We’ll get back to measurement, interference and the double-slit experiment just as soon as I can get my math program to produce pictures of the relevant wave functions reliably. I owe you some further discussion of why measurement (and even interactions without measurement) can partially or completely eliminate quantum interference.
But in the meantime, I’ve gotten some questions and some criticism for arguing that superposition is an OR, not an AND. It is time to look closely at this choice, and understand both its strengths and its limitations, and how we have to move beyond it to fully appreciate quantum physics. [I probably should have written this article earlier — and I suspect I’ll need to write it again someday, as it’s a tricky subject.]
In recent talks at physics departments about my book, I have emphasized that the elementary “particles” of nature — electrons, photons, quarks and so on — are really little waves (or, to borrow a term that was suggested by Sir Arthur Eddington in the 1920s, “wavicles”.) But this notion inevitably generates confusion. That’s because of another wavy concept that arises in “quantum mechanics” —the quantum physics of the 1920s, taught to every physics student. That concept is Erwin Schrödinger’s famous “wave function”.
It’s natural to guess that wave functions and wavicles are roughly the same. In fact, however, they are generally unrelated.
Wavicles Versus Wave Functions
Before quantum physics came along, field theory was already used to predict the behavior of ordinary waves in ordinary settings. Field theory is useful for sound waves in air, seismic waves in rock, and waves on water.
Quantum field theory, the quantum physics that arose out of the 1940s and 1950s, adds something new: it tells us that waves in quantum fields are made from wavicles, the gentlest possible waves. A photon, for instance, is a wavicle of light — the dimmest possible flash of light.
By contrast, a wave function describes a system of objects operating according to quantum physics. Importantly, it’s not one wave function per object — it’s one wave function per system of interacting objects. That’s true whether the objects in the system are particles in motion, or something as simple as particles that cannot move, or something as complex as fields and their wavicles.
One of the points I like to make, to draw the distinction between these two types of waves in quantum physics, is this:
Wavicles can hurt you.
Wave functions cannot.
Daniel Whiteson, the well-known Large Hadron Collider physicist, podcaster and popular science writer, liked this phrasing so much that he quoted the second half on X/Twitter. Immediately there were protests. One person wrote “Everything that has ever hurt anyone was in truth a wave function.” Another posted a video of an unfortunate incident involving the collision between a baseball and a batter, and said: “the wave function of this baseball disagrees.”
It’s completely understandable why there’s widespread muddlement about this. We have two classes of waves floating around in quantum physics, and both of them are inherently confusing. My aim today is to make it clear why a wave function couldn’t hurt a fly, a cat, or even a particle.
The Basic Concepts
Wavicles, such as photons or electrons, are real objects. X-rays are a form of light, and are made of photons — wavicles of light. A strong beam of X-ray photons can hurt you. The photons travel across three-dimensional space carrying energy and momentum; they can strike your body, damage your DNA, and thereby cause you to develop cancer.
The wave function associated with the X-ray beam, however, is not an object. All it does is describe the beam and its possible futures. It tells us what the beam’s energy may be, but it doesn’t have any energy, and cannot inflict the beam’s energy on anything else. The wave function tells us where the beam may go, but itself goes nowhere. Though it describes a beam as it crosses ordinary three-dimensional space, the wave function does not itself exist in three-dimensional space.
In fact, if the X-ray beam is interacting with your body, then the X-ray beam cannot be said to have its own wave function. Instead, there is only one wave function — one that describes the beam of photons, your atoms, and the interactions between your atoms and the photons.
More generally, if a bunch of objects interact with each other,the multiple interacting objects form a single indivisible system, and a single wave function must describe it. The individual objects do not have separate wave functions.
Schrödinger’s Cat
This point is already illustrated by Schrödinger’s famous (albeit unrealistic) example of the cat in a box that is both dead and alive. The box contains a radioactive atom which will, via a quantum process, eventually “decay” [i.e. transform itself into a new type of atom, releasing a subatomic particle in the process], but may or may not have done so yet. If and when the atom does decay, it triggers the poisoning of the cat. The cat’s survival or demise thus depends on a quantum effect, and it becomes a party to a quantum phenomenon.
