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What Is Light?

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What Is Light? What is light? Gordon McCabe September 9, 2020 Abstract The purpose of this paper is to answer the question, `What is light?', from a mathematical and foundational perspective. The paper begins by exploring the relevance of space-time symmetries, and the nature of polarization, before a detailed exposition of the quantized radiation field, and the difficulties created by gauge freedom. Attention then turns to the interaction of light with matter, beginning with the coupled electromagnetic field, then progressing to the representa- tion of scattering and virtual particles in quantum electrodynamics. This provokes an analysis of the Coulomb electrostatic field, and the question of whether longitudinal and scalar photons exist. It is argued that the lon- gitudinal component of the electric field is associated with the state-space of a charged matter field, not the Fock space of the free electromagnetic field, hence the presence of electrostatic fields is consistent with the vac- uum state of the free electromagnetic field. It is also argued that scalar and longitudinal photons do indeed exist as links in the spacelike networks into which the Coulomb interaction can be decomposed. Consideration of the stimulated emission of light leads to a general exposition and analysis of the `coherent states' of the quantized radiation field. As a by-product of this, a novel explanation is proposed for why there is something classical rather than nothing classical. An attempt is made to develop this into a fully-fledged universe creation scenario. The role of fermions and the gravitational degrees of freedom in such a scenario are discussed, and a comparison is drawn with the inflationary cosmological scenario. The role of coherent states in our concept of the classical world is then critically analysed. The notion that the classical states of the radiation field are emergent from the quantum states is rejected. In particular, it is argued that the classical states of light do not emerge in the limit where there are large numbers of photons, and it is pointed out that the putative emergent classical states fail the test of reference-frame independence. The paper concludes by expounding the implications of the nature of light for `decoherence', a way of trying to reconcile quantum theory with the apparent nature of the macroscopic world. 1 Contents 1 Introduction 3 2 What is light? 4 2.1 Classical electromagnetic waves . 4 2.2 Symmetry . 4 2.3 Polarization . 5 3 Quantum Field Theory 11 3.1 Fock space . 11 3.2 The quantum vacuum . 16 3.3 The quantized radiation field . 18 4 Interactions between light and matter 22 4.1 The coupled electromagnetic field . 22 4.2 Scattering, Feynman diagrams and virtual particles . 24 4.3 The interacting field vacuum . 28 4.4 Do longitudinal photons exist? . 31 4.5 Spontaneous and stimulated emission of light . 37 5 Coherent states 39 6 Why is there something classical rather than nothing classical? 46 6.1 Coherent states and the vacuum . 46 6.2 Fermions, Supersymmetry and Grand Unified Theories . 47 6.3 Gravity and gravitons . 49 6.4 Inflation . 53 7 Coherent states and the classical world 56 7.1 Are coherent states classical? . 56 7.2 Coherent scattering . 64 8 Decoherence 65 9 Conclusions 71 2 1 Introduction A plurality of mathematical structures exist for the representation of light: it is represented as bundles, pencils and beams of 1-dimensional rays in geometrical optics; as a tensor field satisfying a wave-equation in Maxwellian electromag- netism; and as the elements of a photonic Fock space in quantum field theory. All of which begs the question, `What is light?' The purpose of this paper is to answer that question from a mathematical and foundational perspective. The second section begins with a concise exposition of light as a classical electromagnetic wave. Noting that such waves can be decomposed into differ- ent modes and polarizations, prompts the question `What is polarization?' To answer this, we digress to consider light as a representation of the space-time symmetry group. Equipped with this perspective, we then consider the detailed mathematical representation of polarization. The third section of the paper begins with a brief recapitulation of the mathematical structure of Fock space and the quantum vacuum. In particular, the isomorphism between Fock space and the space of wavefunctionals on the one-particle space is made explicit. The remainder of the section is devoted to the quantized radiation field, and the problems created by gauge freedom. The fourth section considers how interactions between light and matter are represented, beginning with the coupled electromagnetic field in the classical theory, and progressing to the representation of interacting fields in quantum field theory. This provokes an analysis of the Coulomb electrostatic field, and the question of whether longitudinal and scalar photons