Selected Papers of Abdus Salam

The following sections are included:

• Introduction

• Contents

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By extending considerations given by Dyson, general rules are obtained for isolating divergent parts from integrals corresponding to overlapping graphs, and a proof is obtained for the appearance of an extra factor Z₁⁻¹ from “b divergences.” In the last section the possibility of renormalization for scalar meson-nucleon interactions is demonstrated.

A brief account is given of Dyson's proof of the finiteness after renormalization of the matrix elements for scattering processes (S-matrix elements) in electrodynamics (interaction of photons and electrons). It is shown to which meson interactions this proof can be extended and some of the difficulties which arose in this extension are discussed.

The proofs of the renormalization of the theories of scalar and pseudoscalar mesons in scalar interaction with nucleons and with the electromagnetic field, given previously by the author, depended on a certain subtraction procedure. A general proof is here given that this subtraction procedure, when applied to a divergent integral in the S matrix of a renormalizable field theory, does, in fact, lead to an absolutely convergent, covariant, and unique remainder.

By a consistent use of the concepts of mass and charge renormalization Dyson has demonstrated the possibility of constructing a divergence-free S-matrix for scattering problems in spinor-electrodynamics, valid to all orders in the fine-structure constant. In this paper a proof is given of the possibility of renormalizing theories of charged scalar (or pseudoscalar) particles in the presence of the electromagnetic field. It is found that in addition to the renormalization of mass and charge, an infinite constant of direct interaction δλ has to be introduced in a term δλφ^*2φ² added to the Hamiltonian, in order to cancel consistently all divergences arising from the Møller scattering of one spinless particle by another. This, combined with the fact that theories of charged scalar and pseudoscalar mesons in scalar interaction with the nucleons can be renormalized with the same additional term in the Hamiltonian, seems to be of some significance.

It is shown that Feynman's relativistic solution for the scattering of an electron (or pair creation) by a given external field is the Fredholm resolvent of the related integral equation and is thus the unique and absolutely convergent solution for any strength of field.

A new type of amplitude—Feynman amplitude—is defined for any state. These amplitudes are shown to propagate in the Feynman manner. It is shown further that they satisfy an infinite set of four-dimensional, covariant, coupled integral equations, analogous to the three-dimensional Fock equations between the ordinary probability amplitudes.

A theoretical justification for the infinite subtractions, which have to be made in the renormalization of the S matrix, is given along the lines suggested by Gupta and developed by Takeda. It is shown that this is equivalent to working with the renormalized field variables of Dyson, and that the method deals very simply with overlapping divergences and the “wave function” renormalization associated with external lines. It also gives directly Ward's identities and brings out their essential dependence on gauge invariance. The method is applied to free and bound electrons in electrodynamics and all renormalizable meson theories.

In the later sections the new method is related to the original method of Dyson; the Bethe-Salpeter equation is renormalized and closed forms are derived for the renormalization constants.

Eigenvalues and eigenvectors of the total Hamiltonian in field theory are determined by the method of direct diagonalization, using canonical transformations. The difficulties of this method are discussed in detail.

By defining an analytic continuation method, Wick has been able to elucidate the structure of the relativistic equation for bound states. In particular, the equation acquires an ‘elliptic’ rather than a ‘hyperbolic’ metric. Taking advantage of a gap in the rest-mass spectrum of one-nucleon states, a similar investigation is here carried through for the relativistic equation for two-nucleon scattering, in the energy region 2K < E < 2K + μ, K and μ being nucleon and meson masses.

A relation is derived which connects the real part of the non forward meson-nucleon scattering amplitude, to its imaginary part.

Considerations of an earlier paper are generalised to write down non-forward scattering, spin-flip dispersion relations. The contribution from the « bound state » is discussed.

An attempt is made to clarify the fundamental physical basis of the intrinsic characteristics of unstable elementary particles and to replace the conventional, purely phenomenological, description in terms of decaying states and complex energy levels, by definitions which are consistent with general requirements of relativistic quantum mechanics like Hermiticity, unitarity, and causality.

