arXiv subjects

Leszek Pysiak

Publications and source records attributed to Leszek Pysiak.

12 recordsLinked to original sources

arXiv2025

Functorial Einstein Algebras and the Malicious Singularity

Einstein algebra, the concept due to Geroch, is essentially general relativity in an algebraic disguise. We introduce the concept of Einstein-Grassmann algebra as a superalgebra (defining a supermanifold) which is also an Einstein algebra. We employ this concept to confront the supermanifold structure with the structure of strong singularity, the so-called malicious singularity, in general relativity. Einstein-Grassmann algebras consist of two parts: a part called body and a part called soul. For the body part, the singularity theorems apply and the singularities persist as the conclusions of the classical theorems on the existence of singularities require. We prove that, if we relax algebraical requirements, the soul part of the algebra can survive the malicious singularity. In particular, we study the behaviour of supercurves in the presence of malicious singularity.

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arXiv2023

Functorial differential spaces and the infinitesimal structure of space-time

We generalize the differential space concept as a tool for developing differential geometry, and enrich this geometry with infinitesimals that allow us to penetrate into the superfine structure of space. This is achieved by Yoneda embedding a ring of smooth functions into the category of loci. This permits us to define a category of functorial differential spaces. By suitably choosing various algebras as "stages" in this category, one obtains various classes of differential spaces, both known from the literature and many so far unknown. In particular, if one chooses a Weil algebra, infinitesimals are produced. We study the case with some Weil algebra which allows us to fully develop the corresponding differential geometry with infinitesimals. To test the behavior of infinitesimals, we construct a simplified RWFL cosmological model. As it should be expected, infinitesimals remain latent during the entire macroscopic evolution (regarded backwards in time), and come into play only when the universe attains infinitesimal dimensions. Then they penetrate into the structure of the initial singularity.

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arXiv2022

Einstein Algebras in a Categorical Context

According to the basic idea of category theory, any Einstein algebra, essentially an algebraic formulation of general relativity, can be considered from the point of view of any object of the category of smooth algebras; such an object is then called a stage. If we contemplate a given Einstein algebra from the point of view of the stage, which we choose to be an "algebra with infinitesimals" (Weil algebra), then we can suppose it penetrates a submicroscopic level, on which quantum gravity might function. We apply Vinogradov's notion of geometricity (adapted to this situation), and show that the corresponding algebra is geometric, but then the infinitesimal level is unobservable from the macro-level. However, the situation can change if a given algebra is noncommutative. An analogous situation occurs when as stages, instead of Weil algebras, we take many other smooth algebras, for example those that describe spaces in which with ordinary points coexsist "parametrized points", for example closed curves (loops). We also discuss some other consequences of putting Einstein algebras into the conceptual environment of category theory.

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arXiv2013

A Sheaf of von Neumann Algebras and Its Geometry

It is shown that the differential geometry of space-time, can be expressed in terms of the algebra of operators on a bundle of Hilbert spaces. The price for this is that the algebra of smooth functions on space-time has to be made noncommutative. The generalized differential geometry of space-time is constructed in terms of the algebra A (and its derivations) on a transformation groupoid. Regular representation π of A in the algebra of bounded operators on a bundle of Hilbert spaces leads to the algebra π(A) = M_0 which can be completed to the von Neumann algebra M . The representation πestablishes the isomorphism between A and M_0 which, in turn, implies the isomorphism between moduli of their derivations. In this way, geometry naturally transfers to the algebra M_0 and its derivations. Although geometry, as defined in terms of M_0, is formally isomorphic to that defined in terms of A, it exhibits a strong probabilistic flavour. However, the geometry of M_0 does not prolong to M . This is clearly a serious stumbling block to fully unify mathematical tools of general relativity and quantum theory.

