Lorentz transformations .1 The KG equation

Lorentz Transformations and Discrete Symmetries

In this chapter we shall review various covariances(see appendix D) of the KG and Dirac equations, concentrating mainly on the latter. First, we consider Lorentz transformations (rotations and velocity transformations) and show how the scalar KG wavefunction and the 4-component Dirac spinor must transform in order that the respective equations be covariant under these transformations. Then we perform a similar task for the discrete transforma­

tions of parity, charge conjugation and time reversal. The results enable us to construct ‘bilinear covariants’ having well-defined behaviour (scalar, pseu­

doscalar, vector, etc.) under these transformations. This is essential for later work, for two reasons: first, we shall be able to do dynamical calculations in a way that is manifestly covariant under Lorentz transformations; and secondly we shall be ready to study physical problems in which the discrete transfor­

mations are, or are not, actual symmetries of the real world, a topic to which we shall return in the second volume.

4.1 Lorentz transformations

4.1.1 The KG equation

In order to ensure that the laws of physics are the same in all inertial frames, we require our relativistic wave equations to be covariant under Lorentz trans­

formations – that is, they must have the same form in the two different frames (see appendix D). In the case of the KG equation

(❗ +m ²)φ(x) = −iq[∂_μA^μ(x) + A^μ(x)∂_μ]φ(x) + q ²A²(x)φ(x) (4.1) for a particle of charge q in the field A^μ, this requirement is taken care of, almost automatically, by the notation. Consider a Lorentz transformation such that x → x ′ . A^μ will transform by the usual 4-vector transformation law (i.e. like x^μ), which we write as A^μ(x) → A ^′μ(x ^′ ). Similarly we write the transform of φ as φ(x) → φ ^′ (x ^′ ). Then in the primed coordinate frame physics must be described by the equation

(❗ ^′ +m ²)φ ^′ (x ^′ ) = −iq[∂_μ^′ A ^′μ(x ^′ ) + A ^′μ(x ^′ )∂_μ^′ ]φ ^′ (x ^′ ) + q ²A ^′2(x ^′ )φ ^′ (x ^′ ). (4.2) 87

Now the 4-dimensional dot products appearing in (4.2) are all invariant under the Lorentz transformation, so that (4.2) can be written as

(❗ +m ²)φ ^′ (x ^′ ) = −iq[∂_μA^μ(x) + A^μ(x)∂_μ]φ ^′ (x ^′ ) + q ²A²(x)φ ^′ (x ^′ ), (4.3) and we see that the wavefunction in the primed frame may be identified (up to a phase) with that in the unprimed frame:

φ ^′ (x ^′ ) = φ(x). (4.4) Equation (4.4) is the condition for the KG equation to be covariant under Lorentz transformations. Since x ′ is a known function of x, given by the angles and velocities parametrizing the transformation, equation (4.4) enables one to construct the correct function φ which the primed observers must use, ′

in order to be consistent with the unprimed observers.

By way of illustration, consider a rotation of the coordinate system by an angle αin a positive sense about the x-axis; then the position vector referred

′ ′ ′

to the new system is x = (x , y , z ^′ ) where

( )_′ ( ) ( )

x 1 0 0 x

( y ^′ ) = ( 0 cosα sinα) ( y) , (4.5) z ′ 0 − sinα cosα z

which we shall write as

x′ = R_x(α)x. (4.6)

Correspondingly, equation (4.4) is, in this case,

φ ^′ (R_x(α)x) = φ(x), (4.7) which can also be written as

φ ^′ (x) = φ(R⁻_x ¹(α)x). (4.8) It is convenient to begin with an ‘infinitesimal rotation’, where the angle αin (4.5) is replaced by ∈_x such that cos∈_x ≈1 and sin∈_x ≈∈_x. Then it is easy to verify that (4.5) becomes

x′ = R_x(∈_x)x= x −∈ ×x (4.9) where ∈= (∈_x,0,0). For a general infinitesimal rotation, we simply replace this

∈by a general one, (∈_x, ∈_y, ∈_z). For such a rotation, condition (4.8) becomes φ ^′ (x) = φ(x+∈ ×x). (4.10) Expanding the right hand side to first order in ∈we obtain

φ ^′ (x) = φ(x) + (∈ ×x)·∇φ= φ(x) + ∈ ·(x ×∇)φ

= (1 + i∈ ·L)φ(x) ˆ (4.11)

where Lˆ is the vector angular momentum operator x × −i∇.

