Coherentstates.docx

Measuring Quantum Geometries via Quasi-Coherent States Introduction 

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Structure of spacetime is expected to be quantum at very short distances like at plank scale. One of examples of such geometries is matrix geometries realized by non-trivial solutions by Yang-Mills models like IKKT or BFSS. These geometries (configurations) are defined by set of Hermitian matrices in the target space. These matrices are interpreted as quantized coordinates in target space equipped with a metric. Fuzzy Sphere and Fuzzy Tori are some good examples (Fuzzy Spaces).

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To differentiate non-geometric and geometric configurations And also measure this geometry

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To look for optimally localized states with small dispersion Then X (matrices) can almost be simultaneously measured and their expectation values provide location of some Variety M embedded in the target space R.

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Aim

Method

Quasi-Coherent States  

The ground states of matrix Laplacian or matrix Dirac operators in the presence of point like test brane in the target space. The eigenvalues of matrix Laplacian or matrix Dirac operators are interpreted as displacement energy (energy of the strings stretching between test brane and the background brane).

Hierarchy of Eigenvalues of Hessian 

For matrix backgrounds which define some approximate semi-classical brane geometry, the eigenvalues of H must exhibit a clear hierarchy between small eigenvalues corresponding to tangential directions and eigenvalues O(1) which correspond to directions transversal to the brane in target space.

Comparison between matrix Laplacian or matrix Dirac operators realizations   

For matrix Laplacian operator ground state energy is strictly positive and provide information about local dispersion and geometric uncertainty. For matrix Dirac operator location of state is recovered form exact zero modes.

Main Points  

To determine the coherent states To find Quasi Minimum of displacement energy E