Deviating from the Canonical: Induced Noncommutativity Chryssomalis Chryssomalakos Instituto de Ciencias Nucleares UNAM Joint work with P. Aguilar and H. Hernandez (ICN)

Geometric Phases Hamiltonian H(R), R: parameters R(t): slow curve in parameter space Non-degenerate eigenstate |n; R>: H(R) |n;R> = En(R) |n;R> < n; R|n; R > = 1 SE: i dy(t)/dt = H(R(t)) y(t) y(t=0) = |n; R(t=0)> y(t) ~ |n; R(t)> (adiabaticity)

Geometric Phases II Naive guess where does not work… Berry tried… …and found

QM on Hypersurfaces Intrinsic quantization: coordinates on M, ignore ambient space (unphysical) Confining potential approach: with Frenet-Serret: Induced metric on the surface, from the ambient euclidean one:

QM on Hypersurfaces II Total (3D) hamiltonian giving rise to normal & tangent SE’s (2D harmonic oscillator) Effective hamiltonian for motion on M (Maraner, Destri 93)

Cables with quantum memory Choose: n, b frame, |+>, |-> normal states Effective hamiltonian for motion along the wire Curvature and torsion of the curve become parameters for the 1D particle hamiltonian

Cables with quantum memory II Features of the effective 1D hamiltonian Curvafilia: Induced gauge field, compensating arbitrary rotations of the normal frame:

Cables with quantum memory III Pre-curve: Keep total length 2p Use arclength parametrization 3D position vector in the neighborhood of the curve:

Cables with quantum memory IV Compute hamiltonian Curvature, torsion Zeroth and first order hamiltonian 1D wavefunction 2D normal wavefunction 3D total wavefunction

Cables with quantum memory V (Notice: s, a, b, taken as functions of (X,Y,Z; x,y,z)) Initial state: |2,+> + |2,-> Cyclic change: x=cos t, y=sin t, z=2 Geometric phase causes rotation of the probability profile in the normal plane

Cables with quantum memory VI

Back reaction I Example: Spin coupled to position Adiabatic approximation: Effective hamiltonian: Noncommuting momenta:

Back reaction II What if light particle’s eigenstates depend also on heavy particle’s momentum? Noncommuting effective position operators emerge for the heavy particle… Apply to wire quantization:

Back reaction III Lagrangian density (implement LAP dynamically) In terms of Fourier modes Quantize with canonical commutation relations

Back reaction IV Effective 1D particle hamiltonian: with In the case of the unit circle…

Back reaction V 1D perturbed wavefunction Berry’s curvature

Back reaction VI CPT? Deformed CCR’s: grain of salt advised… Adiabatic parasites noncommutative effective quantum field theory Noncommutativity fixed by standard physics CPT?