Quantum Mechanics, High Energy Physics and Accelerators: Selected Papers of John S Bell (With Commentary)

The following sections are included:

• CONTENTS

• Introduction

• Summary of John S. Bell's Scientific Career

• Notes on Section 1

• Notes on Section 2

• Notes on Section 3

• Bibliography

The axial motion of a particle in a linear accelerator is analysed taking account of the detailed distribution of accelerating field. It is shown that the frequency and rate of damping of small phase oscillations do not depend on the details of the distribution.

It is found that the effect of “flattening” of the electron velocity distribution is to increase the rate of cooling of small betatron oscillations by a factor of 2.4, and not by a factor of 4 as often quoted. This is when the cooler magnetic field is ignored. When it is allowed for, in the usual way, the cooling rate involves a divergent integral whose regulation depends on the details of particular cases.

Simplified formulae for the capture of low-energy electrons by stationary protons are averaged over Maxwellian and “flattened” Maxwellian electron velocity distributions. The latter distribution is more nearly appropriate for electron beams used in accelerator proton-beam cooling experiments. Flattening increases the capture rate by a factor of about two. Similar formulae for the capture of antiprotons by protons are mentioned.

A general formula is presented for the damping of small oscillations about closed orbits in classical mechanics by dissipative perturbations. It is based on the variation of Lagrange invariants. It is applied to rederive the standard results for the effects of classical radiation damping on storage-ring orbits.

The possibility of using accelerated electrons to exhibit the quantum field theoretic relation between acceleration and temperature is considered. In principle, the depolarization of electrons in a magnetic field could be used to give the temperature reading. The effect is examined for linearly accelerated electrons, but the result is that the relevant orders of magnitude are too small for real experiments in linear accelerators. For electrons in storage rings sufficiently large accelerations can be obtained, and the residual depolarization which has been found theoretically and experimentally is shown to be an effect closely related to the thermal effect of linearly accelerated electrons.

The Unruh effect (heating by acceleration in vacuum) is discussed for the case of extended thermometers. An ambiguity in the temperature, corresponding to the well-known ambiguity in temperature of an extended body in thermal equilibrium in a gravitational field, is noted. The generality of the effect is related, following Sewell, to PCT symmetry. Application is made to the case of a spin-half particle in a confining and accelerating electromagnetic field.

The quantum fluctuation of electron orbits in ideal storage rings is a sort of Fulling-Unruh effect (heating by acceleration in vacuum). To spell this out, the effect is analyzed in an appropriate comoving, and so accelerating and rotating, co-ordinate system. The depolarization of the electrons is a related effect, but is greatly complicated by spin-orbit coupling. This analysis confirms the standard result for the polarization, except in the neighbourhood of a narrow resonance.

The Sokolov-Ternov-Matveev-Nikishov-Ritus-Baier-Katkov formula for synchrotron radiation is averaged over the field of a cylindrical space charge, neglecting end effects, in connection with the beamstrahlung problem in linear colliders. The result differs slightly from that given recently by Blankenbecler and Drell. The discrepancy is traced to helicity-flip transitions.

End effects in quantum beamstrahlung, in the case of a uniform deflecting field, in the extreme quantum limit, are calculated. The result is applied to beamstrahlung with uniform cylindrical bunches and small disruption.

Inhomogeneity effects in quantum bremsstrahlung, in the case of a weakly nonuniform deflecting field, are calculated in the extreme quantum limit.

The idea of reversibility in time as applied to quantized fields is expounded from first principles. It is shown that in the usual theories reversibility of a certain kind is a concomitant of relativistic invariance and symmetry in space. Finally the relations between transition amplitudes consequent on the reversibility are derived.

In recent work on the nuclear many-body problem momentum-dependent potentials have been used to describe the motion of individual particles, and this has led to some uncertainty in regard to electromagnetic interactions, especially magnetic moments. In this paper the causes of momentum dependence of the potentials are discussed, and it is shown that some of these do not lead to any change in magnetic moments, whilst others may affect them but not simply in the way indicated by the ‘reduced mass’ approximation to the potentials. The magnetic moment operator is in any case ambiguous in that it involves the unknown functions familiar from studies of exchange currents.

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T. H. R. Skyrme's variation principle for the one-nucleon propagator is cast into a new form which is essentially a functional Fourier transform of the original.

