Murray Gell-Mann

A schematic model of baryons and mesons

The bootstrap model for strongly interacting particles described in terms of the broken eightiold way is discussed to determine algebraic properties of the interactions with scattering amplitudes on the mass shell. A mathematical model based on field theory is described. (R.E.U.)

The axial vector current in beta decay

SummaryIn order to derive in a convincing manner the formula of Goldberger and Treiman for the rate of charged pion decay, we consider the possibility that the divergence of the axial vector current in β-decay may be proportional to the pion field. Three models of the pion-nucleon interaction (and the weak current) are presented that have the required property. The first, using gradient coupling, has the advantage that it is easily generalized to strange particles, but the disadvantages of being unrenormalizable and of bringing in the vector and axial vector currents in an unsymmetrical way. The second model, using a strong interaction proposed bySchwinger and a weak current proposed byPolkinghorne, is renormalizable and symmetrical betweenV andA, but it involves postulating a new particle and is hard to extend to strange particles. The third model resembles the second one except that it is not necessary to introduce a new particle. (Renormalizability in the usual sense is then lost, however). Further research along these lines is suggested, including consideration of the possibility that the pion decay rate may be plausibly obtained under less stringent conditions.RiassuntoAllo scopo di dedurre in maniera convincente la formula di Goldberger e Treiman per il tasso di decadimento dei pioni carichi, prendiamo in considerazione la possibilità che la divergenza della corrente vettoriale assiale nel decadimento β sia proporzionale al campo del pione. Si presentano tre modelli della interazione pionenucleone (e della corrente debole) che hanno la proprietà richiesta. Il primo, che si serve dell’accoppiamento di gradiente, ha il vantaggio di poter essere facilmente generalizzato alle particelle strane, ma gli svantaggi di non essere rinormalizzabile e di introdurre le correnti vettoriale e vettoriale assiale in modo asimmetrico. Il secondo modello, che usa un’interazione forte proposta daSchwinger ed una corrente debole proposta daPolkinhorne, è rinormalizzabile e simmetrico fraV edA, ma comporta la postulazione di una nuova particella ed è difficilmente estensibile alle particelle strane. Il terzo modello è simile al secondo salvo che non è necessario introdurre una nuovo particella. (Si perde, tuttavia, la rinormalizzazione nel senso usuale.) Si suggerisce una ulteriore ricerca su queste linee, compresa la considerazione della possibilità che il tasso di decadimento del pione possa ottenersi in modo plausibile con condizioni meno restrittive.

Behavior of current divergences under SU(3) x SU(3)

We investigate the behavior under SU3×SU3 of the hadron energy density and the closely related question of how the divergences of the axial-vector currents and the strangeness-changing vector currents transform under SU3×SU3. We assume that two terms in the energy density break SU3×SU3 symmetry; under SU3 one transforms as a singlet, the other as the member of an octet. The simplest possible behavior of these terms under chiral transformations is proposed: They are assigned to a single (3,3*)+(3*,3) representation of SU3×SU3 and parity together with the current divergences. The commutators of charges and current divergences are derived in terms of a single constant c that describes the strength of the SU3-breaking term relative to the chiral symmetry-breaking term. The constant c is found not to be small, as suggested earlier, but instead close to the value (-sqrt[2]) corresponding to an SU2×SU2 symmetry, realized mainly by massless pions rather than parity doubling. Some applications of the proposed commutation relations are given, mainly to the pseudoscalar mesons, and other applications are indicated.

Behavior of current divergences under SU3×SU3

We investigate the behavior under SU3×SU3 of the hadron energy density and the closely related question of how the divergences of the axial-vector currents and the strangeness-changing vector currents transform under SU3×SU3. We assume that two terms in the energy density break SU3×SU3 symmetry; under SU3 one transforms as a singlet, the other as the member of an octet. The simplest possible behavior of these terms under chiral transformations is proposed: They are assigned to a single (3,3*)+(3*,3) representation of SU3×SU3 and parity together with the current divergences. The commutators of charges and current divergences are derived in terms of a single constant c that describes the strength of the SU3-breaking term relative to the chiral symmetry-breaking term. The constant c is found not to be small, as suggested earlier, but instead close to the value (-sqrt[2]) corresponding to an SU2×SU2 symmetry, realized mainly by massless pions rather than parity doubling. Some applications of the proposed commutation relations are given, mainly to the pseudoscalar mesons, and other applications are indicated.

Advantages of the Color Octet Gluon Picture

It is pointed out that there are several advantages in abstracting properties of hadrons and their currents from a Yang-Mills gauge model based on colored quarks and color octet gluons.

What Is Complexity

It would require many different concepts to capture all our notions of the meaning of complexity. The concept that comes closest to what we usually mean is effective complexity (EC). Roughly speaking, the EC of an entity is the length of a very concise description of its regularities. A novel is considered complex if it has a great many scenes, subplots, characters, and so forth. An elaborate hierarchy can contribute to complexity, as in the case of nested industrial clusters each composed of a great variety of firms and other institutions. In general, though, what are regularities? We encounter in many different situations the interplay between the regular and the random or incidental: music and static on the radio, specifications and tolerances in manufacturing, etc. But ultimately the distinction between the regular and the incidental depends on a judgment of what is important, although the judge need not be human or even alive. For instance, in the case of songs of a male bird in the nesting season, the identification of regularities is perhaps best left to the other birds of the same species — what features are essential in repelling other males from the territory or attracting a suitable female? A technical definition of EC involves the quantity called algorithmic information content (AIC). The description of an entity is converted to a bit string and a standard universal computer is programmed to print out that string and then halt. The length of the shortest such program (or, in a generalization, the shortest that executes within a given time) is the AIC. The AIC is expressed as the sum of two terms, one (the EC) referring to the regularities and the other to the random features. The regularities of a real entity are best expressed by embedding it conceptually in a set of comparable things, the rest of which are imagined. The EC can then be related to the AIC of the set, the choice of which is restricted by the conditions imposed by the judge. Theorists like to study highly simplified models of complex systems, often by computer modelling. What can be claimed for such models? Overall agreement with observation is hardly to be expected. However, in many cases simple regularities can be found in both the observational data and the model, which may then be helpful in understanding those regularities. Examples are given, involving scaling laws and also “implicational scales”.

