Foundations of Quantum Field Theory. Klaus D Rothe

       ⁵

      ⁶These commutation relations do not fix the sign of the generator Ki of boosts. We follow in (2.22) the convention adopted by S. Weinberg.

      ⁷The minus sign in the last commutation relations reflects the fact that the Lorentz group SO(3, 1) can be considered as the complexification of the rotation group in four dimensions, SO(4). The fact that the commutator of two generators of the boost is given as a linear combination of the generators of rotations is related to the phenomenon of the Thomas precession. From the group theoretic point of view it expresses the fact that, if we perform the sequence of infinitesimal boosts g(−δθ₂)g(−δθ₁)g(δθ₂)g(δθ₁) with , the effect is just an infinitesimal rotation in the frame we started from.

      ⁸We follow closely the notation of S. Weinberg, Phys. Rev. 133 (1964) B1318.

      ⁹We follow again closely the notation of S. Weinberg, Phys. Rev. 133 (1964) B1318 and Phys. Rev. 134 (1964) B882. It is to be kept in mind that, unlike us, S. Weinberg uses the metric gμν = (−1, 1, 1, 1).

      ¹⁰For a more detailed analysis see E. P. Wigner, Ann. Phys. Math. 40 (1939) 149.

      ¹¹E.P. Wigner, Theoretical Physics (International Atomic Energy Vienna, 1963) p. 59.

      ¹²We follow again the notation of S. Weinberg, Phys. Rev. 134 (1964) B882.

Chapter 3

       Search for a Relativistic Wave Equation

      In this chapter we engage in the search for a relativistic wave equation for a spin zero particle moving in an external potential, reducing to the ordinary Schrödinger equation in the non-relativistic limit. We begin by looking at the case of a free particle. We adhere to the familiar quantum mechanical principles, as long as we can. We shall encounter a number of difficulties which will lead us to eventually abandon the usual probability interpretation.

3.1A relativistic Schrödinger equation

      As discussed in Chapter 1, the time evolution of a quantum mechanical state is governed by the equation of motion (ħ = c = 1)

      where H is the Hamiltonian operator of the system obtained from the corresponding classical Hamiltonian by representing the canonically conjugate dynamical variables q and p in H(q, p) by operators satisfying canonical commutation relations. In this and the following two sections we consider the case of a free particle. Since in a relativistic theory the relation between the energy of a free particle and its momentum is given by , it is natural to take for the Hamiltonian

      In the coordinate representation, the canonical commutation relations are as usual realized by

      Defining the wave function associated with the state , one arrives thus at the wave equation

      This equation in turn implies that is also a solution of the Klein–Gordon (KG) equation

      where □ denotes the D’Alembert operator. Since this operator is manifestly Lorentz-invariant, covariance of the physical laws demand that under a Lorentz transformation the wave function transforms like a scalar:

      This is also implied by Eq. (3.1), since on the space of solutions of the Klein–Gordon equation (3.2) the operator appearing on the left- and right-hand sides of Eq. (3.1) transform in the same way:

      Note that in contrast to the non-relativistic case, where the wave function does not transform like a scalar, but rather picks up a phase under Galilei transformations, such a phase is absent in the relativistic case, rendering the covariance of the equation of motion manifest in this case.

      The operator appearing on the rhs of Eq. (3.1) is defined in terms of its Taylor expansion:

      Hence its eigenfunctions are given by the plane waves , with the corresponding eigenvalues . As a consequence, the action of this operator on any absolute integrable function in R³ is defined via its Fourier transform

      as

      The factor 1/2ω(

) has been included in the measure to make it relativistically invariant. In order for to be a solution of Eq. (3.1), we must further have

      where . Hence, the most general solution to Eq. (3.2) has the form

      In order for to be a Lorentz scalar, the Fourier coefficients will themselves have to be Lorentz scalars:

      In the non-relativistic limit

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