Sleeping beauty and quantum mechanics

My newest favorite philosophical dilemma is the Sleeping Beauty problem.  The experiment goes as follows:

1. Sleeping Beauty is put to sleep.
2. We flip a coin.
3. If the coin is heads, then we wake Sleeping Beauty on Monday, and let her go.
4. If the coin is tails, then we wake Sleeping Beauty on Monday.  Then, we put her to sleep and cause her to lose all memory of waking up.  Then we wake her up on Tuesday, and let her go.
5. Now imagine Sleeping Beauty knows this whole setup, and has just been woken up.  What probability should she assign to the claim that the coin was tails?

There are two possible answers.  "Thirders" believe that Sleeping Beauty should assign a probability of 1/3 to tails.  "Halfers" believe that Sleeping Beauty has gained no new relevant information, and therefore should assign a probability of 1/2 to tails.  The thirder answer is most popular among philosophers.

This has deep implications for physics.

There is an argument that Everettian Quantum Mechanics (aka Many Worlds Interpretation, henceforth EQM) requires that you be a halfer.  Say that you tell Sleeping Beauty that you will wake her up on Monday and Tuesday, with a memory wipe in between.  The argument goes that this is exactly analogous to telling her that you will cause her wavefunction to branch, and in one branch she will wake up on Monday and in the other on Tuesday.  In both cases, the Sleeping Beauty on Monday and Sleeping Beauty on Tuesday are both real, and neither has access to the other (either because of the memory wipe or because they are in separate branches).

Therefore, the standard sleeping beauty problem ("Two-Branch-Beauty" on left) is equivalent to the quantum sleeping beauty problem ("Three-Branch-Beauty" on right).

From "Self-Locating Uncertainty and the Origin of Probability in Everettian Quantum Mechanics".  In this paper, they also interpret the very first coin flip as a quantum measurement which causes the wavefunction to branch.

In the Three-Branch-Beauty problem, EQM straightforwardly says that Sleeping Beauty should assign a probability of 1/2 to being in the "tails" branch.  If the Three-Branch-Beauty and Two-Branch-Beauty problems are indeed analogous, then she should also assign a probability of 1/2 in the Two-Branch problem.  And therefore, if we accept EQM, we must accept the unpopular halfer solution.

The problem with this argument, I think, is that assigning probabilities in EQM is... not at all straightforward.  This is actually the major disadvantage of EQM, is that it's not clear why we should interpret the branching worlds with probabilities.  How can we say one branch is more likely than another, if both branches are equally real?

The paper, "Self-Locating Uncertainty and the Origin of Probability in Everettian Quantum Mechanics" purports to answer this question (also see Sean Carroll's blog about it).  When we assign probabilities to branches, these are interpreted as "self-locating" probabilities.  That is, even if all branches are real, we still have to ask the question, which branch am I in now?  You might naively take the principle of "indifference", arguing that all branches are equally likely.  But that gets you the wrong probabilities.

The paper argues for a different principle, the "Epistemic Separability Principle", the idea that Sleeping Beauty assigns probabilities on the basis of only what she observes.  So if there is a second measurement device that Sleeping Beauty does not look at, then the probability she assigns to the first measurement should be independent of the result of the second measurement.  There's a simple argument on page 4 which shows that if there are two branches of the wavefunction with equal amplitude, then we should assign equal probabilities to each branch.

So let's say we have the Three-Branch-Beauty experiment, and Sleeping Beauty has just woken up.  By the Epistemic Separability Principle, she should assign equal probability to waking up in the first branch, and waking up in the second branch, because those two branches have equal amplitude.  The third branch does not have equal amplitude, and based on more mathematical arguments, you can show that the third branch is twice as likely as the other two.  Therefore, she would assign probabilities 1/4, 1/4, and 1/2 to the three locations.

Let's say we have the Two-Branch-Beauty experiment, and Sleeping Beauty has just woken up.  By the Epistemic Separability Principle, she should assign equal probability between waking up on Monday in the first branch, and waking up on Monday in the second branch, since those two branches have equal amplitudes.  Likewise, she should assign equal probability to waking up on Tuesday in the first branch, and waking up on Monday in the second branch.  Therefore, she should assign probabilities of 1/3, 1/3, and 1/3 to the three locations.  This recovers the popular thirder answer without giving up EQM.

In another post I will have further comments on Sleeping Beauty.  (Update: it's written)

More literal spaghetti

Yesterday, I wrote about the spaghetti metaphor for Everettian Quantum Mechanics (aka Many Worlds).  Unfortunately, the metaphor breaks down because different worlds in quantum mechanics can constructively and destructively interfere with each other on a microscopic level, although we generally don't need to worry about this on a macroscopic level.

But there is another interpretation of quantum mechanics for which the spaghetti metaphor is more exact.  I'm speaking of the Bohmian interpretation of quantum mechanics.

The Bohmian interpretation is usually the go-to example for how we can have a deterministic theory of quantum mechanics.  In Bohmian theory, every particle has a well-defined trajectory, and only occupies one position at any given time.  The fact that we can't predict exactly where the particle will be just has to do with the fact that we do not (and cannot) know with certainty the particle's initial position.  Bohmian theory makes all the same predictions as the other major interpretations of quantum mechanics.  But this comes at a cost: faster-than-light information transfer.

I'm largely a Many Worlds partisan, but I give the Bohmian interpretation credit, because it basically starts with the Many Worlds interpretation.  In Bohmian theory there is no wavefunction collapse.  Instead, the wavefunction simply evolves according a single equation, and splits into many worlds just like in the Many Worlds interpretation.

The difference is that the wavefunction is not interpreted as a description of reality (and therefore there aren't really many worlds).  While the wavefunction is a real object, it is seen as distinct from all the particles we see around us.  The wavefunction is interpreted as a pilot wave which merely guides the motion of particles.*  All particles have a definite position, and we just need is this complicated pilot wave object to determine their motion.

*For those who have studied quantum mechanics, it's actually quite simple to understand.  Even in standard quantum mechanics, we speak of the probability current.  We can obtain the probability "velocity" by dividing the probability current by the probability density.  Bohmian theory interprets this literally, by having the particle velocity equal the probability velocity.

To relate this to the spaghetti metaphor, let me consider the classic double slit experiment.  We send light through two slits, and the waves coming from the two slits interfere with each other.

 Waves of light go through two slits, located at the bottom, and travel upwards.  (Technical details: I'm just showing the real part of the wavefunction, with blue positive, red negative, and green zero.)

At some points, the waves interfere constructively, and at other points they interfere destructively.   This creates alternating dark and light fringes.  In more ordinary quantum interpretations, this is because the wavefunction of the light determines the probability that the light is in any particular location.

 The colors here show the probability that light is in any particular location.

But in the Bohmian interpretation, we do not interpret the wavefunction as probability.  Instead, we interpret it as a pilot wave.  The light goes on a well-defined trajectory, it's just hard to predict which particular trajectory it's on.

Here I show many possible trajectories for the light, as calculated using Bohmian theory.  As you can see, particles don't respect conservation of momentum, and even though light only goes through one slit, it is obviously affected by the presence of the other slit.  (You can find lots of similar images on the net, but this whole post is really an excuse for me to write a Bohmian calculator, so I'm giving you my image.)

Perhaps now you can see how this follows the spaghetti metaphor.  Each possible trajectory for the light is a single strand of spaghetti.  When we speak of probabilities in quantum mechanics, we're really talking about our degree of belief that we are in any particular strand of spaghetti.

