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Quantization of the electromagnetic field

The quantization of the electromagnetic field is a foundational concept in quantum physics that applies the principles of quantum mechanics to the classical electromagnetic field described by Maxwell's equations, treating it as a system of infinite harmonic oscillators whose energy is quantized into discrete units called photons.^[1]^[2]^[3] This process transforms the continuous classical field variables—such as the electric field E and magnetic field B—into non-commuting operators, enabling the description of electromagnetic radiation as a stream of massless spin-1 bosons with energy ℏω, where ℏ is the reduced Planck's constant and ω is the angular frequency.^[1]^[2] The resulting theory, known as quantum electrodynamics (QED), accounts for interactions between light and matter at the quantum level, including vacuum fluctuations and spontaneous emission.^[2]^[3] The development of electromagnetic field quantization emerged in the mid-1920s amid efforts to reconcile quantum mechanics with special relativity and classical field theory.^[4] Pioneering work began with Pascual Jordan's 1925 quantization of the free electromagnetic field using matrix mechanics, followed by Paul Dirac's 1927 formulation of QED, which introduced the idea of the field as an operator to describe radiation emission and absorption by atoms.^[4]^[5] Contributions from Max Born, Werner Heisenberg, and Wolfgang Pauli in 1926–1929 further solidified the canonical quantization procedure, treating the field as an infinite collection of degrees of freedom analogous to mechanical oscillators.^[4]^[5] This "second quantization" approach, as it became known, resolved inconsistencies in early quantum theories of radiation and laid the groundwork for modern quantum field theory.^[3]^[6] In the standard quantization procedure, the electromagnetic field is confined to a finite volume, such as a cubic cavity with periodic boundary conditions, to discretize the modes.^[2]^[1] The vector potential A(r, t) is expanded in a Fourier series of plane-wave normal modes u_{kλ}(r), where k denotes the wave vector and λ the polarization, yielding coefficients that serve as canonical coordinates q_{kλ} and momenta p_{kλ}.^[2] These are promoted to operators satisfying commutation relations [Q_{kλ}, P_{k'λ'}] = iℏ δ_{kk'} δ_{λλ'}, and further expressed via boson creation a†_{kλ} and annihilation a_{kλ} operators with [a_{kλ}, a†{k'λ'}] = δ{kk'} δ_{λλ'}.^[2]^[1] The Hamiltonian then takes the form H = ∑{kλ} ℏω_k (a†{kλ} a_{kλ} + 1/2), revealing discrete energy levels and a non-zero vacuum energy (zero-point energy) even in the ground state.^[2]^[3] Gauge choices, such as the Coulomb or radiation gauge, ensure transversality (∇ · A = 0) and compatibility with relativity.^[1] This quantization framework underpins QED, the paradigmatic quantum field theory, and has profound implications for quantum optics, enabling predictions of phenomena like the Casimir effect—arising from vacuum fluctuations between conducting plates, first theorized by Hendrik Casimir in 1948 and experimentally confirmed in 2001.^[1]^[3] It also facilitates the study of photon statistics in Fock states |n_{kλ}⟩, light-matter interactions, and extensions to media with dielectrics or magnetic monopoles.^[2]^[7] Modern applications include quantum information processing and precision measurements, underscoring its enduring role in fundamental physics.^[2]

Classical Electromagnetic Field

Vector Potential and Fields

In the mid-19th century, James Clerk Maxwell introduced the concept of electromagnetic potentials as part of his dynamical theory of the electromagnetic field, building on earlier work by Michael Faraday and William Thomson. Maxwell's seminal 1865 paper utilized a quantity he termed "electromagnetic momentum," which corresponds to the modern vector potential, to express the relations between electric and magnetic forces. The vector potential was mathematically formalized by Thomson in 1851 as a means to represent the magnetic field, providing a foundation that Maxwell later incorporated into his equations.^[8]^[9] The electromagnetic four-potential is a relativistic four-vector A^\mu = (\phi/c, \mathbf{A}), where \phi is the scalar electric potential, \mathbf{A} is the vector potential, and c is the speed of light in vacuum. This formulation combines the scalar and vector potentials into a single entity that transforms covariantly under Lorentz transformations, unifying the description of electromagnetic phenomena in special relativity. The four-potential satisfies the inhomogeneous wave equation \partial_\nu F^{\mu\nu} = \mu_0 J^\mu, where F^{\mu\nu} = \partial^\mu A^\nu - \partial^\nu A^\mu is the electromagnetic field tensor and J^\mu is the four-current.^[10]^[11] The electric and magnetic fields can be derived from the four-potential components. In three-vector notation, the electric field is given by

\mathbf{E} = -\nabla \phi - \frac{\partial \mathbf{A}}{\partial t},

which arises from the antisymmetric nature of the field tensor F^{\mu\nu}, capturing both the conservative part from the scalar potential gradient and the induced part from the time-varying vector potential. The magnetic field is