This paper is a direct continuation of an earlier paper (I) where an attempt was made to set up a field-theoretic foundation for the theory of mean mass and lifetime of an unstable particle. It was argued in I that the decay-time plot of a beam of unstable particles is a concept peculiar to a single-particle theory; that from a field-theoretic point of view, mass (the variable conjugate to proper time) rather than time has the primary significance. Here we show that the spectral function ρ(m²) appearing in the (field-theoretic) one-particle propagator has a direct significance as the probability of finding in production an unstable particle of mass m. This allows us to define a “one-particle” state for the unstable particle as a superposition of its outgoing decay states suitably weighted in mass space [with a factor which is the square-root of ρ(m²)]. The proper-time propagation of this state gives the decay amplitude, and its modulus is ideally the experimentally observed decay-time plot.

The time plot is explicitly evaluated for π decay. Insofar as the distribution of mass values for the π meson starts with the μ mass (assumed stable), the time plot is not merely the conventional decay exponential e_−r/r0. There are additional terms which become important about a hundred lifetimes after the particle is created.

Finally we compare the time plots for particle and antiparticle decays on the basis of CTP invariance.

An attempt is made to correlate the observed asymmetries in hyperon decays and the hyperon decay times using a pole approximation to the decay matrix elements. It is found that a natural correlation is obtained if a near-global relation between pion coupling constants is assumed.

In a recent paper, Chisholm has proved the important theorem that the S-matrix in quantum field theories remains unchanged under any point transformation of field operators. In view of some unsatisfactory features in his argument, another proof of this theorem, within the framework of the conventional canonical formalism of field theories, is given. Further, it is pointed out that on the basis of Chisholm's theorem, most of the ordinary equivalence theorems in field theories can be obtained by rather trivial changes of variables in the Lagrangians. The equivalence theorem for the Yang-Mills field is discussed in detail.

It is shown that the conventional theory of charged spin-one mesons interacting with photons can be renormalized provided meson mass m and charge e are restricted by the relation Z(m², e²)=0, where Z is the meson wave-function renormalization constant.

Methods are developed for constructing momentum-space amplitudes corresponding to nonpolynomial nonderivative interactions of a real scalar field. The methods give rise to a supergraph technique and rules for writing down matrix elements very similar to Feynman techniques. The methods are not established rigorously; at several points the argument requires certain analytic properties of Feynman integrands which, though plausible, can only be demonstrated rigorously for the zero-mass case. Asymptotic behavior, both in spacelike and timelike directions, is discussed. Rough arguments are given that indicate that the singularity structure of the amplitudes is likely to be consistent with unitarity.

The recent result of Coleman and Weinberg deriving the one-loop quantum-corrected effective potential in a self-coupled scalar field theory Is rederived using the Dyson–Schwinger equation.

The following sections are included:

• Present Status of the Standard Model

• Bounds on the Higgs Boson Mass from Partial Wave Unitarity

• Bounds on the Higgs Boson Mass from Triviality Arguments and Lattice Regularization

A new four-dimensional isotopic spin formalism is developed for the heavy fundamental particles. All particles correspond to single-valued representations of the rotation group and there is complete symmetry between the isotopic properties of fermions and bosons. The resulting classification is similar to Gell-Mann's except that the degeneracy in the classification of the K-mesons is removed. The forms of the strong interactions are given explicitly. The physical consequences of the theory are discussed in detail. In particular it predicts the existence of a neutral K-meson of lifetime ~ 10⁻¹⁹s. A new experimental technique for detecting this particle is suggested.

No abstract received.

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The rule is replaced by a conservation law. This new scheme is practically equivalent to the rule just mentioned (with the possibility of introducing in a natural way some admixture). Its main advantage is that it can incorporate lepton interactions.

The postulate of a «local connection» in a [3] charge space leads to the introduction of three spin one fields. One of these can be identified with the electro-magnetic field and the other two can be shown to mediate all known weak interactions, thus unifying these interactions with electro-magnetism. The theory takes account of the fact that weak interactions violate parity and strangeness conservation while electromagnetic interactions do not do so.

One attractive way of formulating the rule is to postulate the existence of "spurions" 1 — neutral isospinor particles carrying no energy and momentum which are emitted or absorbed in decays of strange particles . We wish to point out that conventional field theory contains a perfectly natural realization of spurions. Consider the decay mode . If the Lagrangian for the process is , it is easy to see that the vacuum expectation value of would not be zero. In fact . If we now consider a seemingly isotopic-spin conserving T product of field operators, like ΛN^†τ · πK it not only possesses matrix elements for strong interactions like N + π → Λ + K, but also a matrix element for the decay Λ → N + π (arising from ΛN^†τ · π〈K〉) consistent with the rule: thus gives a realization for the spurion…

A theory of strong as well as weak interactions is proposed using the idea of having only such interactions which arise from generalized gauge transformations.