The difference is that in the Bohmian interpretation, there is only one strand of spaghetti, the one that includes us.  And in my Literal Spaghetti interpretation, there are many strands of spaghetti, of which we are just one.

Convergent spaghetti worlds

3:AM Magazine had an interview with Alastair Wilson, a philosopher who thinks about the Everettian Quantum Mechanics (also known as the Many Worlds Interpretation, henceforth referred to as EQM).  It's nice to know that some philosophers are thinking seriously about it, since physicists generally aren't trained or paid to do so.  Wilson plays to the strengths of his academic discipline by identifying the most interesting questions of Many Worlds.  I'd like to discuss a few of them.

Branching or parallel worlds?

Wilson says:

In informal or popular discussions, people use two metaphors pretty much interchangeably to describe the Everettian multiverse: branching worlds and parallel worlds. Of course, these two metaphors are in tension: the former suggests mereological overlap of worlds, like a branching tree, whereas the latter suggests mereological distinctness of worlds, like a packet of spaghetti.

Wilson goes on to note (correctly) that the many worlds of EQM are an emergent macroscopic structure, from the much more complex fundamental structure of quantum physics.  Therefore, the question of a branching tree vs a packet of spaghetti is really a question of which is more useful.

Wilson prefers the spaghetti metaphor because it partially resolves the "probability problem" of EQM:  If all of the many worlds exist, then how does it make sense to assign probabilities to each of the many worlds?  In the spaghetti metaphor, when we speak of probabilities, the probabilities represents our belief that we are in any particular noodle.

I am agreement with Wilson, but it's worth poking at that answer.  Let's consider the simplest kind of world-splitting.  No, not the double slit experiment, even simpler!  Consider a beam splitter.

A beam splitter takes a beam of light, and reflects half of it while transmitting the other half.  It's a rather standard optic, and we have several of them on the laser table in our lab.

But suppose that instead of a beam of light, we had a single photon of light.  Even though there's only one photon, this single photon will still split paths, becoming a superposition of path 1 and path 2.  We've generated two distinct worlds, one where the photon is on path 1, and one where it's on path 2.

But these worlds are not very different from one another.  Put another way, they are very close to each other in parameter space--all their parameters are exactly the same, except for the one parameter describing the location of the photon.  Because the worlds are very close to each other, it is possible for them to interact, and experimentally feasible to make them interact in a controlled manner.  In the EQM picture, we would say that the worlds have not yet diverged. Note that the distinction between divergent worlds and not-yet-divergent worlds is an emergent distinction rather than a fundamental one, since it partly has to do with what is experimentally feasible.

The beamsplitter in reverse

Yes, it's entirely possible to join these two worlds together again, to perform the beamsplitter experiment in reverse.  Simply place mirrors on the two beam paths.  This is a very common setup, called an interferometer (used in many experiments, but best known for the one that led to Relativity theory).

It's called an interferometer is because the light can actually end up on two paths: path A or path B.  Whether it ends up on path A or B depends on whether the waves of light are in sync or out of sync, which is to say, whether they interfere constructively or destructively.  Let's say that they interfere in such away that they only go down path B.

This presents a problem for the metaphor of the packet of spaghetti.  Let's treat path 1 as a packet of spaghetti, and path 2 as another packet of spaghetti.  If we have just path 1 on its own, half of its spaghetti goes down path A, and half of it goes down path B.  And if we have just path 2 on its own, the same thing happens.  And yet, if we have the photon go down path 1 and path 2 simultaneously, all of the spaghetti ends up on path B.

There is spaghetti from 1 that goes down A, and spaghetti from 2 that goes down A, but rather than adding the quantities of noodles in A, we subtract them.  Here the metaphor of spaghetti just breaks down, since there's no reason you would ever subtract quantities of spaghetti if we took the metaphor seriously.

But we had already admitted that the spaghetti was just a metaphor for an emergent macroscopic structure of EQM, so it's unsurprising to see the metaphor break down on the microscopic level.  It's just good to keep in mind how exactly it happens, and maintain skepticism about our metaphors.

Why divergence?

Now that I've illustrated how in EQM worlds can converge as well as diverge, we can ask why divergence is so much more common than convergence.  Ultimately, it has to do with the Second Law of thermodynamics, and the increase of entropy over time.

If you have two strands of spaghetti, there are many ways for them to be apart from each other, and only a few ways for them to be stuck to each other.  That is to say, pulling strands of spaghetti apart is associated with an increase in entropy.  Thus, strands are more likely to diverge than converge.

But there's also an issue of dimensionality.  Imagine that the spaghetti strands are close together in parameter space, in that they only differ by one parameter: the location of a single photon.  As long as this is the only parameter which distinguishes the strands of spaghetti, it's as if they're trapped in a low-dimensional space.  They're like cars, trapped on a 2d surface, prone to crashing into one another if we're not careful.  But once the worlds differ by more parameters, we add more dimensions.  It's much harder for two airplanes to collide into each other than cars, because they live in a 3-dimensional space.

However, not impossible

Now imagine that the two spaghetti strands differ by billions of billions of parameters.  This can easily happen, if for instance the photon on path 1 hits a screen, a screen which is composed of about 10^23 electrons and nuclei.  So now we're talking about airplanes not in 3-dimensional space, but airplanes in 10^23-dimensional space.  It's not surprising if they hardly ever collide.

And that's why the many worlds metaphor works.  If macroscopic worlds converged as often as they diverged, then we'd constantly face this problem of interfering noodles.  But convergence only tends to occur on the microscopic levels.

Update: I have a followup post talking about the Bohmian interpretation

Quantum Mechanics for skeptics, redux

When I was an undergrad, I gave an informal talk on quantum mechanics for my friends in the skeptical student group.  Unfortunately, the website hosting the slides went defunct, so it's no longer available.  I offered to give a similar talk to the atheist student group at my current university.  So I'm redesigning it as a chalk talk.  It will probably be a bit more serious this time, since I'm not an undergrad, but it's still very informal.

To organize the talk, it helps me to write a blog post along the same lines.  So that's what you're getting.  Apologies if it's a bit rough.

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Quantum Mechanics for Skeptics

I.  Introduction

(These quotes will be handed out on cards)

I think I can safely say that no one understands quantum mechanics.
-Richard Feynman

The physical world, including our bodies, is a response of the observer. We create our bodies as we create the experience of our world.
-Deepak Chopra

The physical process of making a measurement has a very profound effect. 
-David Albert

We're all connected by an energy field.  We swim in a sea of light, basically, which is the zero point field.
-Lynn Mc Taggart

Light and matter are both single entities, and the apparent duality arises in the limitations of our language.
-Werner Heisenberg

I wake up in the morning and I consciously create my day the way I want it to happen... and out of nowhere little things happen that are so unexplainable, I know that they are the process or the result of my creation.
-Joe Dispenza

There's all sorts of universes sitting on top of each other, and they're splitting apart and differentiating as time moves on.
-Sean Carroll

A shift in quantum state brings a parallel lifetime. The relationship to you and your environment is lifted... You are now in a parallel existence.
-Ramtha, channeled by J.Z. Knight

Some of these quotes are from physicists, and some are nonsense. I do not intend for them to be difficult to distinguish.  Most of the nonsense comes from people interviewed in the documentary What the Bleep Do We Know?

