Noncommutative Integration Research Articles

Matrix algebras over algebras of unbounded operators

Let $${{\mathscr {M}}}$$ be a $$II_1$$ factor acting on the Hilbert space $${{\mathscr {H}}}$$ , and $${\mathscr {M}} _{\text {aff}}$$ be the Murray–von Neumann algebra of closed densely-defined operators affiliated with $${{\mathscr {M}}}$$ . Let $$\tau $$ denote the unique faithful normal tracial state on $$\mathscr {M}$$ . By virtue of Nelson’s theory of non-commutative integration, $${\mathscr {M}} _{\text {aff}}$$ may be identified with the completion of $${{\mathscr {M}}}$$ in the measure topology. In this article, we show that $$M_n({\mathscr {M}} _{\text {aff}}) \cong M_n({{\mathscr {M}}})_{\text {aff}}$$ as unital ordered complex topological $$*$$ -algebras with the isomorphism extending the identity mapping of $$M_n({{\mathscr {M}}}) \rightarrow M_n({{\mathscr {M}}})$$ . Consequently, the algebraic machinery of rank identities and determinant identities are applicable in this setting. As a step further in the Heisenberg–von Neumann puzzle discussed by Kadison–Liu (SIGMA Symmetry Integrability Geom. Methods Appl., 10:Paper 009, 40, 2014), it follows that if there exist operators P, Q in $${\mathscr {M}} _{\text {aff}}$$ satisfying the commutation relation $$Q \; {{\hat{\cdot }}} \;P \; {\hat{-}} \;P \; {{\hat{\cdot }}} \;Q = {i\,}I$$ , then at least one of them does not belong to $$L^p({{\mathscr {M}}}, \tau )$$ for any $$0 < p \le \infty $$ . Furthermore, the respective point spectrums of P and Q must be empty. Hence the puzzle may be recasted in the following equivalent manner - Are there invertible operators P, A in $${{\mathscr {M}}}_{\text {aff}}$$ such that $$P^{-1} \; {{\hat{\cdot }}} \;A \; {{\hat{\cdot }}} \;P = I \; {\hat{+}} \;A$$ ? This suggests that any strategy towards its resolution must involve the study of conjugacy invariants of operators in $${{\mathscr {M}}}_{\text {aff}}$$ in an essential way.

Operators on Anti-dual pairs: Self-adjoint Extensions and the Strong Parrott Theorem

AbstractThe aim of this paper is to develop an approach to obtain self-adjoint extensions of symmetric operators acting on anti-dual pairs. The main advantage of such a result is that it can be applied for structures not carrying a Hilbert space structure or a normable topology. In fact, we will show how hermitian extensions of linear functionals of involutive algebras can be governed by means of their induced operators. As an operator theoretic application, we provide a direct generalization of Parrott’s theorem on contractive completion of 2 by 2 block operator-valued matrices. To exhibit the applicability in noncommutative integration, we characterize hermitian extendibility of symmetric functionals defined on a left ideal of a $C^{\ast }$-algebra.

In- and out-states of scalar particles confined between two capacitor plates

In the present article, using a non-commutative integration method of linear differential equations, we, considering the Klein-Gordon equation with the $L$-constant electric field with large $L$ and using the light cone variables, find new complete sets of its exact solutions. These solutions can be related by integral transformations to previously known solutions that were found in Phys. Rev. D. $\textbf{93}$, 045033(2016). Then, using the general theory developed in Phys. Rev. D. $\textbf{93}$, 045002 (2016), we construct (in terms of the new solutions) the so-called in- and out-states of scalar particles confined between two capacitor plates.

Некоммутативное интегрирование уравнения Клейна - Гордона в электромагнитных полях, допускающих функциональный произвол

Некоммутативное интегрирование уравнения Клейна - Гордона в электромагнитных полях, допускающих функциональный произвол

Vacuum quantum effects on Lie groups with bi-invariant metrics

We consider the effects of vacuum polarization and particle creation of a scalar field on Lie groups with a non-stationary bi-invariant metric of the Robertson–Walker type. The vacuum expectation values of the energy momentum tensor for a scalar field determined by the group representation are found using the noncommutative integration method for the field equations instead of separation of variables. The results obtained are illustrated by the example of the three-dimensional rotation group.

Nonholonomic connections, time reparametrizations, and integrability of the rolling ball over a sphere

We study a time reparametrisation of the Newton type equations on Riemannian manifolds slightly modifying the Chaplygin multiplier method, allowing us to consider the Chaplygin method and the Maupertuis principle within a unified framework. As an example, the reduced nonholonomic problem of rolling without slipping and twisting of an n-dimensional balanced ball over a fixed sphere is considered. For a special inertia operator (depending on n parameters) we prove complete integrability when the radius of the ball is twice the radius of the sphere. In the case of symmetry, noncommutative integrability for any ratio of the radii is established.