\mathbf{B} = \nabla \times \mathbf{A},

obtained directly as the curl of the vector potential, ensuring the divergence-free condition \nabla \cdot \mathbf{B} = 0 is automatically satisfied. These expressions demonstrate how the potentials encode the observable fields while introducing redundancy due to gauge invariance.^[12]^[13] The choice of potentials is not unique, reflecting the gauge freedom inherent in Maxwell's equations. A gauge transformation A^\mu \to A^\mu + \partial^\mu \Lambda for an arbitrary scalar \Lambda leaves the field tensor F^{\mu\nu} unchanged, thus preserving the physical fields \mathbf{E} and \mathbf{B}. Common gauge choices include the Lorenz gauge, \partial_\mu A^\mu = 0, which simplifies the wave equations for the potentials to the form \Box A^\mu = -\mu_0 J^\mu and is Lorentz invariant. Another is the Coulomb gauge, \nabla \cdot \mathbf{A} = 0, which decouples the scalar potential to satisfy Poisson's equation instantaneously and is useful in non-relativistic contexts or for radiation problems.^[14]^[10]

Lagrangian and Hamiltonian in Terms of Potentials

The Lagrangian density for the free electromagnetic field in vacuum is given by

\mathcal{L} = -\frac{1}{4} F_{\mu\nu} F^{\mu\nu},

where the electromagnetic field strength tensor is defined as F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu, with A^\mu = (\phi, \mathbf{A}) denoting the four-potential comprising the scalar potential \phi and the vector potential \mathbf{A}. This form ensures Lorentz invariance and gauge invariance under transformations A^\mu \to A^\mu + \partial^\mu \Lambda. In three-vector notation and using Heaviside-Lorentz units (with c = 1), the Lagrangian density simplifies to

\mathcal{L} = \frac{1}{2} (\mathbf{E}^2 - \mathbf{B}^2),

where the electric field is \mathbf{E} = -\nabla \phi - \partial_t \mathbf{A} and the magnetic field is \mathbf{B} = \nabla \times \mathbf{A}. The action functional S = \int \mathcal{L} \, d^4x is stationary under variations of the potentials, leading via the Euler-Lagrange equations \partial_\mu \left( \frac{\partial \mathcal{L}}{\partial (\partial_\mu A_\nu)} \right) - \frac{\partial \mathcal{L}}{\partial A_\nu} = 0 to the homogeneous and inhomogeneous Maxwell equations in vacuum: \partial^\mu F_{\mu\nu} = 0 and \partial_{[\lambda} F_{\mu\nu]} = 0. To transition to the Hamiltonian formulation, the canonical momentum conjugate to the vector potential \mathbf{A} is computed as \boldsymbol{\Pi} = \frac{\partial \mathcal{L}}{\partial (\partial_t \mathbf{A})} = -\mathbf{E}, while the scalar potential \phi has no conjugate momentum, rendering it non-dynamical. The Hamiltonian density then follows from the Legendre transform \mathcal{H} = \boldsymbol{\Pi} \cdot \partial_t \mathbf{A} - \mathcal{L}, yielding

\mathcal{H} = \frac{1}{2} (\boldsymbol{\Pi}^2 + \mathbf{B}^2).

This expression represents the total energy density of the electromagnetic field, with the first term corresponding to the electric field contribution and the second to the magnetic. In the Hamiltonian framework, Hamilton's equations generate the field dynamics, but the absence of a conjugate to \phi implies that the equation \nabla \cdot \mathbf{E} = 0 (Gauss's law in free space) emerges as a primary constraint rather than a dynamical equation. This constraint must be preserved under time evolution, enforcing consistency with the transverse nature of the field in vacuum.

Quantization Procedure

Gauge Choice and Mode Expansion

In the quantization of the electromagnetic field, particularly in non-relativistic contexts, the Coulomb gauge is selected, defined by the transversality condition \nabla \cdot \mathbf{A} = 0. This choice is preferred because it eliminates the unphysical longitudinal modes from the dynamical quantization, restricting the quantum degrees of freedom to the two transverse polarizations that correspond to physical radiation.^[15] In this gauge, the longitudinal electric field arises instantaneously from the charge density via the scalar potential \phi = -\int \frac{\rho(\mathbf{r}')}{4\pi\epsilon_0 |\mathbf{r} - \mathbf{r}'|} d^3\mathbf{r}', treated classically rather than as a propagating quantum mode, simplifying the Hamiltonian structure for matter-field interactions.^[15] The vector potential \mathbf{A} in the Coulomb gauge is decomposed into a Fourier mode expansion suitable for the free field, providing a basis of plane waves that diagonalize the classical wave equation. In free space, this takes the form