Richard Feynman of course was a famous physicist.  But despite what he said, it is clear that some people understand quantum mechanics better than others.  Now, most of you don't study physics (are there any physics majors in the audience?), so you probably don't understand quantum mechanics.  The question is, how can you tell the nonsense from the science?  Can you do it without deferring to an expert?

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II. A quick overview of quantum mechanics

A. The context of quantum mechanics in physics

First I need to give some context.  I am not a quantum physicist.  The fact is that quantum physics is very well established, and isn't a topic of cutting edge research.  Almost every physicist uses quantum physics as the framework to study something else.  I'm a condensed matter physicist; I apply quantum theory to extremely large numbers of atoms.  The fundamental rules of the game are well understood, it's scaling it up that's hard.

But yes, there are some things about quantum theory that are not well understood.



But quantum gravity isn't really relevant to this talk.  Everything here is well understood.  In fact I'll stick mostly to quantum mechanics.

B. The wavefunction and measurement

Quantum mechanics describes matter as made of things that are like particles and also like waves.  Take for instance the electrons in atoms.  The electrons can be in many possible states which we describe with a "wavefunction".  This is usually represented with pictures of orbitals around the nucleus of the atom.  But it's actually just some mathematical function, which I'll plot as a function of position.



The first consequence is that the possible energies of an electron are discrete.  The energy of the electron is roughly related to the number of times this wave wiggles up and down.  It needs to wiggle up and down an integer number of times, so there are discrete energy levels.  In particular, there's a lowest energy level, which is a good thing.  Otherwise the electrons would collapse into lower and lower energies, causing all atoms to implode.

There's the question of what this wavefunction actually represents.  Well say that you had an ultra-precise way of measuring the position of the electron.  The probability of finding the electron in any position is equal to the square of the wavefunction.  So even if you prepare lots of electrons in the same way, you can never predict exactly where they are.

And here's where it gets weirder.  Say that you make two measurements, one right after the other.  The second measurement will agree with the first.  So even though the position was uncertain to begin with, by measuring it you make its position certain.  One way to describe this is by saying that the wavefunction has changed after measuring it.  This is referred to as wavefunction collapse.



C. Quantum uncertainty vs classical uncertainty

The picture I've just drawn sounds a little bit like we just don't know where the electron is.  After we measure it, and then we know where it is.  I call this "classical uncertainty".  But the uncertainty in quantum mechanics is different.

For instance, let's say that we don't know whether an electron is in a 1s state or a 2p state.  But it's not just that we don't know in the classical sense, let's say we don't know in the quantum sense.  In quantum mechanics, you represent this by adding the wavefunctions together.  Now we can take two kinds of measurements of this system.  If you try to measure the energy, sometimes you'll get the 1s energy and sometimes you'll get the 2p energy.  Then the electron will collapse into the 1s or 2p state according to what you measured.



But suppose that we instead measure the position of the electron.  We would mostly see the electron on the left side here and not on the right side.  Now if the electron were really in the 1s state, we'd see it on the right and left sides equally.  And if it were in the 2p state, we'd see it in the right and left sides equally.  But it's not merely that we don't know whether it's in 1s or 2p, it's that in some sense it's in both states.

This, by the way, is entirely a thought experiment.  Practically speaking, we wouldn't be able to control whether the electron was in a 1s + 2p state or a 1s - 2p state.  If it's 1s + 2p, the electron would appear on the left, and if it's 1s - 2p, it would appear on the right.  Since we don't know which one it is, we're back to the situation of classical uncertainty rather than quantum uncertainty.  But there are other experiments that really do demonstrate that quantum uncertainty is special.

D. Entanglement

One of the strange consequences of quantum mechanics is that you can have correlations between particles, even if those particles are far away from each other.  For instance, there's a way to prepare two photons such that they have the same polarization, even though we don't know the polarization of either individual photon.  Again, it's not that we are ignorant of the polarization, it's that it's actually in a superposition of vertical/vertical and horizontal/horizontal



Sometimes people use entanglement to argue that if we think positive, positive things will come to us by the law of entanglement.   But generally, if far-apart particles are correlated at all, there's no particular reason they would be correlated vs anticorrelated.  If the particles are interacting with a random environment, they would switch between correlation and anticorrelation such that, effectively, there's no correlation at all.

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III. How to recognize quantum nonsense

A. Vocabulary

Quantum nonsense uses lots of complicated terminology in order to confuse people.  People also feel afraid to challenge it because maybe they just don't understand.  The problem is that real science also uses lots of terminology, and if you're not an expert in the field, you may not be able to tell the difference.

It's difficult to make a rule of thumb to tell the difference, but here's what I propose:  Look at who the intended audience is.  If scientists are talking to other scientists, they need terminology in order to communicate precisely.  If a scientist is speaking to the public, they may use terminology because they're not really sure how to say it in plain language.  But plain language would be ideal.

In contrast, pseudoscientists are almost always talking to the public, and use scientific terminology intentionally.  It's not that they don't know clearer ways of speaking, they actually want you not to understand.

B. Ignoring scale

In What the Bleep do We Know? there's a clip where they show a basketball bouncing in many places on a court.  Then the basketball player looks at it, and it's only in one place.  This is an okay illustration of quantum mechanics, but they neglected to explain how this only occurs on very small scales.

The appropriate scale is the atomic scale.  When you have electrons in an atom, you don't know where the electron is, but there's an extremely high probability that it's not very far from the nucleus.  The uncertainty of the basketball is on the same scale (smaller, really, since it's a heavier object).

In fact the picture is very much complicated by a system which is made by more than a few particles. As I said earlier, in my research I apply quantum physics to very large numbers of particles, such as what you would find in a grain of dust.  Quantum physics has a big impact (for one thing, the atoms aren't imploding), but large objects do not behave like small ones.  Unless the system is really cold (ie at the very limits of our cooling technology), there's too much randomness.  This randomness turns quantum uncertainty into classical uncertainty.

C. Observers

My favorite part of What the Bleep was the following argument.  In order to cause wavefunction collapse we need conscious observers.  Human cells can cause wavefunction.  Therefore, human cells are conscious beings.  What follows is a computer-animated segment with anthropomorphic human cells.  And when you think negative thoughts, the human cells have decadent parties and destroy your health.  Long story short, you should throw out your medication and just think positive.

But seriously, there's nothing in quantum mechanics that requires conscious observers.  Really what you need is some large complicated system, such as a grain of dust which introduces randomness.  This makes a quantum system behave classically, and that's what wavefunction collapse is, more or less.  Quantum mechanics doesn't say you're special (although you're free to think you're special anyway).

D. Quantum Interpretations

Now there are a few different interpretations of quantum mechanics, as to what it really all means, on the bottom of it.  The most popular interpretations are the Many Worlds Interpretation and the Copenhagen interpretation.

The Copenhagen interpretation is more or less what I've already described.  There's a quantum system which follows certain rules.  And you can measure or observe the system, which causes the system to change.  In the Many Worlds interpretation, there's nothing fundamentally different about the observation process.  The system just interacts with a measurement device, and becomes a superposition of two states.  These two states don't really interact and for all intents and purposes are independently evolving worlds.



These two interpretations are equivalent to each other, at least experimentally.  There is no experiment that can be performed to distinguish between these two.  So if someone says something that makes sense in one interpretation, but totally contradicts the experimental predictions of the other interpretation, then it's probably nonsense.

For example, when someone says that quantum mechanics requires conscious observers, you know that's wrong because there are no observers whatsoever in the Many Worlds Interpretation.  When someone says that you interact with the parallel worlds, you know that's nonsense because in the Copenhagen interpretation there are no parallel worlds to interact with.

