Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
  1. Home
  2. Search

Refine search

Refine search

Access:
Content type:
Publication date:
Apply   From and To filters
Subject: Show more
Journals Show more
Publishers: Show more
Societies Show more
Series: Show more
Collections: Show more

Actions for selected content:

Select all | Deselect all
  • View selected items
  • Export citations
  • Download PDF (zip)
  • Save to Kindle
  • Save to Dropbox
  • Save to Google Drive

Save Search

You can save your searches here and later view and run them again in "My saved searches".

Please provide a title, maximum of 40 characters.
Cancel
×

5 results

  • 4 - The Lorentz Group and the Poincaré Group

  • from PART I - Basics
  • Michel Talagrand
  • Book:
    What Is a Quantum Field Theory?
    Published online:
    22 February 2022
    Print publication:
    17 March 2022, pp 102-131

  • Summary

    It is an extremely well-established experimental fact that the speed of light is the same for all “inertial observers” (those who do not undergo accelerations). The analysis of the consequences of this remarkable fact has forced a complete revision of Newton’s ideas: Space and time are not different entities but are different aspects of one single entity, space-time. Different inertial observers may use different coordinates to describe the points of space-time, but these coordinates must be related in a way that preserves the speed of light. The changes of coordinates between observers form a group, the Lorentz group. To a large extent the mathematics of Special Relativity reduce to the study of this group. Physics appears to respect causality, a strong constraint in the presence of a finite speed of light. We introduce the Poincaré group, related to the Lorentz group. We develop Wigner’s idea that to each elementary particle is associated an irreducible unitary representation of the Poincaré group and we describe the representation corresponding to a spinless massive particle, explaining also how the physicists view these matters.

  • 4 - Review of Special Relativity: Lorentz Tensors

  • from Part I - General Properties of Fields; Scalars and Gauge Fields
  • Horaƫiu Năstase, Universidade Estadual Paulista, São Paulo
  • Book:
    Classical Field Theory
    Published online:
    04 March 2019
    Print publication:
    14 March 2019, pp 36-45

  • Summary

    Here we review special relativity. We define the Lorentz group and Lorentz transformations; then the kinematics of special relativity, defining arbitrary tensors and various common ones; then the relativistic Lorentz force law, for the kinematics of special relativity. We define a relativistically covariant Lagrangian and invariant action for a particle and extend that to a coupling between a current and an electromagnetic gauge field.

  • 11 - Representations of the Lorentz Group

  • from Part I - General Properties of Fields; Scalars and Gauge Fields
  • Horaƫiu Năstase, Universidade Estadual Paulista, São Paulo
  • Book:
    Classical Field Theory
    Published online:
    04 March 2019
    Print publication:
    14 March 2019, pp 97-107

  • Summary

    We find the Lie algebra of the Lorentz group and then extend it to the Poincaré group, the group of symmetries of flat space. We then point out that, as SU(2) is the universal cover of SO(3), for the Lorentz group SO(3,1) the universal cover is SL(2,C).We then use Wigner's method, using the little group in four dimensions, to find massive and massless representations of the Lorentz and Poincaré groups. We thus find various possible fields, corresponding to these representations. We end by explaining how SL(2,C) is the universal cover of SO(3,1).

  • Lie methods for color robot vision

  • Reiner Lenz
  • Journal:
    Robotica / Volume 26 / Issue 4 / July 2008
    Published online by Cambridge University Press:
    01 July 2008, pp. 453-464

  • We describe how Lie-theoretical methods can be used to analyze color related problems in machine vision. The basic observation is that the nonnegative nature of spectral color signals restricts these functions to be members of a limited, conical section of the larger Hilbert space of square-integrable functions. From this observation, we conclude that the space of color signals can be equipped with a coordinate system consisting of a half-axis and a unit ball with the Lorentz groups as natural transformation group. We introduce the theory of the Lorentz group SU(1, 1) as a natural tool for analyzing color image processing problems and derive some descriptions and algorithms that are useful in the investigation of dynamical color changes. We illustrate the usage of these results by describing how to compress, interpolate, extrapolate, and compensate image sequences generated by dynamical color changes.