Suppose you took away the Sun. How long would it take for us to notice, and for the Earth to fly off into interstellar space?
According to Isaac Newton's theory of gravity, you would notice it instantly — because in his theory, gravity is instantaneous. This "action at a distance" bothered Newton considerably.
So great an Absurdity that I believe no Man who has in philosophical Matters a competent Faculty of thinking can ever fall into it.
— Isaac Newton, Letters to Bentley, 1692/3
Nevertheless, the theory was so successful in describing planetary orbits that it was tolerated for nearly 250 years, until Albert Einstein rewrote the gravitational theory completely.
The problem with action at a distance is that it can reverse cause and effect. Although we might see the Sun's disappearance as being the cause of the Earth flying off into space, visiting aliens traveling at an appreciable fraction of the speed of light — with the Sun behind them and the Earth ahead — would see the Earth flying off first, and then the Sun disappear.
That is a consequence of Einstein's 1905 theory of special relativity, which introduced the speed of light as an absolute speed limit. He spent the next ten years developing his general theory of relativity, first for accelerating reference frames, and then for gravity. His final theory, published in 1915, showed that if the Sun were to disappear, we would not notice at all for eight minutes. This ensures that cause and effect remain in the correct order for all observers, human and alien alike.
Einstein had solved the 250-year-old problem of action at a distance and shown that all phenomena propagating through space — including gravity — obey the speed of light. Ten years later, however, Heisenberg, Schrödinger, and Bohr developed the theory of quantum mechanics, explaining the strange phenomena observed in experiments with individual particles.
Heisenberg developed the mathematics first — a version largely lost from popular consciousness, called the Heisenberg Picture. In this picture, particles are modeled with complex, multifaceted properties. Where Newton determined that position, velocity, and mass were the most important properties of a moving body, Heisenberg replaced these familiar quantities with mathematical objects called operators.
Schrödinger offered a different approach. Rather than replacing familiar Newtonian concepts with operators, he invented a new kind of physical object — one based on waves. All the machinery of quantum behavior was absorbed into the wave function and the equations governing how it evolved in time. The wave function became the standard way to describe quantum particles, and remains so today.
"Whereas the playing field of special relativity is space and time, the playing field of quantum mechanics is the Hilbert space — an infinite-dimensional space of possibilities."
About ten years after the wave function's introduction, Einstein pointed out a problem. When a particle emerges from an emitter — whether an electron or a photon — quantum mechanics models it as an expanding wave. Yet when it strikes a detector, the particle appears only at a single point. Einstein asked: how does the rest of the detector know, instantly, that the particle has been detected somewhere?
He called this "spooky action at a distance" in a letter to Max Born in 1947. In quantum mechanics parlance, the wave function "collapses" when it interacts with macroscopic matter — and this happens everywhere all at once, even if the detector is light-years wide.
David Bohm and Hugh Everett later came up with alternative explanations. Bohm proposed there was a single particle, guided by the wave function, which was never directly detected. Everett proposed the wave function never collapsed at all, but rather the detector itself maintained the wave function within itself. Bryce DeWitt used this as the basis for the Many Worlds interpretation of quantum mechanics.
One wonders, however, whether this would have been a real problem had Heisenberg's picture prevailed over Schrödinger's wave mechanics. The two are mathematically equivalent but philosophically very different. One proposes a new kind of entity — the wave function — while the other proposes a new kind of property — the operator.
Schrödinger derived his wave function from an earlier framework called the Hamilton-Jacobi theory, which connects classical mechanics to wave theory. What emerged was an equivalence between the trajectories of bodies in motion — governed by Newton's laws — and wave fronts in motion. Schrödinger saw that individual photons and electrons behaved like waves even when alone, and concluded they were following the Hamilton-Jacobi law — with one crucial addition: a random component proportional to Planck's constant.
The wave function is nothing more than a useful tool — but does not exist in the form we think it does.
— On the ontological status of the wave function
If the Hamilton-Jacobi law shows an equivalence between classical trajectories and wave fronts, does that mean all objects are classical waves? Certainly not. It means only that wave fronts are equivalent to the trajectories of many particles. Likewise, even though we can model particle trajectories with a wave function, that does not mean those particles are waves. The action at a distance Einstein observed was simply a byproduct of the wave function being treated as a real entity — as if we decided to model Jupiter's orbit as a wave and then screamed "action at a distance" when it appeared at a particular location.
Whereas the playing field of special relativity is space and time, the playing field of quantum mechanics is the Hilbert space — an infinite-dimensional space of possibilities. In the Heisenberg picture, particle properties such as position, momentum, and spin are evolving operators acting on a fixed Hilbert space position. In Schrödinger's wave mechanics, particle properties are extracted by means of fixed operators on an evolving Hilbert space position. That is precisely why they are equivalent.
If we fundamentally live not only in space and time but also in a Hilbert space of possibility, then there is no reason to suppose that a Hilbert space position simply collapses into the familiar classical realm when we observe something. When we make a measurement, it is like looking at Saturn's rings. Depending on how Saturn is tilted toward us, we might see a thin line or a broad oval shape. We may not understand all the dynamics involved — but we might simply be incapable of seeing all infinite dimensions at once. Instead, when we observe something, we perceive merely a small subset of the dimensions that are there.
"Quantum reality requires no action at a distance. Instead, we are perceiving an infinite-dimensional reality through a very narrow lens — interpreting it as communicating across vast distances when, in fact, those distances are also part of the Hilbert space."
You can interpret this how you like. Lev Vaidman calls this Many Worlds but without splitting. It could simply be one world where we are perceiving Hilbert spaces from a particular angle — where "angle" is a loose analogy for a particular point in an infinite-dimensional space. All of reality is, indeed, merely a matter of perspective.
If relativity of Hilbert spaces is the correct framework for quantum mechanics, there must be a Hilbert space relativity theory — and indeed there is. In 2006, Alexey Kryukov published a paper proposing a new formulation of quantum theory on Hilbert space manifolds. A manifold is a mathematical surface with shape, such as the surface of the Earth. Spacetime itself is modeled as a manifold; the curvature of space and time — especially time — is why gravity works.
(Yes, the reason the planets orbit the Sun is roughly because time moves at different rates closer to and farther from the Sun — not, as you might have expected, because space is curved, as in the terrible rubber-sheet analogy often inflicted on hapless students.)
The paper proposes that three-dimensional manifolds are embedded inside the Hilbert space, representing the possible positions or momenta of particles. A measuring apparatus constitutes a "hole" in the Hilbert space manifold — a point of equilibrium. When a particle is measured, it falls into the hole. Because we can only perceive the holes and not the rest of the Hilbert space, we see this as a collapse of reality. That is simply our perspective. Reality is much larger — and infinite-dimensional.
While this is not directly analogous to special relativity, it suggests we may have quantum interpretation all wrong because we are not thinking big enough. We are not willing to let go of our desire to view the universe as a classical reality we can make intuitive sense of. The truth may be that our minds are, as it were, stuck in holes in Hilbert space.
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