Fun with Computational Physics: Noncommutative Geometry on the Lattice

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Fun with Computational Physics Non-commutative
Geometry on the Lattice

• Alberto de Campo1, Wolfgang Frisch2, Harald
  Grosse3, Natascha Hörmann2, Harald Markum2

1 Institut für Elektronische Musik und Akustik,
Kunstuniversität Graz 2 Atominstitut, TU Wien 3
Institut für Theoretische Physik, Universität
Wien
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Coordinates become Hermitean operators
Space-time is deformed
deformation parameter
antisymmetric tensor
Non-commutative geometry in continuum product
between fields is replaced by the Moyal product
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Weyl Operator
Functions are mapped on operators All functions
under consideration are assumed to have a
represention as Fourier transforms
function in the continuum
operator
Weyl operator
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Non-commutative Yang-Mills Theory
U(N) gauge field with generating field Ai(x)
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Non-commutative Torus
Periodic boundary conditions expressed by matrix

Momentum space is discretized
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Operators of coordinates
Coordinate operators fulfill the following
commutation relations
dimensionless tensor for non-commutativity in
space-time
Non-commutative torus is discretized by
introducing a shift operator which performs
translations of one lattice spacing a
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Solution of equation
The construction scheme of the matrices is
known
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U(1) Gauge Theory on a 2-dimensional
non-commutative Torus
The coordinate operators and the shift operators
can be constructed from the matrices with
the possible choice
The dimension N of the matrices corresponds to an
NxN lattice!
The gauge fields (links) can be expanded in
a Fourier series
expansion coefficient
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One-Plaquetten action
Monte Carlo simulations of non-commutative
geometry possible
Partition function
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One-Plaquette Action corresponds to the action of
the Twisted Eguchi-Kawai model (TEK)
unitary NxN matrix
twist
continuum limit
constraint finite in continuum
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Observables
Wilson loops
Polyakov lines
area law for Wilson loops
string tension
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W. Bietenholtz, F. Hofheinz, J. Nishimura,
hep-lat/0209021
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Phase of complex Wilson loop
W. Bietenholtz, F. Hofheinz, J. Nishimura,
hep-lat/0209021
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Current Studies Check of the phase structure from
Polyakov lines Definition of the topological
content via monopole and charge density
formulation on the twisted Eguchi-Kawai
model Possible Investigations Topological
structure of U(1) theory on a 4-dimensional
non-commutative torus Topology of Yang-Mills
theory on the fuzzy sphere?