Foundations of Quantum Field Theory. Klaus D Rothe

Foundations of Quantum Field Theory

Foundations of Quantum Field Theory - Klaus D Rothe

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alt="figure"/>, we get

      or in particular

      that is, for an infinitesimal time after t0 one already finds the particle with equal probability anywhere in space; this violates obviously the principle of relativity, as well as causality.

      ________________________

      1In a relativistic theory,

can also contain negative energy states, which then require a particular interpretation or must decouple altogether from the “physical sector” of the theory.

      2In Quantum Mechanics

      In this chapter we use everywhere lower indices, repeated indices being summed over.

      3For notational simplicity we suppose the spectrum to be discrete.

       Lorentz Group and Hilbert Space

      In this chapter we first discuss the realization of the homogeneous Lorentz transformations in four-dimensional space-time, as well as the corresponding Lie algebra. From here we obtain all finite dimensional representations, and in particular the explicit form of the matrices representing the boosts for the case of spin = 1/2, which will play a fundamental role in Chapter 3. The Lorentz transformation properties of massive and zero-mass 1-particle in Hilbert space (and their explicit realization in Chapter 9) lie at the heart of the Fock space representation (second quantization) in Chapter 9. It is assumed that the reader is already familiar with the essentials of the Special Theory of Relativity and of Group Theory.

      Homogeneous Lorentz transformations are linear transformations on the space-time coordinates,1

      leaving the quadratic form

      invariant. Any 4-tuplet transforming like the coordinates in (2.1) is called a contravariant 4-vector. In particular, energy and momentum of a particle are components of a 4-vector

      with

      The same transformation law defines 4-vector fields at a given physical point:

      Note that the quadruple (Λx)μ referred to

is the same point as the quadruple xμ referred to
′. Thus, alternatively

      Notice that (2.4) and (2.5) represent inverse transformations of the reference frame, respectively. Examples are provided by the 4-vector current

and the vector potential
of electrodynamics in a Lorentz-covariant gauge.

      The differential element dxμ transforms like

      Hence it also transforms like a contravariant 4-vector, since

      The partial derivative

, on the other hand, transforms differently. The usual chain rule of differentiation gives

      From the inversion of (2.1) it follows that

      Hence for the partial derivative

we have the transformation law

      Four-tuples which transform like the partial derivative are called covariant 4-vectors. Contravariant and covariant 4-vectors are obtained from each other by raising and lowering the indices with the aid of the metric tensors gμν and gμν, defined by2

      respectively, in terms of which the invariant element of length (2.2) can be written in the form

      The requirement that ds2 be a Lorentz invariant

      now implies

      Thus the metric gμν is said to be a Lorentz-invariant tensor. It is convenient to write this equation in matrix notation by grouping the elements

into a matrix as follows:

      Defining the elements of the transpose matrix ΛT by

      we can write (2.8) as follows:

      From here we obtain for the inverse Λ−1,

      or in terms of components

      Define the dual to

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