The Relativistic Schrödinger’s Paradox
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In its standard form the imaginary experiment that Erwin Rudolf Josef Alexander Schrödinger (1887 Aug 12 - 1961 Jan 04) conceived in 1935 to reveal his quantum paradox involves a Rube Goldberg device that contains a hapless cat and a glass vial filled with enough Prussic acid (HCN, hydrogen cyanide) to kill the cat if and when a hammer smashes the vial. I prefer to apply my imagination to something a little less gruesome, so I will replace the cat and the poison with a light bulb, a battery, and a toggle switch.
We enclose our experimental apparatus within a perfectly opaque, completely soundproof, thermally insulated box. Simply put, the apparatus consists of three parts: the lower part consists of a light bulb, a battery, a toggle switch, and the wires that connect them into a circuit; the middle part consists of a hammer mounted on a pivot and a solenoid against which the hammer rests in such a way that if the solenoid’s core moves, the hammer will lose its balance, swing down, and hit the toggle switch in a way that opens the circuit and turns the light off; and the upper part consists of an unstable particle, such as an atom of tritium, inside the tube of a Geiger counter that feeds its output into an amplifier, whose output goes to the solenoid. With that apparatus assembled inside the box we turn on the light, close up the box, and wait.
An atom of tritium consists of one proton and two neutrons stuck together with a single electron revolving around them. With a half-life of 12.32 years, it decays by spitting out from its nucleus an electron and an electron anti-neutrino and becoming an atom of helium-3. We can re-conceive the tritium atom, rather crudely, as a self-jostling quantum system comprising an atom of helium-3 (which we conceive as a potential-energy cage) containing an electron in its nucleus. This system exists in one of two states - the electron inside the cage (the un-decayed tritium atom) and the electron outside the cage (the decayed tritium atom). In the un-decayed state the system conforms to a state function that has two parts, the probability wave confined inside the cage and the much smaller probability wave outside the cage.
When the atom decays, the state function describing the system must change into one with a very small amplitude inside the potential-energy cage that the helium-3 nucleus represents and a larger amplitude outside the cage. Thus the electron has essentially zero probability of existing inside the nucleus and a probability close to unity of existing outside the nucleus. Taking energy with it, the electron moves rapidly and generates a faint electrical pulse in the Geiger counter. The amplifier strengthens the pulse, which then goes to the solenoid, where it generates a magnetic field that moves the solenoid’s core. That moving core nudges the hammer out of balance and the hammer, in consequence, swings down on its pivot and hits the toggle switch. Flipped by the hammer, the toggle switch breaks the electrical circuit feeding electricity from the battery to the light bulb and thus turns the light off. And there we have a simple nuclear decay detector.
If we assemble this apparatus and turn the light on, we will see when the tritium atom decays by noting when the light goes out (assuming, of course, that no part of the apparatus fails). In that case, although we interpret the state function associated with the tritium atom as representing a mixed state, we can see what state the atom actually occupies. If we put the apparatus into the box and close up the box, we no longer have that certainty.
In Schrödinger’s interpretation of the experiment, once we put the apparatus into the box the tritium atom goes into an actual mixed state; that is, the atom exists as both decayed and un-decayed at the same time. That proposition necessitates that the Geiger counter exist in a mixed state of receiving/not receiving an electrical pulse. Every component of the apparatus must then exist in two different states simultaneously. The hammer must exist as both balanced and fallen; the toggle switch must exist as both flipped and un-flipped; and the light bulb must exist as both off and on (and note that this does not mean that a 100-watt bulb shines out 50 watts: it means that the bulb shines out both 100 watts and zero watts at the same time).
This seems to give us an easy reductio ad absurdum, but we need to go deeper into the analysis of the experiment before we can resolve this paradox. We need to ask why does putting the apparatus into a box change the way we must describe it. Why does making a system un-observable in some way alter the very existence of the system? That question brings to mind George Berkeley’s proof of the existence of God.
At the beginning of the Eighteenth Century George Berkeley (1685 Mar 12 - 1753 Jan 14) developed his doctrine of immaterialism (now called subjective idealism), which he summed up in the statement "esse est percipi" (to exist is to be perceived). He asserted a belief that matter does not exist as a thing-in-itself, does not possess the property of reality, but rather that only perception is real, so that things exist only to the extent that they are perceived. He then argued that the continued and consistent existence of things necessitates the existence of an omnipresent observer to perceive them at all times.
