Quantum Theory Regression under the Classical Limit

In quantum electrodynamics (QED), the electromagnetic field is fundamentally framed as a collection of quantized harmonic oscillators whose excitations correspond to photons. The classical limit typically emerges when the Planck constant approaches zero ($\hbar \to 0$) or when the photon occupation number becomes extraordinarily large ($N \to \infty$), while keeping the total macro-energy finite. Under these specific conditions, quantum fluctuations vanish relative to the background mean field. Consequently, the energy, momentum, and conservation laws predicted by quantum theory seamlessly converge into the familiar structures of classical electrodynamics. Grasping this regression mechanism serves as a vital conceptual bridge uniting microscopic quantum optics with macroscopic classical electromagnetism.
By expanding the electromagnetic field inside a cubic cavity of volume $V$ into plane waves and introducing the corresponding creation and annihilation operators, $a_{\mathbf{k},\lambda}^\dagger$ and $a_{\mathbf{k},\lambda}$, the free-field Hamiltonian and momentum operators are expressed as:

[
H = \sum_{\mathbf{k},\lambda} \hbar \omega_{\mathbf{k}} \left( a_{\mathbf{k},\lambda}^\dagger a_{\mathbf{k},\lambda} + \frac{1}{2} \right)
]

[
\mathbf{P} = \sum_{\mathbf{k},\lambda} \hbar \mathbf{k} , a_{\mathbf{k},\lambda}^\dagger a_{\mathbf{k},\lambda}
]

Here, $\omega_{\mathbf{k}} = c|\mathbf{k}|$ and $\lambda$ denotes the polarization state. In the quantum domain, the classical energy density $u = \frac{\varepsilon_0}{2}E^2 + \frac{1}{2\mu_0}B^2$ and momentum density $\mathbf{g} = \varepsilon_0 \mathbf{E} \times \mathbf{B}$ correspond directly to the expectation values of these quantum operators. The classical regime demands that such expectation values match macroscopic field quantities precisely, with quantum correction terms evaporating entirely.

Coherent States: The Bridge to Classicality

Coherent states $|\alpha\rangle$ are defined as the exact eigenstates of the annihilation operator:

[
a |\alpha\rangle = \alpha |\alpha\rangle, \quad \alpha = |\alpha| e^{i\theta}
]

For a single-mode field, the expected photon number is given by $\langle N \rangle = |\alpha|^2$, accompanied by a statistical fluctuation of $\Delta N = |\alpha|$. The relative fluctuation is thus expressed as $\Delta N / \langle N \rangle = 1/|\alpha|$. As $|\alpha| \to \infty$, this relative uncertainty drops sharply toward zero, rendering the quantum fluctuations of both electric and magnetic fields entirely negligible. This makes coherent states the quintessential model for investigating the classical limit.

The Correspondence Principle and the $\hbar \to 0$ Limit

Bohr’s correspondence principle dictates that quantum mechanics must reproduce classical physics in the limit of large quantum numbers. For electromagnetic fields, we fix the classical field amplitude $E_0$ while taking $N \to \infty$ and $\hbar \to 0$, ensuring the product $N\hbar$ remains invariant. Under this scaling, the energy expectation value of a single-mode coherent state behaves as:

[
\langle H \rangle = \hbar \omega \left( |\alpha|^2 + \frac{1}{2} \right) \approx \hbar \omega |\alpha|^2
]

The vacuum zero-point energy $\frac{1}{2}\hbar\omega$ becomes completely imperceptible compared to the macroscopic macro-energy $N\hbar\omega$. Setting $|\alpha|^2 = N$ yields $\langle H \rangle \to N\hbar\omega$, which aligns perfectly with classical field energy. Similarly, the momentum expectation $\langle \mathbf{P} \rangle = \hbar \mathbf{k} N$ maps directly to classical momentum.

