**Unformatted text preview: **Physics 523 & 524: Quantum Field Theory I
Kevin Cahill
Department of Physics and Astronomy
University of New Mexico, Albuquerque, NM 87131 Contents 1 Quantum fields and special relativity
1.1 States
1.2 Creation operators
1.3 How fields transform
1.4 Translations
1.5 Boosts
1.6 Rotations
1.7 Spin-zero fields
1.8 Conserved charges
1.9 Parity, charge conjugation, and time reversal
1.10 Vector fields
1.11 Vector field for spin-zero particles
1.12 Vector field for spin-one particles
1.13 Lorentz group
1.14 Gamma matrices and Clifford algebras
1.15 Dirac’s gamma matrices
1.16 Dirac fields
1.17 Expansion of massive and massless Dirac fields
1.18 Spinors and angular momentum
1.19 Charge conjugation
1.20 Parity 2 Feynman diagrams
2.1 Time-dependent perturbation theory
2.2 Dyson’s expansion of the S matrix
2.3 The Feynman propagator for scalar fields
2.4 Application to a cubic scalar field theory
2.5 Feynman’s propagator for fields with spin page 1
1
1
3
7
8
9
9
12
13
15
18
18
23
25
27
28
41
44
49
51
53
53
54
56
60
62 ii Contents 2.6
2.7
2.8
2.9
2.10 Feynman’s propagator for spin-one-half fields
Application to a theory of fermions and bosons
Feynman propagator for spin-one fields
The Feynman rules
Fermion-antifermion scattering 63
66
71
73
74 3 Action
3.1 Lagrangians and hamiltonians
3.2 Canonical variables
3.3 Principle of stationary action in field theory
3.4 Symmetries and conserved quantities in field theory 76
76
76
78
79 4 Quantum electrodynamics
4.1 Global U( 1) symmetry
4.2 Abelian gauge invariance
4.3 Coulomb-gauge quantization
4.4 QED in the interaction picture
4.5 Photon propagator
4.6 Feynman’s rules for QED
4.7 Electron-positron scattering
4.8 Trace identities
4.9 Electron-positron to muon-antimuon
4.10 Electron-muon scattering
4.11 One-loop QED
4.12 Magnetic moment of the electron
4.13 Charge form factor of electron 85
85
86
88
89
90
92
93
96
97
103
104
116
124 5 Nonabelian gauge theory
5.1 Yang and Mills invent local nonabelian symmetry
5.2 SU (3) 126
126
127 6 Standard model
6.1 Quantum chromodynamics
6.2 Abelian Higgs mechanism
6.3 Higgs’s mechanism
6.4 Quark and lepton interactions
6.5 Quark and Lepton Masses
6.6 CKM matrix
6.7 Lepton Masses
6.8 Before and after symmetry breaking
6.9 The Seesaw Mechanism
6.10 Neutrino Oscillations 130
130
130
132
138
140
145
148
149
153
155 Contents iii 7 Path integrals
7.1 Path integrals and Richard Feynman
7.2 Gaussian integrals and Trotter’s formula
7.3 Path integrals in quantum mechanics
7.4 Path integrals for quadratic actions
7.5 Path integrals in statistical mechanics
7.6 Boltzmann path integrals for quadratic actions
7.7 Mean values of time-ordered products
7.8 Quantum field theory
7.9 Finite-temperature field theory
7.10 Perturbation theory
7.11 Application to quantum electrodynamics
7.12 Fermionic path integrals
7.13 Application to nonabelian gauge theories
7.14 Faddeev-Popov trick
7.15 Ghosts
7.16 Integrating over the momenta 157
157
157
158
162
167
172
175
177
181
184
188
192
200
200
203
