Stanley Mandelstam

Quantum electrodynamics without potentials

A scheme is proposed for quantizing electrodynamics in terms of the electromagnetic fields without the introduction of potentials. The equations are relativistically covariant and do not require the introduction of unphysical states and an indefinite metric. Calculations carried out according to current quantization methods in the Coulomb or Lorentz gauges are justified in the new formalism. The theory exhibits an analogy between phases of operators and electromagnetic fields on the one hand, and coordinate systems and space curvature on the other. It is suggested that this analogy may be useful in quantizing the gravitational field.

Interacting-string picture of dual-resonance models☆

Dual-resonance models are analyzed by means of operators which act within the physical Hilbert space of positive-metric states. The basis of the method is to extend the relativistic-string picture of Goldstone, Goddard, Rebbi and Thorn to interacting particles. Functional methods are used, but their relation to the operator formalism is evident, and factorization is maintained. An expression is given for the N-point amplitude in terms of physical-particle operators. For the three-point function the Neumann functions which occur in this expression are evaluated, so that we have a formula for the on- and off-energy-shell vertex. We assume that the string has no longitudinal degrees of freedom, and our results are Lorentz invariant and dual only if d=26.

Dual-resonance models

Abstract Dual-resonance models are treated both as S -matrix theories and as systems of interacting strings. We show how Veneziano was able to construct a dual four-point amplitude with narrow resonances and rising Regge trajectories. The construction is generalized to the N -point amplitude in the manifestly dual manner suggested by Koba and Nielsen. We develop the operator formalism which exhibits the factorization property of the above amplitude. The related questions of ghost elimination and null states are discussed. Models with extra degrees of freedom and, in particular, the Neveu-Schwarz-Ramond model with spin, are treated. The latter model has a quark-line spectrum of mesons, but it possesses massless vector mesons and fermions. It is shown how the operator formalism is related to a quantized string. The theory of such a string is developed, with particular emphasis on the ghost-free “Coulomb-gauge” quantization. By constructing theories of interacting strings, we reproduce the dual-model S -matrix. A brief account is given of the theory of loops, in which one attempts to improve on the narrow-resonance model in a perturbative manner.

Interacting-string picture of the Neveu--Schwarz--Ramond model

The interacting-string picture is extended to the Neveu-Schwarz and Ramond models. The picture is manifestly dual, even for the Ramond model, and amplitudes with an arbitrary number of external fermions can be constructed. In all channels the only singularities are poles corresponding to the expected particles. There exist both qq and zero-quark mesons, which differ in the parity of the states with odd Neveu-Schwarz g-parity. Calculations of ff scattering amplitudes are performed. The results are B-functions multiplied by simple kinematical factors, and it is easily verified that the amplitudes possess all the necessary properties.

Dynamical variables in the Bethe-Salpeter formalism

It is shown that a knowledge of the behaviour of the propagators around their singularities enables one to determine not only the masses of bound states, but also the matrix element of any dynamical variable between two bound states. One is thus enabled to find such a matrix element, to any order in the coupling constant, by the integration of certain expressions over the corresponding Bethe-Salpeter wave-functions. As a consequence, it is possible to find normalization and orthogonality properties of these wave-functions, which in turn lead to the condition which must be imposed on their singularities a t the origin. More light is thus shed on Goldstein’s difficulty concerning the existence of a continuous infinity of bound states. The formalism is extended to scattering states in which some of the particles may be composite—in particular, an expression for the S-matrix is obtained

Vortices and quark confinement in non-Abelian gauge theories

Abstract It is shown that finite-length vortices in an SU(n) Nielsen-Olesen model require explicit introduction of monopoles, which are confined in multiples of n by the Meissner effect. The model therefore possesses a natural explanation of quark confinement.

An extension of the Regge formula

Abstract The Regge formula is modified so as to incorporate the analytic properties of the scattering amplitude in the complete l-plane. The asymptotic behavior of the background term (previously z − 1 2 ) can thereby be made arbitrarily small, and a formal asymptotic expansion of the scattering amplitude can be obtained.

Quantization of the gravitational field

Abstract The scheme proposed in the preceding paper for the gauge-independent quantization of the electromagnetic field is here applied to the coordinate independent quantization of the gravitational field. Einsteins theory is first reformulated so as to avoid reference to a coordinate system. The quantization of the resulting theory is then carried through. No mention is made of unphysical variables such as the metric tensor except for the purpose of linking the present formalism with more conventional theories, the gravitational field is described by the Riemann tensor and its connection with space curvature is clear from the outset. First-order perturbation calculations are carried out and give results equivalent to those of “flat-space” theories.

Lorentz properties of the three-string vertex

The three-string vertex is shown to be Lorentz invariant by virtue of the locality properties of the Lorentz generator. The operators cannot be normal ordered and a high-frequency cutoff must be introduced. Anomalous terms thereby appear at the joining point, but they cancel at the critical dimension of space-time. The off-shell vertex changes under a Lorentz transformation as a result of the change of the interaction time, but S-matrix elements are Lorentz invariant. The Neveu-Schwarz-Ramond (NSR) model is also treated; the form of the Lorentz generator may be understood by referring to the anti-ferromagnetic string model of Aharonov, Casher and Susskind. In both the orbital model and the NSR model the Lorentz-transformation properties of the vertex guarantee that the volume element of the dual-model integration is conformally invariant. This last result is extended to loops.

A resonance model for pion production in nucleon-nucleon collisions at fairly low energies

Cross-sections for pion production in nucleon-nucleon collisions are calculated on the assumption that the production takes place into a few angular momentum states only, and that the matrix element for each particular transition is constant except for factors due to the final-state pion-nucleon and nucleon-nucleon interactions. The outgoing pion is assumed to be in a resonant (3/2, 3/2) state with one of the nucleons. The angular momentum states introduced are compared with those of Rosenfeld (1954) and Gell-Mann & Watson (1954). It is found that a three-parameter theory can give a good account of most of the experiments on pion production, both total and differential, up to 660 MeV, except near threshold when non-resonance production is important and further parameters are necessary. Energy distribution and angular correlation experiments bring out the importance of the pion-nucleon resonance. The cross-section for neutral pion production is predicted fairly accurately in terms of that for positive pion production from threshold to 660 MeV; at 660 MeV, the calculated positive:neutral ratio is 3⋅9. The difference between this value and that of Peaslee (1954) is partly due to the mass difference between the positive and neutral pion and partly to the inclusion of interference effects between the outgoing nucleons.