By C. Taylor

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5. Wigner (1939) demonstrated the signiﬁcance of irreducible representations of the Poincar´e group. We brieﬂy review the most important features of the Poincar´e group in this section. A general reference is Bogoliubov, Logunov, Oksak & Todorov (1990), Naimark (1964), Gel’fand et al. (1963), and Barut & R¸aczka (1977). 7. 6. The relation of relativity and causality is discussed in many places. For an extensive discussion of this subject see Frank (1932), and Bunge (1959). A more sophisticated version of causality involving hypothetical superluminal signals is advocated by Bilaniuk & Sudarshan (1969).

For any spacelike vector k µ , one can ﬁnd a Lorentz frame where k µ = (0, k) . 104) that, for an arbitrary timelike or spacelike vector k µ , there are three mutually orthogonal vectors eµ1 , eµ2 , and eµ3 , which are also orthogonal to k µ . 106) is the Minkowski metric. This means that any vector X µ can be expressed in terms of these basis vectors: 3 Xµ = (X · ei ) µ (X · k) µ k + ei . 107) A set of vectors of this kind is known as a vierbein, or tetrad. 108) to a vector X µ to map this vector on 3 i=1 (X · ei ) µ ei .

A 2-form which can be transformed to a single term, the exterior product of two 1-forms is called decomposable. 133) as for d = 2n will arise. 137). We now turn to ﬁelds. For simplicity, we consider here only real-valued ﬁelds. In the following we will deal also with complex-valued ﬁelds, but this generalization is superﬂuous for the present discussion. A scalar ﬁeld φ is a mapping M4 → R. A scalar ﬁeld φ takes a deﬁnite numerical value at each point of Minkowski space, no matter what coordinates are assigned to this point by diﬀerent observers.

### A Non-Mathematical Introduction to X-Ray Crystallography [short article] by C. Taylor

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