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:

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

derived from the 2-form tensored flat metric:

    \[\eta = {\eta _{\mu \nu }}d{\hat x^\mu }{ \otimes _{{{\bar A}_c}}}d{\hat x^\nu }\]

where {\bar A_c} is the Hilbert space representation algebra corresponding to the Schrödinger deformed differential structure, and the corresponding Minkowski metric is given by:

    \[d{s^2} = {\eta _{\mu \nu }}d{\hat x^\mu }d{\hat x^\nu }\]

For an introduction to deformation quantization, this is a good read.

  • The hard part is that, due to the structure of FRW-minisuperspace, the quantization is not uniquely fixed up to isomorphism: there are infinite such transformations. A criterion of acceptability seems imposed on us, namely, that the state and quantization should probabilify the emergence of a classical limit in the topological neighborhood of classical FRW-predictions.

One must be careful in canonical quantization of Einsteinian gravity: the corresponding Hamiltonian is a linear combination of the FRW-constraints, and hence annihilates the physical quantum state, so we lose the time evolution from the theory. An equally grave problem arises if we try and apply canonical quantum gravity to cosmology, since the main mathematical entity is the wave function, thus obtained by Wheeler-DeWitt (WDW) equation or path integral, it must be possible to have an adequate wave packet that would peak around the classical FRW-cosmological model. Standardly, a wave-function collapse: reduction, in the Copenhagen interpretation would be adequate in quantum mechanics via a description of the dynamics of an ensemble of identical systems. In cosmology, there is only one Universe and hence only one system. Thus, the quantum state of the Universe has no well-defined wave-function. Add to that the fact that in quantum cosmology, the ‘observer‘ is an integral element of Universe. In any of the standard interpretations of quantum mechanics, a measurement involving a system, and ‘observer‘ [‘o’], and a measuring ‘device’ [‘m’], the quantum system interacts with a classical domain consisting, in part, of ‘o’, ‘m’, … , and it is this relational interaction that explains the reduction. However, quantum cosmology is a theory of the whole universe, and the measurement problem can have no standard quantum-mechanical solution, as can be attested by the fact that the cosmological quantum state {\left| {{\psi _U}} \right\rangle _{\cos }} cannot satisfy the Lindblad collapse equation:

    \[\begin{array}{*{20}{l}}{d{\psi _t}^{S,m,o} = \left[ { - iHdt + \sum\limits_{k = 1}^n {\left( {{L_k} - {\ell _{k,t}}} \right)} } \right.}\\{d{W_{k,t}} - \frac{1}{2}\sum\limits_{k = 1}^n {\left. {\left( {L_k^\dagger {L_k} - 2{\ell _{k,t}}{L_k} + {{\left| {{\ell _{k,t}}} \right|}^2}} \right)} \right]} }\\{{\psi _t}^{S,m,o}}\end{array}\]

since there can be no ‘m’, ‘o’ with Hilbert-separability with respect to {\left| {{\psi _U}} \right\rangle _{\cos }}, given the unitarity of the Lindblad collapse operators {\ell _{k,t}}:

    \[{\ell _{k,t}} \equiv \frac{1}{2}\left\langle {{\psi _t},\left( {L_k^\dagger + {L_k}} \right){\psi _t}^{S,m,o}} \right\rangle \]

as I showed: the Heisenberg equation with respect to the Lindblad collapse operators {\ell _{k,t}}:

    \[\begin{array}{c}i\hbar \frac{{\rm{d}}}{{{\rm{d}}t}}{\ell _{k,t}}(t) = \left[ {{\ell _{k,t}}(t),\hat H(t)} \right]\\ + \frac{{\partial {\ell _{k,t}}(t)}}{{\partial t}}\end{array}\]

has no solutions for {\left| {{\psi _U}} \right\rangle _{\cos }}, as entailed by the Wheeler-DeWitt equation, and related, {\left| {{\psi _U}} \right\rangle _{\cos }} satisfies the property of the Wheeler-DeWitt equation: namely, time totally disappears, and thus the above Lindblad collapse equation is actually undefinable for {\left| {{\psi _U}} \right\rangle _{\cos }}, since ‘time‘ is essential to the Lindblad collapse operator {\ell _{k,t}}.

  • This is the crucial role of deformation quantization: it transforms quantum cosmology into