In like manner, the København Interpretation of the quantum theory, usually associated with Niels Bohr, asserts that a quantum system does not achieve a definite state until someone subjects it to an observation. Until an observation collapses it onto a single, definite state, the state function describing the system evolves as the description of a mixed state.
But Schrödinger’s apparatus extends the presumed mixed state of the tritium atom into the realm where actions are so vastly greater than the Planck unit that the single Planck unit of indeterminacy fades into complete insignificance. It pushes the blurry physics of the quantum realm into the classical, sharply focused realm of perfectly deterministic physics. In that latter realm the law of the excluded middle, the basis for the reductio ad absurdum, reigns: if we have put a system into one state (hammer balanced upright, light bulb glowing), then the system cannot at the same time manifest itself in a contrary state (hammer fallen, light bulb dark).
If, on the other hand, we interpret the mixed state as representing nothing more than our ignorance of which state the system occupies, then the paradox dissolves readily. In that case the tritium atom always exists in a definite state (un-decayed or decayed) so all of the other components of the apparatus do as well. We simply don’t know what state that is until we open the box and look at the light bulb. And that makes sense in light of our belief that putting the system into a box should diminish our knowledge of the system but certainly should not alter the system itself.
How did we get into this paradox? We misinterpreted the state function that we use to describe the tritium atom. In describing the state function of the electron trapped inside the un-decayed tritium atom’s nucleus I noted that it divides neatly into two parts – the high amplitude part inside the potential-energy cage that represents a helium-3 nucleus and the low amplitude part outside the cage. We can use those two parts via Enrico Fermi’s Golden Rule to calculate the probability that the atom will decay within a certain time interval and thence calculate the atom’s half life. But the fact that we can divide the state function into two parts that we can treat more less independently does not mean that it represents a genuine mixed state, a combination of un-decayed atom and decayed atom.
We can see the truth of that proposition in the fact that when the atom actually decays the state function of the electron changes. Inside the cage the amplitude of the state function goes from high to minuscule and outside the cage the amplitude goes from low to high. An actual mixed state representing a true quantum stochastism would have a description consisting of a linear sum of that state function (both parts) with the original state function (both parts).
By reductio ad absurdum, then, we infer that the atom does not exist in a mixed state. At some time it decays and causes the apparatus to turn off the light and it doesn’t need any witnesses (feline or otherwise) to help it along. In accordance with Heisenberg’s indeterminacy principle, we cannot anticipate when that event will occur, but we can say with complete confidence that it will definitely occur. Thus, we do not need to add to our description of Reality any bizarre contortions, such as the many-worlds hypothesis. We simply accept the fact that the uncertainty that we calculate for this experiment merely reflects our ignorance of its outcome (until we open the box) and nothing more. For all of our theoretical intents and experimental purposes the Schrödingerisch mixed state does not exist.
Consider what Schrödinger had to say:
"It is typical of these cases that an indeterminacy originally restricted to the atomic domain becomes transformed into macroscopic indeterminacy, which can then be resolved by direct observation. That prevents us from so naively accepting as valid a "blurred model" for representing reality. In itself it would not embody anything unclear or contradictory. There is a difference between a shaky or out-of-focus photograph and a snapshot of clouds and fog banks." – E. Schrödinger, 1935.
So does the quantum theory express our own limitations, the imprecision inherent in our apparatus (the blurry photograph), or does it display a fuzziness inherent in Reality (the snapshots of clouds)? If we take the latter possibility as true to Reality, then by bringing that fuzziness into the macroscopic realm, Schrödinger’s thought experiment reduces it to an absurdity. Thus Schrödinger dismissed the "blurred model" of the very theory that he helped to create. In that act he stood with Albert Einstein, who commented to him in a letter he wrote in 1950 that "...one cannot get around the assumption of reality...as something independent of what is experimentally established."