The formalism of the preceding paper is applied to calculate the magnetic moments of nucleons and the neutron-electron interaction in the symmetric PS–PS theory. Vacuum polarization is ignored, and a very simple trial function employed. With a suitable coupling constant, and a reasonable cut-off for the single logarithmically divergent integral that appears, a rather rough fit to the observed values is obtained.

It is found impossible to give a consistent account of mirror transitions involving nuclei with LS-closed cores plus or minus one particle in terms of conventional beta-decay theory and using shell model wave functions. Possible explanations of the discrepancies are discussed and it is concluded that there is some evidence for mesonic effects of an exchange character.

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It has been predicted, by largely classical arguments, that destructive interference of the radiation from many collisions greatly reduces the rate of energy loss by very high energy electrons undergoing multiple scattering. It is observed that this is incompatible with a standard formula in classical theory, and the discrepancy is traced to an inadequate account of the high frequency part of the radiation spectrum. It is concluded that any real effect of this kind is of essentially quantal origin.

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Charge and current distributions are defined for unstable particles in no-recoil theories. They are in general complex, and oscillate with amplitudes increasing exponentially at large distances. Moments can be defined nevertheless. As with stable particles, electric dipole moments are necessarily zero if time reversibility holds.

The following sections are included:

• Introduction

• Theory

• Experiment

• Discussion

Estimates are made of cross-sections for the production of pions or strange particles by high-energy neutrinos incident on nuclei. At 1 GeV the production of pions via the (33) resonance is found to augment the total cross-section by about 50%.

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A theorem of statistical mechanics relates density fluctuations to compressibility. A new derivation of this is given. The theorem is violated in the BCS model of a superconductor. The difficulty is resolved by those same improvements in the theory which lead to a gauge-invariant Meissner effect.

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The following sections are included:

• ν_μ ≠ ν_e

• Neutrino Flip

• Lepton Conservation

• Neutral Lepton Currents

• Elastic Events

• Inelastic Events

• Lepton Pairs

• References

The following sections are included:

• The Coherent Process

• The Incoherent Process

• Total Cross Section

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The theory of the Primakoff plot is developed in a simple way. Corrections advocated by KLEIN and WOLFENSTEIN are shown to be absent when reasonable correlation functions, such as those of the ideal Fermi gas, are used. However, development of the gas model throws doubt on the accuracy of the theory. The excellent agreement with experiment must be in part accidental.

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Models are constructed with double poles in partial-wave scattering amplitudes. The associated unstable particles have nonexponential decay laws which contain parameters dependent on the production and detection arrangements.

The Ũ₁₂ symmetry is developed along lines used for SL(6, C) by Rühl and by Katz. There are 143 components of momentum. The Wigner-Bargmann equations are formally invariant. The little group is U₆×U₆, rather than SU₆, which explains the increased dimensions of multiplets. New « irregular » couplings are permitted. Charge conjugation and the substitution rule are briefly explained.

In this review the emphasis will be on developments associated with CP violation. We will be concerned both with phenomenology and with experiments and experimental results.

An equal-time commutator in the Lee model is examined. It is found that a related sum rule does not have the canonical value. On the other hand, in the corresponding zero-energy theorem the canonical value is correct.

It is observed that in pion beta decay the coupling constants are dictated by current algebra and PDDAC, independently of CVC.

It is urged that the lesson of gauge invariance in quantum electrodynamics implies the irrelevance of « Schwinger term » difficulties in current algebra. The divergence equations of Veltman form the basis of a gauge-variation formalism in which these questions are avoided.

A critical review is given of developments in the theory subsequent to the demonstration that in first approximation η→ 3π is forbidden. It remains difficult to understand the process with conventional ideas without reducing to an accident the success of current algebra, and a simple linear matrix element, in K → 3π.

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The following sections are included:

• Introduction

• Gauge Invariance

• KLWZ formalism

• References

The following sections are included:

• Introduction

• Optical Model for Virtual Mesons

• Nuclear Absorption

• Eikonal Approximation

• Application to Photoproduction of Vector Mesons

• Coupled Propagation

• Neutrino Reactions

• Final Remarks

• References

The effective coupling constant for π° → γγ should vanish for zero pion mass in theories with PCAC and gauge invariance. It does not so vanish in an explicit perturbation calculation in the σ-model. The resolution of the puzzle is effected by a modification of Pauli-Villars-Gupta regularization which respects both PCAC and gauge invariance.