204 8 Landau theory of phase transitions
8.1 First- and second-order phase transitions 206
206 9 Effective field theories and gravity
9.1 Effective field theories
9.2 Is gravity an effective field theory?
9.3 Quantization of Fields in Curved Space
9.4 Accelerated coordinate systems
9.5 Scalar field in an accelerating frame
9.6 Maximally symmetric spaces
9.7 Conformal algebra
9.8 Conformal algebra in flat space
9.8.1 Angles and analytic functions
9.9 Maxwell’s action is conformally invariant for d = 4
9.10 Massless scalar field theory is conformally invariant if
d=2
9.11 Christoffel symbols as nonabelian gauge fields
9.12 Spin connection
References 209
209
210
213
219
221
225
228
229
233
239
240
241
246
248 1
Quantum fields and special relativity 1.1 States
A Lorentz transformation Λ is implemented by a unitary operator U (Λ)
which replaces the state |p, σi of a massive particle of momentum p and spin
σ along the z-axis by the state
s
(Λp)0 X (j)
Ds0 σ (W (Λ, p)) |Λp, s0 i
(1.1)
U (Λ)|p, σi =
p0
0
s where W (Λ, p) is a Wigner rotation
W (Λ, p) = L−1 (Λp)ΛL(p) (1.2) and L(p) is a standard Lorentz transformation that takes (m, ~0) to p. 1.2 Creation operators
The vacuum is invariant under Lorentz transformations and translations
U (Λ, a)|0i = |0i. (1.3) A creation operator a† (p, σ) makes the state |p, σi from the vacuum state
|0i
|pσi = a† (p, σ)|0i. (1.4) The creation and annihilation operators obey either the commutation relation
[a(p, s), a† (p0 , s0 )]− = a(p, s) a† (p0 , s0 ) − a† (p0 , s0 ) a(p, s) = δss0 δ (3) (p − p0 )
(1.5) 2 Quantum fields and special relativity or the anticommutation relation
[a(p, s), a† (p0 , s0 )]+ = a(p, s) a† (p0 , s0 ) + a† (p0 , s0 ) a(p, s) = δss0 δ (3) (p − p0 ).
(1.6)
The two kinds of relations are written together as
[a(p, s), a† (p0 , s0 )]∓ = a(p, s) a† (p0 , s0 ) ∓ a† (p0 , s0 ) a(p, s) = δss0 δ (3) (p − p0 ).
(1.7)
A bracket [A, B] with no signed subscript is interpreted as a commutator.
Equations (1.1 & 1.4) give
s
U (Λ)a† (p, σ)|0i = (Λp)0 X (j)
Ds0 σ (W (Λ, p)) a† (Λp, s0 )|0i.
p0
0 (1.8) s And (1.3) gives
s
† U (Λ)a (p, σ)U −1 (Λ)|0i = (Λp)0 X (j)
Ds0 σ (W (Λ, p)) a† (Λp, s0 )|0i.
p0
0 (1.9) s SW in chapter 4 concludes that
s
U (Λ)a† (p, σ)U −1 (Λ) = (Λp)0 X (j)
Ds0 σ (W (Λ, p)) a† (Λp, s0 ).
p0
0 (1.10) s If U (Λ, b) follows Λ by a translation by b, then
s
U (Λ, b)a† (p, σ)U −1 (Λ, b) = e−i(Λp)·a (Λp)0 X (j)
Ds0 σ (W (Λ, p)) a† (Λp, s0 )
p0
0
s s
= e−i(Λp)·a (Λp)0 X †(j)
Ds0 σ (W −1 (Λ, p)) a† (Λp, s0 )
p0
0
s s
= e−i(Λp)·a (Λp)0 X ∗(j)
Dσs0 (W −1 (Λ, p)) a† (Λp, s0 )
p0
0
s (1.11) 1.3 How fields transform 3 The adjoint of this equation is
s
U (Λ, b)a(p, σ)U −1 (Λ, b) = e i(Λp)·a (Λp)0 X ∗(j)
Ds0 σ (W (Λ, p)) a(Λp, s0 )
p0
0
s s
= ei(Λp)·a (Λp)0 X †(j)
Dσs0 (W (Λ, p)) a(Λp, s0 )
p0
0
s s
= ei(Λp)·a (Λp)0 X (j)
Dσs0 (W −1 (Λ, p)) a(Λp, s0 ).
p0
0
s (1.12)
These equations (1.11 & 1.12) are (5.1.11 & 5.1.12) of SW. 1.3 How fields transform
The “positive frequency” part of a field is a linear combination of annihilation operators
XZ
+
d3 p u` (x; p, σ) a(p, σ).