It would be a mistake to say that “the atom has a wave function” (or even worse, that “the atom is a wave function”) and that this wave function can kill the cat. To do so would miss Schrödinger’s point. Instead, the wave function includes the atom, the killing device, and the cat.
Initially, when the box is closed, the three are independent of one another, and so they have a relatively simple wave function which one may crudely sketch as
Wave Function = (atom undecayed) x (device off) x (cat alive)
This wave function represents our certainty that the atom has not yet decayed, the murder weapon has not been triggered, and the cat is still alive.
But this initial wave function immediately begins evolving into a more complicated form, one which depends on two time-varying complex numbers C and D, with |C|2 + |D|2 = 1:
Wave Function = C(t) x (atom undecayed) x (device off) x (cat alive) + D(t) x (atom decayed) x (device on) x (cat dead)
The wave function is now a sum of two “branches” which describe two distinct possibilities, and assigns them probabilities |C|2 and |D|2, the former gradually decreasing and the latter gradually increasing. [Note these two branches are added together in the wave function; its branches cannot be rearranged into wave functions for the atom, device and cat separately, nor can the two branches ever be separated from one another.]
In no sense has the wave function killed the cat; in one of its branches the cat is dead, but the other branch describes a live cat. And in no sense did the “wave function of the atom” or “of the device” kill the cat, because no such wave functions are well-defined in this interacting system.
A More Explicit Example
Let’s now look at an example, similar to the cat but more concrete, and easier to think about and draw.
Let’s take two particles [not wavicles] A and B. These particles travel only in a one-dimensional line, instead of in three-dimensional space.
Initially, particle B is roughly stationary and particle A comes flying toward it. There are two possible outcomes.
There is a 30% probability that A passes right by B without affecting it, in which case B simply says “hi” as A goes by.
There is a 70% probability that A strikes B head-on and bounces off of it, in which case B, recoiling from the blow, says “ow”.
In the second case, we may indeed say that A “hurts” B — at least in the sense of causing B to recoil suddenly.
The Classical Probabilities
Before we answer quantum questions, let’s first think about how one might describe this situation in a world without quantum physics. There are several ways of depicting what may happen.
Motion in One-Dimensional Physical Space
We could describe how the particles move within their one-dimensional universe, using arrows to illustrate their motions over time. In the figure below, I show both the “hi” possibility and the “ow” possibility.
Figure 1: (Top) With 30% probability, the Hi case: A (in blue) passes by B without interacting with it. (Bottom) With 70% probability, the Ow case: A strikes B, following which A rebounds and B recoils.
Or, using an animation, we can show the time-dependence more explicitly and more clearly. In the second case, I’ve assumed that B has more mass than A, so it recoils more slowly from the blow than does A.
Figure 2: Animation of Fig. 1, showing the Hi case in which A passes B, and the Ow case where A strikes B.
Motion in the Two-Dimensional Space of Possibilities
But we could also describe how the particles move as a system in their two-dimensional space of possibilities. Each point in that space tells us both where A is and where B is; the point’s location along the horizontal axis gives A’s position, and its location along the vertical axis gives B’s position. At each moment, the system is at one point in that space; over time, as A and B change their positions, the location of the system in that two-dimensional space also changes.
The motion of the system for the Hi and Ow cases is shown in Fig. 3. It has exactly the same information as Fig. 1, though depicted differently and somewhat more precisely. Instead of following the two dots that correspond to the two particles as they move in one dimension, we now depict the whole system as a single diamond that tells us where both particles are located.
In the first part of Fig. 3, we see that B’s position is at the center of the space, and remains there, while A’s position goes from the far right to the far left; compare to Fig. 1. In the second part of Fig. 3, A and B collide at the center, following which A moves to positive position, B moves to negative position, and correspondingly, within its space of possibilities, the system as a whole moves down and to the right.
