Eightfold Way Research Articles

Symmetries in particle physics: from nuclear isospin to the quark model

We present a concise pedagogic introduction to group representation theory motivated by the historical developments surrounding the advent of the Eightfold Way. definitions of groups and representations are avoided in favour of the physical intuition of symmetries of the nuclear interaction. The concept of nuclear isospin is used as a physical motivation to introduce SU(2) and discuss the main techniques of representation theory. The discovery of strange particles motivates extending the symmetry group to SU(3), at first in the context of the Sakata model. We highlight the successes in fitting mesons in the SU(3) octet, discuss the drawbacks of the Sakata model for baryonic classifications, and how the Eightfold Way finally led to the quark model. This approach has two major advantages: (i) the main concepts of the theory of Lie groups are introduced and discussed without ever losing touch with its applications in particle physics; (ii) it allows the beginner to study group theory while also becoming acquainted with the historical developments of particle physics that led to the concept of quarks. In particular, in this pedagogical path the quarks appear as yet another class of particles predicted from symmetry principles, rather than being introduced ad hoc for postulating an SU(3) symmetry, as usually done in the literature.

A Glimpse in to the World of Quarks

Quarks are fundamental variety of matter, like leptons. All hadrons (baryons and mesons) are made up of these fundamental units, quarks. The Mendeleev’s periodic table of atomic elements, in 1869, pointed towards a fundamental structure of matter common to all atoms. Nearly 50 years later, the discovery that atoms consisted of electrons revolving a nucleus confirmed this. Similarly, the observation of a periodic pattern among the existing hadrons (called as the eightfold way pattern), during 1960, pointed the possibility of a more fundamental variety of matter-quarks- out of which all hadrons, including protons and neutrons , are formed. With the discovery of the 6th and the last quark –the top quark- in 1995, at Fermi Lab, confirmation about the existence of the all the quarks was proved.

Binary Encoded Recursive Generation of Quantum Space-Times

Real geometric algebras distinguish between space and time; complex ones do not. Space-times can be classified in terms of number n of dimensions and metric signature s (number of spatial dimensions minus number of temporal dimensions). Real geometric algebras are periodic in s, but recursive in n. Recursion starts from the basis vectors of either the Euclidean plane or the Minkowskian plane. Although the two planes have different geometries, they have the same real geometric algebra. The direct product of the two planes yields Hestenes’ space-time algebra. Dimensions can be either open (for space-time) or closed (for the electroweak force). Their product yields the eight-fold way of the strong force. After eight dimensions, the pattern of real geometric algebras repeats. This yields a spontaneously expanding space-time lattice with the physics of the Standard Model at each node. Physics being the same at each node implies conservation laws by Noether’s theorem. Conservation laws are not pre-existent; rather, they are consequences of the uniformity of space-time, whose uniformity is a consequence of its recursive generation.

Eightfold way to dark states in SU(3) cold gases with two-body losses

We study the quantum dynamics of a one-dimensional $\mathrm{SU}(3$)-symmetric system of cold atoms in the presence of two-body losses. We exploit the representation theory of $\mathrm{SU}(3$), the so-called eightfold way, as a scheme to organize the dark states of the dissipative dynamics in terms of generalized Dicke states and show how they are dynamically approached, both in the weakly and strongly interacting and dissipative regimes. Our results are relevant for a wide class of alkaline-earth(-like) gas experiments, paving the way to the dissipative preparation and exploitation of generalized Dicke states.

The eightfold way of teaching psychological type.

For Jung, the discovery of type differences represented the beginning of his individuation journey, and his experiential discovery can model the path of learning about psychological type. A course on psychological type can enable individuals to discover empirically Jung's revelations about the psyche if it is structured like the individuation journey itself: beginning with differentiation of the preferred functions, proceeding to the inferior function at the midpoint of the journey, and concluding with an exploration of the unconscious functions. Such a pedagogical method mandates a non-prescriptive approach that empowers students to lead their own learning, an approach expressed by the Taoist concept of wu wei. Psychological Types references wu wei as a means of balancing consciousness and the unconscious. Sometimes Jung's typology is thought to deal only with consciousness, but in fact it represented Jung's own eightfold way into the unconscious. To illustrate the dynamic interaction of conscious and unconscious functions, John Beebe's eight-function eight-archetype model is an indispensable tool, because it illuminates the unconscious opposites inherent in each type and connects them with archetypal complexes. With the aid of Beebe's model, students can plot a typological trajectory of development for their journey to the Self.

