arXiv subjects

Mathias Pillin

Publications and source records attributed to Mathias Pillin.

7 recordsLinked to original sources

arXiv2025

Autonomous motion in changing environment, fibrations and reaction mechanisms

In this paper we develop further the formalism of fibrations of configuration spaces as a tool for modelling motion of autonomous systems in variable environments. We analyse the situations when the external conditions may change during the motion of the system and analyse two possibilities: (a) when the behaviour of the external conditions is known in advance; and (b) when the future changes of the external conditions are unknown but the autonomous system is able to measure the current state and the current velocity of the external conditions, at every moment of time. We prove that in the case (a) the complexity of the motion algorithm is the same as in the case of constant external conditions. In case (b) we introduce a new concept of a reaction mechanism which allows to take into account unexpected and unpredictable changes in the environment. A reaction mechanism is mathematically an infinitesimal lifting function on a fibre bundle, a nonlinear generalisation of the classical concept of an Ehresmann connection. We illustrate these notions by examples which show that nonlinear infinitesimal lifting function (reaction mechanisms) appear naturally, are inevitable and ubiquitous.

math.AT↗

arXiv1999

Replica-deformation of the SU(2)-invariant Thirring model via solutions of the qKZ equation

The response of an integrable QFT under variation of the Unruh temperature has recently been shown to be computable from an S-matrix preserving (`replica') deformation of the form factor approach. We show that replica-deformed form factors of the SU(2)-invariant Thirring model can be found among the solutions of the rational $sl_2$-type quantum Knizhnik-Zamolodchikov equation at generic level. We show that modulo conserved charge solutions the deformed form factors are in one-to-one correspondence to the ones at level zero and use this to conjecture the deformed form factors of the Noether current in our model.

hep-th↗

arXiv1997

Polynomial Recursion Equations in Form Factors of ADE Toda Field Theories

It is shown that the problem of calculating form factors in ADE affine Toda field theories can be reduced to the nonperturbative recursive calculation of polynomials symmetric in each sort of variables. We determine these recursion equations explicitly for the ADE series and characterize the polynomial solutions by an interplay between the weight space of the underlying Lie algebra and representations of the symmetric group.

hep-th↗

arXiv1997

Form Factors in $D_n^{(1)}$ Affine Toda Field Theories

We derive closed recursion equations for the symmetric polynomials occuring in the form factors of $D_n^{(1)}$ affine Toda field theories. These equations follow from kinematical- and bound state residue equations for the full form factor. We also discuss the equations arising from second and third order forward channel poles of the S-matrix. The highly symmetric case of $D_4^{(1)}$ form factors is treated in detail. We calculate explicitly cases with a few particles involved.

hep-th↗

arXiv1996

Selfdual 2-form formulation of gravity and classification of energy-momentum tensors

It is shown how the different irreducibility classes of the energy-momentum tensor allow for a Lagrangian formulation of the gravity-matter system using a selfdual 2-form as a basic variable. It is pointed out what kind of difficulties arise when attempting to construct a pure spin-connection formulation of the gravity-matter system. Ambiguities in the formulation especially concerning the need for constraints are clarified.

gr-qc↗

arXiv1995

On the deformability of Heisenberg algebras

Based on the vanishing of the second Hochschild cohomology group of the enveloping algebra of the Heisenberg algebra it is shown that differential algebras coming from quantum groups do not provide a non-trivial deformation of quantum mechanics. For the case of a q-oscillator there exists a deforming map to the classical algebra. It is shown that the differential calculus on quantum planes with involution, i.e. if one works in position-momentum realization, can be mapped on a q-difference calculus on a commutative real space. Although this calculus leads to an interesting discretization it is proved that it can be realized by generators of the undeformed algebra and does not posess a proper group of global transformations.

q-alg↗

arXiv1993

q-Deformed Relativistic Wave Equations

Based on the representation theory of the $q$-deformed Lorentz and Poincaré symmeties $q$-deformed relativistic wave equation are constructed. The most important cases of the Dirac-, Proca-, Rarita-Schwinger- and Maxwell- equations are treated explicitly. The $q$-deformed wave operators look structurally like the undeformed ones but they consist of the generators of a non-commu\-ta\-tive Minkowski space. The existence of the $q$-deformed wave equations together with previous existence of the $q$-deformed wave equations together with previous results on the representation theory of the $q$-deformed Poincaré symmetry solve the $q$-deformed relativistic one particle problem.

hep-th↗