the quantum theory of the electromagnetic field.

Quantum electrodynamics: relativistically, gauge-invariant form ← elementary particle physics

light quanta in materials

Light is a quantum object.→ quantum observables, Hamitian operators, states vector or density matrix.

Light is a field. → the observables extends over space and time, they are functions of space and time coordinates.

Light is an electromagnetic wave. → the central physical quantities of light are connected to the electromagnetic field stengths.

SUPPOSE we regard the classical fields as the expectation values of the corresponding quantum observables. $\mathbf B\to \hat {\mathbf {B}},\mathbf B=\lang\psi|\hat{\mathbf B}|\psi\rang=\lang \mathbf B\rang$. In other words, the classical fields are the ensemble averages of quantum fields.

• the quantum field strengths $\hat {\mathbf B}$ must obey Maxwell’s equations as well(not all eigenvalues are zero)

$$ \begin{aligned} \nabla\times\langle\psi|\hat{\mathbf{E}}|\psi\rangle & = -\partial\langle\psi|\hat{\mathbf{B}}|\psi\rangle/\partial t, \\ \langle\psi|\nabla\times{\hat{\mathbf{E}}}+\partial{\hat{\mathbf{B}}}/\partial t|\psi\rangle&=0. \\ \nabla\times\hat{\mathbf{E}}+\partial\hat{\mathbf{B}}/\partial t&=0. \end{aligned} $$

Optical instruments (mirrors, lenses, beam splitters) are often made of dielectric media.[$\rho_f=0$]→ a quantum field theory of light in media.

• The medium responds to the light by becoming electrically polarized or magnetized.
• A regime of linear response where the polarizations and magnetizations are proportional 成正比 to the field strengths. [non-dispersive, isotropic]

  • → Constitutive equations 本构方程：$\mathbf E \to \hat {\mathbf E}$

    $$ \hat{\mathbf{D}}=\varepsilon_{0}\varepsilon\hat{\mathbf{E}},\quad\hat{\mathbf{B}}=\mu_{0}\mu\hat{\mathbf{H}},\quad\varepsilon_{0}\mu_{0}=c^{-2}, $$

  • $\varepsilon_0$ the permittivity of the vacuum

  • $\mu _0$ the permeablility of the vacuum

  • $\varepsilon$ the electric penbvcxzrmittivities, [real, do not depend on time or on frequency]

  • $\mu$ the magnetic permeabilities, [real, do not depend on time or on frequency]

• If the constitutive equations (各常数为实数)are real, which is the case for non-absorptive media

The fields are often expressed in terms of potentials. For quantum electromagnetism, we use the operator of the vector potential $\hat {\mathrm A}$ but $\phi\equiv0$.(dielectric media and $\rho_f\equiv0$):

$$ \hat{\mathbf{E}}=-\frac{\partial\hat{\mathbf{A}}}{\partial t},\quad\hat{\mathbf{B}}=\nabla\times\hat{\mathbf{A}}. $$

Coulomb gauge condition:

$$ \nabla\cdot\varepsilon\hat{\mathbf A}=0, $$

the electromagnetic wave equation:

$$ \frac{1}{\varepsilon}\nabla\times\frac{1}{\mu}\nabla\times\hat{\mathbf{A}}+\frac{1}{c^{2}}\frac{\partial^{2}\hat{\mathbf{A}}}{\partial t^{2}}=0. $$