Experiments on Σ-K production on hydrogen for pion incident energies around 1 GeV seem to indicate that (^1,2)

Some proofs are presented of Goldstone's conjecture, that if there is continuous symmetry transformation under which the Lagrangian is invariant, then either the vacuum state is also invariant under the transformation, or there must exist spinless particles of zero mass.

One of the recurrent dreams in elementary particles physics is that of a possible fundamental synthesis between electro-magnetism and weak interactions [1]. The idea has its origin in the following shared characteristics…

It is shown that a relativistic basis for invariance under SU (6) exists only if the group structure is extended to U⁺(6)⊗U⁻(6) for any interaction terms. The notion of an inhomogeneous extension U_w⁺(6)⊗U_w⁻(6) is introduced. This extension leaves the kinetic-energy terms invariant, though it still does not provide a fully satisfactory theory.

A classification of particles is suggested based on a Ũ(12) symmetry scheme. This is a relativistic generalization of the U(6) symmetry. The spin and baryons are each described by 20-component spinors which satisfy Bargmann–Wigner equations and belong to the 364 representation of the Ũ(12) group while the vector and p.s. mesons belong to the representation 143. The procedure for writing fully relativistic form factors is worked out in detail for baryon–meson and meson–meson cases.

The new results are the following:

(1) where F^C and F^M are (Sachs) electromagnetic form factors.

(2) μ_ρ = 1 + 2m/〈μ〉, where〈μ〉is the mean mass of the 1⁻ multiplet and m the nucleon mass.

(3) μ_ρ,K* = 3.

The conventional U(6) results can be recovered by projecting to the positive energy subspace in the rest system for each particle. To any irreducible representation of the U(6) there corresponds one irreducible representation of Ũ(12) and vice versa.

Scattering amplitudes for particles of arbitrary mass and spin are expanded in terms of homogeneous Lorentz group representations. The expansion satisfies the group theoretic O(3,1) boundary constraints in the forward elastic limit and incorporates the correct threshold factors.

In earlier work the Reggeization of approximate dynamical groups has been presented as a calculational method for classifying particles and evaluating S-matrix elements at high energies. In continuation of this work, an especially simple model is considered where just one invariant of the higher U(6)× U(6) approximate symmetry, quark-plus-antiquark number, is Reggeized. The resulting classification of particles (according to their quark content) into exploding supermultiplets of spin and unitary spin, and the formulas for computing S-matrix elements, are given for high energies, where an exchange of an N-plane trajectory in the cross channel may be expected to dominate the scattering. The hope is that this analysis may help reduce the large number of parameters now used in Regge theory by combining Regge ideas with higher symmetries. The type of Fourier expansion on a higher approximate symmetry group and the Regge technique used here for evaluating asymptotic behavior may possess wider applications than the case considered in this paper.

One of the recurrent dreams in elementary particle physics is that of a possible fundamental synthesis between electromagnetism and weak interaction. The idea has its origin in the following shared characteristics:

1. Both forces affect equally all forms of matter -leptons as well as hadrons.

2. Both are vector in character.

3. Both (individually) possess universal coupling strengths.

We review work done on realization of broken symmetry under the conformal group of space-time in the framework of finite-component field theory. Topics discussed include: Most general transformation law of fields over Minkowski space. Consistent formulation of an orderly broken conformal symmetry in the framework of Lagrangian field theory; algebra of currents and their divergences; Manifestly conformally covariant fields and their couplings.

A method, based upon analogies with the Wigner boost technique, is presented for setting up the nonlinear realizations of any continuous symmetry group. It is argued that such realizations are relevant to and useful in the treatment of cases of spontaneous symmetry breakdown.

