16
Chapter 2 The Klein-Gordon Field
and end of this region. If we restrict our consideration to deformations
that
vanish on the spatial b oundary of the region as well, then the surface term is
zero. Factoring out the
from the rst two terms, we note that, since the
integral must vanish for arbitrary
, the quantity that multiplies
must
vanish at all p oints. Thus we arrive at the Euler-Lagrange equation of motion
for a eld,
@
@
L
@
(
@
)
;
@
L
@
=0
:
(2
:
3)
If the Lagrangian contains more than one eld, there is one such equation for
each.
Hamiltonian Field Theory
The Lagrangian formulation of eld theory is particularly suited to relativistic
dynamics b ecause all expressions are explicitly Lorentz invariant. Nevertheless
we will use the Hamiltonian formulation throughout the rst part of this
bo ok, since it will make the transition to quantum mechanics easier. Recall
that for a discrete system one can dene a conjugate momentum
p
@L=@
_
q
(where _
q
=
@q=@t
) for each dynamical variable
q
. The Hamiltonian is then
H
P
p
_
q
;
L
. The generalization to a continuous system is b est understo o d
by pretending that the spatial p oints
x
are discretely spaced. We can dene
p
(
x
)
@L
@
_
(
x
)
=
@
@
_
(
x
)
Z
L
;
(
y
)
_
(
y
)
d
3
y
@
@
_
(
x
)
X
y
L
;
(
y
)
_
(
y
)
d
3
y
=
(
x
)
d
3
x
where
(
x
)
@
L
@
_
(
x
)
(2
:
4)
is called the
momentum density
conjugate to
(
x
). Thus the Hamiltonian can
be written
H
=
X
x
p
(
x
)
_
(
x
)
;
L:
Passing to the continuum, this b ecomes
H
=
Z
d
3
x
(
x
)
_
(
x
)
;L
Z
d
3
x
H
:
(2
:
5)
We will rederive this expression for the Hamiltonian density
H
near the end
of this section, using a dierent metho d.
As a simple example, consider the theory of a single eld
(
x
), governed
by the Lagrangian
L
=
1
2
_
2
;
1
2
(
r
)
2
;
1
2
m
2
2
=
1
2
(
@
)
2
;
1
2
m
2
2
:
(2
:
6)
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