Let’s recall some notions from classical Lagrangian field theory. Let M be an oriented supermanifold: our spacetime, of dimension n|d. For our purposes, we will most often think about the case where M is either Minkowski space M̌ = R1,n−1, or Super Minkowski space M = M̌ ×ΠS where S is a representation of Spin(1, n− 1). Definition 1.1. Let E → M be a smooth (super) fibre bundle. The associated sp...

We study a one-to-one map between massless scalar fields on d+1 dimensional Minkowski (Euclidean) space time and scalar fields on d+1 dimensional De Sitter (Euclidean AdS) space. We solve the Klein-Gordon equation for massless scalar field in terms of initial data on the hypersurface t=0 which is isomorphic to the boundary of dS (AdS) space. By inserting these solutions into the action of massl...

Salkowski [11], one century ago, introduced a family of curves with constant curvature but non-constant torsion (Salkowski curves) and a family of curves with constant torsion but non-constant curvature (anti-Salkowski curves) in Euclidean 3-space E. In this paper, we adapt definition of such curves to time-like curves in Minkowski 3-space E31. Thereafter, we introduce an explicit parametrizati...

Recently there has been some renewed interest in Laguerre differential geometry [1], [2], [3]. This geometry was to a large extent developed by Blaschke and his school and a large amount of material about it can be found in [4]. Consider the set of oriented spheres in Euclidean space. If we add to this set the space of oriented planes, i.e. the spheres of infinite radii, then we obtain the spac...

We consider the problem of obtaining conformally invariant differential operators in Minkowski space. We show that the conformal electrodynamics equations and the gauge transformations for them can be obtained in the frame of the method of dimensional reduction developed in the first part of the paper. We describe a method for obtaining a large set of conformally invariant differential operator...

We investigate particle detector responses in some topologically non-trivial spacetimes. We extend a recently proposed regularization of the massless scalar field Wightman function in 4-dimensional Minkowski space to arbitrary dimension, to the massive scalar field, to quotients of Minkowski space under discrete isometry groups and to the massless Dirac field. We investigate in detail the trans...

We solve the following problem of Z. Füredi, J. C. Lagarias and F. Morgan [FLM]: Is there an upper bound polynomial in n for the largest cardinality of a set S of unit vectors in an n-dimensional Minkowski space (or Banach space) such that the sum of any subset has norm less than 1? We prove that |S| ≤ 2n and that equality holds iff the space is linearly isometric to l ∞ , the space with an n-c...

If K is a 0-symmetric, bounded, convex body in the Euclidean n-space R (with a fixed origin O) then it defines a norm whose unit ball is K itself (see [12]). Such a space is called Minkowski normed space. The main results in this topic collected in the survey [16] and [17]. In fact, the norm is a continuous function which is considered (in the geometric terminology as in [12]) as a gauge functi...

Minkowski space can be sliced, outside the lightcone, in terms of Euclidean Antide Sitter and Lorentzian de Sitter slices. In this paper we investigate what happens when we apply holography to each slice separately. This yields a dual description living on two spheres, which can be interpreted as the boundary of the light cone. The infinite number of slices gives rise to a continuum family of o...

A solution of the sourceless Einstein's equation with infinite values for the cosmological constant Λ and for the speed of light c is discussed by using Inönü–Wigner contractions of the de Sitter groups and spaces. When Λ → ∞, spacetime becomes a four–dimensional cone, dual to Minkowski space by a spacetime inversion. This inversion relates the four–cone vertex to the infinity of Minkowski spac...