Covariant Schr dinger operator

Janyška, Josef; Modugno, Marco

doi:10.1088/0305-4470/35/40/304

Cited by 26 publications

(34 citation statements)

References 41 publications

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“…On the other hand, it became clear nowadays that an intrinsic, i.e., a frame-independent formulation of the Newtonian dynamics requires affine and not vectorial objects. We refer here to our earlier work [5,6,10,24], to recent papers by Janyška and Modugno [13], and Mangiarotti and Sardanashvily [16].…”

Section: Introductionmentioning

confidence: 99%

The Schrödinger operator as a generalized Laplacian

Grabowska¹,

Grabowski²,

Urbański³

2008

J. Phys. A: Math. Theor.

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The Schrödinger operators on the Newtonian space-time are defined in a way which make them independent on the class of inertial observers. In this picture the Schrödinger operators act not on functions on the space-time but on sections of certain one-dimensional complex vector bundle -the Schrödinger line bundle. This line bundle has trivializations indexed by inertial observers and is associated with an U (1)-principal bundle with an analogous list of trivializations -the Schrödinger principal bundle. If an inertial frame is fixed, the Schrödinger bundle can be identified with the trivial bundle over space-time, but as there is no canonical trivialization (inertial frame), these sections interpreted as 'wave-functions' cannot be viewed as actual functions on the space-time. In this approach the change of an observer results not only in the change of actual coordinates in the space-time but also in a change of the phase of wave functions. For the Schrödinger principal bundle a natural differential calculus for 'wave forms' is developed that leads to a natural generalization of the concept of Laplace-Beltrami operator associated with a pseudoRiemannian metric. The free Schrödinger operator turns out to be the Laplace-Beltrami operator associated with a naturally distinguished invariant pseudo-Riemannian metric on the Schrödinger principal bundle. The presented framework does not involve any ad hoc or axiomatically introduced geometrical structures. It is based on the traditional understanding of the Schrödinger operator in a given reference frame -which is supported by producing right physics predictions -and it is proven to be strictly related to the frame-independent formulation of analytical Newtonian mechanics and Hamilton-Jacobi equations, that makes a bridge between the classical and quantum theory. MSC 2000: 35J10, 70G45.

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Section: Introductionmentioning

confidence: 99%

The Schrödinger operator as a generalized Laplacian

Grabowska¹,

Grabowski²,

Urbański³

2008

J. Phys. A: Math. Theor.

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“…Even more, in some cases, a precise mathematical language on the scales allows us to achieve new results. For instance, in the paper [17] the uniqueness of the Schrödinger operator has been proved under the hypothesis of covariance. Here, the covariance is understood in a broad sense: namely, as invariance not only with respect to observers and coordinates, but also with respect to units of measurement.…”

Section: Advantages Of This Approachmentioning

confidence: 99%

“…The difference between the standard language and our viewpoints is more than a trivial formality. Indeed, several advantages arise when we use our approach in a concrete development of a specific theory; we just quote the result on the uniqueness of the Schrödinger operator proved in [17].…”

Section: Concrete Use Of Positive Spaces In Our Papersmentioning

confidence: 99%

“…Finally, we show how unit spaces can be employed in physical theories by the language of differential geometry. The formalism discussed in this section has been widely used in several papers dealing with physical theories (see, for instance, [8,17,18,25,31,34,35]). In the present paper we analyse the mathematical foundations of this formalism for the first time.…”

Section: Algebraic Model Of Physical Scalesmentioning

confidence: 99%

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An Algebraic Approach to Physical Scales

Janyška

Modugno

Vitolo³

2009

Acta Appl Math

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This paper is aimed at introducing an algebraic model for physical scales and units of measurement. This goal is achieved by means of the concept of "positive space" and its rational powers. Positive spaces are "semi-vector spaces" on which the group of positive real numbers acts freely and transitively through the scalar multiplication. Their tensor multiplication with vector spaces yields "scaled spaces" that are suitable to describe spaces with physical dimensions mathematically. We also deal with scales regarded as fields over a given background (e.g., spacetime).

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“…Related ideas, describing the evolution of a quantum system in terms of a connection on a functional bundle in a Galileian setting, have been introduced in [JM02,CJM95]. and the section of scaled densities…”

Section: Quantum Interactionmentioning

confidence: 99%

Quantum Bundles and Quantum Interactions

Canarutto

2005

Int. J. Geom. Methods Mod. Phys.

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A geometric framework for describing quantum particles on a possibly curved background is proposed. Natural constructions on certain distributional bundles ('quantum bundles') over the spacetime manifold yield a quantum "formalism" along any 1-dimensional timelike submanifold (a 'detector'); in the flat, inertial case this turns out to reproduce the basic results of the usual quantum field theory, while in general it could be seen as a local, "linearized" description of the actual physics.1991 MSC: 58B99, 53C05, 81Q99, 81T20

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Covariant Schr dinger operator

Cited by 26 publications

References 41 publications

The Schrödinger operator as a generalized Laplacian

The Schrödinger operator as a generalized Laplacian

An Algebraic Approach to Physical Scales

Quantum Bundles and Quantum Interactions

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