\mathbf{A}(\mathbf{r}, t) = \int \frac{d^3k}{(2\pi)^3} \sum_{\lambda=1,2} \left[ a_{\mathbf{k},\lambda} \boldsymbol{\epsilon}_{\mathbf{k},\lambda} e^{i(\mathbf{k}\cdot\mathbf{r} - \omega t)} + \mathrm{h.c.} \right],

where \omega = c |\mathbf{k}| relates the angular frequency to the wave vector magnitude, the coefficients a_{\mathbf{k},\lambda} are complex amplitudes (to be promoted to annihilation operators in quantization), and \boldsymbol{\epsilon}_{\mathbf{k},\lambda} (\lambda = 1, 2) are unit polarization vectors.^[16] This expansion originates from the classical Hamiltonian expressed in terms of \mathbf{A} and its conjugate momentum, facilitating the identification of normal modes analogous to harmonic oscillators.^[17] The polarization vectors \boldsymbol{\epsilon}_{\mathbf{k},\lambda} satisfy the transversality condition \mathbf{k} \cdot \boldsymbol{\epsilon}_{\mathbf{k},\lambda} = 0, ensuring the vector potential and resulting fields \mathbf{E} = -\partial_t \mathbf{A} and \mathbf{B} = \nabla \times \mathbf{A} have no longitudinal components, with the two orthogonal polarizations spanning the plane perpendicular to \mathbf{k} such that \boldsymbol{\epsilon}_{\mathbf{k},\lambda} \cdot \boldsymbol{\epsilon}_{\mathbf{k},\lambda'}^* = \delta_{\lambda\lambda'}.^[16] This condition enforces the physical content of Maxwell's equations in vacuum, where only transverse waves propagate. The form of the mode expansion depends on boundary conditions: in unbounded free space, the continuous spectrum of \mathbf{k} yields the volume integral over d^3k, normalized by (2\pi)^3 for delta-function orthogonality of plane waves. In a finite cavity, such as a cubic box of volume V = L^3 with periodic boundary conditions, the modes discretize to \mathbf{k} = \frac{2\pi}{L} (n_x, n_y, n_z) with integers n_i, replacing the integral with a sum \frac{1}{V} \sum_{\mathbf{k}}, which converges to the continuous limit as V \to \infty.^[16]

Canonical Quantization and Commutation Relations

In the canonical quantization approach to the electromagnetic field, the classical vector potential \mathbf{A}(\mathbf{r}, t) and its conjugate momentum density \boldsymbol{\Pi}(\mathbf{r}, t) = -\epsilon_0 \mathbf{E}(\mathbf{r}, t), derived from the Lagrangian, are promoted to non-commuting operators \hat{\mathbf{A}}(\mathbf{r}, t) and \hat{\boldsymbol{\Pi}}(\mathbf{r}, t). This promotion, first systematically applied to the radiation field by Dirac, transforms the classical field theory into a quantum mechanical framework by imposing operator algebra on the dynamical variables.^[17] The fundamental equal-time commutation relations for these operators are [\hat{A}_i(\mathbf{r}, t), \hat{\Pi}_j(\mathbf{r}', t)] = i \hbar \delta_{ij} \delta^3(\mathbf{r} - \mathbf{r}'), with all other commutators among components of \hat{\mathbf{A}} and \hat{\boldsymbol{\Pi}} vanishing at equal times. These relations ensure the consistency of the quantum theory, analogous to the position-momentum commutator in single-particle quantum mechanics, and follow directly from the canonical structure of the field theory.^[18] Building on the mode expansion of the classical field in a chosen gauge, the operator \hat{\mathbf{A}}(\mathbf{r}, t) is expressed in terms of creation \hat{a}^\dagger_{\mathbf{k},\lambda} and annihilation \hat{a}_{\mathbf{k},\lambda} operators as

\hat{\mathbf{A}}(\mathbf{r}, t) \propto \sum_{\mathbf{k},\lambda} \left( \hat{a}_{\mathbf{k},\lambda} \boldsymbol{\epsilon}_{\mathbf{k},\lambda} \, e^{i \mathbf{k} \cdot \mathbf{r} - i \omega_k t} + \hat{a}^\dagger_{\mathbf{k},\lambda} \boldsymbol{\epsilon}^*_{\mathbf{k},\lambda} \, e^{-i \mathbf{k} \cdot \mathbf{r} + i \omega_k t} \right),

where \boldsymbol{\epsilon}_{\mathbf{k},\lambda} denotes the polarization vector for mode \mathbf{k} and helicity \lambda, and the proportionality includes normalization factors involving \hbar, \epsilon_0, \omega_k = c |\mathbf{k}|, and the quantization volume. The creation and annihilation operators obey the bosonic commutation algebra [\hat{a}_{\mathbf{k},\lambda}, \hat{a}^\dagger_{\mathbf{k}',\lambda'}] = \delta_{\mathbf{k},\mathbf{k}'} \delta_{\lambda,\lambda'}, with all other commutators zero, which is derived from the canonical relations to preserve the quantum structure.^[17]^[18] This framework operates in the Heisenberg picture, where the field operators evolve according to the classical field equations while acting on a fixed Hilbert space of states, distinguishing it from first-quantized single-particle quantum mechanics. In the context of second quantization, the field operators directly create and annihilate particles (photons) in the Fock space, providing a many-body description essential for quantum electrodynamics.^[18]

