Canonical forms of tensor representations and spontaneous symmetry breaking

We introduce a Lohe group which is a new class of matrix Lie groups and present a continuous dynamical system for the synchronization of group elements in a Lohe group. The Lohe group includes classical Lie groups such as the orthogonal, unitary, and symplectic groups, and since Lohe groups need not be compact, global existence of ODEs may fail. The proposed dynamical system generalizes the Lohe model (Lohe in J Phys A 43:465301, 2010; Lohe in J Phys A 42:395101–395126, 2009) itself a nonabelian generalization of the Kuramoto model, and alongside we also generalize the analytical framework (Ha and Ryoo in J Stat Phys 163:411–439, 2016) of emergent and unique phase-locked states. For the construction of the phase-locked states, we introduce Lyapunov functions measuring the ensemble diameter and the dissimilarity between two Lohe flows, and derive Gronwall-type differential inequalities for them. The global existence of solutions then become a consequence of the boundedness of these Lyapunov functions. Our sufficient framework for the emergent dynamics is formulated in terms of coupling strength and initial states, and it leads to the global existence of solutions and the formation and uniqueness of a phase-locked asymptotic state. As a concrete example, we demonstrate how our theory can show emergent phenomenon on the Heisenberg group, where all initial configurations tend to a unique phase-locked state exponentially fast.

In the minimal supersymmetric standard madel, it is shown that a radiative correction of the top and stop loops gives a finite, but non-negligible contribution to Higgs scalar masses if m, ' 150-250 GeV. The upper limit to the lightest-scalar mass becomes 70-190 GeV in the range of heavy top quark. 1 The mechanism of electroweak symmetry breaking is one of the most important issues in the present particle physics_ In the standard electroweak model a funda­ mental Higgs doublet is introduceq to cause the spontaneous symmetry breaking. Supersymmetry (SUSY), eliminating all quadratic divergences, may provide a better theoretical basis to describe a fundamental Higgs boson with a relatively small mass to a high energy cutoff scale, say the Planck scale for example. I) In the minimal SUSY extension of the standard electroweak model the Higgs sector consists of two chiral superfields of Higgs doublets (/JHl and (/JH2 with opposite hypercharges. They are required to give masses for all quarks and leptons and to guarantee the ~bsence of the gauge anomaly. Five physical Higgs bosons among them survive the gauge symmetry breaking, namely, there appear two neutral scalars rpa and rpb, a neutral pseudoscalar X and a pair of charged scalars x± as physical particles. It has been, furthermore, shown by many authors 2 ) that there is at least one neutral scalar boson lighter than ZO(m<p< mzo) in the minimal SUSY model. This has strongly motivated many recent analyses of Higgs boson production at LEP energies. 3 ) In this paper, however, we stress that a radiative correctiQn gives a significant contribution to the Higgs mass term if the top quark is 8uficiently heavy as mt ~150-250 GeV. Therefore, the presence of the Higgs scalar lighter than mzo is not an inevitable prediction of the minimal SUSY standard model. Let us discuss the Higgs sedor in the minimal SUSY model. With general soft-breaking terms 4 ) of SUSY the tree-level Higgs potential is given by

We describe non-Abelian generalizations of the Kuramoto model for any classical compact Lie group and identify their main properties. These models may be defined on any complex network where the variable at each node is an element of the unitary group U(n), or a subgroup of U(n). The nonlinear evolution equations maintain the unitarity of all variables which therefore evolve on the compact manifold of U(n). Synchronization of trajectories with phase locking occurs as for the Kuramoto model, for values of the coupling constant larger than a critical value, and may be measured by various order and disorder parameters. Limit cycles are characterized by a frequency matrix which is independent of the node and is determined by minimizing a function which is quadratic in the variables. We perform numerical computations for n = 2, for which the SU(2) group manifold is S3, for a range of natural frequencies and all-to-all coupling, in order to confirm synchronization properties. We also describe a second generalization of the Kuramoto model which is formulated in terms of real m-vectors confined to the (m − 1)-sphere for any positive integer m, and investigate trajectories numerically for the S2 model. This model displays a variety of synchronization phenomena in which trajectories generally synchronize spatially but are not necessarily phase-locked, even for large values of the coupling constant.