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IV. Conclusion

Quantum Mechanics is a little strange.  Quantum uncertainty is fundamentally different from what we usually think of as uncertainty.  We can have correlations between far away particles.  But it does not make conscious observers special, and nobody "chooses" reality.  I hope this helps you to distinguish quantum science and quantum nonsense.  But if not, you can ask an expert.  I can take questions now.

The fluidity uncertainty principle

This was cross-posted on The Asexual Agenda.

This is a silly observation only I would ever make.

Every physicist should know that frequency and time are complementary variables. They’re related by an uncertainty principle, similar to the uncertainty principle in quantum mechanics governing position and momentum. The more precisely you know the position of a particle, the less precisely you can know its momentum. The more precisely you know its momentum, the less precisely you can know its position.

Likewise, if you want to precisely know the frequency of something, you have to average over a long period of time. If you want to know how the frequency changes over short timescales, you must accept an inherent uncertainty in the frequency.

One of the ways in which people are different from each other is in how frequently they’re sexually attracted to other people. However, our sexualities are not always constant throughout our lives. Therefore, you can describe (one aspect of) sexuality with a frequency, and you can say that this frequency changes over time.

But there’s a fundamental limitation to how precisely you can know frequency and time together. If you want to talk about how someone’s frequency of attraction varies from year to year, then it is impossible to pin down this frequency with precision greater than once a year.

Mostly, this doesn’t matter. If you’re attracted to about ten people a year, then what does it matter if you’re not sure if it’s actually nine or eleven people a year? Who can even count up that high anyway?

However, if you’re very infrequently attracted to people, and experience high fluidity, then we enter what I’m going to call the quantum sexuality regime. Here, the fluidity uncertainty principle reigns.

Quantum interpretations are scientific

Quantum Mechanics is famous for having multiple interpretations.  Among them, the two most common interpretations are the Many Worlds Interpretation (MWI) and the Copenhagen interpretation.

According to the Copenhagen interpretation, when you measure a system that is in a mixed quantum state, then the system "collapses" into a definite state (that is, a state that gives you only a single result for your measurement).  There are different probabilities for the system to collapse into different states, but it will always be a definite state.

In contrast, MWI says that there is no collapse.  Rather, when you measure a system in a mixed quantum state, now you are in a mixed quantum state.  One component of your state consists of you having measured one outcome; another component consists of you having measured the other outcome.  These different components don't interact with each other, and evolve independently.  The ultimate consequence is that the entire universe is in a mixed state with many components that don't interact with one another.  Thus the name "many worlds".

MWI and the Copenhagen interpretation give identical predictions in all experiments.  So it's impossible to falsify one in favor of the other.  That's why some contend that the interpretation question is non-scientific.  I do not agree, for two reasons:

1. Different interpretations suggest different directions for future theories.
2. Experiments might have something to say about Copenhagen vs MWI after all.

1. Directions for future theories

I want to quote something Richard Feynman said.  Not because Feynman said it, therefore it was right, but because Feynman put the idea into my head.  This occurred in a lecture series Feynman gave at Cornell (specifically, the second lecture, section 8).  Feynman explained that there are three different ways to state the law of gravitation:

1. Each object senses where all the other objects are, and feels a force towards each object of magnitude GmM/R^2.
2. There is a gravitational potential in every point of space, governed by laws that only look at its surrounding neighborhood, without looking at far away objects.  The gravitational force is determined by this potential.
3. Given a start and end point, an object travels by the path that minimizes a particular quantity.

So the question is, which of these theories is correct?  Is this a scientific question?  Feynman said:

They are equivalent, scientifically; it is impossible to make a decision, because there's no experimental way to distinguish if all the consequences are the same.

Psychologically, they're very different in two ways.  First, philosophically, you like them or don't like them--training is the only thing you can do to beat that disease.  Second, psychologically they're different because they're completely unequivalent when you go to guess at a new law.

As long as physics is incomplete, and we're trying to find out the other laws, and to understand the other laws, then the different possible formulations give clues as to what might happen in another circumstance.  And they become not equivalent in psychologically suggesting to us to guess as to what the laws might look like in a wider situation.

Physics has developed a lot since we discovered the law of gravitation, so we know for a fact that the different interpretations have different uses.  The second theory has helped us understand some of the fundamental character of quantum field theory.  But the third theory gave us Feynman path integrals, which are related to Feynman diagrams, an easy way to represent fundamental particle interactions.  The first theory has not been very useful, and that's that.

MWI and the Copenhagen interpretation are in the same situation as the law of gravity.  They're equivalent in terms of predictions, but they lead to different ways of thinking which suggest different directions for expanding physical theories.  The first thing that comes to mind is that MWI is deterministic and unitary (meaning it is deterministic if time plays backwards too).  That's useful because it suggests that we can continue coming up with fundamental laws that obey time-symmetry.  There may be other uses too.

As I argued in "Multiverses are scientific", scientific ideas can serve many roles.  There are observations, hypotheses, experiments, theories, predictions, and so forth. Quantum interpretations also fulfill a role in science--they suggest different directions for future theories.  They do not fulfill the role of hypotheses which can be tested.  And that is okay, because not all scientific ideas need to fulfill every single role at once.  A hypothesis doesn't also need to be a theory, and an interpretation doesn't also need to be a hypothesis.  So the fact that the interpretations are not falsifiable doesn't necessarily mean it is unscientific.


2. How to (possibly) verify MWI

First, I need to explain how the Copenhagen interpretation and MWI, despite their differences, vary continuously into one another.  Consider a thought experiment where a mechanical device detects whether a radioactive atom decays within a half-life.  If it does, then it turns on a laser pointer, which provokes a cat to run through a hallway.  At the other end of the hallway, I can see if the cat appears or not.  Whatever I see, I publish my results for other scientists to see.

The question is, when does the collapse occur?  Does it collapse when radioactive atom observes itself decaying?  Does it collapse when the mechanical device observes the radioactive atom?  Do the mechanical device and radioactive atom both collapse when the cat sees the laser pointer?  Do the cat, device, and atom collapse when I see the cat (or not)?  Do the cat, device, atom, and I all collapse when other scientists see my results? Etc. etc.

If you answer "no" ad infinitum, you are taking the MWI.  But if you eventually answer "yes", you are taking the Copenhagen interpretation.  To make the Copenhagen interpretation similar to the MWI, all you have to do is say "no" lots of times before eventually saying "yes".

We don't know the answer to all those questions, and cannot know.  But we do know the answer to the first few is "no".  We can verify that small systems are in mixed states because we can measure interference effects.  With more complex systems, it's harder because of the sheer randomness that occurs when you have lots of particles at a non-zero temperature.  In my knowledge, the largest quantum system created was 40 microns in length, and needed to be cooled down to 0.1 K.

It will be hard to push that limit, but we definitely could, in very small steps.  We could push the Copenhagen interpretation closer and closer to MWI, though we will never quite reach it.  Alternatively, MWI could be falsified if it's found that mixed states do not exist past a certain point (ie if no interference is found where it is expected).

Conclusion

People sometimes say that MWI is unscientific, because it posits parallel worlds that cannot be observed.  This is mistaken because it assumes that every single idea in science needs to be verifiable.  Scientific ideas may also serve other roles, such as suggesting future directions for theories which themselves would be experimentally verifiable.  Secondly, it so happens that MWI is partially verifiable in very small increments.

