Light - The Physics of the Photon

LIGHT

The Physics of the Photon

Ole Keller

Aalborg University, Denmark

LIGHT

The Physics of the Photon

Cover image: Courtesy of Esben Hanefelt Kristensen, based on a painting entitled “A Wordless Statement.”

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In memory of my mother, Cecilie Marie Keller

Contents

Preface xiii

Acknowledgments xix

About the author xxi

I Classical optics in global vacuum 1

1 Heading for photon physics 3

2 Fundamentals of free electromagnetic fields 7

2.1 Maxwell equations and wave equations . . . . . . . . . . . . . . . . . . . . 7

2.2 Transverse and longitudinal vector fields . . . . . . . . . . . . . . . . . . . 8

2.3 Complex analytical signals . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.4 Monochromatic plane-wave expansion of the electromagnetic field . . . . . 13

2.5 Polarization of light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.5.1 Transformation of base vectors . . . . . . . . . . . . . . . . . . . . . 14

2.5.2 Geometrical picture of polarization states . . . . . . . . . . . . . . . 15

2.6 Wave packets as field modes . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.7 Conservation of energy, moment of energy, momentum, and angular momen￾tum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.8 Riemann–Silberstein formalism . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.9 Propagation of analytical signal . . . . . . . . . . . . . . . . . . . . . . . . 24

3 Optics in the special theory of relativity 27

3.1 Lorentz transformations and proper time . . . . . . . . . . . . . . . . . . . 27

3.2 Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.3 Four-vectors and -tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.4 Manifest covariance of the free Maxwell equations . . . . . . . . . . . . . . 33

3.5 Lorentz transformation of the (transverse) electric and magnetic fields. Du￾ality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.6 Lorentz transformation of Riemann–Silberstein vectors. Inner-product invari￾ance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

