Uncertainty relations in classical and quantum theories of electromagnetism

This paper derives a universal sharp uncertainty relation, $\Delta r\Delta k \ge 5/2$, that identically constrains the variances of position and momentum in classical light beams, coherent quantum states, and individual photons.

The Gist

The Big Idea: A Universal "Blur" Rule

Imagine you are trying to take a picture of a moving object. There is a fundamental rule in physics that says you can't make the object look perfectly sharp in two different ways at the same time: you can't know exactly where it is (position) and exactly how fast or in what direction it is moving (momentum/wave) simultaneously.

Usually, we think of this as a "Quantum Rule" (Heisenberg's Uncertainty Principle), something that only happens in the weird world of tiny particles like photons.

This paper argues something surprising: This "blur" rule isn't just a quirk of quantum mechanics. It is actually a rule of waves. Whether you are dealing with a giant beam of classical light (like a laser pointer), a coherent quantum beam, or a single individual photon, the math describing this "blur" is exactly the same.

The Three Scenarios

The authors tested this rule in three different "universes" of light:

1. Classical Light: The old-school view where light is just a wave, like ripples in a pond.
2. Coherent Quantum Light: A laser beam treated with quantum rules, but acting like a smooth wave.
3. Single Photons: The tiniest, individual particles of light.

The Result: In all three cases, the "fuzziness" follows the exact same formula:
$$\text{Position Spread} \times \text{Wave Spread} \ge 2.5$$
(The paper writes this as $\Delta r \Delta k \ge 5/2$).

The Analogy: The Perfectly Focused Balloon

To understand what this means, imagine you have a magical balloon filled with light.

• Scenario A (The Classical Wave): You blow up the balloon. If you squeeze it tight to make it very small in one spot (very precise position), the air inside rushes out and the balloon becomes very "wiggly" and spread out in its movement (very imprecise wave).
• Scenario B (The Single Photon): Now, imagine the balloon is just a single grain of sand. If you try to pin that grain of sand to a specific spot, its "wave nature" forces it to be spread out in a very specific way.

The paper proves that the shape of the balloon that is "just right" to minimize the blur is identical whether the balloon is made of a massive ocean wave or a single grain of sand. The "perfectly balanced" shape is a specific mathematical curve (involving a function called the Dawson function, which is a bit like a complex, wavy hill).

Why This Matters (According to the Paper)

For a long time, people debated whether the Uncertainty Principle was a sign that the universe is "quantum" (weird and probabilistic) or if it was just a property of waves.

• The Old View: "Uncertainty is because we can't measure things perfectly without disturbing them."
• This Paper's View: "Uncertainty is because waves naturally behave this way."

The authors show that you don't need to talk about "quantum operators" or "probabilities" to derive this rule. You can derive it using pure math for classical waves, and you get the exact same answer.

The "Planck Constant" Trick:
In quantum physics, the uncertainty rule usually includes a tiny number called Planck's constant ($\hbar$), which makes it look like a "quantum" thing. The authors decided to ignore that number and just look at the wave vector (how wavy the light is) instead of momentum. When they did this, the "quantum" number vanished, and the rule looked exactly like a classical wave rule.

The "Perfect" Shape

The paper doesn't just say the rule exists; it finds the exact shape of the light beam that hits the limit of this rule (the "saturating" function).

• It turns out the "perfect" light beam isn't a simple sphere.
• It has a specific, complex shape involving a mix of exponential curves and special "Dawson" functions.
• If you create a light beam with this specific shape, you have achieved the absolute minimum amount of blur possible for both its position and its wave nature.

Summary

Think of the Uncertainty Principle not as a "Quantum Mystery," but as a "Wave Law of Nature." Just as a guitar string has a limit to how short and how fast it can vibrate at the same time, light has a limit to how small a spot it can be in and how specific its direction can be.

This paper proves that this limit is universal. It applies to the big, classical waves we see every day, and it applies to the tiny, individual photons of the quantum world, using the exact same mathematical recipe. The "weirdness" of quantum mechanics isn't the cause of the uncertainty; the wave nature of light is.

Technical Summary

Technical Summary: Uncertainty Relations in Classical and Quantum Theories of Electromagnetism

Problem Statement
Recent advancements in nanophotonics have highlighted the practical limitations imposed by uncertainty relations on the localization of light. While the Heisenberg uncertainty principle is a cornerstone of quantum mechanics, its applicability and universality across classical and quantum regimes of electromagnetism require clarification. Previous derivations for photons, such as those based on center-of-energy operators, yielded a lower bound of $1 + \sqrt{5}/2$ but suffered from non-commutativity issues that obscured physical interpretation. Furthermore, the specific functional forms of wave packets that saturate these bounds remained elusive. This work addresses the need to derive sharp uncertainty relations for position and momentum (wavevector) variances across three distinct frameworks: classical Maxwell theory, the quantum theory of coherent light beams, and the quantum theory of individual photons.

