discrete equivalence relation on X. The full group [\mathscr{R}] endowed with the uniform topology is shown to be contractible. For an approximately finite \mathscr{R} of type II or III_λ, 0≤λ<1, the normalizer N[\mathscr{R}] of \mathscr{R} furnished with the natural Polish topology is established to be homotopically equivalent to the centralizer of the associated Poincaré flow. These are the measure theoretical analogues of the resent results of S. Popa and M. Takesaki on the topological structure of the unitary and the automorphism group of a factor.

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In this short note, we determine precisely which operators have the property that their (full, symmetric or antisymmetric) second quantisation is an operator which is bounded or belongs to one of the various Schatten ideals; we also note that in ‘the interior’ of the natural domain, the second quantisation is a continuous map.

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Let G be a locally compact abelian group and let ξ be a principal G-bundle:
G→E\xrightarrow{p}B,
where E is path connected and B is a locally finite CW complex. We recall that G acts freely on the right of E. Then we denote by Top (B) the topological group of homeomorphisms of B and by Top_G(E) the group of G-equivariant homeomorphisms of E. Furthermore, let α be a map of B into the classifying space BG whose homotopy class [α] classifies the principal bundle ξ. Then we have
Corollary 4. Let G be a locally compact abelian group, then we have the following Sene fibration:
map(B, G)→Top_G(E)\xrightarrow{Φ}Top^[a](B),
where Top^[α](B) is the subspace of Top(B) consisting of homeomorphisms f : B → B such that f^*([α])=[α].
As a special case, let S¹ → S³ → S² be the Hopf principal bundle. By using Corollary 4 we have
Theorem 5. There exists a weak homotopy equivalence
Top_s¹(S³)\underset{w}≅Spin^c(3).

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