II Light rays and geodesics. Maxwell theory in general

relativity 39

4 The light-particle and wave pictures in classical physics 41

5 Eikonal theory and Fermat’s principle 45

5.1 Remarks on geometrical optics. Inhomogeneous vacuum . . . . . . . . . . . 45

5.2 Eikonal equation. Geometrical wave surfaces and rays . . . . . . . . . . . . 47

5.3 Geodetic line: Fermat’s principle . . . . . . . . . . . . . . . . . . . . . . . . 52

vii

viii Contents

6 Geodesics in general relativity 55

6.1 Metric tensor. Four-dimensional Riemann space . . . . . . . . . . . . . . . 55

6.2 Time-like metric geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

6.3 The Newtonian limit: Motion in a weak static gravitational field . . . . . . 59

6.4 Null geodesics and “light particles” . . . . . . . . . . . . . . . . . . . . . . 61

6.5 Gravitational redshift. Photon in free fall . . . . . . . . . . . . . . . . . . . 62

7 The space-time of general relativity 67

7.1 Tensor fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

7.2 Covariant derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

7.3 Parallel transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

7.4 Riemann curvature tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

7.5 Algebraic properties of the Riemann curvature tensor . . . . . . . . . . . . 73

7.6 Einstein field equations in general relativity . . . . . . . . . . . . . . . . . . 74

7.7 Metric compatibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

7.8 Geodesic deviation of light rays . . . . . . . . . . . . . . . . . . . . . . . . 76

8 Electromagnetic theory in curved space-time 79

8.1 Vacuum Maxwell equations in general relativity . . . . . . . . . . . . . . . 79

8.2 Covariant curl and divergence in Riemann space . . . . . . . . . . . . . . . 80

8.3 A uniform formulation of electrodynamics in curved and flat space-time . . 81

8.3.1 Maxwell equations with normal derivatives . . . . . . . . . . . . . . 81

8.3.2 Maxwell equations with E, B, D, and H fields . . . . . . . . . . . . 83

8.3.3 Microscopic Maxwell–Lorentz equations in curved space-time . . . . 84

8.3.4 Constitutive relations in curved space-time . . . . . . . . . . . . . . 85

8.3.5 Remarks on the constitutive relations in Minkowskian space . . . . . 87

8.3.6 Permittivity and permeability for static metrics . . . . . . . . . . . . 88

8.4 Permittivity and permeability in expanding universe . . . . . . . . . . . . . 89

8.5 Electrodynamics in potential description. Eikonal theory and null geodesics 91

8.6 Gauge-covariant derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

III Photon wave mechanics 97

9 The elusive light particle 99

10 Wave mechanics based on transverse vector potential 105

10.1 Gauge transformation. Covariant and noncovariant gauges . . . . . . . . . 105

10.2 Tentative wave function and wave equation for transverse photons . . . . . 107

10.3 Transverse photon as a spin-1 particle . . . . . . . . . . . . . . . . . . . . . 110

10.4 Neutrino wave mechanics. Massive eigenstate neutrinos . . . . . . . . . . . 113

11 Longitudinal and scalar photons. Gauge and near-field light quanta 119

11.1 L- and S-photons. Wave equations . . . . . . . . . . . . . . . . . . . . . . . 119

11.2 L- and S-photon neutralization in free space . . . . . . . . . . . . . . . . . 120

11.3 NF- and G-photons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

11.4 Gauge transformation within the Lorenz gauge . . . . . . . . . . . . . . . . 124

Contents ix

12 Massive photon field 127

12.1 Proca equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

12.2 Dynamical equations for E and A . . . . . . . . . . . . . . . . . . . . . . . 129

12.3 Diamagnetic interaction: Transverse photon mass . . . . . . . . . . . . . . 130

12.4 Massive vector boson (photon) field . . . . . . . . . . . . . . . . . . . . . . 132

12.5 Massive photon propagator . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

13 Photon energy wave function formalism 143

13.1 The Oppenheimer light quantum theory . . . . . . . . . . . . . . . . . . . . 143

13.2 Interlude: From spherical to Cartesian representation . . . . . . . . . . . . 146

13.3 Photons and antiphotons: Bispinor wave functions . . . . . . . . . . . . . . 150

13.4 Four-momentum and spin of photon wave packet . . . . . . . . . . . . . . . 153

13.5 Relativistic scalar product. Lorentz-invariant integration on the

energy shell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

IV Single-photon quantum optics in Minkowskian space 159

14 The photon of the quantized electromagnetic field 161

15 Polychromatic photons 165

15.1 Canonical quantization of the transverse electromagnetic field . . . . . . . 165

15.2 Energy, momentum, and spin operators of the transverse field . . . . . . . 168

15.3 Monochromatic plane-wave photons. Fock states . . . . . . . . . . . . . . . 171

15.4 Single-photon wave packets . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

15.5 New T-photon “mean” position state . . . . . . . . . . . . . . . . . . . . . 177

15.6 T-photon wave function and related dynamical equation . . . . . . . . . . . 179

15.7 The non-orthogonality of T-photon position states . . . . . . . . . . . . . . 181

16 Single-photon wave packet correlations 183

16.1 Wave-packet basis for one-photon states . . . . . . . . . . . . . . . . . . . . 183

16.2 Wave-packet photons related to a given t-matrix . . . . . . . . . . . . . . . 184

16.3 Integral equation for the time evolution operator in the interaction

picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

16.4 Atomic and field correlation matrices . . . . . . . . . . . . . . . . . . . . . 189

16.5 Single-photon correlation matrix: The wave function fingerprint . . . . . . 194

17 Interference phenomena with single-photon states 197

17.1 Wave-packet mode interference . . . . . . . . . . . . . . . . . . . . . . . . . 197

17.2 Young-type double-source interference . . . . . . . . . . . . . . . . . . . . . 198

17.3 Interference between transition amplitudes . . . . . . . . . . . . . . . . . . 201

17.4 Field correlations in photon mean position state . . . . . . . . . . . . . . . 201

17.4.1 Correlation supermatrix . . . . . . . . . . . . . . . . . . . . . . . . . 202

17.4.2 Relation between the correlation supermatrix and the transverse pho￾ton propagator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

