String Theory and M-Theory

38 The bosonic string
where φ is any field of the theory and ε is an infinitesimal parameter. Such
a transformation is a symmetry of the theory if it leaves the equations of
motion invariant. This is the case if the action changes at most by a surface
term, which means that the Lagrangian density changes at most by a total
derivative. The Noether current is then determined from the change in the
action under the above transformation
L → L + ε∂αJ α . (2.65)
When ε is a constant, this change is a total derivative, which reflects the
fact that there is a global symmetry. Then the equations of motion imply
that the current is conserved, ∂αJ α = 0. The Poincaré transformations
δX µ = a µ νX ν + b µ , (2.66)
are global symmetries of the string world-sheet theory. Therefore, they give
rise to conserved Noether currents. Applying the Noether method to derive
the conserved currents associated with the Poincaré transformation of X µ ,
one obtains
P µ α = T ∂αX µ , (2.67)
J µν
α = T (X µ ∂αX ν − X ν ∂αX µ ) , (2.68)
where the first current is associated with the translation symmetry, and the
second one originates from the invariance under Lorentz transformations.
0
Hamiltonian
World-sheet time evolution is generated by the Hamiltonian
π
H = ˙XµP µ
π
0 − L
where
P µ
0
dσ = T
2
= δS
δ ˙ Xµ
0
˙X 2 + X ′2
dσ, (2.69)
= T ˙ X µ , (2.70)
was previously called P µ (σ, τ). Inserting the mode expansions, the result
for the closed-string Hamiltonian is
H =
+∞
n=−∞
(α−n · αn + α−n · αn) , (2.71)
while for the open string the corresponding expression is
H = 1
2
+∞
n=−∞
α−n · αn. (2.72)