The blind ghost

[dantana]

One would surely be simpler than three, yet all the kinds of particles that we think of as composing matter (all the fundamental fermions) come in three versions, differing only in mass. So there are electrons, which we all know and love, but also muons, which are exactly like electrons except some 200 times more massive, and also tauons, which are nearly 4000 times more massive than electrons but otherwise again identical. The “up” and “down” quarks that make up protons and neutrons (along with a lot of, um, glue) also each have two heavier siblings, with the whimsical names “charm”, “strange”, “top”, and “bottom”.

What I find curious about particles is they are just point particles with no real measurable spatial dimensions. No parts or internal mechanism. Yet they all behave differently when subjected to the same forces or environment. How is it that they have different properties when they have no machinery?

[Physics Guy]

Gravitational waves will literally set your eardrums vibrating as they pass through your head, that's true. An early attempt at detecting them, which is still being pursued with fancier technology, was just to monitor metal bars to see if a gravitational wave might have made them ring. Gravitational waves are generally incredibly faint signals, though. They can in principle be dramatically strong, but it takes a colossal event quite nearby to create that. All the gravitational wave signals that have been observed so far have been far, far too weak to have any noticeable effects on a human ear. Just the Brownian motion from air molecules would completely drown out the gravitational signal.

Most if not all of the gravitational wave bursts that have been detected so far do happen to be within a frequency range which for air pressure waves would be between infrasound and ultrasound, audible to the human ear. So when gravitational wave people give talks on their work, they often play sound clips made from their data; in this somewhat artificial sense you can hear two big black holes merging, a billion-odd years ago, far away. Gravitational waves can in principle have any frequency at all, though, and the kinds of events that might create waves we can detect could produce gravitational waves in a wide range of frequencies. The frequencies emitted in a black hole merger, for example, depend on how big each of the black holes is and exactly how they collide. The frequency range in which the current laser interferometer detectors for gravitational waves are sensitive happens to lie within the human audible range, for technical reasons that have nothing to do with that fact. So it's not a feature of gravitational waves that they have frequencies in our audible range—it's a coincidental feature of our current detectors.

[Physics Guy]

[Particles] all behave differently when subjected to the same forces or environment. How is it that they have different properties when they have no machinery?

That's an insightful question. One should indeed expect that things as simple as points would be unable to show the kind of range of behaviours that more complicated structures can have.

Point particles can have mass. In principle, as far as we know, they can have any mass whatever. According to quantum field theory, there can be different kinds of particles, and there can be arbitrarily many individual particles of each kind; the individual particles of the same kind are all "of the same kind" in a very strict sense. They are not just similar, like two peas or two tigers, but absolutely identical, like the second meter and the seventh meter in a length of ten meters. Switching two particles of the same kind with each other literally is not a change, any more than switching the second meter and the seventh meter changes anything.

As far as we can see now, there are only thirty-one or thirty-eight different kinds of particles (depending on whether we count gluons as one kind or eight). Since every particle of the same kind is identical, every particle of each kind has the same particular mass. All of those eighteen different masses are currently just taken to be constants of nature, determined by measuring. In most cases we have no idea why their values are what they are. (We do think we understand why the photon and the gluon have exactly zero mass.)

Point particles can also have charge. Every particle of the same kind also has to have exactly the same charge, but a priori one might expect that every different kind of particle would have some random different charge, the way they have all different masses. Instead the charges of all the different kinds of particles are all simple multiples of one value, "the elementary charge". It could apparently be anything but is something in particular, which we have measured to high precision, without finding any reason at all why it is what it is.

I give a lecture in which I think I explain fairly convincingly why a simple point particle can have an arbitrary mass, and an arbitrary charge, but no other properties. The lecture is based on the action principle of Lagrangian mechanics, plus Einsteinian relativity. I'm not sure I can translate this into non-technical language because it is a technical point that probably can't be translated. Maybe I'd be able to express it with an analogy to something familiar if I understood it better, or maybe it's really impossible, because the human mind trying to do this kind of physics is kind of like a chess game trying to understand the pawn move. The pawn move is simple but it's not like any of the higher-level principles, like controlling the center or developing the pieces, that a chess game as a whole can understand.

There could still in principle be a lot of different ways for particles to differ from each other, because as far as we can tell there could in principle be many different kinds of charge. Maybe there could be electrical charge, gluten-free charge, rhythm-and-blues charge, and so on, like the ratings of a TV show in different demographics. Just a lot of alternative categories of electrical charge, each one with its own alternative copies of the electric and magnetic fields, and for each kind of particle you could list the specific values of each of its different kinds of charge. But you can forget all that, because apparently there is only one kind of electrical charge. Nobody knows why we don't have any more.

