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CHAPTER 2

Solutions of the Dirac equation

2.1Plane wave solutions

The Dirac equation in the momentum representation (see (1.80))

or in the coordinate representation

defines a set of matrix equations. Since the Dirac matrices, γ^µ, are 4 × 4 matrices, the wave function ψ, in this case, is a four component column matrix (column vector). From the study of angular momentum, we know that multicomponent wave functions suggest a nontrivial spin angular momentum for the particle. (Other nontrivial internal symmetries can also lead to a multicomponent wavefunction, but here we are considering a simple system without any nontrivial internal symmetry.) Therefore, we expect the solutions of the Dirac equation to describe particles with spin. To understand what kind of particles are described by the Dirac equation, let us look at the plane wave solutions of the equation (which are supposed to describe free particles). Let us denote the four component wave function as (x stands for both space and time)

with

Substituting this back into the Dirac equation, we obtain (we define for any four vector A_µ)

where the four component function, u(p), has the form

Let us simplify the analysis by restricting to motion along the z-axis. In other words, let us set

In this case, equation (2.5) takes the form

Taking the particular representation of the γ^µ matrices in (1.91), we can write this explicitly as

This is a set of four linear homogeneous equations (in the four variables u_α(p), α = 1, 2, 3, 4) and a nontrivial solution exists only if the determinant of the coefficient matrix vanishes. Thus, requiring,

we obtain,

Thus, we see that a nontrivial plane wave solution of the Dirac equation exists only for the energy values

Furthermore, we see from (2.11) that each of these energy values is doubly degenerate. Of course, we would expect the positive and the negative energy roots in (2.12) from Einstein’s relation. However, the double degeneracy seems to be a reflection of the nontrivial spin structure of the wave function as we will see shortly.

The energy eigenvalues (and the degeneracy) can also be obtained in a simpler fashion by noting that (in the gamma matrix representation of (1.91))

This is identical to (2.11) and the energy eigenvalues would then correspond to the roots of this equation given in (2.12). (Note that this method of evaluating a determinant is not valid, in general, for matrices involving submatrices that do not commute. In the present case, however, the submatrices are both diagonal and, therefore, commute which is why this simpler method works.)

We can obtain the solutions (wave functions) of the Dirac equation, in this case, by directly solving the set of four coupled equations in (2.9). Alternatively, we can introduce two component wave functions and and write

where

We note that for the positive energy solutions

the set of coupled equations takes the form

Writing out explicitly, (2.17) leads to

The two component function can be solved in terms of and we obtain from the second relation in (2.18)

Let us note here parenthetically that the first relation in (2.18) also leads to the same relation (as it should), namely,

where we have used the property of the Pauli matrices, namely, (in the first line). Note also that if the relation (2.19) obtained from the second equation in (2.18) is substituted into the first relation, it will hold trivially (because of the Einstein relation). Therefore, the positive energy solution is completely determined by the relation (2.19) in terms of

Choosing the two independent solutions for û as

we obtain respectively

and

This determines the two positive energy solutions of the Dirac equation (remember that the energy eigenvalues are doubly degenerate). (The question of which components can be chosen independently follows from an examination of the dynamical equations. Thus, for example, from the second of the two two-component Dirac equations in (2.18), we note that must vanish in the rest frame while remains arbitrary. Thus, can be thought of as the independent solution.)

Similarly, for the negative energy solutions we write

and the set of equations (2.9) becomes

We can solve these as

Choosing the independent solutions as

we obtain respectively

and

and these determine the two negative energy solutions of the Dirac equation.

The independent two component wave functions in (2.21) and (2.27) are reminiscent of the spin up and spin down states of a two component spinor. Thus, from the fact that we can write

the positive and the negative energy solutions have the explicit forms

The notation is suggestive and implies that the wave function corresponds to that of a spin particle. (We will determine the spin of the Dirac particle shortly.) It is because of the presence of negative energy solutions that the wave function becomes a four component column matrix as opposed to the two component spinor we expect in non-relativistic systems. (The correct counting for the number of components of the wave function for a massive, relativistic particle of spin s in the presence of both positive and negative energies follows to be 2(2s + 1), unlike the nonrelativistic counting (2s + 1).)

