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Register a new userThe Kirillov picture for the Wigner particle
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Added by
Joseph C. V\\'arilly
Publication date
2017
fields
Physics
and research's language is
English
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We discuss the Kirillov method for massless Wigner particles, usually (mis)named continuous spin or infinite spin particles. These appear in Wigners classification of the unitary representations of the Poincare group, labelled by elements of the enveloping algebra of the Poincare Lie algebra. Now, the coadjoint orbit procedure introduced by Kirillov is a prelude to quantization. Here we exhibit for those particles the classical Casimir functions on phase space, in parallel to quantum representation theory. A good set of position coordinates are identified on the coadjoint orbits of the Wigner particles; the stabilizer subgroups and the symplectic structures of these orbits are also described.
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It is well known in NSR string theory, that vertex operators can be constructed in various ``pictures. Recently this was discussed in the context of pure spinor formalism. NSR picture changing operators have an elegant super-geometrical interpretation. In this paper we provide a generalization of this super-geometrical construction, which is also applicable to the pure spinor formalism.
The Bargmann-Wigner (BW) framework describes particles of spin-j in terms of Dirac spinors of rank 2j, obtained as the local direct product of n Dirac spinor copies, with n=2j. Such spinors are reducible, and contain also (j,0)+(0,j)-pure spin representation spaces. The 2(2j+1) degrees of freedom of the latter are identified by a projector given by the n-fold direct product of the covariant parity projector within the Dirac spinor space. Considering totally symmetric tensor spinors one is left with the expected number of 2(2j+1) independent degrees of freedom. The BW projector is of the order $partial ^{2j}$ in the derivatives, and so are the related spin-j wave equations and associated Lagrangians. High order differential equations can not be consistently gauged, and allow several unphysical aspects, such as non-locality, acausality, ghosts and etc to enter the theory. In order to avoid these difficulties we here suggest a strategy of replacing the high order of the BW wave equations by the universal second order. To do so we replaced the BW projector by one of zeroth order in the derivatives. We built it up from one of the Casimir invariants of the Lorentz group when exclusively acting on spaces of internal spin degrees of freedom. This projector allows one to identify anyone of the irreducible sectors of the primordial rank-2j spinor, in particular (j,0)+(0,j), and without any reference to the external space-time and the four-momentum. The dynamics is then introduced by requiring the (j,0)+ (0,j) sector to satisfy the Klein-Gordon equation. The scheme allows for a consistent minimal gauging.
It is shown that the quantum ground state energy of particle of mass m and electric charge e moving on a compact Riemann surface under the influence of a constant magnetic field of strength B is E_0=eB/2m. Remarkably, this formula is completely independent of both the geometry and topology of the Riemann surface. The formula is obtained by reinterpreting the quantum Hamiltonian as the second variation operator of an associated classical variational problem.
We review the noncommutative approach to the standard model. We start with the introduction if the mathematical concepts necessary for the definition of noncommutative spaces, and manifold in particular. This defines the framework of spectral geometry. This is applied to the standard model of particle interaction, discussing the fermionic and bosonic spectral action. The issues relating to the calculation of the mass of the Higgs are discussed, as well as the role of neutrinos and Wick rotations. Finally, we present the possibility of solving the problem of the Higgs mass by considering a pregeometric grand symmetry.
In this survey, our aim is to emphasize the main known limitations to the use of Wigner measures for Schrodinger equations. After a short review of successful applications of Wigner measures to study the semi-classical limit of solutions to Schrodinger equations, we list some examples where Wigner measures cannot be a good tool to describe high frequency limits. Typically, the Wigner measures may not capture effects which are not negligible at the pointwise level, or the propagation of Wigner measures may be an ill-posed problem. In the latter situation, two families of functions may have the same Wigner measures at some initial time, but different Wigner measures for a larger time. In the case of systems, this difficulty can partially be avoided by considering more refined Wigner measures such as two-scale Wigner measures; however, we give examples of situations where this quadratic approach fails.
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