By H. Gausterer, H. Grosse, L. Pittner

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**Sample text**

The tensor M µν is not translationally invariant. 52) 20 1 Classical ﬁelds, symmetries and their breaking (where εµνρκ is the totally antisymmetric tensor deﬁned by ε0123 = −ε0123 = 1) which reduces in the rest frame P µ = (m, 0) to the three-dimensional total angular momentum M k ≡ 12 εi jk M i j : M k = W k /m. 2 Classical ﬁeld equations Scalar ﬁeld theory and spontaneous breaking of global symmetries In this section we illustrate our general considerations with the simplest example: the classical theory of scalar ﬁelds (x).

93)). At the quantum level g is relevant since quantum theory contains the Planck constant h¯ and g is dimensionless in units h¯ = 1. Equivalently, at the quantum level the normalization of the ﬁeld is ﬁxed by the normalization of the single particle state. 104) is an inﬁnitesimal gauge parameter in the matrix notation. 105) form a group. 105) that under global transformations (∂µ = 0) the gauge ﬁelds transform according to the adjoint representation of the group with (T α )βγ = −icαβγ . 93); this is convenient for embedding electromagnetism into the electroweak theory.

68)) and bL , dR are expansion coefﬁcients.