From the structure of the wave function, it is also clear that, for the case of general motion, where

the solutions take the forms (with p₀ = E_± = ±E = )

which can be explicitly verified. (The change in the sign in the dependent components in (2.34) compared to (2.30) comes from raising the index of the momentum, namely, p_i = −pⁱ = −(p)_i.)

2.2Normalization of the wave function

Let us note that if we define

then, we can write the solutions (2.34) for motion along a general direction as

Here α and β are normalization constants to be determined. The two component spinors and in (2.21) and (2.27) respectively are normalized as (for the same spin components)

For different spin components, this product vanishes.

Given this, we can now calculate

where we have used the familiar identity satisfied by the Pauli matrices, namely

Similarly, for the negative energy solutions we have

It is worth remarking here that although we have seen in (2.37) that, for the same spin components, we have carried along these factors in (2.38) and (2.40) simply because we have not specified their spin components.

In dealing with the Dirac equation, another wave function (known as the adjoint spinor) plays an important role and is defined to be

Thus, for example,

Thus, we see that the difference between the Hermitian conjugate u^† and the adjoint is in the relative sign in the second of the two-component spinors.

We can also calculate the product

Similarly, we can show that

Our naive instinct will be to normalize the wave function, as in the non-relativistic case, by requiring (for the same spin components)

However, as we will see shortly, this is not a relativistic normalization. In fact, u^†u, as we will see, is related to the probability density which transforms like the time component of a four vector. Thus, a relativistically covariant normalization would be to require (for the same spin components)

(Remember that this will correspond to the probability density and, therefore, must be positive. By the way, the motivation for such a normalization condition comes from the fact that, in the rest frame of the particle, this will reduce to which corresponds to the non-relativistic normalization (2.45).) The independent wave functions for a free particle, ψ_p(x) = e^−ip·xu(p) with p₀ = ±E, with this normalization condition, would give (for the same spin components)

With the requirement (2.46), we determine from (2.38) and (2.40) (for the same spin components when (2.37) holds)

Therefore, with this normalization, we can write the normalized positive and negative energy solutions of the Dirac equation to be

It is also clear that, with this normalization, we will obtain from (2.43) and (2.44) (for the same spin components)

This particular product, therefore, appears to be a Lorentz invariant (scalar) and we will see later that this is indeed true.

Let us also note here that by construction the positive and the negative energy solutions are orthogonal. For example,

Therefore, the solutions we have constructed correspond to four linearly independent, orthonormal solutions of the Dirac equation. Note, however, that

While we will be using this particular normalization for massive particles, let us note that it becomes meaningless for massless particles. (There is no rest frame for a massless particle.) The probability density has to be well defined. Correspondingly, an alternative normalization which works well for both massive and massless particles is given by

This still behaves like the time component of a four vector (m is a Lorentz scalar). In this case, we will obtain from (2.38) and (2.40) (for the same spin components)

Correspondingly, in this case, we obtain

which vanishes for a massless particle. This product continues to be a scalar. Let us note once again that this is a particularly convenient normalization for massless particles.

Let us note here parenthetically that, while the arbitrariness in the normalization of u(p) may seem strange, it can be understood in light of what we have already pointed out earlier as follows. We can write the solution of the Dirac equation for a general motion (along an arbitrary direction) as

where a(p) is a coefficient which depends on the normalization of u(p) in such a way that the wave function would lead to a total probability normalized to unity,

Namely, a particular choice of normalization for the u(p) is compensated for by a specific choice of the coefficient function a(p) so that the total probability integrates to unity. The true normalization is really contained in the total probability.

2.3Spin of the Dirac particle

As we have argued repeatedly, the structure of the plane wave solutions of the Dirac equation is suggestive of the fact that the particle described by the Dirac equation has spin That this is indeed true can be seen explicitly as follows.

