Last time I started exploring questions I have about the Many Worlds Interpretation of Quantum Mechanics (the MWI of QM). Obviously I’m not a fan; quite the opposite. It presents as parsimonious, hung on the single hook of a universal wavefunction, but I think it gets more complicated and cumbersome when examined. I can’t say it’s broken, but I don’t find it very attractive.
I suspect most people, even in physics, don’t care. A few have invested themselves in books or papers, but these interpretations don’t matter to real physics work. The math is the math. But among the philosophical, especially the ontological, it’s food for debate.
Being both philosophical and ontological, I do smell what’s cooking!
Hence these posts. Reading about MWI leads to thinking about MWI, and that leads to notes as I try to sort out the thoughts. Notes lead to posts that seek to solidify those thoughts, so off we go.
Last time I left off with the question of the ontology of the wavefunction in MWI and the more urgent question of how physical matter coincides. The latter is a key objection I have to the MWI. I want to examine it, and the notion of decoherence, in detail.
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The central tenet of MWI insists that “it’s all just the Schrödinger equation.” In fact that specific equation is not quite up to the task, but the phrasing is actually shorthand for the general notion of a universal wavefunction as physical reality.
The Schrödinger equation is a mathematical function that uses complex numbers and lives in Hilbert space (which requires a huge — in some cases an infinite — number of dimensions). Further, there are different versions of it, time-dependent or not, relativistic or not.
Here’s the non-relativistic time-dependent version:
I think it is useful to understand the Schrödinger equation as the quantum version of Newton’s classical second law of motion (that the change in momentum is proportional to the force applied). There is a certain similarity of form (both are derivatives over time):
Note the fundamental importance of the imaginary unit, i, in quantum mechanics. It appears frequently in general physics as a convenience, but it’s absolutely required in quantum mechanics. (It’s part of what makes it hard to apply the math of QM to the physical world.)
Both Schrödinger’s and Newton’s equations are laws of motion. Classical mechanics deals with position and momentum. Quantum mechanics — in the Schrödinger equation, for instance — uses position and energy. (In QM, position and momentum, per Heisenberg Uncertainty, are in some sense orthogonal to each other. Both cannot be definite simultaneously.)
So the big question is how do we go from this math to physical reality? What implements the wavefunction mathematics?
Mathematically, given a solution to the Schrödinger equation for some quantum system, we apply mathematical operators to give us probability amplitudes. These operators represent possible measurements of the system (for instance, of its position, momentum, energy, or spin). Because a probability amplitude is a complex number (which can be negative), we take the square of its absolute value to get a real number — the probability of a given result if we make the measurement implied by the operator.
The critical point here is that the Schrödinger equation gives us potentials. How does that become reality? How does the statement “the electron might be here or might be there” become separate actual worlds with multiple physical electrons?
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Obviously reality itself must implement the wavefunction mathematics. Or rather, reality behaves according what the math describes. Which is the general situation with physics. Our mathematical theorems describe the lawful behavior of physical reality.
But what does it mean to say reality comprises at this moment (and in the past) every possible version of the universe that could exist? And that these are all equally physically real (and occupy the same physical space)?
These questions really boil down to one about quantum superposition, and I’ll come back to that in the next post.
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Another aspect of this is that the wavefunction for a large system is an extremely complicated proposition. I’m not certain the notion is meaningful for large systems, especially those bathed by the environment. The notion of a universal wavefunction may be purely abstract mathematical fantasy.
The math involves a vast (if not infinite) number of dimensions (and uses complex numbers). The precision required of coordinate values seems formidable, even impossible. These values need to encode all of reality, past and future. (And what about holographic limits to information volume?)
I question how useful — or even possible — a wavefunction is for a macro object (a measuring device, for instance). Such would seem to be a mess of the decohered states of the gazillions of particles involved and the environment they interact with.
My sense is that, even if such a thing is meaningful, it’s going to be a complex, massively decohered, hot mess that is not going to be affected by the tiny and pure quantum state of a small quantum system being measured. No more so than an ocean liner is by ping pong balls thrown at it. (Although it might be capable of recording them somehow; with video cameras, for instance. Such would cause insignificant changes to the overall state of the ship.)
