11 research outputs found
On the realization of Symmetries in Quantum Mechanics
The aim of this paper is to give a simple, geometric proof of Wigner's
theorem on the realization of symmetries in quantum mechanics that clarifies
its relation to projective geometry. Although several proofs exist already, it
seems that the relevance of Wigner's theorem is not fully appreciated in
general. It is Wigner's theorem which allows the use of linear realizations of
symmetries and therefore guarantees that, in the end, quantum theory stays a
linear theory. In the present paper, we take a strictly geometrical point of
view in order to prove this theorem. It becomes apparent that Wigner's theorem
is nothing else but a corollary of the fundamental theorem of projective
geometry. In this sense, the proof presented here is simple, transparent and
therefore accessible even to elementary treatments in quantum mechanics.Comment: 8 page
A Supersymmetric D4 Model for mu-tau Symmetry
We construct a supersymmeterized version of the model presented by Grimus and
Lavoura (GL) in [1] which predicts theta_{23} maximal and theta_{13}=0 in the
lepton sector. For this purpose, we extend the flavor group, which is D4 x
Z2^{(aux)} in the original model, to D4 x Z5. An additional difference is the
absence of right-handed neutrinos. Despite these changes the model is the same
as the GL model, since theta_{23} maximal and theta_{13}=0 arise through the
same mismatch of D4 subgroups, D2 in the charged lepton and Z2 in the neutrino
sector. In our setup D4 is solely broken by gauge singlets, the flavons. We
show that their vacuum structure, which leads to the prediction of theta_{13}
and theta_{23}, is a natural result of the scalar potential. We find that the
neutrino mass matrix only allows for inverted hierarchy, if we assume a certain
form of spontaneous CP violation. The quantity |m_{ee}|, measured in
neutrinoless double beta decay, is nearly equal to the lightest neutrino mass
m3. The Majorana phases phi1 and phi2 are restricted to a certain range for m3
< 0.06 eV. We discuss the next-to-leading order corrections which give rise to
shifts in the vacuum expectation values of the flavons. These induce deviations
from maximal atmospheric mixing and vanishing theta_{13}. It turns out that
these deviations are smaller for theta_{23} than for theta_{13}.Comment: 19 pages, 4 figure
Minimal Mass Matrices for Dirac Neutrinos
We consider the possibility of neutrinos being Dirac particles and study
minimal mass matrices with as much zero entries as possible. We find that up to
5 zero entries are allowed. Those matrices predict one vanishing mass state, CP
conservation and U_{e3} either zero or proportional to R, where R is the ratio
of the solar and atmospheric \Delta m^2. Matrices containing 4 zeros can be
classified in categories predicting U_{e3} = 0, U_{e3} \neq 0 but no CP
violation or |U_{e3}| \neq 0 and possible CP violation. Some cases allow to set
constraints on the neutrino masses. The characteristic value of U_{e3} capable
of distinguishing some of the cases with non-trivial phenomenological
consequences is about R/2 \sin 2 \theta_{12}. Matrices containing 3 and less
zero entries imply (with a few exceptions) no correlation for the observables.
We outline models leading to the textures based on the Froggatt-Nielsen
mechanism or the non-Abelian discrete symmetry D_4 \times Z_2.Comment: 32 pages, 3 figures. Comments and references added. To appear in JHE
Non-Abelian Discrete Flavor Symmetries from T^2/Z_N Orbifolds
In [1] it was shown how the flavor symmetry A4 (or S4) can arise if the three
fermion generations are taken to live on the fixed points of a specific
2-dimensional orbifold. The flavor symmetry is a remnant of the 6-dimensional
Poincare symmetry, after it is broken down to the 4-dimensional Poincare
symmetry through compactification via orbifolding. This raises the question if
there are further non-abelian discrete symmetries that can arise in a similar
setup. To this end, we generalize the discussion by considering all possible
2-dimensional orbifolds and the flavor symmetries that arise from them. The
symmetries we obtain from these orbifolds are, in addition to S4 and A4, the
groups D3, D4 and D6 \simeq D3 x Z2 which are all popular groups for flavored
model building.Comment: 12 pages, 4 figure
S4 Flavor Symmetry and Fermion Masses: Towards a Grand Unified theory of Flavor
Pursuing a bottom-up approach to explore which flavor symmetry could serve as
an explanation of the observed fermion masses and mixings, we discuss an
extension of the standard model (SM) where the flavor structure for both quarks
and leptons is determined by a spontaneously broken S4 and the requirement that
its particle content is embeddable simultaneously into the conventional SO(10)
grand unified theory (GUT) and a continuous flavor symmetry G_f like SO(3)_f or
SU(3)_f. We explicitly provide the Yukawa and the Higgs sector of the model and
show its viability in two numerical examples which arise as small deviations
from rank one matrices. In the first case, the corresponding mass matrix is
democratic and in the second one only its 2-3 block is non-vanishing. We
demonstrate that the Higgs potential allows for the appropriate vacuum
expectation value (VEV) configurations in both cases, if CP is conserved. For
the first case, the chosen Yukawa couplings can be made natural by invoking an
auxiliary Z2 symmetry. The numerical study we perform shows that the best-fit
values for the lepton mixing angles theta_12 and theta_23 can be accommodated
for normal neutrino mass hierarchy. The results for the quark mixing angles
turn out to be too small. Furthermore the CP-violating phase delta can only be
reproduced correctly in one of the examples. The small mixing angle values are
likely to be brought into the experimentally allowed ranges by including
radiative corrections. Interestingly, due to the S4 symmetry the mass matrix of
the right-handed neutrinos is proportional to the unit matrix.Comment: 27 pages, published version with minor change
Conformal linear gravity in de Sitter space II
From the group theoretical point of view, it is proved that the theory of
linear conformal gravity should be written in terms of a tensor field of rank-3
and mixed symmetry [Binegar, et al, Phys. Rev. D 27, (1983) 2249]. We obtained
such a field equation in de Sitter space [Takook, et al, J. Math. Phys. 51,
(2010) 032503]. In this paper, a proper solution to this equation is obtained
as a product of a generalized polarization tensor and a massless scalar field
and then the conformally invariant two-point function is calculated. This
two-point function is de Sitter invariant and free of any pathological
large-distance behavior.Comment: 16 pages, no figure, published versio