Let us define a four dimensional generalization of the Pauli matrices as (in this section, we will use the notations of three dimensional Euclidean space since we will be dealing only with three dimensional vectors)

It can, of course, be checked readily that these matrices are related to the α_i matrices defined in (1.98) and (1.101) through the relation

where

We note that so that we can invert the defining relation (2.59) and write

From the structures of the matrices α_i and we conclude that

This shows that satisfies the angular momentum algebra (remember ℏ = 1) and this is why we call the matrices, the generalized Pauli matrices. (Note, however, that define a reducible representation of spin generators since these matrices are block diagonal.) Using (2.59) and (2.60), it can also be checked that [α_i, α_j] =

Let us also note that

Here we have used the fact that (see (1.101))

is block diagonal like

With these relations at our disposal, let us look at the free Dirac Hamiltonian in (1.100) (remember that we are using three dimensional Euclidean notations in this section)

As we will see in the next chapter, the Dirac equation transforms covariantly under a Lorentz transformation. In other words, Lorentz transformations define a symmetry of the Dirac Hamiltonian and, therefore, rotations which correspond to a subset of the Lorentz transformations must also be a symmetry of the Dirac Hamiltonian. Consequently, the (total) angular momentum operators which generate rotations should commute with the Dirac Hamiltonian. Let us recall that the orbital angular momentum operators are given by (repeated indices are summed)

Calculating the commutator of this operator with the Dirac Hamiltonian, we obtain

Here we have used the fact that since β is a constant matrix and m is a constant, the second term in the Hamiltonian drops out of the commutator. Thus, we note that the orbital angular momentum operator does not commute with the Dirac Hamiltonian. Consequently, the total angular momentum which should commute with the Hamiltonian must contain a spin part as well.

To determine the spin angular momentum, we note that (see (2.63))

so that combining this relation with (2.67) we obtain

In other words, the total angular momentum which should commute with the Hamiltonian, if rotations are a symmetry of the system, can be identified with

In this case, therefore, we can identify the spin angular momentum operator with

Note, in particular, that

which has doubly degenerate eigenvalues Therefore, we conclude that the particle described by the Dirac equation corresponds to a spin (fermionic) particle.

2.4Continuity equation

The Dirac equation, written in the Hamiltonian form (see (1.99)), is given by

Taking the Hermitian conjugate of this equation, we obtain

where the gradient is assumed to act on ψ^†. Multiplying (2.73) with ψ^† on the left and (2.74) with ψ on the right and subtracting the second from the first, we obtain

This is the continuity equation for the probability current density associated with the Dirac equation and we note that we can identify

to write the continuity equation as

This suggests that we can write the current four vector as

so that the continuity equation can be written in the manifestly covariant form

This, in fact, shows that the probability density, ρ, is the time component of J^µ (see (2.78)) and, therefore, must transform like the time coordinate under a Lorentz transformation (as we had alluded to earlier). (We are, of course, yet to show that J^µ transforms like a four vector which we will do in the next chapter.) On the other hand, the total probability

is a constant independent of any particular Lorentz frame. It is worth recalling that we have already used this Lorentz transformation property of ρ in defining the normalization of the wave function.

An alternative and more covariant way of deriving the continuity equation is to start with the covariant Dirac equation

and note that the Hermitian conjugate of ψ satisfies

Multiplying this equation with γ⁰ on the right and using the fact that we obtain ( so that )

where we have used the property of the gamma matrices that (for µ = 0, 1, 2, 3, see also (2.102) and (2.103) in section 2.6)

Multiplying (2.81) with on the left and (2.83) with ψ on the right and subtracting the second from the first, we obtain

This is, in fact, the covariant continuity equation for the Dirac equation and we can identify the conserved current density with

Note from the definition in (2.86) that we can identify

which is what we had derived earlier in (2.78).

Let us conclude this discussion by noting that although the Dirac equation has both positive and negative energy solutions, because it is a first order equation (particularly in the time derivative), the probability density is independent of time derivative much like in the Schrödinger equation. Consequently, the probability density, as we have seen explicitly in (2.38) and (2.40), can be defined to be positive definite even in the presence of negative energy solutions. This is rather different from the case of the Klein-Gordon equation that we have studied in chapter 1.

2.5Dirac’s hole theory

We have seen that the Dirac equation leads to both positive and negative energy solutions. In the free particle case, for example, the energy eigenvalues are given by

Thus, even for this simple case of a free particle the energy spectrum has the form shown in Fig. 2.1. We note from Fig. 2.1 (as well as from the equation above, (2.88)) that the positive and the negative energy solutions are separated by a gap of magnitude 2m (remember that we are using c = 1).

