Download Massive Vector Fields and Hamiltonian Dynamics and more Assignments Physics in PDF only on Docsity!
PHY–396 K. Problem set #2. Due September 25, 2000.
- Consider a massive relativistic vector field A
μ
(x) with the Lagrangian density
L = −
1
4
F
μν
F
μν
1
2
m
2
A μ
A
μ
− A
μ
J μ
(in c = 1 units) where F μν
def
μ
A
ν
ν
A
μ
and the current J
μ (x) is a fixed source for the
A
μ (x) field. Note that because of the mass term, the Lagrangian (1) is not gauge invariant.
(a) Derive the Euler–Lagrange field equations for the massive vector field A
μ (x).
(b) Show that this field equation does not require current conservation; however, if the current
happens to satisfy ∂ μ
J
μ = 0, then the field A
μ (x) satisfies
μ
A
μ
= 0 and (∂
2
2
)A
μ
= J
μ
. (2)
Next, consider the Hamiltonian formalism for the massive vector field. Our first step in
deriving this formalism is to identify the canonically conjugate “momentum” fields.
(c) Show that ∂L/∂
A = −E but ∂L/∂
A
0
In other words, the canonically conjugate field to A(x) is −E(x) but the A 0
(x) does not have
a canonical conjugate! Consequently,
H = −
d
3
x ˙A(x) · E(x) − L. (3)
(d) Show that in terms of the A, E and A 0
fields and their space derivatives,
H =
d
3
x
1
2
E
2
(J
0
− ∇ · E) −
1
2
m
2
A
2
0
1
2
(∇ × A)
2
1
2
m
2
A
2
− J · A
Because the A 0
field does not have a canonical conjugate, the Hamiltonian formalism does
not produce an equation for the time-dependence of this field. Instead, it gives us a time-
independent equation relating the A 0
(x, t) to the values of other fields at the same time t.
1
Specifically, we have
δH
δA 0
(x)
∂H
∂A
0
x
∂H
∂∇A
0
x
At the same time, the vector fields A and E satisfy the Hamiltonian equations of motion,
∂t
A(x, t) = −
δH
δE(x)
t
∂t
E(x, t) = +
δH
δA(x)
t
(e) Write down the explicit form of all these equations.
(f) Finally, verify that the equations you have just written down are equivalent to the Euler–
Lagrange equations you derived in question (a).
- Later in this class, we shall learn how to constuct the quantum electromagnetic fields
E(x, t)
and
B(x, t) out of creation and annihilation operators in the photonic Fock space. For the
moment, let us simply take it for granted that they obey the time-independent Maxwell eqs.
E(x, t) = ∇ ·
B(x, t) = 0 (7)
(we assume free EM fields, i.e. no electric charges or currents). In the Heisenberg picture, the
quantum EM fields also obey the time-dependent Maxwell equations
B
∂t
= −∇ ×
E ,
E
∂t
= +∇ ×
B ,
which follow from the free electromagnetic Hamiltonian
H
EM
d
3
x
1
2
E
2
1
2
B
2
and the equal-time commutation relations
[
E
i
(x, t),
E
j
(x
′
, t
′
= t)
]
[
B
i
(x, t),
B
j
(x
′
, t
′
= t)
]
[
E
i
(x, t),
B
j
(x
′
, t
′
= t)
]
Such commutation relations for the electromagnetic fields are completely determined by the
consistency of eqs. (8) with the Hamiltonian (9), so write them down. Make sure your
answer is consistent with the transversality of the fields, i.e., with the time-independent
Maxwell equations (7).
2