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arXiv2011

Semidirect Product of Groupoids, Its Representations and Random Operators

One of pressing problems in mathematical physics is to find a generalized Poincaré symmetry that could be applied to nonflat space-times. As a step in this direction we define the semidirect product of groupoids $Γ_0 \rtimes Γ_1$ and investigate its properties. We also define the crossed product of a bundle of algebras with the groupoid $Γ_1$ and prove that it is isomorphic to the convolutive algebra of the groupoid $Γ_0 \rtimes Γ_1$. We show that families of unitary representations of semidirect product groupoids in a bundle of Hilbert spaces are random operators. An important example is the Poincaré groupoid defined as the semidirect product of the subgroupoid of generalized Lorentz transformations and the subgroupoid of generalized translations.

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arXiv2011

Geometry of Non-Hausdorff Spaces and Its Significance for Physics

Hausdorff relation, topologically identifying points in a given space, belongs to elementary tools of modern mathematics. We show that if subtle enough mathematical methods are used to analyze this relation, the conclusions may be far-reaching and illuminating. Examples of situations in which the Hausdorff relation is of the total type, i.e., when it identifies all points of the considered space, are the space of Penrose tilings and space-times of some cosmological models with strong curvature singularities. With every Hausdorff relation a groupoid can be associated, and a convolutive algebra defined on it allows one to analyze the space that otherwise would remain intractable. The regular representation of this algebra in a bundle of Hilbert spaces leads to a von Neumann algebra of random operators. In this way, a probabilistic description (in a generalized sense) naturally takes over when the concept of point looses its meaning. In this situation counterparts of the position and momentum operators can be defined, and they satisfy a commutation relation which, in the suitable limiting case, reproduces the Heisenberg indeterminacy relation. It should be emphasized that this is neither an additional assumption nor an effect of a quantization process, but simply the consequence of a purely geometric analysis.

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arXiv2010

Imprimitivity theorem for groupoid representations

We define and investigate the concept of the groupoid representation induced by a representation of the isotropy subgroupoid. Groupoids in question are locally compact transitive topological groupoids. We formulate and prove the imprimitivity theorem for such representations which is a generalization of the classical Mackey's theorem known from the theory of group representations.

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arXiv2008

General Relativity on Random Operators

We present a mathematical structure which unifies mathematical structures of general relativity and quantum mechanics. It consists of the noncommutative algebra of compactly supported, complex valued functions ${\mathcal A}$, with convolution as multiplication, on a groupoid $Γ$ the base of which is the total space $E$ of the frame bundle over space-time $M$. A differential geometry based on derivations of ${\mathcal A}$ suitably generalizes the standard differential geometry of space-time, and the algebra ${\mathcal A}$, when represented in a bundle of Hilbert spaces, defines a von Neumann algebra ${\mathcal M}$ of random operators that generalizes the usual quantum mechanics. The main result of the present paper is that there exists a space ${\mathcal M_0}$, dense in ${\mathcal M}$, that is isomorphic with the algebra ${\mathcal A}$. This isomorphism allows us to transfer all differentially geometric constructions, generalized Einstein's equations including, made with the help of ${\mathcal A}$ (and its derivations) to the space ${\mathcal M_0}$. In this way, we obtain a generalization of general relativity in terms of random operators on a bundle of Hilbert spaces. However, this generalization cannot be extended to the whole of ${\mathcal M}$, and this is the main mathematical obstacle, at least in this approach, to fully unify theory of gravity with physics of quanta.

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arXiv2007

Anatomy of Malicious Singularities

As well known, the b-boundaries of the closed Friedman world model and of Schwarzschild solution consist of a single point. We study this phenomenon in a broader context of differential and structured spaces. We show that it is an equivalence relation $ρ$, defined on the Cauchy completed total space $\bar{E}$ of the frame bundle over a given space-time, that is responsible for this pathology. A singularity is called malicious if the equivalence class $[p_0]$ related to the singularity remains in close contact with all other equivalence classes, i.e., if $p_0 \in \mathrm{cl}[p]$ for every $p \in E$. We formulate conditions for which such a situation occurs. The differential structure of any space-time with malicious singularities consists only of constant functions which means that, from the topological point of view, everything collapses to a single point. It was noncommutative geometry that was especially devised to deal with such situations. A noncommutative algebra on $\bar{E}$, which turns out to be a von Neumann algebra of random operators, allows us to study probabilistic properties (in a generalized sense) of malicious singularities. Our main result is that, in the noncommutative regime, even the strongest singularities are probabilistically irrelevant.