Entangled minds

Sometimes people use quantum entanglement as an explanation for psychic communication between minds.  The idea is that my neurons are entangled with yours, and whatever my neurons see would be the same as yours.

Being a physicist, I know that's wrong.  Even if psychic communication exists, quantum entanglement would utterly fail to explain it.  But at the same time, as a physicist, I feel compelled to get the technicalities right.  There's a nagging technicality here: Yes, our neurons are in fact entangled.  But that doesn't mean what you think it means.

Most people are told that entanglement is about correlations over large distances.  What someone measures here will be the same as what someone measures over there.  But what you may not have known is that entanglement is also about anti-correlations over large distances.  What someone measures here could be the opposite of what someone measures over there.

So suppose that I feel a twinge of sadness.  I interpret this to mean that you are experiencing sadness right now, because my sad neuron must be correlated with your sad neuron.  Except, maybe they're anti-correlated.  Maybe you are feeling the exact opposite way, and your neuron is happy or whatever?

Or suppose that I think happy thoughts.  Surely happy thoughts are correlated with happy things in the world via entanglement, so if I think positive thoughts I will attract good things to me.  Except, maybe my happy thoughts are correlated with the opposite, and are actually attracting bad things to me?

So if we have two neurons, are they correlated or anti-correlated?  Under carefully-controlled experimental conditions, involving small numbers of particles (ie much fewer than the number of particles in a neuron), and isolating those particles from the random interactions with the outside world, physicists can produce things that are definitely correlated or anti-correlated.  But under any less-controlled conditions, it's essentially random.  There's no way to know.  Every time it happens, it will be different.  It will be completely unpredictable.  On average there will be no correlation at all.  We say that the neurons are decoherent.

(Because there is no observable correlation, some physicists would say that the neurons are not entangled.  This appears to contradict what I said, that the neurons are entangled but decoherent.  We're not really saying different things, we're just using different words to communicate the same ideas.)

Trying to explain psychic ability by quantum mechanics is no better than trying to explain it by thermodynamics.  Thermodynamics is all about the random motion of molecules.  Thermodynamics says that there is a miniscule chance that the molecules in my neuron will just happen to move in the same way as molecules in your neuron!  But out of all the ways molecules could move, why should we single out this particular way, except that it makes us feel fuzzy inside?

Some other gross simplifications I've made:

1. To speak of "same" and "opposite" results presumes that there are only two possible configurations.  I'm sure that neurons are complicated enough that they have a very large number of configurations.  If each neuron has three possible configurations, then we have nine total possible combinations.  Because the neurons are entangled, some of these combinations are more likely than others, but all the probabilities themselves are essentially random.

2. There's no reason to think that entanglement should only involve two things.  In general, entanglement involves every possible configuration of everything.  But imagine that we have just three things: two neurons and my aunt's left foot.  And imagine that each of these has two possible configurations.  Now there are 8 possible combinations.  Some are more likely than others, but the probabilities themselves are essentially random.

QM broke logic!

Stand to Reason is some sort of Christian apologetics organization that I don't know much about.  They have this program for high school students where they come up to UC Berkeley and meet godless university students, presumably to "inoculate" them.  They did it last year, and it was all quite civil.  I didn't blog about it probably because nothing piqued my interest.  They're doing it again tonight, but I probably won't blog about it.  In fact I might not go at all, because I'm feeling a little sick.

But I was amused to have a glimpse of their YouTube channel.  For fun...



The main problem with this video is... who the hell claims that quantum mechanics casts doubt on the laws of logic?  I'm not sure if he actually met an atheist who said that, or if he misunderstood something an atheist said, or if it's made up from whole cloth.  The rest of the argument is well-reasoned enough, but he kind of missed the crucial step of citing his opponent.

Also, while it is true that we shouldn't be dazzled by complicated jargon, this is only heuristic reasoning.  Clearly there exist valid arguments which are difficult to understand and involve lots of jargon!  One possible resolution is to ask an expert.  As a physicist, I can tell you that quantum mechanics does not cast doubt on logic, and does nothing of the sort.

Physics and causality

In some previous posts, I discussed the notion of causality.  I talked about how difficult causality is to prove, and how the very concept of causality breaks down in some situations.  For example, if a particular trend has two causes, there is no objective way to define the relative importance of those two causes.

In my humble opinion, the fact that "causality" breaks down in some situations indicates that it is not a fundamental property of the universe.  If causality were fundamental, it would apply in all situations.  You are free to disagree.  I think it's an arguable point.

Some readers may think that this is dissonant with my other views.  I'm a physicist after all.  Isn't the whole of physics based on the idea of cause and effect?  In fact, you might even say I have it all backwards.  On an everyday level, we have to deal with these free-acting humans, who appear to defy cause and effect.  But the reductionists among us know that the free will is an illusion, and that on a fundamental level, it's all cause and effect.  Right?

(Image not original; found all over the 'net.)

But let me tell you what physics really says.  The fundamental laws of physics never mention cause and effect.  It looks a little more like this:

|Ψ> represents the universal state at some point in time.
U(t) represents the time-evolution operator.
U(t)|Ψ> represents the universal state after a length of time t.  t may be positive or negative.

Where do you see causality here?  Let's break it down.

• What if the universe were different?  If instead of |Ψ>, we had |Φ> (that is, if the universe were in a different state), then it is indeed true that after time t, the universe would still be different.  Changing the initial state changes the outcome.  Sounds a little like causality!  So perhaps we could say that |Ψ> causes U(t)|Ψ>.
• The past causes the future.  Not the other way around.  Therefore, if t is positive, we would say |Ψ> causes U(t)|Ψ>, but if t is negative, we would say U(t)|Ψ> causes |Ψ>.
• Is the present caused by all stages of the past?  If we say |Ψ> causes U(1)|Ψ>, should we also say that U(-1)|Ψ> causes U(1)|Ψ>?  Should we say U(-2)|Ψ> also causes U(1)|Ψ>?  We might conclude that any given universal state has an infinite number of causes, corresponding to all the universal states preceding it.
• Are the laws of physics also a cause?  We could also talk about U(t)|Ψ> being caused by the nature of U(t).  That is, we could talk about the state of the universe being caused by the laws of physics which govern how it changes over time.  But this seems to be a distinct sense of causation.  U(t) does not cause U(t)|Ψ> the same way |Ψ> causes U(t)|Ψ>.
• What about objects within the universe?  It is very clunky to restrict causality to the state of the entire universe.  We would have to add more machinery in order to speak of one small part of the universe causing another small part of the universe.  In fact, this is possible.  Because of relativity, no information travels faster than light.  And so the events at one point can only be caused by past events which are a finite distance away.  The further back in the past we look, the further away the cause may be.

The past light cone is a way of illustrating what events at what distance could have caused an event at one point in space and time.  Image from Wikipedia.

• Each event has infinite causes.  I already suggested that the universe at any point in time is caused by all previous stages in the past.  This is also true of each individual event within the universe.  Additionally, at any particular past point in time, there are infinite points in space which might be considered causes.
• Are you caused by faraway stars?  Betelgeuse is a star that is 640 lightyears away, meaning that the state of Betelgeuse 640 years ago may have "caused" my present state.  But I usually see Betelgeuse as having no significance to my life.  And yet, if we are considering all possible states of the universe |Ψ>, there must be one in which the state of Betelgeuse was different 640 years ago, and where this has an effect on my life.  For instance, if 640 years ago, Betelgeuse went supernova, and released a gamma ray burst that killed me, then Betelgeuse would have "caused" my death.  Why do we say a supernova has an effect on me, while an absence of a supernova does not?  Is it because supernovaes which harm life on earth are rare, while absence is common?