The Ihara zeta function for infinite graphs

We put forward the concept of measure graphs. These are (possibly uncountable) graphs equipped with an action of a groupoid and a measure invariant under this action. Examples include finite graphs, periodic graphs, graphings, and percolation graphs. Making use of Connes’s noncommutative integration theory, we construct a zeta function and present a determinant formula for it. We further introduce a notion of weak convergence of measure graphs and show that our construction is compatible with it. The approximation of the Ihara zeta function via the normalized version on finite graphs in the sense of Benjamini and Schramm follows as a special case. Our framework not only unifies corresponding earlier results occurring in the literature. It likewise provides extensions to rich new classes of objects such as percolation graphs.

INTEGRAL OPERATOR APPROACH OVER OCTONIONS TO SOLUTION OF NONLINEAR PDE

Integration of nonlinear partial differential equations with the help of the non-commutative integration over octonions is studied. An apparatus permitting to take into account symmetry properties of PDOs is developed. For this purpose formulas for calculations of commutators of integral and partial differential operators are deduced. Transformations of partial differential operators and solutions of partial differential equations are investigated. Theorems providing solutions of nonlinear PDEs are proved. Examples are given. Applications to PDEs of hydrodynamics and other types PDEs are described.

Around Trace Formulas in Non-commutative Integration

Trace formulas are investigated in non-commutative integration theory. The main result is to evaluate the standard trace of Takesaki duals and, for this, we introduce the notion of interpolator and accompanied boundary objects. The formula is then applied to explore a variation of Haagerup's trace formula.

Integration of the Dirac Equation on Lie Groups in an External Electromagnetic Field Admitting a Noncommutative Symmetry Algebra

Noncommutative integration of the Dirac equation on Lie groups with a right-invariant metric and an invariant electromagnetic field tensor is considered. An electromagnetic field, admitting a noncommutative reduction of the Dirac equation, but not admitting either diagonalization of the squared Dirac equation or separation of variables, is found.

Integrable reductions of the Bogoyavlenskij-Itoh Lotka-Volterra systems

Given a constant skew-symmetric matrix A, it is a difficult open problem whether the associated Lotka-Volterra system is integrable or not. We solve this problem in a special case when A is a Toeplitz matrix where all off-diagonal entries are plus or minus one. In this case, the associated Lotka-Volterra system turns out to be a reduction of Liouville integrable systems, whose integrability was shown by Bogoyavlenskij and Itoh. We prove that the reduced systems are also Liouville integrable and that they are also non-commutative integrable by constructing a set of independent first integrals, having the required involutive properties (with respect to the Poisson bracket). These first integrals fall into two categories. One set consists of polynomial functions that are restriction of the Bogoyavlenskij-Itoh integrals; their involutivity was already pointed out by Bogoyavlenskij. The other set consists of rational functions which are obtained through a Poisson map from the first integrals of some recently discovered superintegrable Lotka-Volterra systems. The fact that these polynomial and rational first integrals, combined, have the required properties for Liouville and non-commutative integrability is quite remarkable; the quite technical proof of functional independence of the first integrals is given in detail.

Noncommutative Integrability of the Klein–Gordon and Dirac Equations in (2+1)-Dimensional Spacetime

Noncommutative integration of the Klein–Gordon and Dirac relativistic wave equations in (2+1)-dimensional Minkowski space is considered. It is shown that for all non-Abelian subalgebras of the (2+1)-dimensional Poincare algebra the condition of noncommutative integrability is satisfied.

Virtual billiards in pseudo–euclidean spaces: Discrete hamiltonian and contact integrability

The aim of the paper is to unify the efforts in the study of integrable billiards within quadrics in flat and curved spaces and to explore further the interplay of symplectic and contact integrability. As a starting point in this direction, we consider virtual billiard dynamics within quadrics in pseudo-Euclidean spaces. In contrast to the usual billiards, the incoming velocity and the velocity after the billiard reflection can be at opposite sides of the tangent plane at the reflection point. In the symmetric case we prove noncommutative integrability of the system and give a geometrical interpretation of integrals, an analog of the classical Chasles and Poncelet theorems and we show that the virtual billiard dynamics provides a natural framework in the study of billiards within quadrics in projective spaces, in particular of billiards within ellipsoids on the sphere ${\mathbb{S}^{n - 1}}$ and the Lobachevsky space $\mathbb H^{n-1}$.

Noncommutative Integration and Symmetry Algebra of the Dirac Equation on the Lie Groups

The algebra of first-order symmetry operators of the Dirac equation on four-dimensional Lie groups with right-invariant metric is investigated. It is shown that the algebra of symmetry operators is in general not a Lie algebra. Noncommutative reduction mediated by spin symmetry operators is investigated. For the Dirac equation on the Lie group SO(2,1) a parametric family of particular solutions obtained by the method of noncommutative integration over a subalgebra containing a spin symmetry operator is constructed.

