Prof. Dr. Lust
Summer 2006
Assignment # 1
(Due May 8,2006)
1) Consider the ation of a massive relativisti point partile:
S = m
Z
1
0 d
p
_ x
_ x
:
a) Using x 0
t, show the equivalene of S to the ation
^
S = Z
t
1
t
0
dtL(t) m 2
Z
t
1
t
0 dt
r
1
~v 2
2
;
where~v =~v(t) denotes the ordinaryveloity ofthe partile with
respet to the physial time t.
b) Verify that for small veloities, j~vj , L(t) redues to the
standard form of a Lagrange funtion, i.e., kineti minus poten-
tial energy. What plays the r^ole of the potential energy in this
ase?
2) The advantage of the ation S over the ation
^
S is that it
treats time x 0
and the spae oordinates ~x on an equal footing,
makingPoinareinvarianemanifest. Thisomesatthe expense
of a new, unphysial, parameter, . Verify that the ovariant
ation S is indeed invariant under hanges of this unphysial
parameter, i.e., under reparameterizations
!~():
S 0
= 1
2 Z
1
0
d(e 1
_ x
_ x
em 2
2
):
a) How does e have to transform under the reparameterization
!~() in order to ensure the reparameterization invariane
of S 0
?
b) Find the equation of motion for e by varying S 0
. Insert the
resulting equation into S 0
and verify that S 0
is lassially equiv-
alent to the ation S of Problem 1).
4) The Nambu-Gotoation of a one-dimensional objet is given
by:
S
NG
= T Z
dd q
det(
a X
b X
):
a) Chek the invariane under Poinare transformations of the
\target spae", i.e. under
X 0
(;) =
X
(;)+a
;
where
denotes a onstant (pseudo-)orthogonal matrix with
respet to the metri
, and a
is a onstant vetor.
b) Show the invariane of S
NG
under arbitrary reparameteriza-
tions of the worldsheet
(;) ! (~(;);(~ ;)):