Even when the probability density is consistently defined, the presence of negative energy solutions leads to many conceptual difficulties. First of all, in such a case, we note that the energy spectrum is unbounded from below. Since physical systems have a tendency to go to the lowest energy state available, this implies that any such physical system (of Dirac particles) would make a transition to these unphysical energy states thereby leading to a collapse of all stable systems such as the Hydrogen atom. Classically, of course, we can restrict ourselves to the subspace of positive energy solutions. But as we have argued earlier within the context of the Klein-Gordon equation, quantum mechanically this is not acceptable. Namely, even if we start out with a positive energy solution, any perturbation would cause the energy to lower, destabilizing the physical system and leading to an ultimate collapse.

Figure 2.1: Energy spectrum for a free Dirac particle.

In the case of Dirac particles, however, there is a way out of this difficulty. Let us recall that the Dirac particles carry spin and are, therefore, fermions. To be specific, let us assume that the particles described by the Dirac equation are the spin electrons. Since fermions obey Pauli exclusion principle, any given energy state can accommodate at the most two electrons with opposite spin projections. Taking advantage of this fact, Dirac postulated that the physical ground state (vacuum) in such a theory should be redefined for consistency. Namely, Dirac postulated that the ground state in such a theory is the state where all the negative energy states are filled with electrons. Thus, unlike the conventional picture of the ground state as being the state without any particle (quantum), here the ground state, in fact, contains an infinite number of negative energy particles. Furthermore, Dirac assumed that the electrons in the negative energy states are passive in the sense that they do not produce any observable effect such as charge, electromagnetic field etc. (Momentum and energy of these electrons are also assumed to be unobservable. This simply means that one redefines the values of all these observables with respect to this ground state.)

This redefinition of the vacuum automatically prevents the instability associated with matter. For example, a positive energy electron can no longer drop down to a negative energy state without violating the Pauli exclusion principle since the negative energy states are already filled. (Note that this would not work for a bosonic system such as particles described by the Klein-Gordon equation. It is only because fermions obey Pauli exclusion principle that this works for the Dirac equation.) On the other hand, such a redefinition of the ground state does predict some new physical phenomena which are experimentally observed. For example, if enough energy is provided to such a ground state, a negative energy electron can make a transition to a positive energy state and can appear as a positive energy electron. Furthermore, the absence of a negative energy electron can be thought of as a “hole” which would have exactly the same mass as the particle but otherwise opposite internal quantum numbers. This “hole” state is what we have come to recognize as the antiparticle – in this case, a positron – and the process under discussion is commonly referred to as pair creation (production). Thus, the Dirac theory predicts an anti-particle of equal mass for every Dirac particle. (The absence of a negative energy electron in the ground state can be thought of as the ground state plus a positive energy “hole” state with exactly opposite quantum numbers to neutralize its effects. The amount of energy necessary to excite a negative energy electron to a positive energy state is E ≥ 2m.)

This is Dirac’s theory of electrons and works quite well. However, we must recognize that it is inherently a many particle theory in the sense that the vacuum (ground state) of the theory is defined to contain infinitely many negative energy particles. (This unconventional definition of the vacuum state can be avoided in a second quantized field theory which we will study later.) In spite of this, the Dirac equation passes as a one particle equation primarily because of the Pauli exclusion principle. On the other hand, this is a general feature that combining quantum mechanics with relativity necessarily leads to a many particle theory.

2.6Properties of the Dirac matrices

The Dirac matrices, γ^µ, were crucial in taking a matrix square root of the Einstein relation and, thereby, in defining a first order equation. In this section, we will study some of the useful properties of these matrices. As we have seen, the four Dirac matrices satisfy (in addition to the Clifford algebra)

Since these are 4 × 4 matrices, a complete set of Dirac matrices must consist of 16 such matrices. Of course, the identity matrix will correspond to one of them.

