I got to know Charles “Chip” Sebens back in 2012, when he emailed to ask if he could spend the summer at Caltech. Chip is a graduate student in the philosophy department at the University of Michigan, and like many philosophers of physics, knows the technical background behind relativity and quantum mechanics very well. Chip had funding from NSF, and I like talking to philosophers, so I said why not?
We had an extremely productive summer, focusing on our different stances toward quantum mechanics. At the time I was a casual adherent of the Everett (many-worlds) formulation, but had never thought about it carefully. Chip was skeptical, in particular because he thought there were good reasons to believe that EQM should predict equal probabilities for being on any branch of the wave function, rather than the amplitude-squared probabilities of the real-world Born Rule. Fortunately, I won, although the reason I won was mostly because Chip figured out what was going on. We ended up writing a paper explaining why the Born Rule naturally emerges from EQM under some simple assumptions. Now I have graduated from being a casual adherent to a slightly more serious one.
But that doesn’t mean Everett is right, and it’s worth looking at other formulations. Chip was good enough to accept my request that he write a guest blog post about another approach that’s been in the news lately: a “Newtonian” or “Many-Interacting-Worlds” formulation of quantum mechanics, which he has helped to pioneer.
In Newtonian physics objects always have definite locations. They are never in two places at once. To determine how an object will move one simply needs to add up the various forces acting on it and from these calculate the object’s acceleration. This framework is generally taken to be inadequate for explaining the quantum behavior of subatomic particles like electrons and protons. We are told that quantum theory requires us to revise this classical picture of the world, but what picture of reality is supposed to take its place is unclear. There is little consensus on many foundational questions: Is quantum randomness fundamental or a result of our ignorance? Do electrons have well-defined properties before measurement? Is the Schrödinger equation always obeyed? Are there parallel universes?
Some of us feel that the theory is understood well enough to be getting on with. Even though we might not know what electrons are up to when no one is looking, we know how to apply the theory to make predictions for the results of experiments. Much progress has been made―observe the wonder of the standard model―without answering these foundational questions. Perhaps one day with insight gained from new physics we can return to these basic questions. I will call those with such a mindset the doers. Richard Feynman was a doer:
“It will be difficult. But the difficulty really is psychological and exists in the perpetual torment that results from your saying to yourself, ‘But how can it be like that?’ which is a reflection of uncontrolled but utterly vain desire to see it in terms of something familiar. I will not describe it in terms of an analogy with something familiar; I will simply describe it. … I think I can safely say that nobody understands quantum mechanics. … Do not keep saying to yourself, if you can possibly avoid it, ‘But how can it be like that?’ because you will get ‘down the drain’, into a blind alley from which nobody has yet escaped. Nobody knows how it can be like that.”
-Feynman, The Character of Physical Law (chapter 6, pg. 129)
In contrast to the doers, there are the dreamers. Dreamers, although they may often use the theory without worrying about its foundations, are unsatisfied with standard presentations of quantum mechanics. They want to know “how it can be like that” and have offered a variety of alternative ways of filling in the details. Doers denigrate the dreamers for being unproductive, getting lost “down the drain.” Dreamers criticize the doers for giving up on one of the central goals of physics, understanding nature, to focus exclusively on another, controlling it. But even by the lights of the doer’s primary mission―being able to make accurate predictions for a wide variety of experiments―there are reasons to dream:
“Suppose you have two theories, A and B, which look completely different psychologically, with different ideas in them and so on, but that all consequences that are computed from each are exactly the same, and both agree with experiment. … how are we going to decide which one is right? There is no way by science, because they both agree with experiment to the same extent. … However, for psychological reasons, in order to guess new theories, these two things may be very far from equivalent, because one gives a man different ideas from the other. By putting the theory in a certain kind of framework you get an idea of what to change. … Therefore psychologically we must keep all the theories in our heads, and every theoretical physicist who is any good knows six or seven different theoretical representations for exactly the same physics.”