18 Free-field operators: Time evolution and commutation relations 205

18.1 Maxwell operator equations. Quasi-classical states . . . . . . . . . . . . . . 205

18.2 Generalized Landau–Peierls–Sudarshan equations . . . . . . . . . . . . . . 207

18.3 Commutation relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

18.3.1 Commutation relations at different times (τ 6= 0) . . . . . . . . . . . 209

18.3.2 Equal-time commutation relations . . . . . . . . . . . . . . . . . . . 210

x Contents

V Photon embryo states 213

19 Attached photons in rim zones 215

20 Evanescent photon fields 221

20.1 Four-potential description in the Lorenz gauge . . . . . . . . . . . . . . . . 221

20.2 Sheet current density: T-, L-, and S-parts . . . . . . . . . . . . . . . . . . . 223

20.3 Evanescent T-, L-, and S-potentials . . . . . . . . . . . . . . . . . . . . . . 225

20.4 Four-potential photon wave mechanics . . . . . . . . . . . . . . . . . . . . . 229

20.5 Field-quantized approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

20.6 Near-field photon: Heisenberg equation of motion and coherent state . . . . 234

21 Photon tunneling 237

21.1 Near-field interaction. The photon measurement problem . . . . . . . . . . 237

21.2 Scattering of a wave-packet band from a single current-density sheet . . . . 238

21.3 Incident fields generating evanescent tunneling potentials . . . . . . . . . . 243

21.4 Interlude: Scalar propagator in various domains . . . . . . . . . . . . . . . 246

21.5 Incident polychromatic single-photon state . . . . . . . . . . . . . . . . . . 247

21.6 Photon tunneling-coupled sheets . . . . . . . . . . . . . . . . . . . . . . . . 250

22 Near-field photon emission in 3D 255

22.1 T-, L-, and S-potentials of a classical point-particle . . . . . . . . . . . . . 255

22.1.1 General considerations on source fields . . . . . . . . . . . . . . . . . 255

22.1.2 Point-particle potentials . . . . . . . . . . . . . . . . . . . . . . . . . 257

22.2 Cerenkov shock wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260 ˘

22.2.1 Four-potential of point-particle in uniform motion in vacuum . . . . 260

22.2.2 Transverse and longitudinal response theory in matter . . . . . . . . 263

22.2.3 The transverse Cerenkov phenomenon . . . . . . . . . . . . . . . . . 266 ˘

22.2.4 Momenta associated to the transverse and longitudinal parts of the

Cerenkov field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 ˘

22.2.5 Screened canonical particle momentum . . . . . . . . . . . . . . . . . 272

VI Photon source domain and propagators 275

23 Super-confined T-photon sources 277

24 Transverse current density in nonrelativistic quantum mechanics 283

24.1 Single-particle transition current density . . . . . . . . . . . . . . . . . . . 283

24.2 The hydrogen 1s ⇔ 2pz transition . . . . . . . . . . . . . . . . . . . . . . . 286

24.3 Breathing mode: Hydrogen 1s ⇔ 2s transition . . . . . . . . . . . . . . . . 289

24.4 Two-level breathing mode dynamics . . . . . . . . . . . . . . . . . . . . . . 292

25 Spin-1/2 current density in relativistic quantum mechanics 297

25.1 Dirac matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297

25.2 Covariant form of the Dirac equation. Minimal coupling. Four-current density 299

25.3 Gordon decomposition of the Dirac four-current density . . . . . . . . . . . 301

25.4 Weakly relativistic spin current density . . . . . . . . . . . . . . . . . . . . 303

25.5 Continuity equations for spin and space four-current densities . . . . . . . 306

Contents xi

26 Massless photon propagators 309

26.1 From the Huygens propagator to the transverse photon propagator . . . . 309

26.2 T-photon time-ordered correlation of events . . . . . . . . . . . . . . . . . . 311

26.3 Covariant correlation matrix . . . . . . . . . . . . . . . . . . . . . . . . . . 313

26.4 Covariant quantization of the electromagnetic field: A brief review . . . . . 314

26.5 The Feynman photon propagator . . . . . . . . . . . . . . . . . . . . . . . 316

26.6 Longitudinal and scalar photon propagators . . . . . . . . . . . . . . . . . 318

VII Photon vacuum and quanta in Minkowskian space 321

27 Photons and observers 323

28 The inertial class of observers: Photon vacuum and quanta 329

28.1 Transverse photon four-current density . . . . . . . . . . . . . . . . . . . . 329

28.2 Boosts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332

28.2.1 Lorentz and Lorenz-gauge transformations of the four-potential . . . 332

28.2.2 Plane-mode decomposition of the covariant potential . . . . . . . . . 333

28.2.3 Mode functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336

28.3 Physical (T-photon) vacuum . . . . . . . . . . . . . . . . . . . . . . . . . . 337

29 The non-inertial class of observers: The nebulous particle concept 345

29.1 Bogolubov transformation. Vacuum states . . . . . . . . . . . . . . . . . . 345

29.2 The non-unique vacuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348

29.3 The Unruh effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 352

29.3.1 Rindler space and observer . . . . . . . . . . . . . . . . . . . . . . . 352

29.3.2 Rindler particles in Minkowski vacuum . . . . . . . . . . . . . . . . . 354

30 Photon mass and hidden gauge invariance 363

30.1 Physical vacuum: Spontaneous symmetry breaking . . . . . . . . . . . . . . 363

30.2 Goldstone bosons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366

30.3 The U(1) Higgs model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368

30.4 Photon mass and vacuum screening current . . . . . . . . . . . . . . . . . . 372

30.5 ’t Hooft gauge and propagator . . . . . . . . . . . . . . . . . . . . . . . . . 373

