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4) and summing over n we find on the whole Fock space3 : okl a† (k)a(l). 15) kl This has a simple interpretation: The one-particle operator destroys a particle in state l and creates a particle in state k. The “weight” with which this happens is given by the one-particle matrix element okl . We give now a very simple example for this formalism. As the one-particle basis we chose the generalised position-spin basis |xσ . As a typical one-particle operator we take the momentum of the particle. 16) which is already known from elementary quantum mechanics.
The Jacobian is in first order in δx: det ∂x µ ∂xν = det (δνµ + ∂ν δxµ ) = 1 + ∂µ δxµ . 32) This can be seen easily if one thinks of the definition of a determinant as the sum over permutations of matrix elements. In first order in δx only the product of the diagonal elements 51 Chapter 3 · Canonical Field Quantisation are important. All other products appearing in the definition of the determinant contain at least two factors δx. Now we have to take into account that variation and differentiation of the fields do not commute since we are varying the space-time variables as well as the fields: δ(∂µ φ) = ∂µ φ (x ) − ∂µ φ = ∂ν (φ + δφ) ∂x µ ∂xν −1 − ∂µ φ = ∂µ (δφ) − (∂µ δxν )∂ν φ.
The origin of the primed system is given by x = 0 and is thus moving with respect to the other system with velocity: v=c x1 = c tanh λ = cβ. 12) Here again we set c for the velocity of light. This is an important physical conclusion of this whole calculation: The relative velocity of two inertial frames can not be greater than the velocity of light. Since we can express the hyperbolic functions with help of tanh, we have cosh λ = 1 1 − β2 , sinh λ = 45 β 1 − β2 . 12) we find the more familiar form of this physically most important sort of proper Lorentz transformations, namely the so called Lorentz boosts.