Figure 3: How the A/B system moves through the space of possibilities. (Top) In the Hi case, A moves while B remains fixed at its central position. (Botttom) The Ow case is the same as the Hi case until A’s position reaches B’s position at the center; a collision then causes A to reverse course to a positive position, while B is driven to a negative position (which is downward in this graph.)The system as a whole thus moves down and to the right in the space of possibilities.
And finally, let’s look at an animation in the two-dimensional space of possibilities. Compare this to Fig. 3, and then to Fig. 2, noting that it has the same information.
Figure 4: Animation of Figure 3, showing (top) the Hi case in which A passes B and (bottom) the Ow case where A strikes B.
In Fig. 4, we see that
the system as a whole is represented as a single moving point in the space of possibilities
each of the two futures for the system are represented as separate time-dependent paths across the space of possibilities
The Quantum System
Now, what if the system is described using quantum physics? What’s up with the system’s wave function?
As noted, we do not have a wave function for particle A and a separate wave function for particle B, and so we do not have a collision of two wave functions. Instead, we have a single wave function for the A/B system, one which describes the collision of the two particles and the aftermath thereof.
It is impossible to depict the wave function using the one-dimensional universe that the particles live in. The wave function itself only exists in the space of possibilities. So in quantum physics, there are no analogues to Figs. 1 and 2.
Meanwhile, although we can depict the wave function at any one moment in the two dimensional space, we cannot simply use arrows to depict how it changes over time. This is because we cannot view the “Hi” and “Ow” cases as distinct, and as something we can draw in two separate figures, as we did in Figs. 3 and 4. In quantum physics, we have to view both possibilities as described by the same wave function; they are not distinct outcomes.
The only option we have is to do an animation in the two-dimensional space of possibilities, somewhat similar to Fig. 4, but without separating the Hi and Ow outcomes. There’s just one wave function that shows both the “Hi” and “Ow” cases together. The square of this wave function, which gives the probabilities for the system’s possible futures, is sketched in Fig. 5.
[Note that what is shown is merely a sketch! It is not the true wave function, which requires a complete solution of Schrödinger’s wave equation. While the solution is well known, it is tricky to get all the details of the math exactly right, and I haven’t had the time. I’ll try to add the complete and correct solution at a later date.]
Compare Fig. 5 with Fig. 4, recalling that the probability of the Hi case is 30% and the probability of the Ow case is 70%. Both possibilities appear in the wave function, with the branch corresponding to the Ow case carrying larger weight than the branch corresponding to the Hi case.
Figure 5: A rough sketch of what the square of the wave function of the A/B system looks like; small-scale details are not modeled correctly. Note both Hi and Ow possibilities, and their relative probabilities, appear in the wave function. Compare with Fig. 4 and with the example of Schrödinger’s cat.
In contrast to Fig. 4, the key differences are that
the system is no longer represented as a point in the space of possibilities, but instead as a (broadened) set of possibilities
the wave function is complicated during the collision, and develops two distinct branches only after the collision
all possible futures for the system exist within the same wave function
this has the consequence that distinct future possibilities of the system could potentially affect each other at a later time — a concept which makes no sense in non-quantum physics
the probabilities of those distinct futures are given by the relative sizes of the wave function within the two branches.
Notice that even though particle A has a 70% probability of “hurting” B, the wave function itself does not, and cannot, “hurt” B. It just describes what may happen; it contains both A and B, and describes both the possibility of Hi and Ow. The wave function isn’t a part of the A/B system, and doesn’t participate in its activities. Instead, it exists outside the system, as a means for understanding that system’s behavior.
Summing Up
A system has a wave function, but individual objects in the system do not have wave functions. That’s the key point.