To obtain the other basis matrices, let us define the following sets of matrices. Let

where

represents the four-dimensional generalization of the Levi-Civita tensor. Note that in our particular representation for the γ^µ matrices given in (1.91), we obtain

where we have used the property of the Pauli matrices

We recognize from (2.92) that we can identify this with the matrix ρ defined earlier in (2.60). Note that, by definition,

and that, since it is the product of all the four γ^µ matrices, it anti-commutes with any one of them. Namely,

Given the matrix γ₅, we can define four new matrices as

Since we know the explicit forms of the matrices γ^µ and γ₅ in our representation, let us write out the forms of γ₅γ^µ also in this representation.

Finally, we can also define six anti-symmetric matrices, σ^µν, as (µ, ν = 0, 1, 2, 3)

whose explicit forms in our representation can be worked out to be (i, j, k = 1, 2, 3)

Here we have used the three dimensional notation ϵ_ijk = ϵ^ijk. We have already seen in (2.71) that the matrices represent the spin operators for the Dirac particle. From (2.99) we conclude, therefore, that the matrices

can be identified with the spin operators for the Dirac particle. (This relation can be obtained from (2.99) using the identity for products of Levi-Civita tensors, namely, ϵ_ijkϵ_ℓjk = 2 δ_iℓ.)

We have thus constructed a set of sixteen Dirac matrices, namely,

where the numbers on the right denote the number of matrices in each category and these, in fact, provide a basis for all the 4 × 4 matrices. Here, the notation is suggestive and stands for the fact that transforms like a scalar under Lorentz and parity transformations. Similarly, and behave respectively like a vector, tensor, axial-vector and a pseudo-scalar under Lorentz and parity transformations as we will see in the next chapter.

Let us note here that each of the matrices, even within a given class, has its own Hermiticity property. However, it can be checked that except for γ₅, which is defined to be Hermitian, all other matrices satisfy

In fact, it follows easily that

where we have used the fact that γ₅ is Hermitian and it anti-commutes with γ^µ. Finally, from

it follows that

The Dirac matrices satisfy nontrivial (anti) commutation relations. We already know that

We can also calculate various other commutation relations in a straightforward and representation independent manner. For example,

In this derivation, we have used the fact that

We note here parenthetically that the commutator in (2.108) can also be expressed in terms of commutators (instead of anti-commutators) as

However, since γ^µ matrices satisfy simple anti-commutation relations, the form in (2.108) is more useful for our purpose.

Similarly, for the commutator of two σ^µν matrices, we obtain

Thus, we see that the σ^µν matrices satisfy an algebra in the sense that the commutator of any two of them gives back a σ^µν matrix. We will see in the next chapter that they provide a representation for the Lorentz algebra.

The various commutation and anti-commutation relations also lead to many algebraic simplifications in dealing with such matrices. This becomes particularly useful in calculating various amplitudes involving Dirac particles. Thus, for example, (these relations are true only in 4-dimensions)

where we have used (γ_µ = η_µνγ^ν)

and it follows now that,

Similarly,

and so on.

The commutation and anti-commutation relations also come in handy when we are evaluating traces of products of such matrices. For example, we know from the cyclicity of traces that

Therefore, it follows (in 4-dimensions) that

Here in the second relation we have used the fact that γ₅ anti-commutes with γ^µ in addition to the cyclicity of trace. Even more complicated traces can be evaluated by using the basic relations we have developed so far. For example, we note that

and so on. We would use all these properties in the next chapter to study the covariance of the Dirac equation under a Lorentz transformation.

To conclude this section, let us note that we have constructed a particular representation for the Dirac matrices commonly known as the Pauli-Dirac representation. However, there are other equivalent representations possible which may be more useful for a particular system under study. For example, there exists a representation for the Dirac matrices where γ^µ are all purely imaginary. This is known as the Majorana representation and is quite useful in the study of Majorana fermions which are charge neutral fermions. Explicitly, the matrices have the forms

It can be checked that the Dirac matrices in the Pauli-Dirac representation and the Majorana representation are related by the similarity (unitary) transformation (see (1.93))

Similarly, there exists yet another representation for the γ^µ matrices, namely,

where

This is known as the Weyl representation for the Dirac matrices and is quite useful in the study of massless fermions. It can be checked that the Weyl representation is related to the standard Pauli-Dirac representation through the similarity (unitary) transformation

2.6.1 Fierz rearrangement. As we have pointed out in (2.101), the sixteen Dirac matrices Γ^(a), a = S, V, T, A, P define a complete basis for 4 × 4 matrices. This is easily demonstrated by showing that they are linearly independent which is seen as follows.

