From Special Relativity to Feynman Diagrams.pdf

If we are to achieve a lorentz invariant description

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If we are to achieve a Lorentz invariant description of the process, we need to find a maximal number of combinations of the 3 N 4 which are not affected by Lorentz transformations of the frame of reference (Lorentz-invariant quantities). We have already taken into account N of them, namely the rest-masses m 2 i , with the mass- shell condition. The remaining Lorentz-invariant quantities are obtained by requiring generic functions of the 3 N 4 variables to be invariant under each of the six indepen- dent infinitesimal Lorentz transformations. This implies six further conditions which reduce the number of variables to a total of 3 N 10 Lorenz-invariant quantities. The above counting therefore accounts for the 10 conserved Noether charges associated with each Lorentz symmetry generator, which reduce the number of independent momentum components to 3 N – 10. For particles with spin, this number should be further multiplied by the number of spin states. 12.2.1 Decay Processes Each elementary decay process consists of a single particle decaying into two or more particles, like, for instance, a neutron which decays into a proton, an electron and an anti-neutrino: n p + + e + ¯ ν e . (12.5) Consider a system of identical unstable particles prepared in a same initial state | ψ in at t → −∞ . With the passing by of time a number of the initial particles will decay. If N ( t ) 1 is the number of particles in a small volume dV 3 at a time t , so that ρ( t ) = N ( t ) dV is the corresponding particle density, and if dN ( in ) N ( t ) denotes the number of these particles decaying between t and t + dt , we can write 3 Here we denote by dV a volume which is infinitesimal but still macroscopic in size, so as to contain a considerable number of particles.
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12.2 Kinematics of Interaction Processes 437 the probability of a decay per unit time as follows: d P ( in ) dt = dN ( in ) N ( t ) dt = dN ( in ) ρ( t ) dV dt . (12.6) Experimentally one finds that this quantity is a constant, depending only on the initial state | ψ in and related to the probability of a single decay event to occur. Such constant is expressed in terms of a decay width ( in ) , which has the dimension of an energy, divided by : dN ( in ) ρ( t ) dV dt = ( in ) . (12.7) When computed in the rest-frame of the particle, the inverse of the above quantity gives the mean life-time τ ( in ) , which is a characteristic feature of the particle itself. The mean life-time of an isolated neutron, for instance, is about 15min. while that of the muon μ is of the order of 10 6 sec. (see Chap. 1) Let us discuss now the relativistic covariance of ( 12.7 ). Suppose the quantities in ( 12.7 ) are referred to an inertial RF S and let us consider the same decay process as described from a different inertial RF S (primed quantities being referred to the latter). The space–time volume element dV dt is Lorentz invariant, and so is the number of events contained therein: dV dt = dV dt , dN = dN . Equation ( 12.7 ) implies that the product ρ ( in ) is Lorentz invariant: ρ ( in ) = ρ ( in ), where in refers to the initial state | ψ in of the decaying particles as seen from S .
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