-Feynman, The Character of Physical Law (chapter 7, pg. 168)
In the spirit of finding alternative versions of quantum mechanics―whether they agree exactly or only approximately on experimental consequences―let me describe an exciting new option which has recently been proposed by Hall, Deckert, and Wiseman (in Physical Review X) and myself (forthcoming in Philosophy of Science), receiving media attention in: Nature, New Scientist, Cosmos, Huffington Post, Huffington Post Blog, FQXi podcast… Somewhat similar ideas have been put forward by Böstrom, Schiff and Poirier, and Tipler. The new approach seeks to take seriously quantum theory’s hydrodynamic formulation which was developed by Erwin Madelung in the 1920s. Although the proposal is distinct from the many-worlds interpretation, it also involves the postulation of parallel universes. The proposed multiverse picture is not the quantum mechanics of college textbooks, but just because the theory looks so “completely different psychologically” it might aid the development of new physics or new calculational techniques (even if this radical picture of reality ultimately turns out to be incorrect).
Let’s begin with an entirely reasonable question a dreamer might ask about quantum mechanics.
“I understand water waves and sound waves. These waves are made of particles. A sound wave is a compression wave that results from particles of air bunching up in certain regions and vacating other. Waves play a central role in quantum mechanics. Is it possible to understand these waves as being made of some things?”
There are a variety of reasons to think the answer is no, but they can be overcome. In quantum mechanics, the state of a system is described by a wave function Ψ. Consider a single particle in the famous double-slit experiment. In this experiment the one particle initially passes through both slits (in its quantum way) and then at the end is observed hitting somewhere on a screen. The state of the particle is described by a wave function which assigns a complex number to each point in space at each time. The wave function is initially centered on the two slits. Then, as the particle approaches the detection screen, an interference pattern emerges; the particle behaves like a wave.
There’s a problem with thinking of the wave as made of something: the wave function assigns strange complex numbers to points in space instead of familiar real numbers. This can be resolved by focusing on |Ψ|2, the squared amplitude of the wave function, which is always a positive real number.
We normally think of |Ψ|2 as giving the probability of finding the particle somewhere. But, to entertain the dreamer’s idea about quantum waves, let’s instead think of |Ψ|2 as giving a density of particles. Whereas figure 2 is normally interpreted as showing the evolution of the probability distribution for a single particle, instead understand it as showing the distribution of a large number of particles: initially bunched up at the two slits and later spread out in bands at the detector (figure 3). Although I won’t go into the details here, we can actually understand the way that wave changes in time as resulting from interactions between these particles, from the particles pushing each other around. The Schrödinger equation, which is normally used to describe the way the wave function changes, is then viewed as consequence of this interaction.
In solving the problem about complex numbers, we’ve created two new problems: How can there really be a large number of particles if we only ever see one show up on the detector at the end? If |Ψ|2 is now telling us about densities and not probabilities, what does it have to do with probabilities?
Removing a simplification in the standard story will help. Instead of focusing on the wave function of a single particle, let’s consider all particles at once. To describe the state of a collection of particles it turns out we can’t just give each particle its own wave function. This would miss out on an important feature of quantum mechanics: entanglement. The state of one particle may be inextricably linked to the state of another. Instead of having a wave function for each particle, a single universal wave function describes the collection of particles.
The universal wave function takes as input a position for each particle as well as the time. The position of a single particle is given by a point in familiar three dimensional space. The positions of all particles can be given by a single point in a very high dimensional space, configuration space: the first three dimensions of configuration space give the position of particle 1, the next three give the position of particle 2, etc. The universal wave function Ψ assigns a complex number to each point of configuration space at each time. |Ψ|2 then assigns a positive real number to each point of configuration space (at each time). Can we understand this as a density of some things?
A single point in configuration space specifies the locations of all particles, a way all things might be arranged, a way the world might be. If there is only one world, then only one point in configuration space is special: it accurately captures where all the particles are. If there are many worlds, then many points in configuration space are special: each accurately captures where the particles are in some world. We could describe how densely packed these special points are, which regions of configuration space contain many worlds and which regions contain few. We can understand |Ψ|2 as giving the density of worlds in configuration space. This might seem radical, but it is the natural extension of the answer to the dreamer’s question depicted in figure 3.
