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This work presents a brief discussion and a plan towards the analytical solving of Partial Differential Equations (PDEs) using symbolic computing, as well as an implementation of part of this plan as the PDEtools software-package of commands.
The Bisognano-Wichmann property on the geometric behavior of the modular group of the von Neumann algebras of local observables associated to wedge regions in Quantum Field Theory is shown to provide an intrinsic sufficient criterion for the existence of a covariant action of the (universal covering of) the Poincaré group. In particular this gives, together with our previous results, an intrinsic characterization of positive-energy conformal pre-cosheaves of von Neumann algebras. To this end we adapt to our use Moore theory of central extensions of locally compact groups by polish groups, selecting and making an analysis of a wider class of extensions with natural measurable properties and showing henceforth that the universal covering of the Poincaré group has only trivial central extensions (vanishing of the first and second order cohomology) within our class.
Given an irreducible inclusion of infinite von-Neumann-algebras $\cn \subset \cm$ together with a conditional expectation $ E : \cm \rightarrow \cm $ such that the inclusion has depth 2, we show quite explicitely how $\cn $ can be viewed as the fixed point algebra of $\cm$ w.r.t. an outer action of a compact Kac-algebra acting on $\cm$. This gives an alternative proof, under this special setting of a more general result of M. Enock and R. Nest, [E-N], see also S. Yamagami, [Ya2].
A study of Hilbert $C^*$-bimodules over commutative $C^*$-algebras is carried out and used to establish a sufficient condition for two quantum Heisenberg manifolds to be isomorphic.
We derive for a pair of operators on a symplectic space which are adjoints of each other with respect to the symplectic form (that is, they are sympletically adjoint) that, if they are bounded for some scalar product on the symplectic space dominating the symplectic form, then they are bounded with respect to a one-parametric family of scalar products canonically associated with the initially given one, among them being its ``purification''. As a typical example we consider a scalar field on a globally hyperbolic spacetime governed by the Klein-Gordon equation; the classical system is described by a symplectic space and the temporal evolution by symplectomorphisms (which are symplectically adjoint to their inverses). A natural scalar product is that inducing the classical energy norm, and an application of the above result yields that its ``purification'' induces on the one-particle space of the quantized system a topology which coincides with that given by the two-point functions of quasifree Hadamard states. These findings will be shown to lead to new results concerning the structure of the local (von Neumann) observable-algebras in representations of quasifree Hadamard states of the Klein-Gordon field in an arbitrary globally hyperbolic spacetime, such as local definiteness, local primarity and Haag-duality (and also split- and type III_1-properties). A brief review of this circle of notions, as well as of properties of Hadamard states, forms part of the article.
We show three Hahn-Banach type extension criteria for (sets of) bounded C*-linear maps of Hilbert C*-modules to the underlying C*-algebras of coefficients. One criterion establishes an alternative description of the property of (AW*-) C*-algebras to be monotone complete or additively complete.
A certain representation for the Heisenberg algebra in finite-difference operators is established. The Lie-algebraic procedure of discretization of differential equations with isospectral property is proposed. Using $sl_2$-algebra based approach, (quasi)-exactly-solvable finite-difference equations are described. It is shown that the operators having the Hahn, Charlier and Meixner polynomials as the eigenfunctions are reproduced in present approach as some particular cases. A discrete version of the classical orthogonal polynomials (like Hermite, Laguerre, Legendre and Jacobi ones) is introduced.
Translationally invariant symmetric polynomials as coordinates for $N$-body problems with identical particles are proposed. It is shown that in those coordinates the Calogero and Sutherland $N$-body Hamiltonians, after appropriate gauge transformations, can be presented as a {\it quadratic} polynomial in the generators of the algebra $sl_N$ in finite-dimensional degenerate representation. The exact solvability of these models follows from the existence of the infinite flag of such representation spaces, preserved by the above Hamiltonians. A connection with Jack polynomials is discussed.
Explicit examples of quasi-exactly-solvable $N$-body problems on the line are presented. These are related to the hidden algebra $sl_N$, and they are of two types -- containing up to $N$ (infinitely-many eigenstates are known, but not all) and up to 6 body interactions only (a finite number of eigenstates is known). Both types degenerate to the Calogero model.
