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What is the signed particle formulation of quantum mechanics?
The signed particle formulation of quantum mechanics is a recently published novel formalism which describes quantum systems in terms of (signed) Newtonian field-less particles. These particles interact with a given external potential by means of creation/annihilation of signed particles. An introduction to this novel theory can be found here.

What is the Wigner formalism?
The Wigner formalism is an approach to quantum mechanics which is completely equivalent to the (standard) Schroedinger formalism [1.1], [1.2]. Indeed, an invertible Wigner-Weyl transform exists which converts wave functions into quasi-distribution functions and vice-versa. In this perspective, the situation is not any different than classical mechanics where different formalisms exist (Newtonian, Lagrangian, Hamiltonian, etc) and can be utilized according to the system under investigation.

[1.1] E. Wigner. On the quantum correction for thermodynamic equilibrium. Phys.
Rev., 40:749, 1932.

[1.2] Moyal, J. E.; Bartlett, M. S. (1949). "Quantum mechanics as a statistical theory". Mathematical Proceedings of the Cambridge Philosophical Society 45: 99.

What are the advantages of the Wigner formalism?
In the Wigner formalism a system is described in terms of a (quasi) distribution function defined in the phase-space of n-particles. It is, thus, a very intuitive approach which is closer to the way experimentalists perform their experiments. The Wigner equation allows the simulation of many-body quantum systems in a time-dependent, multi-dimensional fashion. This allows scientists to simulate ground and excited states. Furthermore, Monte Carlo techniques for this formalism nowadays exist which scale incredibly well on parallel machines, allowing the simulation of very complex systems.

Does the Wigner function have an experimental meaning?
The Wigner quasi-distribution function can be measured. Indeed quantum tomographical techniques exist nowadays. Using the tools of quantum tomography, one can reconstruct the Wigner quasi-distribution function along with its negative part (Radon transform). For example, the following paper gives a very good description on how this is achieved:
"Shadows and Mirrors: Reconstructing Quantum States of Atom Motion", D. Leibfried, T. Pfau, C. Monroe, Physics Today, April 1998.

Can the Wigner approach be utilized in practical situations?
The Wigner formalism is nowadays applied to a plethora of different situations. For example, recently it has been utilized in the field of simulation of Semiconductor devices [4.1], post-CMOS design [4.2], Quantum Chemistry (Wigner DFT and many-body techniques) [4.3], [4.4], just to mention a few. See the page of publications for a list of selected papers.

[4.1] D. Querlioz, P. Dollfus, The Wigner Monte Carlo Method for Nanoelectronic Devices—A Particle Description of Quantum Transport and Decoherence, ISTEWiley, 2010.

[4.2] J.M. Sellier, I. Dimov, The Wigner–Boltzmann Monte Carlo method applied to electron transport in the presence of a single dopant, Computer Physics Communications, Volume 185, Issue 10, October 2014, Pages 2427–2435.

[4.3] J.M. Sellier, I. Dimov, A Wigner Monte Carlo approach to density functional theory, J. Comput. Phys. 270 (2014) 265–277.

[4.4] J.M. Sellier, I. Dimov, The many-body Wigner Monte Carlo Method for time-dependent Ab-initio quantum simulations, J. Comp. Phys., Volume 273, 15 September 2014, Pages 589–597.

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(C) 2012-2017, Jean Michel Sellier.