145
5.3. Generalizations: four dimensions, relativity and mass
will be relativistically invariant if L is, since the volume element d
4 x is invariant. Thus, to construct a relativistic field theory, we have to construct
an invariant density L and use the already given covariant Euler–Lagrange
equation. Thus our previous string Lagrangian
( ) 2
( ) 2
1
∂φ
1
∂φ
L ρ = ρ
− ρc
2
(5.149)
2
∂t
2
∂x
with ρ = c = 1 generalizes to
1
L = ∂ μ φ∂
μ φ
(5.150)
2
and produces the invariant wave equation
(
)
∂
2
∂ μ ∂
μ φ =
− ∇
2 φ = 0.
(5.151)
∂t 2
All of this goes through just the same when the fields are quantized.
This invariant Lagrangian describes a field whose quanta are massless.
To find the Lagrangian for the case of massive quanta, we need to find the
Lagrangian that gives us the Klein–Gordon equation (see section 3.1)
(❗ + m
2 )φ(x, t) = 0
(5.152)
via the Euler–Lagrangian equations.
The answer is a simple generalization of (5.150):
1
2 φ
2
L KG = ∂ μ φ∂
μ φ −
1 m .
(5.153)
2
2
The plane-wave solutions of the field equation – now the KG equation – have
frequencies (or energies) given by
2
ω
2 = k
2 + m
(5.154)
which is the correct energy–momentum relation for a massive particle.
How do we quantize this field theory? The four-dimensional analogue of
the Fourier expansion of the field φ takes the form
∫ ∞
d
3
k
φ ˆ (x) =
√ [ˆ a(k)e
−ik·x + ˆ
a
† (k)e
ik·x ]
(5.155)
−∞ (2π) 3 2ω
˙
ˆ
with a similar expansion for the ‘conjugate momentum’ π ˆ = φ:
∫ ∞
d
3
k
π ˆ(x) =
√ (−iω)[ˆ a(k)e
−ik·x
− a ˆ
† (k)e
ik·x ].
(5.156)
−∞ (2π) 3 2ω
Here k · x is the four-dimensional dot product k · x = ωt − k · x, and ω =
+(k
2 + m
2 )
1/2 . The Hamiltonian is found to be
∫
∫ ∞
1
H ˆ KG = d
3
xH ˆ KG =
d
3
x 2 [ˆ π
2 + ∇φ ˆ · ∇φ ˆ + m
2 φ ˆ 2 ]
(5.157)
−∞