110
4. Lorentz Transformations and Discrete Symmetries
4.2 In this problem, use the representation (3.40) for the Dirac matrices, as
in section 4.1.2.
(a) Using the rule (4.19) for the transformation of the spinor φ under
an infinitesimal rotation of the coordinate system, verify that φ
†
σφ
′
transforms as a 3-vector. [Hint : you need to show that φ
′†
σφ =
φ
†
σφ − ∈ × φ
†
σφ; use the results of problem 3.4(a).] Show also that
the free-particle Dirac probability current density is a 3-vector.
(b) Using the rule (4.42) for the transformation of φ and χ under an
infinitesimal boost, verify that j = φ
†
σφ − χ
†
σχ transforms as the
3-vector part of the 4-vector (ρ, j). [Hint : you need to show that
′
j = j − ηρ.]
4.3
(a) Defining the four ‘γ matrices’
γ
μ = (γ
0 , γ)
where γ
0 = β and γ = βα, show that the Dirac equation can
be written in the form (iγ
μ ∂ μ − m)ψ = 0. Find the anticommutation relations of the γ matrices. Show that the positive energy
spinors u(p, s) satisfy ( / p − m)u(p, s) = 0, and that the negative
energy spinors v(p, s) satisfy ( / p + m)v(p, s) = 0, where / p = γ
μ p μ
(pronounced ‘p-slash’).
(b) Define the conjugate spinor
¯
ψ(x) = ψ
† (x)γ
0
and use the previous result to find the equation satisfied by ψ ¯ in γ
matrix notation.
(c) The Dirac probability current may be written as
j
μ = ψ ¯ (x)γ
μ ψ(x).
Show that it satisfies the conservation law
∂ μ j
μ = 0.
4.4
¯
(a) Verify that, under P, ψ(x, t)γ
0 ψ(x, t) is a scalar, and that ψ ¯ (x, t)γψ(x, t)
is a polar vector.
(b) Verify that a
μ (x, t) = ψ ¯ (x, t)γ 5 γ
μ ψ(x, t) transforms under infinitesimal rotations and boosts as a 4-vector; and that under P a
0 (x) is
a pseudoscalar, and a(x, t) is an axial vector.
∗
(c) Show that σ 2 φ transforms under rotations and boosts as a χ-type
∗
spinor, and that σ 2 χ transforms as a φ-type spinor.
4.5 Verify that ψ ¯ (x, t)Σψ(x, t) · E of (4.132) is odd under T.
4. Lorentz Transformations and Discrete Symmetries
4.2 In this problem, use the representation (3.40) for the Dirac matrices, as
in section 4.1.2.
(a) Using the rule (4.19) for the transformation of the spinor φ under
an infinitesimal rotation of the coordinate system, verify that φ
†
σφ
′
transforms as a 3-vector. [Hint : you need to show that φ
′†
σφ =
φ
†
σφ − ∈ × φ
†
σφ; use the results of problem 3.4(a).] Show also that
the free-particle Dirac probability current density is a 3-vector.
(b) Using the rule (4.42) for the transformation of φ and χ under an
infinitesimal boost, verify that j = φ
†
σφ − χ
†
σχ transforms as the
3-vector part of the 4-vector (ρ, j). [Hint : you need to show that
′
j = j − ηρ.]
4.3
(a) Defining the four ‘γ matrices’
γ
μ = (γ
0 , γ)
where γ
0 = β and γ = βα, show that the Dirac equation can
be written in the form (iγ
μ ∂ μ − m)ψ = 0. Find the anticommutation relations of the γ matrices. Show that the positive energy
spinors u(p, s) satisfy ( / p − m)u(p, s) = 0, and that the negative
energy spinors v(p, s) satisfy ( / p + m)v(p, s) = 0, where / p = γ
μ p μ
(pronounced ‘p-slash’).
(b) Define the conjugate spinor
¯
ψ(x) = ψ
† (x)γ
0
and use the previous result to find the equation satisfied by ψ ¯ in γ
matrix notation.
(c) The Dirac probability current may be written as
j
μ = ψ ¯ (x)γ
μ ψ(x).
Show that it satisfies the conservation law
∂ μ j
μ = 0.
4.4
¯
(a) Verify that, under P, ψ(x, t)γ
0 ψ(x, t) is a scalar, and that ψ ¯ (x, t)γψ(x, t)
is a polar vector.
(b) Verify that a
μ (x, t) = ψ ¯ (x, t)γ 5 γ
μ ψ(x, t) transforms under infinitesimal rotations and boosts as a 4-vector; and that under P a
0 (x) is
a pseudoscalar, and a(x, t) is an axial vector.
∗
(c) Show that σ 2 φ transforms under rotations and boosts as a χ-type
∗
spinor, and that σ 2 χ transforms as a φ-type spinor.
4.5 Verify that ψ ¯ (x, t)Σψ(x, t) · E of (4.132) is odd under T.