(1.13)
ψ` (x) =
σ The “negative frequency” part of a field is a linear combination of creation
operators of the antiparticles
XZ
ψ`− (x) =
d3 p v` (x; p, σ) b† (p, σ).
(1.14)
σ To have the fields (1.13 & 1.14) transform properly under Poincar´e transformations
X
U (Λ, a)ψ`+ (x)U −1 (Λ, a) =
D``¯(Λ−1 )ψ`+
¯ (Λx + a)
`¯ = X U (Λ, a)ψ`− (x)U −1 (Λ, a) = X = X D``¯(Λ −1 ) `¯ XZ d3 p u`¯(Λx + a; p, σ) a(p, σ) σ D``¯(Λ−1 )ψ`−
¯ (Λx + a) `¯ `¯ D``¯(Λ−1 ) XZ d3 p v`¯(Λx + a; p, σ) b† (p, σ) σ (1.15)
the spinors u` (x; p, σ) and v` (x; p, σ) must obey certain rules which we’ll now
determine. 4 Quantum fields and special relativity First (1.12 & 1.13) give
U (Λ, a)ψ`+ (x)U −1 (Λ, a) = U (Λ, a) XZ d3 p u` (x; p, σ) a(p, σ)U −1 (Λ, a) σ = XZ d3 p u` (x; p, σ) U (Λ, a)a(p, σ)U −1 (Λ, a) σ (1.16)
= XZ s
d3 p u` (x; p, σ) ei(Λp)·a σ (Λp)0 X (j)
Dσs0 (W −1 (Λ, p)) a(Λp, s0 ).
p0
0
s Now we use the identity
d3 p
d3 (Λp)
=
p0
(Λp)0 (1.17) to turn (1.16) into
U (Λ, a)ψ`+ (x)U −1 (Λ, a) = XZ d3 (Λp) u` (x; p, σ) ei(Λp)·a σ s
× p0 X (j)
D 0 (W −1 (Λ, p)) a(Λp, s0 ).
(Λp)0 0 σs (1.18) s Similarly (1.11, 1.14, & 1.17) give
U (Λ, a)ψ`− (x)U −1 (Λ, a) = U (Λ, a) XZ d3 p v` (x; p, σ) b† (p, σ)U −1 (Λ, a) σ = XZ d p v` (x; p, σ) U (Λ, a)b† (p, σ)U −1 (Λ, a)
3 (1.19) σ s (Λp)0 X ∗(j)
Dσs0 (W −1 (Λ, p)) b† (Λp, s0 )
0
p
σ
s0
s
Z
X
p0 X ∗(j)
=
d3 (Λp) v` (x; p, σ) e−i(Λp)·a
Dσs0 (W −1 (Λ, p)) b† (Λp, s0 ).
