This quantum theory may be Bohmian mechanics done right
Imagine that Alice is piloting a space ship light years away from Earth. Bob steers his space ship in the opposite direction, also light years away.
Back on Earth, scientists prepare in a lab two particles from a single reaction, such as the decay of a pion. These particles are sent out into space, one towards Alice and the other towards Bob. The particles are traveling fast enough to overtake their ships and, years later, each receives their particle and measures it.
If I were to draw a picture of what this looks like on paper, we might imagine a set of coordinates with space on the horizontal axis and time on the vertical. Each particle is a line traveling at an angle, say 45 degrees. In velocity units where the speed of light is c=1, a 45 degree angle means they are massless particles traveling at the speed of light in vacuum such as photons.
The space ships are traveling at an angle somewhat more than that since they are massive and cannot reach the speed of light.
They intersect the particle lines, meaning that they can each capture a particle in a detector at that point.
Earth, meanwhile, is stationary in the center, a vertical line through time.
When the particles reach their destinations and the measurements are made, we can ask, does the way in which one particle is measured of one affect the outcome of the measurement of the other?
Before Einstein’s revelation in 1905 that the speed of light is a cosmic speed limit, we could reasonably suppose that they might. After all, Newton’s universal law of gravitation said that two massive bodies would affect one another instantaneously.
After 1905, however, we learned that all forces and matter obey relativity which means that nothing can travel faster than light. After 1915, with the introduction of general relativity, we know this to be a local effect. Nevertheless, if we ignore strongly curving gravitational fields, the speed of light still holds globally.
For event A to affect event B, therefore, event A has to be in B’s past light cone.
A light cone is simply a higher dimensional version of a cone, so instead of each cross-section being a circle, it is a sphere. For a drawing with only one dimension of space like the above, the cone is a triangle like this:
Any events that are under the diagonal lines can affect what happens at the red asterisk. Any events above those lines cannot.
Now, it’s not hard to see that, for two events light years apart, such as the measurements taking place on the two rockets, if those measurements are taking place at sufficiently close times, there is no way that one can be in the past light cone of the other. Physicists, therefore, say that they have spacelike separation.
Two events with spacelike separation cannot affect one another, and this is known as locality.
It should therefore be reasonable to assume that the way Alice chooses to measure her particle should not affect the outcome of Bob’s measurement.
In classical physics this is true, but, in quantum physics, this is not true. Alice and Bob will find that if they carry out many, many measurements of many particles, over time, Bob will see a correlation between what he measures from his particles and how Alice measured her particles.
An example is if Alice measures the polarization of photons produced from a laser at some angle with respect to Bob’s detector, that angle will be reflected in his measurements.
This in no way allows for Alice to transmit any information to Bob because Alice and Bob must reunite in order for him to understand how Alice’s measurement affected his. He cannot do this from his measurements alone.
For this reason, some physcists want to define relativity as a limit on the transmitting of information rather than cause and effect, but that gives up locality, and that is not the only choice.
If you don’t want to give up locality because it means that you have to reintroduce action-at-a-distance which Einstein had so miraculously removed from physics, you can instead assume that Alice and Bob’s behavior was correlated. In other words, their choices are conspired.
This is the conspiracy assumption: Alice and Bob never had any choice on how to conduct their measurements to begin with. All is predetermined and the connection between how Alice did her measurement and the outcome of Bob’s measurement is simply a feature of all things being correlated within the universe, including our choices.
If we assume this, which is called superdeterminism, however, all is not well. We now have to give up all of science because it means that measurements are not independent of one another. Much of science depends on the assumption of independence. That is: we cannot know things if the way in which we measure them is biased in some way.
If everything in the universe is conspiring together, then there may be no order to anything.
Another choice is we can believe that the universe isn’t real until it is measured. Alice and Bob can make local, independent measurements, but nothing exists until they observe it. They aren’t so much measuring reality as creating it.
Once Alice or Bob makes a measurement, only then does the particle gain the characteristic that was measured, not before.
If we accept that measurement or observation creates reality, we reject counterfactual definiteness. This gets its name because a “counterfactual” world is one where we make a measurement prior to the measurement we factually did make.