Most current theories of particle symmetries assume that violations of these symmetries come about through a spontaneous breaking mechanism which produces non zero expectation values for certain scalar (elementary or composite) fields. As is well known (refs 1–7 and Lebedev Institute Reprint No. 101) these expectation values may make a phase transition to a zero value for certain critical temperatures and possibly also for certain critical external magnetic field strengths H_c, H_c1, H_c2, … Here we point out that it is conceivable that the charge asymmetry (associated with CP violation ) in decays may disappear for fields of ~ 8 × 10¹⁰ gauss if CP violation is milli-weak in character, and that the Cabibbo angle may be reduced to zero—leading to suppression of certain hyperon decays—in fields of the order of 10¹⁶ gauss. These estimates are so strongly model-dependent that it may be worthwhile in any case , to make a systematic phenomenological search for effects on particle asymmetries of strong magnetic fields of 10⁶ gauss upwards…

We analyse the effects of the intensity-dependent mass shift predicted for elections in an electromagnetic plane wave on bound states. It is shown that the null result of the experiment of Mowat et al. on the ¹³³Cs ground-state hyperfine splitting was to be expected, although their hypothetical explanation of it is incorrect. The recent suggestion that an intense laser beam might effect the restoration of a spontaneously broken symmetry is examined in more detail, and it is demonstrated that although a pure plane will not suffice, a superposition of plane waves, for example a standing wave, would do so. At present, however, the effect is unobservable small.

The following sections are included:

• I. Fundamental Particles, Fundamental Forces, and Gauge Unification

• II. The Emergence of Spontaneously Broken SU(2)×U(1) Gauge Theory

• III. The Present and Its Problems

• IV. Direct Extrapolation from the Electroweak to the Electronuclear
  • A. The three ideas
  • B. Tests of electronuclear grand unification

• V. Elementarity: Unification with Gravity and Nature of Charge
  • A. The quest for elementarity, prequarks (preons and pre-preons
  • B. Post–Planck physics, supergravity, and Einstein's dreams
  • C. Extended supergravity, SU(8) preons, and composite gauge fields

• Appendix A: Examples of Grand Unifying Groups

• Appendix B: Does the Grand Plateau really exist

• References

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An attempt is made to unify the fundamental hadrons and leptons Into a common Irreducible representation F of the same symmetry group G and to generate a gauge theory of strong, electromagnetic, and weak interactions. Based on certain constraints from the hadronic side, it is proposed that the group G is SU(4′) ×SU(4″), which contains a Han-Nambu-type SU(3′) ×SU(3″) group for the hadronic symmetry, and that the representation F is (4,4*). There exist four possible choices for the lepton number L and accordingly four possible assignments of the hadrons and leptons within the (4,4*). Two of these require nine Han-Nambu-type quarks, three “charmed” quarks, and the observed quartet of leptons. The other two also require the nine Han-Nambu quarks, plus heavy leptons in addition to observed leptons and only one or no “charmed” quark. One of the above four assignments is found to be suitable to generate a gauge theory of the weak, electromagnetic, and SU(3″) gluonlike strong interactions from a selection of the gauges permitted by the model. The resulting gauge symmetry is SU(2′)_L × U(1) × SU(3″)_L+R. The scheme of all three interactions is found to be free from Adler-Bell-Jackiw anomalies. The normal strong interactions arise effectively as a consequence of the strong gauges SU(3″)_L+R. Masses for the gauge bosons and fermions are generated suitably by a set of 14 complex Higgs fields. The neutral neutrino and ΔS = 0 hadron currents have essentially the same strength in the present model as in other SU(2)_L ×U(1) theories. The mixing of strong- and weak-gauge bosons (a necessary feature of the model) leads to parity–violating nonleptonic amplitudes, which may be observable depending upon the strength of SU(3″) symmetry breaking. The familiar hadron symmetries such as SU(3′) and chiral SU(3′)_L × SU(3′)_R are broken only by quark mass terms and by the electromagnetic and weak interactions, not by the strong interactions. The difficulties associated with generating gauge interactions in the remaining three assignments are discussed in Appendix A. Certain remarks are made on the question of proton and quark stability in these three schemes.

We suggest that baryon-number conservation may not be absolute and that an integrally charged quark may disintegrate into two leptons and an antilepton with a coupling strength G _Bm_p²≲ 10⁻⁹. On the other hand, if quarks are much heavier than low–lying hadrons, the decay of a three-quark system like the proton is highly forbidden (proton lifetime ≳ 10²⁸ y). Motivation for these ideas appears to arise within a unified theory of hadrons and leptons and their gauge interactions. We emphasize the consequences of such a possibility for real quark searches.