0
(Λp)
0
σ
= XZ d3 p v` (x; p, σ) e−i(Λp)·a s So to get the fields to transform as in (1.15), equations (1.18 & 1.19) say 1.3 How fields transform 5 that we need
X D``¯(Λ −1 )ψ`+
¯ (Λx + a) = X = X D``¯(Λ −1 ) `¯ `¯ d3 p u`¯(Λx + a; p, σ) a(p, σ) σ D``¯(Λ−1 ) `¯ = XZ
XZ d3 (Λp) u`¯(Λx + a; Λp, σ) a(Λp, σ) σ XZ d3 (Λp) u` (x; p, σ) ei(Λp)·a (1.20) σ s p0 X (j)
D 0 (W −1 (Λ, p)) a(Λp, s0 )
(Λp)0 0 σs
s
XZ
=
d3 (Λp) u` (x; p, s0 ) ei(Λp)·a
× s0 s
× p0 X (j)
Ds0 σ (W −1 (Λ, p)) a(Λp, σ)
(Λp)0 σ and
X D``¯(Λ −1 )ψ`−
¯ (Λx + a) = X = X `¯ D``¯(Λ −1 ) `¯ d3 p v`¯(Λx + a; p, σ) b† (p, σ) σ D``¯(Λ−1 ) `¯ = XZ
XZ d3 (Λp) v`¯(Λx + a; Λp, σ) b† (Λp, σ) σ XZ d3 (Λp) v` (x; p, σ) e−i(Λp)·a (1.21) σ s p0 X ∗(j)
D 0 (W −1 (Λ, p)) b† (Λp, s0 )
(Λp)0 0 σs
s
XZ
=
d3 (Λp) v` (x; p, s0 ) e−i(Λp)·a
× s0 s
× p0 X ∗(j)
Ds0 σ (W −1 (Λ, p)) b† (Λp, σ).
(Λp)0 σ Equating coefficients of the red annihilation and blue creation operators, we
find that the fields will transform properly if the spinors u and v satisfy the 6 Quantum fields and special relativity rules s
X D``¯(Λ−1 ) u`¯(Λx + a; Λp, σ) = `¯ p0 X (j)
D 0 (W −1 (Λ, p))u` (x; p, s0 ) ei(Λp)·a
(Λp)0 0 s σ
s (1.22)
s
X D``¯(Λ−1 )v`¯(Λx + a; Λp, σ) = `¯ p0 X ∗(j)
D 0 (W −1 (Λ, p))v` (x; p, s0 )e−i(Λp)·a
(Λp)0 0 s σ
s (1.23) which differ from SW’s by an interchange of the subscripts σ, s0 on the
rotation matrices D(j) . (I think SW has a typo there.) If we multiply both
sides of these equations (1.22 & 1.23) by the two kinds of D matrices, then
we get first X D`0 ` (Λ)D``¯(Λ−1 ) u`¯(Λx + a; Λp, σ) = u`0 (Λx + a; Λp, σ) ¯
`,` s
= p0 X (j)
Ds0 σ (W −1 (Λ, p))D`0 ` (Λ)u` (x; p, s0 ) ei(Λp)·a
(Λp)0 0 (1.24) s ,` X D`0 ` (Λ)D``¯(Λ −1 )v`¯(Λx + a; Λp, σ) = v`0 (Λx + a; Λp, σ) ¯
`,` s
= p0 X ∗(j)
Ds0 σ (W −1 (Λ, p))D`0 ` (Λ)v` (x; p, s0 )e−i(Λp)·a (1.25)
(Λp)0 0
s ,` 1.4 Translations 7 and then with W ≡ W (Λ, p)
X (j)
Dσ¯s (W )u`0 (Λx + a; Λp, σ)
σ s
= p0 X (j)
(j)
Ds0 σ (W −1 )Dσ¯s (W )D`0 ` (Λ)u` (x; p, s0 ) ei(Λp)·a
(Λp)0 0
s ,σ,` s
= p0 X
D`0 ` (Λ)u` (x; p, s¯) ei(Λp)·a
(Λp)0 (1.26) ` X ∗(j) Dσ¯s (W )v`0 (Λx + a; Λp, σ) σ s
= p0 X ∗(j)
∗(j)
Ds0 σ (W −1 )Dσ¯s (W )D`0 ` (Λ)v` (x; p, s0 )e−i(Λp)·a
(Λp)0 0
σ,s ,` s
= p0 X
D`0 ` (Λ)v` (x; p, s¯)e−i(Λp)·a
(Λp)0 (1.27) ` which are equations (5.1.13 & 5.1.14) of SW:
s
X
p0 X
(j)
i(Λp)·a
u`¯(Λx + a; Λp, s¯)Ds¯σ (W (Λ, p)) =
D``
¯ (Λ)u` (x; p, σ) e
0
(Λp)
s¯
`
s
X
p0 X
∗(j)
−i(Λp)·a
v`¯(Λx + a; Λp, s¯)Ds¯σ (W (Λ, p)) =
D``
.