To be fair, it is true that when objects or groups of objects in a system interact weakly enough, we may imagine the system’s full wave function as though it were a simple combination of wave functions for each object or group of objects. That is true of the initial Schrödinger cat wave function, which is a product of separate factors for the atom, device and cat, and is also true of the wave function in Fig. 5 before the collision of A and B. But once significant interactions occur, this is no longer the case, as we see in the later-stage Schrödinger cat wave function and in Fig. 5 after the collision.
A wave function expresses how the overall system moves through the full space of its possibilities, and grows ever more complex when there are many possible paths for a system to take. This is completely unrelated to wavicles, which are objects that move through physical space and create physical phenomena, forming parts of a system that itself is described by a wave function.
A Final Note on Wave Functions
As a final comment: I’ve given this simple example because it’s one of the very few that one can draw start to finish.
Wave functions of systems with just one particle are misleading, because they make it easy to imagine that there is one wave function per particle. But with more than one particle, the only wave functions that can easily be depicted are those of two particles moving in one dimension, such as the one I have given you. Such examples offer a unique opportunity to clarify what a wave function is and isn’t, and it’s therefore crucial to appreciate them.
Any wave function more complicated than this becomes impossible to draw. Here are some things to consider.
I have only drawn the square of the wave function in Fig. 5. The full wave function is a complex function (i.e. a complex number at each point in the space of possibilities), and the contour plot I have used in Fig. 5 could only be used to draw its real part, its imaginary part, or its square. Thus even in this simple situation with a two-dimensional space of possibilities, the full wave function cannot easily be represented.
If we had four particles moving in one dimension instead of two, with positions x1, x2, x3 and x4 respectively, then the wave function would be a function of the four-dimensional space of possibilities, with coordinates x1, x2, x3, x4 . [The square of the wave function at each point in that space tells us the probability that particle 1 is at position x1, particle 2 is at position x2, and similarly for 3 and 4.] A function in four dimensions can be handled using math, but is impossible to draw.
If we had two particles moving in three dimensions, the first with position x1, y1, z1, and the second with position x2, y2, z2, the space of possibilities would be six-dimensional — x1, y1, z1, x2, y2, z2 . Again, this cannot be drawn.
These difficulties explain why one almost never sees a proper discussion of wave functions of complicated systems, and why wave functions of fields are almost never described and are never depicted.
Once we clear away the hype (see the previous posts 1, 2, 3, 4), and realize that no one is doing anything as potentially dangerous as making real wormholes (ones you could actually fall into) in a lab, or studying how to send dogs across the galaxy, we are left with a question. Why bother to do wormhole research at all?
The answer is that it has nothing to do with actually making wormholes… at least, not in the sense of science fiction portals that you and I could use to travel from here to some faraway place across the universe. It has to do with potentially gaining new insight into the quantum physics of gravity, space and time.
How Do We Study Black Holes, and Why?
Why do scientists do research on black holes? There are at least two very different reasons.
Large black holes can be observed in nature. These black holes, which astronomers and gravitational wave experimenters study, are well-described by non-quantum physics — “classical” physics, where the future is (in principle) truly predictable from the past.
Small black holes are a window into quantum gravity — the unknown quantum physics of spacetime, where space itself is governed by the uncertainty principle, meaning that the very shape of spacetime can’t be precisely specified. This is relevant for black holes far too small for us to discover using astronomy, yet far too difficult for us to produce experimentally. They are important because they pose conceptual problems and puzzles for quantum gravity. Theoretical physicists think about black holes, and study their math, in hopes of uncovering quantum gravity’s secrets.
To gain more insight into their workings, scientists also simulate black holes on computers, and study analogues to black holes in laboratories.
Why Wormholes?
In contrast to black holes, there may be no wormholes worthy of the name anywhere in our universe. Though recent research clearly shows that there’s no principle that forbids wormholes from existing, it also shows it’s unlikely that large wormholes can be produced or can endure in our universe. While black holes are a generic outcome of the collapse of a huge star, wormholes are relatively delicate, and difficult to create and maintain.
But wormholes may be even more interesting than black holes for the problems of quantum gravity. This was only appreciated, slowly at first, over the past 10 years.
