arXiv subjects
Publications and source records attributed to J. M. Isidro.
We study the eikonal approximation to quantum mechanics on the Moyal plane. Instead of using a star product, the analysis is carried out in terms of operator-valued wavefunctions depending on noncommuting, operator-valued coordinates.
We elaborate on the existing idea that quantum mechanics is an emergent phenomenon, in the form of a coarse-grained description of some underlying deterministic theory. We apply the Ricci flow as a technical tool to implement dissipation, or information loss, in the passage from an underlying deterministic theory to its emergent quantum counterpart. A key ingedient in this construction is the fact that the space of physically inequivalent quantum states (either pure or mixed) has positive Ricci curvature. This leads us to an interesting thermodynamical analogy of emergent quantum mechanics.
We obtain Schroedinger quantum mechanics from Perelman's functional and from the Ricci flow equations of a conformally flat Riemannian metric on a closed 2-dimensional configuration space. We explore links with the recently discussed "emergent quantum mechanics".
We construct the classical mechanics associated with a conformally flat Riemannian metric on a compact, n-dimensional manifold without boundary. The corresponding gradient Ricci flow equation turns out to equal the time-dependent Hamilton-Jacobi equation of the mechanics so defined.
The Ricci flow equation of a conformally flat Riemannian metric on a closed 2-dimensional configuration space is analysed. It turns out to be equivalent to the classical Hamilton-Jacobi equation for a point particle subject to a potential function that is proportional to the Ricci scalar curvature of configuration space. This allows one to obtain Schroedinger quantum mechanics from Perelman's action functional: the quantum-mechanical wavefunction is the exponential of $i$ times the conformal factor of the metric on configuration space. We explore links with the recently discussed emergent quantum mechanics.
We establish a 1-to-1 relation between metrics on compact Riemann surfaces without boundary, and mechanical systems having those surfaces as configuration spaces.
It has been argued that, underlying any given quantum-mechanical model, there exists at least one deterministic system that reproduces, after prequantisation, the given quantum dynamics. For a quantum mechanics with a complex d-dimensional Hilbert space, the Lie group SU(d) represents classical canonical transformations on the projective space CP^{d-1} of quantum states. Let R stand for the Ricci flow of the manifold SU(d-1) down to one point, and let P denote the projection from the Hopf bundle onto its base CP^{d-1}. Then the underlying deterministic model we propose here is the Lie group SU(d), acted on by the operation PR. Finally we comment on some possible consequences that our model may have on a quantum theory of gravity.
We prove that the effective low-energy, nonlinear Schroedinger equation for a particle in the presence of a quasiperiodic potential is the potential-free, nonlinear Schroedinger equation on noncommutative space. Thus quasiperiodicity of the potential can be traded for space noncommutativity when describing the envelope wave of the initial quasiperiodic wave.
Quantum mechanics rests on the assumption that time is a classical variable. As such, classical time is assumed to be measurable with infinite accuracy. However, all real clocks are subject to quantum fluctuations, which leads to the existence of a nonzero uncertainty in the time variable. The existence of a quantum of time modifies the Heisenberg evolution equation for observables. In this letter we propose and analyse a generalisation of Heisenberg's equation for observables evolving in real time (the time variable measured by real clocks), that takes the existence of a quantum of time into account. This generalisation of Heisenberg's equation turns out to be a delay-differential equation.
Dirichlet branes are objects whose transverse coordinates in space are matrix-valued functions. This leads to considering a matrix algebra or, more generally, a Lie algebra, as the classical phase space of a certain dynamics where the multiplication of coordinates, being given by matrix multiplication, is nonabelian. Further quantising this dynamics by means of a star-product introduces noncommutativity (besides nonabelianity) as a quantum h-deformation. The algebra of functions on a standard Poisson manifold is replaced with the universal enveloping algebra of the given Lie algebra. We define generalised Poisson brackets on this universal enveloping algebra, examine their properties, and conclude that they provide a natural framework for dynamical setups (such as coincident Dirichlet branes) where coordinates are matrix-valued, rather than number-valued, functions.