Universal strong, weak, and electromagnetic Interactions of leptons and hadrons are generated by gauging a non–Abelian renormalizable anomaly-free subgroup of the fundamental symmetry structure SU(4)_L × SU(4)_R × SU(4′), which unites three quartets of “colored” baryonic quarks and the quartet of known leptons into 16-folds of chiral fermionic multiplets, with lepton number treated as the fourth “color” quantum number. Experimental consequences of this scheme are discussed. These include (1) the emergence and effects of exotic gauge mesons carrying both baryonic as well as leptonic quantum numbers, particularly in semileptonic processes, (2) the manifestation of anomalous strong interactions among leptonic and semileptonic processes at high energies, (3) the independent possibility of baryon-lepton number violation In quark and proton decays, and (4) the occurrence of (V+A) weak–current effects.

The neutral-current weak-interaction sector of the left–right–symmetric unified theory of quarks and leptons based on the symmetry structure SU(2)_L × SU(2)_R × SU(4)′_{L + R} and its subgroup SU(2)_L × SU(2)_R × SU(3)′_{L + R} × U(1)_L+R (suggested in earlier papers) is studied in detail here. The theory admits in general of two weak neutral gauge bosons N₁ and N₂ [and two sets of left and right charged W's (W^±_L, W^±_R)]. As Pointed out earlier, there are two distinct possibilities for the pattern of spontaneous symmetry breaking which lead to the mass relations (i) (ii) Case (i) is identical to the familiar SU(2) × U(1) theory for all predictions. Case (ii), on the other hand, leads to departures from SU(2) × U(1). One of the crucial theorems following from our investigation is that suchdepartures occur only for the electron-induced atomic parity experiments and not for neutrino-induced weak processes in the chiral γ₅-lnvariant limit, in which W_L–W_R mixing as well as fermion masses vanish. In this manner (for the chiral limit defined above), SU(2) × U(1) becomes an integral and stable ingredient of the left–right–symmetric theory for both cases (i) and (ii) insofar as their predictions for neutrino–induced reactions are concerned; the differences between the two cases (in this chiral limit) lie solely in their predictions regarding atomic parity violation and, of course, the masses of the two neutral particles N₁ and N₂. We study the chiral as well as the nonchlral cases and compare their predictions with experiments, observing that approximate or exact global chiral symmetry of the left–right–symmetric theory is a point of good agreement between theory and all the present neutral–current experiments. We exhibit the dependence of the atomic parity–violation parameter Q_w on the masses of the two neutral particles N₁ and N₂ and remark in particular that (depending upon the sign and magnitude of Q_w), there is the exciting experimental possibility within the left–right–symmetric theory that the mass of one of the neutral gauge particles is sensibly smaller than that of the charged W^±_L. This particular possibility does not materialize, if one insists on exact atomic parity conservation.

The distinct advantage which a lunar detector would have over a terrestrial one in searching for proton decays corresponding to rather long nucleon lifetimes ≳ 6× 10³² yrs is noted.

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Computations of gravity-modified quantum electrodynamics are performed using nonpolynominal Lagrangian field-theory techniques. The inverse of the gravitational constant appears as an effective cutoff mass, and, in particular, it is shown that to order e² the electron and photon self-energies are finite. The cutoff can be interpreted as if the electron had an intrinsic radius equal to its Schwarzschild radius. A central feature is the construction of the tensor gravity superpropagator.

A simple-minded perturbation calculation to order G (the Newtonian gravitational constant) of the two-graviton coupling to a pseudoscalar (P) or axial (A) current via a fermion loop gives an anomalous contribution which can be added to the Adler term in the form ∂_αA_α = 2m P + e²ε_kλμν F_kλF_μν/16π² + ε_kλμνR_kλρσR_μνρσ/768π². The anomalous terms can be interpreted as arising from an infinite-mass regulating loop.

A systematic method for constructing Wess–Zumino supergauge transformations is exhibited.

A method is given for constructing some of the unitary irreducible representations of the Wess–Zumino super-gauge symmetry. Application of this symmetry to the analysis of S-matrix elements is considered. A new super-gauge symmetry which includes isospin is introduced and some of its representations are constructed.

Spontaneous violation of super-gauge symmetries is considered. One consequence must be the emergence of a particle with spin 1/2 and mass zero.

We show that the conventional Yang-Mills Lagrangian with the Yang-Mills field interacting with Majorana fermions belonging to the adjoint representation of an internal symmetry group like SU(n) is super-gauge invariant.

