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Register a new userBosonic Fields in Causal Set Theory
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Authors
Roman Sverdlov
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In this paper we will define a Lagrangian for scalar and gauge fields on causal sets, based on the selection of an Alexandrov set in which the variations of appropriate expressions in terms of either the scalar field or the gauge field holonomies around suitable loops take on the least value. For these fields, we will find that the values of the variations of these expressions define Lagrangians in covariant form.
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A while ago a proposal have been made regarding Klein Gordon and Maxwell Lagrangians for causal set theory. These Lagrangian densities are based on the statistical analysis of the behavior of field on a sample of points taken throughout some small region of spacetime. However, in order for that sample to be statistically reliable, a lower bound on the size of that region needs to be imposed. This results in unwanted contributions from higher order derivatives to the Lagrangian density, as well as non-trivial curvature effects on the latter. It turns out that both gravitational and non-gravitational effects end up being highly non-linear. In the previous papers we were focused on leading order terms, which allowed us to neglect these nonlinearities. We would now like to go to the next order and investigate them. In the current paper we will exclusively focus on the effects of higher order derivatives in the flat-space toy model. The gravitational effects will be studied in another paper which is currently in preparation. Both papers are restricted to bosonic fields, although the issue probably generalizes to fermions once Grassmann numbers are dealt with in appropriate manner.
This is the second paper in a series on the dynamics of matter fields in the causal set approach to quantum gravity. We start with the usual expression for the Lagrangian of a charged scalar field coupled to a SU(n) Yang-Mills field, in which the gauge field is represented by a connection form, and show how to write it in terms of holonomies between pairs of points, causal relations, and volumes or timelike distances, all of which have a natural correspondence in the causal set context. In the second part of the paper we present an alternative model, in which the gauge field appears as the result of a procedure inspired by the Kaluza-Klein reduction in continuum field theory, and the dynamics can be derived simply using the gravitational Lagrangian of the theory.
The goal of this paper is to define fermionic fields on causal set. This is done by the use of holonomies to define vierbines, and then defining spinor fields by taking advantage of the leftover degrees of freedom of holonomies plus additional scalar fields. Grassmann nature is being enforced by allowing measure to take both positive and negative values, and also by introducing a vector space to have both commutting dot product and anticommutting wedge product.
The goal of this paper is to propose an approach to the formulation of dynamics for causal sets and coupled matter fields. We start from the continuum version of the action for a Klein-Gordon field coupled to gravity, and rewrite it first using quantities that have a direct correspondent in the case of a causal set, namely volumes, causal relations, and timelike lengths, as variables to describe the geometry. In this step, the local Lagrangian density $L(f;x)$ for a set of fields $f$ is recast into a quasilocal expression $L_0(f;p,q)$ that depends on pairs of causally related points $p prec q$ and is a function of the values of $f$ in the Alexandrov set defined by those points, and whose limit as $p$ and $q$ approach a common point $x$ is $L(f;x)$. We then describe how to discretize $L_0(f;p,q)$, and use it to define a discrete action.
Thought experiments in the antitelephone concept with superluminal communication do not have causal contradiction.
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