Now that we have moved to a theory with many worlds, the first problem above can be answered: The reason that we only see one particle hit the detector in the double-slit experiment is that only one of the particles in figure 3 is in our world. When the particles hit the detection screen at the end we only see our own. The rest of the particles, though not part of our world, do interact with ours. They are responsible for the swerves in our particle’s trajectory. (Because of this feature, Hall, Deckert, and Wiseman have coined the name “Many Interacting Worlds” for the approach.)
No matter how knowledgeable and observant you are, you cannot know precisely where every single particle in the universe is located. Put another way, you don’t know where our world is located in configuration space. Since the regions of configuration space where |Ψ|2 is large have more worlds in them and more people like you wondering which world they’re in, you should expect to be in a region of configuration space where|Ψ|2 is large. (Aside: this strategy of counting each copy of oneself as equally likely is not so plausible in the old many-worlds interpretation.) Thus the connection between |Ψ|2 and probability is not a fundamental postulate of the theory, but a result of proper reasoning given this picture of reality.
There is of course much more to the story than what’s been said here. One particularly intriguing consequence of the new approach is that the three sentence characterization of Newtonian physics with which this post began is met. In that sense, this theory makes quantum mechanics look like classical physics. For this reason, in my paper I gave the theory the name “Newtonian Quantum Mechanics.”
This allows a pleasing level of intuition to interpreting the wavefunction and its time evolution.
I wonder if Einstein might have found this formulation quite palatable?
That was a very interesting paper. I have two comments:
1. Given that it’s an F=ma kind of theory, it seems odd to me to constrain the many worlds to have velocities that are precisely determined by positions. In a classical fluid, there would be random motion on top of the average velocity fields. Of course, maybe I just shouldn’t expect it to look like a classical fluid.
2. The biggest concern is one that I didn’t even think of until the paper brought it up: Newtonian QM doesn’t necessarily satisfy the quantization condition. While it’s easy enough to posit that in our particular universe, the quantization condition is satisfied, it’s not obvious to me that this is a stable situation. Why does it continue to be satisfied for every new particle created? And if Newtonian QM deviates slightly from MWI (due to there being a finite number of worlds), shouldn’t the quantization condition drift over time?
Interesting! Nice graphic animations.
Sean and Chip,
You know how, in high school algebra, there are problems that make you solve for the length of a rectangle, if you know its area, by solving a quadratic equation? Usually, you obtain a positive solution and a negative solution. Why do we always throw out the negative solution?
So anyway, I have this theory that asks, “What if there is another universe we cannot see where such negative lengths exist?” So when we get a negative solution for the length of the rectangle, it’s because there are two possible lengths, corresponding to the two universes!
Why not just take the math literally and postulate the existence of this other, parallel universe?
Sounds promising, doesn’t it?? Oh sure, we cannot observe this other universe…but so what, right?
David Deutsch also seems to believe something very similar to this. I don’t know if he has put forward a technical description, but in his popular-level book The Fabric of Reality he describes essentially the same picture of wavefunction interference as interactions between copies of the same state in otherwise separate branches.
Edit: Okay, looking at your paper I now see that it is more different than the standard multi-verse formulation than I realized- especially, you don’t have any branching of the universal wavefunction. Interesting. I guess I will have to actually read the thing :).
Sean, Chip, or anyone who knows more about this than me,
this paragraph intrigued me.
“There’s a problem with thinking of the wave as made of something: the wave function assigns strange complex numbers to points in space instead of familiar real numbers. This can be resolved by focusing on |Ψ|2, the squared amplitude of the wave function, which is always a positive real number.”
Why are strange complex numbers a problem? Are they just a problem from the perspective of a “dreamer?” It seems to me the dreamers are looking for a familiar interpretation of QM. Could it be that the current understanding is too poor, yet the better understanding lies in equally/more unfamiliar territory? If you will, a doers solution to the dreamers problem?
