Phenomenological quantum mechanics: II. Deducing the formalism from experimental observations

In the vast landscape of modern physics, quantum mechanics stands as the most successful theory ever devised, yet it remains the most puzzling. It governs the behavior of the smallest particles in the universe, from the electrons orbiting an atom to the photons of light that allow us to see. For nearly a century, scientists have relied on a specific mathematical framework to make predictions about these tiny systems. This framework works with stunning accuracy, allowing us to build lasers, transistors, and MRI machines. However, the theory comes with a strange baggage: it requires a special, often confusing distinction between the quantum world and the everyday world we live in. It suggests that particles exist in a haze of possibilities until they are measured, at which point they seem to suddenly "choose" a single reality. This abrupt change, known as the collapse of the wave function, has never been observed directly, and it leaves many physicists wondering if the standard mathematical tools they use are the only way to describe nature, or just the most convenient one.

A team of researchers from Poland, Germany, and China has taken a different approach. Instead of starting with the familiar equations of quantum mechanics, they asked a fundamental question: what if we tried to rebuild the entire theory from scratch, using only the raw data of what we actually see in the laboratory? Their goal was to see if the strange mathematical structures of quantum theory, such as the abstract spaces where particles "live" in the standard model, would naturally emerge from the simple act of recording sequences of measurements. They wanted to know if the theory's most famous features were inevitable laws of nature or merely artifacts of the specific way humans have chosen to write the math.

The researchers began by looking at how quantum systems behave when probed repeatedly over time. In a typical experiment, a scientist might measure a property of a particle, wait a moment, measure it again, and continue this sequence. The standard theory predicts the probabilities of these outcomes, but it treats the act of measurement as a special event that interrupts the natural flow of time. The team, however, focused on the raw patterns found in the data itself. They noticed that while the results of these sequential measurements follow a clear cause-and-effect order—where the past influences the future but never the reverse—they also exhibit a deep inconsistency that defies classical logic. In the classical world, if you know the path a ball took, you can predict where it will be next. In the quantum world, the path taken depends on how you look at it, and the data shows that the system does not follow a single, continuous track like a ball on a hill.

To make sense of this, the team introduced a new way of thinking about the data. Instead of assuming the system follows one path, they considered the possibility that the system is simultaneously exploring two different paths at once, one moving forward in time and one moving backward. They called this a "bi-trajectory." By treating the experimental data as a combination of these two opposing flows, they found that the complex, multi-time probabilities recorded in the lab could be described with a surprising simplicity. They discovered that these probabilities could be mapped onto a geometric structure known as a Hilbert space. In everyday terms, this is a mathematical space where every possible outcome of an experiment corresponds to a specific direction or point. The researchers showed that this space is not an arbitrary invention of physicists, but a necessary consequence of how the data behaves. If you try to describe the sequence of quantum measurements without this structure, the math simply does not add up.

As they continued to build their model, the team found that the familiar tools of quantum mechanics began to appear naturally. The "projectors" used in standard theory to represent the act of measurement emerged as the fundamental building blocks of their new framework. The "state" of a system, which in the standard view is a snapshot of a particle's condition, was reinterpreted as a record of how the system was initialized. Most importantly, the laws of motion that govern how a system changes over time were derived directly from the data. The team found that the system evolves according to a rule that looks exactly like the standard Schrödinger equation, driven by a quantity called the Hamiltonian. This confirmed that their new approach, which started with no assumptions about the underlying math, inevitably leads to the same predictive power as the established theory.

However, the new formalism is not just a rephrasing of the old one; it changes the story of what is actually happening. In the standard view, the quantum state is the master object, the central character that dictates everything. In this new bi-trajectory picture, the master object is the measure of the two paths themselves. The "state" is demoted to a mere description of how the experiment began. This shift has profound implications for one of the oldest problems in physics: the measurement problem. The standard theory struggles to explain how a quantum system becomes a classical object when observed, often requiring an artificial cut between the two. The new framework, by treating the system as a continuous interplay of forward and backward paths, removes the need for this sharp divide. It suggests that the transition from the quantum world to the classical world is not a sudden collapse, but a gradual process where the two paths of the bi-trajectory begin to overlap and align, effectively merging into a single, classical track.

The researchers also explored how this new view handles systems that interact with one another. They found that when two quantum systems interact, their bi-trajectories become entangled, meaning the path of one system depends on the path of the other. This provides a natural way to describe how a quantum system can influence a measuring device, which is often treated as a classical object. By modeling the measuring device as a system that has been pushed into a classical limit, the team showed how information about the quantum system is encoded into the device's behavior without requiring a mysterious, instantaneous collapse. This offers a promising path toward building realistic models of how measurements actually work, treating the observer and the observed as part of a single, continuous physical process.

While the new formalism agrees perfectly with all the experimental predictions of the standard theory, it offers a fresh perspective on the nature of reality. It suggests that the strange features of quantum mechanics, such as the inability to define a single path for a particle, are not flaws in our understanding but fundamental properties of how the universe operates. The bi-trajectory approach does not claim to have solved every mystery, nor does it prove that the standard theory is wrong. Instead, it provides a different lens through which to view the same data, one that might help physicists tackle problems that have remained stubbornly unsolved for decades. By showing that the complex machinery of quantum mechanics can be deduced from the simple act of observing sequences of events, the team has opened a door to a deeper understanding of the quantum world, one that may eventually allow us to see the machinery behind the magic.