If we assume the particle would have the property we measured later in that counterfactual world, if we measured it earlier, then we believe in counterfactual definiteness. Some people also call this “realism” in the sense that particles have real properties.
Both of the most popular interpretations of quantum mechanics, “Many Worlds” and “Copenhagen” reject counterfactual definiteness while retaining locality and independence of measurements. (Despite this, you will see countless articles suggesting that locality has been “disproved” but this is merely the interpretation of the people who write such articles.)
Many Worlds rejects both counterfactual and factual definiteness since it posits multiple worlds where measurements have different outcomes. Copenhagen only rejects counterfactual definiteness since it assumes that when a measurement is made there is a “collapse” of probable measurements into one factual one.
One early pioneer who attempted to push back against the Copenhagen interpretation was David Bohm, who, in 1952, introduced his own interpretation of the phenomenon we are describing known as entanglement.
Bohm wanted to preserve realism. He wanted particles to be there and have definite properties counterfactually, so he proposed an alternative ontology of quantum mechanics where particles are “guided” by a non-local field that we cannot measure directly but only probablistically.
There is a difference between an interpretation and an ontology of quantum mechanics. The former seeks to interpret quantum mechanics as it is. The latter attempt to turn it into a more complete theory that establishes what is real, which is what “ontology” means.
The guiding or pilot wave steers the particle both in space and time as well as steering other properties such as spin and polarization. Therefore, when Alice makes a measurement of her particle, the particle is really there but the effect of her measurement is instantly transmitted via the guiding wave to Bob’s particle, and he, therefore, measures the effect of Alice’s measurement as well as his particle.
Bohm created a mathematical framework for his theory called Bohmian mechanics.
Bohmian mechanics had no real practical use and created additional headaches for people trying to do quantum mechanics since it was extra math on top of what was already there. In addition it was very difficult to turn it into a relativistic theory and, even now, there is no clear merging of Bohm’s non-relativistic theory with general relativity although there have been attempts.
Physicists do not like Bohmian mechanics in general while philosophers seem to love it, and the reason is pretty simple as expressed in the words of Stephen Weinberg:
[T]he basic reason for not paying attention to the Bohm approach is not some sort of ideological rigidity, but much simpler—it is just that we are all too busy with our own work to spend time on something that doesn’t seem likely to help us make progress with our real problems.
As physicists, in fact, we don’t need to ask for an interpretation of quantum mechanics. It works just fine the way it is. But as physicist-philosophers, we desire an interpretation that is appealing both philosophically and scientifically.
One that may satisfy these needs is called the Deutsch-Hayden approach which was published in 2000. Although on its face it doesn’t have much resemblance mathematically to Bohm’s mechanics, philosophically it has a lot in common with it, which is why I call it Bohmian mechanics done right.
Deutsch-Hayden is the fruition of a vision Heisenberg had when he introduced his mathematical description of quantum mechanics in 1924. This 3 years before he would publish his famous Heisenberg principle and 1 year before Schroedinger would publish his famous Schroedinger’s equation. (Schroedinger’s cat wouldn’t come along till 1935.)
In that year, Heisenberg published what would later be known as the Heisenberg picture, a vision of quantum mechanics as a redefinition of the facts of nature.
In his picture, Heisenberg described particles with properties that were not individual numbers like speed nor vectors like velocity, but matrices which represented the realm of possibility for those properties. These matrices are called observables. Every particle, while having a real and definite existence, nevertheless, had properties that could only be condensed into individual numbers when measured. These observables would evolve over time.
Heisenberg’s picture was strange and confusing to physicists who couldn’t understand how anything could have properties represented as high dimensional matrices.
A year later Schroedinger introduced his wave mechanics. Schroedinger’s interpretation was favored because physicists understood how waves worked quite well. It was easy to visualize and more over it was mathematically equivalent to Heisenberg’s.
Neils Bohr, who was a mentor to Heisenberg, took the two interpretations to indicate that reality didn’t have a single, concrete description. It was tomay-to, tomah-to, and he wanted to call the whole thing off.