We investigate the phenomenology of spontaneous symmetry breaking (SSB) and spontaneous antisymmetry breaking (SASB) in the framework of a system of coupled nonlinear Schrödinger equations(NLSEs) with a combination of linear and nonlinear double-well potentials (DWPs), including self- and cross-phase-modulation (SPM and XPM) nonlinear terms. The DWP structure is represented by a symmetric pair of delta functions. This setup can be realized in optics and for matter waves in Bose-Einstein condensates. It is commonly known that SSB states are completely stable in the single one-dimensional NLSE with a combination of DWP and attractive SPM nonlinearity. The objective of the present work is to elaborate on the SSB and SASB phenomenology in the two-component system. One factor that affects the phenomenology (actually, the stability of stationary states) in this context is the relative sign of the two components, the states with identical/opposite signs defined as ones with in-phase/out-of-phase components. We demonstrate that the SSB/SASB, with both components originating from spatially symmetric or antisymmetric modes, of the out-of-phase type, in the coupled NLSEs including the SPM and XPM terms with opposite signs, are similar to their counterparts in the single NLSE. On the other hand, the SSB/SASB of in-phase states in the coupled NLSEs, including the SPM and XPM terms with opposite signs, are quite different from the ones in the single NLSE. In particular, we find that the in-phase SSB/SASB states are unstable in a part of the intervals of their propagation constantif the system includes SPM and XPM terms with opposite signs. As another feature that is possible only in the two-component system, we address mixed-parity states, originally built of spatially symmetric and antisymmetric components, and find that such states, featuring SSB or SASB, are stable in a part of their interval of the propagation constant. We clarify that the spatially antisymmetric component is a dominant one in determining the asymmetry of the established state if it is coupled to the symmetric one, in the case of the repulsive SPM. On the other hand, the symmetric component dominates if it is coupled to the antisymmetric one in the case of the attractive SPM.

A natural embedding for most time-continuous systems is presented. A set of nonlinear transformations is shown to split the whole family of dynamical systems into equivalence classes. The degree of nonlinearity is not a relevant characteristic of these classes of ordinary differential equations (ODEs). In each class there exist two particular simple canonical forms into which any such ODE can be cast. We show on an example how to use these canonical forms to find nontrivial integrability conditions. © 1989 The American Physical Society.

We study the complex-time Segal-Bargmann transform Bs,τKN on a compact type Lie group KN, where KN is one of the following classical matrix Lie groups: the special orthogonal group SO(N,R), the special unitary group SU(N), or the compact symplectic group Sp(N). Our work complements and extends the results of Driver, Hall, and Kemp on the Segal-Bargman transform for the unitary group U(N). We provide an effective method of computing the action of the Segal-Bargmann transform on trace polynomials, which comprise a subspace of smooth functions on KN extending the polynomial functional calculus. Using these results, we show that as N→∞, the finite-dimensional transform Bs,τKN has a meaningful limit Gs,τ which can be identified as an operator on the space of complex Laurent polynomials.

Let \(G\) be a connected and simply connected Lie group with Lie algebra \(\mathfrak g \). We say that a subset \(X\) in the set \(\mathfrak g ^\star / G\) of coadjoint orbits is convex hull separable when the convex hulls differ for any pair of distinct coadjoint orbits in \(X\). In this paper, we define a class of solvable Lie groups, and we give an explicit construction of an overgroup \(G^+\) and a quadratic map \(\varphi \) sending each generic orbit in \(\mathfrak g ^\star \) to a \(G^+\)-orbit in \(\mathfrak{g ^+}^\star \), in such a manner that the set \(\varphi (\mathfrak g ^\star _{gen}){/ G^+}\) is convex hull separable. We then call \(G^+\) a weak quadratic overgroup for \(G\). Thanks to this construction, we prove that any nilpotent Lie group, with dimension at most 7 admits such a weak quadratic overgroup. Finally, we produce different examples of solvable Lie groups, having weak quadratic overgroups, but which are not in our class of Lie groups and for which usual constructions fail to hold.