From Periods to Anabelian Geometry and Quantum Amplitudes

Periods are algebraic integrals, extending the class of algebraic numbers, and playing a central, dual role in modern Mathematical-Physics: scattering amplitudes and coefficients of de Rham isomorphism. The Theory of Periods in Mathematics, with their appearance as scattering amplitudes in Physics, is discussed in connection with the Theory of Motives, which in turn is related to Conformal Field Theory (CFT) and Topological Quantum Field Theory (TQFT), on the physics side. There are three main contributions. First, building a bridge between the Theory of Algebraic Numbers and Theory of Periods, will help guide the developments of the later. This suggests a relation between the Betti-de Rham theory of periods and Grothendieck’s Anabelian Geometry, towards perhaps an algebraic analog of Hurwitz Theorem, relating the algebraic de Rham cohomology and algebraic fundamental group, both pioneered by A. Grothendieck. Second, a homotopy-homology refinement of the Theory of Periods will help explain the connections with quantum amplitudes. The novel approach of Yves Andre to Motives via representations of categories of diagrams, relates from a physical point of view to generalized TQFTs. Finally, the known “universality” of Galois Theory, as how symmetries “grow”, controlling the structure of the objects of study, is discussed, in relation to the above several areas of research, together with ensuing further insight into the Mathematical-Physics symbiosis. To better understand and investigate Kontsevich-Zagier conjecture on abstract periods, the article ponders on the case of algebraic Riemann Surfaces representable by Belyi maps. Reformulation of cohomology of cyclic groups as a discrete analog of de Rham cohomology and the Arithmetic Galois Theory will provide a purely algebraic toy-model of the said algebraic homology/homotopy group theory of Grothendieck as part of Anabelian Geometry. The corresponding Platonic Trinity 5,7,11/TOI/E678 leads to connections with ADE-correspondence, and beyond, e.g. Theory of Everything (TOE) and ADEX-Theory. In perspective of the “Ultimate Physics Theory”, quantizing “everything”, i.e. cyclotomic quantum phase and finite Platonic-Hurwitz geometry of qubit frames as baryons, could perhaps be “The Eightfold (Petrie polygon) Way” to finally understand what quark flavors and fermion generations really are.

Hidden SU(2) symmetries, the symmetry hierarchy, and the emergent eightfold way in spin-1 quantum magnets

The largest allowed symmetry in a spin-1 quantum system is an $SU(3)$ symmetry rather than the $SO(3)$ spin rotation. In this work, we reveal some $SU(2)$ symmetries as subgroups of $SU(3)$ that, to the best of our knowledge, have not previously been recognized. Then, we construct $SU(2)$ symmetric Hamiltonians and explore the ground-state phase diagram in accordance with the $SU(3)\supset SU(2)\times U(1)$ symmetry hierarchy. It is natural to treat the eight generators of the $SU(3)$ symmetry on an equal footing; this approach is called the eight-fold way. We find that the spin spectral functions and spin quadrupole spectral functions share the same structure, provided that the elementary excitations are flavor waves at low energies, which serves as a clue to the eight-fold way. An emergent $S=1/2$ quantum spin liquid is proposed to coexist with gapful spin nematic order in one of the ground states. In analogy to quantum chromodynamics, we find the gap relation for hydrodynamic modes in quantum spin-orbital liquid states, which is nothing but the Gell-Mann-Okubo formula.

A Schematic Model of Baryons and Mesons

If we assume that the strong interactions of baryons and mesons are correctly described in terms of the broken "eightfold way", we are tempted to look for some fundamental explanation of the situation. A highly promised approach is the purely dynamical "bootstrap" model for all the strongly interacting particles within which one may try to derive isotopic spin and strangeness conservation and broken eightfold symmetry from self-consistency alone. Of course, with only strong interactions, the orientation of the asymmetry in the unitary space cannot be specified; one hopes that in some way the selection of specific components of the F-spin by electromagnetism and the weak interactions determines the choice of isotopic spin and hypercharge directions.