¯ (Λ)v` (x; p, σ)e
0
(Λp)
s¯
` (1.28)
These are the equations that determine the spinors u and v up to a few
arbitrary phases. 1.4 Translations
When Λ = I, the D matrices are equal to unity, and these last equations
(1.28) say that for x = 0
u` (a; p, σ) = u` (0; p, σ) eip·a
v` (a; p, σ) = v` (0; p, σ)e−ip·a . (1.29) Thus the spinors u and v depend upon spacetime by the usual phase e±ip·x
u` (x; p, σ) = (2π)−3/2 u` (p, σ) eip·x
v` (x; p, σ) = (2π)−3/2 v` (p, σ)e−ip·x (1.30) 8 Quantum fields and special relativity in which the 2π’s are conventional. The fields therefore are Fourier transforms:
XZ
+
−3/2
d3 p eip·x u` (p, σ) a(p, σ)
ψ` (x) = (2π)
σ ψ`− (x) = (2π)−3/2 XZ (1.31)
d3 p e−ip·x v` (p, σ) b† (p, σ) σ and every field of mass m obeys the Klein-Gordon equation
(∇2 − ∂02 − m2 ) ψ` (x) = (2 − m2 ) ψ` (x) = 0. (1.32) Since exp[i(Λp · (Λx + a))] = exp(ip · x + iΛp · a), the conditions (1.28)
simplify to
s
X
p0 X
(j)
u`¯(Λp, s¯)Ds¯σ (W (Λ, p)) =
D``
¯ (Λ)u` (p, σ)
(Λp)0
s¯
`
(1.33)
s
0
X
X
p
∗(j)
v`¯(Λp, s¯)Ds¯σ (W (Λ, p)) =
D``
¯ (Λ)v` (p, σ)
(Λp)0
s¯
` for all Lorentz transformations Λ. 1.5 Boosts
Set p = k = (m, ~0) and Λ = L(q) where L(q)k = q. So L(p) = 1 and
W (Λ, p) ≡ L−1 (Λp)ΛL(p) = L−1 (q)L(q) = 1. (1.34) Then the equations (1.33) are
r mX
D``
0, σ)
¯ (L(q))u` (~
q0 r mX
D``
0, σ).
¯ (L(q))v` (~
q0 u`¯(q, σ) = ` v`¯(q, σ) = (1.35) ` Thus a spinor at finite momentum is given by a representation D(Λ) of
the Lorentz group (see the online notes of chapter 10 of my book for its
finite-dimensional nonunitary representations) acting on the spinor at zero
3-momentum p = k = (m, ~0). We need to find what these spinors are. 1.6 Rotations 9 1.6 Rotations
Now set p = k = (m, ~0) and Λ = R a rotation so that W = R. For rotations,
the spinor conditions (1.33) are
X
X
(j)
u`¯(~0, s¯)Ds¯σ (R) =
D``
0, σ)
¯ (R)u` (~
s¯ X `
∗(j)
v`¯(~0, s¯)Ds¯σ (R) = X s¯ 0, σ).
