Carrollian limits of ModMax.
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Date
2024
Journal Title
Journal ISSN
Volume Title
Publisher
Universidad de Concepción
Abstract
Los límites Carrollianos de teorías de campos Lorentzianas se han encontrado recientemente estudiadas, en los últimos años, como resultado de un renovado interés en teorías y geometría Carrolliana desde el lado de física teórica. De ahí nace la decisión de estudiar los límites Carrollianos de la teoría de Maxwell Modificado (ModMax, por su escritura en inglés), que es la única extensión no-linear de la teoría de Maxwuell con invarianza conforme y de dualidad en el vacío. El presente trabajo contiene tanto una derivación de ambos limites a nivel de las ecuaciones de movimiento como una construcción de una formulación Hamiltoniana para cada uno. Se encontró que el limite magnético tiene una contribución no-linear no-nula a las ecuaciones de movimiento controlada por el parámetro de ModMax γ y que esta admite una biyección con el límite Carrolliano magnético de la teoría de Maxwell.
Cabe destacar que estos límites no son equivalentes pues existen configuraciones que son solución de uno de ellos y no así del otro. En particular, existen soluciones que muestran una dependencia explícita del parámetro de ModMax γ. Se encontró que el límite Carrolliano eléctrico de ModMax es equivalente al de Maxwell, siendo carente de contribución no-lineal.
Las simetrías de los límites Carrollianos de Maxwell fueron obtenidas empleando el método de simetrías puntuales de Lie y se probó que constituyen también simetrías de sus correspondientes contrapartes en los límites Carrollianos de ModMax mediante la anteriormente mencionada biyección. Estas simetrías incluyen desplazamientos finitos tanto temporales como espaciales, rotaciones espaciales, impulsos Carrollianos, dilaciones temporales, dilaciones espaciales, transformaciones conformes especiales Carrollianas de nivel k = 2, súpertraslaciones temporales, dilataciones de campo y una simetría interna que surge como legado de la simetría de dualidad en la versión Lorentziana. Debido a la separación de las dilaciones espacio-temporales en dilaciones espaciales y temporales, estas simetrías no caben dentro de ninguna clasificación de grupos conformes Carrollianos. Sin embargo, al tomar el sub-álgebra diagonal se encontró que esta satisface los criterios necesarios para pertenecer al álgebra conforme Carrolliana de nivel 2.
Carrollian limits of Lorentzian field theories have recently found themselves studied in the last few years as a result of renewed interest on Carrollian theories and geometry on the theoretical part of physics. Thus the decision to study the Carrollian limits of Modified-Maxwell (ModMax) theory, the unique conformal and duality invariant non-linear extension of Maxwell theory, was taken. The present work contains a derivation of these limits at the level of the equations of motion and the construction of a Hamiltonian formulation of them. It was found that the magnetic limit has a non-vanishing non-linear contribution to the equations of motion controlled by the ModMax parameter γ and that it admits a bijection with the magnetic limit of Maxwell theory, however, these two are not equivalent since there exists solutions on one side that are not solutions of the other and, in particular, there exists solutions with explicit dependence on the ModMax parameter γ. The electric limit of ModMax was found to be equivalent of that of Maxwell theory, having no non-linear contribution. The symmetries of the Carrollian limits of Maxwell theory were obtained through the use of Lie point symmetry method and are proven to also be symmetries of the Carrollian limits of ModMax theory by use of the Maxwell-ModMax bijection. These symmetries include time translations, space translations, Carrollian boosts, spatial rotations, time dilations, space dilations, special conformal transformations, field dilations, super-translations on the temporal part and an internal symmetry that corresponds to a legacy of duality invariance of the Lorentzian theory. Because of the separation of the space-time dilation into space and time dilations, the resulting algebra of symmetries does not belong in any categorization of conformal Carrollian algebras, however, by taking the diagonal sub-algebra it was found these belong to a subset of the conformal Carrollian algebra of level 2.
Carrollian limits of Lorentzian field theories have recently found themselves studied in the last few years as a result of renewed interest on Carrollian theories and geometry on the theoretical part of physics. Thus the decision to study the Carrollian limits of Modified-Maxwell (ModMax) theory, the unique conformal and duality invariant non-linear extension of Maxwell theory, was taken. The present work contains a derivation of these limits at the level of the equations of motion and the construction of a Hamiltonian formulation of them. It was found that the magnetic limit has a non-vanishing non-linear contribution to the equations of motion controlled by the ModMax parameter γ and that it admits a bijection with the magnetic limit of Maxwell theory, however, these two are not equivalent since there exists solutions on one side that are not solutions of the other and, in particular, there exists solutions with explicit dependence on the ModMax parameter γ. The electric limit of ModMax was found to be equivalent of that of Maxwell theory, having no non-linear contribution. The symmetries of the Carrollian limits of Maxwell theory were obtained through the use of Lie point symmetry method and are proven to also be symmetries of the Carrollian limits of ModMax theory by use of the Maxwell-ModMax bijection. These symmetries include time translations, space translations, Carrollian boosts, spatial rotations, time dilations, space dilations, special conformal transformations, field dilations, super-translations on the temporal part and an internal symmetry that corresponds to a legacy of duality invariance of the Lorentzian theory. Because of the separation of the space-time dilation into space and time dilations, the resulting algebra of symmetries does not belong in any categorization of conformal Carrollian algebras, however, by taking the diagonal sub-algebra it was found these belong to a subset of the conformal Carrollian algebra of level 2.
Description
Tesis presentada para optar al al grado de Magíster en Ciencias con Mención en Física
Keywords
Teoría de campos (Física)