Quantized Field Hamiltonian

Expansion in Normal Modes

Following gauge fixing to the Coulomb gauge, where the scalar potential is set to zero and the vector potential \mathbf{A} is transverse (\nabla \cdot \mathbf{A} = 0), the constraints are satisfied, and the Hamiltonian for the free electromagnetic field takes the form

\hat{H} = \int d^3 r \left[ \frac{1}{2} \hat{\boldsymbol{\Pi}}^2(\mathbf{r}) + \frac{1}{2} (\nabla \times \hat{\mathbf{A}}(\mathbf{r}))^2 \right],

with \hat{\boldsymbol{\Pi}}(\mathbf{r}) = -\dot{\hat{\mathbf{A}}}(\mathbf{r}) serving as the canonical momentum conjugate to \hat{\mathbf{A}}, in units where \epsilon_0 = \mu_0 = 1. The vector potential is expanded in terms of transverse plane-wave normal modes as

\hat{\mathbf{A}}(\mathbf{r}) = \sum_{\mathbf{k},\lambda} \sqrt{\frac{\hbar}{2 \omega_{\mathbf{k}} V}} \left[ \hat{a}_{\mathbf{k},\lambda} \boldsymbol{\epsilon}_{\mathbf{k},\lambda} e^{i \mathbf{k} \cdot \mathbf{r}} + \hat{a}^\dagger_{\mathbf{k},\lambda} \boldsymbol{\epsilon}^*_{\mathbf{k},\lambda} e^{-i \mathbf{k} \cdot \mathbf{r}} \right],

where \mathbf{k} labels the wave vector, \lambda = 1,2 denotes the two transverse polarization states with unit vectors \boldsymbol{\epsilon}_{\mathbf{k},\lambda}, \omega_{\mathbf{k}} = c |\mathbf{k}| is the mode frequency, V is the quantization volume, and \hat{a}_{\mathbf{k},\lambda}, \hat{a}^\dagger_{\mathbf{k},\lambda} are the annihilation and creation operators satisfying bosonic commutation relations from canonical quantization. Substituting this mode expansion into the Hamiltonian, along with the corresponding expansion for \hat{\boldsymbol{\Pi}}(\mathbf{r}), yields a diagonal form

\hat{H} = \sum_{\mathbf{k},\lambda} \hbar \omega_{\mathbf{k}} \left( \hat{a}^\dagger_{\mathbf{k},\lambda} \hat{a}_{\mathbf{k},\lambda} + \frac{1}{2} \right),

where the orthogonality of the modes ensures no cross terms between different \mathbf{k} and \lambda. The term \sum_{\mathbf{k},\lambda} \frac{1}{2} \hbar \omega_{\mathbf{k}} represents the zero-point energy of the vacuum, which diverges due to the infinite number of modes in continuous space. This infinity is regularized through normal ordering, redefining the Hamiltonian as

\hat{H} = \sum_{\mathbf{k},\lambda} \hbar \omega_{\mathbf{k}} \hat{a}^\dagger_{\mathbf{k},\lambda} \hat{a}_{\mathbf{k},\lambda},

which sets the ground-state energy to zero while preserving energy differences observable in phenomena like the Casimir effect. In the ground state, known as the vacuum, the expectation value of the field operators vanishes (\langle 0 | \hat{\mathbf{A}}(\mathbf{r}) | 0 \rangle = 0), yet vacuum fluctuations persist, manifesting as non-zero variances \langle 0 | \hat{\mathbf{A}}^2(\mathbf{r}) | 0 \rangle > 0 that arise from the commutation relations and contribute to effects such as spontaneous emission and the Lamb shift.

Analogy to Quantum Harmonic Oscillators

In the classical description of the free electromagnetic field, the vector potential satisfies the wave equation derived from Maxwell's equations in vacuum, allowing a Fourier decomposition into plane-wave normal modes labeled by wave vector \mathbf{k} and polarization index \lambda. These modes represent independent transverse oscillations, decoupled in Fourier space, where each behaves dynamically as a classical harmonic oscillator with frequency \omega_k = c k (with k = |\mathbf{k}| and c the speed of light).^[19] Upon canonical quantization, this structure maps directly to quantum mechanics, treating each mode (\mathbf{k}, \lambda) as an independent one-dimensional quantum harmonic oscillator. The canonical variables of the mode—the "position" coordinate q_{\mathbf{k}\lambda} (proportional to the mode amplitude of the vector potential) and conjugate "momentum" p_{\mathbf{k}\lambda} (proportional to the mode's electric field component)—become operators satisfying [q_{\mathbf{k}\lambda}, p_{\mathbf{k}'\lambda'}] = i\hbar \delta_{\mathbf{k}\mathbf{k}'} \delta_{\lambda\lambda'}. These are expressed in terms of creation \hat{a}^\dagger_{\mathbf{k}\lambda} and annihilation \hat{a}_{\mathbf{k}\lambda} operators via

