Kiefer C. Quantum gravity

THE FREE BOSONIC STRING 81
render the action functional unnecessarily complicated. Therefore, since δN =˙ɛ,
see (3.32), one can choose the ‘non-canonical gauge’
Ṅ = χ(p, x, N) . (3.36)
In electrodynamics, A 0 plays the role of N. Therefore, the Lorenz gauge ∂ µ A µ =
0 is a non-canonical gauge, whereas the Coulomb gauge ∂ a A a =0wouldbean
example of a canonical gauge.
Some final remarks are in order; cf. Henneaux and Teitelboim (1992). First,
the restriction ɛ(τ 1 )=0=ɛ(τ 2 ) only holds if the action is an integral over the
Lagrangian without additional boundary terms. If appropriate boundary terms
are present in the action principle, one can relax the condition on ɛ (but to
determine these boundary terms, one has to solve first the equations of motion).
Second, if such boundary terms are present, one can even choose τ-independent
canonical gauges (an extreme choice would be x 0 (τ) = 0 for all τ).
3.2 The free bosonic string
Nowadays superstring theory (or ‘M-theory’) is considered to be a candidate for
a unified theory of all interactions including quantum gravity. This aspect will be
discussed in Chapter 9. In this section, we shall consider the free bosonic string
as a model for (canonical) quantum gravity. However, the bosonic string (where
no supersymmetry is included) is also used in a heuristic way in the development
of superstring theory itself.
In the case of the relativistic particle, the action is proportional to the proper
time; see (3.21). A string is a finite one dimensional object that can either be
open (with two ends) or closed (having no ends). A straightforward generalization
to the string would thus be to use an action proportional to the area of the
worldsheet M, called the Nambu–Goto action,
S = − 1 √
2πα
∫M
′ d 2 σ |detG αβ | . (3.37)
Here, d 2 σ ≡ dσdτ denotes the integration over the parameters of the worldsheet
(with both the space part σ and the time part τ chosen to be dimensionless),
and G αβ is the metric on the worldsheet. The string tension is (2πα ′ ) −1 ,that
is, there is a new fundamental parameter α ′ with dimension length/mass. In the
quantum version, the fundamental string length
l s = √ 2α ′ (3.38)
will occur. The string propagates in a higher dimensional space–time, and the
worldsheet metric G αβ (α, β =1, 2) is induced by the metric of the embedding
space–time. In the following, we shall assume that the string propagates in D-
dimensional Minkowski space, with the worldsheet given by X µ (σ, τ), where