John,
I think the biggest issue with complex numbers is just that we represent everything that we actually observe with real numbers. As far as I know, mathematically it doesn’t make a difference using complex numbers over real numbers. But whenever we use complex numbers in physics, such as to generate a wave via Euler’s relation, what we actually observe in nature is either the real or imaginary part of the wave, giving us a real-valued answer. Or a material having a complex-valued index of refraction is a tool we can use to describe materials that cause the electric field amplitude of a wave to decay as it travels through.
Generally when you start with real-valued conditions, you want your answer to be real-valued. Personally I’m not sure if there is an issue with interpreting complex numbers beyond this, but this has been my experience with it.
In the Newtonian QM version of Newton’s law in the paper, the potential term depends on the “density of worlds (ρ)”, which itself is a function of Φ…so the wavefunction is still not eliminated is it?
And while we’re in the business of quoting Feynman, I found this:
“There’s a school of thought that cannot believe that the atomic behavior is so different than large scale behavior. I think that’s a deep prejudice… and they’re always waiting for the day we discover that underneath the quantum mechanics there’s some mundane ordinary balls hitting or particles moving…I think they’re gonna be defeated”
This new interpretation appears to me like some sort of a blend of MWI and dBB. Actually, it is precisely dBB, with the difference that “all other” particles are in “different worlds”, as opposed to our world.
Given MWI and dBB as two extreme interpretations of QM, some people might find an interpolated “middle ground” between those two more digestible than each of the extremes on their own. But I fail to see any actual improvement in understanding QM.
“How can there really be a large number of particles if we only ever see one show up on the detector at the end?”
Because there isn’t a large number of particles, because one particle is a wave in space. When you detect it at one slit something akin to an optical Fourier transform occurs, and it becomes pointlike, so it goes through that slit only. Otherwise it goes through both slits. Then when you detect it at the screen something akin to an optical Fourier transform occurs, and it becomes pointlike, so you see a dot on the screen. No multiverses are required. See weak measurement work by Aephraim Steinberg et al and by Jeff Lundeen et al.
The paper mentions that the number of worlds is finite, or at least taken to be finite since a continuum would cause conceptual problems. How finite? How large are these particle clouds expected to be? Is every particle cloud exactly equally large?
The animations touch on something beyond words, the inner experience of a dreamer 🙂 The swerves in fig 4 are interesting.
… When the particles hit the detection screen at the end we only see our own. The rest of the particles, though not part of our world, do interact with ours. They are responsible for the swerves in our particle’s trajectory …
I dont understand this
… When the particles hit the detection screen at the end we only see our own. The rest of the particles, though not part of our world, do interact with ours. They are responsible for the swerves in our particle’s trajectory …
How come those other particles interact with our particle, but dont interact with the detection screen? If our particle interacts with the screen, shouldnt those other interact too? I mean the screen is also made from particles.
So it appears that our multiverse contains a finite number of universes. But perhaps our multiverse is just one of many.
The fair probability of multi-universe being true. If you observe one would the wave function change to produce just the one that is observed? I am sure this would be explain with a helpful link. Thanks
Can anyone help me with researching an effect that I observed many years ago in my teens? In the last few years it looks like the buckling Euler’s column and contained column studies. This buckling effect under increasing load produces waves that relate to Eigen values of quantum particles. As of late, it’s starting to look like a model of the “particle in a box” i.e., (Infinite well) wave solution
The effect in the video uses the new variation of Euler’s Contained Column Theory. Am I wrong to say this is an “a new variation of his contained column theory”?
You tube: Particle in a box-model-using- Guthrie variation of Euler contained column theory
http://youtu.be/wrBsqiE0vG4
Jared,
Yes, I see. That makes a lot of sense.
I feel that we may need to stop thinking entirely of real and imaginary parts of things. For example, it is natural to think of the particle as more real than it’s wave function, yet their doesn’t appear to be any good reason to think that way.
So to me it does seem handy to use real numbers to represent real things, and imaginary numbers for imaginary things. However, it may lock us into wanting to find a more real explanation for what is going on, When the thing supposed to be imaginary is the most real and existential thing in the whole system.
