Richard A. Brandt

Stability analysis for singular non-Abelian magnetic monopoles

Abstract The Dirac magnetic monopole potential g A(r) = (g/4π)( r ×^n )(r− r· n ) −1 is a stable solution to the Abelian Maxwell equations. The simple generalization A = M A is a solution to the classical non-Abelian generalized Yang-Mills equations, where M is a matrix in the Lie algebra G of the gauge group. In this paper, the stability problem for these non-Abelian monopoles is posed and solved. Although A is essentially Abelian in that A × A = 0 , the stability analysis is non-trivial because it involves the full non-Abelian structure of the theory. It is first shown that the potential A leads to a rotationally invariant classical theory only if the quantization condition g β = 1 2 n β (g β = eigenvalue of M /4π; n β = integer = 0, ±1, ±2,…; 0⩽β⩽ dim G ; gauge field coupling constant = e = 1) is satisfied. (In contrast, the Abelian Dirac quantization condition g/4π = 1 2 n is necessary only in quantum mechanics.) The stability analysis is performed by solving the linearized equations for the perturned potentials A (r) + a (r , t) . Thus the existence of a solution a (r ,t) which grows exponentially in time t is equivalent to the instability of A . Using a convenient choice for the basis of G , and the background gauge condition, the equation for the Fourier transform ψ( r ,ω) of a is seen to be equivalent to the Schrodinger equation for a particle of unit spin, unit charge and unit anamolous magnetic moment moving in the potential 4πg β A . This equation is separated using the Wu-Yang monopole harmonics, generalized to include unit spin. The radial equation is then solved in terms of the eigenvalues of an operator related to the spin and orbital angular momentum operators. The result is that A is stable if and only if each integer n β is either 0 or ±1. The monopoles with |g β | 1 2 are thus unstable and therefore have no quantum-mechanical significance. This conclusion is used to speculate about the empirical absence of monopoles, the stability of the non-singular (t Hooft-Polyakov) monopoles, and the existence of magnetic confinement.

Conservation of energy in competitive swimming

Energy conservation in swimming is formulated in terms of four functions of the swim speed v: the consumed power K, the mechanical power P used for horizontal propulsion, the remaining expended mechanical power N, and the thermal power loss H. K is well-known (from VO2) and P = FDv, where the drag force FD is not known with certainty but represents a small effect. Estimates of the nonpropulsive components of a swimmers body motions reveal that N is small, less than 3 watts per kg of body mass. H is estimated by using the theory of convective heat transfer to express the heat loss in terms of the difference between the swimmers skin temperature and the water temperature. This temperature difference was measured for the NYU mens swim team and found to be 0.033 K per kg of body mass, giving H approximately 34 W kg-1. The results are seen to be in good agreement with the asymptotic (large swim time tf) form of energy conservation. The further requirement that energy conservation is valid to order 1/tf, and use of the 1993 world record race times, is shown to imply that the initial energy available in a swimmers body is 963 +/- 231 joules per kg of body mass.

Magnetic monopoles in SU(N) gauge theories

The potential A(r) ≡ M(r×n)(r−r·n)−1 is a static solution to the classical theory of non-abelian gauge fields coupled to a point magnetic source, for any matrix M in the Lie algebra of the gauge group G. This solution is rotationally invariant if the eigenvalues of M in the adjoint representation of G are quantized in half-integer units, but is stable to small perturbations only if all non-vanishing eigenvalues are ±12. In this paper, for the gauge groups G = SU(N), it is shown which sets of eigenvalues of M are consistent with the group structure, which consistent sets are gauge inequivalent, and which consistent gauge inequivalent sets correspond to stable monopoles. It is found that there are N inequivalent stable monopoles, including the trivial case M = 0. Equivalence here is with respect to non-singular gauge transformations—the symmetry transformations of the classical theory. Singular gauge transformations are, in contrast, not symmetries but they are nevertheless useful for classifying solutions and for relating the above concept of local stability to the global, or topological, stability associated with the Dirac strings. In this context, it is shown that there are N distinct topological classes of monopoles, with the group structure of the center ZN≊Π1(SU(N)/ZN) of SU(N), that each class contains exactly one stable monopole, and that any other monopole in the same class has a strictly larger value of the magnetic charge magnitude trM2. This leads to an interesting physical picture of local stability as a consequence of the minimization of magnetic energy. The paper concludes with some comments on related topics: the empirical absence of magnetic charge, `t Hoofts calculation of magnetic energy, magnetic confinement, and spontaneously broken theories.