We have explicitly constructed the sixteen matrices to correspond to the set

From the properties of the γ^µ matrices, it can be easily checked that

where “Tr” denotes trace over the matrix indices. As a result, given this set of matrices, we can construct the inverse set of matrices as

such that

Explicitly, we can write the inverse set of matrices as

With this, the linear independence of the set of matrices in (2.123) is straightforward. For example, it follows now that if

then, multiplying (2.128) with Γ_(b), where b is arbitrary, and taking trace over the matrix indices and using (2.126) we obtain

for any b = S, V, T, A, P . Therefore, (2.128) implies that all the coefficients of expansion must vanish which shows that the set of sixteen matrices Γ^(a) in (2.123) are linearly independent. As a result they constitute a basis for 4 × 4 matrices.

Since the set of matrices in (2.123) provide a basis for the 4 × 4 matrix space, any arbitrary 4 × 4 matrix M can be expanded as a linear superposition of these matrices, namely,

Multiplying this expression with Γ_(b) and taking trace over the matrix indices, we obtain

Substituting (2.131) into the expansion (2.130), we obtain

Introducing the matrix indices explicitly, this leads to

Here α, β, γ, η = 1, 2, 3, 4 and correspond to the matrix indices of the 4 × 4 matrices and we are assuming that the repeated indices are being summed.

Equation (2.133) describes a fundamental relation which expresses the completeness relation for the sixteen basis matrices. Just like any other completeness relation, it can be used effectively in many ways. For example, we note that if M and N denote two arbitrary 4 × 4 matrices, then using (2.133) we can derive (for simplicity, we will use the standard convention that the repeated index (a) as well as the matrix indices are being summed)

Using the relations in (2.134), it is now straightforward to obtain

The two relations in (2.135) are known as the Fierz rearrangement identities which are very useful in calculating cross sections. In deriving these identities, we have assumed that the spinors are ordinary functions. On the other hand, if they correspond to anti-commuting fermion operators, the right-hand sides of the identities in (2.135) pick up a negative sign which arises from commuting the fermionic fields past one another.

Let us note that using the explicit forms for Γ^(a) and Γ_(a) in (2.123) and (2.125) respectively, we can write the first of the Fierz rearrangement identities in (2.135) as (assuming the spinors to be ordinary functions and not anti-commuting fermion fields which will introduce an overall negative sign, for example, in commuting ψ₂ past )

Since this is true for any matrices M, N and any spinors, we can define a new spinor = Nψ₄ to write the identity in (2.136) equivalently as

which is often calculationally simpler. Thus, for example, if we choose

then using various properties of the gamma matrices derived earlier as well as (2.111) and (2.114), we obtain from (2.137)

This is the well known fact from the weak interactions that the V − A form of the weak interaction Hamiltonian proposed by Sudarshan and Marshak is form invariant under a Fierz rearrangement (the negative sign is there simply because we are considering spinor functions and will be absent for anti-commuting fermion fields).

2.7References

1.J. D. Bjorken and S. Drell, Relativistic Quantum Mechanics, McGraw-Hill, New York (1964).

2.A. Das, Lectures on Quantum Mechanics, Hindustan Publishing, New Delhi, India and World Scientific, Singapore (2011).

3.A. Das and S. Okubo, Lie Groups and Lie Algebras for Physicists, Hindustan Publishing, New Delhi and World Scientific, Singapore (2014).

4.C. Itzykson and J.-B. Zuber, Quantum Field Theory, McGraw-Hill, New York (1980).

5.S. Okubo, Real representations of finite Clifford algebras. I. Classification, Journal of Mathematical Physics 32, 1657 (1991).

6.L. I. Schiff, Quantum Mechanics, McGraw-Hill, New York (1968).

7.E. C. G. Sudarshan and R. E. Marshak, Proceedings of Padua-Venice conference on mesons and newly discovered particles, (1957); Physical Review 109, 1860 (1958).