• 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.

  • Clifford-Kähler Algebras and M-Theory Compactification

    Continuing from my work on the relation between Clifford algebraic symmetries and M-theory, here I will initiate an analysis of compactification via the derived Kähler-Atiyah bundle associated with Clifford-Kähler manifolds. Recall that whenever 2 or more D-branes coincide, there is a Clifford algebraic symmetry whose generators allow us to derive the total action:

    \begin{array}{*{20}{l}}{{S^{Total}} = \frac{1}{{2\pi {\alpha ^\dagger }12}}\int\limits_{{\rm{world - volumes}}} {{d^{26}}} x{\mkern 1mu} d{\mkern 1mu} \Omega {{\left( {{\phi _{Inst}}} \right)}^2}\sqrt {\frac{{ - {g_{\mu \nu }}}}{{ - \gamma }}} {\mkern 1mu} {e^{ - {c_{2n}}/{\Upsilon _\kappa }(\cos \varphi )}} \cdot }\\{\left( {{R_{icci}} - 4{{\left( {{{\not D}^{SuSy}}\left( {{\phi _{Inst}}} \right)} \right)}^2}} \right) + \frac{1}{{12}}H_{3,\mu \nu \lambda }^bH_3^{b,\mu \nu \lambda }/A_\mu ^H + \sum\limits_{D - p - branes} {S_{Dp}^{WV}} }\end{array}

    and since D-p-branes are metaplectic solitons in closed string-theory, by the von Neumann boundary condition, there is a natural coupling of the super-Higgs field A_\mu ^H to the world-sheet of a string through its boundary:

    {S_{open}} = {S_{cld}} + \int\limits_{{\rm{end - points}}} {d\tau } A_\mu ^H{\bar X^\mu }{e^{ - H_3^b}}d{\mkern 1mu} \Omega {\left( {{\phi _{Inst}}} \right)^{\exp {\kern 1pt} ({c_{2n}}/{\Upsilon _\kappa }(\cos \varphi )}}

    Hence, A_\mu ^H lives on a p+1 dimensional subspace with a {\Upsilon _\kappa }(\cos \varphi ) contribution, yielding the world-volume action:

    S_{Dp}^{WV} = {S_{cld}} + \int\limits_{{\rm{end - points}}} {d\tau } A_\mu ^H{\bar X^\mu }d{\mkern 1mu} \Omega {\left( {{\phi _{Inst}}} \right)^2}{e^{ - \left( {H_3^b} \right)/{\Upsilon _\kappa }(\cos \varphi )}} + {e^{{c_{2n}}/{\Upsilon _\kappa }(\cos \varphi )}}/H_3^b

    and since world-volumes have conformal invariance, by solving the n-loop level equation of motion