The picture is additionally complicated by the fact that there is more than one way to view the same physical laws.  In the Lagrangian formulation of physics, we start with an initial state A and a final state B.  The Lagrangian formulation tells you what path the universe must take to get from A to B.  In this view, causation is an alien concept.  I've discussed this years ago, opining that it is meaningless to ask whether the universe is governed by cause and effect.  If causation appears in one formulation of physics, but not in another, and if each formulation predicts the same results in every experiment... that means a universe with causation and a universe without causation could look exactly the same.

I do not wish to convince the reader that there is no such thing as causality.  I argue that the concept of causality is not contained within physics.  To the degree that physics does contain causality, it does not entirely match our colloquial understanding of "cause".

Of course you won't find many physicists going around saying that there is no such thing as cause and effect.  To a physicist, it is obvious that our everyday lives are quite different from the world of subatomic particles and fundamental physics.  It is clear that baseball exists, even if it is not contained within fundamental physics.  If baseball can exist, why not causality?  For that matter, why not free will?

Other posts in this mini-series:
Colds and Causality
Women and Causality
Responsibility and Causality
Nature/nurture and Causality
Physics and Causality
Math and Causality

New theorem: Quantum states are real

There's a new paper on arxiv called "The quantum state cannot be interpreted statistically".  It has a theorem which proves that, given a few basic assumptions, the quantum state (ie the wavefunction) must be real, rather than a merely statistical object.  Nature has an article which mostly just harps on how "seismic" the paper is. 

Nature (correction: the article's author, not Nature itself) compares its importance to Bell's Theorem, which is a very important result indeed from 1964.  Bell's theorem proved that if there were "hidden variables" underneath the quantum state, then entangled particles must be communicating with each other faster than light.  I've explained Bell's theorem in the past.

I felt the news coverage left a lot of unanswered questions.  What do they even mean by the "statistical interpretation" of quantum mechanics?  Roughly how is it proven?  What is the difference between this and Bell's theorem?  I found the answers in the arxiv print, and will attempt to summarize them.

What does the "statistical interpretation" mean?

Let's say that we have two ways of flipping a coin.  The first method leads to a 50% chance of heads, and a 50% chance of tails.  The second method rigs it so the coin always comes up heads.  Let's say that I flipped a coin by one of these two methods, and showed you the result.  If the coin was heads, then you would not know which method I used.

Now say that I have two ways of preparing an electron.  And suppose that you measured the vertical spin component of the electron.  If I use the first method, there is a 50% chance the electron is spin up, and 50% chance spin down.  If I use the second method, the electron will always be spin up.  If I prepared the electron by one of these two methods, and you found that the electron is spin up, you would not know which method I used.

But electron spin is a little trickier than coin flips, because you can measure the spin component in any direction.  Suppose you had tried to measure the horizontal spin component, would you always be able to tell which method I used then?  The answer is no.  But perhaps there is yet another way to measure it?

The authors equate the "statistical interpretation" with the following: Given any two distinct ways to prepare a quantum state, there is a nonzero probability that the result is consistent with either method of preparation.  In other words, no matter what kind of measurement we make, there is a chance that we'll get an outcome that doesn't tell us anything.

What's the difference between this theorem and Bell's Theorem?

Bell's theorem requires that you take many measurements and compile statistics of these measurements.  Once you are confident enough in your statistics, you can show that the probabilities are incompatible with the "hidden variable" view of quantum mechanics.

This new theorem requires only one measurement.  One measurement, and you're done.  (Of course, if you have a noisy experiment, you may need to repeat it to build confidence in your result.)

Of course, the new theorem and Bell's theorem also have a slightly different set of assumptions, and slightly different conclusions.  But I think the primary difference is that the new theorem requires one measurement, while Bell's theorem requires compiling statistics.

Roughly how is it proven?

As an example, let's take the two methods of preparing an electron that I described above.  It turns out that no matter what measurement I make, there is a chance of an outcome that is consistent with either method A or method B.

But we can be tricky.  Let's duplicate the machine that prepares the electrons, and assume that these machines are independent of each other.  Now there are four methods of preparation:

1. A and A (ie both machines use method A)
2. A and B
3. B and A
4. B and B

Suppose that there is a chance that the first machine will produce an electron that is consistent with either method A or method B.  There is also a chance that the second machine will produce an electron that is consistent with either method A or method B.  Therefore, there is a chance that both machines produce electrons which are consistent with any of the four methods.

But it turns out that there is a measurement we can make with four possible outcomes. And each outcome is inconsistent with one of the methods.

• Outcome 1: inconsistent with method 1
• Outcome 2: inconsistent with method 2
• Outcome 3: inconsistent with method 3
• Outcome 4: inconsistent with method 4

What is this special measurement?  It's not straightforward.  In quantum mechanics, we can measure things like position, momentum, and spin.  But we can also measure things like helicity, which tells you whether the spin and momentum are in the same direction, without telling you what direction that is.  Similarly, we can measure whether the electrons have spin in the same direction or opposite directions.  The measurement described in the paper is sort of like that, but more complicated.

The same theorem can be generalized to any two methods of preparing a quantum state.  Suppose that one method always produces a spin up electron, and the other produces a spin up electron 99% of the time.  All you have to do is have N duplicates of the electron-producing machine (in this case, N=15 suffices), and take a special measurement.  No matter the outcome of this measurement is, it is inconsistent with one of the 2^N possible methods of preparation.

The conclusion is that any two distinct quantum states are not just "probably" different, but always different.  You just need a tricky measurement to show it.

Is this paper as groundbreaking as Nature claims?

I don't know.

God as quantum observer

Today we will look at the intersection of quantum nonsense and terrible arguments for God.  How fun!

This argument is based on something Michio Kaku said to Deepak Chopra in an interview. It wasn't very clearly stated, so I'm filling in the blanks.

1. Quantum mechanics says that things do not exist unless there is an observer to observe them.
2. The universe exists.
3. There must be an omniscient conscious being to observe the universe.
4. That being is God.

 As with most arguments for God, the last step is troublesome, because there are all sorts of things the being could be.  There's no compelling reason to think that the object in question can appropriately be called "God", much less the Judeo-Christian god.  For example, why should we think that it's a single object observing the entire universe rather than multiple objects?  But since Deepak Chopra is not part of any western religion, he would skip step 4, thus avoiding the problem.  Unfortunately for him, the rest of the argument is crap too.

Rebutting the rest of the argument is really an exercise in applying the advice in my presentation on Quantum Mechanics for Skeptics.

Error #1: Elementary misunderstanding of what Quantum Mechanics actually says

According to step 1, things don't exist until we observe them.  This is incorrect.   Observation does not cause things to come into existence, it just causes a mixed state to become a definite state.  For example, if we have an electron in two places at once, and we observe its position, then it "collapses" into a single position.  The electron existed before you observed it.  The mixed state existed before you observed it.

Getting a little more advanced: Even after you observe an object, in some sense it's still in a mixed state.  This is because all states are mixed states.  So even if you mistakenly believed that "mixed state" = "does not exist", then you would be forced to conclude that nothing exists with or without observers.