Spontaneous chiral symmetry breaking in QCD with massless quarks at infinite volume can be seen in a finite box by studying, for instance, the dependence of the chiral condensate from the volume and the quark mass. We perform a feasibility study of this program by computing the quark condensate on the lattice in the quenched approximation of QCD at small quark masses. We carry out simulations in various topological sectors of the theory at several volumes, quark masses and lattice spacings by employing fermions with an exact chiral symmetry, and we focus on observables which are infrared stable and free from mass-dependent ultraviolet divergences. The numerical calculation is carried out with an exact variance-reduction technique, which is designed to be particularly efficient when spontaneous symmetry breaking is at work in generating a few very small low-lying eigenvalues of the Dirac operator. The finite-size scaling behaviour of the condensate in the topological sectors considered agrees, within our statistical accuracy, with the expectations of the chiral effective theory. Close to the chiral limit we observe a detailed agreement with the first Leutwyler-Smilga sum rule. By comparing the mass, the volume and the topology dependence of our results with the predictions of the chiral effective theory, we extract the corresponding low-energy constant.

In the first two volumes of this book we have considered special functions related to most degenerate series of irreducible representations of classical simple Lie groups and of corresponding inhomogeneous groups. In the next chapters we study special functions related to non-degenerate series of representations. These special functions depend on many variables and in some cases it is convenient to consider them as functions of matrix argument or as functions with matrix indices. In this chapter we give necessary information from the theory of semisimple Lie groups and related homogeneous spaces. We describe also invariant measures and invariant differential operators on these groups and spaces.

Spontaneous chiral symmetry breaking is a ubiquitous property in nature and diverse fields of modern physics. However, such symmetry breaking has been elusive experimentally in the optical systems, which usually demands multiple identical subsystems [1]. As a prominent photonic device, the ultrahigh-Q whispering-gallery mode (WGM) microresonator supports clockwise (CW) and counterclockwise (CCW) propagating waves coupled to each other, leading to symmetric and antisymmetric standing-wave modes with equal CW and CCW amplitudes. The demonstrations of such overall chirality have to rely on external perturbations to a resonator, either by breaking the parity or time-reversal symmetry [2,3]. The chirality with unbalanced CW and CCW components not only attracts general interest in physics, but also is of importance in novel devices [2,3]. Here, we experimentally demonstrate the spontaneous chirality in a single WGM microresonator (Fig. 1) without any explicit breaking of parity or time-reversal symmetry.

A radiative electroweak symmetry breaking in the minimal supergravity model is discussed, with assumptions of an 0(10 TeV) gravitino and the Nambu-Goldstone Higgs bosons. The electroweak symmetry is broken if a fine tuning of initial parameters for the renormalization group equations is made. This model predicts one light (::;; 0(10 GeV)) neutral Higgs boson. 23 Supergravity theories necessarily involve a massive spin 3/2 particle, the gravitino, and the mass of a gravitino is expected to be 0(100 GeV) in many models/} in order to relate the supersymmetry (SUSY) breaking to the electroweak gauge symmetry breaking. However, the garvitino with a mass between 0(1 ke V) and 0(10 TeV) is forbidden in the standard big bang cosmology/} since such a gravitino has a lifetime longer than the cosmic time at the primordial nucleosynthesis, and disastrously alter the abundances of light elements. Although some works in antici­ pation of solving this problem (the gravitino problem) by changing the cosmological scenario (inflation, for example) exist,3} a satisfactory solution is not obtained yet. If the gravitino is assumed to be sufficiently heavy, one may expect that all sparticles and Higgs bosons are very heavy and such a model is not phenome­ nologically appealing at all. However if an assumption that Higgs scalars are (pseudo-) Nambu-Goldstone bosons of a certain broken global symmetry4} is added, one might hope the existence of some light Higgs bosons. In this paper, we analyze a possibility of the electroweak symmetry breaking under the two assumptions, namely, the gravitino mass m3/2=0(10 TeV) and the Nambu-Goldstone hypothesis. We find that the electroweak symmetry can be broken successfully by renormalization effects if a fine tuning of some parameters is allowed, and there exists indeed one light Higgs scalar while other scalar paticles, namely, squarks, sleptons and other Higgs bosons are quite heavy. The model used in our calculation is the minimal supersymmetric 5U(3) X 5U(2) X U(l) gauge model with soft and explicit SUSY breaking terms which are obtained from a spontaneous local SUSY breaking in a hidden sector as usua1. 5