89 4.1. Lorentztransformations

The rule for finite rotations may be obtained from the infinitesimal form by using the result

e = lim (1 + A A/n)ⁿ (4.12)

n→∞

generalized to differential operators (the exponential of a matrix being un­

1 A²

derstood as the infinite series expA= 1+ A+₂ + . . . ). Let ∈ = α/n, where α= (α_x, α_y, α_z ) are three real finite parameters; we may think of the direction of α as representing the axis of the rotation, and the magnitude of α as representing the angle of rotation. Then applying the transformation (4.11) n times, and letting n tend to infinity, we obtain for the finite rotation

iα· ˆ

φ ^′ (x) = e Lφ(x) ≡ Uˆ_R(α)φ(x). (4.13) Note that Uˆ_R(α) is a unitary operator, since Uˆ_R† is the inverse rotation.

Equation (4.13) is, of course, the familiar rule for rotations of scalar wave-functions, exhibiting the intimate connection between rotations and angular momentum in quantum mechanics. We recall that if a Hamiltonian is invari­

ant under rotations, then the operators Lˆ commute with the Hamiltonian and angular momentum is conserved.

A similar calculation may be done for velocity transformations (‘boosts’), leading to corresponding operators Kˆ – see problem 4.1.

4.1.2 The Dirac equation

The case of the Dirac equation is more complicated, because (unlike the KG φ) the wavefunction has more than one component, corresponding to the fact that it describes a spin-1/2 particle. There is, however, a direct connection between the angular momentum associated with a wavefunction, and the way that the wavefunction transforms under rotations of the coordinate system. To take a simple case, the 2p wavefunctions mentioned in section 3.2 correspond to l = 1 on the one hand and, on the other, to the components of a vector – indeed the most basic vector of all, the position vector x= (x, y, z) itself. If we rotate the coordinate system in the way represented by (4.5), the components in the primed system transform into simple linear combinations of the components in the original system.

Very much the same thing happens in the case of spinor wavefunctions, except that they transform in a way different from – though closely related to – that of vectors. In the present section we shall discuss how this works for three-dimensional rotations of the spatial coordinate system, and explain how it generalizes to boosts, which include transformations of the time coordinate as well. It will be convenient to use the alternative representation (3.40) for the Dirac matrices. In this representation, the components φ, χ of the free-particle 4-spinor ω of (3.43) satisfy

Eφ = σ · pφ +mχ (4.14)

Eχ = −σ · pχ +mφ (4.15)

rather than (3.45) and (3.46).

As before, we start with the infinitesimal rotation (4.9). Since pis a vector, it transforms in the same way as x, so that under an infinitesimal rotation p becomes p′ where

p′ = p−∈×p. (4.16)

The question for us now is: how do the spinors φ and χ transform under this same rotation of the coordinate system?

The essential point is that in the new coordinate system the defining equa­

tions (4.14) and (4.15) should take exactly the same form, namely

′ φ ^′

Eφ ^′ = σ· p +mχ ^′ (4.17)

′ χ ^′

Eχ ^′ = −σ·p +mφ ^′ (4.18) where φ ^′ and χ ^′ are the spinors in the new coordinate system, and we have used the fact that both E and m do not change under rotations. Our task is to find φ ^′ and χ ^′ in terms of φ and χ.

Since both φ and χ are 2-component spinors, we might guess from (4.11) that the answer is

φ ^′ = (1 + iσ·∈/2)φ, χ ^′ = (1 + iσ· ∈/2)χ, (4.19) since the σ/2 are the spin-1/2 matrices, taking the place of Lˆ . To check that this is, in fact, the correct transformation law, we proceed as follows.¹ First, multiply (4.14) from the left by the matrix (1 + iσ· ∈/2): then, since E and m commute with all matrices, the result is

Eφ ^′ = (1 + iσ· ∈/2)σ·pφ +mχ ^′ (4.20)

= (1 + iσ· ∈/2)σ·p(1 −iσ· ∈/2)φ ^′ +mχ ^′ (4.21) where we have used

(1 + iσ·∈/2)⁻¹ ≈ (1 −iσ·∈/2) (4.22) to first order in ∈. Keeping only first order terms in ∈, the first term on the right hand side of (4.21) is

