I) The closest cosmetic resemblance between the Nambu-Goto action and the Polyakov action is achieved if we write them as
SNG = −T0c∫d2vol det(M)12,
and
SP = −T0c∫d2vol tr(M)2,
respectively. Here hab is an auxiliary world-sheet (WS) metric of Lorentzian signature (−,+), i.e. minus in the temporal WS direction;
d2vol := √−h dτ∧dσ
is a diffeomorphism-invariant WS volume-form (an area actually);
Mac := (h−1)abγbc
is a mixed tensor; and
γab := (X∗G)ab := ∂aXμ ∂bXν Gμν(X)
is the induced WS metric via pull-back of the target space (TS) metric Gμν with Lorentzian signature (−,+,…,+).
Note that the Nambu-Goto action (1) does actually not depend on the auxiliary WS metric hab at all, while the Polyakov action (2) does.
II) As is well-known, varying the Polyakov action (2) wrt. the WS metric hab leads to that the 2×2 matrix
Mab ≈ tr(M)2δab ∝ δab
must be proportional to the 2×2 unit matrix on-shell. This implies that
det(M)12 ≈ tr(M)2,
so that the two actions (1) and (2) coincide on-shell, see e.g. the Wikipedia page. (Here the ≈ symbol means equality modulo eom.)
III) Now, let us imagine that we only know the Nambu-Goto action (1) and not the Polyakov action (2). The the only diffeomorphism-invariant combinations of the matrix Mab are the determinant det(M), the trace tr(M), and functions thereof.
If furthermore the TS metric Gμν is dimensionful, and we demand that the action is linear in that dimension, this leads us to consider action terms of the form
S = −T0c∫d2vol det(M)p2(tr(M)2)1−p,
where p∈R is a real power. Alternatively, Weyl invariance leads us to consider the action (8). Obviously, the Polyakov action (2) (corresponding to p=0) is not far away if we would like simple integer powers in our action.
This post imported from StackExchange Physics at 2014-03-12 15:52 (UCT), posted by SE-user Qmechanic