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

2. 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)

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

4. 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:

5. 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)

6. 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

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

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

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

11. 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

12. Cables with quantum memory VI

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

15. 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:

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

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

18. Back reaction V
1D perturbed wavefunction
Berry’s curvature

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