The Cosmological Quantum State from Deformation Quantization

I will show that the Weyl-Wigner-Groenewold formalism of the cosmological quantum state is described by the Moyal-Wheeler-DeWitt deformation quantization equation with symplectic solutions in the Moyal-Wigner phase space: this is philosophically of foundational importance since quantum gravity necessitates that spacetime is quantized in a way described by the Moyal-Wheeler-DeWitt equation:

    \[\tilde H\left( {x + \frac{i}{2}{{\overrightarrow \partial }_x},{\Pi _x} - \frac{i}{2}{{\overrightarrow \partial }_x}} \right)W\left( {x,{\Pi _x}} \right) = 0\]

where in the deformation quantization Hilbert space formalism, the deformed operator in the scalar product relative to \tilde H is given by:

    \[\begin{array}{*{20}{l}}{\left\langle {\Psi ,{{\left( {d{X^\mu }d{X^\mu }} \right)}_\Theta }\Psi } \right\rangle = \left\langle {\Psi ,{{\left( {d{X^\mu }} \right)}^2}_\Theta \Psi } \right\rangle }\\{ = {{\left( {2\pi } \right)}^{ - d}}\mathop {\lim }\limits_{\varepsilon \to 0} \int {\int {dx{\mkern 1mu} dy{\mkern 1mu} {e^{ - ixy}}} } \chi \left( {\varepsilon x,\varepsilon y} \right) \cdot }\\{\left\langle {\Psi ,U(y)\alpha {\mkern 1mu} {\Theta _x}\left( {d{X^\mu }} \right)} \right\rangle }\\{ = {{\left( {2\pi } \right)}^{ - d}}\mathop {\lim }\limits_{\varepsilon \to 0} \int {\int {dx{\mkern 1mu} dy{\mkern 1mu} {e^{ - ixy}}} } \chi \left( {\varepsilon x,\varepsilon y} \right){b^\mu }\left( {x,y} \right)}\end{array}\]

with \Theta a skew-symmetric matrix on {\mathbb{R}^d}, and \chi \in f_{DQ}^{ps}\left( {{\mathbb{R}^d} \times {\mathbb{R}^d}} \right), \chi \left( {0,0} \right) = 1, and the deformation quantization differential operator is given by:

    \[{\left( {d{X^\mu }} \right)^2}_\Theta = \int {{e^{ - 2{a_\mu }{{\left( {\Theta X} \right)}_\mu }}}} {\left( {d{X^\mu }} \right)^2}\]

The central property of the cosmological quantum state is that it must entail the emergence of a classical universe satisfying all of the observable properties induced by the Friedmann-Robertson-Walker space-time, where such a FRW-space-time flat line-element, curved by quantum deformation, is:

    \[{\left( {d{s^2}} \right)_\Theta } = d{\hat t^2} - {e^{H\hat t}}d{\hat x_ \bot }^2\]

with curved space-time metric:

    \[{\left( {{\eta _{\mu \nu }}} \right)_\Theta } = {e^{ - 2{a_\mu }{{\left( {\Theta \hat x} \right)}_\mu }}}{\eta _{\mu \nu }}\]

Deriving 4D De Sitter Space from T-Branes via D7-Brane Action

Building on my earlier work on T-branes and F-theory, here I will show how De Sitter space emerges from an expansion of the D7-brane action around a T-brane background in the presence of 3-form supersymmetry breaking fluxes: this is crucial since de Sitter space is unstable in quantum gravity, while a D7-brane action resolves the instability. “T-branes are a non-abelian generalization of intersecting branes in which the matrix of normal deformations is nilpotent along some subspace”, and it is truly remarkable that “the simplest heterotic string compactifications are dual to T-branes in F-theory.” Recalling that the pull-back on the D7-brane worldvolume is given by:

    \[{\rm{P}}{\left[ {{V_\mu }{\rm{d}}{z^\mu }} \right]_\alpha } = {V_\alpha } + \lambda {V_i}{\not \partial _\alpha }{\Phi ^i}\]

where \alpha is a coordinate on \tilde S and the second quantized integral of the D-brane partition function for closed strings is given by:

    \[P_{{\rm{int}}}^{Dp} \equiv \not Z = \sum\limits_{\gamma = 0}^\infty {\underbrace {\int {{D^K}\gamma {{D'}^K}X{e^{S_{cld}^s}}} }_{{\rm{Topologies}}}} \]

with a non-Abelian D-term:

    \[D_{\hat A}^K = \int_{\tilde S} {\rm{P}} \left[ {{\rm{Im}}{e^{iJ}} \wedge {e^{ - B}}} \right] \wedge {e^{\lambda F}} \wedge \sqrt {{{\rm{A}}^\prime }(\Gamma )/{{\overline {\rm{A}} }^\prime }({\rm{N}})} {\rm{ }}\]

and

    \[\sqrt {{\rm A}'(\Gamma )/\bar {\rm A}'({\rm N})} \]

is the first Pontryagin class-term, and J is the flat space Kähler form:

    \[J = \underbrace {\frac{i}{2}{\rm{dx}} \wedge {\rm{d\bar x + }}\frac{i}{2}{\rm{dy}} \wedge {\rm{d\bar y}}}_{ = :\omega } + 2i{\rm{dz}} \wedge {\rm{d\bar z}}\]

where S_{cld}^s in:

    \[P_{{\rm{int}}}^{Dp} \equiv \not Z = \sum\limits_{\gamma = 0}^\infty {\underbrace {\int {{D^K}\gamma {{D'}^K}X{e^{S_{cld}^s}}} }_{{\rm{Topologies}}}} \]

is given by:

    \[S_{cld}^s = - \frac{1}{{4\pi {\alpha ^\dagger }}}\int_{\partial E_S^5} {{d^2}\sigma d} \Omega {\left( {{\phi _{INST}}} \right)^2}\sigma \sqrt { - \gamma } \left( {\phi \left( {{{\bar X}^\mu }} \right)} \right.{R_{icci}} + {\gamma ^{\alpha \beta }}{\not \partial _\alpha }{X^\mu }{g_{\mu \nu }}\left( {{{\bar X}^\nu }} \right) + \frac{1}{{\sqrt { - \gamma } }}{\varepsilon ^{ - {c_{2n}}/{Y_k}\left( {{{\cos }^2}\varphi } \right)}}{\not \partial _\beta }{\bar X^\nu }{b_{\mu \nu }}{\left( {\bar X} \right)^2}\]

The non-Abelian profiles for \Phi and A must satisfy the 7-brane functional equations of motion. Non-Abelian generalisation of:

    \[{W^O} = \int_{{\Sigma _5}} {\rm{P}} \left[ {{\Omega _0} \wedge {e^{ - B}}} \right] \wedge {e^{\lambda F}}\]

    \[{D^K} = \int_{\tilde S} {\rm{P}} \left[ {{\mathop{\rm Im}\nolimits} {e^{iJ}} \wedge {e^{ - B}}} \right] \wedge {e^{\lambda F}}\]

are built up as follows. Write locally:

    \[{\Omega _0} \wedge {e^B} = d\gamma \]

and localize the integral in:

    \[{W^O} = \int_{{\Sigma _5}} {\rm{P}} \left[ {{\Omega _0} \wedge {e^{ - B}}} \right] \wedge {e^{\lambda F}}\]

as:

    \[\int_{\tilde S} {P\left[ \gamma \right]} \wedge {e^{\lambda F}}\]