Mindfulness and Self-deliverance to Pure Presence

In the contemporary (Western) literature on mindfulness many authors present it as only a mental practice, which may bring one to a more successful and effective working of the mind, as well as different kinds of mental concentration. However, at least in part of Buddhist literature mindfulness is taken as an inseparable part of the Eightfold Way, and not as a means to achieve a separate mental aim. Another important emphasis of Mahayana Buddhism is that mindfulness does not aim at something new, but instead leads our awareness towards a deeper origin, which has already been present with us. While the initial form of mindfulness clings to various methods and achievements, the higher form lies bare in the present moment, always ready to reveal itself. When we are ready to let go of all that we achieve and do, we can surrender our being to the here and now.

One-baryon spectrum and analytical properties of one-baryon dispersion curves in 3 + 1 dimensional strongly coupled lattice QCD with three flavors

Considering a 3 + 1 dimensional lattice quantum chromodynamics (QCD) model defined with the improved Wilson action, three flavors, and 4 × 4 Dirac spin matrices, in the strong coupling regime, we reanalyze the question of the existence of the eightfold way baryons and complete our previous work where the existence of isospin octet baryons was rigorously solved. Here, we show the existence of isospin decuplet baryons which are associated with isolated dispersion curves in the subspace of the underlying quantum mechanical Hilbert space with vectors constructed with an odd number of fermion and antifermion basic quark and antiquark fields. Moreover, smoothness properties for these curves are obtained. The present work deals with a case for which the traditional method to solve the implicit equation for the dispersion curves, based on the use of the analytic implicit function theorem, cannot be applied. We do not have only one but two solutions for each one-baryon decuplet sector with fixed spin third component. Instead, we apply the Weierstrass preparation theorem, which also provides a general method for the general degenerate case. This work is completed by analyzing a spectral representation for the two-baryon correlations and providing the leading behaviors of the field strength normalization and the mass of the spectral contributions with more than one-particle. These are needed results for a rigorous analysis of the two-baryon and meson-baryon particle spectra.

Adiabatic hydrodynamics: the eightfold way to dissipation

We provide a complete solution to hydrodynamic transport at all orders in the gradient expansion compatible with the second law constraint. The key new ingredient we introduce is the notion of adiabaticity, which allows us to take hydrodynamics off-shell. Adiabatic fluids are such that off-shell dynamics of the fluid compensates for entropy production. The space of adiabatic fluids is quite rich, and admits a decomposition into seven distinct classes. Together with the dissipative class this establishes the eightfold way of hydrodynamic transport. Furthermore, recent results guarantee that dissipative terms beyond leading order in the gradient expansion are agnostic of the second law. While this completes a transport taxonomy, we go on to argue for a new symmetry principle, an Abelian gauge invariance that guarantees adiabaticity in hydrodynamics. We suggest that this symmetry is the macroscopic manifestation of the microscopic KMS invariance. We demonstrate its utility by explicitly constructing effective actions for adiabatic transport. The theory of adiabatic fluids, we speculate, provides a useful starting point for a new framework to describe non-equilibrium dynamics, wherein dissipative effects arise by Higgsing the Abelian symmetry.

The eightfold way model, the SU(3)-flavor model and the medium–strong interaction

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.

The eight-fold way for optical quasicrystals

In a recent paper we proposed a means to realize a quasicrystal with eight-fold symmetry by trapping particles in an optical potential created by four lasers. The quasicrystals obtained in this way, which are closely related to the well-known octagonal tiling, offer unique possibilities to study the effects of quasiperiodicity on physical properties. This method allows, furthermore, to transform the structures, to inflate or deflate them, include interactions or disorder and thus realize a large variety of theoretical models, both classical and quantum. In this paper we present details of the model, derive a number of interesting geometrical properties of the optical quasicrystals, as well as some results obtained by numerical calculations.