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arXiv2005

Noncommutative Unification of General Relativity and Quantum Mechanics

In Gen. Rel. Grav. (36, 111-126 (2004); in press, gr-qc/0410010) we have proposed a model unifying general relativity and quantum mechanics based on a noncommutative geometry. This geometry was developed in terms of a noncommutative algebra A defined on a transformation groupoid given by the action of a group G on a space E. Owing to the fact that G was assumed to be finite it was possible to compute the model in full details. In the present paper we develop the model in the case when G is a noncompact group. It turns out that also in this case the model works well. The case is important since to obtain physical effects predicted by the model we should assume that G is a Lorentz group or some of its representations. We show that the generalized Einstein equation of the model has the form of the eigenvalue equation for the generalized Ricci operator, and all relevant operators in the quantum sector of the model are random operators; we study their dynamics. We also show that the model correctly reproduces general relativity and the usual quantum mechanics. It is interesting that the latter is recovered by performing the measurement of any observable. In the act of such a measurement the model ``collapses'' to the usual quantum mechanics.

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arXiv2004

Observables in a Noncommutative Unification of Quanta and Gravity. A Final Model

We further develop a noncommutative model unifying quantum mechanics and general relativity proposed in {\it Gen. Rel. Grav.} (2004) {\bf 36}, 111-126. Generalized symmetries of the model are defined by a groupoid $Γ$ given by the action of a finite group on a space $E$. The geometry of the model is constructed in terms of suitable (noncommutative) algebras on $Γ$. We investigate observables of the model, especially its position and momentum observables. This is not a trivial thing since the model is based on a noncommutative geometry and has strong nonlocal properties. We show that, in the position representation of the model, the position observable is a coderivation of a corresponding coalgebra, "coparallelly" to the well known fact that the momentum observable is a derivation of the algebra. We also study the momentum representation of the model. It turns out that, in the case of the algebra of smooth, quickly decreasing functions on $Γ$, the model in its "quantum sector" is nonlocal, i.e., there are no nontrivial coderivations of the corresponding coalgebra, whereas in its "gravity sector" such coderivations do exist. They are investigated.

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arXiv2004

Noncommutative Dynamics of Random Operators

We continue our program of unifying general relativity and quantum mechanics in terms of a noncommutative algebra ${\cal A}$ on a transformation groupoid $Γ= E \times G$ where $E$ is the total space of a principal fibre bundle over spacetime, and $G$ a suitable group acting on $Γ$. We show that every $a \in {\cal A}$ defines a random operator, and we study the dynamics of such operators. In the noncommutative regime, there is no usual time but, on the strength of the Tomita-Takesaki theorem, there exists a one-parameter group of automorphisms of the algebra ${\cal A}$ which can be used to define a state dependent dynamics; i.e., the pair $({\cal A}, ϕ)$, where $ϕ$ is a state on ${\cal A}$, is a ``dynamic object''. Only if certain additional conditions are satisfied, the Connes-Nikodym-Radon theorem can be applied and the dependence on $ϕ$ disappears. In these cases, the usual unitary quantum mechanical evolution is recovered. We also notice that the same pair $({\cal A}, ϕ)$ defines the so-called free probability calculus, as developed by Voiculescu and others, with the state $ϕ$ playing the role of the noncommutative probability measure. This shows that in the noncommutative regime dynamics and probability are unified. This also explains probabilistic properties of the usual quantum mechanics.

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