It is only a slight idealization to say that all theories of physics begin by stating the types of objects that exist, which is done by listing their fundamental properties. The ways objects evolve and interact are the basis of physical laws—statements about how the “values” of the fundamental physical properties change and affect each other. For example, if “mass” and “position” and “speed” are fundamental properties of a rock, then “a two-ton rock is located directly above your head and falling at a hundred miles per hour” is a statement about the initial values of those properties, from which you can deduce the exciting future consequences.
Originally our physical laws were based on properties we have personal experience with—we know rocks have mass, speed, position, color, texture, and so on. As time went on, we found that there are properties whose value cannot be experienced directly by a human quite so easily—energy for example. Typically, the properties are all interdependent—you can deduce the speed of the rock from its mass and energy and current position, but equivalently you can deduce the energy of the rock from its mass and speed and current position. Making good choices about the properties to take as fundamental is crucial to doing good physics—the theory of seismic wave propagation should not begin with considerations of the locations of individual grains of sand on the beach.
Correspondence between mathematics and physical properties
Our physical theories cover incredible scales, from the tiniest particles to the whole universe. The properties of the things at the smallest scales—the subatomic particles—include properties like mass, charge, position, energy, spin, etc. These would seem to be the “absolutely most fundamental” properties of the stuff that makes up the world. But the set of “most fundamental” physical properties keeps undergoing revision as we deepen our understanding, and this changes the terms in which we couch our smallest-scale theories. Often things that seemed absolutely fundamental turn out to be built up from more fundamental things, and so previously absolutely fundamental properties are now seen as derivative.
We infer new properties—and even the existence of new things—that we cannot directly experience, either by realizing that a good explanation of some phenomenon requires them, or because the mathematical consistency of some physical laws requires them. In either case the laws of physics are ultimately mathematical, and so for every object and every associated physical property “out there” in the world there is some kind of mathematical counterpart in our physical theory. The converse, however, is not true. Our theories can contain mathematical objects that we do not believe necessarily have a direct counterpart in the physical world. A subtle example, relevant to what comes later, occurs when you’re about to flip a coin. “50% probability” is a mathematical object you assign to the coin’s landing on “heads.” It is not a property of the coin, it is not something that affects the coin’s trajectory through the air, but it’s manifestly useful to you predicting the outcome of a future observation. It is a state of your knowledge or information. While it may seem that such a subjective thing surely cannot be an integral part of a physical law, in fact the laws of thermodynamics have just such an “ignorance-quantifying” mathematical object indispensably hardwired in, known as entropy. Without entropy in our physical laws we would not be able to understand how the engine of your (future?) motorcycle works.
Sometimes, and this is where our story becomes interesting, explaining what we observe can only be done using some particular mathematical object, but we are unsure if that object is or is not in correspondence with some physical property. The misty states are the example we will examine in detail, because our whole notion of “what is real” and “what is really going on” changes dramatically according to whether or not they are a physical property of the world.
In order to appreciate the full ramifications of what we are going to talk about, I want to emphasize that if our current physical understanding of the universe based on misty states is correct, then everything you have learned about them applies equally well to all physical properties of everything that makes up the universe. The black and white balls (via which I introduced you to misty states) are typically composed from many particles, and their color is a derivative property. Yet their color can be in a misty state. To date we do not know of any physical property, fundamental or derivative, that in principle cannot be put into a misty state, and it would be mysterious if such a non-misterious property existed.
To refine some concepts and obtain some useful language we first consider a case about which there is no controversy, namely the distinction between states that are in your mind and the real states of the world when you flip a coin.
A deeper description of the rocky state of a coin
You might assign a coin you have just flipped the rocky state:
This is a more diagrammatic representation of the “50% probability of heads” you assign to the coin. It would be implausible to suggest that what happens to the coin when you toss it in the air is a result of the coin thinking to itself: “Now I should fly over there and land this way because I can sense that the weird-looking human over there thinks I play fair and is thus assigning me that particular rocky state.”
A rocky state of the coin is therefore unarguably a state of your mind: a state of your knowledge or your information. It is a tool that lets you summarize your best predictions as to what you will subsequently observe. The connection between the rocky state in your mind, and the dynamics of the coin, is indirect. The coin responds to the “actually real” physical circumstances—the same ones that you believe are relevant but realize you are ignorant about, causing you to assign fair odds in the first place.