Another Word on “God and the Twenty-First Century”

Another Word on “God and the Twenty-First Century”

ON THE ABSENCE OF PENTAQUARK STATES FROM DYNAMICS IN STRONGLY COUPLED LATTICE QCD

We consider an imaginary time functional integral formulation of a two-flavor, 3+1 lattice QCD model with Wilson's action and in the strong coupling regime (with a small hopping parameter, κ > 0, and a much smaller plaquette coupling, [Formula: see text], so that the quarks and glueballs are heavy). The model has local SU (3)c gauge and global SU (2)f flavor symmetries, and incorporates the corresponding part of the eightfold way particles: baryons (mesons) of asymptotic mass ≈-3 ln κ(≈-2 ln κ). We search for pentaquark states as meson–baryon bound states in the energy–momentum spectrum of the model, using a lattice Bethe–Salpeter equation. This equation is solved within a ladder approximation, given by the lowest nonvanishing order in κ and β of the Bethe–Salpeter kernel. It includes order κ2 contributions with a [Formula: see text] exchange potential together with a contribution that is a local-in-space, energy-dependent potential. The attractive or repulsive nature of the exchange interaction depends on the spin of the meson–baryon states. The Bethe–Salpeter equation presents integrable singularities, forcing the couplings to be above a threshold value for the meson and the baryon to bind in a pentaquark. We analyzed all the total isospin sectors, I = 1/2, 3/2, 5/2, for the system. For all I, the net attraction resulting from the two sources of interaction is not strong enough for the meson and the baryon to bind. Thus, within our approximation, these pentaquark states are not present up to near the free meson–baryon energy threshold of ≈-5 ln κ. This result is to be contrasted with the spinless case for which our method detects meson–baryon bound states, as well as for Yukawa effective baryon and meson field models. A physical interpretation of our results emerges from an approximate correspondence between meson–baryon bound states and negative energy states of a one-particle lattice Schrödinger Hamiltonian.

Interview with Murray Gell-Mann

An interview in two sessions, July 1997, with Murray Gell-Mann, Robert Andrews Millikan Professor of Theoretical Physics, emeritus. Dr. Gell-Mann was on the faculty of Caltech’s Division of Physics, Mathematics, and Astronomy from 1955 until 1993. In this anecdotal interview tracing his career to 1960, he begins by recalling his Manhattan childhood during the Depression, family background, early education at Columbia Grammar School. Discusses his undergraduate years at Yale, graduate work at MIT with Victor Weisskopf, courses at Harvard with Norman Ramsey and Julian Schwinger—followed in 1951 by two terms at Institute for Advanced Study, working with Francis Low on a problem in quantum field theory. Summer 1951, University of Illinois, works on complex systems with Keith Brueckner; interaction with John von Neumann. Joins University of Chicago’s Institute for Nuclear Studies, headed by Enrico Fermi; recalls such colleagues as M. L. Goldberger, Leo Szilard, Harold Urey, Gerald Wasserburg; works on dispersion relations and pseudoscalar meson theory with Goldberger. At University of Illinois, summer 1953, works with Low on elementary-particle field theory, invents the renormalization group; comments on later contributions of Petermann & Stueckelberg, his student Kenneth Wilson. His early work at Caltech on what was later called S-matrix theory; comments on contribution to superstring theory. Meets future wife, Margaret Dow; travels in Scotland with her, 1954; their marriage. Recruited to Caltech by R. P. Feynman; life in Pasadena; visits Bohr Institute, Copenhagen, summer 1955; Spain, France, and the U.K. Back at Caltech fall 1956, teaches quantum mechanics course. Recollections of Robert and Kitty Oppenheimer, Stewart Harrison. Comments on undergraduate education at Caltech and vain efforts to promote behavioral and social sciences there. Work at RAND, 1956; paper with Brueckner; objections by Brueckner and Tatsuro Sawada; contributions of Bill Karzas, Don DuBois, Jeffrey Goldstone. Annual Review of Nuclear Science article on “last stand of the universal Fermi Interaction” with Arthur Rosenfeld; related work by Marshak & Sudarshan; Feynman’s approach; their collaboration; later work by Yang & Lee. Comments on origins of the Eightfold Way. Preoccupation with symmetry, supermultiplets, weak and strong interactions, Yang-Mills theory. Collaboration with Maurice Levy et al., in France, 1959, on the axial vector current in beta decay.

