Govind S. Krishnaswami

A Critique of Recent Semi-Classical Spin-Half Quantum Plasma Theories

Certain recent semi-classical theories of spin-half quant um plasmas are examined with regard to their internal consistency, physical applicabilit y and relevance to fusion, astrophysical and condensed matter plasmas. It is shown that the derivations and some of the results obtained in these theories are internally inconsistent and contradi ct well-established principles of quantum and statistical mechanics, especially in their treatment o f fermions and spin. Claims of large semiclassical effects of spin magnetic moments that could dominate the plasma dynamics are found to be invalid both for single-particles and collectively. Lar mor moments dominate at high temperature while spin moments cancel due to Pauli blocking at low temperatures. Explicit numerical estimates from a variety of plasmas are provided to demonstrate that spin effects are indeed much smaller than many neglected classical effects. The analysis presented suggests that the aforementioned ‘Spin Quantum Hydrodynamic’ theories are not rel evant to conventional laboratory or astrophysical plasmas.

A KdV-like advection-dispersion equation with some remarkable properties

Abstract We discuss a new non-linear PDE, u t + ( 2 u xx / u ) u x = ϵ u xxx , invariant under scaling of dependent variable and referred to here as SIdV. It is one of the simplest such translation and space–time reflection-symmetric first order advection–dispersion equations. This PDE (with dispersion coefficient unity) was discovered in a genetic programming search for equations sharing the KdV solitary wave solution. It provides a bridge between non-linear advection, diffusion and dispersion. Special cases include the mKdV and linear dispersive equations. We identify two conservation laws, though initial investigations indicate that SIdV does not follow from a polynomial Lagrangian of the KdV sort. Nevertheless, it possesses solitary and periodic travelling waves. Moreover, numerical simulations reveal recurrence properties usually associated with integrable systems. KdV and SIdV are the simplest in an infinite dimensional family of equations sharing the KdV solitary wave. SIdV and its generalizations may serve as a testing ground for numerical and analytical techniques and be a rich source for further explorations.

A model of interacting partons for hadronic structure functions

Abstract We present a model for the structure of baryons in which the valence partons interact through a linear potential. This model can be derived from QCD in the approximation where transverse momenta are ignored. We compare the baryon structure function predicted by our model with those extracted from global fits to Deep Inelastic Scattering data. The only parameter we can adjust is the fraction of baryon momentum carried by valence partons. Our predictions agree well with data except for small values of the Bjorken scaling variable.

Phase transition in matrix model with logarithmic action: toy-model for gluons in baryons

We study the competing effects of gluon self-coupling and their interactions with quarks in a baryon, using the very simple setting of a hermitian 1-matrix model with action trA4−log det (ν+A2). The logarithmic term comes from integrating out N quarks. The model is a caricature of 2d QCD coupled to adjoint scalars, which are the transversely polarized gluons in a dimensional reduction. ν is a dimensionless ratio of quark mass to coupling constant. The model interpolates between gluons in the vacuum (ν = ∞), gluons weakly coupled to heavy quarks (large ν) and strongly coupled to light quarks in a baryon (ν→0). Its solution in the large-N limit exhibits a phase transition from a weakly coupled 1-cut phase to a strongly coupled 2-cut phase as ν is decreased below νc = 0.27. Free energy and correlation functions are discontinuous in their third and second derivatives at νc. The transition to a two-cut phase forces eigenvalues of A away from zero, making glue-ring correlations grow as ν is decreased. In particular, they are enhanced in a baryon compared to the vacuum. This investigation is motivated by a desire to understand why half the protons momentum is contributed by gluons.

Parton model from bi-local solitonic picture of the baryon in two-dimensions

Abstract We study a previously introduced bi-local gauge invariant reformulation of two-dimensional QCD, called 2d hadron dynamics. The baryon arises as a topological soliton in hadron dynamics. We derive an interacting parton model from the soliton model, thus reconciling these two seemingly different points of view. The valence quark model is obtained as a variational approximation to hadron dynamics. A succession of better approximations to the soliton picture are obtained. The next simplest case corresponds to a system of interacting valence, ‘sea’ and anti-quarks. We also obtain this ‘embellished’ parton model directly from the valence quark system through a unitary transformation. Using the solitonic point of view, we estimate the quark and anti-quark distributions of 2d QCD. Possible applications to deep inelastic structure functions are pointed out.

