From Special Relativity to Feynman Diagrams.pdf

We observe that the expression v f on the right hand

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being the velocity of the particle. We observe that the expression v · F on the right hand side of the first of ( 2.71 ), is the power of the force F acting on the moving particle. The time component of ( 2.67 ) then reads f 0 = d P 0 d τ = γ c dE dt ≡= γ c v · F , (2.72) and is the familiar statement that the rate of change of the energy in time equals the power of the force.
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2.2 Space–Time and Four-Vectors 61 If no force is acting on the particle the equation of the motion reduces to: d P μ d τ = 0 , (2.73) or, using ( 2.42 ), dU μ d τ = d 2 x μ d τ 2 = 0 , (2.74) Being d τ = 1 v 2 c 2 dt , it is easy to see that this equation implies v = const . Thus ( 2.74 ) is the Lorentz covariant way of expressing the principle of inertia. 2.2.2 Relativistic Theories and Poincaré Transformations We have seen that the conservation of the four-momentum and equation ( 2.67 ) defin- ing the four-force are equations between four-vectors and therefore they automati- cally satisfy the principle of relativity being covariant under the Lorentz transforma- tions implemented by the matrix ( μ ν ). It follows that the laws of mechanics discussed in this chapter, excluding the treatment of the gravitational forces , satisfy the principle of relativity, implemented in terms of general Lorentz transformations. We may further extend the covariance of relativistic dynamics by adding trans- formations corresponding to constant shifts or translations x μ = x μ + b μ , (2.75) b μ being a constant four-vector . This transformation is actually the four-dimen- sional transcription of time shifts and space translations already discussed for the extended Galilean transformations ( 1.15 ). However, differently from the Galilean case, there is no need in the relativistic context to add three-dimensional rotations, since, as mentioned before, they are actually part of the general Lorentz transforma- tions implemented by the matrix . It is easy to realize that the four-dimensional translations do not affect the proper time or proper distance definitions, nor the fundamental equations of the relativistic mechanics, ( 2.46 ) and ( 2.67 ). We conclude that relativistic dynamics is covariant under the following set of transformations x μ = μ ν x ν + b μ . (2.76) This set of transformations is referred to as Poincaré transformations . Both the Lorentz and Poincaré transformations will be treated in detail in Chap.4 . Furthermore in Chap.5 it will be shown that also the Maxwell theory is covariant under ( 2.76 ),
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62 2 Relativistic Dynamics thus proving that the whole of the relativistic physics, namely relativistic dynamics and electromagnetism, is invariant under Poincaré transformations. One could think that the invariance under translations and time shifts should not play an important role on the interpretation of a physical theory. On the contrary we shall see that such invariance implies the conservation of the energy and momen- tum in the Galilean case and of the four-momentum in the relativistic case (see Chap.8 ).
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