In the four-dimensional effective theory from string compactification discrete flavor symmetries arise from symmetric structure of the compactified space and generally contain both the R symmetry and non-R symmetry. We point out that a new type of non-Abelian flavor symmetry can also appear if the compact space is non-commutative. Introducing the dihedral group D 4 as such a new type of flavor symmetry together with the R symmetry and non-R symmetry in SU (6) × SU (2) R model, we explain not only fermion mass hierarchies but also hierarchical energy scales including the breaking scale of the GUT-type gauge symmetry, intermediate Majorana masses of R-handed neutrinos and the scale of µ-term.

We study the unitarity bounds of the scattering amplitudes in the extra dimensional gauge theory where the gauge symmetry is broken by the boundary condition. The estimation of the amplitude of the diagram including four massive gauge bosons in the external lines shows that the asymptotic power behavior of the amplitude is canceled. The calculation will be done in the 5 dimensional standard model and the SU (5) grand unified theory, whose 5th dimensional coordinate is compactified on S 1 /Z 2 . The broken gauge theories through the orbifolding preserve the unitarity at high energies similarly to the broken gauge theories where the gauge bosons obtain their masses through the Higgs mechanism. We show that the contributions of the Kaluza-Klein states play a crucial role in conserving the unitarity. * There are some discussions about unitarity in these theories in Ref. [8].

We introduce the flavor symmetry M × N ×D 4 into the SU(6)×SU(2) R string-inspired model. The cyclic group M and the dihedral group D 4 are R symmetries, while N is a non-R symmetry. By imposing the anomaly-free conditions on the model, we obtain a viable solution under many phenomenological constraints coming from the particle spectra. For the neutrino sector, we find a LMA-MSW solution but no SMA-MSW solution. The solution includes phenomenologically acceptable results concerning fermion masses and mixings and also concerning hierarchical energy scales including the GUT scale, the µ scale and the Majorana mass scale of R-handed neutrinos. * )

A generic unital positive operator-valued measure (POVM), which transforms a given stationary pure state to an arbitrary statistical state with perfect decoherence, is presented. This allows one to operationally realize thermalization as a special case. The loss of information due to randomness generated by the operation is discussed by evaluating the entropy. Thermalization of the bipartite spin-1/2 system is discussed as an illustrative example.PACS number(s): 05.30.Ch

The Casimir effect is a physical manifestation of zero point energy of quantum vacuum. In a relativistic quantum field theory, Poincaré symmetry of the theory seems, at first sight, to imply that non-zero vacuum energy is inconsistent with translational invariance of the vacuum. In the setting of two uniform boundary plates at rest, quantum fields outside the plates have (1+2)-dimensional Poincaré symmetry. Taking a massless scalar field as an example, we have examined the consistency between the Poincaré symmetry and the existence of the vacuum energy. We note that, in quantum theory, symmetries are represented projectively in general and show that the Casimir energy is connected to central charges appearing in the algebra of generators in the projective representations. * Since the pioneering work of Casimir [1], vacuum energy of quantum fields has been the subject of intense investigations from both experimental and theoretical sides [2,3,4,5]. Experimental measurements of the Casimir forces, by using an atomic force microscope or micro-electromechanical system, reach the high precision at the level within 1% and agreement with the theoretical prediction is also at the same level at least for zero temperature. Theoretical investigation of the Casimir effects extends a variety of fields of physics such as particle physics, atomic physics, astrophysics and cosmology, and condensed matter physics. In particle physics, for example, the Casimir energy of quark and gluon fields inside a hadron makes essential contributions to its mass. The Casimir force offers one of the effective mechanisms for spontaneous compactification of extra spatial dimensions in the Kaluza-Klein theories.This paper discuss more theoretical issue, i.e. we examine the consistency between the existence of the Casimir energy and the Poincaré symmetry in the setting of two uniform perfectly-reflecting parallel boundary planes at rest. In this configuration, the quantum field theory is invariant under the time-translation, the translations and boosts along the plane, and under the rotation in the plane. As a result of these invariances of the theory, it seems that, if require the translational invariance of the vacuum (vanishing total momentum of the field), then the vacuum energy should vanish. This argument has a loophole as expected. We pay our attention to the representation of symmetries in quantum theory and to the fact that, in order to compare the zero point energies, we have to consider time-dependent Hamiltonian connecting the different static configurations.The paper is organize as follows. In Sec. 2, we set up the problem in the example of a massless scalar field. Sec. 3 summarizes the projective representation and linear representations of symmetries in quantum theory. In Sec. 4, an adiabatic process connecting two static configuration is analyzed. The final section is devoted to the conclusion.

We investigate high-energy behavior of scattering amplitudes in extra dimensional gauge theory where the gauge symmetry is broken by the boundary conditions. We study, in particular, the 5D SU(5) grand unified theory whose 5th-dimensional coordinate is compactified on S 1 /Z 2 . We pay attention to the gauge symmetry compatible with the boundary conditions on an orbifold and present the BRST formalism of the 4D theory that is obtained through integration of the 5D theory along the extra dimension. We derive the 4D equivalence theorem on the basis of the Slavnov-Taylor identities. We also calculate the amplitudes of the process including four massive gauge bosons in the external lines and compare them with those for the connected reactions in which the gauge fields are replaced by the corresponding would-be Nambu-Goldstone (NG) fields. We explicitly confirm that the equivalence theorem holds. §1. IntroductionIt is well-known that in 4D gauge theories with explicit gauge symmetry breakings the tree-level amplitudes of four massive gauge bosons in the external lines exhibit bad high-energy behavior, ∼ O(E 4 /m 4 ) and ∼ O(E 2 /m 2 ), which breaks the unitarity at tree level. (Unitarity at the tree level requires that the tree amplitude of n external particles at high-energy increase with the energy no more rapidly than O(E 4−n ). If this bound holds, we sometimes say that 'unitarity is maintained'.) Contrastingly, when the gauge bosons obtain their masses through the Higgs mechanism, the power-law behavior ∼ O(E 4 /m 4 ) and ∼ O(E 2 /m 2 ) is canceled by the contribution of the Higgs bosons. 1)-4) The cancellation is guaranteed by the equivalence theorem which states that the amplitude of massive gauge bosons in the external lines is the same, up to some constant factor, as that of the connected reaction in which the gauge fields are replaced by the corresponding would-be Nambu-Goldstone (NG) fields. 5)-10)With the above considerations, it is natural to inquire what the situation is regarding unitarity in the extra-dimensional gauge theories in which the gauge symmetries are broken by the boundary conditions. In the higher-dimensional gauge theory, the reduction of gauge symmetry is realized through the boundary conditions of the extra-dimensional coordinate (see, for example, Ref. 11)). In this case, * )

The time-dependent oscillator describing parametric oscillation, the concept of invariant and Noether's theorem are important issues in physics education. Here, it is shown how they can be interconnected in a simple and unified manner.

No abstract