EXACT DYNAMICAL EIGHTFOLD WAY BARYON SPECTRUM AND CONFINEMENT IN STRONGLY COUPLED LATTICE QCD

We obtain from the quark–gluon dynamics the eightfold way baryon spectrum exactly in an imaginary time functional integral formulation of 3+1 lattice QCD with Wilson's action in the strong coupling regime (small hopping parameter 0 < κ ≪ 1 and much smaller plaquette coupling [Formula: see text]). The model has SU(3)c local gauge and global SU(3)f flavor symmetries. A decoupling of the hyperplane method naturally unveils the form of the baryon composite fields. In the subspace of the physical Hilbert space of vectors with an odd number of quarks, the baryons are associated with isolated dispersion curves in the energy–momentum spectrum. Spectral representations are derived for the two-baryon correlations, which allow us to detect the energy–momentum spectrum and particles as complex momentum space singularities. The spin 1/2 octet and spin 3/2 decuplet baryons have asymptotic mass -3ln κ and for each baryon there is an antibaryon with identical spectral properties. An auxiliary function method is used to obtain convergent expansions for the masses after subtracting the singular part -3ln κ. The nonsingular part of the mass is analytic in κ and β, i.e. the expansions are controlled to all orders. For β = 0, all the masses have the form M = -3ln κ - 3κ3/4 + κ6r(κ), with r(κ) real analytic. Although we have no Lorentz symmetry in our lattice model, we show that there is a partial restoration of the continuous rotational symmetry at zero spatial momentum, which implies that for all members of the octet (decuplet) r(κ) is the same. So, there is no mass splitting within the octet and within the decuplet. However, there is an octet–decuplet mass difference of [Formula: see text] at β = 0; the splitting persists for β ≠ 0. We also obtain the (anti)baryon dispersion curves which admit the representation [Formula: see text], where [Formula: see text] and [Formula: see text] is of [Formula: see text]. For the octet, [Formula: see text] is jointly analytic in κ and in each pj, for small [Formula: see text]. A new local symmetry, which we call spin flip, is used to establish constraints for the matrix-valued two-baryon correlation and show that all the octet dispersion curves are the same and that the four decuplet dispersion curves are pairwise-identical and depend only on the modulus of the spin z-component. Using a correlation subtraction method we show that the spectrum generated by the baryon and antibaryon fields is the only spectrum, in the odd quark subspace of physical states, up to near the baryon–meson threshold of ≈ -5ln κ. Combining this result with a similar result for the mesons, with mass ≈ -2ln κ, shows that the only spectrum in the entire space of states, up to near the two-meson threshold of ≈ -4ln κ, is generated by the eightfold way hadrons. Hence, for 0 < κ ≪ β ≪ 1, we have shown confinement up to near this threshold.

Ontic Structuralism and the Symmetries of Particle Physics

According to structural realism, in mature science there is structural continuity along theoretical change. A major counterexample to this thesis is the transition from the Eightfold Way to the Standard Model in particle physics. Nevertheless, the notion of structure is significantly important in comprehending the theoretical picture of particle physics, where particles change and undergo transmutations, while the only thing which remains unchanged is the basic structure, i.e. the symmetry group which controls the transmutations. This kind of view agrees with the paradigmatic case where the structure is an internal symmetry and the instantiations are the elementary particles. The metaphysical view which reflects this situation is a version of ontic structuralism.

Exact Mesonic Eightfold Way From Dynamics and Confinement in Strongly Coupled Lattice QCD

We review our results on the exact determination of the mesonic eightfold way from first principles, directly from the quark-gluon dynamics. For this, we consider an imaginary-time functional integral formulation of 3 + 1 dimensional lattice QCD with Wilson action, three flavors, SU(3)f flavor symmetry and SU(3)c local gauge symmetry. We work in the strong coupling regime: a small hopping parameter κ > 0 and a much smaller plaquette coupling β > 0 . By establishing a Feynman-Kac formula and a spectral representation to the two-meson correlation, we provide a rigorous connection between this correlation and the one-meson energy-momentum spectrum. The particle states can be labeled by the usual SU(3)f quantum numbers of total isospin I and its third-component I3, the quadratic Casimir C2 and, by a partial restoration of the continuous rotational symmetry on the lattice, as well as by the total spin J and its z–component Jz. We show that, up to near the two-meson energy threshold of ≈ − 4 ln κ , the spectrum in the meson sector is given only by isolated dispersion curves of the eightfold way mesons. The mesons have all asymptotic mass of − 2 ln κ and, by deriving convergent expansions for the masses both in κ and β, we also show a κ 4 mass splitting between the J = 0 , 1 states. The splitting persists for β ≠ 0 . Our approach employs the decoupling of hyperplane method to uncover the basic excitations, complex analysis to determine the dispersion curves and a correlation subtraction method to show the curves are isolated. Using the latter and recalling our similar results for baryons, we also show confinement up to near the two-meson threshold.

Mesonic eightfold way from dynamics and confinement in strongly coupled lattice quantum chromodynamics

We show the existence of all the 36 eightfold way mesons and determine their masses and dispersion curves exactly, from dynamical first principles such as directly from the quark-fluon dynamics. We also give a proof of confinement below the two-meson energy threshold. For this purpose, we consider an imaginary time functional integral representation of a 3+1 dimensional lattice QCD model with Wilson action, SU(3)f global and SU(3)c local symmetries. We work in the strong coupling regime, such that the hopping parameter κ>0 is small and much larger than the plaquette coupling β>1/g02⩾0 (β⪡κ⪡1). In the quantum mechanical physical Hilbert space H, a Feynman-Kac type representation for the two-meson correlation and its spectral representation are used to establish an exact rigorous connection between the complex momentum singularities of the two-meson truncated correlation and the energy-momentum spectrum of the model. The total spin operator J and its z-component Jz are defined by using π∕2 rotations about the spatial coordinate axes, and agree with the infinitesimal generators of the continuum for improper zero-momentum meson states. The mesons admit a labelling in terms of the quantum numbers of total isospin I, the third component I3 of total isospin, the z-component Jz of total spin and quadratic Casimir C2 for SU(3)f. With this labelling, the mesons can be organized into two sets of states, distinguished by the total spin J. These two sets are identified with the SU(3)f nonet of pseudo-scalar mesons (J=0) and the three nonets of vector mesons (J=1,Jz=±1,0). Within each nonet a further decomposition can be made using C2 to obtain the singlet state (C2=0) and the eight members of the octet (C2=3). By casting the problem of determination of the meson masses and dispersion curves into the framework of the the anaytic implicit function theorem, all the masses m(κ,β) are found exactly and are given by convergent expansions in the parameters κ and β. The masses are all of the form m(κ,β=0)≡m(κ)=−2lnκ−3κ2/2+κ4r(κ) with r(0)≠0 and r(κ) real analytic; for β>0,m(κ,β)+2lnκ is jointly analytic in κ and β. The masses of the vector mesons are independent of Jz and are all equal within each octet. All isospin singlet masses are also equal for the vector mesons. For each nonet and β=0, up to and including O(κ4), the masses of the octet and the singlet are found to be equal. But there is a pseudoscalar-vector meson mass splitting given by 2κ4+O(κ6) and the splitting persists for β>0. For β=0, the dispersion curves are all of the form w(p⃗)=−2lnκ−3κ2∕2+(14)κ2∑j=132(1−cospj)+κ4r(κ,p⃗), with ∣r(κ,p⃗)∣⩽const. For the pseudoscalar mesons, r(κ,p⃗) is jointly analytic in κ and pj, for ∣κ∣ and ∣Impj∣ small. We use some machinery from constructive field theory, such as the decoupling of hyperplane method, in order to reveal the gauge-invariant eightfold way meson states and a correlation subtraction method to extend our spectral results to all He, the subspace of H generated by vectors with an even number of Grassmann variables, up to near the two-meson energy threshold of ≈−4lnκ. Combining this result with a previously similar result for the baryon sector of the eightfold way, we show that the only spectrum in all H≡He⊕Ho (Ho being the odd subspace) below ≈−4lnκ is given by the eightfold way mesons and baryons. Hence, we prove confinement up to near this energy threshold.