PHY431 Homework Set 7
Reading:
Lecture Notes
Homework:
See below:
Due date:
Wednesday April 19
Hints and Solutions
Problem VII.1
Show that a mass term of the form ½
m
²
A
μ
A
μ
does break the usual gauge invariance of the
electromagnetic field in that a "shift" transformation of the field
A
μ
→
A
μ
'
=
A
μ
+
∂
μ
χ
/
q
does not leave the term invariant.
Hints:
no hints
Solution:
no solution yet
Problem VII.2
Consider the gauge transformation
φ
→
φ
'
=
e
i
α
φ
and
A
μ
→
A
μ
'
=
A
μ
+
∂
μ
α
/
q
. If
φ
satisfies
the equation (
D
μ
D
μ
+
m
²)
φ
= 0, with
D
μ
=
∂
μ

iqA
μ
, derive the equation that is satisfied by
φ
'
.
Hints:
no hints
Solution:
no solution yet
Problem VII.3
Consider the complex scalar field Schrödinger Lagrangian
L
=
i
(
∂
φ
/
∂
t
)
φ
* 
∇
φ
·
∇
φ
*/(2
m
).
Treating the fields
φ
and
φ
* as independent, derive the equations of motion for these fields
and show that they are indeed the usual Schrödinger equations, and that they are consistent,
i.e. they are each other's complex conjugates.
Hints:
no hints
Solution:
no solution yet
phy431_s99_hw07
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 Spring '01
 Rijssenbeek
 Physics, Mass, Work, Photon, Quantum Field Theory, Aµ Aµ

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