The realization of supergauge transformations on fields defined over an 8-dimensional space whose points are labeled by x_μ and the anticommuting Majorana spinor θα is described. The covariant derivative is defined and applied to the problem of decomposing superfields into irreducible (chiral) parts and to the problem of constructing “supersymmetric” Lagrangians. Further, it is shown how to build internal symmetries (both global and local) into these Lagrangians. An example is discussed in which the internal (global) symmetry is spontaneously violated, giving rise to a supermultiplet of Goldstone particles (including fermions). When a local symmetry is broken the Higgs mechanism (for bosons and fermions) is shown to be operative. A possible solution to the problem of defining a conserved fermion number is indicated.

The notion of functional differentiation with respect to superfields is defined and used to setup formal rules for computing Green functions and scattering amplitudes in supersymmetric models.

We remark that exact classical Schwarzschild-like solutions to Einstein's (and possibly f gravity) equations provide examples of realistic solitons.

Assuming the compactification of 4 + K-dimensional space-time implied in Kaluza–Klein-type theories, we consider the case in which the internal manifold is a quotient space, G/H. We develop normal mode expansions on the internal manifold and show that the conventional gravitational plus Yang–Mills theory (realizing local G symmetry) is obtained in the leading approximation. The higher terms in the expansions give rise to field theories of massive particles. In particular, for the original Kaluza–Klein 4 + 1-dimensional theory, the higher excitations describe massive, charged, purely spin-2 particles. These belong to infinite dimensional representations of an O(1, 2).

A discrete set of solutions to the classical Einstein–Maxwell equations in six-dimensional space-time is considered. These solutions have the form of a product of four-dimensional constant curvature space-time with a 2-sphere. The Maxwell field has support on the 2-sphere where it represents a monopole of magnetic charge, n = ±1, ±2,…. The spectrum of massless and massive states is obtained for the special case of flat 4-space, and the solution is shown to be classically stable. The limiting case where the radius of the 2-sphere becomes small is considered and a dimensionally reduced effective lagrangian for the long range modes is derived. This turns out to be an SU(2) × U(1) gauge theory with chiral couplings.

The question of feimion chirality in Kaluza–Klein theories with coupling to Yang–Mills fields is discussed. The argument is illustrated in eight dimensions where an SU(2) Yang–Mills field assumes the one-instanton form on the internal space. This serves not only to trigger spontaneous compactification of the internal space but will ensure the emergence of zero modes in an irreducible eight-spinor belonging to the (2t + 1)-dimensional representation of SU(2).

We show that a gauged supergravity theory based on E₆ × E₇ × U(1) is free of gauge and gravitational anomalies in six dimensions. It compactifies to (Minkowski)⁴ × S² by the standard monopole mechanism. With a monopole of strength n in E₆, the resulting four-dimensional theory exhibits chiral SO(10) × U(1) with 2 |n| families (and no antifamilies). Supersymmetry is broken.

We construct an N = 1 locally supersymmetric σ-model with a Wess-Zumino term coupled to supergravity in two dimensions. If one takes the σ-model manifold to be the product of d-dimensional Minkowski space M^d and a group manifold G, and if the radius of G is quantized in appropriate units of the string tension, then the model describes a Neveu-Schwarz-Ramond (NSR)-type string moving on M_d × G. (Our model generalizes earlier work of refs. [1,2] which do not contain a Wess-Zumino term and that of refs. [5,6] which is not locally supersymmetric.) The zweibein and the gravitino field equations yield constraints which generalize those of the NSR model to the case of a non-abelian group manifold. In particular, the fermionic constraint contains a new term trilinear in the fermionic fields. We quantize the theory in the light-cone gauge and derive the critical dimensions. We compute the mass spectrum of a closed string moving on M_d × G and show that massless fermions do not arise for non-abelian G for the spinning string, in agreement with the result of Friedan and Shenker [22].

The propagation of closed bosonic strings interacting with background gravitational and dila-ton fields is reviewed. The string is treated as a quantum field theory on a compact 2-dimensional manifold. The question is posed as to how the conditions for the vanishing trace anomaly and the ensuing background field equations may depend on global features of the manifold. It is shown that to the leading order in σ-model perturbation theory the string loop effects do not modify the gravitational and the dilaton field equations. However for the purely bosonic strings new terms involving the modular parameter of the world sheet are induced by quantum effects which can be absorbed into a re-definition of the background fields. We also discuss some aspects of several regularization schemes such as dimensional, Pauli-Villars and the proper-time cut off in an appendix.