Particle kinds do differ in more ways than just mass and electrical charge, but the additional properties are not just alternative versions of electrical charge. They're also not counter-examples to the argument in my lecture about how only mass and charge are allowed for simple point particles, because these additional properties are properties of particles that are not actually just simple points. None of the particles we know really is just a simple point. They are points, but not simple: they all have some minimal form of "internal machinery".

The form of internal machinery that point particles have really only makes sense in quantum field theory. Fields have different components, like the x-, y-, and z-components of the electric field. So there's this three-fold structure to the electric field, at every point in space. All the different kinds of particles are quantised excitations of different kinds of fields, and all those fields have different kinds of multi-component structures. Even though the quantised excitations of the fields carry momentum and energy just like particles, so that we call them "particles", they also carry the multi-component structures of the fields.

If you want you could try to visualise this by imagining the particles as tiny specks that somehow have little arrows inside that can point in different directions, but it's not that we start with that assumption as our idea and express it in math. Instead we start with the quantum fields and it all just comes out: particles have a few internal degrees of freedom. One of these possibilities is a form of internal angular momentum, called spin. There could in principle be a lot more as well, besides spin, but as far as we know there seem to be at most two. Some kinds of particle only have one of these, and the photon has neither.

One of these internal properties is called "color", because it coincidentally happens to share a mathematical structure with the trichromatic structure of human color vision. Particles with color participate in the so-called "strong force" interaction; the force is called "strong" because it is able to bind many protons tightly together in an atomic nucleus even though they repel each other electrically. The strong force is so strong, paradoxically, that it can very often be ignored: everything that feels it gets bound together tightly into a neutral composite. The world is, as it were, made of ferocious dragons that have claimed all their treasures and settled down on their gold piles to sleep.

The other internal property that most particles have is a bit tricky, because it's actually the difference between an electron and a neutrino, or between an "up" and "down" quark. Normally we don't think of this difference as an internal feature of the same kind of particle, because electrons and neutrinos, and up and down quarks, have different masses and charges. It's a bit of a long story. The bottom line, though, is that the internal nature of the difference between electrons and neutrinos allows one to change into the other, like Clark Kent changing into Superman, if that tiny internal arrow gets bumped. This possibility is called the "weak interaction" because it is indeed quite a weak effect. We normally only notice it in certain forms of radioactive decay.

So dantana's intuition is quite correct: particles show a wide range of different behaviours because they actually do have a certain minimal amount of internal machinery. They have this in spite of being mere points, because quantum field theory.

[dantana]

Thanks Physics Guy! Hearing you lecture is well worth the price of admission. (being schooled by Dean Robbers.)

[Doctor CamNC4Me]

Point particles can have mass. In principle, as far as we know, they can have any mass whatever.

That was an awesome write-up, PG. Are these point particles responsible for the fundamental forces of nature, or are fundamental forces their own category in physics? In my caveman brain way of thinking the base line reality or the base line make up of the universe is composed of forces acting upon each other. In other words, at its fundamental core the universe doesn’t really exist in a material state. How wrong is that thinking?

- Doc

[Physics Guy]

I guess it depends what you mean by "responsible for". A lot of the time we think of matter as being made up of particles, which interact through force fields. But the particles are quantised excitations of other fields. There is an electron field and a positron field, as well as an electric field and a magnetic field. And the force fields also have quantised excitations that walk and talk just like particles, too. In the case of the electromagnetic field, these quantised excitations are called photons.

The electron-positron field differs in several important ways from the electromagnetic field, and so electrons are different from photons. In particular they differ in ways that make it easier to overlook the field nature of electrons and the particle nature of electromagnetic fields. Photons appear and disappear easily, and tend to travel in such dense swarms that one can scarcely notice the individual quanta. With electrons, on the other hand, it's harder to see field-like properties like wavelength and frequency.

So everything is particles and everything is also fields. Which one you consider primary is possibly a matter of taste. If you want to think of everything being made out of particles, though, you're probably going to want to focus on electrons and quarks, because if you start talking about photons and gluons people are pretty soon going say, "Wait a minute, those don't sound like what I had in mind when I heard you say 'particle'." If on the other hand you want to say that everything is fields, you'll likely want to talk mostly about the fundamental force fields, because if you talk too long about electron and quark fields people will begin to think that you're really just using a weird sort of code to talk about particles.

