Introduction to the Standard Model of Particle Physics

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Part II Kinematics 39 10/23/2000 Part II Kinematics 1 Special Relativity Before going any further, we present a quick overview of notations for the kinematics of spe- cial relativity. 1.1 Four Vectors and Lorentz Transformations Four-vectors are denoted by x µ , with the greek index µ =0,1,2,3; a single time-like component 0, followed by the three space-like components 1,2 and 3: x µ = ( x 0 , x 1 , x 2 , x 3 ). The space-like indices 1,2, and 3 are habitually represented by lower case roman characters: e.g. x i = x = ( x 1 , x 2 , x 3 ). In (special) relativity a space-time distance (four-length) t 2 x 2 y 2 z 2 is constant and invariant under Lorentz trans- formations: i.e. t 2 x 2 y 2 z 2 = t 2 x 2 y 2 z 2 , a direct consequence of the constancy of the speed of light for all observers. The space-time fourvector coordinate is denoted by x µ = ( x 0 , x 1 , x 2 , x 3 ) = ( x 0 , x i ) = ( t , x ) = ( t , x , y , z ). The space-time components mix between themselves by Lorentz transformations from one inertial sys- tem to another. A Lorentz transformation from an (unprimed) system to another (primed) system mov- ing at relative velocity β// x with respect to the first, can be expressed in terms of speed β and relativis- tic factor γ (1 β 2 ) ½ as: 2 2 2 0 0 0 1 0 0 , with det( ) 1 0 0 1 0 0 0 µ µ ν ν γ βγ γ β γ βγ γ Λ = Λ = = (II.1) Note the lower and upper indices: this facilitates the definition of a summation convention of same in- dices: 3 0 1 2 3 0 1 2 3 0 1 2 3 0 ' x x x x x x x t x y z µ µ ν µ ν µ µ µ µ µ µ µ µ ν ν ν = = Λ Λ = Λ + Λ + Λ + Λ = Λ + Λ + Λ + Λ (II.2) The invariance of the length of x µ is indeed satisfied: t 2 x 2 y 2 z 2 = ( γ t βγ z ) 2 x 2 y 2 – ( βγ t + γ z ) 2 = ( γ 2 β 2 γ 2 ) t 2 x 2 y 2 – ( γ 2 β 2 γ 2 ) z 2 = t 2 x 2 y 2 z 2 . This leads to a definition of an invariant four-dimensional dot-product: x µ x µ = ( x 0 ) 2 – ( x 1 ) 2 – ( x 2 ) 2 ( x 3 ) 2 = g µν x ν x µ , where the metric tensor g µν in special relativity is defined as: 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 g g µν µν = = (II.3) Other commonly used fourvectors, contravariant (= index high) or covariant (= index low), are: ( , ) ( , ) , ( , ), ( , ), ( , ), ( , ) p E i i i j A t t t µ µ µ µ µ µ ρ φ = = − ∇ = ∂ = −∇ = +∇ = = p j A # # # (II.4)
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Part II Kinematics 40 10/23/2000 1.2 Lorentz Scalars The length of a Lorentz fourvector is invariant under Lorentz transformations, as a special case of the invariance of the Lorentz fourvector dot product. The four-momentum transfer (= difference) q is defined as q p 3 p 1 , where the quantities q and p are implied fourvectors. The labels 1 and 3 are particle labels, with the labeling convention shown in the accompanying Figure. Note that the four-momentum transfer q is defined using external lines only; i.e. it is independent of the details of the interaction taking place in the volume.
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