Burnett and Kroll's extension of Low's theorem is proved for particles of arbitrary spin.

For elastic scattering of particles of arbitrary spin, an invariant amplitude is constructed which has the « positivity » required for Martin's proof of Froissart bounds.

The possibility of bounding the hadronic vacuum polarization contribution to the muon magnetic moment anomaly is considered. A proposed theoretical bound, based on the hypothesis of field current identity, is found not to add usefully to the vector meson dominance estimate. However, it is observed that improved electron–electron scattering experiments will yield a useful empirical bound.

A simple model is considered for near-forward neutrino reactions on nuclear targets in the conditions of the CERN experiment. The model accounts for the failure to observe the shadow effect, which is compensated by the elastic interaction and masked by unshadowed terms which grow rapidly away from the forward direction.

It is argued that final–state interaction effects can become appreciable only if certain CP violating amplitudes are unexpectedly large. If such large hitherto hidden violations exist, then variation of the asymmetry over the Dalitz plot can yield otherwise unobtainable information about them.

Calculations of the Pauli exclusion effect are made in a simple shell model. Shell structure effects, associated with spin–orbit splitting, appear, but are not large and decrease rapidly with increasing momentum transfer. The shell model remains less exclusive on the whole than the standard Fermi gas, but not sufficiently so to explain the apparent experimental absence of the exclusion effect.

The following sections are included:

• Introduction

• Coherent Meson Propagation

• Neutrino Reactions

• The Pion Pole

• Goldberger-Treiman Conspiracy

• Adler's Formula

• Experiment

• Generalized Optical Model

• 33 Optical Bar Model

• Longitudinal and Transverse Inelastic

• Quasi-Elastic

• Conclusion

• References

Tree diagram scattering amplitudes may behave extra badly at high energy when vector mesons are involved. The cancellation, among bad terms, implied by gauge invariance is discussed. It is shown that in the renormalizable spontaneously broken gauge theories such cancellation is complete.

The formalism of light-plane wave functions is applied to the problem of diffractive excitation. Suggestions in the literature, for removing an experimentally embarrassing forward zero in the single scattering approximation to Glauber theory, are not borne out.

The following sections are included:

• Introduction

• Charge and Symmetries

• Appendix

• References

Identification of SU(6)W, Currents with SU(6)W, Constituents is shown to exclude the existence of the 35(L_z = 0) mesons.

In reactions such as ēe → anything, , the final state (neglecting secondary isospin violating decays) is customarily supposed to have isospin zero or one. We show that for such states the average fraction (X) of the energy carried away by neutral pions is bounded by , when I₃ = 0 (e.g., for ee, pp, nn) and by , when I₃ = ± 1 (e.g., for pn or np).

Assuming a non-derivative point interaction, and Born approximation, there are some simple relations between neutrino and antineutrino scattering on electrons or partons. They have been observed already, for some special cases, in the results of explicit calculations. Here they are obtained from simple general considerations.

The null-plane dynamics of hydrogen-like atoms is studied in approximations depending on c, the velocity of light, being large. Neglecting terms in the Hamiltonian of order c⁻³ (relative to electron rest energy) a symmetry SU(2)_W appears which is analogous to the SU(6)_W of hadron classification. This symmetry, if accurate, would dictate zero ground state magnetic moment. The symmetry is broken by terms of third order, which can, however, be transformed away by the appropriate approximation to the Melosh transformation. There then emerges a better symmetry, SU(2)_M broken only at fourth order. The ground state magnetic moment acquires its usual non-relativistic value. The symmetry SU(2)_M corresponds to a subgroup of a symmetry [U(2) × U(2)]_FW which appears in the old Foldy-Wouthuysen approach when spin-orbit coupling is neglected. As well as “current” and “constituent” pictures, “classification” pictures are distinguished; it is to one of the latter that the Melosh transformation transforms.

