Methods of information geometry

a ) p. 131 The discussion between eqs. (5.14) and (5.15) is incorrect (dA should be made as large as possible!). b ) p. 256 In the figure, the numbers 6) and 7) occur twice. c ) p. 292 At the end of section 12.5, it should be the space of isospectral 0Hermitian matrices. d ) p. 306 A ”Tr” is missing in eq. (13.43). e ) p. 327, Eq. (14.64b) is 〈Trρ〉B = N(14N+10) (5N+1)(N+3) should be 〈Trρ〉B = 8N+7 (N+2)(N+4)

Geometry of Quantum States: Frontmatter

Kaluza-Klein Supergravity

In the classical theory black holes can only absorb and not emit particles. However it is shown that quantum mechanical effects cause black holes to create and emit particles as if they were hot bodies with temperature\(\frac{{h\kappa }}{{2\pi k}} \approx 10^{ - 6} \left( {\frac{{M_ \odot }}{M}} \right){}^ \circ K\) where κ is the surface gravity of the black hole. This thermal emission leads to a slow decrease in the mass of the black hole and to its eventual disappearance: any primordial black hole of mass less than about 1015 g would have evaporated by now. Although these quantum effects violate the classical law that the area of the event horizon of a black hole cannot decrease, there remains a Generalized Second Law:S+1/4A never decreases whereS is the entropy of matter outside black holes andA is the sum of the surface areas of the event horizons. This shows that gravitational collapse converts the baryons and leptons in the collapsing body into entropy. It is tempting to speculate that this might be the reason why the Universe contains so much entropy per baryon.

/pdf/particle-creation-by-black-holes-1htfzwsw5k.pdf

Particle Creation by Black Holes

https://chaos.if.uj.edu.pl/~karol/erratum10.pdf

Geometry of Quantum States

A theory of graded Banach modules over a Banach-Grassmann algebra is developed and applied to differential geometry of super-manifolds. The explicit structure of superspaces carrying Poincare supersymmetry and extended supersymmetry, including central charges, is described.

/pdf/superspaces-and-supersymmetries-3id0obgx09.pdf

Superspaces and supersymmetries

LetG be a compact group of transformation (global symmetry group) of a manifoldE (multidimensional universe) with all orbits of the same type (one stratum). We studyG invariant metrics onE and show that there is one-to-one correspondence between those metrics and triples (gμv,Aμa,hαβ), wheregμv is a (pseudo-) Riemannian metric on the space of orbits (space-time),Aμa is a Yang-Mills field for the gauge groupN|H, whereN is the normalizer of the isotropy groupH inG, andhαβ are certain scalar fields characterizing geometry of the orbits (internal spaces). The scalar curvature ofE is expressed in terms of the component fields onM. Examples and model building recipes are also given. The results generalize those of non-abelian Kaluza-Klein theories to the case where internal spaces are not necessarily group manifolds.

/pdf/geometry-of-multidimensional-universes-56u992oh3q.pdf

Geometry of Multidimensional Universes

http://arkadiusz-jadczyk.org/papers/blaja93a.pdf

On the interaction between classical and quantum systems

The standard formalism of quantum theory is enhanced and definite meaning is given to the concepts of experiment, measurement and event. Within this approach one obtains a uniquely defined piecewise deterministic algorithm generating quantum jumps, classical events and histories of single quantum objects. The wave-function Monte Carlo method of Quantum Optics is generalized and promoted to the level of a fundamental process generating all the real events in Nature. The already worked out applications include SQUID-tank model and generalized cloud chamber model with GRW spontaneous localization as a particular case. Differences between the present approach and quantum measurement theories based on environment-induced master equations are stressed. Questions: what is classical, what is time, and what observers are addressed. Possible applications of the new approach are suggested, among them connection between the stochastic commutative geometry and Connes' noncommutative formulation of the Standard Model, as well as potential applications to the theory and practice of quantum computers.

Arkadiusz Jadczyk

Papers

Superspaces and supersymmetries

Geometry of Multidimensional Universes

On the interaction between classical and quantum systems

Event enhanced quantum theory and piecewise deterministic dynamics

Strongly coupled quantum and classical systems and Zeno's effect