A new 2-parameter quadratic deformation of the quantum oscillator algebra and its 1-parameter deformed Heisenberg subalgebra are considered. An infinite dimensional Fock module representation is presented which at roots of unity contains null vectors and so is reducible to a finite dimensional representation. The cyclic, nilpotent and unitary representations are discussed. Witten's deformation of $sl_2$ and some deformed infinite dimensional algebras are constructed from the $1d$ Heisenberg algebra generators. The deformation of the centreless Virasoro algebra at roots of unity is mentioned. Finally the $SL_q(2)$ symmetry of the deformed Heisenberg algebra is explicitly constructed.
We study the connection between $ζ$- and cutoff-regularized Casimir energies for scalar fields. We show that, in general, both regularization schemes lead to divergent contributions, and to finite parts which do not coincide. We determine the relationships among the various coefficients appearing in one approach and the other. As an application, we discuss the case of scalar fields in $d$-dimensional boxes under periodic boundary conditions.
On the basis of the classical theory of envelopes, we formulate the renormalization group (RG) method for global analysis, recently proposed by Goldenfeld et al. It is clarified why the RG equation improves things.
We consider hilbert spaces of holomorphic functions in Cartan domains (in particular in ball and polydisk) and operator of restriction of holomorphic function to a submanifold in Shilov boundary. We discuss conditions when this operator exists. Using such 'trace theorems' it is possible to construct discrete increments to spectra of some unitary representation and to catch singular unitary representations in the spectra. We also discuss spectral problems related to Berezin type kernels on riemann symmetric spaces.
For W*-algebras A and self-dual Hilbert A-modules M we show that every self-adjoint, ''compact'' module operator on M is diagonalizable. Some specific properties of the eigenvalues and of the eigenvectors are described.
A new approach to the Lance-Blecher theorem is presented resting on the interpretation of elements of Hilbert C*-module theory in terms of multiplier theory of operator C*-algebras: The Hilbert norm on a Hilbert C*-module allows to recover the values of the inducing C*-valued inner product in a unique way, and two Hilbert C*-modules {M_1, <.,.>_1}, {M_2, <.,.>_2} are isometrically isomorphic as Banach C*-modules if and only if there exists a bijective C*-linear map S: M_1 --> M_2 such that the identity <.,.>_1 \equiv _2 is valid. In particular, the values of a C*-valued inner product on a Hilbert C*-module are completely determined by the Hilbert norm induced from it. In addition, we obtain that two C*-valued inner products on a Banach C*-module inducing equivalent norms to the given one give rise to isometrically isomorphic Hilbert C*-modules if and only if the derived C*-algebras of ''compact'' module operators are *-isomorphic. The involution and the C*-norm of the C*-algebra of ''compact'' module operators on a Hilbert C*-module allow to recover its original C*-valued inner product up to the following equivalence relation: <.,.>_1 \sim <.,.>_2 if and only if there exists an invertible, positive element $a$ of the center of the multiplier C*-algebra M(A) of A such that the identity <.,.>_1 \equiv a \cdot <.,.>_2 holds.
Let $\cal M$ be a Banach C*-module over a C*-algebra $A$ carrying two $A$-valued inner products $< .,. >_1$, $<.,. >_2$ which induce equivalent to the given one norms on $\cal M$. Then the appropriate unital C*-algebras of adjointable bounded $A$-linear operators on the Hilbert $A$-modules $\{ {\cal M}, < .,. >_1 \}$ and $\{ {\cal M}, < .,. >_2 \}$ are shown to be $*$-isomorphic if and only if there exists a bounded $A$-linear isomorphism $S$ of these two Hilbert $A$-modules satisfying the identity $< .,. >_2 \equiv < S(.),S(.) >_1$. This result extends other equivalent descriptions due to L.~G.~Brown, H.~Lin and E.~C.~Lance. An example of two non-isomorphic Hilbert C*-modules with $*$-isomorphic C*-algebras of ''compact''/adjointable bounded module operators is indicated.