From Special Relativity to Feynman Diagrams.pdf

Μ is defined as x μ 2 x 2 x 1 2 x 2 2 x 3 2 c 2 τ

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μ is defined as x μ 2 = ( x 0 ) 2 ( x 1 ) 2 ( x 2 ) 2 ( x 3 ) 2 = c 2 τ 2 = s 2 . (2.44) Note, however, that the square of the Lorentzian norm is not positive definite , that is, it is not the sum of the squared components of the vector (see ( 2.43 )) as the Euclidean norm | x | 2 is. Consequently a non-vanishing four vector can have a vanishing norm. In analogy with the relative position four-vector, we define the norm of the energy- momentum vector as p μ 2 = ( p 0 ) 2 ( p 1 ) 2 ( p 2 ) 2 ( p 3 ) 2 = ( p 0 ) 2 − | p | 2 . From ( 2.38 ) it follows that this norm is precisely the (Lorentz invariant) squared rest mass of the particle times c 2 : p μ 2 = m 2 c 2 . Using the notation of four-vectors, we may rewrite the results obtained so far in a more compact way. Consider once again a collision between two particles with initial energies and momenta E 1 , E 2 and p 1 , p 2 , respectively, from which two new particles are pro- duced, with energies and momenta E 3 , E 4 , p 3 , p 4 . The conservation laws of energy and momentum read: E 1 + E 2 = E 3 + E 4 , p 1 + p 2 = p 3 + p 4 . (2.45) If we now introduce the four-vectors p μ n , n = 1 , 2 , 3 , 4 associated with the initial and final particles 9 Alternatively also the denominations Lorentzian or Minkowskian distance are used.
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56 2 Relativistic Dynamics p μ n = E n / c p nx p ny p nz , and define the total energy-momentum as the sum of the corresponding four-vectors associated with the two particles before and after the process, we realize that the conservation laws of energy and momentum are equivalent to the statement that the total energy momentum four-vector is conserved. To show this we note that ( 2.45 ) can be rewritten in a simpler and more compact form as the conservation law of the total energy-momentum four-vector : p μ tot = p μ 1 + p μ 2 = p μ 3 + p μ 4 . (2.46) Indeed the 0th component of this equation expresses the conservation of energy, while the components μ = 1 , 2 , 3 (spatial components) express the conservation of linear momentum. Note that for each particle the norm of the four-vector gives the corresponding rest mass: p μ n 2 = E n c 2 − | p n | 2 = m 2 n c 2 . Until now we have restricted ourselves to Lorentz transformations between frames in standard configuration . For next developments it is worth generalizing our setting to Lorentz transformations with generic relative velocity vector V , however keeping, for the time being, the three coordinate axes parallel and the origins coincident at the time t = t = 0 . Consider two events with relative position four-vector x ( x μ ) = ( c t , x ) with respect to a frame S . We start decomposing the three-dimensional vector x as follows x = x + x , where x and x denote the components of x orthogonal and parallel to V , respectively. Consider now the same events described in a RF S moving with respect to S at a velocity V . It is easy to realize that the corresponding Lorentz transformation can be written as follows x = x + γ ( V ) ( x V t ) , (2.47) t = γ ( V ) t x · V c 2 . (2.48) Indeed they leave invariant the fundamental ( 1.49 ) or, equivalently, the proper time (and thus the proper distance) c 2 t 2 − | x | 2
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