Lectures on Quantum Field Theory. Ashok Das

transforms in an inverse manner compared with the contravariant vector.) Therefore, we can think of the Lorentz boost along the 1-axis (x-axis) as a rotation in the 0-1 plane with an imaginary angle (so that we have hyperbolic functions instead of ordinary trigonometric functions). (That these rotations become complex is related to the fact that the metric has opposite signature for time and space components.) Furthermore, as we have seen, the “angle” of rotation, ω, can take any real value and, as a result, Lorentz boosts correspond to noncompact transformations unlike space rotations.

      Let us finally note that if we are considering an infinitesimal Lorentz boost along the 1-axis (or a rotation in the 0-1 plane), then we can write (see (3.13) with ω = ϵ = infinitesimal)

      where,

      It follows from this that

      In other words, the matrix representing the change under an infinitesimal Lorentz boost is anti-symmetric just like the case of an infinitesimal rotation. In a general language, therefore, we note that we can combine rotations and Lorentz boosts into what are known as the homogeneous Lorentz transformations, which can be thought of as rotations in the four dimensional space-time.

      General Lorentz transformations are defined as transformations

      which leave the length of the vector invariant, namely,

      where we have used the fact that the metric, η_µν, remains invariant under a Lorentz transformation. Equation (3.22) is, of course, what we have seen before in (3.16). Lorentz transformations define the maximal symmetry of the space-time manifold which leaves the origin invariant.

      Choosing ρ = σ = 0, we can write out the relation (3.22) explicitly as

      Therefore, we conclude that

      If Λ⁰₀ ≥ 1, then the transformation is called orthochronous. (The Greek prefix “ortho” means straight up. Thus, orthochronous means straight up in time. Namely, such a Lorentz transformation does not change the direction of time. Incidentally, “gonia” in Greek means an angle or a corner and, therefore, orthogonal means the corner that is straight up (perpendicular). In the same spirit, an orthodontist is someone who can make your teeth straight.) Note also that since (see (3.16))

      we obtain

      where we have used det Λ^T = det Λ. The set of homogeneous Lorentz transformations satisfying

      are known as the proper orthochronous Lorentz transformations and constitute a set of continuous transformations that can be connected to the identity matrix. (Just to emphasize, we note that the set of transformations with det Λ = 1 are known as proper transformations and the set for which Λ⁰₀ ≥ 1 are called orthochronous.) In general, however, there are four kinds of Lorentz transformations, namely,

      Given the proper orthochronous Lorentz transformations, we can obtain the other Lorentz transformations by simply appending space reflection or time reflection or both (which are discrete transformations). Thus, if Λ_prop denotes a proper orthochronous Lorentz transformation, then by adding space reflection, x → −x, we obtain a Lorentz transformation

      This would correspond to having Λ⁰₀ ≥ 1, det Λ = −1 (which is orthochronous but no longer proper). If we add time reversal, t → −t, to a proper orthochronous Lorentz transformation, then we obtain a Lorentz transformation

      satisfying Λ⁰₀ ≤ −1 and det Λ = −1 (which is neither proper nor orthochronous). Finally, if we add both space and time reflections, xµ → −x^µ, to a proper orthochronous Lorentz tranformation, we obtain a Lorentz transformation

      with Λ⁰₀ ≤ −1 and det Λ = 1 (which is proper but not orthochronous). These additional transformations, however, cannot be continuously connected to the identity matrix since they involve discrete reflections. In these lectures, we would refer to proper orthochronous Lorentz transformations as the Lorentz transformations.

3.2Covariance of the Dirac equation

      Given any dynamical equation of the form

      where L is a linear operator, we say that it is covariant under a given transformation provided the transformed equation has the form

      where ψ′ represents the transformed wavefunction and L′ stands for the transformed operator (namely, the operator L with the transformed variables). In simple terms, covariance implies that a given equation is form invariant under a particular transformation (has the same form in different reference frames).

      With this general definition, let us now consider the Dirac equation

      Under a Lorentz transformation

      if the transformed equation has the form

      where ψ′(x′) is the Lorentz transformed wave function, then the Dirac equation would be covariant under a Lorentz transformation. Note that the Dirac matrices, γ^µ, are a set of four space-time independent matrices and, therefore, do not change under a Lorentz transformation.

      Let us assume that, under a Lorentz transformation, the transformed wavefunction has the form