Narrow Results

This paper gives a detailed review on symmetry and symmetry breaking of quasicrystals. The symmetry groups of solid and soft-matter quasicrystals observed so far are summarized and the symmetry breakings of quasicrystals are systematically discussed. The present review connects several branches of mathematics and physics and related applications in science and technology are also discussed.

In 2021, Li and Yan considered a more general and improved anisotropic version of the Caffarelli–Kohn–Nirenberg celebrated interpolation inequalities

In this paper, we study the Li–Yan inequality with $\delta =1$, $p=2$, and answer some fundamental questions concerning these inequalities such as the existence and symmetry breaking region of extremal functions, and their symmetry properties.

We study a variant of the ferromagnetic Potts model, recently introduced by Tamura, Tanaka and Kawashima, consisting of a ferromagnetic interaction among q "visible" colors along with the presence of r non-interacting "invisible" colors. We introduce a random-cluster representation for the model, for which we prove the existence of a first-order transition for any q > 0, as long as r is large enough. When q > 1, the low-temperature regime displays a q-fold symmetry breaking. The proof involves a Pirogov–Sinai analysis applied to this random-cluster representation of the model.

This paper is devoted to a collection of results on nonlinear interpolation inequalities associated with Schrödinger operators involving Aharonov–Bohm magnetic potentials, and to some consequences. As symmetry plays an important role for establishing optimality results, we shall consider various cases corresponding to a circle, a two-dimensional sphere or a two-dimensional torus, and also the Euclidean spaces of dimensions 2 and 3. Most of the results are new and we put the emphasis on the methods, as very little is known on symmetry, rigidity and optimality in the presence of a magnetic field. The most spectacular applications are new magnetic Hardy inequalities in dimensions 2 and 3.

We consider Yang–Mills theory with a compact structure group $G$ on a Lorentzian 4-manifold $M=ℝ\times \mathrm{\Sigma }$ such that gauge transformations become identity on a submanifold $S$ of $\mathrm{\Sigma }$ (framing over $S\subset \mathrm{\Sigma }$). The space $S$ is not necessarily a boundary of $\mathrm{\Sigma }$ and can have dimension $k\le 3$. Framing of gauge bundles over $S\subset \mathrm{\Sigma }$ demands introduction of a $G$-valued function ${\varphi }_S$ with support on $S$ and modification of Yang–Mills equations along $ℝ\times S\subset M$. The fields ${\varphi }_S$ parametrize non-equivalent flat connections mapped into each other by a dynamical group ${{𝒢}}_S$ changing gauge frames over $S$. It is shown that the charged condensate ${\varphi }_S$ is the Stueckelberg field generating an effective mass of gluons in the domain $S$ of space $\mathrm{\Sigma }$ and keeping them massless outside $S$. We argue that the local Stueckelberg field ${\varphi }_S$ can be responsible for color confinement. We also briefly discuss local breaking of symmetries in gravity. It is shown that framing of the tangent bundle over a subspace of spacetime makes gravitons massive in this subspace.

We study holographic operators associated with Rankin–Cohen brackets which are symmetry breaking operators for the restriction of tensor products of holomorphic discrete series of the universal covering of $S{L}_2(ℝ)$. Furthermore, we investigate a geometrical interpretation of these operators and their relations to classical Jacobi polynomials.

In this paper, a numerical study of nonlinear flow phenomena in two-dimensional symmetric channels using the lattice-Boltzmann equation method is presented. The results are compared with both experimental results and other numerical results using some traditional methods. Comparisons are found to be quantitatively accurate.

Spatially extended systems yield complex patterns arising from the coupled dynamics of its different regions. In this paper we introduce a matrix computational operator, , for the characterization of asymmetric amplitude fragmentation in extended systems. For a given matrix of amplitudes this operation results in an asymmetric-triangulation field composed by L points and I straight lines. The parameter (I-L)/L is a new quantitative measure of the local complexity defined in terms of the asymmetry in the gradient field of the amplitudes. This asymmetric fragmentation parameter is a measure of the degree of structural complexity and characterizes the localized regions of a spatially extended system and symmetry breaking along the evolution of the system. For the case of a random field, in the real domain, which has total asymmetry, this asymmetric fragmentation parameter is expected to have the highest value and this is used to normalize the values for the other cases. Here, we present a detailed description of the operator and some of the fundamental conjectures that arises from its application in spatio-temporal asymmetric patterns.

We conjecture that the extra dimensions are physical noncompact at high energy scale or high temperature; after the symmetry breaking or cosmological phase transition, the bulk cosmological constant may become negative, and then, the extra dimensions may become physical compact at low energy scale. We show this in a five-dimensional toy brane model with three parallel three-branes and a real bulk scalar whose potential is temperature-dependent. We also point out that after the global or gauge symmetry breaking, or the supersymmetry breaking in supergravity theory, the spontaneous physical compactification of the extra dimensions might be realized.

We study the spontaneous symmetry breaking (SSB) induced by a scalar field and its non-minimal interaction with gravity in the space–time of an arbitrary dimension (D > 2), where the gravitational field is treated as a dynamical field. We explore mainly the possibility of implementing SSB after introducing the non-minimal coupling with such dynamical gravitational field.

