From Special Relativity to Feynman Diagrams.pdf

# Is no longer represented as an action at a distance

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is no longer represented as an action at-a-distance between the two charges but as mediated by the electromagnetic field, and can be divided into two moments (see the Fig. 2.1 ): (a) a charge q 1 generates an electromagnetic field; (b) The field, which is a physical quantity defined everywhere in space, propagates until it reaches the charge q 2 located at some point and acts on it by means of a force (the Lorentz forces). This mechanism is apparent when one of the two charges (say q 1 ) is moving at a very high speed. One then observes that the information about the position of the moving charge is transmitted to q 2 at the speed of light through the electromagnetic field, causing the force acting on it to be adjusted accordingly with a characteristic delay which depends on the distance between the two charges. In this action-by- contact picture the interacting parts are three instead of just two: the two charges and

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2.1 Relativistic Energy and Momentum 39 the field. The force acting on q 2 is the effect of the action of the field generated by q 1 on q 2 . This implies that the action and the resulting force occur at the same time and place (the position of q 2 ) and this property is now Lorentz-invariant. Indeed if t = | x | = 0 , (2.1) in a given frame, using the Lorentz transformations ( 1.57 )–( 1.60 ), we also have t = | x | = 0 in any other frame. Thus the action-by-contact representation is consistent with the principles of relativity and causality. As for the electromag- netic interaction, we would also expect the gravitational one to be mediated by a gravitational field. However, as we have mentioned earlier, a correct treatment of the gravitational interaction requires considering non-inertial frames of reference which goes beyond the framework of special relativity. In order to discuss how classical mechanics should be generalized in order to be compatible with Lorentz transformations (relativistic mechanics), we shall therefore refrain from considering gravitational interactions. Even in classical mechanics we can consider processes in which the interaction is localized in space and time, so that the locality condition ( 2.1 ) is satisfied and we can avoid the inconsistencies discussed above, related to Newton’s second law. These are typically collisions in which two or more particles interact for a very short time and in a very small region of space. Since the strength of the interaction is much higher than that of any other external force acting on the particles, the system can be regarded as isolated, so that the total linear momentum is conserved, and its initial and final states are described by free particles. Let us focus on this kind of processes in order to illustrate how one of the fundamental laws of classical mechanics, the conservation of linear momentum, can be made consistent with the principle of relativity, as implemented by the Lorentz transformations.
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