From Special Relativity to Feynman Diagrams.pdf

Old and new coordinates the formalism is readily

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old and new coordinates, the formalism is readily extended to more general transfor- mations relating curvilinear coordinate systems, and thus also to curved spaces where Cartesian coordinates cannot be defined. We shall then study rotations in Euclidean space and show that they close an object called a Lie group , whose properties are locally captured by a Lie algebra . This will lead us to the important concept of covariance of an equation of motion with respect to rotations. The generalization of all these notions from Euclidean to Minkowski space will be straightforward. As anticipated in an earlier chapter, points in Minkowski space are described by a Cartesian system of four coordinates x 0 , x 1 , x 2 , x 3 and the distance between two points is defined by a metric with Minkowskian (or Lorentzian) signature. Poincaré transformations will then be introduced as Cartesian coordinate transformations which leave the coordinate dependence of the distance between two points invariant. These linear transformations include, as the homogeneous part, the Lorentz trans- formations, which generalize the notion of rotation to Minkowski space. Poincaré transformations also comprise, as their inhomogeneous part, the space–time trans- lations, and close a Lie group called the Poincaré group . The principle of special relativity is now restated as the condition that the equations of motion be covariant with respect to Poincaré transformations. 4.1 Linear Vector Spaces Let us briefly recall some basic facts about vector spaces. Consider the three- dimensional Euclidean space E 3 . We can associate with each couple of points R. D’Auria and M. Trigiante, From Special Relativity to Feynman Diagrams , 91 UNITEXT, DOI: 10.1007/978-88-470-1504-3_4, © Springer-Verlag Italia 2012
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92 4 The Poincaré Group A, B in E 3 a vector −→ AB originating in A and ending in B . If we arbitrarily fix an origin O in E 3 , any other point A in E 3 will be uniquely identified by its position vector r −→ O A . We can then consider the collection of vectors, each associated with couples of points in E 3 , which constitutes a vector space associated with E 3 and denoted by V 3 . Indeed the space V 3 is endowed with a linear structure, which means that an operation of sum and multiplication by real numbers is defined on its elements: we can consider a generic linear combination of two or more vectors V 1 , . . . , V k in V 3 with real coefficients and the result V : V = a 1 V 1 + · · · a k V k , (4.1) is still a vector, namely an element of V 3 , that is there is a couple of points A, B in E 3 such that V = −→ AB . If A and B coincide the corresponding vector is the null vector 0 −→ AA . A set of three linearly independent vectors { u 1 , u 2 , u 3 } = { u i } 1 defines a basis for V 3 and any vector V can be expressed as a unique linear combination of { u i } : V = V 1 u 1 + V 2 u 2 + V 3 u 3 = i V i u i , (4.2) where V 1 , V 2 , V 3 (with the upper index), are the components of V in the basis { u i } .
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