We prove that Wigner functions contain a symplectic connection. The latter covariantises the symplectic exterior derivative on phase space. We analyse the role played by this connection and introduce the notion of local symplectic covariance of quantum-mechanical states. This latter symmetry is at work in the Schroedinger equation on phase space.
Any quantum-mechanical system possesses a U(1) gerbe naturally defined on configuration space. Acting on Feynman's kernel exp(iS/h), this U(1) symmetry allows one to arbitrarily pick the origin for the classical action S, on a point-by-point basis on configuration space. This is equivalent to the statement that quantum mechanics is a U(1) gauge theory. Unlike Yang-Mills theories, however, the geometry of this gauge symmetry is not given by a fibre bundle, but rather by a gerbe. Since this gauge symmetry is spontaneously broken, an analogue of the Higgs mechanism must be present. We prove that a Heisenberg-like noncommutativity for the space coordinates is responsible for the breaking. This allows to interpret the noncommutativity of space coordinates as a Higgs mechanism on the quantum-mechanical U(1) gerbe.
We study the selfadjoint time operator recently constructed by one of the authors. We will show that this time operator must be interpreted as a ``selfadjoint variant'' of the time operator.
We regard classical phase space as a generalised complex manifold and analyse the B-transformation properties of the *-product of functions. The C*-algebra of smooth functions transforms in the expected way, while the C*-algebra of holomorphic functions (when it exists) transforms nontrivially. The B-transformed *-product encodes all the properties of phase-space quantum mechanics in the presence of a background magnetic field.
In symplectic mechanics, the magnetic term describing the interaction between a charged particle and an external magnetic field has to be introduced by hand. On the contrary, in generalised complex geometry, such magnetic terms in the symplectic form arise naturally by means of B-transformations. Here we prove that, regarding classical phase space as a generalised complex manifold, the transformation law for the symplectic form under the action of a weak magnetic field gives rise to Dirac's prescription for Poisson brackets in the presence of constraints.
We construct a U(1) gerbe with a connection over a finite-dimensional, classical phase space P. The connection is given by a triple of forms A,B,H: a potential 1-form A, a Neveu-Schwarz potential 2-form B, and a field-strength 3-form H=dB. All three of them are defined exclusively in terms of elements already present in P, the only external input being Planck's constant h. U(1) gauge transformations acting on the triple A,B,H are also defined, parametrised either by a 0-form or by a 1-form. While H remains gauge invariant in all cases, quantumness vs. classicality appears as a choice of 0-form gauge for the 1-form A. The fact that [H]/2iπis an integral class in de Rham cohomology is related with the discretisation of symplectic area on P. This is an equivalent, coordinate-free reexpression of Heisenberg's uncertainty principle. A choice of 1-form gauge for the 2-form B relates our construction with generalised complex structures on classical phase space. Altogether this allows one to interpret the quantum mechanics corresponding to P as an Abelian gauge theory.
An Abelian gerbe is constructed over classical phase space. The 2-cocycles defining the gerbe are given by Feynman path integrals whose integrands contain the exponential of the Poincare-Cartan form. The U(1) gauge group on the gerbe has a natural interpretation as the invariance group of the Schroedinger equation on phase space.
It has been argued that, underlying M-theoretic dualities, there should exist a symmetry relating the semiclassical and the strong-quantum regimes of a given action integral. On the other hand, a field-theoretic exchange between long and short distances (similar in nature to the T-duality of strings) has been shown to provide a starting point for quantum gravity, in that this exchange enforces the existence of a fundamental length scale on spacetime. In this letter we prove that the above semiclassical vs. strong-quantum symmetry is equivalent to the exchange of long and short distances. Hence the former symmetry, as much as the latter, also enforces the existence of a length scale. We apply these facts in order to classify all possible duality groups of a given action integral on spacetime, regardless of its specific nature and of its degrees of freedom.