exist. As a special case of the interaction between light and matter, the spontaneous emission of light is explained. This is then contrasted with the production of laser light by stimulated emission. Laser light is represented mathematically by coherent states in photonic Fock space, and the fifth section is devoted to an exposition and analysis of these privileged states of the radiation field. This analysis leads to a number of important philosophical points. Section six is devoted to the first of these, a possible explanation for why there is something classical rather than nothing classical. An attempt is made to develop this into a fully-fledged universe creation scenario. The role of fermions and the gravitational degrees of freedom in such a scenario are discussed, and a comparison is drawn with the inflationary cosmological scenario. The seventh section provides a general discussion of the implications of co- herent states for the concept of classicality. A number of the common beliefs, in particular the notion that the classical world emerges from the quantum world, are shown to be false. The paper concludes with a section on the implications of the nature of light for `decoherence', a way of trying to reconcile quantum theory with the apparent nature of the macroscopic world. 3 2 What is light? 2.1 Classical electromagnetic waves According to classical electromagnetism, light is a solution of the free-field Maxwell equations. Under a Lorentz choice of gauge, the electromagnetic po- tential is mapped to a real vector potential A satisfying r · A = 0. Given a real vector potential A, the electromagnetic field strength is F = dA. The Maxwell equations for a vector potential obtained with a Lorentz choice of gauge are simply: r · dA = 0; r · A = 0 : These equations are equivalent to, (Derdzinski 2002, Appendix 48): A = 0; r · A = 0 : The first equation here is the wave equation on Minkowski space-time M, 2 2 2 2 @ @ @ @ µ ( 2 − 2 − 2 − 2 )A = A = (Aµ)dx = 0 ; @x0 @x1 @x2 @x3 where is the d'Alembertian. Such solutions can be subjected to a two-fold decomposition: (i) a Fourier decomposition into monochromatic modes of different wavelengths; and (ii) a decomposition of monochromatic light into different polarizations. Both of these decompositions were part of classical optics, theoretically and in terms of the historical experiments which explored their nature. Yet whilst the Fourier decomposition is clearly classical, the polarization decomposition introduces a quantum-mechanical substructure into classical theory. Moreover, whilst numerous physicists are fond of claiming that there is no analogue of quantum spin in classical physics, the case of polarization provides a counter- example to such claims. To gain a better understanding of what polarization is, the next subsection considers the role of space-time symmetries. 2.2 Symmetry Wigner used the method of induced group representation to obtain an irre- ducible unitary Hilbert space representation of the universal cover of the local space-time symmetry group for every type of elementary particle. The largest group of space-time symmetries satisfied by all the elementary particles is the 3;1 restricted Poincar´egroup, SO0(3; 1)nR . This group does not contain time re- versal operations, parity transformations, or combinations thereof. The univer- sal cover of the restricted Poincar´egroup is SL(2; C) n R3;1, and the irreducible unitary representations of this group can be classified by two parameters, (m; s), mass and spin, hence each elementary particle is associated with a Hilbert space Hm;s. The photon is a particle of mass 0 and spin 1. The Wigner approach obtains + a Hilbert space H0;1 of square-integrable cross-sections of a vector bundle E0;s 4 + over the forward light cone V0 in energy-momentum space, with typical fibre isomorphic to C1. However, the electromagnetic force respects symmetries such as parity trans- formations. As a consequence, it is conventional to treat the interaction car- rier of the electromagnetic force, the photon, as an irreducible representation of O"(3; 1) n R3;1, the isochronous Poincar´egroup, (also called the orthochronous Poincar´egroup). This group consists of both the identity component of the Poincar´egroup, and the component which contains the operation of parity re- versal, P :(x0; x1; x2; x3) 7! (x0; −x1; −x2; −x3). Whilst the irreducible representations of the restricted Poincar´egroup are 1 parameterized by s 2 2 Z, the irreducible representations of the isochronous 1 Poincar´egroup are parameterized by t 2 2 Z+. For t 6= 0, the t-representation is a space of cross-sections of a bundle over the forward light cone with typical fibre isomorphic to C2. It can be decomposed into a direct sum of the s = t and s = −t representations. In the case of a mass zero particle, the discrete parameter s is often alternatively referred to as the `helicity' or the `polarization' of the particle, and the spin is treated as the absolute value jsj.
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