Torsional deformation of nonrelativistic string theory

Nonrelativistic string theory is a self-contained corner of string theory, with its string spectrum enjoying a Galilean-invariant dispersion relation. This theory is unitary and ultraviolet complete, and can be studied from first principles. In these notes, we focus on the bosonic closed string sector. In curved spacetime, nonrelativistic string theory is defined by a renormalizable quantum nonlinear sigma model in background fields, following certain symmetry principles that disallow any deformation towards relativistic string theory. We review previous proposals of such symmetry principles and propose a modified version that might be useful for supersymmetrizations. The appropriate target-space geometry determined by these local spacetime symmetries is string Newton-Cartan geometry. This geometry is equipped with a two-dimensional foliation structure that is restricted by torsional constraints. Breaking the symmetries that give rise to such torsional constraints in the target space will in general generate quantum corrections to a marginal deformation in the worldsheet quantum field theory. Such a deformation induces a renormalization group flow towards sigma models that describe relativistic strings.

Taking Up Superspace

Supersymmetry (SUSY) is a proposed symmetry between bosons and fermions. The structure of the space of SUSY generators is such that the distinction between internal and spacetime symmetries is blurred. As a result, there are two viable candidates for the spacetime setting for a flat supersymmetric field theory—Minkowski spacetime and superspace, an extension of Minkowski spacetime to include (at least) four new dimensions, coordinatized by ‘supernumbers’, i.e. numbers with nontrivial commutation properties. This chapter argues for two theses: first, that one standard set of arguments, related to universality of symmetry behaviour, that motivate a particular choice of spacetime structure in familiar spacetime theories motivates the choice of superspace as the appropriate spacetime for SUSY field theories; and second, that the metaphysical utility of the concept of spacetime requires more than just the satisfaction of this universality condition; in supersymmetric theories, the spacetime concept is not as useful as in special relativity.

Nilpotent symmetries as a mechanism for Grand Unification

In the classic Coleman-Mandula no-go theorem which prohibits the unification of internal and spacetime symmetries, the assumption of the existence of a positive definite invariant scalar product on the Lie algebra of the internal group is essential. If one instead allows the scalar product to be positive semi-definite, this opens new possibilities for unification of gauge and spacetime symmetries. It follows from theorems on the structure of Lie algebras, that in the case of unified symmetries, the degenerate directions of the positive semi-definite invariant scalar product have to correspond to local symmetries with nilpotent generators. In this paper we construct a workable minimal toy model making use of this mechanism: it admits unified local symmetries having a compact (U(1)) component, a Lorentz (SL(2, ℂ)) component, and a nilpotent component gluing these together. The construction is such that the full unified symmetry group acts locally and faithfully on the matter field sector, whereas the gauge fields which would correspond to the nilpotent generators can be transformed out from the theory, leaving gauge fields only with compact charges. It is shown that already the ordinary Dirac equation admits an extremely simple prototype example for the above gauge field elimination mechanism: it has a local symmetry with corresponding eliminable gauge field, related to the dilatation group. The outlined symmetry unification mechanism can be used to by-pass the Coleman-Mandula and related no-go theorems in a way that is fundamentally different from supersymmetry. In particular, the mechanism avoids invocation of super-coordinates or extra dimensions for the underlying spacetime manifold.

ON WICK-ROTATIONS AND QUATERNIONS: The Game of Symmetry between Space and Time (SOBRE ROTAÇÕES DE WICK E QUATERNIONS: O Jogo da Simetria entre Espaço e Tempo)

<p><strong>Abstract:</strong> Despite so many advances, modern physics remains entangled in a number of open questions, some of which are responsible for considerable obstacles in the last 30-40 years. I belong to the group of those who consider that the biggest problem comes from instrumental and conceptual limitations. Our ways of understanding the universe are still extremely limited by the heritage of classical-positivist thinking. Moreover, it is not simple to break the constraints of a brain whose functional design has developed over millions of years in three-dimensional interactive evolution basically conditioned by the imperatives of survival. This natural resistance to a broad theoretical reconstruction leads us to advance very slowly through the innermost essence of the cosmos, having evident reexes on our motivations and expectations. Present essay examines a preliminary model of spacetime structure in an attempt to oer new conceptual support for the study of the quantum entanglement. Wick-rotations are applied on a quaternionic basis to establish the theoretical foundations of the proposed spacetime symmetries.</p><p><strong>Key words</strong>: Wick-rotation, quaternion, Clifford algebra, quantum entanglement, spacetime symmetries.</p><p><span>===========================================================================</span></p><p><strong>Resumo:</strong> Apesar de tantos avanços, a física moderna permanece enredada em várias questões abertas, algumas das quais responsáveis por obstáculos consideráveis nos últimos 30-40 anos. Eu pertenço ao grupo daqueles que consideram que o maior problema vem das limitações instrumentais e conceituais. Nossos modos de entender o universo ainda são extremamente restringidos pela herança do pensamento positivista clássico. Além disso, não é simples quebrar as restrições de um cérebro cujo design funcional se desenvolveu ao longo de milhões de anos de evolução interativa tridimensional, basicamente condicionada pelos ditames da sobrevivência. Essa resistência natural a uma ampla reconstrução teórica nos leva a avançar muito lentamente através da essência mais íntima do cosmos, tendo reexos evidentes em nossas motivações e expectativas. O presente ensaio examina um modelo preliminar de estrutura de espaço-tempo numa tentativa de oferecer novos subsídios conceituais para o estudo do emaranhamento quântico. Rotações de Wick são aplicadas a uma base quaterniônica para estabelecer os fundamentos teóricos da simetria proposta entre espaço e tempo.</p><p><strong>Palavras-chave</strong>: Rotação de Wick, quatérnio, álgebra de Clifford, emaranhamento quântico, simetrias do espaço-tempo.</p>