String-Theory and the String/Non-Local-‘Particles’ Duality

One of the deepest result in quantum physics is that the Wave-Particle-Duality-Relations correspond to a modern formulation of the Heisenberg uncertainty principle stated in terms of entanglement entropies: a Bell-type argument in the context of T-duality’s winding modes, shows the action describing any particle implies an ontological non-locality intrinsic to the notion of ‘particle’ and in this series of posts, I will show how that entails that ‘non-local-particles’ can be symmetrically identified with strings in string-theory. Consider, as I showed, the non-commutative action for the dynamics of N-branes of type J in the string regime:

with the D-1-action:

    \[S_1^D = {\mkern 1mu} - {T_1}\int\limits_{{\rm{worldvolumes}}} {{d^{1 + 1}}} \xi \frac{{\not D_{\mu \nu }^{susy}L}}{{{{\not \partial }_{{v_a}}}}}{e^{ - {\Phi _{bos}}}}{\rm{de}}{{\rm{t}}^{1/2}}G_{ab}^{\exp \left( {H_{1 + 1}^{{\rm{array}}}} \right)}\]

and we get from:

    \[{e^{ - {\Phi _{bos}}}}{\rm{de}}{{\rm{t}}^{1/2}}G_{ab}^{\exp \left( {H_{1 + 1}^{{\rm{array}}}} \right)}\]

the d-1 mass-term:

    \[{T_1}{e^{ - {\Phi _{bos}}}}\prod\limits_{i = 1}^1 {\left( {2\pi nR} \right)} \]

String-Theory and Calabi-Yau Fourfolding of M-Theory

In this post, I will carry a Calabi-Yau fourfold compactification of M-theory in a topologically smooth way. Since M-theory is the only quantum theory of gravity that provably has a finite renormalization group and is the only complete self-consistent GUT, such 4-D compactifications are essential in order to have a correspondence with 4-D spacetime. Recall I derived, via Clifford algebraic symmetry, the total action:

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

which is deep since Clifford algebras are a quantization of exterior algebras, and applying to the ‘Einstein-Minkowski’ tangent bundle, we get via Gaussian matrix elimination, an expansion of {\not D^{SuSy}} via Green-functions, yielding M-Theory’s action:

    \[{S_M} = \frac{1}{{{k^9}}}\int\limits_{{\rm{world - volumes}}} {{d^{11}}} \sqrt {\frac{{ - {g_{\mu \nu }}}}{{ - \gamma }}} {T_p}^{10}{\mkern 1mu} d{\mkern 1mu} \Omega {\left( {{\phi _{INST}}} \right)^{26}}\left( {{R_{icci}} - A_\mu ^H\frac{1}{{48}}G_4^2} \right) + \sum\limits_{Dp} {\not D_\mu ^{SuSy}} {e^{ - H_3^b}}/S_{Dp}^{WV} + \sum\limits_{Dp} {\not D_\nu ^{SuSy}} {e^{H_3^b}}/S_{Dp}^{SV}\]

with k the kappa symmetry gravitonic term and the supergravitational Hamiltonian term being:

    \[\sum\limits_{Dp} {\not D_\mu ^{SuSy}} {e^{ - H_3^b}}/S_{Dp}^{WV} + \sum\limits_{Dp} {\not D_\nu ^{SuSy}} {e^{H_3^b}}/S_{Dp}^{SV}\]

Let {Y_4} be a smooth Calabi-Yau fourfold and start with the bosonic 11-D SuGra sector