(σ·p+ 1

iσ·∈ σ· p− 1

iσ· p σ·∈)φ ^′ . (4.23)

2 2

This can be simplified using the result from problem 3.4(b):

σ·a σ· b= a·b+ iσ· a×b, (4.24) provided all the components of a and b commute. Applying (4.24), (4.23) becomes

[σ·p+ i

(∈·p+ iσ· ∈× p) − i

(∈· p+ iσ·p×∈)]φ ^′ (4.25)

2 2

′ φ ^′

= (σ· p−σ·∈×p)φ ^′ = σ·p . (4.26)

1We shall derive (4.19), and the corresponding rule for velocity transformations, equation (4.42)below,inappendixMofvolume2usinggrouptheory.

91 4.1. Lorentztransformations

Hence (4.21) is just

′ φ ^′

Eφ ^′ = σ · p +mχ ^′ (4.27) as required in (4.17). We can similarly check the correctness of the transfor­

mation law (4.19) for χ.

The transformation rule for a finite rotation may be obtained from the infinitesimal form by using the result (4.12) applied to matrices. Then for a finite rotation we obtain the result

φ ^′ = exp(iσ ·α/2) φ, χ ^′ = exp(iσ · α/2) χ. (4.28) We note that the behaviour of φ and χ under rotations is the same: equation (4.28) is the way all 2-component spinors transform under rotations.

By way of an illustration, consider the case of the finite rotation (4.5).

Here α= (α,0,0), and the transformation matrix is exp(iσ_xα/2) = 1 + iσ_xα/2 + 1

(iσ_xα/2)²+. . . . (4.29) 2

Multiplying out the terms in (4.29) and remembering that σ_x ² = 1, we see that the transformation matrix is

( )

cosα/2 i sinα/2

cosα/2 + iσ_x sinα/2 = . (4.30) i sinα/2 cosα/2

This means that the components φ1, φ2 of the spinor φ transform according to the rule

φ ^′ ₁ = cosα/2 φ1+ i sinα/2 φ2 (4.31) φ ^′ ₂ = i sinα/2 φ1+ cosα/2 φ2, (4.32) for this particular rotation. The transformed components are linear combina­

tions of the original components, but it is the half-angle α/2 that enters, not α.

Let us denote the finite transformation matrix by U, so that U^†

U = exp(iσ · α/2) and = exp(−iσ ·α/2). (4.33) It follows that

U U^† = U^†U= 1, (4.34)

since the rotation parametrized by −αclearly undoes the rotation parametrized by α. So U is a 2 × 2 unitary matrix. It follows that the normalization of φ and χ is preserved under rotations: φ^′†φ ^′ = φ^†φ, and χ^′†χ ^′ = χ^†χ. The free-particle Dirac probability density ρ = ψ^†ψ = φ^†φ + χ^†χ is therefore also (as we expect) invariant under rotations.

More interestingly, we can examine the way the free-particle current den­

sity

j= ψ^†αψ = φ^†σφ −χ^†σχ (4.35)

transforms under rotations. Of course, it should behave as a 3-vector, and this is checked in problem 4.2(a).

We now turn to the behaviour of the spinors φ and χ under boosts, which mix x and t, or equivalently p and E. For example, consider a Lorentz velocity transformation (boost) from a frame S to a frame S ^′ which is moving with speed u with respect to S along the common x-axis. Then the energy E and momentum p_x of a particle in S are transformed to E ^′ and p ′ _x in S ^′ where (cf (D.1))

E ^′ = coshϑ E −sinhϑ p_x (4.36) p ′ _x = coshϑ p_x −sinhϑ E, (4.37)

2)^−1/2

where coshϑ = (1 − u ≡ γ(u), and sinhϑ = γ(u)u. As before, we start with an infinitesimal transformation, where ϑ is replaced by η_x such that coshη_x ≈ 1 and sinhη_x ≈ η_x. Then (4.36) and (4.37) become E ^′ = E − η_xp_x, p ′ _x = p_x − η_xE. For the general infinitesimal boost parametrized by η= (η_x, η_y, η_z ), the transformation law for (E,p) is

E ^′ = E−η ·p (4.38)

p′ = p −ηE. (4.39)

Once again, we have to determine φ ^′ and χ ^′ such that the transformed versions of (4.14) and (4.15) are

(E ^′ −σ · p^′ )φ ^′ = mχ ^′ (4.40) (E ^′ +σ ·p^′ )χ ^′ = mφ . ′ (4.41) Note that this time E does transform, according to (4.38).