We’re talking about a single wavefunction vector that sums all the particles involved. That means individual particles — or even largish groups of them — cannot have a large effect on the vector. If the object is a photo detector, consider how small the state change is when detecting a photon compared to, say, the detector being heated in an oven or crushed in a press or any other massive change. Given the huge total space that state vector can occupy, tiny normal operational changes involve a very tiny slice of it.
I want to emphasize this, because I see it as a key objection to MWI: Even accepting the notion of a wavefunction for a particle detector, that wavefunction would be a hot mess of environmentally decohered particles. More importantly, it would be due to the contributions of billions or trillions of particles. The single quantum state of a measured quantum system is not likely to affect it greatly, let alone put the whole detector into superposition.
The math might work somehow, but I don’t think the physics does. Why would the measured quantum system have any special privilege over all the surrounding environmental systems? (I’m dubious the math exists or even can exist. I think a lot of this is abstract fantasy.)
Bottom line, I question the meaningfulness of a wavefunction for any object with billions, if not trillions or more, particles. I think the idea becomes all the more meaningless with larger systems such as planets (let alone the universe).
For me, key support for MWI would come from a clear demonstration of the reality of a meaningful wavefunction for, say, a toaster (let alone a car, let alone a city).
§ §
Currently, my biggest question for the MWI is how physical matter can coincide. This applies to both the one-cat-splits-into-two and the two-identical-cats-diverge versions. (Call them, respectively, the branching and superposed versions.)
The canonical answer is “decoherence” and I get what’s being suggested, but there seems to me a disconnect. (In general decoherence seems misunderstood and misused to me.)
In interferometer and two-slit experiments, photons have two available paths that meet at some point. The interference effect comes from the phase information of the photons (which, in flight, behave like waves) interacting at the meeting point. If we introduce smoke particles to one path — and photons interact with them and shift their phase — the interference pattern disappears (or is altered).
The smoke particles cause decoherence between the paths. The phase relationship is no longer consistent from particle to particle. Individual particles still take both paths and still interfere, but since phase varies randomly, they no longer sum to an overall pattern. We can accomplish a similar effect in single-particle systems by slightly altering the length of one path between particles.
In quantum computing, decoherence refers to a similar loss of phase information, but of a single qubit to the environment. In this case the qubit’s phase becomes shifted due to phase information from various environment states, not smoke particles, but the principle is the same.
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Mathematically, coherence comes from the interaction between two states in superposition:
Where the coefficients, c, are complex numbers. This means the superposition can be restated:
Where normalization coefficients, r, are real numbers. The phase angles, θ (theta), are the values referred to by “coherence” and “decoherence” — respectively, whether that angle remains true or has been altered by interaction.
The probability, P, of a measurement is calculated using:
Those two last terms cause constructive or destructive interference. The first two will always be positive, but the last two can be negative, which can amplify, cancel, or reduce, the total value. In two-slit experiments, mathematically, this is where the interference comes from — these terms interacting.
I believe this is what MWI refers to with “decoherence” — the idea apparently being that this prevents worlds from interacting and allows physical matter to coincide.
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But… how?
For one thing, generally speaking, how does a universal wavefunction decohere? Due to what? There’s no environment; the wavefunction is the environment.
Presumably what’s meant is that worlds are in superposition. The alive and dead cats are superposed and don’t interact.
But what the Schrödinger equation superposes is multiple possibilities — not actualities. It gives us the probability of finding the cat alive or dead. Or of finding the electron reflecting off, or tunneling through, a barrier. Here the superposition is no more mysterious than saying, “Either I will order onion rings or I won’t.” Those two coincide just fine until one actually orders food. Then one of them, and only one of them, is true. Physically, only one of them can be true.
I’ll talk more about superposition next time, but I should point out now that larger and larger physical objects have been put in states that can be described as superposition. In one case, if I recall correctly, a vibrating micro-beam was in two physical states (positions) at once. That said, these are very difficult experiments to accomplish and require isolation from the environment. Still, the reality is that quantum systems — physical quantum systems, not just math — can be in superposition.
