The Photomagneton and Quantum Field Theory

The following sections are included:

• PREFACE

• CONTENTS

Magnetization by light, an experimentally observed phenomenon, is interpreted to mean that the photon has three polarizations, right and left circular and longitudinal. The third is accompanied by a photomagneton which has no energy or linear momentum and which is generated from photon spin. This is consistent with photon mass, however small in magnitude, and with Planck's radiation law. It implies that there is an anti-photon, whose kinematics are the same as those of the photon, but whose concomitant fields are changed in sign. The photon is therefore assigned a number F, akin to charge, baryon and lepton number and strangeness. Reversing the sign of F produces the anti-photon.

The concepts are introduced of the longitudinal ghost field B⁽³⁾ and photomagneton of electromagnetism: where B⁽⁰⁾ is the magnetic flux density amplitude and Ĵ the angular momentum operator of a photon beam. The major implication is that the individual photon has three degrees of polarization, the longitudinal one being accompanied by the ghost field B⁽³⁾ which has no energy or linear momentum, and is generated from the angular momentum of the photon.

It is shown that the antisymmetric part of the standard light intensity tensor is proportional in vacuo to a longitudinal magnetic field, denoted B⁽³⁾. This means that the cross product of electric or magnetic components of the usual transverse electromagnetic waves are linked to B⁽³⁾. The latter is a real, physically meaningful, uniform magnetic field. It is shown by considerations of energy conservation that its concomitant electric field is the imaginary iE⁽³⁾, directed in the same, propagation, axis of the transverse plane waves. Thus, B⁽³⁾ and iE⁽³⁾ are shown to be consistent with Poynting's theorem and with the inverse Faraday effect, the magnetization of material by circularly polarized light. The net contribution of B⁽³⁾ and iE⁽³⁾ to electromagnetic energy density, U, is zero,

It is shown that the effect of a circularly polarized laser field on an electron can be described in terms of a latent or equivalent magnetic field B⁽³⁾, defined through the vector product E⁽¹⁾ × E⁽²⁾ of the rotating electric field E⁽¹⁾ and its complex conjugate E⁽²⁾. The study of the intensity dependence of the magnetization in a one electron plasma reveals whether the field B⁽³⁾ has an intrinsic meaning in free space, or whether it becomes effective only when the laser beam interacts with matter. In the former case, there is a square root intensity dependence present in addition to the intensity dependence normally expected through the inverse Faraday effect. In both cases, the field B⁽³⁾ is the fundamental property responsible for magnetization by light, and is defined through the product E⁽¹⁾ × E⁽²⁾.

It is shown that there exist real magnetic and imaginary electric longitudinal components of electromagnetism in free space. These phase independent electric and magnetic fields are shown to be rigorously consistent in the Lorentz gauge with the concepts of relativistic quantum field theory, in which admixtures of the time-like and longitudinal space-like photon polarizations may be physically meaningful.

Experimental observation of the inverse Faraday effect (phase free magnetization by light) is shown to be proportional to the conjugate product of oscillating magnetic field components B⁽¹⁾ × B⁽²⁾ = iB⁽⁰⁾B⁽³⁾, where B⁽³⁾ is a longitudinal magnetic field associated with free space electromagnetism and where B⁽⁰⁾ is the scalar amplitude. It is demonstrated rigorously that the field B⁽³⁾ can be expanded in terms of irreducible compound tensors, the vector spherical harmonics, which are eigenvectors of the rotation operator for a vector field. Such an analysis implies that the neglect of B⁽³⁾ in standard electrodynamics is equivalent to a violation of fundamental group theory. The inclusion of B⁽³⁾ implies that in the quantum field, the photon becomes a rigorously well defined boson, with three eigenvalues, as for any boson of unit spin angular momentum. The field B⁽³⁾ is linked geometrically to the transverse wave fields B⁽¹⁾ and B⁽²⁾ in free space, but is phase free and has no Planck energy. It is therefore a ghost field of electromagnetism which cannot be absorbed or re-emitted in field-matter interactions.