So probably the only way to really get it right is to go beyond mysterious statements like mine, about everything being both fields and particles, and learn enough math to understand exactly what those statements are trying to express: the fields are operator-valued, and their operations are to create and destroy quantised amounts of energy and momentum. It may not be clear what that means, but it means something quite precise that is in no way self-contradictory or paradoxical. It's just quite abstract. The impression of paradox is entirely an artefact of trying to find analogies with more familiar things.

If I try to find a familiar metaphor for the problem with familiar metaphors, it's like trying to explain a bagel by saying that it's both a bun and a donut. Up to a point that's a helpful analogy, but somebody who has never seen a bagel is bound to think of the donut-bun as sugary, maybe even frosted, and they'll blink hard if you mention sesame seeds. Unless, that is, they pick up more on the bun part and think of the bagel as just basically a bun; then they may have a hard time remembering the hole. Either way, they're going to have no idea that bagels are boiled before being baked.

Metaphors can mislead as much as they explain, and "particle" and "field" are really both just more-familiar metaphors trying to describe different aspects of a quantum field, which is really a little harder to grasp than either of them. It's not a mystery, though. It's just math.

[Physics Guy]

I remembered one other metaphorical explanation I used to use. Some wargame-like computer games give you a board composed of a large grid of tiles, between which units can move and on which structures can be built. The tiles can also usually have different kinds of terrain, or maybe different kinds of resources.

How does the game's computer code represent that data? There seem to be two basic choices. The game is usually coded so that every tile has to have some kind of terrain, and the question is which kind of terrain each tile has. So there's a database organised by tile, in which each tile's terrain type is stored: water, water, forest, forest, mountain—and so on. If it's one of those games in which you can build cities, though, these are usually represented in a database organised by city, with location being one property of each city.

You could in principle do it the other way around. You could have a list of all the forest tiles, saying where each piece of forest was located, then a list of water tiles, saying where each tile of water was located, and so on. Or you could add a field to your tile database, "City", in which you enter "0" for every tile that has no city on it, and for every tile that has a city you record which city is there.

That would really be the inefficient way to do it, though. For instance if there are N tiles on the board and C cities in the game, then usually C is much smaller than N. Having to say, for each of the N tiles, which of C cities is there, is going to take N log(C) bits. Saying where each city is, in contrast, will only take C log(N) bits, which is much smaller.

In the physics of the real world, describing something as a field is treating it the way those computer games treat terrain. Every point in space has to have some local value of the field, and the question is what the field strength is at each point. Describing something as a particle, in contrast, is treating it the way those games treat cities. The things exist, and the question is where those things are.

Just as with the computer game, though, one could in principle represent the same information either way. As in the computer game, one approach or the other might be more appropriate for different kinds of information. Terrain is an inherently field-like property, while cities are inherently particle-like. Or maybe what we really mean is that fields are inherently terrain-like features of the simulation that is the real world, while particles are inherently city-like features.

Unlike the computer game, however, the real world is quantum mechanical. Unlike cities in Sid Meyer's Civilisation games, particles can be in superpositions of being in many different places at once, and field strengths can likewise be in superpositions of many different values, with weird correlations ("entanglement") between their values in different places. With this vast wild range of additional possibilities, the distinction between the field picture and the particle picture really dissolves: either one can be represented as an appropriately correlated superposition of the other.

And so even if some features of the real world are inherently more terrain-like while others are inherently more city-like, the distinction is never really that sharp. The things that are usually more terrain-like can sometimes be more city-like, and vice versa. Particles are also fields and fields are also particles, and sometimes that really matters.

[Rivendale]

Unlike the computer game, however, the real world is quantum mechanical. Unlike cities in Sid Meyer's Civilisation games, particles can be in superpositions of being in many different places at once, and field strengths can likewise be in superpositions of many different values, with weird correlations ("entanglement") between their values in different places. With this vast wild range of additional possibilities, the distinction between the field picture and the particle picture really dissolves: either one can be represented as an appropriately correlated superposition of the other.

The entanglement comment you made reminds me of a paradox regarding gravitational lensing where two entangled photons take separate journeys around a massive globular cluster. When I say paradox it just exposes my lack of knowledge about entanglement. When Feynman said that a photon takes all possible journeys from his book qed does that imply that if the properties of one photon is measured does it follow that its companion may have had to take the journey over again?

[Physics Guy]

No, I don't think it means that.