It’s hard to define the quantum state of a black hole. [In quantum physics, objects don’t just have locations and motions; roughly speaking, they have “states”, in which they have a combination of many locations and motions all at once.] The basic obstacle is entropy, a measure of missing information. The air in your room has entropy, because although you may know its temperature and pressure, you do not know where every atom of air is; that’s missing information. It turns out that a black hole has entropy too, which means that our usual description of a black hole is intrinsically missing some crucial information. That prevents us from knowing precisely what its state is.
But surprisingly, in some circumstances the quantum state of a wormhole can be sharply defined — in which case its entropy is zero. (Such a wormhole is not missing any information. But if you take either half of this wormhole and ignore the other half, you find a black hole. That black hole has entropy precisely because you’re ignoring all the information included in the other half of the wormhole!) To obtain and understand such a wormhole involves giving it two apparently different but actually interchangeable descriptions, one in terms of space-time and gravity, where the wormhole’s geometric shape is clear, and one in terms of what one might call a gravity-independent auxiliary quantum system, in which its quantum state is precisely defined.
The Power of Duality: A Rosetta Stone for Quantum Gravity
One physical object, two quantum descriptions — one with gravity, one without; the first with more space dimensions than the latter. It’s like being able to read the same text in two completely different languages. It’s an example of what physicists often call “a duality.” (I’ve gone into more detail about this in recent posts here and here.)
This is the message of what goes by the mantra of “ER=EPR”, referring to two famous and apparently unrelated papers from 1935 by Einstein and Rosen, with the second having Podolsky as a co-author. ER=EPR asserts that two apparently different things,
a tangible bridge across curved, extra-dimensional space between two regions, and
a less tangible bridge, established with quantum entanglement between objects in the same two regions, without any use of gravity,
are literally the same thing.
An conceptual illustration of the proposal that two perfectly entangled quantum systems (EPR) are equivalent to a wormhole connecting the locations of those systems (ER), and represent two languages for describing exactly the same thing. The wormhole is empty; the slabs shown merely indicate how distances are shrinking as one proceeds to the wormhole’s midpoint. Not shown is that the wormhole changes shape over time; for the situation in this picture, this wormhole is “non-traversable”, because there’s insufficient time to cross from one side of the bridge to the other before it shrinks down to nothing.
Discovering that spacetime is related to quantum entanglement, and that ER and EPR involve the same issues, is somewhat like discovering that two poorly understood and partially readable texts in completely different languages are actually two translations of exactly the same document. It’s a Rosetta stone. Parts of the document can easily be read in one language, other parts in the second language; and putting them together, we find we can read more and more.
Similarly, the math of a wormhole (ER) looks completely different from the math of two quantum-entangled non-gravitational systems (EPR). But in particular cases, Juan Maldacena and Lenny Susskind argued, they are two languages describing the same object. We can combine these two partial views of this single object to learn more and more about it.
Moreover, because we’re using math, not text, we can go a step further. Even in regimes where we cannot “read the document” in either language, we can use computers to explore. Scientists can try to simulate the math of the entangled auxiliary quantum systems on a computer, ideally a quantum computer so that it keeps track of all quantum effects, to learn more about the wormhole’s behavior in regimes where we have no idea how it works — in regions where the quantum uncertainty principle affects space and time.
Even more remarkable would be to actually make — not merely simulate — this entangled pair of auxiliary quantum systems. Then we would be closer to making a wormhole, with laws of nature different from ours and with its own gravity, that connects on to our world. But that’s a long ways off, and not the story for the present.
From ER=EPR to Traversable Wormholes
A further breakthrough, beyond the original ER=EPR idea, came with the work of Gao, Jafferis and Wall (see also here and here) in which it was demonstrated for the first time that “traversable wormholes” — ones that can truly serve as bridges across which objects can be transported — do make physical sense. Astonishingly, they are related by duality to an important and exciting research area in quantum information, called “quantum teleportation.” That’s the process by which, using two entangled quantum systems, quantum information can be brought to one of the systems, destroyed in that system, and recreated in that second system some distance away. Again, don’t expect anyone to be teleporting your dog, but simple information and ultra-microscopic objects might be transportable.