A generalization of Nekhoroshev’s theorem

Nekhoroshev discovered a beautiful theorem in Hamiltonian systems that includes as special cases not only the Poincare theorem on periodic orbits but also the theorem of Liouville–Arnol’d on completely integrable systems [7]. Sadly, his early death precluded him publishing a full account of his proof. The aim of this paper is twofold: first, to provide a complete proof of his original theorem and second a generalization to the noncommuting case. Our generalization of Nekhoroshev’s theorem to the nonabelian case subsumes aspects of the theory of noncommutative complete integrability as found in Mishchenko and Fomenko [5] and is similar to what Nekhoroshev’s theorem does in the abelian case.

Extensions of the Noncommutative Integration

In this paper we will continue the analysis undertaken in Bagarello et al. (Rend Circ Mat Palermo (2) 55:21–28, 2006), Bongiorno et al. (Rocky Mt J Math 40(6):1745–1777, 2010), Triolo (Rend Circ Mat Palermo (2) 60(3):409–416, 2011) on the general problem of extending the noncommutative integration in a *-algebra of measurable operators. As in Aiena et al. (Filomat 28(2):263–273, 2014), Bagarello (Stud Math 172(3):289–305, 2006) and Bagarello et al. (Rend Circ Mat Palermo (2) 55:21–28, 2006), the main problem is to represent different types of partial *-algebras into a *-algebra of measurable operators in Segal’s sense, provided that these partial *-algebras posses a sufficient family of positive linear functionals (states) (Fragoulopoulou et al., J Math Anal Appl 388(2):1180–1193, 2012; Trapani and Triolo, Stud Math 184(2):133–148, 2008; Trapani and Triolo, Rend Circolo Mat Palermo 59:295–302, 2010; La Russa and Triolo, J Oper Theory, 69:2, 2013; Triolo, J Pure Appl Math, 43(6):601–617, 2012). In this paper, a new condition is given in an attempt to provide a extension of the non commutative integration.

Noether symmetries and integrability in time-dependent Hamiltonian mechanics

We consider Noether symmetries within Hamiltonian setting as transformations that preserve Poincar?-Cartan form, i.e., as symmetries of characteristic line bundles of nondegenerate 1-forms. In the case when the Poincar?-Cartan form is contact, the explicit expression for the symmetries in the inverse Noether theorem is given. As examples, we consider natural mechanical systems, in particular the Kepler problem. Finally, we prove a variant of the theorem on complete (non-commutative) integrability in terms of Noether symmetries of time-dependent Hamiltonian systems.

The Dirac equation in an external electromagnetic field: symmetry algebra and exact integration

Integration of the Dirac equation with an external electromagnetic field is explored in the framework of the method of separation of variables and of the method of noncommutative integration. We have found a new type of solutions that are not obtained by separation of variables for several external electromagnetic fields. We have considered an example of crossed electric and magnetic fields of a special type for which the Dirac equation admits a nonlocal symmetry operator.

Higher abelian gauge theory associated to gerbes on noncommutative deformed M5-branes and S-duality

We enhance the action of higher abelian gauge theory associated to a gerbe on an M5-brane with an action of a torus ${\mathbb T}^n (n\ge 2)$, by a noncommutative ${\mathbb T}^n$-deformation of the M5-brane. The ingredients of the noncommutative action and equations of motion include the deformed Hodge duality, deformed wedge product, and the noncommutative integral over the noncommutative space obtained by strict deformation quantization. As an application we then introduce a variant model with an enhanced action in which we show that the corresponding partition function is a modular form, which is a purely noncommutative geometry phenomenon since the usual theory only has a $\mathbb Z_2$-symmetry. In particular, S-duality in this 6-dimensional higher abelian gauge theory model is shown to be, in this sense, on par with the usual 4-dimensional case.

Non-Commutative Integration, Zeta Functions and the Haar State for SU q (2)

We study a notion of non-commutative integration, in the spirit of modular spectral triples, for the quantum group $SU_{q}(2)$. In particular we define the non-commutative integral as the residue at the spectral dimension of a zeta function, which is constructed using a Dirac operator and a weight. We consider the Dirac operator introduced by Kaad and Senior and a family of weights depending on two parameters, which are related to the diagonal automorphisms of $SU_{q}(2)$. We show that, after fixing one of the parameters, the non-commutative integral coincides with the Haar state of $SU_{q}(2)$. Moreover we can impose an additional condition on the zeta function, which also fixes the second parameter. For this unique choice the spectral dimension coincides with the classical dimension.