In fact, Schrödinger presented his 1935 paper describing the cat experiment as part of a discussion of the incompleteness of quantum theory asserted by Einstein, Podolsky, and Rosen earlier that year. Where the Købnhavn Interpretation echoes Berkeley in asserting that nothing is definite until it is perceived/observed, EPR asserted that if we can predict with certainty the outcome of an observation, then we can say that the data in that observation emanate from some element of physical reality. But if things must be perceived in order to have a definite existence, then what do we perceive that gives us the sensations that tell us about the object of our perception? What anchors those sensations in Reality? Modern science does not accept Berkeley’s theological solution of the problem, so we must perforce accept as a fact of Reality the existence of matter independent of any interaction we may have with it. But that necessitates in turn that we accept the existence of the events in which that matter participates, again independent of any interaction we may have with the system in which those events occur. The quantum theory blurs the locations of those events in space and time, but it does not blur the fact that they occur.
But in that analysis I have made a tacit assumption that leaves out of the analysis the other great advance in Twentieth Century physics. I have assumed that we stay close to the box and that time has elapsed smoothly around us. Let’s change that assumption: now I assume that the box exists far enough away from us that our various motions bring the temporal offset of Special Relativity into play. What happens to our analysis then?
In the essay on Bennett’s clock paradox I ended with an imaginary experiment in which stop and go traffic on Earth alters the history of a planet in the Great Galaxy in Andromeda (Messier 31, NGC 224), roughly two million lightyears from Earth. I contemplated the idea of stop-and-go traffic on the Santa Monica Freeway convulsing the histories of entire civilizations across the full length, breadth, and depth of the Great Galaxy in Andromeda (at a distance of two million lightyears a delta-vee of fifty-five miles per hour creates a temporal displacement of 216 days) and then asked whether I and my history are equally phantasms to the inhabitants of one of that galaxy's worlds. That experiment implies a confirmation of Hugh Everett III’s many-worlds interpretation of the quantum theory.
Does shifting back and forth between inertial frames change the Universe that we occupy? At a distance of two million lightyears the temporal offset (xv/c2) equals 26 hours per mile per hour. So consider an intergalactic version of Schrödinger’s cat paradox in its light-bulb-in-a-box form. I sit reading and on the planet Pfft, occupying the same inertial frame that I occupy in my chair, a team of scientists has performed the experiment and opened the box. As Andromeda rises in the east, I get up and walk west to get a cup of tea and thereby go into a frame in which the box has not yet been opened, then I return to my chair and the alien experimenters have opened the box a second time. Is that possible? Can I actually manipulate distant events by the simple act of pacing the floor? Does the Universe outside my lightcone have an indeterminate existence that I can control? More to my point, in the experiment the box has been opened, then unopened, then opened again: was the outcome the same the second time we opened the box?
If we say, "Yes, it must have been", then the temporal offset term in the Lorentz Transformation necessarily entails that Reality be absolutely, perfectly deterministic. That entailment would seem to be falsified by Werner Heisenberg's indeterminacy principle (and it is indeed indeterminacy, not uncertainty, according to Heisenberg himself) and by the Born interpretation of the wave function as representing a probability distribution.
If, on the other hand, we say, "No, it might have been different", then the temporal offset must entail something like Hugh Everett's many-worlds interpretation of the quantum theory. Imagine the history of the Universe as resembling a railroad marshaling yard. We have gone down one track, stopped, and backed up, then moved forward again. In the time that it took us to back up, several switches were thrown at random and we end up going forward on a track different from the one we originally followed.
So is it yes or no? How can we tell? I can’t know the outcome for two million years: during my wait the situation exists in a superposition. I can, of course, accelerate my ship into a frame in which Andromeda is closer and advanced in time, but then I come out two million years in Earth’s future.
Schrödinger intended his cat experiment to take a quantum effect from the submicroscopic realm of the atom and to move it, step by step, into the macroscopic realm of classical physics, thereby creating an absurdity showing that the quantum theory is incomplete. In our imaginary experiment we have extended Schrödinger’s quantum effect into the cosmological realm and brought Relativity into play.
But even Relativity doesn’t help us. We still have superpositions existing as real things, even though, as in Schrödinger’s original experiment, we can’t see them. We still have before us the possibility that the theory of quantum mechanics is incomplete, that there must exist some hidden variable that will make the theory more deterministic.
But in 1964 John Bell presented a theorem which states that quantum mechanics is complete, that there are no hidden variables. And unlike Schrödinger’s presentation, Bell’s theorem has been tested through actual experiments (not merely the imaginary kind).
So how did Schrödinger go wrong and how can we resolve his paradox? To answer that question we must look at Bell’s theorem and then re-examine Schrödinger’s paradox. Those topics will be the subjects of subsequent essays in this series.
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