Regression of Energy and Momentum

For a single-mode plane wave, the classical electric field takes the form:

[
\mathbf{E}(\mathbf{r},t) = E_0 \hat{\mathbf{e}} \sin(\mathbf{k}\cdot\mathbf{r} - \omega t + \theta)
]

Within a coherent state $|\alpha\rangle$, the expectation value of the electric field operator mirrors this exact configuration, where the amplitude satisfies:

[
E_0 = \sqrt{\frac{2\hbar\omega N}{\varepsilon_0 V}}
]

As $N \to \infty$ and $\hbar \to 0$ while keeping $N\hbar$ constant, $E_0$ stays rigorously finite. Consequently, the energy expectation evaluates to:

[
\langle H \rangle = \int_V \left( \frac{\varepsilon_0}{2} \langle E^2 \rangle + \frac{1}{2\mu_0} \langle B^2 \rangle \right) dV \to \int_V \left( \frac{\varepsilon_0}{2} E_0^2 \sin^2(\cdots) + \frac{1}{2\mu_0} B_0^2 \cos^2(\cdots) \right) dV
]

This integral retrieves the standard classical electromagnetic energy. Likewise, the momentum expectation transforms into the volume integral of the Poynting vector:

[
\langle \mathbf{P} \rangle = \int_V \varepsilon_0 \langle \mathbf{E} \times \mathbf{B} \rangle dV \to \int_V \varepsilon_0 \mathbf{E} \times \mathbf{B} , dV
]

Quantum uncertainties, scaling as $\Delta H / \langle H \rangle \sim 1/\sqrt{N}$, completely vanish in the macroscopic limit.

Transition of Conservation Laws

The Hamiltonian and momentum operators for a free electromagnetic field are constants of motion, satisfying Heisenberg's equations of motion:

[
\frac{dH}{dt} = \frac{i}{\hbar}[H,H] = 0, \quad \frac{d\mathbf{P}}{dt} = \frac{i}{\hbar}[H,\mathbf{P}] = 0
]

When evaluated within coherent states, these expectation values are identically conserved. In classical electrodynamics, energy and momentum conservation are governed by local continuity equations:

[
\frac{\partial u}{\partial t} + \nabla \cdot \mathbf{S} = 0, \quad \frac{\partial \mathbf{g}}{\partial t} + \nabla \cdot \overleftrightarrow{\mathbf{T}} = 0
]

where $\mathbf{S} = \mathbf{E} \times \mathbf{H}$ represents the Poynting vector and $\overleftrightarrow{\mathbf{T}}$ is the Maxwell stress tensor. In the fully quantum treatment, these relations manifest as operator expectation values laced with higher-order quantum corrections. In the classical limit, these corrective terms decay as powers of $1/\sqrt{N}$ or $\hbar$, yielding the pristine classical conservation laws.

Concrete Scenario: The Classical Limit in a Laser Field

Consider a single-mode field characterized by frequency $\omega$ and wave vector $k$ in a coherent state $|\alpha\rangle$, with $\alpha = \sqrt{N} e^{i\theta}$. The expected electric field reads:

[
\langle \mathbf{E}(z,t) \rangle = E_0 \hat{\mathbf{x}} \sin(kz - \omega t + \theta)
]

where $E_0 = \sqrt{2\hbar\omega N / (\varepsilon_0 V)}$. Substituting typical macroscopic parameters—such as $N = 10^{12}$ photons, $\hbar = 1.054 \times 10^{-34} , \text{J·s}$, $\omega = 10^{15} , \text{rad/s}$, and $V = 1 , \text{cm}^3$—yields an electric field amplitude $E_0$ on the order of $10^6 , \text{V/m}$, closely mirroring standard laboratory laser intensities. Under these conditions:

• The energy expectation $\langle H \rangle \approx N\hbar\omega$, with a negligible relative fluctuation of $1/\sqrt{N} \sim 10^{-6}$;
• The momentum expectation $\langle P \rangle \approx N\hbar k$, directed squarely along the wave vector;
• The quantum correlations between electric and magnetic components approach a deterministic classical correlation.

Pushing this framework further by letting $N \to \infty$ and $\hbar \to 0$ eradicates all traces of quantum fuzziness, compelling the quantum description to revert completely to classical electrodynamics.

Conclusion

The regression of quantum theory to the classical limit is not merely an arbitrary truncation of quantum phenomena. Instead, it relies on the cooperative dynamics of coherent states and macroscopic occupation numbers, ensuring that quantum expectation values track classical field profiles while fluctuations and high-order corrections fade away. The natural alignment of field energy and momentum operators with classical energy densities and Poynting vectors validates the self-consistency of quantum electrodynamics, solidifying a robust, transparent bridge between quantum mechanics and classical field theory.