Feynman Diagrams For Pedestrians - Herbstschule Maria Laach

1.2 Lorentz TransformationsBasic principle of special relativity:• the velocity of light c is the same in each inertial system.∴ the wavefronts of a spherical light wave is for every observer located at|⃗x| = ct (2)• introducing the notation x 0 = ct, this means that the solutions ofare the same in every inertial reference framex 2 0 − ⃗x 2 = 0 (2 ′ )• adding homogeneity and isotropy of space, this means thatmust be the same in every inertial reference frame.• useful notations:– 3 vectors: (covariant w. r. t. rotations)x 2 = x 2 0 − ⃗x 2 (3)⃗x = (x 1 , x 2 , x 3 ) (4)– 4 vectors: (covariant w. r. t. rotations and boosts into a moving inertial frame)x = (x 0 ;⃗x) = (x 0 ; x 1 , x 2 , x 3 ) = (x 0 ; −x 1 , −x 2 , −x 3 ) (5)• introduce Minkowski metric⎛⎞1 0 0 0g µν = g µν = ⎝0 −1 0 0 ⎠0 0 −1 0(6)0 0 0 −1to shift indices down or upx µ =3∑g µν x ν , x µ =ν=0• convenient summation conventionxp =3∑g µν x ν (7)ν=03∑x µ p µ = x µ p µ = x µ p µ = g µν x µ p ν = g µν x µ p νµ=0= x 0 p 0 −23∑x i p i = x 0 p 0 − x i p i = x 0 p 0 − ⃗x⃗p (8)i=1