Error #2: Misunderstanding observers

Step 1 just talks about observers, while step 3 suddenly jumps to conscious observers.  Observers do not need to be conscious.  The chair I'm sitting on functions as an observer.  A piece of white paper functions as an observer.  Anything with enough atoms to be visible to the naked eye probably functions as an observer.  When you realize the banality of observers, it makes much more sense to conclude that the universe is full of non-conscious observers, none of which need be omniscient.

Error #3: Misunderstanding quantum interpretations

This whole time we've been talking about observers.  But observers are specific to one interpretation of quantum mechanics, the Copenhagen interpretation.  Under many other interpretations, there is no such thing as an observer.

The thing is with quantum interpretations, they're all or nothing.  All interpretations make the same predictions.  Therefore, all of them predict God, or none of them do.  If you've formulated an argument that only works for the Copenhagen interpretation, but not for others, then this is a sign that you made an error somewhere.  I already showed errors in the argument, so the problem is resolved.

A summary:
#1: The universe does not need to be observed to exist.
#2: There are probably non-conscious observers all over the universe.
#3: A valid argument would make sense under all quantum interpretations, but this one does not.

I've also heard an argument from quantum mechanics that God does not exist.  It is also a crap argument.  I will cover it next time.

Michio Kaku and Deepak Chopra

Some people have asked me what I think about Michio Kaku, the string theorist who popularizes physics and futurism.  His last book, Physics of the Impossible: A Scientific Exploration Into the World of Phasers, Force Fields, Teleportation, and Time Travel, discusses three classes of "impossible" technologies.  Class I impossibilities may become possible within a century or two.  Class II impossibilities may become possible in thousands or millions of years.  Class III impossibilities will never become possible unless there is some fundamental shift in our understanding of physics.

I think Michio Kaku and I have very different philosophies about popularizing science.  Kaku likes to reach for the amazing and sensational potentials of the world.  I like to bring things down to earth, grounding them in simple concepts.  Or something like that.  Clearly there are differences even if I cannot articulate them.

But that doesn't mean we have to clash!  I would probably agree with Michio Kaku on most of his futurist predictions, at least when properly translated.  Allow XKCD to do the translating.

Michio Kaku says that class I impossibilities may come in a century or so.  The translation is "It has not been conclusively proven impossible."  I totally agree!  See, no clashing.

But recently, Uncertain Principles pointed out an interview of Michio Kaku by Deepak Chopra.  If there is a time to clash, now is it.  Let me pull out a bunch of the worst quotes.

DC [Deepak Chopra]: What the basis for your book is, that if it does not violate the laws of mathematics or physics then it is in the realm of possibility, really?
MK [Michio Kaku]: That's right. If it's not forbidden by the laws of physics, it's mandatory.

This is an example of type 2 technobabble.  It's true that there is a principle in physics that states, "If it's not forbidden, then it's mandatory," but this is really not the right context for it.  The correct context is particle physics, because all possible particle interactions will mathematically contribute to the result.  Of course, some interactions contribute more than others, and most interactions are just impossible.  But don't let that get you down on life, because this only relates to the context of particle physics.

MK: Right. Think about this: if you were to push a button and the force field has knowledge of how to construct walls and floors and sidewalks, with a push of a button you could create an entire city.

If I had an infinite lever and an immovable place to stand on, then we could move the world around.  It's all a matter of getting the technical details sorted out. (That's a Terry Pratchett reference btw.)

DC: Is our conversation affecting something in another galaxy right now?
MK: In principle. What we're talking about right is affecting another galaxy far, far beyond the Milky Way Galaxy. Now when the Big Bang took place we think that most of the matter probably was vibrating in unison.

I guess if we take a strict Many Worlds Interpretation, I suppose it is true "in principle" that things here are correlated with things in other galaxies.  But this is very misleading.  Really, it will be a mix of correlations, anti-correlations, and everything in between, which is to say that on average there is no correlation at all.  This is the very important principle of quantum decoherence.

MK: We actually demonstrated it right on TV cameras. We went to the University of Maryland outside Baltimore and we showed an atom being teleported right across the room. You can actually see two chambers, an atom in one being zapped across the room.

Chad explained why this is misleading over at Uncertain Principles.  One major error is that there is already an atom in each chamber.  Quantum teleportation only transfers a quantum state from one atom to another, not the atoms themselves.  Also, the atoms were very carefully prepared by experimenters, not by the Big Bang.

This post is longer than I expected, so I should insert another visual.

DC: Every cell is instantly correlated with every cell. A human body can think thoughts, play a piano, kill germs, remove toxins, make a baby all at once.
...
To my mind the human body is an example or for that matter a leaf for any biological, of quantum entanglement. Everything is correlated with everything instantly. What would you say to that?

MK: Yes, things are entangled so in some sense messages can travel faster than light instantaneously, however the messages that go faster than light are random messages. You can't send Morse code or information through these things and sometimes we de-cohere from matter so that we can no longer communicate with other forms of matter.

This time, it's Deepak Chopra who is saying something deeply silly, and Kaku just lets most of it go.

Chopra is proposing that correlations between different parts of our body is caused by quantum entanglement.  I'm no expert, but I thought it was caused by electrical signals moving along neurons.  It certainly is not instant, it's just instant for all practical purposes.  It is not faster than light, so no quantum entanglement is necessary.  And as Kaku points out, if it were faster than light, then only random messages would be communicated.

DC: But let's come back to a biological system. That's not random, that's very coherent, you know this biological system or a system like say when you have morphogenesis and differentiation, when a cell divides, keeps dividing so that you know in first year applications it has become the hundred trillion cells which is more than all the stars in the Milky Way Galaxy. That requires some kind of non-local correlation to my mind, theoretically.
MK: Well these non-local correlations are going to be extremely important in the next few decades coming from the computer realm of things.

Auuggh, no!  Michio Kaku just conceded that the biological correlations are non-local (ie faster than light).  They aren't!  That's why the messages can be non-random, as Chopra observes.

DC: To me a rose is rainbows and sunshine, earth, water, and wind, air, and the infinite void and the Big Bang all rolled into one.
MK: Mmm hmm. And Einstein was wrong in this one. We measured this every day in the laboratory. That electrons can dance in between multiple states and then the question is why can't I dance between multiple states?

If someone says, "a rose is rainbows and sunshine," the correct answer is "No it isn't," no matter what kind of dance the electrons are doing.

Michio Kaku also said something very silly about how quantum mechanics might prove the existence of an omniscient being, but I'll leave that one for another post.  (ETA: It is done)

I agree with Chad, someone ought to be ashamed for this interview.

Quantum Mechanics for skeptics

I've made a PowerPoint presentation called "Quantum Mechanics for the skeptic: How to recognize quantum nonsense without being a physicist".  The title pretty much explains the target audience and purpose.

The presentation features appearances by: Professor Farnsworth, Professor X, the Great A’Tuin, Richard Feynman, and Deepak Chopra.

A lot of the source material for the presentation comes from my review of What the Bleep!?

Holes and antiparticles

In the Theory of Relativity, there is a famous equation involving energy.

E = mc²

No, not that one!  That equation doesn't include kinetic energy.  Here's a more complete equation.

E² = (mc²)² + (pc)²

Okay, but if you don't know what all those letters mean, it's just a bunch of math gibberish.  For now, there is only one part of the equation which is important, the E².  E is, of course, the energy.  But because we squared the energy, it could be negative or positive.  Just looking at the math, both the positive and negative energy solutions make just as much sense.