We consider U(1)-symmetric scalar quantum field theories at zero temperature. At nonzero charge densities, the ground state of these systems is usually assumed to be a superfluid phase, in which the global symmetry is spontaneously broken along with Lorentz boosts and time translations. We show that, in d > 2 spacetime dimensions, this expectation is always realized at one loop for arbitrary non-derivative interactions, confirming that the physically distinct phenomena of nonzero charge density and spontaneous symmetry breaking occur simultaneously in these systems. We quantify this result by deriving universal scaling relations for the symmetry breaking scale as a function of the charge density, at low and high density. Moreover, we show that the critical value of μ above which a nonzero density develops coincides with the pole mass in the unbroken, Poincaré invariant vacuum of the theory. The same conclusions hold non-perturbatively for an O(N) theory with quartic interactions in d = 3 and 4, at leading order in the 1/N expansion. We derive these results by computing analytically the zero-temperature, finite-μ one-loop effective potential, paying special attention to subtle points related to the iε terms. We check our results against the one-loop low-energy effective action for the superfluid phonons in λϕ4 theory in d = 4 previously derived by Joyce and ourselves, which we further generalize to arbitrary potential interactions and arbitrary dimensions. As a byproduct, we find analytically the one-loop scaling dimension of the lightest charge-n operator for the λϕ6 conformal superfluid in d = 3, at leading order in 1/n, reproducing a numerical result of Badel et al. For a λϕ4 superfluid in d = 4, we also reproduce the Lee-Huang-Yang relation and compute relativistic corrections to it. Finally, we discuss possible extensions of our results beyond perturbation theory.

Spontaneous discrete symmetry breaking during inflation and the NMSSM domain wall problem

The question of whether Bose-Einstein condensation involves spontaneous symmetry breaking is surprisingly controversial. We review the theory of spontaneous symmetry breaking in ferromagnets, compare it with the theory of symmetry breaking in condensates, and discuss the different viewpoints on the correspondence to experiments. These viewpoints include alternative perspectives in which we can treat condensates with fixed particle numbers, and where coherence arises from measurements. This question relates to whether condensates of quasiparticles such as polaritons can be viewed as “real” condensates. Introduction Spontaneous symmetry breaking is a deep subject in physics with long historical roots. At the most basic level, it arises in the field of cosmology. Physicists have long had an aesthetic principle that leads us to expect symmetry in the all of the basic equations of physical law. Yet the universe is manifestly full of asymmetries. How does a symmetric system acquire asymmetry merely by evolving in time? Starting in the 1950s, cosmologists began to borrow the ideas of spontaneous symmetry breaking from condensed matter physics, which were originally developed to explain spontaneous magnetization in ferromagnetic systems. Spontaneous coherence in all its forms (e.g., Bose-Einstein condensation [BEC], superconductivity, and lasing) can be viewed as another type of symmetry breaking. The Hamiltonian of the system is symmetric, yet under some conditions, the energy of the system can be reduced by putting the system into a state with asymmetry, namely, a state with a common phase for a macroscopic number of particles. The symmetry of the system implies that it does not matter what the exact choice of that phase is, as long as it is the same for all the particles. It is not obvious, however, whether the symmetry breaking which occurs in spontaneous coherence of the type seen in lasers or in Bose-Einstein condensation is the same as that seen in ferromagnetic systems. There are similarities in the systems which encourage the same view of all types of symmetry breaking, but there are also differences. In fact, there is a substantial school of thought that symmetry breaking in Bose-Einstein condensates with ultracold atoms is a “convenient fiction” (a term applied to optical coherence by Molmer [1]).

In this paper we prove isomorphisms between 5 Lie groups (of arbitrary dimension and fixed signatures) in Clifford algebra and classical matrix Lie groups - symplectic, orthogonal and linear groups. Also we obtain isomorphisms of corresponding Lie algebras.