Space-Time Uncertainty and Non-Locality in String-Theory

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:

    \[I\left[ {\Phi ,X} \right] \equiv \frac{{{\mu _0}}}{2}\int\limits_\Sigma {{d^2}} \sigma \left[ {\frac{{\det \left( {{\gamma _{mn}}} \right)}}{{\Phi \left( \sigma \right)}} + \Phi \left( \sigma \right)} \right]\]

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:

    \[{S_{ngs}} = - \int\limits_\Sigma {{d^2}} \xi \left\{ {\frac{1}{e}\left[ { - \frac{1}{{2{{\left( {4\pi \alpha '} \right)}^2}}}{{\left( {{\varepsilon ^{ab}}{\partial _a}{X^\mu }{\partial _b}{X^\nu }} \right)}^2}} \right] + e} \right\}\]

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:

    \[\left\{ {\begin{array}{*{20}{c}}{{b_{\mu \nu }}\left( \xi \right)}\\{{P^2} + \frac{1}{{4\pi \alpha '}}{{\hat X}^2} = 0,\;P \cdot \hat X = 0}\end{array}} \right.\]

to yield:

Before proceeding, let us get some clarity.

Probing Multi-Field Inflation with Supersymmetric Quantum Mechanics

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:


    \[{{{\bar X}_{ij}} = \frac{1}{2}{\partial _a}{\pi _i}{\partial ^a}{\pi _j}}\]

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:

    \[S = \int {{d^D}} x\hat L\left( {{\pi _i},{\partial _a}{\pi _j},{\partial _b}{\partial _c}{\pi _k}} \right)\]

whose Euler-Lagrange equations are given by:

    \[\frac{{\partial \hat L}}{{\partial {\pi _i}}} - {\partial _a}\left( {\frac{{\partial \hat L}}{{\partial {\pi _{ia}}}}} \right) + {\partial _a}{\partial _b}\left( {\frac{{\partial \hat L}}{{\partial {\pi _{iab}}}}} \right) = 0\]

with a fourth derivatives constraint:

    \[\frac{{\partial \hat L}}{{\partial {\pi _{icd}}\partial {\pi _{iab}}}}{\pi _{i,abcd}}\]

Thus, the universal multi-field action is:

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

    \[\begin{array}{l}{G_{\alpha \beta }} + {g_{\alpha \beta }}\Lambda = + \frac{1}{2}\left( {{\nabla _\alpha }\phi {\nabla _\beta }\phi - \frac{1}{2}{g_{\alpha \beta }}{g^{\mu \nu }}{\nabla _\mu }\phi {\nabla _\nu }\phi } \right)\\ + \frac{1}{2}\left( {{\nabla _\alpha }\sigma {\nabla _\beta }\sigma - \frac{1}{2}{g_{\alpha \beta }}{g^{\mu \nu }}{\nabla _\mu }\sigma {\nabla _\nu }\sigma } \right)\\ - \frac{1}{2}{g_{\alpha \beta }}V\left( {\phi ,\sigma } \right) - 8\pi G{{\rm T}_{\alpha \beta }}\end{array}\]