Relativistic Quantum Mechanics pp 313-338 | Cite as

# Lorentz Invariance and Relativistic Symmetry Principles

Chapter

## Abstract

We consider the four-dimensional space with coordinates and the length the absolute value should remain unchanged by this transformation (this is the fundamental, defining condition for orthogonal transformations), i.e. .

*x*_{ μ }(*μ*= 0, 1, 2, 3), which — assuming the most general case — may be complex numbers. The absolute value of the position vector is given by (summation convention)$$s = \sqrt {{g_{\mu v}}{x^\mu }{x^v}} = \sqrt {{x_\mu }{x^\mu }} ,$$

(16.1)

*s*can also take complex values. Examining an orthogonal transformation*a*_{ μν }, which relates each point with coordinates*x*_{ μ }to new ones*x*_{ ′μ }$${x'^v} = {a^v}_\mu {x^\mu }$$

(16.2)

$$s' = \sqrt {{{x'}_\mu }{{x'}^\mu }} = \sqrt {{x_\mu }{x^\mu }} = s$$

(16.3)

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### Biographical Notes

**POINCARÉ**, Henri Jules, French mathematician and philosopher, *29.4.1854 Nancy, †l7.7.1912 Paris, was a cousin of Raymond Poincaré, President of the French Republic during World War I. Between 1879 and 1881 at the University of Caen, and from 1881 at the University of Paris, P. worked in the field of pure mathematics (automorpheous functions), made important contributions to the theory of equilibrium properties of rotating fluids and achieved — independently of Einstein — a number of results of the special theory of relativity in his famous paper on the dynamics of the electron, published in 1906.Google Scholar**CAYLEY**, Arthur, British mathematician, * 16.08.1821 Richmond, †26.01.1895 Cambridge. C. was first a lawyer in London and from 1863 a professor at Cambridge. With JJ. Sylvester he founded the “theory of invariants” and algebraic geometry. By using complex coordinates C. showed that metric geometry is contained in projective geometry: with his projective measure (1859) he gave a new foundation to geometry which allowed the treatment of euclidian and noneuclidian geometry from a common point of view. He invented matrix calculus and was the first to formulate group theory (the representation of finite groups by multiplication tables or permutations) in an abstract way. C. also worked on conformai mappings, elliptic and hyperelliptic functions, the theory of differential equations, theoretical mechanics, the motion of the moon, and spherical astronomy. [BR]Google Scholar**KLEIN**, Felix, German mathematician, * 25.04.1849 Düsseldorf, 122.06.1925 Göttingen. K. studied from 1865 to 1870 in Bonn. During an educational stay 1870 in Paris he came into contact with the rapidly developing group theory. From 1871 K. taught at Göttingen and became professor at Erlangen in 1872, at München in 1875, at Leipzig in 1880 and at Göttingen in 1886. He made fundamental contributions to function theory, geometry, and algebra. He was especially interested in group theory and its applications. In 1872 K. published the Erlanger program. When he was older he occupied himself more intensely with pedagogical and historical problems.Google Scholar

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© Springer-Verlag Berlin Heidelberg 1990