q_{\mathbf{k}\lambda} \propto \sqrt{\frac{\hbar}{2\omega_k}} \left( \hat{a}_{\mathbf{k}\lambda} + \hat{a}^\dagger_{\mathbf{k}\lambda} \right), \quad p_{\mathbf{k}\lambda} \propto i \sqrt{\frac{\hbar \omega_k}{2}} \left( \hat{a}^\dagger_{\mathbf{k}\lambda} - \hat{a}_{\mathbf{k}\lambda} \right),

with the commutation relations [\hat{a}_{\mathbf{k}\lambda}, \hat{a}^\dagger_{\mathbf{k}'\lambda'}] = \delta_{\mathbf{k}\mathbf{k}'} \delta_{\lambda\lambda'}.^[16] The resulting Hamiltonian for each mode mirrors that of a quantum harmonic oscillator, summing over all modes to yield the total quantized field energy.^[17] The energy spectrum for each mode consists of discrete levels E_n = \hbar \omega_k \left( n + \frac{1}{2} \right), where n = 0, 1, 2, \dots labels the excitation number, introducing the zero-point energy \frac{1}{2} \hbar \omega_k even in the ground state. This quantization framework provides intuitive insight into the field's particle-like excitations (photons) while preserving its wavelike mode structure. The analogy originated in Paul Dirac's seminal 1927 paper, which first applied quantum rules to the radiation field by treating its monochromatic components as quantized oscillators to reconcile emission and absorption processes with quantum mechanics.^[17]

Photon States

Fock States and Photon Number Operator

In the quantized electromagnetic field, the photon number operator for a specific mode labeled by wave vector \mathbf{k} and polarization \lambda is defined as \hat{N}_{\mathbf{k},\lambda} = \hat{a}^\dagger_{\mathbf{k},\lambda} \hat{a}_{\mathbf{k},\lambda}, where \hat{a}_{\mathbf{k},\lambda} and \hat{a}^\dagger_{\mathbf{k},\lambda} are the annihilation and creation operators, respectively.^[17]^[20] This operator has eigenvalues n_{\mathbf{k},\lambda} = 0, 1, 2, \dots , corresponding to the number of photons occupying that mode, reflecting the discrete nature of the field's energy quanta.^[17]^[21] The eigenstates of the total number operator \hat{N} = \sum_{\mathbf{k},\lambda} \hat{N}_{\mathbf{k},\lambda} form the Fock basis for the field, constructed as multi-mode tensor products. A general Fock state is expressed as

| \{ n_{\mathbf{k},\lambda} \} \rangle = \prod_{\mathbf{k},\lambda} \frac{ (\hat{a}^\dagger_{\mathbf{k},\lambda})^{n_{\mathbf{k},\lambda}} }{\sqrt{n_{\mathbf{k},\lambda}!}} |0\rangle,

where the product runs over all modes, and |0\rangle denotes the vacuum state.^[21]^[20] These states provide a complete orthonormal basis for the Hilbert space of the field, with fixed total photon number N = \sum_{\mathbf{k},\lambda} n_{\mathbf{k},\lambda}, analogous to the energy eigenstates of a system of independent harmonic oscillators.^[17]^[21] The vacuum state |0\rangle is uniquely defined as the state annihilated by all annihilation operators, satisfying \hat{a}_{\mathbf{k},\lambda} |0\rangle = 0 for every mode \mathbf{k},\lambda.^[17]^[20] Despite containing zero photons, this state exhibits non-zero expectation values for the squares of the electric and magnetic field operators, \langle 0| \hat{\mathbf{E}}^2 |0 \rangle > 0 and \langle 0| \hat{\mathbf{B}}^2 |0 \rangle > 0, arising from the infinite sum over zero-point fluctuations across all modes.^[20] These vacuum fluctuations underscore the quantum nature of the electromagnetic field even in the absence of excitations.^[20] The bosonic character of photons stems from the commutation relations [\hat{a}_{\mathbf{k},\lambda}, \hat{a}^\dagger_{\mathbf{k}',\lambda'}] = \delta_{\mathbf{k},\mathbf{k}'} \delta_{\lambda,\lambda'}, which enforce symmetric statistics under particle exchange.^[17] This leads to indistinguishable photons within the same mode, consistent with the foundational treatment of radiation quanta as bosons obeying Bose-Einstein statistics.^[22]^[17]

Coherent States (brief introduction)