It may make mathematical sense, but is it real?

Let's go back to the early 1900s.  Einstein published Special Relativity Theory in 1905 and General Relativity Theory in 1915.  Quantum Mechanics became established in the mid-1920s.  If you read popular physics literature, you already know that there is a bit of a conflict between Relativity theory and quantum theory.  String Theory, for instance, is an attempt to combine the two.  But you have to be careful.  The conflict is between quantum theory and General Relativity.  Quantum theory and Special Relativity have already been combined.

One of the big steps towards combining the two occurred in 1928.  Paul Dirac formulated the Dirac Equation, which is basically a quantum version of the above equation.  As it happens, the Dirac Equation has the same issue: the energy can be positive or negative.  This issue was hard to ignore, because the theory predicted that particles would jump between the positive and negative energies.

So in 1929, Dirac proposed a solution called the Dirac sea.  The negative energies are real, but they're already filled up with a "sea" of particles.  As I explained before, it's not possible for two fermions to occupy the same state.  And in quantum theory, the possible energy levels are discrete, meaning you can count them one by one.  If all the negative energy levels are filled, then it's impossible for any more electrons to fall into the negative energy levels.

Each horizontal black line represents an energy level.  The higher the line, the higher the energy.  The red circles represent electrons which fill the levels.

What happens when you give the system a good kick, and one of those electrons in the Dirac sea jumps up to a positive energy?

We got two things out of that kick.  First, we got a brand new particle from nothing.  Second, we got a "hole", an empty energy level which would normally be filled.  The hole is not a particle per se, but a lack of a particle.  It has the opposite charge, opposite spin, opposite everything of a normal particle.  It even has the opposite energy, but since it's in a negative energy level, the opposite of a negative energy is a positive energy.

We can basically treat the hole as a new kind of particle, called an antiparticle.  If a particle is a fermion, then it must have a corresponding antiparticle.  The antiparticle has the opposite charge but the same energy.  The antiparticle of an electron is called a positron.

Antiparticles are awesome because you can create them with nothing but a kick of energy.  This is called pair production.  Also, if a particle and antiparticle meet, they can disappear, leaving energy in their place.  The technical term for this is annihilation.

Motivated by Dirac's new idea, the positron was discovered in 1932 by Carl D. Anderson, earning him the Nobel Prize in 1936.  Incidentally, the positron had been observed in 1923, but no one really knew what it was back then.

Positrons have real applications too.  The P in PET scan stands for positron.  First you inject a positron-emitting tracer into the subject, and then a device detects the positrons.  PET scans can do all sorts of useful things, like find tumors and image brains.  But, hey, don't give physicists all the credit, I'm sure engineers and doctors deserve some too.

Positrons are also a vital component of the positronic brains of sentient androids.

Clearly, antiparticles have many present and future applications.

Going back to our first question, is it real, or is it just math?  Are antiparticles really just holes in the Dirac sea?  Perhaps, but there are problems with the theory.  In most systems, there would be infinitely many energy states, going up higher and higher.  Therefore, there are also infinite negative energy states, going lower and lower.  Are we willing to accept that there is an infinite sea of particles filling those states all the way down?

And if we're talking about electrons (or any other charged particle), wouldn't that mean that a pure vacuum has an infinite charge?  Antiparticles clearly exist, but some of the other implications of the theory are just weird.  So how can the Dirac sea be real?

The answer, I'm afraid is not that satisfying.  Modern quantum theory interprets antiparticles as real particles all to themselves, rather than holes.  The theory is mathematically equivalent to the Dirac sea, and makes the same predictions.  But there isn't any sea anymore, as pretty as it was to look at.  Unfortunately, the goal of a physicist isn't always to make things pretty...

Identical particles in Quantum Mechanics

It's been a while since I last discussed Quantum Mechanics. Let's do some more!

One of the aspects of Quantum Mechanics is the existence of identical particles. Suppose I'm holding a photon in my right hand, and a photon in my left hand. I can't physically hold a photon in my hand, of course, but consider it as an abstract example that I can illustrate with pretty pictures.


These two photons are identical. They are exactly alike in every respect. And I do mean every respect. Furthermore, this has profound physical consequences. By profound, I mean that you wouldn't be here if it weren't for identical particles.

Now, I could start talking about the mystical oneness and sameness discovered by scientists in the 20th century, but that would be quite meaningless. There is in fact some meaning I intend to convey here. I say that the photons are "identical" because if you switch photon A and photon B, nothing will change.


But as shown above, it sure seems like something does change when we switch the photons. Namely, before the switch, photon A is in my left hand, and photon B is in my right hand. After the switch, photon A is in my right hand, and photon B is in my left hand. Arguably, this is a very similar situation, but I said that the particles are identical in every respect, identical in a deep quantum mechanical sense.

Therefore, it is not the case that photon A is in my left hand while photon B is in my right hand. Instead, we have a mixed quantum state, with both photons in both hands. Being in a mixed state means adding or subtracting the wavefunctions of two or more states.


Recall that addition is commutative (ie x+y = y+x). And that's how switching the two photons leaves us with the exact same quantum state that we started with.

Okay, so that's nice, you say, but what physical consequence does that have? In this case, there is no consequence whatsoever, because the photons in my two hands are too far apart. But imagine that the photons were very close together, so that their wavefunctions were overlapping. Then we'd have some constructive interference in the overlap.

More impressively, some kinds of particles will have destructive interference. Let's say that instead of photons, we had electrons in our two hands. The two electrons are identical in every respect. But unlike photons, if we switch two electrons, one thing does change. That is, after switching, the wavefunction will be the negative of what it was before. Therefore, the quantum state of the two electrons will have subtraction instead of addition.


Aside: I am representing photons with wavy things, and electrons with fuzzy circles, but the intelligent reader will realize that these are just artistic representations with little basis in reality.

Now imagine that the two electrons weren't in two different hands, but were both in the exact same spot, with the exact same momentum and spin and everything. Then the wavefunction would be something like this:


But I just took a wavefunction and subtracted it from itself. That's just zero! A wavefunction of zero is not possible. Therefore, it is not possible for the two electrons to occupy the same state.

Consequences? To start, this is why atoms have electrons in a complex orbital structure. If the electrons weren't identical, then they would all fall to the lowest orbital around the nucleus. Chemistry would be very different, and you can just forget about biology. But because they're identical, electrons cannot all occupy into the same state. Instead, electrons are forced into higher energy states around the atom, with the highest energy electrons doing all the chemical reactions.

A more astronomical consequence: White dwarfs. White dwarfs are small stars that are held up by so-called electron degeneracy pressure. The electrons can't compress into a smaller volume, because that would require them to fill up the higher energy states. The gravitational force is not enough to supply this energy. However, if the star accumulates enough mass, perhaps from a nearby star, it reaches the Chandrasekhar limit, where the gravitational force is strong enough to supply the extra energy. Upon reaching the Chandrasekhar limit, the white dwarf will collapse, causing a supernova.

Neutron stars are very similar, except they're held up by neutron degeneracy pressure.

Neutrons, like electrons, will interfere destructively with each other. Particles which interfere destructively (like electrons) are called fermions. Particles which interfere constructively (like photons) are called bosons. Fermions include electrons, neutrons and protons. Bosons include photons, gravitons, and certain atomic nuclei, like Helium-4.

Identical particles are all around us. They exhibit weird and important properties that have no analogue in classical mechanics. This is why quantum mechanics matters.