A circularly polarized pump laser applied to atomic or molecular matter rotates the plane of polarization of a linearly polarized probe laser whose propagation direction is parallel to that of the pump. We refer to this as the optical Faraday effect, and develop a theory based on the B⁽³⁾ field of the pump, a theory which shows that the molecular property tensors responsible for the optical Faraday effect (OFE) are the same as those which are used to describe the usual Faraday effect. This means that the effect of the B⁽³⁾ field of a pump laser is to shift an MCD spectrum upwards or downwards in frequency, depending on the sense of circular polarization of the pump laser. This effect occurs without distortion of the original MCD spectrum, and is precisely what has been observed experimentally recently by Sanford et. al. [1] in cadmium chromium selenium ferromagnetics. This is strong evidence for first order effects based on the B⁽³⁾ field of the pump laser. These first order OFE effects are distinguished from second order equivalents based on the conjugate product of the pump laser. The latter are mediated by different molecular property tensors, and would distort as well as shift the normal MCD profile of a sample treated with a circularly polarized pump laser. Such distortions were not observed by Sanford et. al. [1].

The classical relativistic theory is developed of electric and magnetic fields in terms of boost and rotation generators, respectively, of the Lorentz group of space-time. This development shows that Minkowski geometry requires that there be three states of polarization of radiation in free space. The magnetic components in a circular basis are right and left circular and longitudinal. The longitudinal component is real and physical, and proportional to one of the three, non-zero, rotation generators of the Lorentz group. The longitudinal electric component is pure imaginary, and proportional to one of the three boost generators. These theoretical arguments conform with experimental data from the Planck radiation law and from magnetic effects of light such as the inverse Faraday effect.

The relation is developed between rotation generators of the Lorentz group and the magnetic fields of free space electromagnetism. Using these classical relations, it is shown that in the quantum field theory there exists a longitudinal photomagneton, a quantized magnetic flux density operator which is directly proportional to the photon spin angular momentum. Commutation relations are given in the quantum field between the longitudinal photomagneton and the usual transverse magnetic components of quantized electromagnetism. The longitudinal component is phase free, the transverse components are phase dependent. All three components can magnetize material in general, but only the transverse components contribute to Planck's law. The photon therefore has three, not two, relativistically invariant degrees of polarization, an axial, longitudinal, polarization, and the usual right and left circular transverse polarizations. Since the longitudinal polarization is axial, it is a phase free magnetic field.

Some consequences of finite photon mass are reviewed in electromagnetic field theory, using as a conceptual framework the Einstein-de Broglie interpretation of wave particle dualism, recently supported by experiment. It is demonstrated that finite photbn rest mass (m₀) embodied in the Proca field equation, is consistent with Special relativity and with gauge transformation of the second kind provided , where A_μ is the (complex) potential four-vector. Finite m₀ is, however, inconsistent with the transverse, radiation, or Coulomb gauge, this being a subsidiary condition which is used routinely, but which is inconsistent with special relativity. Finite photon mass leads to longitudinal magnetic and electric fields in free space, fields which conserve the fundamental discrete symmetries of nature, which remain finite in the limit m₀ → 0, and which are related to the corresponding transverse field components through Lie algebra. The limit m₀ → 0 represents the transition from the Proca to the Maxwell formalism. The longitudinal fields B⁽³⁾ and iE⁽³⁾ respectively are frequency and phase independent components of four-vectors and B_μ and E_μ, so that electromagnetic energy density is represented by Poincaré invariants such as E_μE_μ and B_μB_μ. Although B⁽³⁾ and iE⁽³⁾ have no Planck energy (i.e. correspond to zero frequency) they are expected to produce a variety of novel spectral effects in the laboratory, effects which if observed, would provide evidence for finite photon mass.

