The Heisenberg Uncertainty Principle
What Happens If We Apply the de Broglie Wave-Particle Duality Principle to the Wavefunction Itself?
Part 3:
disclaimer:
This is a sketch of an idea that is far-fetched but worth exploring. I enjoyed entertaining its potential and pitfalls and thought sharing it might be interesting and maybe even helpful. This is not a claim of scientific rigor but rather an invitation to consider a new perspective.
Introduction:
I was continuing my read of David Bohm’s book when I stumbled upon a chapter discussing the Heisenberg Uncertainty Principle and the EPR paradox. That got me thinking: how would these concepts look through the lens of infons? If you wonder what this is about read the first two parts first: part 1, part 2.
Before diving into that, though, let’s refresh what the Heisenberg Uncertainty Principle says about reality. Unlike classical mechanics, quantum mechanics deals with observables that don’t always commute - meaning you can’t measure certain properties of a system simultaneously with complete precision. In simple terms, when you try to measure one property of a quantum system, you inevitably disturb another. Focusing on position (x) and momentum (p), the uncertainty relation is mathematically expressed as:
This equation tells us there’s a fundamental limit to how precisely we can know both the position and momentum of a particle at the same time.
Let’s break this down into two cases:
For a single-particle system, the uncertainty principle is fairly intuitive. If you try to measure the particle’s position (x), you inevitably disturb its momentum (p) because the act of measurement (like bouncing a photon off the particle) introduces random changes to the system. This understanding dates back to the mid-1920s when Heisenberg first proposed the principle. Bohr and Heisenberg famously defended this interpretation against Einstein’s early critiques, even using special relativity to argue that any measurement must disturb the system, forcing it to obey the uncertainty principle. In Bohr’s words: measurement is messy.
But things get complicated with the introduction of the EPR paradox in 1937, presented by Einstein, Podolsky, and Rosen. Their thought experiment suggested that it might be possible to measure a system’s properties without disturbing it -if you consider a system of two (or more) particles. If these particles are far enough apart, measuring one shouldn’t affect the other (locality principle). This directly challenges Bohr’s defense of "messy measurement" and forces a confrontation with the unsettling idea of non-locality -or what Einstein famously dismissed as “spooky action at a distance.” Bohr responded by embracing non-realism to preserve the standard interpretation of quantum mechanics.
In short:
Bohmian mechanics agrees with standard quantum mechanics for single-particle systems—it accepts the messiness of measurement. But with two-particle systems, Bohm took a different route since his interpretation is rooted in realism. Bohm proposed sub-quantum randomness, suggesting there could be hidden mechanisms beneath the quantum level that appear as random noise. These mechanisms might be responsible for the randomness and uncertainty we observe, especially in systems of two entangled particles.
Bohm even explored experiments that could potentially challenge standard quantum predictions - suggesting that this isn’t just philosophical speculation but something testable in principle.
The Infons Perspective: A New Take on Uncertainty
Now, let’s shift to the infons view. For a single-particle system, the Heisenberg Uncertainty Principle can be adopted without any issues. But with two-particle systems, things take a sharp turn. To make it compatible with the infons framework, the principle could be extended like this:
Here, d represents the distance between particles (or more generally, the system’s characteristic length). The function f(d) behaves such that:
For a two-particle system separated by a significant distance, this extended uncertainty principle starts to diverge. Imagine two entangled photons connected through a long fiber optic cable. If you alter the path so that the effective distance becomes infinite (say, by introducing loops or knots in the cable), the connection between the particles breaks, and the uncertainty principle vanishes.
The Problem: Breaking Conservation Laws
One can immediately see that the extended version of Heisenberg’s principle could result in violations of conservation laws. In fact, the Heisenberg uncertainty appears in two-particle systems because of conservation laws -it's the preservation of physical quantities across distances that manifests as the EPR paradox. In other words, entanglement is the conservation law imposed on superposition of two-particle system which share a history somewhere in their past.
Let’s dig deeper. Imagine a system in superposition with a total spin of zero. When the system splits into two parts. Let's assume that the process of breaking up has nothing to do with the spin of the system and hence conservation of spin should be enforced. The conservation of spin dictates that the total spin must remain zero. However, because the system was initially in superposition, each part should also exist in a superposition of spin states. Yet, due to the conservation law, these two parts must remain perfectly coordinated: if one collapses to -0.5 upon measurement, the other must collapse to +0.5 to maintain the balance demanded by Mother Nature.
But the extended Heisenberg principle challenges this coordination. Imagine two photons traveling through a long thin fiber-optic cable. If we somehow manipulate the path to become effectively infinite, perhaps by introducing a knot or loop that stretches their connection. According to the extended uncertainty relation, the operators now commute (the right side of the equation becomes zero), breaking their connection. The particles could then behave as if they no longer share a common history or any obligation to conserve spin. If measured, they could produce results that blatantly violate the conservation of spin.
This sounds outrageous. If conservation laws break down, what’s left of physics? Interestingly, this isn’t the first time this idea has surfaced. Bohr himself once suggested that conservation laws might only hold statistically (BKS theory)—meaning they could break for individual systems but remain valid when averaged across many experiments. Imagine repeating the above experiment countless times: while individual outcomes might violate spin conservation, their averages could still respect the law.
From the infons perspective, distance matters because entanglement exists within the physical world, not some abstract Hilbert space. Infons actively shuttle information between particles, keeping them in sync. When the distance becomes effectively infinite, infons can’t complete their loop—they get stuck—and the particles lose their connection, leading to violations of conservation laws.
Imagine this process:
For this process to keep up with what experiments suggest, the speed of information transfer would need to surpass the speed of light by several orders of magnitude. In standard quantum mechanics, though, asking about the speed of entanglement transfer is considered meaningless. If one insists the best answer is: it’s effectively infinite.
What We Gain, What We Lose
By considering the infons framework, we gain two things:
First, we regain a sense of realism. Entanglement stops being an abstract, non-local phenomenon and becomes something tangible -a real, physical connection in 3D space. This isn’t too surprising since the whole point of Bohmian mechanics, and by extension the infons framework built on it, is to preserve realism.
Second, we introduce a measurable parameter: the speed of infons. This can bring the concept of infons into the mathematical framework, a parameter that represents infons in equations. However, there are many questions about it: Could the speed of infons be constant, or might it depend on properties of the wavefunction, like its frequency or wavelength?
But, of course, this comes at a hefty price: it challenges the very foundation of conservation laws. We'd be left accepting Bohr’s earlier idea that conservation might only hold statistically, rather than being strictly enforced in every instance.
Is this trade-off worth it? Maybe that’s a conversation for another day.