In quantum optics, coherent states provide a quantum description of light that closely resembles classical electromagnetic waves, particularly those produced by lasers, where the field exhibits stable amplitude and phase.^[23] These states are constructed as superpositions of Fock states, the number eigenstates from the photon basis, and are defined as the eigenstates of the annihilation operator \hat{a}, satisfying \hat{a} |\alpha\rangle = \alpha |\alpha\rangle, where \alpha is a complex eigenvalue.^[23] Explicitly, the coherent state is given by

|\alpha\rangle = e^{-|\alpha|^2/2} \sum_{n=0}^\infty \frac{\alpha^n}{\sqrt{n!}} |n\rangle,

which normalizes the state and weights the Fock components |n\rangle according to a Poisson distribution.^[23] The photon number operator \hat{N} = \hat{a}^\dagger \hat{a} in a coherent state yields an expectation value \langle \hat{N} \rangle = |\alpha|^2, with the variance equal to the mean, \Delta N^2 = |\alpha|^2, characteristic of the Poissonian statistics that minimize quantum noise relative to the intensity.^[23] Under free evolution governed by the harmonic oscillator Hamiltonian \hat{H} = \hbar \omega (\hat{N} + 1/2), the coherent state parameter evolves as \alpha(t) = |\alpha| e^{-i \omega t}, preserving the shape of the wave packet while rotating its phase in the complex plane.^[23] Coherent states, introduced by Roy J. Glauber in 1963, form the foundation of modern quantum optics, enabling the analysis of coherent phenomena like interference and superposition in the quantized electromagnetic field.^[23]

Photon Properties

Energy and Frequency Relation

In the quantized electromagnetic field, the energy of a single photon associated with a plane-wave mode of wave vector \mathbf{k} and polarization \lambda is E = \hbar \omega_k, where \omega_k = c |\mathbf{k}| is the angular frequency and c is the speed of light in vacuum. This relation follows from the dispersion relation for electromagnetic waves in free space, \omega_k = c k, yielding the equivalent form E = \hbar c k. Equivalently, in terms of the frequency f = \omega_k / 2\pi, the energy is E = h f, where h = 2\pi \hbar is Planck's constant. This energy-frequency duality was first proposed by Max Planck in 1900 as part of his derivation of the blackbody radiation spectrum, where he postulated that the energy of electromagnetic oscillators is quantized in discrete units to resolve the ultraviolet catastrophe predicted by classical Rayleigh-Jeans theory. Planck's hypothesis, formalized in his 1901 paper, introduced the constant h such that the average energy of a resonator at frequency \nu (equivalent to f) is \langle E \rangle = h \nu / (e^{h \nu / kT} - 1), leading to the spectral energy density that matched experimental observations. Although Planck initially viewed quantization as a mathematical trick for classical oscillators rather than a fundamental property of radiation, this relation laid the groundwork for the photon concept later developed by Einstein in 1905. For multi-photon states in the free field, the total energy is additive across modes due to the non-interacting nature of photons. In the Fock basis, the Hamiltonian acting on a state |\{n_{\mathbf{k},\lambda}\}\rangle with occupation numbers n_{\mathbf{k},\lambda} yields \hat{H} |\{n\}\rangle = \left( \sum_{\mathbf{k},\lambda} \hbar \omega_k n_{\mathbf{k},\lambda} \right) |\{n\}\rangle, excluding the infinite zero-point energy \sum_{\mathbf{k},\lambda} \frac{1}{2} \hbar \omega_k which is typically renormalized away in free-field theory. This spectrum reflects the bosonic statistics of photons, allowing arbitrary occupation numbers per mode while preserving energy as a linear sum over individual photon contributions.

Momentum and Wave Vector

In the quantized description of the electromagnetic field, the total linear momentum operator is expressed as

\hat{\mathbf{P}} = -\int d^3 r \, \hat{\mathbf{E}}(\mathbf{r}) \times \hat{\mathbf{B}}(\mathbf{r}),

where \hat{\mathbf{E}}(\mathbf{r}) and \hat{\mathbf{B}}(\mathbf{r}) are the quantum electric and magnetic field operators evaluated at position \mathbf{r}. This form arises from the classical electromagnetic momentum density, adapted to the quantum regime.^[24] Expanding the field operators in the normal mode basis of plane waves with wave vector \mathbf{k} and polarization \lambda, the momentum operator simplifies to

\hat{\mathbf{P}} = \sum_{\mathbf{k},\lambda} \hbar \mathbf{k} \, \hat{a}^\dagger_{\mathbf{k},\lambda} \hat{a}_{\mathbf{k},\lambda},

where \hat{a}^\dagger_{\mathbf{k},\lambda} and \hat{a}_{\mathbf{k},\lambda} are the creation and annihilation operators for photons in mode (\mathbf{k}, \lambda). The number operator \hat{n}_{\mathbf{k},\lambda} = \hat{a}^\dagger_{\mathbf{k},\lambda} \hat{a}_{\mathbf{k},\lambda} thus weights each mode's contribution by its occupation number, reflecting the additive nature of photon momenta.^[16] A single-photon state |1_{\mathbf{k},\lambda}\rangle is an eigenstate of \hat{\mathbf{P}} with eigenvalue \mathbf{p} = \hbar \mathbf{k}, so the magnitude is |\mathbf{p}| = \hbar k = E/c, where E = [\hbar \omega_k](/page/H-bar) is the photon energy and \omega_k = c k relates frequency to wave number in vacuum. This linear relation underscores the massless, relativistic character of photons.^[16] In free space, the field Hamiltonian commutes with \hat{\mathbf{P}}, conserving the total momentum during time evolution without sources or boundaries. This conservation holds for processes like photon propagation or scattering, where the net momentum remains unchanged unless external interactions occur.^[24] The momentum borne by field modes manifests in radiation pressure, where photons transfer linear momentum to absorbing or reflecting matter, exerting a force F = \Delta p / \Delta t proportional to the rate of momentum change; for perfect absorption, the pressure is I/c with intensity I. This effect enables applications such as optical trapping and levitation.^[25] For transverse electromagnetic modes, the momentum aligns with the propagation direction specified by \mathbf{k}, as the divergence-free and transversality conditions ensure no longitudinal component; this directional helicity ties the photon's motion to its wave vector in the quantized field.^[16]