Feynman's "path integral" formulation of quantum mechanics works, but it's just one way of representing the theory, not the theory itself. It sounds weird but simple when you're talking about, say, a single electron. It gets less elegant for more complicated systems than that, and so there are good reasons to use other representations of the theory instead. I'm not sure how straightforward the path integral formulation is for photons. It works for light, but one usually constructs it in terms of electromagnetic fields, which fluctuate through all possible values everywhere, rather than for photons that travel different paths. And even in terms of the fields, it's not so straightforward for light.

In any case the path integral picture, of particles taking all possible paths, doesn't really have anything to say about entanglement in this case. The path integral is about how to get from an arbitrary initial state to the resulting final state, over time. In this case the entanglement is in the initial state, from the way the two photons were created together; it is not generated as the photons fly through space. So Feynman's path integral picture will just pass along the initial weird entanglement into the final state, without doing anything to explain what it means or why it is there.

[Physics Guy]

I can try to give a version of my "entanglement for dummies" lecture. It's an illustration of one specific case of an entangled state, known as the Greenberger-Horne-Zeilinger ("GHZ") state. The state itself is a bit more complicated than the more commonly cited Bell states, but it shows the weirdness of quantum entanglement much more clearly. In the Bell state the violation of classical logic is only a small discrepancy in probabilities, but in the GHZ state it's a black-versus-white contradiction, with no probabilities involved.

You have three objects, 1, 2, and 3. For each one, you can measure one of two binary properties: X and Z. So you can measure, say, X2 or Z3 or Z1. Each of these six properties is binary, in the sense that if you measure them you only ever find either +1 or -1 for their values. You can also decide to measure products of these properties, rather than measuring them individually. If you measure the product X1*Z2*X3, for instance, you again find either +1 or -1, as you would expect when X1, Z2, and X3 individually must be either +1 or -1.

First think about this situation according to classical logic and arithmetic, without quantum mechanics. A priori each of the six properties can have either value, so you would expect there to be 2^6 = 64 different possible scenarios for the six properties of the three objects. Consider the special case(s) in which X1*Z2*Z3, Z1*X2*Z3, and Z1*Z2*X3 are all -1. Given those three product values, what is the value of X1*X2*X3?

By applying logic it's easy to conclude that it must be -1. You can just go through the 64 possible cases in general, finding first of all that there are eight sub-cases of our assumed special case in which the three given triple products are all -1. Then you can just check that in all eight of those possible sub-cases, the product X1*X2*X3 is also -1. You can also be clever and note that if you multiply out the big product (X1*Z2*Z3)*(Z1*X2*Z3)*(Z1*Z2*X3) then in our case this is (-1)*(-1)*(-1)=-1. If you just regroup the values, though, the same big product is also X1*X2*X3*(Z1*Z1)*(Z2*Z2)*(Z3*Z3). The products (Z1*Z1) and so on are all 1 in all cases, regardless of whether the Z is +1 or -1, and so our big product is in all cases the same as just X1*X2*X3. So anyway, yeah: if X1*Z2*Z3, Z1*X2*Z3, and Z1*Z2*X3 are all -1, then X1*X2*X3 must also be -1, by pure logic. No ambiguities or probabilities are involved.

But now see how this same situation of three objects with six binary properties can appear in quantum mechanics. In quantum mechanics we also speak of properties and measurements, and we can measure products of properties. It may be that neither "property" nor "measurement" really means what we normally think they should mean, though, because in quantum mechanics one does not apply logic. Instead one just does some arithmetic.

In quantum mechanics we can represent one of these objects with a single binary condition, either [a] or [c], even though two different properties X and Z can be measured. In quantum mechanics, measuring something doesn't necessarily just mean looking at it. In general it means changing something.

In this case what X does is to leave [a] unchanged as [a], but change [c] into (-1)*[c]. So what does this have to do with measuring whether X is +1 or -1? The first rule of quantum measurement is that if the change that is made amounts to just multiplying the state by a number, then that number is the result of the measurement. So this means that in condition [a] we always measure X to be +1, while in condition [c] we always measure X to be -1.

What does Z do, then? Z does this: it changes [a] into [c] and [c] into [a]. This is not the same as just multiplying [a] or [c] by a number, so the first rule of quantum measurement doesn't say what we will find if we measure Z in condition [a] or [c]. For that we need further rules, which introduce probabilities; we won't end up needing those further rules for the GHZ state, however, so I'm not going to discuss them.

