By P. Hoffman, R. Piccinini, D. Sjerve

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**Example text**

92) As in electrodynamics, gauge ﬁelds which are gauge transforms of Aµ = 0 are called pure gauges (cf. (8)) and are, according to (90), given by Apg µ (x) = U (x) 1 ∂µ U † (x) . ig (93) Physical observables must be independent of the choice of gauge (coordinate system in color space). e. their value does not change under local gauge transformations. One also introduces non-local quantities which, in generalization of the transformation law (91) for the ﬁeld strength, change homogeneously under gauge transformations.

2]) is π3 (S 2 ) ∼ Z , (45) a result which is useful in the study of Yang–Mills theories in a certain class of gauges (cf. [20]). The integer k labeling the equivalence classes has a geometric interpretation. e. the diﬀerential df is 2-dimensional in y1 and y2 . The preimages of these points M1,2 = f −1 (y1,2 ) are curves C1 , C2 on S 3 ; the integer k is the linking number lk{C1 , C2 } of these curves, cf. (1). It is called the Hopf invariant. 3 F. Lenz Quotient Spaces Topological spaces arise in very diﬀerent ﬁelds of physics and are frequently of complex structure.

Similar concepts are used for a proper description of the topological space of the degrees of freedom in gauge theories. e. variables which are related to each other by gauge transformations. This suggests to deﬁne an equivalence relation in the space of gauge ﬁelds (cf. e. elements of an equivalence class can be transformed into each other by gauge transformations U , they are gauge copies of a chosen representative. The equivalence classes O = A[U ] |U ∈ G (72) with A ﬁxed and U running over the set of gauge transformations are called the gauge orbits.