The real state of the coin—as opposed to its rocky state—consists of physical properties, like its mass, shape, color, height above the floor, speed and rotation through the air, and so on. We can make a simple diagram that captures all the possible real states of a coin:
The square plane in the diagram represents every possible real state of the coin— there are an infinite number of possibilities. I have labelled two of them. You can imagine a piece of paper at each point which lists all the relevant physical properties and their values corresponding to that particular real state—the mass, height, speed and so on. One of the real states of the coin is whether the upwards face of the coin is heads or tails. (While it is spinning in the air there are fractions of a second for which the coin is “on its side,” but to keep things simple I will ignore those.) Although I have represented the set of all real states by a square plane, this is just a schematic diagram. As more properties of the coin are included on each piece of paper that defines a real state physicists prefer to use higher-dimensional shapes to represent the set of all real states. This is because they like the representation to have the feature that two nearby points in the set of real states list physical properties with values very similar to each other. My drawing skills are not up to that, so you should not take the particular square plane I have drawn too literally.
Some of the properties of the coin—its mass, for example—stay the same when you flip it. Other properties change—for example, its height above the ground; its speed; which side faces upwards. In terms of the diagram of the real states of the coin, the whole procedure maps out a line through the set of real states:
The set of all possible real states can be divided in two, according to whether the real state corresponds to the coin having heads facing upwards or tails facing upwards:
Flipping a coin is a game where you are being asked to predict in which of these two regions the final real state of the coin (i.e. after it has landed) will end up. If you knew the actual real state of the coin at the instant you flipped it, as well as all the physical laws governing how it flies through the air, then you would know whether it was going to land heads or tails. In practice, you do not know the initial real state perfectly, nor do you have the capability of calculating the trajectory of the coin fast enough, and so you cann
ot predict in which of the two regions the final real state of the coin will be.
However, at any instant of time, you do know some things about the real state of the coin. You know its height in the air is not a hundred meters; you know its mass is not a kilogram. All of those ridiculous options exist within the set of all possible real states depicted above. You do not, therefore, assign equal likelihood to all of the possible real states—that is not why you say the coin is fair. The reason you assign the coin the rocky state of a “fair flip,” is that you are saying: “Amongst all the potential real states that are consistent with what I know about the coin once it has landed, half of them are in the region corresponding to heads, and half are in the region corresponding to tails.” For simplicity, let’s assume that—amongst all the real states consistent with what you know about the coin—you assign an equal likelihood to the coin “actually being” in any particular one of them once it has landed. To all the rest, you assign probability zero.
This state of affairs, then, is the more precise meaning of the rocky state—your state of knowledge. We can depict it:
In the diagram, half of the real states consistent with your knowledge lie in the region where the coin lands heads, and half lie in the region where it lands tails. This is the reason you use the rocky state
The final lesson we can learn from the coin requires us to consider a situation where, for whatever reason, you think the coin is biased—say more likely to land heads. For the sake of sticking with a simple diagrammatic language, imagine the special case where you believe heads to be twice as likely as tails (i.e., the probability of heads is 2/3 and tails is 1/3). We could represent this new state, as well as the old one, in terms of both real and rocky states as follows:
Because the rocky states are unquestionably representing our knowledge, it is fine to avoid the “squaring” rule that the misty states forced upon us in Part II—so, for this new rocky state, the two heads and one tail means heads is twice as likely as tails.
In the diagram, you can see a feature that will be essential to understand for much of what follows, so if you’ve started to drift off it’s time to refocus. The feature is that there are some real states that are common to both of the rocky states. That is, there is a nontrivial overlap between the real states consistent with the two different (rocky) states of your knowledge.
This is not particularly strange in and of itself. To give a completely different example where a real state of the world might be consistent with two different states of knowledge, consider someone draws a random playing card and tells one person the card is red and another that the card is an ace. The two people have completely different states of knowledge about the card, but the “real states” Ace of Hearts and Ace of Diamonds are in the overlap region.
The most important point about such overlap regions is as follows. Imagine someone comes along and is powerful enough to determine the real state of the coin; that is, they find out exactly all the physical properties of the coin, including whether heads or tails faces upwards. If it so happens that the real state is in the overlap region, then they cannot tell for sure which rocky state you are assigning to the coin. The rocky state is in your head, and no matter how much they know about the physical properties of the coin, they cannot know what is your state of knowledge about the coin. This is true even if you narrowed it down in advance for them to just the two rocky states depicted above.
More subtly, the very fact that two different rocky states can correspond to the same real state indicates that the rocky states themselves are not a physical property of the coin. Remember, the real states are like a sheet of paper listing all physical properties of the coin and their particular values for a coin that happens to be in that real state. So, by definition, if a rocky state was a physical property then we would be able to look at the real state (read the sheet of paper) and know what the “value of the rocky state” was. Since there is more than one rocky state associated with real states in the overlap region, the value of the rocky state cannot be “on the paper” as it were—a rocky state is not a physical property; it is not (part of) the “real physical state” of a coin.