The null-plane dynamics of positronium-like systems is studied in a quasipotential approach, in approximations depending on c, the velocity of light, being large. Neglecting terms in the quasi-potential of order c⁻³ (relative to particle rest energy) symmetries appear whose generators are analogous to the good null-plane charges of relativistic SU(6) hadron classification schemes. These symmetries are broken by terms of order c⁻³, which can, however, be transformed away by an appropriate approximation to the Melosh transformations. There then emerge better symmetries, broken only at order c⁻⁴. When both internal and overall velocities of a composite system are small of first order, and terms of second and higher order are neglected, these better symmetries coincide with those that appear in a Foldy-Wouthuysen description when spin-orbit coupling is neglected. More generally the better symmetries remain good for such a composite system boosted arbitrarily in the three-direction. For this reason we find the original, boost invariant, “first” Melosh transformation more appropriate than the “second” Melosh transformation.

I am indebted to Professor F. Krienen for this problem: work out the basic formula for transition radiation in the case of particles with spin. In deriving this formula a corresponding result is obtained, in passing, for Čerenkov radiation. In that case comparison can be made with some results already in the literature; there is sufficient agreement, except at one point, in connection with dispersion.

When a particle decays in an electric field, and parity is violated, the mean angular momentum of the products is tilted with respect to the angular momentum of the parent. It is in this way that the Zel'dovich electric-dipole moment manifests itself; there is no linear Stark effect, and no precession of the spin of undecayed particles, when time reversal is respected.

The valence and sea quark parton distributions are calculated for deep inelastic scattering from a one-dimensional Dirac box. The sea contribution corresponds explicitly to the excitation of quark–antiquark pairs from the “vacuum” within the box. The connection between coordinate-space wave functions and null-plane parton distributions is explicitly given and we comment on the relevance of our results to those of other recent calculations.

Calculation of partem distributions in the “cavity approximation” to the MIT bag model gives a divergent sum of positive terms. This suggests that Bjorken scaling does not hold for the deep inelastic scattering in this version of the model.

The semi classical formula due to Krammer and Leal-Ferreira, and Quigg and Rosner, for the S wave bound state wave function at the origin, is generalized to all partial waves, and the relation to duality is traced to a common dependence on short time behaviour.

The oscillatory and non-oscillatory regions of radial wave functions are connected by the use of spherical Bessel functions or Coulomb wave functions. The resulting formulae for the magnitude of the wave function at the origin in terms of the energy spectrum are used to exhibit the appropriate form of bound-state/free-state duality.

The process of electron–positron annihilation into quark–antiquark is considered with non-relativistic potential account of final state interaction. Calculations are made for confining potentials, with discrete bound final states, and for corresponding non-confining potentials, with continua of finally free quark–antiquark pairs. When suitably averaged over energy the two ways of calculating agree closely, illustrating well the local duality concept.

The moment method of Shifman, Vainshtein and Zakharov, for calculating bound-state energies in QCD, is tested in the context of potential models. For simple power-law potentials of low degree a refined version of the method works surprisingly well. The cruder version actually used by Shifman, Vainshtein and Zakharov for charmonium works less well, and the composite potentials usually envisaged for charmonium are less accurately dealt with than simple power potentials. We conjecture then that the magnitude of their confinement parameter ϕ has been substantially underestimated by those authors.

From the SVZ moments, for heavy quark–antiquark systems, we extract an “equivalent” non-relativistic potential. It increases as the fourth power of the distance and linearly with the quark mass. Numerical integration of the “equivalent” Schrödinger equation leads to much higher estimates of the gluon condensate parameter φ₁ than the original inverse power moment method or even the Borel transformed exponential version.

Can certain soliton states, with half integral expectation value of charge, be also eigenstates of charge X with half integral eigenvalue? It can be so only with a somewhat sophisticated definition of charge.

Attempts to represent the “gluon condensate”, of Shifman, Vainshtein and Zakharov, by a quark-antiquark potential, are critically reviewed. In particular, it is noted that the Leutwyler-Voloshin level shifts, in hydrogen-atom-like heavy quarkonia, are well reproduced by a static potential. But it is mass dependent. No adequate bridge is found between the field theory of Shifman, Vainshtein, and Zakharov, on the one hand, and popular potential models on the other.

In certain molecular models, and related one-dimensional field theories, localized objects appear with half-integral expectation values of charge. We consider whether these states are eigenstates of charge, with half-integral eigenvalue. We find that it is indeed so for a suitably diffuse definition of the charge operator in question. This diffuse charge operator has a spectrum which approaches a continuum. The analysis is made on a lattice, to avoid divergence ambiguities, and on a finite length, which is only subsequently made large. The half-integral charge phenomenon is not tied to solitons, but can also arise as an end effect.