Classical equations for quantum systems.

The origin of the phenomenological deterministic laws that approximately govern the quasiclassical domain of familiar experience is considered in the context of the quantum mechanics of closed systems such as the universe as a whole. A formulation of quantum mechanics is used that predicts probabilities for the individual members of a set of alternative coarse-grained histories that decohere, which means that there is negligible quantum interference between the individual histories in the set. We investigate the requirements for coarse grainings to yield decoherent sets of histories that are quasiclassical, i.e., such that the individual histories obey, with high probability, effective classical equations of motion interrupted continually by small fluctuations and occasionally by large ones. We discuss these requirements generally but study them specifically for coarse grainings of the type that follows a distinguished subset of a complete set of variables while ignoring the rest. More coarse graining is needed to achieve decoherence than would be suggested by naive arguments based on the uncertainty principle. Even coarser graining is required in the distinguished variables for them to have the necessary inertia to approach classical predictability in the presence of the noise consisting of the fluctuations that typical mechanisms of decoherence produce. We describe the derivation of phenomenological equations of motion explicitly for a particular class of models. Those models assume configuration space and a fundamental Lagrangian that is the difference between a kinetic energy quadratic in the velocities and a potential energy. The distinguished variables are taken to be a fixed subset of coordinates of configuration space. The initial density matrix of the closed system is assumed to factor into a product of a density matrix in the distinguished subset and another in the rest of the coordinates. With these restrictions, we improve the derivation from quantum mechanics of the phenomenological equations of motion governing a quasiclassical domain in the following respects: Probabilities of the correlations in time that define equations of motion are explicitly considered. Fully nonlinear cases are studied. Methods are exhibited for finding the form of the phenomenological equations of motion even when these are only distantly related to those of the fundamental action. The demonstration of the connection between quantum-mechanical causality and causality in classical phenomenological equations of motion is generalized. The connections among decoherence, noise, dissipation, and the amount of coarse graining necessary to achieve classical predictability are investigated quantitatively. Routes to removing the restrictions on the models in order to deal with more realistic coarse grainings are described.

Information measures, effective complexity, and total information

This article defines the concept of an information measure and shows how common information measures such as entropy, Shannon information, and algorithmic information content can be combined to solve problems of characterization, inference, and learning for complex systems. Particularly useful quantities are the effective complexity, which is roughly the length of a compact description of the identified regularities of an entity, and total information, which is effective complexity plus an entropy term that measures the information required to describe the random aspects of the entity. Mathematical definitions are given for both quantities and some applications are discussed. In particular, it is pointed out that if one compares different sets of identified regularities of an entity, the ‘best’ set minimizes the total information, and then, subject to that constraint, minimizes the effective complexity; the resulting effective complexity is then in many respects independent of the observer.

Series of hadron energy levels as representations of non-compact groups

It has recently been emphasized [1] that in an approximate symmetry theory it is by no means necessary that a consistent relativistic model be available in which the symmetry is exact, with the violations then treated as perturbations. It is sufficient that a set of operators be found that obey the equal-time commutation relations characteristic of some algebra, that the energy operator be decomposed into terms transforming according to various irreducible representations of the algebra, and that the stationary and quasi-stationary quantum-mechanical states have a tendency to fall approximately into irreducible representations of the algebra as a result of dynamics. within each irreducible representations of the algebra as a result of dynamics.

Asymptotically scale-invariant occupancy of phase space makes the entropy Sq extensive.

Phase space can be constructed for N equal and distinguishable subsystems that could be probabilistically either weakly correlated or strongly correlated. If they are locally correlated, we expect the Boltzmann-Gibbs entropy SBG ≡ -k Σi pi ln pi to be extensive, i.e., SBG(N) ∝ N for N → ∞. In particular, if they are independent, SBG is strictly additive, i.e., SBG(N) = NSBG(1), ∀N. However, if the subsystems are globally correlated, we expect, for a vast class of systems, the entropy Sq ≡ k[1 - Σi pqi]/(q - 1) (with S1 = SBG) for some special value of q ≠ 1 to be the one which is extensive [i.e., Sq(N) ∝ N for N → ∞]. Another concept which is relevant is strict or asymptotic scale-freedom (or scale-invariance), defined as the situation for which all marginal probabilities of the N-system coincide or asymptotically approach (for N → ∞) the joint probabilities of the (N - 1)-system. If each subsystem is a binary one, scale-freedom is guaranteed by what we hereafter refer to as the Leibnitz rule, i.e., the sum of two successive joint probabilities of the N-system coincides or asymptotically approaches the corresponding joint probability of the (N - 1)-system. The kinds of interplay of these various concepts are illustrated in several examples. One of them justifies the title of this paper. We conjecture that these mechanisms are deeply related to the very frequent emergence, in natural and artificial complex systems, of scale-free structures and to their connections with nonextensive statistical mechanics. Summarizing, we have shown that, for asymptotically scale-invariant systems, it is Sq with q ≠ 1, and not SBG, the entropy which matches standard, clausius-like, prescriptions of classical thermodynamics.