From Special Relativity to Feynman Diagrams.pdf

74 representation of a group on a field let us

Info icon This preview shows pages 175–177. Sign up to view the full content.

View Full Document Right Arrow Icon
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.
Image of page 175

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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.
Image of page 176
Image of page 177
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern