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8.323 Relativistic Quantum Field Theory I
Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
8.323 Lecture Notes 1: Quantization of the Free Scalar Field, p. 1.
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Physics Department
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�� ��� �����������
�
L=
where
� = 12 ∂
=
µφ ∂
µ
d3 x
φ−
�
1 2 2
m φ
2
1
1 2 1
φ̇ − ∇i φ ∇i φ − m2 φ2
2
2
2
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–1–
8.323 Lecture Notes 1: Quantization of the Free Scalar Field, p. 2.
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L = L(qi , q̇i , t)
∂L
∂q̇i
pi ≡
�
H=
pi q̇i − L
i
[qi , pj ] = ih̄ δij
Schrödinger picture:
ih̄
∂
|ψ(t) = H |ψ(t) .
∂t
If H is independent of time,
|ψ(t) = e−iHt/h̄ |ψ(0) .
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–2–
Heisenberg picture:
� (t) = e
iHt/h̄
� �
�
ψ�
so that
� �
� (t)�� ψ
�e
−iHt/h̄
�
�
�
= ψ (t) �
�
,
� �� ψ(t)
�
.
Relativistic quantum field theory is usually done in the
Heisenberg picture: puts space and time on equal footing.
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–3–
8.323 Lecture Notes 1: Quantization of the Free Scalar Field, p. 3.
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Motivation:
1) Can use canonical formulation exactly, instead of as a starting point for
a “natural” generalization. Note that δij might naturally generalize to
δ 3 (x − x ), but the two expressions do not even have the same units. δij
is dimensionless, while δ 3 (x − x ) has units of 1/L3 .
2) For interacting theories, the lattice formulation is the easiest way to understand
renormalization. When strong coupling is essential, as in QCD (Quantum
Chromodynamics) and the strong interactions, the lattice is even the best way
to calculate.
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–4–
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Start with discrete lattice, spacing = a, and finite volume. Later take limits
a → 0 and V → ∞. For the free theory, these limits are trivial.
L=
�
k
�
k
∆V ,
where
∆V = a 3
and
�
1
1
1 2
φ̇k − ∇i φk ∇i φk − m2 φ2k ,
2
2
2
and the lattice derivative is defined by
k
=
∇i φk ≡
φk� (k,i) − φk
a
,
where k (k, i) denotes the lattice site that is a distance a in the ith direction
from the lattice site k .
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–5–
8.323 Lecture Notes 1: Quantization of the Free Scalar Field, p. 4.
Canonical momenta:
pk =
∂L
∂φ̇k
= φ̇k ∆V .
Define a canonical momentum density:
πk ≡
Hamiltonian:
H=
�
�
pk
= φ̇k .
∆V
�
pk φ̇k − L =
k
��
πk φ̇k −
k
�
�
∆V .
k
Canonical commutation relations:
¯ δk � k .
[φk� , φk ] = 0 , [pk� , pk ] = 0 , and [φk� , pk ] = ih
In terms of the canonical momentum densities,
[φk� , φk ] = 0 , [πk� , πk ] = 0 , and [φk� , πk ] =
ih
¯ δk � k
.
∆V
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–6–
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Continuum limit:
k
πk ≡
�
1 ∂L
∂ k
=
∆V ∂φ̇k
∂φ̇k
=⇒
=⇒
x
π(x , t) =
∂
�
∂φ̇(x , t)
= φ̇(x , t) .
Hamiltonian becomes
H=
��
πk φ̇k −
k
�
H=
�
�
d3 x πφ̇ −
�
k
�
∆V =⇒
�
.
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–7–
8.323 Lecture Notes 1: Quantization of the Free Scalar Field, p. 5.
Canonical commutation relations:
�
�
�
�
φ(x , t) , φ(x , t) = 0 and π(x , t) , π(x , t) = 0 ,
[φk� , πk ] =
ih̄ δk� k
∆V
�
=⇒
�
φ(x , t) , π(x , t) = ih̄ δ (x − x ) ,
since
�
� δk � k
∆V = 1 if k ∈
∆V
0 otherwise
k∈ �
�
�
=⇒
x∈
�
d3 x δ 3 (x − x )
Note: Dirac delta function is defined by
�
�
3
d x f (x ) δ (x − x ) ≡ f (x )
x∈�
0
if x ∈
�
otherwise.
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–8–
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Choose m ≡ 1, so
1 2 1 2 2
q̇ − ω q ,
2
2
∂L
p=
= q̇ ,
∂q̇
1
1
H = pq̇ − L = p2 + ω2 q2 .
2
2
Canonical commutation relation:
L=
[q , p] = ih̄ .
Define creation and annihilation operators
�
�
i
ω
i
ω
†
a=
q+
p, a =
q−
p,
2h̄
2h̄ω
2h̄
2h̄ω
so that
�
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�
a , a† = 1 .
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–9–
8.323 Lecture Notes 1: Quantization of the Free Scalar Field, p. 6.
The Hamiltonian can be rewritten as
�
�
1
†
H = h̄ω a a +
,
2
with eigenstates |n with eigenvalues
�
�
1
En = n +
hω
¯ .
2
Then
�
�
H , a† = hω
¯ a,
¯ a† , [H , a] = −hω
which implies that a lowers, and a† raises, the eigenvalues of H by h̄ω.
a |0 = 0 ,
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–10–
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Normalized nth excited state:
1 � † �n
|n = √
|n ,
a
n!
Solve for q and p:
�
q=
�
h̄ �
a + a† , p = −i
2ω
�
�
hω
¯ �
a − a† .
2
In the Heisenberg picture,
�
q (t) =
�
h̄ � −iωt
ae
+ a† eiωt
2ω
�
and
p(t) = q̇(t) = −i
�
hω
¯ � −iωt
ae
− a† eiωt .
2
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–11–
8.323 Lecture Notes 1: Quantization of the Free Scalar Field, p. 7.
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Continuum limit of the lattice:
∂
π(x , t) =
�
= φ̇(x , t) ,
∂φ̇(x , t)
�
�
�
3
d x πφ̇ −
,
H=
�
�
�
�
�
φ(x , t) , φ(x , t) = 0 and π(x , t) , π(x , t) = 0 ,
and
�
�
φ(x , t) , π(x , t) = ih
¯ δ (x − x ) .
Use these relations to build the quantum theory.
Fourier transform:
φ̃(k , t) ≡
and
π̃(k , t) ≡
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�
�
d3 x e−ik ·
x φ(x , t) ,
d3 x e−ik ·
x π(x , t) ,
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–12–
�
Fourier inversion:
φ(x , t) =
�
and
π(x , t) =
d3 k i
k ·
x e
φ̃(k , t)
(2π)3
d3 k i
k ·
x e
π̃ (k , t) .
(2π)3
φ(x , t) real =⇒ Hermitian quantum operator =⇒
φ̃(−k , t) = φ† (k , t) ,
π̃ (−k , t) = π† (k , t) .
Heisenberg equations of motion:
∂ 2φ
2 φ + m2 φ = 0 .
−∇
∂t2
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–13–
8.323 Lecture Notes 1: Quantization of the Free Scalar Field, p. 8.
Since h̄ =
1, m has the units of an inverse length, not a mass. Fourier transform
obeys
∂ 2 φ̃ (k , t) + (k 2 + m2 ) φ̃(k , t) = 0 .
∂t2
General solution:
φ̃(k , t) = φ1 (k )e−iωp t + φ2 (k )eiωp t ,
�
ωp = k 2 + m2 .
where
Reality φ̃(−k , t) = φ† (k , t) implies
φ2 (k ) = φ†1 (−k ) ,
so
φ̃(k , t) = φ1 (k )e−iωp t + φ†1 (−k )eiωp t .
Since π = φ̇,
π̃ (k , t) = −iωp φ1 (k )e−iωp t + iωp φ†1 (−k )eiωp t .
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–14–
These two equations can be solved simultaneously to give:
�
�
i
1
φ1 (k ) =
φ̃(k , t) +
π̃ (k , t) eiωp t
ωp
2
�
�
�
i
1
3
−i(
x −ωp t)
k ·
=
φ(x , t) +
π(x , t) ,
d xe
ωp
2
Using canonical commutation relations,
�
�
� 1�
3
φ1 (k ) , φ1 (q ) =
d3 y e−i(k ·
x −ωk t) e−i(
q ·
y −ωq t)
dx
4
�
�
i
i
π(x , t) , φ(y , t) +
π(y , t)
× φ(x , t) +
ωk
ωq
�
�
1
3
d3 y e−i(k ·
x −ωk t) e−i(
q ·
y −ωq t)
dx
=
4
�
�
i
i
3
3
×
ih
¯ δ (x − y ) −
ih
¯ δ (x − y )
ωq
ωk
= ... .
–15–
8.323 Lecture Notes 1: Quantization of the Free Scalar Field, p. 9.
Using canonical commutation relations,
�
�
� 1�
3
d3 y e−i(k ·
x −ωk t) e−i(
q ·
y −ωq t)
φ1 (k ) , φ1 (q ) =
dx
4
�
�
i
i
× φ(x , t) +
π(x , t) , φ(y , t) +
π(y , t)
ωk
ωq
�
�
1
3
=
d3 y e−i(k ·
x −ωk t) e−i(
q ·
y −ωq t)
dx
4
�
�
i
i
3
3
×
ih̄ δ (x − y ) −
ih̄ δ (x − y )
ωq
ωk
�
�
�
��
1
h̄
1
−i (
q )·
x −(ωk +ωq )t
k +
3
d xe
−
.
=
4
ωk
ωq
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–16–
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�
But
d3 x e−i(k +
q )·
x = (2π)3 δ 3 (k + q ) ,
which requires q = −k , but then the factor in curly brackets vanishes. Thus,
�
�
φ1 (k ) , φ1 (q ) = 0 ,
as expected if φ1 (k ) is proportional to an annihilation operator.
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–17–
8.323 Lecture Notes 1: Quantization of the Free Scalar Field, p. 10.
Similarly,
�
�
��
� h̄ �
�
1
1
†
−i (
x −(ωk −ωq )t
q )·
k −
3
φ1 (k ) , φ1 (q ) =
+
.
d xe
4
ωk
ωq
=
h̄
h̄
2
(2π)3 δ 3 (k − q )
=
(2π)3 δ 3 (k − q ) .
4
ωk
2ωk
Continuum normalization convention:
�
Define
a(k ) =
2ωk
φ1 (k ) .
h̄
so that
�
�
�
�
a(k ) , a(q ) = 0 , a(k ) , a† (q ) = (2π)3 δ 3 (k − q ) ,
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–18–
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Then
�
φ(x , t) =
d3 k
(2π)3
�
�
h̄ � i(
k ·x−ωk t)
a( k ) e
+ a† (k )e−i(k ·
x −ωk t) .
2ωk
Note: in 2nd term, I changed variables of integration k → −k ,
Canonical momentum density:
π(x , t) = φ̇(x , t)
�
�
�
d3 k
h̄ωk � i(
k ·x−ωk t)
† −i(
x −ωk t)
k ·
= −i
a
(
)
e
−
a
(
)
e
.
k
k
(2π)3
2
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–19–
8.323 Lecture Notes 1: Quantization of the Free Scalar Field, p. 11.
Creation and annihilation operators:
a(k ) =
�
a (k ) =
†
ωk
2h̄
�
�
ωk
2h̄
−i(
x −ωp t)
k ·
d xe
3
�
i(
x −ωp t)
k ·
d xe
3
�
�
i
φ(x , t) +
π(x , t) .
ωp
�
i
φ(x , t) −
π(x , t)
ωp
�
.
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–20–
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