From Special Relativity to Feynman Diagrams.pdf

Let us now come back to the case in which g is a

Info icon This preview shows pages 178–180. Sign up to view the full content.

View Full Document Right Arrow Icon
Let us now come back to the case in which G is a transformation group acting on the space-time RF. The simplest instance of field is the scalar field in which D is the trivial representation 1 , defining a type-(0,0) tensor, with p = 1 that is V p = R (real scalar field) or C (complex scalar field). A complex scalar field ( r ) can be 8 Note that the analogous of the Poincaré group in the three dimensional Euclidean space E 3 is the known group of congruences of Euclidean geometry, acting on the space coordinates as in ( 4.102 ).
Image of page 178

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

View Full Document Right Arrow Icon
196 7 Group Representations and Lie Algebras described as a couple of real scalar fields 1 ( r ), 2 ( r ) , defined at each point r by the real and imaginary parts of ( r ) : ( r ) = 1 ( r ) + i 2 ( r ) ( 1 ( r ), 2 ( r )) . (7.50) The transformation law ( 7.45 ) reduces, for a scalar field, to ( r ) = ( r ( r )) = ( R 1 ( r + r 0 )). (7.51) If M n = M 4 and G is the Poincaré group, r is the space-time coordinate vector ( x μ ) and R = = ( μ ν ) SO ( 1 , 3 ) . In general, as discussed in Sect.4.3 , the coordinates r = ( x i ) and r = ( x i ) refer to the same point P of M n , therefore the numerical value of the scalar field must be the same, even if, when substituting x i = x i ( x j ) = ( R 1 ) i j ( x j + x j 0 ) the functional form changes from to . Writing ( r ) = ( r ) we are considering the transformation r = Rr r 0 from a passive point of view since space-points are considered fixed while only the coordinate frame is changed. However the same transformation can be also considered from a different point of view, namely as a change in the functional form of ( r ) ( r ) ( r ), (7.52) with ( r ) = ( R 1 ( r r 0 )) . In this case we consider the transformation as an active transformation, since the emphasis is on the functional change of . The given change of coordinate in this case is thought of as due to a change of the geometric point. 9 When considering the change in the functional form from an active point of view it is sometimes convenient to denote the new functional form taken by as consequence of the coordinate change induced by an element g G , as the action of an operator O g on . 10 Equation ( 7.51 ) takes the following form: O g ( r ) = ( R 1 ( r r 0 )), (7.53) where, as usual, R = R ( g ) and r 0 = r 0 ( g ) . Consider, for the sake of simplicity, a group G acting in a homogeneous way on the coordinates (i.e. r 0 0 ) and apply in succession two transformations g 1 , g 2 G , the resulting transformation corresponding to the product g 2 · g 1 G . We have: 9 Note that in the discussion of the vector and tensor calculus in Sect.4.1 the emphasis was on the passive point of view since the reference frame was changed by the transformations. Therefore the whole of the vector and tensor calculus was developed taking this point of view. The active point, as previously mentioned, will be actually adopted in Chap.9 when discussing the action of a group on the Hilbert space of states in quantum mechanics.
Image of page 179
Image of page 180
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