D``
¯ (R)v` (~ (1.36) `
(j) The representations Ds¯σ (R) of the rotation group are (2j + 1) × (2j + 1)dimensional unitary matrices. For a rotation of angle θ about the θ~ = θ axis,
they are the ones taught in courses on quantum mechanics (and discussed
in the notes of chapter 10)
h
i
(j)
(j)
(1.37)
Ds¯σ (θ) = e−iθ·J
s¯σ where [Ja , Jb ] = iabc Jc . The representations D``
¯ (R) of the rotation group
are finite-dimensional unitary matrices. For a rotation of angle θ about the
θ~ = θ axis, they are
i
h
−iθ·J
(1.38)
D``
¯ (θ) = e
¯
`` in which [Ja , Jb ] = iabc J . For tiny rotations, the conditions (1.36) require
(because of the complex conjugation of the antiparticle condition) that the
spinors obey the rules
X
X
u`¯(~0, s¯)(Ja(j) )s¯σ =
(Ja )``
0, σ)
¯ u` (~
s¯ X ` ∗(j)
v`¯(~0, s¯)(−Ja )s¯σ ) s¯ = X (Ja )``
0, σ)
¯ v` (~ (1.39) ` for a = 1, 2, 3. 1.7 Spin-zero fields
Spin-zero fields have no spin or Lorentz
p indexes. So the boostpconditions
(1.210) merely require that u(q) = m/q 0√
u(0) and v(q) = √
m/q 0 v(0).
The conventional normalization is u(0) = 1/ 2m and v(0) = 1/ 2m. The
spin-zero spinors then are
u(p) = (2p0 )−1/2 and v(p) = (2p0 )−1/2 . (1.40) For simpicity, let’s first consider a neutral scalar field so that b(p, s) = 10 Quantum fields and special relativity a(p, s). The definitions (1.13) and (1.14) of the positive-frequency and negativefrequency fields and their behavior (1.30) under translations then give us
Z
d3 p
+
p
φ (x) =
a(p) eip·x
(2π)3 2p0
(1.41)
Z
d3 p
†
−ip·x
−
p
a (p) e
.
φ (x) =
(2π)3 2p0
Note that
± †
φ (x) = φ∓ (x). (1.42) Since [a(p), a(p0 )]± = 0, it follows that
[φ+ (x), φ+ (y)]∓ = 0 and [φ− (x), φ− (y)]∓ = 0 (1.43) whatever the values of x and y as long as we use commutators for bosons
and anticommutators for fermions.
But the commutation relation
[a(p, s), a† (q, t)]∓ = δst δ (3) (p − q) (1.44) makes the commutator
+ − d3 pd3 p0
0
p
eip·x e−ip ·y δ 3 (p − p0 )
3
0
00
(2π) 2p 2p
Z
d3 p
=
eip·(x−y) = ∆+ (x − y)
(2π)3 2p0
Z [φ (x), φ (y)]∓ = (1.45) nonzero even for (x − y)2 > 0 as we’ll now verify.
For space-like x, the Lorentz-invariant function ∆+ (x) can only depend
upon x2 > 0 since the time x0 and its sign are √
not Lorentz invariant. So we
0
choose a Lorentz frame with x = 0 and |x| = x2 . In this frame,
Z
d3 p
p
∆+ (x) =
eip·x
3
2
2
(2π) 2 p + m
(1.46)
Z
p2 dp d cos θ
ipx cos θ
p
=
e
(2π)2 2 p2 + m2
where p = |p| and x = |x|. Now
Z
d cos θ eipx cos θ = eipx − e−ipx /(ipx) = 2 sin(px)/(px), (1.47) so the integral (1.46) is
1
∆+ (x) = 2
4π x Z
0 ∞ sin(px) pdp
p
p2 + m2 (1.48) 1.7 Spin-zero fields 11 with u ≡ p/m
∆+ (x) = m
4π 2 x Z
0 ∞ m
sin(mxu) udu
√
= 2 K1 (mx2 )
4π x
u2 + 1 (1.49) a Hankel function.
