We describe a set of periodic lattices in -dimensional Minkowski space, where each lattice has an associated symmetry group consisting of inhomogeneous Lorentz transformations that map the lattice onto itself. Our results show how ideas of crystal structure in Euclidean space generalize to Minkowski space and provide an example that illustrates basic concepts of spacetime symmetry.
REFERENCES
1.
An overview of crystal structure is given in
N. W.
Ashcroft
and N. D.
Mermin
, Solid State Physics
(Harcourt Brace
, New York
, 1976
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–82
, 111
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.2.
See also
A.
Nussbaum
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A.
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and E.
Ascher
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45
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A derivation of the inhomogeneous Lorentz group for -dimensional Minkowski space is given in
K. A.
Dunn
, “Poincaré group as reflections in straight lines
,” Am. J. Phys.
49
(1
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–55
(1981
).5.
A discussion of the homogeneous Lorenz group for -dimensional Minkowski space is given in
A. A.
Ungar
, “The abstract Lorentz transformation group
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60
(9
), 815
–828
(1992
).6.
A discussion of the inhomogeneous Lorentz group for -dimensional Minkowski space is given in
S.
Weinberg
, The Quantum Theory of Fields I
(Cambridge U. P.
, Cambridge
, 1995
), pp. 55
–62
.7.
The inhomogeneous Lorentz group for -dimensional Minkowski space is also discussed in
R.
Scurek
, “Understanding the group in particle physics: Standard and nonstandard representations
,” Am. J. Phys.
72
(5
), 638
–643
(2004
);J. A.
Morgan
, “Spin and statistics in classical mechanics
,” Am. J. Phys.
72
(11
), 1408
–1417
(2004
).8.
A.
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,” Can. J. Math.
1
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H. R.
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Here “field” is meant in the algebraic sense, as a commutative ring with multiplicative inverses, rather than in the geometric sense, as a quantity that depends on spacetime position.
11.
Technically, the group we have defined is the restricted inhomogeneous Lorentz group because it does not include space inversion or time reversal transformations. If these transformations are included, the resulting group is the general inhomogeneous Lorentz group, also known as the Poincaré group.
12.
13.
Here we have used the fact that a Lorentz transformation has unit determinant and thus preserves spacetime areas.
14.
The analogous problem for Galilean transformations has a simple solution: any ratio is allowed, and the corresponding velocity is .
15.
Because of the relativity of simultaneity, flashes that are simultaneous in the train frame are not simultaneous in the track frame.
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2010
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