Non-abelian gauge invariant classical lagrangian formalism for point electric and magnetic charge

Abstract The classical electrodynamics of electrically charged point particles has been generalized to include non-Abelian gauge groups and to include magnetically charged point particles. In this paper these two distinct generalizations are unified into a non-Abelian gauge theory of electric and magnetic charge. Just as the electrically charged particles constitute the generalized source of the gauge fields, the magnetically charged particles constitute the generalized source of the dual fields. The resultant equations of motion are invariant to the original “electric” non-Abelian gauge group, but, because of the absence of a corresponding “magnetic” gauge group, there is no “duality” symmetry between electric and magnetic quantities. However, for a class of solutions to these equations, which includes all known point electric and magnetic monopole constructions, there is shown to exist an equivalent description based on a magnetic, rather than electric, gauge group. The gauge potentials in general are singular on strings extending from the particle position to infinity, but it is shown that the observables are without string singularities, and that the theory is Lorentz invariant, provided a charge quantization condition is satisfied. This condition, deduced from a stability analysis, is necessary for the consistency of the classical non-Abelian theory, in contrast to the Abelian case, where such a condition is necessary only for the consistency of the quantum theory. It is also shown that in the classical theory the strings cannot be removed by gauge transformations, as they sometimes can be in the quantum theory. An action principle is presented which leads to the particle and field equations of motion except for extra contributions arising from the possibility that the strings carry electric charge. Only solutions with electrically neutral strings are acceptable as monopoles. The action includes unphysical electrically charged fields defined on particle trajectories and on strings. The Lagrangian is gauge invariant and leads to electric current conservation via Noethers theorem.

Gauge invariance and renormalization

Abstract The renormalization of Abelian and non-Abelian local gauge theories is discussed. It is recalled that whereas Abelian gauge theories are invariant to local c-number gauge transformations δAμ(x) = ∂μ,…, with □Λ = 0, and to the operator gauge transformation δAμ(x) = ∂μφ(x), …, δφ(x) = α−1∂·A(x), with □φ = 0, non-Abelian gauge theories are invariant only to the operator gauge transformations δA μ (x) ∼ μ C(x) , …, introduced by Becchi, Rouet and Stora, where μ is the covariant derivative matrix and C is the vector of ghost fields. The renormalization of these gauge transformation is discussed in a formal way, assuming that a gauge-invariant regularization is present. The naive renormalized local non-Abelian c-number gauge transformation δA μ (x) = (Z 1 /Z 3 )gA μ (x) × Λ (x)+∂ μ Λ (x) , …, is never a symmetry transformation and is never finite in perturbation theory. Only for Λ (x) = (Z 3 /Z 1 )L with L finite constants or for Λ (x) = Ω z 3 C(x) with Ω a finite constant does it become a finite symmetry transformation, where z 3 is the ghost field renormalization constant. The renormalized non-Abelian Ward-Takahashi (Slavnov-Taylor) identities are consequences of the invariance of the renormalized gauge theory to this formation. It is also shown how the symmetry generators are renormalized, how photons appear as Goldstone bosons, how the (non-multiplicatively renormalizable) composite operator Aμ × C is renormalized, and how an Abelian c-number gauge symmetry may be reinstated in the exact solution of many asymptotically fr ee non-Abelian gauge theories.

Conserved and partially conserved currents on the light cone

Abstract We show how the constraints of current conservation and partial current conservation can be incorporated on light-cone operator-product expansions. For example, if J μ ( x ) is conserved and [ J 0 ( x ), K ( y )] δ ( x 0 − y 0 ) = L ( x ) δ 4 ( x − y ), then we show how the expansion for the T-product T[ J μ ( x ) K ( y )] can be made manifestly consistent with the operator Ward identity ∂ x μ T [ J μ ( x ) K ( y )] = L ( x ) δ 4 ( x − y ). Our constructions are presented in the framework of a canonical gluon model, the properties of which are discussed in detail. We show that the usual use of canonical commutation relations and field equations is inconsistent with naive scale invariance and the existence of simple local products of fields and discuss ways of resolving this difficulty. As examples, we exhibit the conserved forms for the expansions of the products V μ a ( x ) S b ( y ) and V μ a ( x ) V ν b ( y ) in the gluon model. As applications of our results, we discuss the K l 3 and π l 3 form factors and derive the scaling laws and scaling relations in deep inelastic neutrino-nucleon scattering.