The required φ ^′ and χ ^′ are

φ ^′ = (1 −σ · η/2)φ, χ ^′ = (1 +σ · η/2)χ. (4.42) The spinors φ and χ behaved the same under rotations, but they transform differently under boosts. There are two kinds of 2-component spinors, φ-type and χ-type, in the representation (3.40), which are distinguished by their behaviour under boosts. The group theory behind this will be explained in appendix M of volume 2.

To verify the rule (4.42), take equation (4.14) in the form (4.40) and mul­

tiply from the left by the matrix (1 + σ ·η/2), to obtain

(1 + σ · η/2)(E−σ · p)φ= mχ ^′ , (4.43) or equivalently

(1 + σ ·η/2)(E−σ ·p)(1 + σ · η/2)φ ^′ = mχ ^′ , (4.44) where we have used (1−σ ·η/2)⁻¹ ≈ (1 +σ·η/2). For (4.44) to be consistent with (4.40) we require

(1 + σ · η/2)(E−σ · p)(1 + σ ·η/2) = E ^′ −σ · p′ . (4.45)

93 4.1. Lorentztransformations

Keeping only first order terms in η, the left hand side of (4.45) is

E −σ· p+Eσ· η− 1 (σ·p σ· η+σ· η σ·p) (4.46) 2

= E −η·p−σ·(p− ηE) (4.47)

E ^′ ′

= − σ·p (4.48)

as required for the right hand side of (4.45).

For a finite boost φ and χ transform by the ‘exponentiation’ of (4.42), namely

φ ^′ = exp(−σ· ϑ/2) φ, χ ^′ = exp(σ·ϑ/2) χ (4.49) where the three real parameters ϑ = (ϑ_x, ϑ_y, ϑ_z) specify the direction and magnitude of the boost. In contrast to (4.28), the transformations (4.49) are not unitary. If we denote the matrix exp(−σ· ϑ/2) by B, we have B= B†

−1 †

rather than B = B . So Bdoes not leave φ^†φ and χ^†χ invariant. Actually this is no surprise. We already know from section 4.1.2 that the density φ^†φ + χ^†χ ought to transform as the fourth component ρ of the 4-vector

as required by (4.38). Similarly, it may be verified (problem 4.2(b)) that j transforms as the 3-vector part of the 4-vector j^μ, under this infinitesimal boost.

On the other hand, the products φ^†χ and χ^†φ are clearly invariant under the transformation (4.49), since the exponential factors cancel. This means that the quantity ω^†βω is a Lorentz invariant.

At this point it is beginning to be clear that a more ‘covariant-looking’

notation would be very desirable. In the case of the KG probability current, the 4-vector index μwas clearly visible in the expression on the right-hand side of (3.20), but there is nothing similar in the Dirac case so far. In problem 4.3 the four ‘γ matrices’ are introduced, defined by γ^μ = (γ⁰,γ) with γ⁰= β and γ = βα, together with the quantity ψ¯ ≡ ψ^†γ⁰, in terms of which the Dirac

¯ ¯

ρ of (3.51) and j of (3.57) can be written as ψ(x)γ⁰ψ(x) and ψ(x)γψ(x) respectively. The complete Dirac 4-current is then

j^μ = ψ¯(x)γ^μψ(x). (4.51) For free particle solutions, we (and problem 4.2) have established that j^μ of (4.51) indeed transforms as a 4-vector under infinitesimal rotations and boosts. We have also just seen that the quantity ¯ψψ is an invariant.

We end this section by illustrating the use of the finite boost transforma­

tions (4.49). Consider two frames S and S ^′ , such that in S a particle is at rest

with E = m,p= 0, and with spin up along the z-axis; in S ^′ , the particle has

where the normalization N is determined (since ¯uu is invariant) from the condition ¯u_S ' u_S ' = 2m to be N = (E ^′ +p ^′ )^1/2, giving

But we can also calculate u_S' by applying the transformation (4.49) with tanhϑ ^′ = −v ′ to u_S . Then the upper two components become

ϑ ^' ϑ ^'

φ ^′ = √

m e ^{σz /2}φ+= √

m e ^/2φ+, (4.57) while the lower two components become

−ϑ ^'

95

Im Dokument GAUGE THEORIES (Seite 103-111)