This has the following consequence: if, for some reason, we were unsure about whether rocky states were a physical property of a coin or not, then a proof that multiple different rocky states could correspond to the same real state would prove to us that they were not. Similar reasoning will form a crucial part of one of the arguments that misty states are not a physical property.
First, however, we delve into the simplest arguments that misty states are, in one way or another, a physical property of a ball.
Revisiting our first conundrum
When I introduced you to misty states I emphasized that the one thing every physicist can agree on is that they are, at a minimum, a practical tool for calculation. They work incredibly well—much of modern technology (transistors/computers, lasers/the internet) is built upon unbelievably precise calculations that misty states enable us to make about concrete physical stuff, that we frantically push together in order to bring you a better cellphone each year.
Let us revisit the first conundrum we encountered with misty states, namely that the output of a single PETE box is random, but of two stacked PETE boxes is not.
If the misty state [W,B] that we use to describe the ball emerging from the first PETE box (in the case where we don’t observe its color) is an “actual real thing” or, alternatively, “an actual physical property of a ball,” then it is plausible that the second PETE box can “see” that the ball is in a misty state—and, in fact, which particular misty state it is in.
Perhaps, unlike ourselves, PETE boxes do not destroy the mist by looking at it? If so, then the second PETE box can distinguish whether the mist is [W,B] or [W,–B], and evolve the color of the output ball to white or black accordingly:
Thus, by taking the misty states to be real physical things—that is, actual physical properties of the balls—we immediately obtain a potential explanation of (some of) the weirdness of the PETE boxes. It seems like such a simple and natural explanation of the PETE boxes experiment that you may be wondering: why is there any controversy at all?
Two variations on a misty state “being real”
The only requirement for the explanation in the preceding section to work is that the PETE boxes be able to “see,” one way or another, exactly what the misty state of the ball is. We would like to say this sort of explanation relies on the misty state “being real,” or “being a physical property” such that the PETE box can respond to (or interact with) the mist.
At this point there is considerable potential for confusion, some of which you even overhear amongst physicists who get paid to argue about such things. The confusion arises when we are not clear about whether we are considering the mist to be the full and only real state itself, or whether we take it to be just one, amongst many, physical properties. Both options are, in fact, tenable, so let me describe each.
As mentioned, although technically difficult, there is no reason to believe we cannot make a superposition of any given physical properties of a ball. There is nothing special about the color. This suggests the option of just using a mist as the one and only physical property of the ball. In this view the mist is the full real state:
In the figure we see two different mists of a single ball, and the mists contain not only the ball colors but also their mass, perhaps whether their surface is rough or smooth and potentially much more. The details are unimportant for our considerations, but for completeness let me say that that the manner in which a mist is constructed to incorporate multiple different physical properties has many precise rules, and is not quite as arbitrary as the above figure implies. Also I should point out that under this view the real state of more than one ball is just a suitably enlarged mist.
The second option is that we allow for the real states to
include the misty state as one of the physical properties, but to potentially include a whole bunch of other things that we do not (yet) know about as well. Under this more conservative view we do not commit to knowing everything about the underlying real states of the world per se. We just say that whatever the real states of the world are, they unambiguously let us (well, the PETE boxes) determine what the misty state is—we can read the misty state off the piece of paper if we know the real state, but perhaps there are other physical properties to read off as well. That is, we use the analogy of the heads-or-tails property of the coin—unarguably real, but just one of a long list of the physical properties. In the case of the coin we divided all the real states into two regions according to whether its real state included heads or tails. For the real states of the ball (whatever they may be) we would need at least four regions like this:
This is not sufficient, however, because which real states correspond to [W,W,B]? Or [–W,B,B]? Or.... Well, you get the picture: we would need to divide the set of real states up into an infinite number of distinct regions, one for each misty state. Note that if you know which point in the plane describes the ball, you know which region it is in, and so you know which misty state describes it.
Both of the options discussed above—one real state per misty state, or many real states per misty state—have the feature that if someone knows the real state of the ball then they also know the misty state. In particular, for both these options there is nothing like the overlap region discussed above for a rocky state. For this reason I will lump them together as options in which we “take the mist as a real physical property,” or more simply just that “the mist is real”. We just mean that at any point in the set of real states, you can read on the piece of paper uniquely what the appropriate misty state is.
Q is for Quantum Page 8