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.

Recalling the relations:

    \[\left\{ {\begin{array}{*{20}{c}}{\left[ {{\sigma ^m}} \right] = {\rm{length}}}\\{\left[ {{X^\mu }} \right] = {\rm{length}}}\\{\left[ \Phi \right] = 1}\end{array}} \right.\]

and the world-manifold Poisson Bracket:

    \[\left\{ {\begin{array}{*{20}{c}}{\det \left( {{\gamma _{mn}}} \right) = {{\left\{ {{X^\mu },{X^\nu }} \right\}}^2}}\\{{{\left\{ {{X^\mu },{X^\nu }} \right\}}_{{\rm{PB}}}} \equiv {\varepsilon ^{mn}}{\partial _m}{X^\mu }{\partial _n}{X^\nu }}\end{array}} \right.\]

Hence, the action:

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

is reparametrization invariant only if the auxiliary field \Phi \left( \sigma \right) transforms as a world-sheet scalar density:

    \[\Phi \left( {\sigma '} \right) = \left| {\frac{{\partial \sigma }}{{\partial \sigma '}}} \right|\Phi \left( \sigma \right)\]

By implementing reparametrization invariance, \Phi \left( \sigma \right) can be transformed to unity and the Schild action can be recovered as a gauge-fixed form:

    \[I\left[ {\Phi = 1,X} \right] = \frac{{{\mu _0}}}{2}\int\limits_\Sigma {{d^2}} \sigma '\left[ {\det \left( {{\gamma _{mn}}} \right) + 1} \right] = {\rm{Schild }} + {\rm{ const}}\]

Thus, by solving \Phi \left( \sigma \right) in terms of X from:

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

one recovers, on-shell, the Nambu–Goto action:

    \[\begin{array}{c}\frac{{\delta I}}{{\delta \,\Phi }} = 0 \to \phi = \sqrt {\det \left( {{\gamma _{mn}}} \right)} \to \\I = {\mu _0}\int\limits_\Sigma {{d^2}} \sigma \sqrt { - \det \left( {{\gamma _{mn}}} \right)} \end{array}\]

The inverse equivalence relation can be deduced by starting from the Schild action:

    \[{I_S} \equiv {\mu _0}\int\limits_\Xi {{d^2}} \varphi \det \left[ {{\gamma _{ab}}\left( \varphi \right)} \right]\]

and we lift the world–sheet coordinates {\varphi ^m} to the role of dynamical variables via the reparametrization {\varphi ^m} \to {\sigma ^m} = {\sigma ^n}\left( \varphi \right):

    \[{I_{rep}} \equiv {\mu _0}\int\limits_\Sigma {{d^2}} \sigma {\Phi ^{ - 1}}\det \left[ {{\gamma _{ab}}\left( \sigma \right)} \right]\]

    \[{\Phi ^{ - 1}} \equiv {\varepsilon _{ij}}{\varepsilon ^{mn}}{\partial _m}\,{\phi ^i}{\partial _n}\,{\phi ^j}\]

By a {I_{rep}}/{\varphi ^i}-variation, we get the field equation:

    \[{\varepsilon _{ij}}{\varepsilon ^{mn}}{\partial _n}{\phi ^j}\,{\partial _m}\left( {\frac{{\det \left[ {{\gamma _{ab}}\left( \sigma \right)} \right]}}{{{\Phi ^2}}}} \right) = 0\]


    \[\frac{{\det \left[ {{\gamma _{ab}}\left( \sigma \right)} \right]}}{{{\Phi ^2}}} = {\rm{const}} \equiv \frac{1}{{4{\mu _0}}}\]

which allows us to regain the Nambu–Goto action.

Space-Time Uncertainty

Note that the fully reparametrization invariant Nambu–Goto action:

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

is a special case of the general two-parameter family of p-brane actions:

    \[I_n^p \equiv \frac{{\mu _0^{\left( {p + 1} \right)/2}}}{n}\int\limits_\Sigma {{d^{p + 1}}} \sigma e\left( \sigma \right)\left[ {\frac{{{{\left( {\det {\gamma _{mn}}} \right)}^{n/2}}}}{{e{{\left( \sigma \right)}^n}}} + n - 1} \right]\]

Thus, the key notion is the geometric structure of the p-brane world volume and string world-sheet topological embedding. The whole notion then, given Witten’s results on supersymmetric quantum mechanics, for space-time uncertainty relation comes from a simple analogy concerning the nature of string quantum mechanics. The central necessary condition of string perturbation theory is world-sheet conformal invariance, and one of the key insights of string theory as a unified theory is due to conformal invariance. The elimination of the ultraviolet divergences in the presence of gravity is essentially due to modular invariance, and that is part of conformal symmetry. From the viewpoint of generic two-dimensional field theory, conformal invariance forces us to choose a very narrow class of all possible two-dimensional field theories corresponding to the fixed points of the Wilsonian renormalization group. In the final formulation of quantum mechanics, the quantization condition is replaced by the more universal framework of Hilbert spaces and the corresponding operator algebras representations. This analogy suggests the importance of reinterpreting the conformal invariance requirement by promoting it to a universal form that ultimately can be formulated in a way that does not depend on perturbative methods.

Modular invariance can be expressed as the string-reciprocity-relation of the extremal length which is a conformally invariant notion of length corresponding to families of curves on Riemann surfaces. If we take some finite region \Omega and a set \Gamma of arcs on \Omega, the extremal length of \Gamma is defined by:

    \[\left\{ {\begin{array}{*{20}{c}}{{\lambda _\Omega }\left( \Gamma \right) = {{\min }_s}^\rho \frac{{L{{\left( {\Gamma ,\rho } \right)}^2}}}{{A\left( {\Omega ,\rho } \right)}}}\\{L\left( {\Gamma ,\rho } \right) = {{\inf }_{\gamma \in \Gamma }}L\left( {\gamma ,\rho } \right)}\\{A\left( {\Omega ,\rho } \right) = \int_\Omega {{\rho ^2}dzd\bar z\,} }\\{{\ell _\Omega }^\Gamma \left( {L\left( {\gamma ,\rho } \right)} \right) \equiv \int_\gamma {\rho \left| {dz} \right|} }\end{array}} \right.\]

in the conformal gauge. Since any Riemann surface can be composed of a set of quadrilaterals pasted along the boundaries, it is sufficient to consider the extremal length for an arbitrary quadrilateral …