Non-anomalous `Ward' identities to supplement large-N multi-matrix loop equations for correlations

This work concerns single-trace correlations of Euclidean multi-matrix models. In the large-N limit we show that Schwinger-Dyson equations (SDE) imply loop equations (LE) and non-anomalous Ward identities (WI). LE are associated to generic infinitesimal changes of matrix variables (vector fields). WI correspond to vector fields preserving measure and action. The former are analogous to Makeenko-Migdal equations and the latter to Slavnov-Taylor identities. LE correspond to leading large-N SDE. WI correspond to 1/N2 suppressed SDE. But they become leading equations since LE for non-anomalous vector fields are vacuous. We show that symmetries at N = ∞ persist at finite N, preventing mixing with multi-trace correlations. For 1 matrix, there are no non-anomalous infinitesimal symmetries. For 2 or more matrices, measure preserving vector fields form an infinite dimensional graded Lie algebra, and non-anomalous action preserving ones a subalgebra. For Gaussian, Chern-Simons and Yang-Mills models we identify up to cubic non-anomalous vector fields, though they can be arbitrarily non-linear. WI are homogeneous linear equations. We use them with the LE to determine some correlations of these models. WI alleviate the underdeterminacy of LE. Non-anomalous symmetries give a naturalness-type explanation for why several linear combinations of correlations in these models vanish.

Algebra and geometry of Hamilton’s quaternions

Inspired by the relation between the algebra of complex numbers and plane geometry, William Rowan Hamilton sought an algebra of triples for application to three-dimensional geometry. Unable to multiply and divide triples, he invented a non-commutative division algebra of quadruples, in what he considered his most significant work, generalizing the real and complex number systems. We give a motivated introduction to quaternions and discuss how they are related to Pauli matrices, rotations in three dimensions, the three sphere, the group SU(2) and the celebrated Hopf fibrations.

Schwinger???Dyson operator of Yang???Mills matrix models with ghosts and derivations of the graded shuffle algebra

We consider large-N multi-matrix models whose action closely mimics that of Yang-Mills theory, including gauge-fixing and ghost terms. We show that the fac- torized Schwinger-Dyson loop equations, expressed in terms of the generating series of gluon and ghost correlations G(�), are quadratic equations S i G = Gi G in con- catenation of correlations. The Schwinger-Dyson operator S i is built from the left annihilation operator, which does not satisfy the Leibnitz rule with respect to con- catenation. So the loop equations are not differential equations. We show that left annihilation is a derivation of the graded shuffle product of gluon and ghost corre- lations. The shuffle product is the point-wise product of Wilson loops, expressed in terms of correlations. So in the limit where concatenation is approximated by shuffle products, the loop equations become differential equations. Remarkably, the Schwinger-Dyson operator as a whole is also a derivation of the graded shuffle prod- uct. This allows us to turn the loop equations into linear equations for the shuffle reciprocal, which might serve as a starting point for an approximation scheme.

Higgs mechanism and the added-mass effect

In the Higgs mechanism, mediators of the weak force acquire masses by interacting with the Higgs condensate, leading to a vector boson mass matrix. On the other hand, a rigid body accelerated through an inviscid, incompressible and irrotational fluid feels an opposing force linearly related to its acceleration, via an added-mass tensor. We uncover a striking physical analogy between the two effects and propose a dictionary relating them. The correspondence turns the gauge Lie algebra into the space of directions in which the body can move, encodes the pattern of gauge symmetry breaking in the shape of an associated body and relates symmetries of the body to those of the scalar vacuum manifold. The new viewpoint is illustrated with numerous examples, and raises interesting questions, notably on the fluid analogues of the broken symmetry and Higgs particle, and the field-theoretic analogue of the added mass of a composite body.

Possible large-N fixed points and naturalness for O(N) scalar fields

We try to use scale-invariance and the large-N limit to find a non-trivial 4d O(N) scalar field model with controlled UV behavior and naturally light scalar excitations. The principle is to fix interactions by requiring the effective action for space-time dependent background fields to be finite and scale-invariant when regulators are removed. We find a line of non-trivial UV fixed-points in the large-N limit, parameterized by a dimensionless coupling. They reduce to classical la phi^4 theory when hbar -> 0. For hbar non-zero, neither action nor measure is scale-invariant, but the effective action is. Scale invariance makes it natural to set a mass deformation to zero. The model has phases where O(N) invariance is unbroken or spontaneously broken. Masses of the lightest excitations above the unbroken vacuum are found. We derive a non-linear equation for oscillations about the broken vacuum. The interaction potential is shown to have a locality property at large-N. In 3d, our construction reduces to the line of large-N fixed-points in |phi|^6 theory.