From Special Relativity to Feynman Diagrams.pdf

74 representation of a group on a field let us

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7.4 Representation of a Group on a Field Let us consider an n -dimensional flat space of points M n and its associated vec- tor space V n described by the vectors −→ AB connecting couples of points in M n . 7 By structure we mean the correspondence between any two elements of G and the third element representing their product.
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7.4 Representation of a Group on a Field 193 The space M n can be the Euclidean space E n if the metric tensor defined on it is δ i j (in this case we shall be mainly interested in our three-dimensional Euclid- ean space E 3 ), or, for n = 4, the Minkowski space M 4 of special relativity if the metric is η μν . It is useful at this point to recall the notations used in Chap.4 for describing Cartesian coordinates in the various spaces: The collection of generic Cartesian coordinates on M n is denoted by r = ( x i ) while our familiar Carte- sian rectangular coordinates are also denoted by x = ( x , y , z ) and the space- time coordinates of an event in Minkowski space are also collectively denoted by x = ( x μ ) = ( ct , x ) . Let us introduce a second p -dimensional vector space V p and let us we consider a map α : M n V p which associates with each point P M n , labeled by Cartesian coordinates x i , i = 1 , . . ., n , a vector in V p , of components α ( x i ), α = 1 , . . ., p , α : x i M n α ( x i ) V p . (7.42) This function is called a field , defined on M n , with values in V p . The index α is called the internal index since it labels the internal components α of the field, which are degrees of freedom not directly related to its space-time propagation. An example is the index α = 1 , 2 labeling the physical polarizations of a photon. V p is consequently called the internal space . If, as V p , we take the space V n k + l of type- ( k , l ) tensors, the corresponding field i 1 ... i k j 1 ... j l ( x i ) is called a tensor field . We have already intro- duced the notion of tensor fields in Chap.4 , Sect.4.3 , and illustrated their transforma- tionpropertiesunderachangeintheCartesiancoordinates(affinetransformations)on M n . There we discussed, as an example, the case of a tensor field T i j k ( x i ) which has values in the n 3 -dimensional vector space V p = V n 3 of type- ( 2 , 1 ) tensors. Its trans- formation law is given by ( 4.73 ), its indices transforming under the homogeneous part D = ( D i j ) (element of GL ( n ) ) of the affine transformation, according to their posi- tions. Thinking of T i j k as the p = n 3 components of a vector in V p , their are subject to the linear action of the matrix ( D D D T ) i js lmk defining the representation the GL ( n ) transformation on (2,1) tensors. This transformation property is general- ized in a straightforward way to generic type- (k,l) tensor fields. If we wish to restrict to transformations preserving the Euclidean or Lorentzian metrics on E 3 or M 4 , as we shall mostly do in the following, we need to restrict the homogeneous part of the affine transformation to O ( 3 ) or to O ( 1 , 3 ) , respectively.
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