Which isn’t really a surprise. That was always the case with particles and two-slit experiments. Somehow, however, it wasn’t quite as compelling with lone particles as it is with physical structures (perhaps due to the clearly wave-like behavior of particles; vibrating beams is a whole other thing).
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Regardless of what can be done with micro objects in an isolated lab experiment, superposition is generally not a property of the physical world. Things can’t be in two places at once, nor can two different objects coincide.
Yet MWI takes the Schrödinger equation’s statement about multiple possibilities and turns it into a statement about multiple realities. Multiple physical realities that occupy the same space. Or that require an object be in two places at once.
Per MWI there are countless copies of myself sitting in roughly the same position doing approximately the same thing. Only some details vary. Further away from this instance, copies of me diverge. Some are dead; some are married (happily, I hope); some pursued music or filmmaking or standup comedy; some (I suspect) are in jail. Further away are realities where my parents never met, so I never existed. Which means this space isn’t just occupied by copies of me, but by all the other things that might have happened here.
All of it somehow coinciding. Because decoherence? I don’t see how.
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In the branching version of MWI, the presumption is decoherence quickly makes two worlds that have branched become inaccessible to each other.
This seems like physics magic to me. What does that even mean?
In the superposed version of MWI, worlds are always inaccessible, so decoherence doesn’t happen in consequence of diverging (which has no effect on either world). To the extent that “decoherence” explains world inaccessibility, in this version worlds must always be “decohered” from each other.
Regardless, I don’t see what “decoherence” is supposed to suggest. There is no sense of the word that, to me, ever means “inaccessible” — it just means phases are no longer correlated with each other.
How that translates to myriad coincident physical realms not interacting through gravity, electromagnetism, and the other forces, is beyond me. Frankly, it sounds like the sort of thing you read in comic books about coincident parallel dimensions being “out of tune” with ours.
(And indeed, outside of fantasy physics, one only encounters multiple world theories in comic books and science fiction.)
§ §
This is long enough, so I’ll leave off until next time. There’s more to say about superposition, and I have some random thoughts from my notes.
As a final thought, MWI doesn’t have measurement in the usual sense — the wavefunction never collapses. It just evolves. But without measurement, can there be decoherence? If the wavefunction constantly evolves, where does decoherence come from?
Another thing to consider: What we perceive as “particles” are wavefunction-collapsing observations of wave-based quantum systems. How does that translate to MWI?
Stay singular, my friends! Go forth and spread beauty and light.
∇
Logos con carne 
July 8th, 2021 at 9:47 am
FWIW, I really do think the keys to understanding quantum mechanics lie in understanding superposition, entanglement, and interference. I suspect an understanding of measurement will come from from understanding those, but if not, then add measurement and decoherence to the list.
July 8th, 2021 at 1:42 pm
With regard to quantum interference, it’s important to understand that it’s different from classical wave interference in the following way:
If we perform the two-slit experiment with sound or water waves (two slightly different types of classical waves — the former are longitudinal, the latter are traverse), the interference pattern that develops is a consequence of wave energy reinforcing or canceling. Critically, the null nodes — spots of no apparent energy — are the result of positive and negative waves cancelling. Wave energy is present, but has the value zero.
But in the quantum two-slit experiment, the dark zones are not the result of energy cancelation. It’s the result of probability cancelation due to phase interference. Photons don’t vanish by being cancelled out in dark zones. They avoid them in favor of the light zones where probability is high of finding them.
July 8th, 2021 at 9:48 am
FWIW, I explored what I think might be happening with quantum measurement last year in the posts Wave-Function Collapse and Wave-Function Story.
My post about Philip Ball’s book, Beyond Weird: Why Everything You Thought You Knew About Quantum Physics Is Different (2018) might also be of interest to those interested in quantum interpretations.
July 8th, 2021 at 9:59 am
The post was long enough that I didn’t go into any details involving the formulas, particularly the probability one. I know it’s one I stumbled over originally, and it might be worth expanding on what’s going on there.
Start by seeing the complex number plane as a complex vector space with an inner product:
Which is also the absolute square of a complex number. Note that the absolute value (not squared) is the magnitude (length) of the vector:
Which is essentially the Pythagorean Theorem. Given that a complex number is:
Then:
Multiplying the terms gives us:
Which again reduces to the Pythagorean Theorem:
Now, when we want to combine two complex amplitudes:
Multiplying the terms gives us:
Which reduces to:
As shown in the post.
