From Special Relativity to Feynman Diagrams.pdf

When we shall consider four dimensional spacetime

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When we shall consider four-dimensional space–time instead of the three- dimensional Euclidean space, among all the possible transformations on a RF, of particular interest are the Lorentz transformations, on which Einstein’s principle of relativity is based. We shall show, at the end of this chapter, that Lorentz transforma- tions close a group, the Lorentz group . If we also include space–time translations, this group enlarges to the Poincaré group . If physical laws are expressed as an equal- ity between tensors of the same type with respect to the Lorentz group, we will be guaranteed that the principle of relativity holds. 4.7 Minkowski Space–Time and Lorentz Transformations In discussing special relativity, we have seen that space–time can be regarded as a four-dimensional space M 4 whose points are described by a set of four Cartesian coordinates ( x μ ) = ( x 0 , x 1 , x 2 , x 3 ), μ = 0 , 1 , 2 , 3 , (4.140) three of which ( x i ) = ( x 1 , x 2 , x 3 ) = ( x , y , z ) are spatial coordinates of our Euclid- ean space E 3 , and one x 0 = ct is related to time. A point on M 4 describes an event taking place at the point ( x , y , z ) , at the time t . Just as for Euclidean space, we can
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4.7 Minkowski Space–Time and Lorentz Transformations 123 define vectors connecting couples of points in M 4 , like the infinitesimal displacement vector connecting two infinitely close events: dx ( dx μ ) = ( dx 0 , dx 1 , dx 2 , dx 3 ). (4.141) These vectors span a four-dimensional linear vector space on which a symmetric scalar product is defined by means of the metric g μν = η μν , where 12 : η μν = 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 . (4.142) Given two 4-vectors P ( P μ ) and Q ( Q μ ) , their scalar product reads: P · Q = P μ η μν Q ν = P 0 Q 0 3 i = 1 P i Q i . (4.143) This scalar product, in contrast to the one defined on the Euclidean space, is not positivedefinite,namelydoesnotsatisfyproperty( c )of( 4.7 ),sincethecorresponding metrichasonepositiveandthreenegativediagonalentries(indefiniteorMinkowskian signature). As a consequence of this the squared norm of a 4-vector P ( P μ ) , defined using this scalar product: P 2 P · P P μ η μν P ν = ( P 0 ) 2 3 i = 1 ( P i ) 2 , (4.144) can vanish even if P is not zero. In particular a non-vanishing 4-vector can have positive, zero or negative squared norm, in which cases we talk about a time-like, null or space-like 4-vector, respectively. We can take, as 4-vector, the displacement vector dx , whose squared norm measures the squared space–time distance ds 2 between two infinitely close events: ds 2 = dx 2 = dx μ η μν dx ν = ( dx 0 ) 2 ( dx 1 ) 2 ( dx 2 ) 2 ( dx 3 ) 3 . (4.145) As pointed out when discussing about relativity, the distance ds in ( 4.145 ) should be interpreted as the infinitesimal proper-time interval times the velocity of light: ds 2 = c 2 d τ 2 . A four-dimensional space on which the metric ( 4.142 ) is defined, is called Minkowski space (or better space–time).
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