From Special Relativity to Feynman Diagrams.pdf

It is useful to describe the vectors u i as column

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It is useful to describe the vectors { u i } as column vectors in the following way: u 1 1 0 0 ; u 2 0 1 0 ; u 3 0 0 1 . (4.3) This allows to describe a generic vector V as a column vector having as entries the components of V with respect to the basis { u i } . V V 1 1 0 0 + V 2 0 1 0 + V 3 0 0 1 = V 1 V 2 V 3 . (4.4) This formalism will allow us to reduce all operations among vectors to matrix oper- ations. We shall also use the boldface to denote the matrix representation of a given quantity. A Cartesian coordinate system in E 3 is defined by an origin O and a basis { u i } of V 3 and it allows to uniquely describe each point P in E 3 by means of three coordinates x, y, z, which are the components of the corresponding position vector r = −→ O P , that is the parallel projections along u i , see Fig. 4.1 a: r = −→ O P = x u 1 + y u 2 + z u 3 x y z . (4.5) 1 Recall that this property means that a 1 u 1 + a 2 u 2 + a 3 u 3 = 0 if and only if a 1 = a 2 = a 3 = 0.
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4.1 Linear Vector Spaces 93 Fig.4.1 a Generic Cartesian coordinate system; b Cartesian rectangular coordinate system It will be convenient to rename the coordinates as follows: x 1 = x , x 2 = y , x 3 = z . This will allow us to use the following short-hand description of the position vector: r 3 i = 1 x i u i ≡ { x i } . (4.6) A frame of reference (RF) in Euclidean space will be defined by a Cartesian coordi- nate system. In V 3 a scalar product is defined which associates with each couple of vectors V , W a real number V · W R and which satisfies the following properties: (a) V · W = W · V (symmetry) , (b) ( a V 1 + b V 2 ) · W = a ( V 1 · W ) + b ( V 2 · W ) (distributivity) , (c) V · V 0 ; V · V = 0 V = 0 (positive definiteness) (4.7) With respect to a basis u i , i = 1 , 2 , 3, a scalar product can be described by means of a symmetric non-singular matrix called metric : g = ( g i j ) ( u i · u j ) i , j = 1 , 2 , 3 , (4.8) in terms of which the scalar product between two generic vectors V , W V = 3 i = 1 V i u i , W = 3 i = 1 W i u i , can be written as follows: V · W = ( V 1 u 1 + V 2 u 2 + V 3 u 3 ) · ( W 1 u 1 + W 2 u 2 + W 3 u 3 ) (4.9) = 3 i = 1 3 j = 1 V i W j u i · u j = 3 i = 1 3 j = 1 V i W j g i j (4.10) = ( V 1 , V 2 , V 3 ) g 11 g 12 g 13 g 21 g 22 g 23 g 31 g 32 g 33 W 1 W 2 W 3 = V T gW , (4.11)
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94 4 The Poincaré Group where we have applied properties ( a ) and ( b ). Property ( c ) is specific to Euclidean space and expresses the positive definiteness of its metric, namely that for any vector V V 3 , different from the null vector 0 = ( 0 , 0 , 0 ) , the quantity V 2 V · V = V i g i j V j , called the norm squared of V , is positive. This in turn implies that the symmetric matrix g i j has only positive eigenvalues. This property will not hold for the metric in Minkowski space, which has three negative and one positive eigenvalues.
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