July 8th, 2021 at 4:27 pm
Here are two examples:
First the superposition:
If we combine the coefficients, and plug them into the equation we get:
Which is:
Note the result! The coefficients combine to give a positive value.
Now consider a similar superposition with a different relative phase:
Here combining the coefficients gives us:
Which is:
Zero! The coefficients in this superposition cancel.
As to the phase difference between them, the first superposition can be expressed as:
Of course:
And both coefficients have the same phase, so they reinforce.
The second superposition can be expressed as:
And, since:
This evaluates to the version shown first. This version makes it more clear the coefficients cancel due to being 180° out of phase. (Because π radians is 180° of a circle.)
July 14th, 2021 at 7:37 pm
Synchronicity strikes again. This latest video from PBS SpaceTime is right on point:
July 17th, 2021 at 1:16 pm
@Peter Moran:
Branching off to another world to discuss decoherence…
“I think decoherence as a description of how events happen in different places works well enough,…”
I would very much like to explore that! The main point of this part two was that I don’t see how any notion of decoherence, as I understand it (which may not be saying much), allows physical objects to coincide.
If anything, decoherence seems, if not the opposite, at least opposed to superposition, which to me seems related to coherence.
Even mathematically it’s not clear to me how decoherence allows physical coincidence.
July 17th, 2021 at 3:28 pm
I’m not sure I’ll say anything different here. I think of decoherence as just a discussion of the mathematics of Hilbert spaces that we use to model lots of atoms, such as one finds in what we call a detector, interacting with a world outside it, which we model with a different Hilbert space. We only record events when we put a detector in place (pace GRW-type collapse events, which we don’t record), so that there being “lots of atoms” is, I suppose, important. Notoriously, decoherence only gets us from a superposition to something that is close to a mixture, which is catastrophic if we worry about the precise form of a state.
If we take a state as in a partnership with the measurement operators we use as a model of an experiment, however, then it’s only the recorded times and places of events that justify a particular state+measurements that matter, and for them there is no collapse of the wave function, by decoherence or otherwise, they just are the baseline for whatever theoretical stories we tell.
Why are there discrete events? I think it’s only because we’ve discovered that there are devices that report discrete events, at random intervals, and we’ve found ways to use them effectively to investigate the world. There’s an effective 19th Century language of phase transitions for such events. Other devices, which don’t report discrete events, are not as immediately interesting.
Imagine, however, that the raw output from a device looks very close to a sine-wave, but when we put that data through multiple nonlinear processes and low-pass filters we discover that the processed data stream occasionally gives us discrete events. Now suddenly we have a different kind of detector, of phonons or whatever-ons, and a few years later we would be able to buy a new black box “whatever-on detector” that incorporates those multiple nonlinear processes and band-pass filters that someone sweated blood to discover in a relatively cheap package (perhaps with some of those nonlinear processes implemented in software and others in hardware) — but in a way it’s still an elaborate nonlinear transformation of the output that would look continuous if we looked at it closely before that processing. In this perspective, recording discrete events is not necessarily more than an effective compression of what otherwise is a continuum. In the early days of QM, they had discrete structure dropping into their laps (though the discrete structure of the light spectrum from the sun and from flames was only noticed when someone looked closely, …), so we tend to think of everything as discrete, but I think we can look more closely and find various kinds of continuity and discrete structure, which very often are statistically incompatible processes.
Once we put many such devices in place as part of an experiment, then there will be some events. We’ve designed such devices so that events happen and there’s an automated way to record them, so yes, there will be events and there will be records of them. When we change the environment of such devices, the statistics will change, sometimes in superficially bizarre ways (this story is already in the first video on my YouTube account). It becomes a game to discover just how extensively bizarre we can engineer the statistics to be and to see whether we can use that something-bizarre usefully and cheaply.
July 20th, 2021 at 9:58 am
My question was about how physical matter can coincide under MWI. I’m not understanding how discrete events or the detectors we use applies. Either you wandered down a different path, or what you said was too far over my head for me to keep up. (Admittedly, probably the latter.)
