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.
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?
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.
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.
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.
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).
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.
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.