It is shown that because of axial-vector coupling and vector coupling between boson and fermions of different masses, the boson field has four independent components. The spin-0 component has problem of negative probability. The unitarity of the theory is broken. These results are applied to the SM. The masses are m_ϕ⁰ = 3.78×10¹⁴GeV and m_ϕ^± = 9.31×10¹³GeV respectively. The unitarity of the SM is broken at about ~ 10¹⁴GeV. The masses of W and Z can be generated by two types of interactions mentioned above without Higgs mechanism. The masses of the W and the Z bosons are obtained to be and with ρ≃1. .

Based on possible, albeit tiny, admixtures of mirror matter in ordinary mesons we study the K_L→γγ transition. We find that this process can be described with a small SU(3) symmetry breaking of only 3%. We also determine the η–η' mixing angle and the pseudoscalar decay constants. The results for these parameters are consistent with those obtained in the literature. They favor three recent determinations; the first is based on two analytical constraints, the second one based on next-to-leading order power corrections, and the third one on a phenomenological determination of the mixing parameters in the framework of the quark-flavor mixing scheme.

Simultaneous nonlinear realizations of spontaneously broken supersymmetry in conjunction with other spontaneous and/or explicitly broken symmetries including R symmetry, global chiral symmetry, dilatations and the superconformal symmetries is reviewed.

The energy density of the universe today may be dominated by the vacuum energy of a slowly rolling scalar field. Making a quantum expansion around such a time-dependent solution breaks fundamental symmetries of quantum field theory. We call this mechanism cosmological symmetry breaking and argue that it is different from the standard phenomenon of spontaneous symmetry breaking. We illustrate this with a toy scalar field theory, whose action displays a U(1) symmetry. We identify a symmetry, called pseudo-scale invariance, which sets the cosmological constant exactly equal to zero, both in classical and quantum theories. This symmetry is also broken cosmologically and leads to a nonzero vacuum or dark energy. The slow-roll condition along with the observed value of dark energy leads to a value of the background scalar field of the order of Planck mass. We also consider a U(1) gauge symmetry model. Cosmological symmetry breaking, in this case, leads to a nonzero mass for the vector field. We also show that a cosmologically broken pseudo-scale invariance can generate a wide range of masses.

We consider a generalization of Einstein's general theory of relativity such that it respects local scale invariance. This requires the introduction of a scalar and a vector field in the action. We show that the theory naturally displays both dark energy and dark matter. We solve the resulting equations of motion assuming an FRW metric. The solutions are found to be almost identical to those corresponding to the standard ΛCDM model.

The low-energy spectrum of a one-component, spontaneously broken Φ⁴ theory is generally believed to have the same simple massive form as in the symmetric phase where 〈Φ〉 = 0. However, in lattice simulations of the 4D Ising limit of the theory, the two-point connected correlator and the connected scalar propagator show deviations from a standard massive behavior that do not exist in the symmetric phase. As a support for this observed discrepancy, we present a variational, analytic calculation of the energy spectrum E₁(p) in the broken phase. This analytic result, while providing the trend at large |p|, gives an energy gap E₁(0)< m_h, even when approaching the infinite-cutoff limit Λ→∞ with that infinitesimal coupling λ ~ 1/ln Λ suggested by the standard interpretation of "triviality" within leading-order perturbation theory. We also compare with other approaches and discuss the more general implications of the result.

Most attractive channel is studied for SU(5) grand unified theory in five dimensions where 5* and 10 fermions and a 24 gauge boson propagate in the bulk. If there are bulk fermions with zero mode for the chirality opposite to quark and lepton multiplets, they can contribute to forming 5, 24, 50 and 75 scalar bound states. We find that the inclusion of a simple anomaly-free set for zero mode yields scalar bound states whose binding strengths are the largest for 24 and the second largest for 5. This corresponds to the breaking pattern of the SU(5) to color and electromagnetism via the standard model gauge group.

This paper was originally created for and printed in the "Proceedings of seminar on unified theories of elementary particles" held in Feldafing, Germany from July 5 to 16, 1965 under the auspices of the Max-Planck-Institute for Physics and Astrophysics in Munich. It details and expands upon the 1964 Guralnik, Hagen, and Kibble paper demonstrating that the Goldstone theorem does not require physical zero mass particles in gauge theories.

Lack of any baryon number in the eightfold way model, and its intrinsic presence in the SU(3)-flavor model, has been a puzzle since the genesis of these models in 1961–1964. First we show that the conventional popular understanding of this puzzle is actually fundamentally wrong, and hence the problem being so old, begs urgently for resolution. In this paper we show that the issue is linked to the way that the adjoint representation is defined mathematically for a Lie algebra, and how it manifests itself as a physical representation. This forces us to distinguish between the global and the local charges and between the microscopic and the macroscopic models. As a bonus, a consistent understanding of the hitherto mysterious medium–strong interaction is achieved. We also gain a new perspective on how confinement arises in quantum chromodynamics.

One kind of spontaneous (2 + 1)-dimensional Lorentz symmetry breaking is discussed. The symmetry breaking pattern is SO(2, 1) $\to$ SO(1, 1). Using the coset construction formalism, we derive the Goldstone covariant derivative and the associated covariant gauge field. Finally, the two-derivative low-energy effective action of the Nambu–Goldstone bosons is obtained.