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

ISBN-10: 3540671129

ISBN-13: 9783540671121

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**Example text**

Therefore, the symmetry transformation on the quantum ﬁelds is the analogue of the transformation on the classical ﬁelds interpreted as the wave-functions of one-particle states. The Feynman propagator For the sake of deﬁniteness, we again consider scalar ﬁeld theory with abelian U (1) symmetry and q = 1. One-particle states a (Q = +1) and b (Q = −1) at † The value of the charge q relative to the charges of other ﬁelds can only be ﬁxed in the presence of interactions. 149) This amplitude can also be interpreted as the creation of the particle a at (t, x) and its reabsorption into the vacuum at (t , x ).

17). It also implies absence of the gauge ﬁeld mass term m 2 Aµ Aµ . But such a term does not break the global U (1) symmetry. Exactly analogous considerations apply to scalar ﬁeld theories with U (1) gauge symmetry and to theories with Weyl fermions. Non-abelian gauge symmetry To construct a non-abelian gauge ﬁeld lagrangian we repeat the same steps. 11). 98). Let us now consider the extension of the group G to a group of local gauge transformations. 101) Thus we need gauge ﬁelds in the number given by the number of generators of the group.

E. 3) which vanish on the boundary of ≡ ( t, V ). 4) (summation over all ﬁelds and for each ﬁeld over its Lorentz and ‘internal’ indices is always understood). 6) is a surface term which vanishes and, therefore, the condition δS = 0 gives us the Euler–Lagrange equations of motion for the classical ﬁelds: ∂L ∂L − ∂ν =0 ∂ iµ ∂(∂ν iµ ) ν, µ = 0, 1, 2, 3 i = 1, . . 7) (here we keep the indices explicitly, with taken as a Lorentz vector; is are internal quantum number indices). 8) give the same classical equations of motion (due to the vanishing of variations of ﬁelds on the boundary of ).

### Gauge Field Theories by H. Gausterer, H. Grosse, L. Pittner

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