VIII Two-photon entanglement in space-time 377

31 The quantal photon gas 379

32 Quantum measurements 385

32.1 Tensor product space (discrete case) . . . . . . . . . . . . . . . . . . . . . . 385

32.2 Definition of an observable (discrete case) . . . . . . . . . . . . . . . . . . . 386

32.3 Reduction of the wave packet (discrete case) . . . . . . . . . . . . . . . . . 387

32.4 Measurements on only one part of a two-part physical system . . . . . . . 387

32.5 Entangled photon polarization states . . . . . . . . . . . . . . . . . . . . . 390

33 Two-photon wave mechanics and correlation matrices 393

33.1 Two-photon two times wave function . . . . . . . . . . . . . . . . . . . . . 393

33.2 Two-photon Schr¨odinger equation in direct space . . . . . . . . . . . . . . 396

33.3 Two-photon wave packet correlations . . . . . . . . . . . . . . . . . . . . . 397

33.3.1 First-order correlation matrix . . . . . . . . . . . . . . . . . . . . . . 397

33.3.2 Second-order correlation matrix . . . . . . . . . . . . . . . . . . . . . 399

xii Contents

34 Spontaneous one- and two-photon emissions 401

34.1 Two-level atom: Weisskopf–Wigner theory of spontaneous emission . . . . 401

34.1.1 Atom-field Hamiltonian in the electric-dipole approximation. RWA￾model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401

34.1.2 Weisskopf–Wigner state vector . . . . . . . . . . . . . . . . . . . . . 406

34.2 Two-level atom: Wave function of spontaneously emitted photon . . . . . . 409

34.2.1 Photon wave function in q-space . . . . . . . . . . . . . . . . . . . . 409

34.2.2 The general photon wave function in r-space . . . . . . . . . . . . . 410

34.2.3 Genuine transverse photon wave function . . . . . . . . . . . . . . . 411

34.2.4 Spontaneous photon emission in the atomic rim zone . . . . . . . . . 413

34.3 Three-level atom: Spontaneous cascade emission . . . . . . . . . . . . . . . 417

34.3.1 Two-photon state vector . . . . . . . . . . . . . . . . . . . . . . . . . 417

34.3.2 Two-photon two-times wave function . . . . . . . . . . . . . . . . . . 420

34.3.3 The structure of Φ2,T (r1, r2, t1, t2) . . . . . . . . . . . . . . . . . . . 422

34.3.4 Far-field part of Φ

(1)

2,T (r1, r2, t1, t2) . . . . . . . . . . . . . . . . . . . 425

Bibliography 429

Index 441

Preface

I have often been asked what is a photon? In order to attempt to answer this question,

as communicating human beings, above all we must learn how to use the word is in an

unambiguous manner. The learning process takes us on a journey into deep philosophical

questions, and many of us end up being bewildered before we finally are snowed under

with philosophical thinking. In my understanding, the is-problem is like a Gordian knot.

In physics we replace the word is by characterizes, although in everyday discussion among

physicists we do not need to distinguish between is and characterizes, in general. So, I take

the liberty to replace the original question with what characterizes a photon? If someone asks

you who is this person you will “only” be able to answer by mentioning as many features,

traits, etc., as you are aware of about the given person. In a sense, a good characterization

of a phenomenon in physics means to look at the phenomenon from various perspectives

(through different windows). In the case of the photon, we approach the original question

what is a photon by looking at the phenomenon through as many windows as possible. Only

in the never attainable limit, where the number (N) of windows [photon perspectives (PP)]

approaches infinity, has one captured the photon phenomenon, at least in my understanding.

Mathematically,

Observational possibility ≡

X

N

i=1

(P P)i

⇒

X∞

i=1

(P P)i ≡ The photon phenomenon.

In this book I take a look at the photon phenomenon from a personal selection of a few

perspectives. The insight obtained by looking through some of the windows may already be

familiar to the reader.

Above I have made use of the word phenomenon, and replaced photon with photon

phenomenon. The concept phenomenon was introduced in the physical literature by Niels

Bohr, and the definition he first formulated publicly at a meeting in Warsaw in 1938,

arranged by the International Institute of Intellectual Co-operation of the League of Nations.

