Foundations of Quantum Field Theory. Klaus D Rothe
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Название: Foundations of Quantum Field Theory

Автор: Klaus D Rothe

Издательство: Ingram

Жанр: Физика

Серия: World Scientific Lecture Notes In Physics

isbn: 9789811221941

isbn:

СКАЧАТЬ generators of boosts. We have

      so that different “boosts” commute with each other. The Galilei transformations thus correspond to an abelian Lie group. In particular

       Boosts

      Let S and S′ be two inertial frames whose clocks are synchronized in such a way, that their respective origins coincide at time t = 0. Then we have for an eigenstate of the momentum operator, as seen by observers O and O′ in S and S′

      respectively. In particular, for a particle at rest in system S we obtain, from the point of view of O′,

      Define

      where

stands for a Galilei transformation taking
and
. U[B(
)] is thus an operator which takes a particle at rest into a particle with momentum
. One refers to this as a “boost” (active point of view).

      We have the following property of Galilei transformations not shared by Lorentz transformations (compare with (2.23)): boosts and rotations separately form a group. Indeed one easily checks that

      (4)Causality

      We next want to show that NRQM violates the principle of causality. We have for any interacting theory,

      Let |En

be a complete set of eigenstates of H:3

      Then

      Define

      as well as

      The kernel

satisfies a heat-like equation:

      In terms of this kernel we have from above,

      We now specialize to the case of a free point-like particle. In that case

      and correspondingly we have with (1.8),

      Notice that the kernel K0 satisfies the desired initial condition (1.9).

      From (1.10), for the initial condition

, 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 СКАЧАТЬ