“I think of decoherence as just a discussion of the mathematics of Hilbert spaces that we use to model lots of atoms, such as one finds in what we call a detector, interacting with a world outside it, which we model with a different Hilbert space.”
What I want to understand is how that translates to physical reality and the MWI claim that “decoherence” allows multiple physical realities (lots and lots of them) to coincide. Right now, under MWI, there are myriad physically real versions of me, my condo, my neighborhood, etc, physically coinciding. How is that possible?
The Pauli Exclusion Principle would seem to exclude this. Say Frank the detector gets cross, and under MWI Frank also doesn’t get cross. In some accounts, myriad slightly different versions of Frank get cross and also don’t. These all physically coincide. The only difference between them is that one set of Franks are cross, which is a classical behavior. Within the sets, the difference would be that one Frank got cross at a slightly different time or in a slightly different way, but it ultimately led to being classically cross. (It’s the “how many damn cats are in the box” question all over again.)
In order for all the electrons in all the versions of Frank to coincide, don’t they need to be in different states? We might assign a different state to an entire branch, but does that overall state translate down to all the leptons involved such that they have different states? But what, among possible electron states, can that be? Certainly not spin, position, or momentum. The bulk of all Franks are identical, but in some a small classical sub-section is different (because Frank is cross).
Under MWI all these Franks are allowed to be (a) physically real and (b) physically coincident. This apparent violation of physical reality is allowed under MWI because “decoherence”.
What I’m after, I guess, is how a knowledgeable proponent of MWI would explain that. To me (and perhaps I need to learn more) decoherence (or coherence) is about phase information, and I don’t understand how phase information can allow physical coincidence. Mathematically, sure, but physically?
July 21st, 2021 at 9:02 am
I feel I have to go sideways to respond to this comment. I think we’re in a place where it would be good to have a better description of the world as physics sees it. That’s quite instrumental, but if we can get that more in hand, then I suppose we would be at a better starting point for explaining what physics sees of the world. I’m personally happy to think of the world as being there and being what it is whether or not physics and physicists or anyone else see it.
What I think physics “sees”, in a formal sense, are records of experiments, both written descriptions of the components used and how an experiment was built and (usually quite long) lists of recorded results. I think we call the entries in those long lists “measurement results”, which we can only interpret relative to the descriptions. Those descriptions are themselves measurement results, such as “a lens of measured focal length 2.9706(1) cm was measured to be 1.01341(2) cm from the output of the laser, with no measurable drift over the course of the experiment”, but there is a qualitative difference that makes it possible to call it the “meta-data” for the experiment.
Formal algorithms, “transformations”, are applied to select subsets of the recorded measurement results, which results in various statistics. Every recorded measurement result Rᵢ has to be placed relative to the experimental meta-data, otherwise it’s useless and will have to be discarded, so that for every i we must be able to associate a description Dᵢ of how and when the corresponding Rᵢ was collected. We can think of the Rᵢ as part of the initial data on the tape for a Turing machine, with another part of the tape specifying the algorithms to generate the statistics Sₙ, each of which we can think of as a compression of the recorded measurement results Rᵢ. Even for an arbitrarily large but finite collection of recorded measurement results, a Turing machine can be programmed to generate a set of statistics that is massively underdetermined by the Rᵢ and Dᵢ, with each “statistic” effectively interpolating and extrapolating in its own way from the measurement results and all the associated meta-data.
In this Turing machine model, each Sₙ can be regarded as adding a new entry onto the end of the list of measurement results, together with its own description Dₙ of how that new entry was constructed. But there’s another aspect to this: the Rᵢ’s themselves can also be thought of as the result of a hardware and software algorithm applied to numbers that were never recorded, which is what the Dᵢ associated with each Rᵢ has to describe.
Backing away from that just a little, I suppose the role of MWI, as of any interpretation of QM, is to decide what statistics are most operationally and intuitively helpful. We can generate as “outputs” enormous 3-D simulations, photographs, schematics, graphs, or a single vector in a 3-dimensional space, each of which is informative in its own way. To understand an academic paper well, we have to understand all the pipelines from the initial idea for the experiment to the warts-and-all details of the final construction to the results and all the algorithms that give us some final graphs and other outputs.