Spin and Polarization

Photons, as quanta of the electromagnetic field, are massless spin-1 bosons, possessing intrinsic angular momentum characterized primarily by their spin degree of freedom in the free field approximation.^[26] The total angular momentum operator for the field decomposes into orbital and spin contributions, \mathbf{J} = \mathbf{L} + \mathbf{S}, where for plane-wave modes propagating along a well-defined direction, the orbital part \mathbf{L} vanishes, leaving the spin \mathbf{S} as the dominant component aligned with the propagation axis.^[27] In the quantized description, the spin angular momentum for a specific wave vector mode \mathbf{k} is represented by the operator \hat{\mathbf{S}}_{\mathbf{k}} = \sum_{\lambda,\lambda'} \hbar \boldsymbol{\sigma}_{\lambda\lambda'} \hat{a}^\dagger_{\mathbf{k},\lambda} \hat{a}_{\mathbf{k},\lambda'}, where \hat{a}^\dagger_{\mathbf{k},\lambda} and \hat{a}_{\mathbf{k},\lambda} are the creation and annihilation operators for photons in mode \mathbf{k} with polarization index \lambda = 1,2, and \boldsymbol{\sigma} denotes the vector of Pauli matrices acting in the two-dimensional polarization space.^[28] This formulation treats the two transverse polarization states as a pseudospin system, capturing the vectorial nature of the photon's spin. The eigenvalues of the spin projection along the propagation direction \mathbf{k}, known as helicity, are \pm \hbar for single-photon states, corresponding to the two possible transverse polarizations.^[26] These helicity states align with circular polarizations: the +\hbar eigenvalue corresponds to right-handed circular polarization (where the electric field vector rotates in the right-hand sense along \mathbf{k}), and -\hbar to left-handed circular polarization, reflecting the intrinsic handedness of the photon's angular momentum.^[28] Due to the transversality condition imposed by gauge invariance in the free electromagnetic field, photons exhibit no scalar (spin-0) component nor longitudinal (helicity-0) polarization modes, restricting the degrees of freedom to the two helicity states.^[26] This exclusion arises from the Lorentz gauge and the physical state condition, which eliminate unphysical timelike and longitudinal degrees of freedom, ensuring only transverse modes contribute to observable photon states.^[26]

Rest Mass and Effective Mass

In the quantization of the electromagnetic field, photons are massless particles in vacuum, as dictated by the relativistic energy-momentum dispersion relation E^2 = p^2 c^2 + m^2 c^4, where E is the energy, p is the momentum, c is the speed of light, and m is the rest mass. For photons, Maxwell's equations in vacuum yield the linear relation E = p c, which requires m = 0 to satisfy the dispersion for all frequencies.^[29] This zero rest mass is a fundamental consequence of the gauge invariance of the electromagnetic field and ensures that photons propagate at the invariant speed c in vacuum, distinguishing them from massive particles. Experimental efforts to detect any nonzero photon rest mass have yielded extraordinarily tight upper bounds, primarily from astrophysical observations. Analyses of solar wind magnetohydrodynamics and galactic magnetic fields constrain the photon rest mass to m_\gamma < 10^{-18} eV/c^2 at 95% confidence level, with no evidence for a nonzero value.^[29] More recent constraints from fast radio bursts and pulsar timing in 2023–2024 maintain or slightly tighten this limit, confirming consistency with m_\gamma = 0 across diverse scales from laboratory tests to cosmic phenomena.^[29] These bounds underscore the photon's role as a truly massless excitation in the quantized field. Although the intrinsic rest mass of photons is zero, concepts of effective mass arise in certain environments, such as dielectric media or confined geometries, where the propagation is modified without altering the vacuum dispersion. In a linear dielectric medium with refractive index n > 1, the phase velocity v_p = c/n leads to an effective mass m^* = \hbar \omega / v_p^2 = n^2 \hbar \omega / c^2, reflecting the reduced speed and increased momentum relative to vacuum for a given energy \hbar \omega. This effective mass describes the photon's quasiparticle behavior, arising from interactions with the medium's polarizable constituents, but it does not imply a true rest mass; the photon remains relativistic with zero invariant mass. In optical cavities, boundary conditions quantize the modes, introducing a cutoff wavevector that curves the dispersion relation near zero in-plane momentum, thereby endowing photons with an effective mass proportional to the inverse cavity size, enhancing light-matter coupling in cavity quantum electrodynamics.^[30] The distinction is crucial: the true rest mass remains zero, preserving the vacuum speed c, while effective masses are context-dependent and vanish in the absence of the medium or confinement.