The accuracy of the Shifman–Vainshtein–Zakharov use of QCD sum rules for charmonium, to determine the value of their gluon condensate parameter, is further examined. The method does not work well for a potential which gives very similar “moments”.

The demonstrations of von Neumann and others, that quantum mechanics does not permit a hidden variable interpretation, are reconsidered. It is shown that their essential axioms are unreasonable. It is urged that in further examination of this problem an interesting axiom would be that mutually distant systems are independent of one another.

The following sections are included:

• Introduction

• Formulation

• Illustration

• Contradiction

• Generalization

• Conclusion

• References

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The following sections are included:

• Motivation

• The absence of disperse-free states in various formalisms derived from quantum mechanics

• A simple example

• A difficulty

• References

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The quantum mechanical measurement problem is considered in a model due to Hepp and Coleman. Whereas Hepp emphasized a ‘rigorous “reduction of the wave packet”’, in a certain mathematical limit, it is emphasized here that no such reduction ever actually occurs. Some general remarks are made on the advantages of the Heisenberg picture for such considerations, especially in connection with extension to relativistic theories. The non-reduction of the wave packet is directly related to the deterministic character of Heisenberg equations of motion.

The following sections are included:

• Local Determinism

• Local Causality

• Quantum mechanics is not locally causal

• Locality inequality

• Quantum mechanics

• Experiments

• Messages

• Reservations and Acknowledgements

• References

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The following sections are included:

• Why worry?

• But is not all this just as in classical physics?

• But is it really true?

• Appendix. Einstein and Hidden Variables

• Notes and References

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In recent years there have been several experiments on polarization correlation between photons emitted in atomic cascades. They are supposed to bear on the notion that the consequences of events do not propagate faster than light. This notion is difficult to reconcile with quantum-mechanical predictions for idealized versions of the experiments in question. The present Comment offers a brief introduction to the situation.

The de Broglie–Bohm version of quantum mechanics is applied to the delayed-choice double-slit experiment. The role of the density matrix is considered.

The following sections are included:

• Introduction

• Common Ground

• The Problem

• The Pilot Wave

• Everett (?)

• References

The Einstein-Podolsky-Rosen correlations are very like many ordinary occurrences of everyday life. So it is a little difficult for the man in the street to understand immediately why there has been so much fuss about them. It must be recalled that the founding fathers of quantum mechanics had convinced themselves that it was necessary to abandon the idea of an objective reality at the microphysical level. But the correlations in question, together with the idea of local causality, were a formidable argument for such a reality. The founding fathers offered counter-arguments (neither very clear nor very convincing in my opinion) and each side held to its position. Since then it has been possible to push the analysis a little further, considering especially situations just a little different from those considered before. Then correlations appear, according to quantum mechanics, which are not at all like those of everyday life. As a result it is not now easy to believe, with Einstein, that quantum mechanical predictions are reconcilable with the notion of a Lorentz invariant objectively real microphysical world.

The strange story of the von Neumann impossibility proof is recalled, and the even stranger story of later impossibility proofs, and how the impossible was done by de Broglie and Bohm. Morals are drawn.

The following sections are included:

• Introduction

• Local Beables

• Dynamics

• OQFT and BQFT

• Concluding Remarks

• Notes and References

In the case of two free spin-zero particles, the wave function originally considered by Einstein, Podolsky and Rosen to exemplify EPR correlations has a non-negative Wigner distribution. This distribution gives an explicitly local account of the correlations. For an irreducible non-locality, more elaborate wave functions are required, with Wigner distributions which are not non-negative.

If we have to go on with these damned quantum jumps, then I'm sorry that I ever got involved. E.Schrödinger.

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The following sections are included:

• Introduction

• What cannot go faster than light?

• Local beables

• No signals faster than light

• Local commutativity

• Who could ask for anything more

• Principle of local causality

• Ordinary quantum mechanics is not locally causal

• Locally explicable correlations

• Quantum mechanics cannot be embedded in a locally causal theory

• But still, we cannot signal faster than light

• Conclusion

• Appendix: History

• References

• Appendix

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