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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:20210624T092100Z
UID:indico-contribution-1350944@indico.cern.ch
DESCRIPTION:Speakers: Daniel Ferrante (Brown University)\, Gerald Guralnik
(Brown University)\nLately\, it is becoming increasingly clear that exten
ding the\nFeynman Path Integral into the Complex domain yields desirable\n
properties. A first hint in support of 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 M
echanical systems (which share the same \nalgebra of observables). Secondl
y\, new results by E. Witten have \nbrought forward new results in 3-dimen
sional Chern-Simons theory \n(in the form of Complex geometries) and Supe
r Yang-Mills theory in \nfour dimensions\, as well as the relation between
Khovanov homology \nand systems of branes. \n\nGiven a system of D0-brane
s\, it is possible to understand it in terms \nof a Fourier Transform. As
such\, we can extend this system into the \nComplex plane\, reinterpreting
the Path Integral as a Fourier Transform \nover a certain integration cyc
le\, which results in a Fractional Fourier \nTransform. This can be furthe
r understood in terms of the Phase Space \nof this system\, where the Frac
tional Fourier Transform is related to the \nWigner function. In this way\
, we realize that the label of our Fractional \nFourier Transform\, which
is the Path Integral quantizing the system\, acts \nas the parameter dete
rmining the vacuum state. Therefore\, the allowed \nvalues of this label\,
for which the Path Integral converges\, \ndetermine the quantum phases of
the system. This can be immediately \nextended to Matrix models and Lie a
lgebra-valued ones \n(known as Group Field Theory): the same results hold\
, so long as\ncertain properties of the Action are satisfied\, guaranteein
g the convergence \nof the Path Integral.\n\nThese results can be dimensio
nally extended to systems of Dp-branes\, \nshowing some relations with the
Geometric Langlands Duality and \nMirror Symmetry. Furthermore\, they can
also be understood in terms of \ncoherent state quantization\, which open
s a window into quantum\ntomography\, and quantum chaos.\nhttps://indico.c
ern.ch/event/129980/contributions/1350944/
LOCATION:Rhode Island Convention Center
RELATED-TO:indico-event-129980@indico.cern.ch
URL:https://indico.cern.ch/event/129980/contributions/1350944/
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