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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 Λprop 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 → −xµ, 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, γµ, 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
where S(Λ) is a 4 × 4 matrix, since ψ(x) is a four component spinor. Parenthetically, what this means is that we are finding a representation of the Lorentz transformation on the Hilbert space. In the notation of other symmetries that we know from studies in non-relativistic quantum mechanics, we can define an operator L(Λ) to represent the Lorentz transformation on the coordinate states as (with indices suppressed)
However, since the Dirac wavefunction is a four component spinor, in addition to the change in the coordinates, the Lorentz transformation can also mix up the spinor components (much like angular momentum/rotation does). Thus, we can define the Lorentz transformation acting on the Dirac Hilbert space (Hilbert space of states describing a Dirac particle) as, (with S(Λ) representing the 4 × 4 matrix which rotates the matrix components of the wave function)
where the wave function is recognized to be
so that, from (3.39) we obtain (see (3.37))
Namely, the effect of the Lorentz transformation, on the wave function, can be represented by a matrix S(Λ) which depends only on the parameter of transformation Λ and not on the space-time coordinates. A more physical way to understand this is to note that the Dirac wave function simply consists of four functions which do not change, but get rotated by the S(Λ) matrix.
Since the Lorentz transformations are invertible, the matrix S(Λ) must possess an inverse so that from (3.37) we can write
Let us also note from (3.35) that
define a set of real quantities. Thus, we can write
where we have used (3.42).
Therefore, we see from (3.44) that the Dirac equation will be form invariant (covariant) under a Lorentz transformation provided there exists a matrix S(Λ), generating Lorentz transformations (for the Dirac wavefunction), such that
Let us note that if we define
then,
where we have used the orthogonality of the Lorentz transformations (see (