The following sections are included:

• Introduction

• References

The molecular theory of optical NMR spectroscopy is developed and illustrated with an example of an optical NMR mechanism in which a magnetic electronic dipole moment is induced in a molecule by an applied laser field. This results in bulk and site specific shifts, both types of shift being induced simultaneously, leading to a new analytical technique of general utility. Irreducible spherical tensorial representations are given for the mediating hyperpolarizability molecular property tensor relevant to this mechanism of induction by light of a magnetic electronic dipole moment. Non-vanishing elements of are given for all the molecular point groups, and examples of point group character tables containing scalar elements of are given for a few point groups. The theory is worked out explicitly for the group 0, the octahedral group without inversion, in which examples of point group character tables containing scalar elements of are given for a few point groups. The theory is worked out explicitly for the group 0, the octahedral group without inversion, in which Gruffydd coefficients are available, showing that the laser shifts the original nuclear resonances in this point group in the context of the mechanism considered here. The site specific name of ONMR is developed semi-qualitatively by considering atomic and bond susceptibility and hyperpolarizability models.

Optical NMR spectroscopy is a technique which has been demonstrated experimentally and theoretically. Laser irradiation of a sample in an NMR spectrometer produces chemical shifts which are different for each resonating nucleus in a molecule. It is shown that optical NMR and ESR spectroscopy can be interpreted generally as a shielding phenomenon, in which the applied laser generates an effective shielding constant σ^(ind). The calculation of σ^(ind) involves several different mechanisms of induction in general, and σ^(ind) is a function of the molecular structure, varying from chromophore to chromophore or atom to atom. In principle therefore, ONMR spectroscopy provides a uniquely new fingerprint of any sample, because the laser induced shielding is unique to that sample.

It is shown that the conjugate product of transverse electric field solutions of Maxwell's equations produces a magnetic field, which is longitudinally polarized in non-conducting and conducting media. The electromagnetic field in these media is therefore manifestly covariant, in that there are three physically meaningful space-like polarizations and one time-like polarization. In non-conducting media, the longitudinal magnetic field is uniform, independent of time, and solenoidal. In conducting media it is independent of time, but is damped exponentially. The phenomena of optical reflection and refraction are considered at an interface between two non-conducting media, and Snell's law is expressed in terms of these novel longitudinal magnetic fields.

The conservation or violation of natural discrete symmetries such as , and Ĉ must be deduced from the mathematical equations describing physical laws. It is shown that recent attempts to by-pass these equations with simple diagrams lead to spurious and as often as not, erroneous, results.

It has recently been deduced theoretically that the photon generates a magnetostatic flux quantum , a concept which appears to be fundamental to physical optics. Considering the classical B⁽³⁾ it is shown that it is directly responsible for several well known phenomena, including circular dichroism and the development of beam ellipticity in the electrical Kerr effect. The reason is that B⁽³⁾ is directly proportional to the well known Stokes parameter S₃, a parameter whose origin resides therefore in the photon's elementary magnetic field.

In the Einstein-de Broglie theory of electromagnetic radiation, the photon has a rest mass of 10⁻⁶⁸ kg (the Einstein photon mass). Within the framework of this theory, it is shown that there exist longitudinal electromagnetic fields which are analytically related to the corresponding transverse components, implying that electric and magnetic fields in vacuo are four-vectors, as first proposed by Einstein and de Broglie. The observation of these longitudinal fields would support the Einstein-de Broglie theory, and experimental arrangements are proposed.

The following sections are included:

• Appendix 1: THE EFFECT OF B⁽³⁾ AND iE⁽³⁾ ON THE FUNDAMENTALS OF THE OLD QUANTUM THEORY

• Appendix 2: LONGITUDINAL FIELDS IN FREE SPACE AND THE DUAL TRANSFORM OF SPECIAL RELATIVITY

• Appendix 3: THE VECTOR POTENTIAL OF B⁽³⁾

• Appendix 4: THE COMPLETE SET OF CYCLICALLY SYMMETRIC RELATIONS IN VACUO