What we can do is see what kind of condition will have Z = +1 or -1. In quantum mechanics we can have superpositions of different conditions. So what happens if we measure Z when our object is in the superposed condition [a]+[c]? Well, it turns into [c]+[a], which is the same as [a]+[c], which is (+1)*([a]+[c]). So the superposed condition [a]+[c] is the condition in which Z can be measured as +1. Conversely, the superposed condition [a]-[c] is changed by Z into [c]-[a], which is the same as (-1)*([a]-[c]). So the condition in which Z is measured as -1 is the superposition [a]-[c].

This is all we need to know to see the entanglement of the GHZ state, because the GHZ state is the following superposed condition of all three objects: [aaa]-[acc]-[cac]-[cca]. What do we find in this state when we measure the four different triple product properties that we discussed above?

Well, applying the rules in succession we have that X1*Z2*Z3 changes [aaa] into [acc], changes [acc] into [aaa], changes [cac] into -[cca], and changes [cca] into -[cac]. The net effect is that X1*Z2*Z3 changes the GHZ state into itself times (-1). So if our three objects are in the GHZ state, whenever we measure X1*Z2*Z3 we find the result -1. And in exactly the same way we find that Z1*X2*Z3 and Z1*Z2*X3 also just swap the four superposed conditions in the GHZ state among each other in such a way that the entire state is just multiplied by (-1). So in the GHZ state we indeed measure our three triple product properties to be -1.

But now what does the fourth triple product X1*X2*X3 do to the GHZ state? It leaves [aaa] unchanged as [aaa], but it turns [acc] into (+1)*(-1)*(-1)[acc], which is just [acc] again. Likewise it ends up leaving [cac] and [cca] unchanged, too. So it leaves the whole GHZ state simply multiplied by +1. By the first rule of quantum measurement this means that what we find for X1*X2*X3 is +1, not -1 as we deduced it had to be by classical logic.

This is sometimes called a paradox. GHZ states have been realised, though, with electron spins or photon polarisations as the three objects and their properties, and the measurements come out as predicted by quantum mechanics, not as predicted by logic.

So whatever measurements really are in quantum mechanics, they are not simply verifications of true-or-false propositions from which other true-or-false propositions can be deduced using logic. They are something more complicated than that, in the sense that they are at least potentially changing something and not just recording it. This shouldn't really be so surprising, however. Measuring the properties of atomic-scale things is a heck of a lot trickier than just laying a ruler onto them and reading off the number.

Quite complex apparatus is needed to magnify those tiny things by amplifying tiny signals. When you realize that, the wonder isn't that measuring submicroscopic things is so weird, but that it is possible at all. A reasonable expectation would be that training a massive amplification device onto a single electron would produce all kinds of weird unstable signals, among which the effects of the target electron were just a drop in the ocean. The astonishing miracle of quantum mechanics is that it introduces a kind of digitalisation which makes possible the massive amplification of extremely faint signals, so that gigantic creatures like us actually can see the effects of single electrons.

Even though what we are doing when we measure electron properties is so much more complex than just looking at a traffic light and seeing which of three colours it is, the correspondence between target object and resulting signals is still sufficiently strong in the complex experiments that we use familiar terms like "measure" or "observe" for the whole complex process. Calling quantum measurements "measurements" is still really a metaphor, though. It shouldn't really be surprising that these processes are not actually just a matter of confirming or ruling out propositions about microscopic properties.

What can definitely be surprising is the specific way in which quantum measurements differ from the things we normally think of as measurements. Even though the experimental measurement processes are complex and indirect, and their results don't follow the rules of logic, the rules for what quantum measurements do can be some very simple mathematical operations. They really are things like multiplying by -1 or swapping between some [a] and some [c]. These complex amplification processes don't follow the simple logical rules of observing traffic lights, but they do follow surprisingly simple rules. The simple rules just aren't the rules of simple logic. Nobody knows why the simple quantum rules have the form that they do. They are not consequences that we have deduced from any glorious underlying ideas. They are the axioms from which the theory begins; we have induced them from Nature empirically without gaining any understanding of what they may mean, if anything.

There are a number of ways in which the simple rules of quantum mechanics are different from the simple rules that human intuition expects. There can be probabilities and minimum values and stuff. Entanglement is one of the more complicated kinds of divergence between quantum mechanics and human intuition, and it can be an especially dramatic divergence, as it is in the GHZ case. The entanglement is the feature that we have a superposition of conditions of multiple objects, which does not factorise into a simple product of states of the individual objects.

Entanglement is to superposition what correlation is to probability. This time, finally, that's not just an analogy, but an exact definition.