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SUMMARY:Complex Path Integral as a Fractional Fourier Transform
DTSTART;VALUE=DATE-TIME:20110809T163200Z
DTEND;VALUE=DATE-TIME:20110809T163300Z
DTSTAMP;VALUE=DATE-TIME:20200928T220847Z
UID:indico-contribution-1350944@indico.cern.ch
DESCRIPTION:Speakers: Daniel Ferrante (Brown University)\nLately\, it is b
ecoming increasingly clear that extending the\nFeynman Path Integral into
the Complex domain yields desirable\nproperties. A first hint in support o
f such construction can be\nseen from the connection between SUSY Quantum
Mechanics and the\nLangevin dynamics: analytically continuing the Langevin
leads into\ndifferent SUSY Quantum Mechanical systems (which share the sa
me \nalgebra of observables). Secondly\, new results by E. Witten have \nb
rought forward new results in 3-dimensional Chern-Simons theory \n(in the
form of Complex geometries) and Super Yang-Mills theory in \nfour dimensi
ons\, as well as the relation between Khovanov homology \nand systems of b
ranes. \n\nGiven a system of D0-branes\, it is possible to understand it i
n terms \nof a Fourier Transform. As such\, we can extend this system into
the \nComplex plane\, reinterpreting the Path Integral as a Fourier Trans
form \nover a certain integration cycle\, which results in a Fractional Fo
urier \nTransform. This can be further understood in terms of the Phase Sp
ace \nof this system\, where the Fractional Fourier Transform is related t
o the \nWigner function. In this way\, we realize that the label of our Fr
actional \nFourier Transform\, which is the Path Integral quantizing the s
ystem\, acts \nas the parameter determining the vacuum state. Therefore\,
the allowed \nvalues of this label\, for which the Path Integral converge
s\, \ndetermine the quantum phases of the system. This can be immediately
\nextended to Matrix models and Lie algebra-valued ones \n(known as Group
Field Theory): the same results hold\, so long as\ncertain properties of t
he Action are satisfied\, guaranteeing the convergence \nof the Path Integ
ral.\n\nThese results can be dimensionally extended to systems of Dp-brane
s\, \nshowing some relations with the Geometric Langlands Duality and \nMi
rror Symmetry. Furthermore\, they can also be understood in terms of \ncoh
erent state quantization\, which opens a window into quantum\ntomography\,
and quantum chaos.\n\nhttps://indico.cern.ch/event/129980/contributions/1
350944/
LOCATION:Rhode Island Convention Center
URL:https://indico.cern.ch/event/129980/contributions/1350944/
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