Classical Limit

Semiclassical Approximation

The semiclassical approximation in the quantization of the electromagnetic field arises primarily through the use of coherent states, which minimize quantum fluctuations and allow the expectation values of field operators to closely mimic classical fields. In these states, the expectation value of the vector potential operator \langle \hat{\mathbf{A}} \rangle approximates the classical vector potential \mathbf{A}_{cl}(t), such that the derived electric and magnetic fields satisfy the classical Maxwell equations in the presence of sources.^[31] This approximation holds because coherent states are eigenstates of the annihilation operator, leading to displaced vacuum states where the field behaves like a classical wave with added quantum noise. The application of Ehrenfest's theorem further justifies this reduction to classical dynamics by governing the time evolution of expectation values for the field operators. Specifically, the theorem yields equations such as \frac{d}{dt} \langle \hat{\mathbf{E}} \rangle = c \langle \nabla \times \hat{\mathbf{B}} \rangle (in Gaussian units) and \frac{d}{dt} \langle \hat{\mathbf{B}} \rangle = -c \langle \nabla \times \hat{\mathbf{E}} \rangle , mirroring the source-free Maxwell equations when averaged over coherent states. These relations demonstrate how quantum field commutators translate into the classical curl identities for the mean fields, provided the state supports non-vanishing expectation values. This approximation is valid in the regime of large mean photon number \langle N \rangle \gg 1, where the relative quantum fluctuations become negligible. The energy uncertainty \Delta E \sim \sqrt{\langle N \rangle} \hbar \omega scales as the square root of the intensity, making the fractional fluctuation \Delta E / E \approx 1 / \sqrt{\langle N \rangle} small compared to the classical signal. However, it breaks down for low-intensity scenarios, such as single-photon states where \langle N \rangle \approx 1 and quantum effects dominate, or vacuum states exhibiting phenomena like the Casimir force due to zero-point fluctuations.

Correspondence to Classical Fields

In the classical regime of the quantized electromagnetic field, the two-point correlation functions of the field operators approach those of a classical field. Specifically, the expectation value \langle \hat{\mathbf{E}}(\mathbf{r},t) \hat{\mathbf{E}}(\mathbf{r}',t') \rangle approximates \mathbf{E}_{cl}(\mathbf{r},t) \mathbf{E}_{cl}^*(\mathbf{r}',t'), where \mathbf{E}_{cl} represents the classical field amplitude, reflecting the factorization typical of coherent or high-intensity states where quantum fluctuations become negligible relative to the mean field.^[32]^[33] This correspondence arises in quantum optics through the analysis of normally ordered correlation functions, as developed by Glauber, where coherent states minimize uncertainty and mimic deterministic classical waves.^[34] In the high-temperature or high-occupation-number limit, the quantum statistical description of the electromagnetic field transitions to classical behavior via the Bose-Einstein distribution. When the thermal energy k_B T greatly exceeds the photon energy \hbar \omega (i.e., h\nu / kT \ll 1), the occupation number \langle n \rangle = 1 / (e^{\hbar \omega / k_B T} - 1) approximates k_B T / \hbar \omega, recovering the classical Rayleigh-Jeans law for the spectral energy density u(\nu) = (8\pi \nu^2 / c^3) k_B T.^[35] This limit demonstrates how the quantized field's thermal equilibrium aligns with classical equipartition of energy among modes, resolving the ultraviolet catastrophe by ensuring the quantum description asymptotes to the classical Rayleigh-Jeans regime for long wavelengths or elevated temperatures.^[35] Decoherence plays a crucial role in suppressing quantum superpositions for macroscopic electromagnetic fields, facilitating the emergence of classical-like behavior. Interactions with environmental modes, such as scattering off atoms or absorption in media, rapidly entangle the field with the surroundings, leading to the exponential decay of off-diagonal density matrix elements and the effective selection of pointer states that resemble classical field configurations.^[36] For instance, in cavity quantum electrodynamics experiments with microwave fields, decoherence times on the order of microseconds or less prevent coherent superpositions of macroscopic field amplitudes, enforcing a classical appearance even though the underlying dynamics remain quantum mechanical.^[36] Historically, the correspondence between quantum radiation and classical fields was advanced by Pascual Jordan in the 1920s through his application of matrix mechanics to electromagnetic fluctuations. In the 1926 Dreimännerarbeit with Born and Heisenberg, Jordan quantized the field modes as harmonic oscillators, deriving fluctuation formulas that unified wave and particle aspects and recovered classical limits via commutator contributions that vanish in the high-energy regime.^[37] This work built on Einstein's 1909 fluctuation analysis, establishing an early correspondence principle for radiation that emphasized the quantum theory's recovery of classical predictions under appropriate limits.^[37]

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