Operator product expansions at large distances and Regge asymptotic behavior

Abstract The class of theories is considered in which the simplest high-energy scattering processes are described by the exchange of a Regge pole or cut and in which light-cone behavior is described by operator product expansions. The scattering processes are those in which an initial particle interacts with an initial cluster of other particles to produce a final particle and final cluster. The high energy limit is that in which the energy of the initial particle relative to the energy of any particle in the clusters becomes large with fixed masses, momentum transfers and cluster subenergies. It is shown that these limits can be completely characterized by an operator statement on the product of the interpolating fields of the initial and final particles, independently of the initial and final clusters. Given then that these limits correspond to large distance limits in space-time, operator product expansions are obtained which describe in an operator manner the behavior of a product A ( x ) B (0) of local fields in the large distance limit x 0 → ∞ with x 2 fixed. This provides an exact operator picture of a Regge pole being built out of an infinite set of elementary spin exchanges. The crucial ingredient in the derivation is the factorization property of Regge poles or Regge cut discontinuities. The leading light-cone contribution to the scattering amplitude will then have the same factorization properties. The operator expansion is derived first for single particle matrix elements, first using commutativity of the Regge and scaling limits and second using diagonalized Bethe-Salpeter equations for the matrix elements of the local fields in the light-cone expansion. It is then deduced for single-double particle matrix elements using the second method where now helicity sums play an important role. It is finally derived for arbitrary matrix elements. Non-leading Regge poles and Regge cuts can be similarly described by using also non-leading light-cone singularities.

Null Plane Restrictions of Current Commutators

We consider some mathematical aspects of the problems of defining restrictions of quantum fields and commutators of such fields to null planes. We give precise meanings to these restrictions and discuss how these lead to unambiguous derivations of the usual formal results. We discuss and relate various definitions of null plane charges and derive some of their properties such as vacuum annihilation. We define and exhibit finite null plane restrictions for causal solutions of the Klein‐Gordon equation. Commutator functions defined by integral representations with various spectral functions are then considered. Light cone operator product expansions are used to calculate some null plane current commutators. In this way we can give precise derivations of Fubini sum rules and electroproduction structure function asymptotic behavior.

Finite field equation of Yang–Mills theory

We consider the finite local field equation −{[1+1/α (1+f4)]gμν⧠−∂μ∂ν}Aνa =−(1+f3) g2N[AcνAaμAνc] +⋅⋅⋅+(1−s)2M2Aaμ, introduced by Lowenstein to rigorously describe SU(2) Yang–Mills theory, which is written in terms of normal products. We also consider the operator product expansion Acν(x+ξ) Aaμ(x) Abλ(x−ξ) ∼ΣMcabνμλc′a′b′ν′μ′λ′ (ξ) N[Aν′c′Aμ′a′Aλ′b′](x ), and using asymptotic freedom, we compute the leading behavior of the Wilson coefficients M......(ξ) with the help of a computer, and express the normal products in the field equation in terms of products of the c‐number Wilson coefficients and of operator products like Acν(x+ξ) Aaμ(x) Abλ(x−ξ) at separated points. Our result is −{[1+(1/α)(1+f4)]gμν⧠−∂μ∂ν }Aνa =−(1+f3) g2limξ→0{ (lnξ)−0.28/2b[Acν (x+ξ) Aaμ(x) Aνc(x−ξ) +eabcAμc(x+ξ) ∂νAbν(x)+⋅⋅⋅] +⋅⋅⋅}+(1−s)2M2Aaμ, where β (g) =−bg3, and so (lnξ)−0.28/2b is the leading behavior of the c‐number coefficient multiplying the operator products in the field equation.