Uncertainty relations in classical and quantum theories of electromagnetism

Abstract: Sharp uncertainty relations restricting the values of variances in the position space and in the momentum (wavevector) space are derived. They have the same form $ΔrΔk\ge 5/2$ in the classical theory of light beams, in the quantum theory of coherent light beams, and in the quantum theory of individual photons.

• The paper derives an exact lower bound for the product of spatial and spectral variances (Δr Δk ≥ 5/2) valid for classical fields, coherent quantum states, and individual photons.
• It employs the Riemann-Silberstein vector and a variational method to transform the problem into a three-dimensional harmonic oscillator eigenvalue challenge.
• The analysis demonstrates that localization limits are inherent to wave dynamics rather than quantization, impacting photonics and nano-optics design.

Universal Uncertainty Relations in Electromagnetic Theory

Abstract and Motivation

The paper "Uncertainty relations in classical and quantum theories of electromagnetism" (2605.28906) develops and rigorously proves an exact lower bound for the product of position and wavevector variances in electromagnetic fields. This lower bound, $\Delta r \Delta k \geq 5/2$, holds identically in three theoretical settings: (1) classical Maxwell theory, (2) quantum coherent states, and (3) single-photon quantum states. The result is notable for its universality—transcending the quantum-classical divide—and for its precise analytical characterization of the fields which saturate the bound.

Theoretical Foundations and Derivation

The key innovation lies in the use of the Riemann-Silberstein (RS) vector, a complex vector field unifying the electric displacement and magnetic field, as the foundational entity for both classical and quantum cases. The uncertainty relation is formulated between the spatial variance ($\Delta r^2$), measured via the energy density of the electromagnetic field, and the variance in wavevector space ($\Delta k^2$).

In the classical regime, energy density acts as the distribution function to measure spatial localization—avoiding ambiguous operator-based measures such as the center-of-energy operator, which presents interpretational difficulties due to non-commutativity. The variational procedure applied to the RS vector’s Fourier amplitude yields—after non-trivial calculation—an eigenvalue problem for a three-dimensional isotropic harmonic oscillator. The resulting minimum eigenvalue is $\gamma = 5/2$, leading directly to the sharp uncertainty bound.

Crucially, the entire derivation is framed in terms of the wavevector $\bm{k}$, decoupling the result from Planck’s constant and thus demonstrating that the form of the relation is not tied to quantum mechanics or operator formalism.

Extension to Quantum Theory: Coherent States and Single Photons

For quantized fields, the derivation is preserved. Coherent states are constructed with arbitrary spectral amplitudes, and expectation values in these states simply reproduce the classical variances due to the properties of the Glauber displacement operator. As a result, the uncertainty bound and saturating functions are unchanged.

The quantum case for individual photons leverages the RS vector formalism, restricting to positive-frequency solutions to ensure compliance with physical photon states. The mathematical structure of the uncertainty relation remains invariant, and the photon’s wavefunction that saturates the bound is computed explicitly—an element missing in previous work by the authors and others.

The previous photon uncertainty bound in [Bialynicki-Birula & Bialynicka-Birula, 2012] involved a higher lower limit due to the use of different localization operators, but the present method, using energy density, yields the universal result for both photons and classical fields.

Mathematical Characterization of the Saturating States

The function that saturates the uncertainty bound is analytically given by $g_0(\kappa) = \kappa \exp(-\kappa^2/2)$, where $\kappa$ is a properly normalized modulus of the wavevector. In real space, the field configuration that saturates the bound is shown to be proportional to $(y, -x, 0)\exp(-r^2/2a^2)$, a vector function with circular symmetry and Gaussian envelope. Its Fourier transform exhibits similar symmetry.

Explicit calculations confirm that this field (and its quantum and photon analogs) achieves $\Delta r^2 = 5a^2/2$ and $\Delta k^2 = 5/(2a^2)$. The set of states saturating the bound forms a degenerate family, including all possible mixtures of positive and negative helicity.

Implications and Theoretical Significance

A central claim, substantiated by the analysis, is the universality of the uncertainty relation for fields and photons: the lower bound and its saturating states are completely independent of quantization. This suggests that the uncertainty principle, at least in this context, should be viewed as a property of wave dynamics rather than a distinctively quantum phenomenon.

By entirely eliminating reference to Planck’s constant and providing a bound in purely classical terms, the result clarifies long-standing debates about the classical limit of uncertainty relations and whether "quantumness" plays a role in localization limits for fields.

Additionally, the explicit connection of the bound with the photon’s helicity generalizes to all massless particles, with the minimal bound depending monotonically on helicity.

Practical Ramifications and Outlook

The results have immediate implications for photonics and nano-optics, especially in fields where the spatial confinement of electromagnetic energy is critical (e.g., the formation of subwavelength optical structures). The universality of the bound sets an absolute limitation on the spatial concentration and spectral bandwidth of electromagnetic field modes, regardless of whether the regime is classical or quantum.

On the theoretical front, the explicit construction of saturating states opens avenues for optimizing field localization in applied settings, and the RS formalism is positioned as a unifying language for both classical and quantum electromagnetism. Future research may examine the extension of these concepts to interacting quantum fields, guided ultrafast photonic wavepackets, or other gauge bosons.

Conclusion

This work rigorously establishes that the uncertainty relations between spatial and spectral localization in electromagnetic theory have a universal lower bound, $\Delta r^2$0, valid for classical fields, coherent quantum states, and individual photons. The mathematical derivation is transparent, relying on wave-theoretic arguments and the RS vector formalism, and all saturating states are characterized analytically. The findings drive home that wave properties—not quantization—determine the fundamental limits of field localization, setting a reference point for future explorations in both foundational and applied electromagnetic theory.