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Lorentz Invariance of Dirac Equation

Consider two inertial frames, and . Let the and be the space-time coordinates of a given event in each frame, respectively. These coordinates are related via a Lorentz transformation, which takes the general form

 (11.52)

where the are real numerical coefficients that are independent of the . We also have

 (11.53)

Now, because [see Equation (11.5)]

 (11.54)

it follows that

 (11.55)

Moreover, it is easily shown that

 (11.56) (11.57)

By definition, a 4-vector, , has analogous transformation properties to the . Thus,

 (11.58) (11.59)

et cetera.

In frame , the Dirac equation is written

 (11.60)

Let be the wavefunction in frame . Suppose that

 (11.61)

where is a transformation matrix that is independent of the . (Hence, commutes with the and the .) Multiplying Equation (11.60) by , we obtain

 (11.62)

Hence, given that the and are the covariant components of 4-vectors, we obtain

 (11.63)

Suppose that

 (11.64)

which is equivalent to

 (11.65)

Here, we have assumed that the commute with and the (because they are just numbers). If Equation (11.64) holds then Equation (11.63) becomes

 (11.66)

A comparison of this equation with Equation (11.60) reveals that the Dirac equation takes the same form in frames and . In other words, the Dirac equation is Lorentz invariant. Incidentally, it is clear from Equations (11.60) and (11.66) that the matrices are the same in all inertial frames.

It remains to find a transformation matrix that satisfies Equation (11.65). Consider an infinitesimal Lorentz transformation, for which

 (11.67)

where the are real numerical coefficients that are independent of the , and are also small compared to unity. To first order in small quantities, Equation (11.55) yields

 (11.68)

Each of the six independent non-vanishing generates a particular infinitesimal Lorentz transformation. For instance,

 (11.69)

for a transformation to a coordinate system moving with a velocity along the -direction. Furthermore,

 (11.70)

for a rotation through an angle about the -axis, and so on.

Let us write

 (11.71)

where the are matrices. To first order in small quantities,

 (11.72)

Moreover, it follows from Equation (11.68) that

 (11.73)

To first order in small quantities, Equations (11.65), (11.67), (11.71), and (11.72) yield

 (11.74)

Hence, making use of the symmetry property (11.68), we obtain

 (11.75)

where . Because this equation must hold for arbitrary , we deduce that

 (11.76)

Making use of the anti-commutation relations (11.29), it can be shown that a suitable solution of the previous equation is

 (11.77)

(See Exercise 7.) Hence,

 (11.78) (11.79)

Now that we have found the correct transformation rules for an infinitesimal Lorentz transformation, we can easily find those for a finite transformation by building it up from a large number of successive infinitesimal transforms [9]. (See Exercises 8 and 9.)

Making use of Equation (11.34), as well as , the Hermitian conjugate of Equation (11.78) can be shown to take the form

 (11.80)

Hence, Equation (11.65) yields

 (11.81)

It follows that

 (11.82)

or

 (11.83)

which implies that

 (11.84)

where the are defined in Equation (11.46). This proves that the transform as the contravariant components of a 4-vector.

Next: Free Electron Motion Up: Relativistic Electron Theory Previous: Dirac Equation
Richard Fitzpatrick 2016-01-22