Heisenberg would later come to promote the Copenhagen interpretation, leaving behind his youthful idea that reality might be more complex under the hood than we think. The Heisenberg picture would remain, however, and later the Dirac picture would join it as a melding of Schroedinger and Heisenberg.
In 1999, David Deutsch and Patrick Hayden, however, introduced to the world a new theory of quantum computation in Heisenberg’s picture that challenged the prevailing interpretations.
They wanted to show that information flow in a quantum computational network, which the picture I’ve drawn of entanglement could be considered to be, was local, real, and still preserved independence of measurement.
The key to Deutsch-Hayden’s approach is observing that the Heisenberg picture is local while the Schroedinger is not, even though they are mathematically equivalent. In the Schroedinger’s wave mechanics the wave mediates information instantaneously between particles, acting as a global repository of information, while in the Heisenberg picture the particles carry with them all the information in an inaccessible form.
You can think of it in this way: Imagine that scientists doing measurements are like customers to an information repository and particles are employees to that repository.
Schroedinger’s wave is like having a central bookkeeping facility, the wavefunction. Employees, in order to access information to give to customers, must go to the central facility. They can also deposite information about the customers there. All the information is always instantly available to all the employees everywhere. Once they give information away to their customers, they lose access to part of the facility pertaining to that information.
In the Heisenberg picture, by contrast, every employee carries a locked briefcase full of information with them, and anything that happens to them they record, and it goes in the briefcase. Although they have all the information they need, only some of it can be given to customers. If customer Alice wants to compare information with that of customer Bob, the employee must go with Alice and confer with an employee who resides with Bob. The two employees get their story straight in secret and then are able to give consistent information to the two customers. Once given away, they lose access to that particular information but retain their briefcases forever.
Deutsch-Hayden avoids sacrificing to any of the usual suspects: locality, realism (counterfactual definiteness), or independence because they assume that when you make a measurement of a particle the outcome of that measurement is not, as most people would assume, the particle having a single, random value for that quantity. Rather, they assume that is just the only information the particle is willing to divulge to you, but it keeps to itself a great deal more which it can pass on despite being measured.
It is this information that allows Bob’s measurement to appear to depend on how Alice measures her particle.
In the Deutsch-Hayden picture the ability for particles to carry locally inaccessible information is crucial because Alice and Bob cannot compare their measurements and determine that they have a correlation between them without coming back into contact again or sending some kind of matter or energy to one another. It is in this matter or energy that the quantum information is carried from one to the other.
Deutsch and Hayden refer to this as “quantum information flowing through a classical channel”.
If this seems bizarre, that is because it is a large departure from how we understand quantum physics to work. It is indeed assumed that once a measurement of a particle is made that it loses all quantum information related to that measurement. This is a process called decoherence. The assumption is that decoherence destroys quantum information and makes it classical. Deutsch and Hayden say not so. Only observable information gets destroyed, while unobservable information is left intact to be transmitted for as long as necessary.
It is not clear that Einstein would have liked this state of affairs since Deutsch and Hayden effectively do away with the concept of a particle having a single real, measureable state. They exchange that for a large collection of unobservable information that is not only unknown but impossible to specify from any number of measurements.
The particle has a real state, but we can’t know what it is.
This means that Deutsch-Hayden achieves what Bohm was looking for: an interpretation grounded in a primitive ontology. A primitive ontology is a philosophy that says that the metaphysics of what exists (ontology) is inherent in the variables of physics. For Bohm, the guiding wave and the particle trajectories formed a primitive ontology but for Deutsch-Hayden it is the descriptors of quantum information, real, local, but inaccessible, that contain the ontology.
This means that Heisenberg’s initial instinct was right and Bohr’s was wrong. His picture contains the real ontology of quantum physics while Schroedinger’s may be mathematically equivalent, but it is metaphysically a mere abstraction.
Deutsch, David, and Patrick Hayden. "Information flow in entangled quantum systems." Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 456.1999 (2000): 1759-1774.
Timpson, Christopher G. "Nonlocality and information flow: The approach of Deutsch and Hayden." Foundations of Physics 35 (2005): 313-343.