Methodology
The authors employ the Riemann-Silberstein (RS) vector, $\mathbf{F}(\mathbf{r}, t) = \mathbf{D}(\mathbf{r}, t)/\sqrt{2\epsilon_0} + i\mathbf{B}(\mathbf{r}, t)/\sqrt{2\mu_0}$, as a universal mathematical tool. This vector formulation allows for a consistent description across all three cases, despite differing physical interpretations.

1. Definition of Variances: Instead of using a conserved four-vector current (which does not exist for the electromagnetic field), the authors utilize the field energy density, proportional to $\mathbf{F}^* \cdot \mathbf{F}$, to define spatial and spectral spreads. The position variance $\Delta r^2$ and wavevector variance $\Delta k^2$ are defined as the second moments of the energy density in real space and Fourier space, respectively, evaluated at $t=0$ to minimize wave packet spreading.
2. Variational Calculus: To find the minimal uncertainty product, the authors vary the product $\Delta r^2 \Delta k^2$ with respect to the Fourier amplitudes $f_{\pm}(\mathbf{k})$. By replacing the position operator $\mathbf{r}$ with derivatives in $k$-space, the problem is transformed into an eigenvalue equation.
3. Reduction to Harmonic Oscillator: Assuming spherical symmetry to find the lowest eigenvalue, the variational equation is reduced to a three-dimensional harmonic oscillator equation in terms of a dimensionless variable $\kappa$.
4. Quantum Extensions:
   • For coherent states, the classical amplitudes are replaced by expectation values of field operators in a Glauber coherent state. Due to the properties of coherent states, the expectation values of the energy density reduce exactly to the classical forms.
   • For individual photons, the RS vector components are interpreted as relativistic wave functions for positive and negative helicity states. The authors explicitly construct the wave functions that saturate the bound, resolving a gap in previous work that utilized relativistic spinors.

Key Contributions and Results
The paper derives a sharp uncertainty relation valid for all three cases:
$$\Delta r \Delta k \ge 5/2$$
where $\Delta r$ and $\Delta k$ are the square roots of the variances in position and wavevector space.

• Universality of Form: The lower bound is identical for classical electromagnetic waves, coherent quantum states, and individual photons.
• Identification of Saturating Functions: The paper explicitly identifies the functions that saturate the inequality. The eigenfunctions correspond to the ground state of the derived harmonic oscillator, involving generalized Laguerre polynomials.
  • In the simplest case, the saturating field at $t=0$ is given by $\mathbf{F}(\mathbf{r}) = C \exp(-r^2/2a^2) [y, -x, 0]^T$.
  • The corresponding Fourier transform is a Gaussian modulated by the transverse wavevector components.
  • For individual photons, the authors provide the explicit closed-form expressions for the positive and negative helicity wave functions ($F_+$ and $F_-$) involving Dawson functions and exponential terms, which were previously unknown.
• Elimination of Planck's Constant: By formulating the relation in terms of the wavevector $\mathbf{k}$ rather than momentum $\mathbf{p}$, the Planck constant $\hbar$ is removed from the fundamental inequality, emphasizing the classical-quantum universality. The quantum form $\Delta r \Delta p \ge 5\hbar/2$ is recoverable via $\mathbf{p} = \hbar \mathbf{k}$.

Significance and Claims
The authors claim that these findings provide a new perspective on the "quantumness" of the Heisenberg uncertainty relations. The primary conclusion is that the uncertainty relations in electromagnetism characterize the wave properties of the field rather than intrinsic quantum properties.

• The derivation does not require the apparatus of quantum mechanics (Hilbert space operators) for the mathematical proof; the same variational logic applies to classical fields.
• The fact that the functional forms saturating the inequality are identical across classical and quantum regimes serves as proof of this universality.
• The work clarifies that while the probabilistic interpretation of wave functions is essential for the physical meaning of uncertainty in quantum theory, the mathematical derivation of the bounds relies on the wave nature of the field, which is common to both classical and quantum descriptions.

The paper modestly positions itself as a clarification of the role of uncertainty relations in nanophotonics and a theoretical unification of these bounds, rather than proposing new experimental applications or future implications beyond the scope of the derived relations.