To get a Lorentz-invariant, causal theory, we use the arbitrary parameters
κ and λ setting
φ(x) = κφ+ (x) + λφ− (x) (1.50) Now the adjoint rule (1.42) and the commutation relations (1.45 and 1.45)
give
[φ(x), φ† (y)]∓ = [κφ+ (x) + λφ− (x), κ∗ φ− (y) + λφ+ (y)]∓
= |κ|2 [φ+ (x), φ− (y)]∓ + |λ|2 [φ− (x), φ+ (y)]∓
= |κ|2 ∆+ (x − y) ∓ |λ|2 ∆+ (y − x)
[φ(x), φ(y)]∓ = [κφ+ (x) + λφ− (x), κφ+ (y) + λφ− (y)]∓
= κλ [φ+ (x), φ− (y)]∓ + [φ− (x), φ+ (y)]∓ (1.51) = κλ (∆+ (x − y) ∓ ∆+ (y − x)) .
But when (x − y)2 > 0, ∆+ (x − y) = ∆+ (y − x). Thus these conditions are
[φ(x), φ† (y)]∓ = |κ|2 ∓ |λ|2 ∆+ (x − y)
(1.52)
[φ(x), φ(y)]∓ = κλ∆+ (x − y)(1 ∓ 1).
The first of these equations implies that we choose the minus sign and so
that we use commutation relations and not anticommutation relations for
spin-zero fields. This is the spin-statistics theorem for spin-zero fields.
SW proves the theorem for arbitrary massive fields in section 5.7.
We also must set
|κ| = |λ|. (1.53) The second equation then is automatically satisfied. The common magnitude
and the phases of κ and λ are arbitrary, so we choose κ = λ = 1. We then
have
φ(x) = φ+ (x) + φ− (x) = φ+ (x) + φ+† (x) = φ† (x). (1.54) Now the interaction density H(x) will commute with H(y) for (x − y)2 > 0,
and we have a chance of having a Lorentz-invariant, causal theory.
The field (1.54)
Z
h
i
d3 p
p
φ(x) =
a(p) eip·x + a† (p) e−ip·x
(1.55)
(2π)3 2p0 12 Quantum fields and special relativity obeys the Klein-Gordon equation
(∇2 − ∂02 − m2 ) φ(x) ≡ (2 − m2 ) φ(x) = 0. (1.56) 1.8 Conserved charges
If the field φ adds and deletes charged particles, an interaction H(x) that is
a polynomial in φ will not commute with the charge operator Q because φ+
will lower the charge and φ− will raise it. The standard way to solve this
problem is to start with two hermitian fields φ1 and φ2 of the same mass.
One defines a complex scalar field as a complex linear combination of the
two fields
1
φ(x) = √ (φ1 (x) + iφ2 (x))
2
Z
d3 p
1
1 †
†
−ip·x
ip·x
p
√ (a1 (p) + ia2 (p)) e
.
=
+ √ a1 (p) + ia2 (p) e
2
2
(2π)3 2p0
(1.57)
Setting
1
a(p) = √ (a1 (p) + ia2 (p))
2
1
and b† (p) = √ a†1 (p) + ia†2 (p)
2 (1.58)
1
and a† (p) = √ a†1 (p) − ia†2 (p)
2 (1.59) so that
1
b(p) = √ (a1 (p) − ia2 (p))
2
we have
Z
φ(x) = i
h
d3 p
p
a(p) eip·x + b† (p) e−ip·x
(2π)3 2p0 (1.60) h
i
d3 p
p
b(p) eip·x + a† (p) e−ip·x .
(2π)3 2p0 (1.61) and
† φ (x) = Z Since the commutation relations of the real creation and annihilation operators are for i, j = 1, 2
[ai (p), a†j (p0 )] = δij δ 3 (p − p0 ) and [ai (p), aj (p0 )] = 0 = [a†i (p), a†j (p0 )]
(1.62)
the commutation relations of the complex creation and annihilation operators are
[a(p), a† (p0 )] = δ 3 (p − p0 ) and [b(p), b† (p0 )] = δ 3 (p − p0 ) (1.63) 1.9 Parity, charge conjugation, and time reversal 13 with all other commutators vanishing.
