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J. M. Isidro

Publications and source records attributed to J. M. Isidro.

At least 73 records · Page 4Linked to original sources
arXiv2006

Gerbes, Quantum Mechanics and Gravity

We prove that invariance of a quantum theory under the semiclassical vs. strong-quantum duality $S/\hbar\longleftrightarrow\hbar/S$, where S is the classical action, is equivalent to noncommutativity (of the Heisenberg-algebra type) of the coordinates of the space on which S is defined. We place these facts in correspondence with gerbes and Neveu-Schwarz B-fields and discuss their implications for a quantum theory of gravity. Feynman's propagator turns out to be closely related to the trivialisation of a gerbe on configuration space.

hep-th↗
arXiv2006

Gerbes and Heisenberg's Uncertainty Principle

We prove that a gerbe with a connection can be defined on classical phase space, taking the U(1)-valued phase of certain Feynman path integrals as Cech 2-cocycles. A quantisation condition on the corresponding 3-form field strength is proved to be equivalent to Heisenberg's uncertainty principle.

hep-th↗
arXiv2005

Magnetic Backgrounds from Generalised Complex Manifolds

The magnetic backgrounds that physically give rise to spacetime noncommutativity are generally treated using noncommutative geometry. In this article we prove that also the theory of generalised complex manifolds contains the necessary elements to generate B-fields geometrically. As an example, the Poisson brackets of the Landau model (electric charges on a plane subject to an external, perperdicularly applied magnetic field) are rederived using the techniques of generalised complex manifolds.

hep-th↗
arXiv2005

Moduli of Quanta

The classical phase of the matrix model of 11-dimensional M-theory is complex, infinite-dimensional Hilbert space. As a complex manifold, the latter admits a continuum of nonequivalent, complex-differentiable structures that can be placed in 1-to-1 correspondence with families of coherent states in the Hilbert space of quantum states. The moduli space of nonbiholomorphic complex structures on classical phase space turns out to be an infinite-dimensional symmetric space. We argue that each choice of a complex differentiable structure gives rise to a physically different notion of an elementary quantum.

quant-ph↗
arXiv2004

Bypassing Pauli's theorem

We define a quantum-mechanical time operator that is selfadjoint and compatible with the energy operator having a spectrum bounded from below. On their common domain, the operators of time and energy satisfy the expected canonical commutation relation. Pauli's theorem is bypassed because the correspondence between time and energy is not given by the standard Fourier transformation, but by a variant thereof known as the holomorphic Fourier transformation.

quant-ph↗
arXiv2004

Complex Moduli of Physical Quanta

Classical mechanics can be formulated using a symplectic structure on classical phase space, while quantum mechanics requires a complex-differentiable structure on that same space. Complex-differentiable structures on a given real manifold are often not unique. This letter is devoted to analysing the dependence of the notion of a quantum on the complex-differentiable structure chosen on classical phase space.

quant-ph↗
arXiv2004

Generalised Complex Geometry and the Planck Cone

Complex geometry and symplectic geometry are mirrors in string theory. The recently developed generalised complex geometry interpolates between the two of them. On the other hand, the classical and quantum mechanics of a finite number of degrees of freedom are respectively described by a symplectic structure and a complex structure on classical phase space. In this letter we analyse the role played by generalised complex geometry in the classical and quantum mechanics of a finite number of degrees of freedom. We identify generalised complex geometry as an appropriate geometrical setup for dualities. The latter are interpreted as transformations connecting points in the interior of the Planck cone with points in the exterior, and viceversa. The Planck cone bears some resemblance with the relativistic light-cone. However the latter cannot be traversed by physical particles, while dualities do connect the region outside the Planck cone with the region inside, and viceversa.

hep-th↗
arXiv2004

Noncommutative Manifolds from the Higgs Sector of Coincident D-Branes

The Higgs sector of the low-energy physics of n of coincident D-branes contains the necessary elements for constructing noncommutative manifolds. The coordinates orthogonal to the coincident branes, as well as their conjugate momenta, take values in the Lie algebra of the gauge group living inside the brane stack. In the limit when n=\infty (and in the absence of orientifolds), this is the unitary Lie algebra u(\infty). Placing a smooth manifold K orthogonally to the stack of coincident D-branes one can construct a noncommutative C*-algebra that provides a natural definition of a noncommutative partner for the manifold K.

hep-th↗
arXiv2004

Quantum States from Tangent Vectors

We argue that tangent vectors to classical phase space give rise to quantum states of the corresponding quantum mechanics. This is established for the case of complex, finite-dimensional, compact, classical phase spaces C, by explicitly constructing Hilbert-space vector bundles over C. We find that these vector bundles split as the direct sum of two holomorphic vector bundles: the holomorphic tangent bundle T(C), plus a complex line bundle N(C). Quantum states (except the vacuum) appear as tangent vectors to C. The vacuum state appears as the fibrewise generator of N(C). Holomorphic line bundles N(C) are classified by the elements of Pic(C), the Picard group of C. In this way Pic(C) appears as the parameter space for nonequivalent vacua. Our analysis is modelled on, but not limited to, the case when C is complex projective space.

