ACADSTAFF UGM
| Title | : | Geometrical Portrait of Quantum Mechanics in Wasserstein Spaces |
| Author | : |
Dr.rer.nat. Muhammad Farchani Rosyid, M.Si. (1) |
| Date | : | 0 2022 |
| Keyword | : | Wasserstein space,quantum dynamics,signed measures,optimal transport Wasserstein space,quantum dynamics,signed measures,optimal transport |
| Abstract | : | A sample space (called the spectrum) has been assumed to be available to each physi- cal observable. The spectral measure (projection-valued measure) corresponding to each (sharp) observable and all pure quantum states have been involved to construct certain elements of the Wasserstein space over the spectrum of each observable which leads to a one-to-one correspondence between the set of all pure quantum states and the set of all distribution functions defined on the spectrum of each observable. Thanks to the correspondence, every pure quantum state could be represented by a probability mea- sure associated with the corresponding distribution function. We have proved that the Wasserstein space over the spectrum of an observable is homeomorphic to the Wasser- stein space over the spectrum of another observable. Thanks to the involvement of the spectral measures of observables in this consideration, the relation between the conver- gence of a sequence of state vectors in the Hilbert space (with respect to the norm) and the convergence of the corresponding sequence in the Wasserstein space with respect to Wasserstein distance has been investigated. A combination of elements of the Wasser- stein space which is in line with the states superposition rule in quantum mechanics is introduced. The combination reduces to the usual convex combination for mutually ”orthogonal” states. The combination can be regarded as a kind of deformation of the classical convex combination. Furthermore, we have shown that the involvement of the spectral measures in the investigation here also leads to an interesting discussion on the quantum dynamics in Wasserstein space. That the rate of change of the probability that the measurement of an observable gives an outcome in a measurable set depends on the measurable sets and bounded by the so-called numerical radius of the Hamiltonian operator has been proved. That the first order equation of quantum dynamics in the Wasserstein space over an observable can be regarded as a vector field defined on the Wasserstein space which is determined by the Hamiltonian operator of the system, where the time evolution of states is an integral curve of the vector field, has also been shown. For certain cases where the numerical radius of the Hamiltonian is finite, the second order equation of the quantum dynamics is proposed. It is shown that the Wasserstein distance is the greatest lower bound for the action functional of the kinetic energy if the quantum dynamics is regarded as a transport phenomenon. The quantum dynamics is an optimal transport with certain cost function if the the action functional of the kinetic energy is equal to the Wasserstein distance. Then, the results obtained in the case of sharp observables have been extended to the case of unsharp quantum observables which are represented by normalised positive-operator-valued measures. The positive-operator- valued measures considered here were determined from the projective-valued measures representing sharp observables by making use of a smearing procedure. In the case of unsharp observable, it also has been shown that the quantum dynamics is also described by the integral curves of a certain vector field defined on the Wasserstein space which is determined by the Hamiltonian operator of the system. The vector field was obtained from the vector field generating the quantum dynamics in the case of sharp observables also by making use of a smearing procedure. |
| Group of Knowledge | : | Fisika |
| Original Language | : | English |
| Level | : | Internasional |
| Status | : |
Draft
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| No | Title | Action |
|---|---|---|
| 1 |
GeometricalPortrait.pdf
Document Type : Dokumen Pendukung Karya Ilmiah (Hibah, Publikasi, Penelitian, Pengabdian)
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