Quantum mechanics with respect to different reference frames

Mangiarotti, Luigi; Sardanashvily, G.

doi:10.1063/1.2769147

Cited by 9 publications

(14 citation statements)

References 19 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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“…. , n − m, such that: (i) the angle coordinates (y λ ) are those on a toroidal cylinder, i.e., fibre coordinates on the fibre bundle (8.17 Forthcoming Theorem 8.9 provides the sufficient conditions of the existence of global actionangle coordinates of a CIS on a symplectic manifold (Z, Ω) [10,18,25,31]. It generalizes the well-known result for the case of compact invariant submanifolds [2,5].…”

Section: Definition 82mentioning

confidence: 76%

“…We are based on the fact that the Hamilton equation (6.17) also is a Lagrange equation of the characteristic Lagrangian L H (6.22). Therefore one can study conservation laws in Hamiltonian mechanics on a phase space V * Q similarly to those in Lagrangian mechanics on a configuration space V * Q [10,18,32].…”

Section: Consequently Integrals Of Motion Of a Hamiltonian System (Vmentioning

confidence: 99%

Noether's first theorem in Hamiltonian mechanics

Sardanashvily

2015

Preprint

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Non-autonomous non-relativistic mechanics is formulated as Lagrangian and Hamiltonian theory on fibre bundles over the time axis R. Hamiltonian mechanics herewith can be reformulated as particular Lagrangian theory on a momentum phase space. This facts enable one to apply Noether's first theorem both to Lagrangian and Hamiltonian mechanics. By virtue of Noether's first theorem, any symmetry defines a symmetry current which is an integral of motion in Lagrangian and Hamiltonian mechanics. The converse is not true in Lagrangian mechanics where integrals of motion need not come from symmetries. We show that, in Hamiltonian mechanics, any integral of motion is a symmetry current. In particular, an energy function relative to a reference frame is a symmetry current along a connection on a configuration bundle which is this reference frame. An example of the global Kepler problem is analyzed in detail.

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“…One can think of its right-hand side as being a general expression of an inertial force in non-relativistic mechanics. Note that Hamiltonian time-dependent mechanics with respect to an arbitrary reference frame has been formulated [5,7].…”

Section: Introductionmentioning

confidence: 99%

Relative non-relativistic mechanics

Sardanashvily

2007

Preprint

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Quantum mechanics with respect to different reference frames

Cited by 9 publications

References 19 publications

The Schrödinger operator as a generalized Laplacian

The Schrödinger operator as a generalized Laplacian

Noether's first theorem in Hamiltonian mechanics

Relative non-relativistic mechanics

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