quant-ph↗
arXiv2004

A Quantum is a Complex Structure on Classical Phase Space

Duality transformations within the quantum mechanics of a finite number of degrees of freedom can be regarded as the dependence of the notion of a quantum, i.e., an elementary excitation of the vacuum, on the observer on classical phase space. Under an observer we understand, as in general relativity, a local coordinate chart. While classical mechanics can be formulated using a symplectic structure on classical phase space, quantum mechanics requires a complex-differentiable structure on that same space. Complex-differentiable structures on a given real manifold are often not unique. This article is devoted to analysing the dependence of the notion of a quantum on the complex-differentiable structure chosen on classical phase space.

hep-th↗
arXiv2004

Quantum-Mechanical Dualities on the Torus

On classical phase spaces admitting just one complex-differentiable structure, there is no indeterminacy in the choice of the creation operators that create quanta out of a given vacuum. In these cases the notion of a quantum is universal, i.e., independent of the observer on classical phase space. Such is the case in all standard applications of quantum mechanics. However, recent developments suggest that the notion of a quantum may not be universal. Transformations between observers that do not agree on the notion of an elementary quantum are called dualities. Classical phase spaces admitting more than one complex-differentiable structure thus provide a natural framework to study dualities in quantum mechanics. As an example we quantise a classical mechanics whose phase space is a torus and prove explicitly that it exhibits dualities.

quant-ph↗
arXiv2003

Effective mass of a radiating charged particle in Einstein's universe

The effective gravitational mass as well as the energy and momentum distributions of a radiating charged particle in Einstein's universe are evaluated. The Moller's energy-momentum complex is employed for this computation. The spacetime under study is a generalization of Bonnor and Vaidya spacetime in the sense that the metric is described in the cosmological background of Einstein's universe in lieu of the flat background. Several spacetimes are limiting cases of the one considered here. Particularly for the Reissner-Nordstrom black hole background, our results are exactly the same with those derived by Cohen and Gautreau using Whittaker's theorem and by Cohen and de Felice using Komar's mass. Furthermore, the power output for the spacetime under consideration is obtained.

hep-th↗
arXiv2003

Duality and Quantum Mechanics

This Festschrift in honour of J. A. de Azcarraga gives an introduction to the concept of duality, i.e., to the relativity of the notion of a quantum, in the context of the quantum mechanics of a finite number of degrees of freedom. Although the concept of duality arises in string and M--theory, Vafa has argued that it should also have a counterpart in quantum mechanics, before moving on to second quantisation, fields, strings and branes. We illustrate our analysis with the case when classical phase space is complex projective space, but our conclusions can be generalised to other complex, symplectic phase spaces, both compact and noncompact.

quant-ph↗
arXiv2003

Quantum Mechanics in Infinite Symplectic Volume

We quantise complex, infinite-dimensional projective space CP(H). We apply the result to quantise a complex, finite-dimensional, classical phase space C whose symplectic volume is infinite, by holomorphically embedding it into CP(H). The embedding is univocally determined by requiring it to be an isometry between the Bergman metric on C and the Fubini-Study metric on CP(H). Then the Hilbert-space bundle over C is the pullback, by the embedding, of the Hilbert-space bundle over CP(H).

hep-th↗
arXiv2003

Quantum-Mechanical Dualities from Classical Phase Space

The geometry of the classical phase space C of a finite number of degrees of freedom determines the possible duality symmetries of the corresponding quantum mechanics. Under duality we understand the relativity of the notion of a quantum with respect to an observer on C. We illustrate this property explicitly in the case when classical phase space is complex n-dimensional projective space. We also provide some examples of classical dynamics that exhibit these properties at the quantum level.

quant-ph↗
arXiv2003

Duality, Quantum Mechanics and (Almost) Complex Manifolds

The classical mechanics of a finite number of degrees of freedom requires a symplectic structure on phase space C, but it is independent of any complex structure. On the contrary, the quantum theory is intimately linked with the choice of a complex structure on C. When the latter is a complex-analytic manifold admitting just one complex structure, there is a unique quantisation whose classical limit is C. Then the notion of coherence is the same for all observers. However, when C admits two or more nonbiholomorphic complex structures, there is one different quantisation per different complex structure on C. The lack of analyticity in transforming between nonbiholomorphic complex structures can be interpreted as the loss of quantum-mechanical coherence under the corresponding transformation. Observers using one complex structure perceive as coherent the states that other observers, using a different complex structure, do not perceive as such. This is the notion of a quantum-mechanical duality transformation: the relativity of the notion of a quantum.

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