Niels Bohr, one of the monumental figures in the establishment of quantum mechanics,

throughout his life, with ever-increasing force of the argument, emphasized that we must

learn to use the words of the common language in an unambiguous manner, because after

all, we as physicists essentially have only the common language when we discuss with

each other what we have learned in our field of study. According to Bohr, no elementary

quantum phenomenon is a phenomenon until it is a registered (observed) phenomenon. For

Bohr quantum mechanics was a rational generalization of classical physics, and his definition

of the phenomenon concept made it possible to unite the seemingly incompatible particle

and wave aspects of the photon phenomenon, e.g., the single- and double-slit experiments

with photons. Bohr’s phenomenon concept, as well as another of his central points, viz.,

that the functioning of the measuring apparatus always must (and only can) be described

in the language of classical physics, will be important for us to remember. To Bohr, every

atomic phenomenon is closed in the sense that its observation is based on registrations

xiii

xiv Light—The Physics of the Photon

obtained by means of suitable macroscopic devices (with irreversible functioning). Bohr

considered the closure of fundamental significance not only in quantum physics but in the

whole description of nature, and he often stressed in discussions that “reality” is a word in

our language and that we must learn to use it correctly. Kalckar, in the 1967 book Niels

Bohr: His Life and Work as Seen by His Friends and Colleagues (edited by S. Rozental)

quoted Bohr for the following statement: I am quite prepared to talk of the spiritual life of

an electronic computer, to say that it is considering or that it is in a bad mood. What really

matters is the unambiguous description of its behaviour, which is what we observe. The

question as to whether the machine really feels, or whether it merely looks as though it did,

is absolutely as meaningless as to ask whether light is “in reality” waves or particles. We

must never forget that “reality” too is a human word just like “wave” or “consciousness.”

Our task is to learn to use these words correctly − that is, unambiguously and consistently.

It will be well to remember the fundamental (central) points of Niels Bohr throughout the

reading of this book.

Notwithstanding the fact that field–matter interaction is needed for a photon to appear

as a photon phenomenon, it is nevertheless indispensable to study the photon as a concept

belonging to global vacuum (matter-free space). Although the photon of the vacuum is an

abstraction of our mind, the photon concept must be firmly connected to the electromagnetic

field concept in free space. The autonomy of the classical electromagnetic field in free space is

solely connected with the vacuum speed of light (c): The classical electromagnetic field is an

intermediary describing the delayed (with speed c) interaction between electrically charged

particles in nonuniform motion. Although there is no room for accommodating the photon

concept in the framework of classical electrodynamics, it is of value to investigate how far

one may proceed toward the introduction of a classical light particle concept in a classical

framework. The autonomy of the electromagnetic field increases in an essential manner with

the introduction of the quantum of action (Planck’s constant, h) in electrodynamics. The

photon concept then flourishes, and the photon-free vacuum appears with its own autonomy.

The modern era of the light particle (based on h) began when Einstein in 1905 concluded

that monochromatic (frequency: ν) radiation of low density (within the range of validity of

Wien’s radiation formula) behaves thermodynamically as though it consisted of a number of

independent energy quanta of magnitude hν.

In Part I, we prepare ourselves for photon physics by studying certain aspects of classical

optics in a global vacuum on the basis of the free-space Maxwell equations. Since the photon

in global vacuum (T-photon) is a transversely (T) polarized object belonging to the positive￾frequency part of the electromagnetic spectrum, studies of transverse (longitudinal) vector

fields, complex analytical signals, and the various polarization states of light are central.

With an eye to the point-like Einstein light particle we also describe how the electromagnetic

field can be resolved into a complete set of wave-packet modes. Because the massless photon

necessarily is a relativistic object propagating with the vacuum speed of light, it is important

to consider the fields of classical optics from the perspective of special relativity. Our brief

account of optics in special relativity culminates with a demonstration of the manifest

covariance of the Maxwell equations, and a discussion of the Lorentz transformation of the

transverse and longitudinal parts of the electromagnetic field.

In Part II, we study light rays and geodesics, and we also present a brief account of the

Maxwell theory in general relativity. In the framework of classical electrodynamics there

is no hope for considering light as consisting of some sort of particles, in general. This

is so because (wave) interference effects cannot occur in classical particle dynamics. In a

corner of the classical field theory, known as geometrical optics, the wavelength (λ) of light

plays no role; however, in the short wavelengths limit and here (λ → 0) a geometrization

of the field description in the form of light rays appears. The eikonal equation is the basic

equation of geometrical optics. A classical particle moves along a trajectory, and in the