Just my two uneducated cents.
God plays dice with the universe!
I have always been a dreamer, and those dreams lead me to believe that quantum uncertainty is a fundamental aspect of nature that can’t be avoided.
If you imagine Minkowski spacetime from the frame of reference of a particle traveling the speed of light, from the frame of reference of that particle, spacetime would contract to zero if it assumed it was at rest. It would then be in a state of “superposition” along everywhere on it’s world line at once. Then to pull that particle out of that frame by an act of observation to a definite speed and position would then have to be truly random. There would be no way to deal with zero and infinity in a classical manner, physically or mathematically. I think that is why the hidden variable problem could never be solved really.
ok, so every particle gets 3 dimensions in config space– what about energy & momentum? Aren’t they just as real? If not, what happens to config space in nuclear reactions?
…Professor Hawking, feel free to weigh in.
I have to say the whole Everett hypothesis has always troubled me whether in its original version or this new interactive one. With regard to this version of the theory I take it that all of these alternate universes already exist and that is how their particles come to interact with ours. How does it happen that there happen to be parallel universes of just the right type and number to interact as they do and not in some other way? As I understand the wave function the probability of a given particle position is proportional to the amplitude of the wave at that position squared. In the other universes, presumably those probabilities would be different. Wouldn’t that require a virtual infinity of alternates given that to integrate all the probabilities to 1 you have to account for the probability at every possible position in space and even that only for a given time. In the original
Everette interpretation the question becomes not “how is it that all these universes happen to exist” but from whence comes the energy to split the universe on every measurement — which would actually mean every interaction, every decoherence! Examples are always given in terms of the slit experiment with 2 slits. But what if there are 10 slits? In that case 10 universes must spring into existence…
I could go on here, but the bottom line seems to be that this is nothing more than a mathematical convenience, a helpful way of understanding what is going on with the wave function, but I don’t see that this could be a description of a genuine ontology.
Since I have the expertise (physics grad student, read the paper), and time to waste, I could answer some of the questions above…
@Wizard, In Newtonian QM, the wavefunction is eliminated, but there’s still the question of what takes its place to represent the same information. That is ρ, the density of worlds. There’s a function to translate from the Everettian Ψ to the Newtonian QM ρ, but that’s just a translation, and doesn’t mean Ψ is required.
@Peter, Each “particle” here is one of the many worlds, and thus contains all information about everything in that particular universe, including the particles in the screen.
@Walker Guthrie, I suspect the mathematics of buckling is ungodly complicated. You’d have to find a curve which minimizes curvature but maintains the total length. Not really related to QM, just has similar solutions.
@Tin Man, The energy and momentum are determined by the position. In standard QM, the momentum is given by the variation of the wavefunction over space. Thus, you only need to specify the wavefunction as a function of position, and the momentum is a freebie. (In Newtonian QM, the details are different, but the end result should be the same.)
@Matthew Rapaport, In Everettian QM, the energy is calculated by sandwiching the Hamiltonian with the universal bra and ket. End result is a weighted average over all worlds, not a sum over all worlds. So increasing the number of worlds conserves energy. (I didn’t understand your question about Newtonian QM, because the probabilities are not the property of any particular universe, they’re properties of the multiverse in aggregate.)
A lot of people like to think classically, even when it comes to modern physics. I am sure the theory would be given every opportunity to become successful.
Is NQM a new theory? Is NQM even a new hypothesis? At this point, it appears as a new formulation of the existing theory.
A new hypothesis would propose new predictions that differ from the existing theory. If those predictions were testable, it would qualify for a new theory. I haven’t seen either criteria be met yet, so I submit that this is a new formulation of the existing theory.
I’d be interested to see how this formulation explains the test results of Bell’s inequalities.
Nobody knows what is a ‘particle’. It is point-like in our experiments, but how does it ‘moves’? Make a Fourier transform from ‘point’ impact (delta function) and you will get a lot of waves.
Does this new take offer any insight to entanglement or delayed-choice experiments?