A generalization of Friedmann–Robertson–Walker cosmology to 4 + K dimensions is considered. The space-time manifold R¹ × S³ × S^K is characterized by two time-dependent scales, R (t) and a (t). The equations of motion for R and a are derived from the 4 + K-dimensional Einstein action supplemented by a one-loop thermal term, corresponding to a gas of non-interacting scalar particles. It is shown that in the approximation when T < 1/a the equations of motion admit a solution in which the internal space has a constant radius a while the external R(t) evolves in the usual manner.

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A simple gauge theory discussed recently in the literature as a model of high-temperature superconductors is examined. The model contains a Maxwell field and a Chern–Simons field coupled to fermions in (2 + 1)-dimensional spacetime. This model has been shown to exhibit a kind of Meissner effect at zero temperature, which originates in the 1-loop mixing between the two gauge fields. We use a euclidean effective action formulation to show that the effect persists at finite temperatures. Although a long-range magnetic type interaction arises at non-zero temperatures, in competition with the finite range forces which dominate the zero temperature interaction, the effect varies smoothly with temperature.

We study the coupling of an abelian Chern–Simons field to fermions in space–times of the form R × M², where M² is a compact riemannian manifold. Upon integrating out the non-zero modes of the Chern–Simons field, an effective N-particle hamiltonian is constructed, which involves a term representing the effects of the zero modes. We also study the transformation to the fractional statistics (anyon) basis. It is shown that unlike the case of the flat euclidean M² the anyon wave equation involves some residual metric dependent interactions, and the wave function is multivalued.

The role of chirality in the theories that determine the origin of life are reemphasized—in particular the fact that almost all amino acids utilized in living systems are of the L type. Starting from Z⁰ interactions, I speculate on an explanation of the above fact in terms of quantum mechanical cooperative and condensation phenomena (possibly in terms of an e–n condensate where the e–n system has the same status as Cooper-pairing), which could give rise to second-order phase transitions (including D to L transformations) below a critical temperature T_c. As a general rule, T_c is a low temperature. From this, it is conceivable that the earth provided too hot a location for the production of L amino acids. I suggest laboratory testing of these ideas by looking for the appropriate phase transitions.

“Atoms such as carbon, oxygen, nitrogen and hydrogen, the major constituents of biological molecules, are less than 0.4 nm in diameter…. The behaviour of small molecules is a reflection of the intrinsic properties of the constituent atoms. Hence it might be expected that the behaviour of large macromolecules can be explained by a knowledge of atomic properties. Since organelles, whole cells and organisms are essentially macromolecular assemblies, it may be possible in time to derive an atomic theory of life” [ A.R. Rees and M.J.E. Sternberg, From cells to atoms – An illustrated introduction to molecular biology (Blackwell, Oxford, 1984) p. 3 ]. It has been suggested that chirality among the twenty amino acids which make up the proteins may be a consequence of a phase transition which is analogous to that due to BCS superconductivity [A. Salam, J. Mol. Evol. 33 (1991) 105]. We explore these ideas in this paper and show, following Lee and Drell [I.H. Lee and S.D. Drell, in: Fermion masses in the standard model, M.A.B. Bég Memorial Volume, eds. A. Ali and P. Hoodbhoy (World Scientific, Singapore, 1991) p. 13], that a crucial form for the transition temperature T_c involves dynamical symmetry breaking. The t-quarks or supersymmetry (or something similar which ensures a heavy mass) appear to be essential if such mechanisms are to hold.

A generalization of the SU(2) spin systems on a lattice and their continuum limit to an arbitrary compact group G is discussed. The continuum limits are, in general, nonrelativistic σ-model-type field theories targeted on a homogeneous space G/H, where H contains the maximal torus of G. In the ferromagnetic case the equations of motion derived from our continuum Lagrangian generalize the Landau-Lifshitz equations with quadratic dispersion relation for small wave vectors. In the antiferro-magnetic case the dispersion law is always linear in the long-wavelength limit. The models become relativistic only when G/H is a symmetric space. Also discussed are a generalization of the Holstein-Primakoff representation of the SU(N) algebra, the topological term, and the existence of the instanton-type solutions in the continuum limit of the antiferromagnetic systems.

The following sections are included:

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