Stable Chaos:
Around any configuration leading to one such multiple collision, the ordering of the two-body
collisions changes abruptly. The non-commutativity of the collision itself induces therefore a discontinuity in the dynamics that is conceptually similar to those postulated in the coupled-map lattices....The discontinuity is due to the fact that only one of them inhibits the other. It is therefore interesting to verify that, in agreement with
the past observations, the presence of discontinuities in the phase-space is a condition for the onset of a “stable chaos” dynamics.
So "Stable Chaos" is an attempt to convert the noncommutativity back into the symmetric standard math.
electronic density of states or the vibrational density of states (phonon modes). Once the
Hull is constructed, it leads to the construction of the Noncommutative Brillouin zone
(NCBZ) and its Geometry. Then the description of electrons in the one-particle approx-
imation, or of the phonons in the harmonic approximation follows easily. No attempt
to account for the large number of results obtained in the eighties and later concerning
the spectral properties for both electrons and phonons will be made here. The reader is
invited to look at [43, 33, 86] concerning spectral results on disordered or quasiperiodic
systems or at [18] concerning transport properties with anomalous spectra or diffusion.
A special emphasis will be put upon recent results obtained to compute the K-theory of
the NCBZ, especially in the context of the so-called gap labelling theorem (see Section 4).
This theorem was formulated in the early eighties in its most general form [12] and has
been given many illustrations in the case of one-dimensional systems during the eighties
[16]. It required however another decade to get precise results for systems in higher di-
mensions. The present notes will conclude on a short description of what is still today
the most spectacular application of the Noncommutative Geometry to realistic physics,
namely the integer quantum Hall effect [22, 41]
Hull is constructed, it leads to the construction of the Noncommutative Brillouin zone
(NCBZ) and its Geometry. Then the description of electrons in the one-particle approx-
imation, or of the phonons in the harmonic approximation follows easily. No attempt
to account for the large number of results obtained in the eighties and later concerning
the spectral properties for both electrons and phonons will be made here. The reader is
invited to look at [43, 33, 86] concerning spectral results on disordered or quasiperiodic
systems or at [18] concerning transport properties with anomalous spectra or diffusion.
A special emphasis will be put upon recent results obtained to compute the K-theory of
the NCBZ, especially in the context of the so-called gap labelling theorem (see Section 4).
This theorem was formulated in the early eighties in its most general form [12] and has
been given many illustrations in the case of one-dimensional systems during the eighties
[16]. It required however another decade to get precise results for systems in higher di-
mensions. The present notes will conclude on a short description of what is still today
the most spectacular application of the Noncommutative Geometry to realistic physics,
namely the integer quantum Hall effect [22, 41]
Another way to describe this aperiodicity is to see the magnetic field acting as an effective Planck constant that makes the ordinary space noncommutative from the point of view of quantum charged particles [13, 15, 95].
We show that in the presence of the torsion tensor Sk
ij , the quantum commutation relation for
the four-momentum, traced over spinor indices, is given by [pi, pj ] = 2i~Sk
ijpk. In the Einstein–
Cartan theory of gravity, in which torsion is coupled to spin of fermions, this relation in a coordinate
frame reduces to a commutation relation of noncommutative momentum space, [pi, pj ] = iǫijkUp3pk,
where U is a constant on the order of the squared inverse of the Planck mass. We propose that this
relation replaces the integration in the momentum space in Feynman diagrams with the summation
over the discrete momentum eigenvalues.
An interesting idea for such a regularization was explored in [10], where ultraviolet divergences in quantum field
theory might be avoided by curving momentum space. The idea that momentum space might be curved was first
suggested by Born [11]. Curved phase space may have several novel consequences on the motion of particles [12].
Snyder pointed out that the curvature of momentum space implies the noncommutativity of spacetime coordinates
[13]. This observation has led to the development of noncommutative geometry, most notably by Connes [14], and
then to noncommutative field theory [15]. Quantum geometry in which a curved momentum space is dual to a
noncommutative spacetime was explored in [16].
The theory of relativity postulates that spacetime is the invariant arena for nonquantum physics. A novel principle
of relative locality was suggested in [17], according to which a phase space is the invariant area for nonquantum
physics. In this scenario, both coordinate space and momentum space are curved. Propagating and interacting
particles are observed in spacetime constructed by observers, but observers at different locations construct different
spacetime projections from the invariant phase space. The curvature, torsion, and nonmetricity of momentum space
can manifest themselves in various deformations of the additivity of the momentum and energy, modifying the energymomentum
conservation laws [17]. In addition, curved momentum space is related to the invariant length scale [18].
If a curved momentum space implies a noncommutative coordinate spacetime, then a curved coordinate spacetime
should imply a noncommutative momentum space. Quantum mechanics with noncommutative momentum was explored
in [19]. The integration in noncommutative momentum space must be replaced with the summation over all
eigenvalues of the momentum [20].
In this article, we postulate that a realistic regularization, expected by Dirac, may come from the noncommutativityIn the noncommutative momentum space, the integration over the momentum components must
of momentum. Moreover, we argue that the noncommutative momentum is a consequence of spacetime torsion. The
existence of torsion [21, 22] is required by the consistency of the conservation law for the total (orbital plus spin)
angular momentum of a Dirac particle curved spacetime with relativistic quantum mechanics [23]. Therefore, we
refer to this postulate as torsional regularization. We propose the commutation relation for the momentum that is
consistent with the Einstein–Cartan (EC) theory of gravity, in which torsion couples to spin [24]. The momentum
operators do not commute (uncertainty principle for momentum), which becomes significant at larger momenta where
regularization is needed.
We propose a prescription for the summation in noncommutative momentum space (replacing the integration
in commutative momentum space) that gives a correct continuous limit for convergent integrals.
be replaced with the summation over the momentum eigenvalues that form a discrete spectrum. Since torsion increases
with the magnitude of the four-momentum, the separation between adjacent eigenvalues also increases. Consequently,
the sum over the momentum eigenvalues converges. We derived a prescription for this summation that gives a
correct continuous limit for convergent integrals. We extended this prescription to tensor integrals. We showed
that ultraviolet-divergent integrals turn into convergent sums, naturally eliminating the ultraviolet divergence in loop
diagrams at all orders.
No comments:
Post a Comment