Žan Grad | University of Ljubljana (original) (raw)
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Papers by Žan Grad
arXiv (Cornell University), Feb 10, 2023
We introduce the basic notions and present examples and results on Lie categories-categories inte... more We introduce the basic notions and present examples and results on Lie categories-categories internal to the category of smooth manifolds. Demonstrating how the units of a Lie category C dictate the behavior of its invertible morphisms G(C), we develop sufficient conditions for G(C) to form a Lie groupoid. We show that the construction of Lie algebroids from the theory of Lie groupoids carries through, and ask when the Lie algebroid of G(C) is recovered. We reveal that the lack of invertibility assumption on morphisms leads to a natural generalization of rank from linear algebra, develop its general properties, and show how the existence of an extension C → G of a Lie category to a Lie groupoid affects the ranks of morphisms and the algebroids of C. Furthermore, certain completeness results for invariant vector fields on Lie monoids and Lie categories with well-behaved boundaries are obtained. Interpreting the developed framework in the context of physical processes, we yield a rigorous approach to the theory of statistical thermodynamics by observing that entropy change, associated to a physical process, is a functor.
arXiv (Cornell University), Feb 10, 2023
Teorija umeritvenih polj ponuja geometrijsko bogat teoretičnofizikalni skelet, v katerem simetrij... more Teorija umeritvenih polj ponuja geometrijsko bogat teoretičnofizikalni skelet, v katerem simetrije narekujejo interakcije. V pričujočem delu začnemo pri osnovah glavnih svežnjev in natančno predstavimo splošen pojem povezave na njih. Pokažemo, da povezava naravno porodi pojma ukrivljenosti in kovariantnega odvoda na pridruženem vektorskem svežnju glede na dano upodobitev strukturne Liejeve grupe, raziščemo njune lastnosti ter natančno definiramo pojem umeritvenega polja ter njegove jakosti. Pri tem izrazimo vpliv umeritvenih transformacij nanju in pokažemo, kako lahko ukrivljenost na glavnem svežnju identificiramo z diferencialno formo, ki ima vrednosti v adjungiranem svežnju. Ukrivljenost povezave je -- skupaj s Hodge-* operatorjem, vnanjim kovariantnim odvodom in variacijskim principom -- osnova naše razprave o Yang--Millsovi teoriji, ki jo interpretiramo kot teorijo interakcije polja umeritvenih bozonov (tj. nosilcev fundamentalnih fizikalnih sil) s samim sabo. V tem kontekstu po...
arXiv (Cornell University), Feb 10, 2023
We introduce the basic notions and present examples and results on Lie categories-categories inte... more We introduce the basic notions and present examples and results on Lie categories-categories internal to the category of smooth manifolds. Demonstrating how the units of a Lie category C dictate the behavior of its invertible morphisms G(C), we develop sufficient conditions for G(C) to form a Lie groupoid. We show that the construction of Lie algebroids from the theory of Lie groupoids carries through, and ask when the Lie algebroid of G(C) is recovered. We reveal that the lack of invertibility assumption on morphisms leads to a natural generalization of rank from linear algebra, develop its general properties, and show how the existence of an extension C → G of a Lie category to a Lie groupoid affects the ranks of morphisms and the algebroids of C. Furthermore, certain completeness results for invariant vector fields on Lie monoids and Lie categories with well-behaved boundaries are obtained. Interpreting the developed framework in the context of physical processes, we yield a rigorous approach to the theory of statistical thermodynamics by observing that entropy change, associated to a physical process, is a functor.
arXiv (Cornell University), Feb 10, 2023
Teorija umeritvenih polj ponuja geometrijsko bogat teoretičnofizikalni skelet, v katerem simetrij... more Teorija umeritvenih polj ponuja geometrijsko bogat teoretičnofizikalni skelet, v katerem simetrije narekujejo interakcije. V pričujočem delu začnemo pri osnovah glavnih svežnjev in natančno predstavimo splošen pojem povezave na njih. Pokažemo, da povezava naravno porodi pojma ukrivljenosti in kovariantnega odvoda na pridruženem vektorskem svežnju glede na dano upodobitev strukturne Liejeve grupe, raziščemo njune lastnosti ter natančno definiramo pojem umeritvenega polja ter njegove jakosti. Pri tem izrazimo vpliv umeritvenih transformacij nanju in pokažemo, kako lahko ukrivljenost na glavnem svežnju identificiramo z diferencialno formo, ki ima vrednosti v adjungiranem svežnju. Ukrivljenost povezave je -- skupaj s Hodge-* operatorjem, vnanjim kovariantnim odvodom in variacijskim principom -- osnova naše razprave o Yang--Millsovi teoriji, ki jo interpretiramo kot teorijo interakcije polja umeritvenih bozonov (tj. nosilcev fundamentalnih fizikalnih sil) s samim sabo. V tem kontekstu po...