Now φ(x) lowers the charge of a state by q if a† adds a particle of charge
q and if b† adds a particle of charge −q. Similarly, φ† (x) raises the charge of
a state by q
[Q, φ(x)] = − qφ(x) and [Q, φ† (x)] = qφ† (x). (1.64) So an interaction with as many φ(x)’s as φ† (x)’s conserves charge. 1.9 Parity, charge conjugation, and time reversal
If the unitary operator P represents parity on the creation operators
Pa†1 (p)P−1 = η a†1 (−p) and Pa†2 (p)P−1 = η a†2 (−p) (1.65) and Pa2 (p)P−1 = η ∗ a2 (−p) (1.66) with the same phase η. Then
Pa1 (p)P−1 = η ∗ a1 (−p)
and so both
Pa† (p)P−1 = ηa† (−p) and Pa(p)P−1 = η ∗ a(−p) (1.67) Pb† (p)P−1 = ηb† (−p) and Pb(p)P−1 = η ∗ b(−p). (1.68) and Thus if the field
d3 p Z
φ1 (x) = p h (2π)3 2p0 a1 (p) eip·x + a†1 (p) e−ip·x i (1.69) or φ2 (x), or the complex field (1.60) is to go into a multiple of itself under
parity, then we need η = η ∗ so that η is real. Then the fields transform under
parity as
Pφ1 (x)P−1 = η ∗ φ1 (x0 , −x) = ηφ1 (x0 , −x)
Pφ2 (x)P−1 = η ∗ φ(x0 , −x) = ηφ2 (x0 , −x)
−1 Pφ(x)P ∗ 0 (1.70) 0 = η φ(x , −x) = ηφ(x , −x). Since P2 = I, we must have η = ±1. SW allows for a more general phase
by having parity act with the same phase on a and b† . Both schemes imply
that the parity of a hermitian field is ±1 and that the state
Z
|abi =
d3 p f (p2 ) a† (p) b† (−p) |0i
(1.71)
has even or positive parity, P|abi = |abi. 14 Quantum fields and special relativity Charge conjugation works similarly. If the unitary operator C represents
charge conjugation on the creation operators
Ca†1 (p)C−1 = ξa†1 (p) and Ca†2 (p)C−1 = − ξa†2 (p) (1.72) with the same phase ξ. Then
Ca1 (p)C−1 = ξ ∗ a1 (p) and Ca2 (p)C−1 = − ξ ∗ a2 (p)
√
√
and so since a = (a1 + ia2 )/ 2 and b = (a1 − ia2 )/ 2 (1.73) Ca(p)C−1 = ξ ∗ b(p) and Cb(p)C−1 = ξ ∗ a(p)
√
√
and since a† = (a†1 − ia†2 )/ 2 and b† = (a†1 + ia†2 )/ 2 (1.74) Ca† (p)C−1 = ξ b† (p) and Cb† (p)C−1 = ξa† (p). Thus under charge conjugation, the field (1.60) becomes
Z
i
h
d3 p
−1
p
Cφ(x)C =
ξ ∗ b(p) eip·x + ξ a† (p) e−ip·x
(2π)3 2p0 (1.75) (1.76) and so if it is to go into a multiple of itself or of its adjoint under charge
conjugation then we need ξ = ξ ∗ so that ξ is real. We then get
Cφ(x)C−1 = ξ ∗ φ† (x) = ξ φ† (x). (1.77) Since C2 = I, we must have ξ = ±1. SW allows for a more general phase
by having charge conjugation act with the same phase on a and b† . Both
schemes imply that the charge-conjugation parity of a hermitian field is ±1
and that the state
Z
|abi =
d3 p f (p2 ) a† (p) b† (p) |0i
(1.78)
has even or positive charge-conjugation parity, C|abi = |abi.
The time-reversal operator T is antilinear and antiunitary. So if
Ta1 (p)T−1 = ζ ∗ a1 (−p)
Ta†1 (p)T−1 = ζa†1 (−...

View
Full Document