I stepped back to such an abstract level because at first I thought that I don’t have a real answer to give you. I’m trying to brainstorm something, anything! I still doubt I have a real answer, but perhaps what decoherence claims is that there are statistics that we could compute if we collected much more detailed experimental results (measurements of the device at the nanometer level), which would justify considering the “event” transitions of the devices to be natural because of the way the device has been constructed.
Having written all the above, I went back to your comment to see just how badly I didn’t answer it. So, this, “Say Frank the detector gets cross, and under MWI Frank also doesn’t get cross,” seems in the above formulation to mess with the individual Rᵢ’s. What would be the consequence if we change the time of one event, or the times of more than a few events, or remove some of the events from the list altogether, or added more events into the list? For most such changes, most statistics would barely change at all, but many systematic changes would have systematic effects. Decoherence, however, like thermodynamics, is about typicality: what statistics of experimental results are relatively more stable than others under incoherent changes of the measurement results?
I surely wouldn’t put the above in an academic paper, but it doesn’t seem totally unhelpful!?!
July 21st, 2021 at 7:13 pm
Still processing (and distracted by other stuff) but some questions:
“…perhaps what decoherence claims is that there are statistics that we could compute if we collected much more detailed experimental results (measurements of the device at the nanometer level), which would justify considering the “event” transitions of the devices to be natural because of the way the device has been constructed.”
I’m not sure I follow. Are you saying small scale measurements would generate results showing Frank in a superposition of (the classical large-scale states) calm and cross?
“Decoherence, however, like thermodynamics, is about typicality: what statistics of experimental results are relatively more stable than others under incoherent changes of the measurement results?”
I definitely don’t follow. I thought it had to do with phase information and accounted for the two-slit pattern or what happens in interferometers?
I do follow the Turing Machine abstraction of experimental science fine, but don’t follow what it has to do with the question of decoherence and physical coincidence? You later seem to imply you were just vamping there, so maybe no connection, but if you intended dots to MWI, I failed to connect them!
Going sideways with you for a moment, should there be a “meta” description that expresses the intent and purpose of the collection? Essentially the theory being tested. It would also be what unifies the collection and binds it to the physical world.
(FWIW: Along those lines I see interpretations of QM as different physical accounts for the same math. They literally interpret what it all means with regard to the real world. It’s one of the strange things about QM — that it needs to be interpreted at all. Nearly all our other theories don’t.)
July 22nd, 2021 at 10:27 am
To me, it’s crucial that unless we perform incompatible measurements, and we know what measurements we’ve performed, we can’t determine whether a state is a superposition or a mixture. People throw around that “a state is a superposition” or that “a system is in a superposition”, but this is something that has to be empirically determined (determining the state, called quantum state tomography, is not easy). It gets much less easy to determine the quantum state as the complexity of what is measured increases, particularly because it becomes hard to know exactly what measurements we’re performing. We perform very few measurements of Frank. At some point I’m not sure what it means in the world to ask plaintively “but how does the superposition finally become exactly a mixture?”
With apologies, apart from that cryptic comment I think I’m gonna cry off just now. I’m working on interacting QFTs and I need to put my head in that game more fully. I’d stick around longer, but I also feel that I’ve started repeating myself too much. I try to vary what I say each time, for my own good as much as for anyone else, but it seems better to repeat what I’ve written over the last few days at some future time, when it might be more varied, instead of right now. I need to mull.
PS: two papers popped up on arXiv that might be relevant to the question of measurement as a thermodynamic process, 2107.09932 and 2107.10222, though I haven’t yet looked at more than the abstracts.
July 22nd, 2021 at 12:29 pm
I can absolutely appreciate the pull of IRL and other concerns. I hope it hasn’t been entirely unprofitable; I’ve found it quite interesting. I agree we don’t seem able to move this decoherence thing forward much. I don’t know if I’m just too ignorant to follow, or maybe a hard-core realist and an instrumental anti-realist just aren’t on the same wavelength enough? In any event, I have indeed found much of it cryptic. There is so much yet to learn!