Be warned though; the teleportation only works if additional non-quantum information is traded between the two systems. In the wormhole language, that means you can only get through the wormhole if information is also passed outside the wormhole from the departure region to the arrival region. This makes it impossible to go someplace that you haven’t already been sending messages to, and to use any such wormhole as a shortcut — i.e., to get to your destination faster than could a near-light-speed spacecraft traveling outside the wormhole. Not only do portals to ultra-distant places remain science fiction, they now seem even more likely to stay that way.
Still, with these caveats, there’s still something amazing here: we can now imagine using the Rosetta stone of duality to simulate a traversable wormhole, and learn how it works in quantum gravity. That would be fantastic!
The Dream of Simulating Quantum Gravity
This is a dream, yet to be fulfilled. Computers are nowhere near being able to handle the questions we’d like to answer about the gravity we live with in our “four-dimensional space-time” (our familiar three space dimensions, plus one more for time). But by simplifying the problem in several steps (see the last figure of this post), we can at least hope to answer some early questions in a much simpler sort of wormhole in a simpler sort of gravity. This is what I’d prefer to call an artificially-simulated cartoon wormhole— rather than a “baby” wormhole, because unlike a baby, it isn’t a small version of an adult, nor has it any hope of growing into one. It’s more like a stick figure. It’s in two-dimensional space-time — one space and one time. That’s a big simplification — there’s nothing like normal gravity there! [Worse, we don’t have an exact duality in that case; the auxiliary quantum system we need isn’t really the same “text” as the wormhole. These are two systems, not one, with a limited but useful overlap.]
But cartoons aren’t to be mocked. Don’t underestimate them; cartoons are a powerful tool for educators everywhere, and subversive political cartoons have helped take down governments. For decades, famous physicists — Schwinger, ‘t Hooft, Gross and Neveu, Kogut and Susskind, and many more — have studied cartoon versions of real physics, especially ones in which our four space-time dimensions are replaced with just two. They’ve often learned interesting lessons from doing so, sometimes even profound ones.
[Note: Stick figure physics also can be a very good description of real stick-figure systems, for example a one-dimensional chain of atoms inside a material.]
I hasten to caution you that this technique does not always work. Not all of the lessons learned from stick-figure physics turn out to apply to the corresponding real-world problem. But this method has had enough success that we should take cartoon studies seriously.
This is why exploration of one-dimensional wormholes, and of some sort of auxiliary quantum problem to which they might be approximately related, may be worthwhile. And this is why it’s important to learn to simulate these auxiliary quantum systems on quantum computers, as was done in the paper that generated all the hype, based on proposals made in this paper and this one. Even if we can’t hope soon to understand how three-dimensional quantum space emerges from quantum entanglement, we can perhaps hope to learn more about one-dimensional quantum space, using quantum computer simulation. Maybe what we learn there would already teach us a deep and universal truth about quantum gravity, or at least suggest new ways to think about its subtleties.
The experiment done in the recent paper is a baby step in this direction. Others have attempted something along similar lines, but this is the first experiment that seems to have focused on the truly wormhole-like regime, and found some evidence for what was expected already of wormholes (from direct calculation and from classical computers…I’ll write about those details in a future post.) That seems like a real step forward. But let’s keep things in perspective. No new knowledge was created by this experiment; its achievements were technical and technological. It’s not a conceptual breakthrough. (I’m not alone in this view; Lenny Susskind, Dan Harlow and Scott Aaronson all expressed the same opinion in the New York Times and elsewhere.)
But nevertheless, this experiment represents a little arrow that points to a possible big future… not a future of a new Elon Musk, building wormholes for fun and profit, but one of a future Einstein, comprehending the quantum nature of spacetime itself.
