Electromagnetism and Geometry : A plane bundle description

In this paper we aim to study a small step in the physicist marathon that is the unifica- tion of two field theories that, while similar in some senses, are rather different in other, namely gravity and electromagnetism. The man on the starting block was none other than Albert Einstein who, although ultimately unsuccessful, spent the later part of his life trying to formulate a unified theory of the two field theories. Having made the monu- mental achievement of General Relativity [1] and thus geometrized gravity, he sought to also geometrize electromagnetism. Some might object that, since we have the Einstein- Maxwell equations, there is already a unification of the two field theories. But this is not really the case, as these equations are more or less obtained by taking the Einstein equa- tions and slapping the Maxwell equations on top. Hardly a very profound nor intimate unification. Thus what one really is seeking, is a deeper, more fundamental unification. This search has a long tradition, and we are not the first since Einstein to attempt to investigate this. A few years after Einstein’s paper on General Relativity, Kaluza and Klein [2, 3] presented their theory of Einstein’s equations on a 5-dimensional space-time where this ”extra” dimension is curled up in a circle. This in fact leads to Einstein’s equations in 4D space-time together with Maxwell’s equations. But also together with a massless spin-0 particle that, sadly, doesn’t appear to exist. Our approach is to show that by imbuing ordinary 4D Minkowski space-time with a second property in addition to the metric, we obtain the theory of electromagnetism by geometric considerations. This second property is a field of two-dimensional spacelike planes. At each point in space-time we fix such a plane, which can be described by an anti-symmetric tensor of rank 2. The first works on this model are found in [4, 5, 6, 7, 8]. In Chapter 1 we will give a short introduction to the formalism of differential forms and tensors, and briefly discuss why this formalism is better suited for the more advanced calculations than vector formalism. For a more in-depth discussion on this formalism, [9] is a good place to look. In Chapter 2 we examine the model in three-dimensional Euclidean space, investigate some of its properties, and we also look at some examples of configurations for various electric and magnetic fields. This chapter is based on the work in [5]. What is new here is that we explicitly include electric fields in addition to magnetic fields, and discuss also examples of some specific electric field configurations. In Chapter 3 we extend the model to four-dimensional Minkowski space, examining the various properties and also how they relate to the three-dimensional model already dis- cussed. This is partially based on [6] as well as the matrix formulation of frame bundles found in [10]. We study in greater detail the properties of some of the general new struc- tures which emerge in four-dimensional space-time than what has been done before and how they may or may not relate to the electromagnetic field. As an example we take a look at the field from a single monopole moving at constant velocity, and various inter- pretations of the quantities in the model with respect to this specific configuration. In Chapter 4 we examine the field from a We try to understand and interpret the various quantities and properties of the model in this context and if possible give a more de- tailed and explicit derivation. Finally in Chapter 5 we investigate the motion of charged particles in the presence of electromagnetic fields described by the plane-model we have introduced earlier, and how we can think of such a particle as a neutral, rotating particle with a constraint on its rotation. We first examine the case of the particle moving non- relativistically, which can be found in [5], and then we try to extend this to also cover particles moving relativistically in an electromagnetic field

Unextendible product bases and extremal density matrices with positive partial transpose

In bipartite quantum systems of dimension 3x3 entangled states that are positive under partial transposition (PPT) can be constructed with the use of unextendible product bases (UPB). As discussed in a previous publication all the lowest rank entangled PPT states of this system seem to be equivalent, under special linear product transformations, to states that are constructed in this way. Here we consider a possible generalization of the UPB constuction to low-rank entangled PPT states in higher dimensions. The idea is to give up the condition of orthogonality of the product vectors, while keeping the relation between the density matrix and the projection on the subspace defined by the UPB. We examine first this generalization for the 3x3 system where numerical studies indicate that one-parameter families of such generalized states can be found. Similar numerical searches in higher dimensional systems show the presence of extremal PPT states of similar form. Based on these results we suggest that the UPB construction of the lowest rank entangled states in the 3x3 system can be generalized to higher dimensions, with the use of non-orthogonal UPBs.Comment: 23 pages, 2 figures, 1 table. V2: Fixed fig.1 not showin

Low rank positive partial transpose states and their relation to product vectors

It is known that entangled mixed states that are positive under partial transposition (PPT states) must have rank at least four. In a previous paper we presented a classification of rank four entangled PPT states which we believe to be complete. In the present paper we continue our investigations of the low rank entangled PPT states. We use perturbation theory in order to construct rank five entangled PPT states close to the known rank four states, and in order to compute dimensions and study the geometry of surfaces of low rank PPT states. We exploit the close connection between low rank PPT states and product vectors. In particular, we show how to reconstruct a PPT state from a sufficient number of product vectors in its kernel. It may seem surprising that the number of product vectors needed may be smaller than the dimension of the kernel.Comment: 29 pages, 4 figure

Learning through modelling in science: Reflections by pre-service teachers

This study analyses pre-service science teachers’ (PSTs’) experiences of working with models and modelling and their ideas about their usefulness in science education. Although several studies have investigated pre- and in-service teachers’ views on models and modelling, research is lacking in the Norwegian context. This study addresses this gap by exposing PSTs to a one-day course on modelling in chemistry and exploring their ideas through focus-group interviews. We found that teaching using modelling-related activities promoted PSTs’ understanding of models and modelling, especially relating to the scope and limitations of models. Additionally, the PSTs increased their understanding of why such learning activitiesare important and how to incorporate them while teaching science. Norwegian PSTs responded positively to modelling-based teaching, which seemed to promote metacognition and critical thinking. Therefore, modelling-based teaching could be an effective tool for educating science teachers in how to promote such skills in their classrooms

Learning through modelling in science: Reflections by pre-service teachers

This study analyses pre-service science teachers’ (PSTs’) experiences of working with models and modelling and their ideas about their usefulness in science education. Although several studies have investigated pre- and in-service teachers’ views on models and modelling, research is lacking in the Norwegian context. This study addresses this gap by exposing PSTs to a one-day course on modelling in chemistry and exploring their ideas through focus-group interviews. We found that teaching using modelling-related activities promoted PSTs’ understanding of models and modelling, especially relating to the scope and limitations of models. Additionally, the PSTs increased their understanding of why such learning activities are important and how to incorporate them while teaching science. Norwegian PSTs responded positively to modelling-based teaching, which seemed to promote metacognition and critical thinking. Therefore, modelling-based teaching could be an effective tool for educating science teachers in how to promote such skills in their classrooms