Among the many truly remarkable properties of M-theory, that it is a unified theory of all interactions, including quantum gravity, and gives a completely well-defined analytic S-matrix satisfying all the axioms for a physically acceptable theory entailing Lorentz invariance, macro-causality and unitarity is perhaps the deepest, and to boot, the only quantum gravity paradigm that has that essential feature. Here, I will discuss some key aspects of nonlocality and space-time uncertainty in string theory. Let us start with an action smoothly interpolating between the area preserving Schild action and the fully reparametrization invariant Nambu–Goto action:

where $\Phi \left( \sigma \right)$ is an auxiliary world-sheet field, ${\gamma _{mn}} \equiv {\eta _{\mu \nu }}{\partial _m}{X^\mu }{\partial _n}{X^\nu }$ the induced metric on the string Euclidean world-sheet ${x^\mu } = {X^\mu }\left( \sigma \right)$, and ${\mu _0} \equiv 1/2\pi \alpha '$ is the string tension. Combining, we get the Nambu-Goto-Schild action:

And to make the Nambu-Goto-Schild action quadratic in space-time coordinates, we use the Virasoro constraint and an auxiliary field that transforms as a world-sheet scalar and as an anti-symmetric tensor with respect to the space-time indices:

to yield:

Before proceeding, let us get some clarity.

Multi-scalar field cosmology is essential for solving the Wheeler-DeWitt equation in the context of quantum gravity. Here, I will test MFI with supersymmetric quantum mechanics based on Witten's axiomatic approach. One can axiomatize multi-scalar field theory by the following conditions: 1) a Lagrangian containing up to second order derivatives of the fields, and 2) field equations that contain up to second order derivatives of the fields obeying:

with:

and $\frac{{\partial {A^{{i_1}...{i_m}}}}}{{\partial {X_{kl}}}}$ symmetric in all of its indices ${i_1}...{i_m},k,l$

With the multi-field action in D dimensions having the form:

whose Euler-Lagrange equations are given by:

with a fourth derivatives constraint:

Thus, the universal multi-field action is:

for the multi-fields $\left( {\sigma ,\phi } \right)$, hence the corresponding field equations: