Undivided Looking

Strictly speaking, most of the other rules about QM are already implicit in what I've already said.  But a few implications of this setup are worth pointing out.

First note that, in QM, the "state" includes information about every single object in the system.  So, when you add up the different histories, they only interfere if the final states are exactly the same in every respect.  If even one tiny particle is in a different place than it otherwise would be, then they don't interfere.  In that case, you just add up the probabilities normally.

This is why measurement is such a significant thing in QM.  If you try to catch out Nature by explicitly measuring which slit the particle went through, then YOU are now different as a result of you knowing which slit it went through.  As a result, the two histories don't interfere.  But it needn't be a person which does the "measurement".  Even if you refuse to look at it, the detector being different still prevents the interference from happening.  As far as we know experimentally, there is no special relationship between consciousness and QM (although some people have proposed interpretations of QM in which there is a connection between the two.).

Usually, once histories become sufficiently different from each other, for a long enough period of time, their random interactions with the environment will tend to be different, so that the chances of getting everything perfectly the same become tiny, and the histories won't interfere anymore.  This phenomenon is called decoherence.  People argue about what this tells us about the interpretation of QM, but the phenomenon itself can be studied in the laboratory, so my use of this word should not be regarded as an endorsement of any particular interpretation.

Secondly, if you have two or more distinct states, then it's possible to take a quantum superposition of the two states, formed by adding them up with complex coefficients.  For example, if X and Y are two distinct states, then

$$(\mathbf{X} + \mathbf{Y}) / \sqrt{2}$$

or

$$(\mathbf{X} - \mathbf{Y}) / \sqrt{2}$$

or

$$(2\mathbf{X} +i \mathbf{Y}) / \sqrt{5}$$

are all equally valid states!  (The reason for the square root in the denominator, is to make it so that, by the Born Rule, the total probability of the state is still 1.)  These states are just as much valid states as X or Y themselves would be.

The possibility of quantum superpositions is implicit in the quantum probability rules, since if you start with a particular state A, in general it will evolve to a superposition of different states as time passes.  And there's no particularly good reason you couldn't also have started out the experiment with a quantum superposition.

(Note that if we take any state like $(\mathbf{X} + \mathbf{Y}) / \sqrt{2}$, and we multiply it by a phase (a number on the unit circle of complex numbers, e.g. $i$, or $-1$, or $(1+i)/\sqrt{2}$) then we can't tell the difference between that and the original state in any way!  That's because, when we work out the patterns of interference, we only care about the relative phases between different histories, not the absolute phase of the whole system.  So it's good to remember that there is a slight redundancy in our description here: two states that differ by a phase are really the same state.)

Now if we have a system with N possible states, then we can imagine a higher dimensional geometry consisting of all possible superpositions of these N possible states (including, for mathematical convenience, those for which the probability doesn't add to 1).  This is called the Hilbert Space of that system.  It is a kind of vector space with N complex dimensions, which means in terms of real numbers it's a 2N-dimensional space.  But don't worry about these details for the moment.

(It's kind of hard to visualize a Hilbert space when N is greater than about 2, but it's still very useful mathematically!)

The simplest nontrivial Hilbert Space is the one with N = 2 states.  (I'll give a physical example in a moment.)  This would normally involve a 4-dimensional space, but to keep things as simple as possible, I give you permission to ignore the bit about complex numbers and just think about a 2-dimensional plane.  (This is the space of all states of the form

$$a\mathbf{X} + b\mathbf{Y}$$

where $a$ and $b$ are now real numbers.)  Then we can think of X as a unit vector pointing along the x-axis, and Y as a unit vector pointing along the (wait for it...) y-axis.

Perhaps a picture will help:

The Hilbert space for a system with 2 states.

As you can see, the Hilbert space has an origin, which is the point in the middle which represents "zero".  Each state is a represented by a vector coming out of the origin, pointing in some direction.  (But remember that $-X$ is really the same state as $+X$, since they differ by a -1 phase.  I didn't draw $-X$ on the picture, but if I had it would be 180º around from $X$.)  The Born Rule tells us that length = total probability squared.  That means that in order for a vector to be a state-in-good-standing, it needs to be length 1.  (In other words, by the Pythagorean Theorem, the sum of the squares of its $(x,y)$ coordinates needs to add up to 1).  So don't ask me what the physical meaning of the "zero" vector is, since it doesn't have one.

A physical example of an $N = 2$ state system would be the polarization of a photon coming straight at you from your computer screen.  Light can be either horizontally polarized (the X state, corresponding to an electric field that points in the $x$ direction) or it can be vertically polarized (the Y state, corresponding to an electric field that points in the $y$ direction).  Now since physics is rotationally symmetric, it's obvious that if light can be horizontal or vertical, it can also be diagonal.  So you might have naïvely thought the photon would have infinitely many possible states.  And in a sense this is true, but each of these diagonal states is really just a quantum superposition of the X and Y states.

Yet on a plane, the choice of axes is arbitrary.  You can rotate the coordinate system by 45º, and it would be just as good as the original coordinate axis.  In the same way, we are currently thinking of X and Y as the two possible states of the system (with every other state being a superposition of X and Y)—but this is an arbitrary choice!  We could just as well say that every state is a superposition of $(\mathbf{X} + \mathbf{Y}) / \sqrt{2}$ and $(\mathbf{Y} - \mathbf{X}) / \sqrt{2}$!  So actually every state is a quantum superposition, of certain other states.

Although the choice of coordinate axis is arbitrary, it is important that the states you pick are all "orthogonal" to each other (i.e. at right angles in the Hilbert space).  That is what tells you that it represents a set of  mutually exclusive possibilities.  Any such set of N orthogonal states is called a basis of the Hilbert space.  (The plural of "basis" is "bases", pronounced BASE-EES.  Just like the plural of "index" is "indices".)  A basis gives the possible set of outcomes for some particular way to measure the system.

For example, suppose we start with a diagonal photon in the $(\mathbf{X} + \mathbf{Y}) / \sqrt{2}$ state, and we measure it to see whether it is horizontally or vertically polarized.  (Maybe by passing it through some kind of material in which these two polarizations follow different trajectories.)  What happens?

Well, people disagree about interpretation (what is ultimately going on), but everyone agrees on the practical set of rules you'd use in the laboratory.  We just look at the state $(\mathbf{X} + \mathbf{Y}) / \sqrt{2}$.  It has an amplitude of $1/\sqrt{2}$ to be $X$, and also $1/\sqrt{2}$ to be $Y$.  By the Born Rule, we've got to square these numbers, so we get a 1/2 chance for it to be horizontal, and a 1/2 chance for it to be vertical.

Let's suppose it turns out to be vertical (the Y state).  Then from now on, the particle behaves just as if it had been in the Y state all along.  (This is called "projection" or sometimes "collapse of the wavefunction"; but see my remarks on decoherence earlier in this post.)  For example, if we measure it a second time to see if it is in the Y state.  If we check to see whether it is in the X state, it is definitely not.

But now we can ask a separate question: is it in the $(\mathbf{X} + \mathbf{Y}) / \sqrt{2}$ state, or the $(\mathbf{Y} - \mathbf{X}) / \sqrt{2}$ state?  This corresponds to sending it through a different kind of filter, which discriminates between the two 45° diagonal polarization choices.  We would then find a 1/2 chance of it being the former, and a 1/2 chance of it being the latter.

Supposing it turns out to be $(\mathbf{Y} - \mathbf{X}) / \sqrt{2}$, this is a bit paradoxical.  Since if we had just started off asking whether the $(\mathbf{X} + \mathbf{Y}) / \sqrt{2}$ photon was in the $(\mathbf{Y} - \mathbf{X}) / \sqrt{2}$ state, Nature's answer would have been "Nope.  Definitely not.  Those states are orthogonal and therefore if it's the one, it's not the other!"

But somehow, merely by answering a series of questions about the photon's polarization, we managed to trick Nature into converting the photon from its original polarization to one 90° away, which is inconsistent with the first.  By measuring the photon we have affected it!

So we see that, somehow, we can get the photon to be definitely - or | polarized, or definitely / or \ polarized.  But we can't get both of these things to be definite simultaneously.  This is an uncertainty relationship.  It's analogous to the "Heisenberg uncertainty principle" where you can't measure position and momentum at the same time; so that measuring one makes the other uncertain.  (Although it's not exactly the same, since position and momentum are continuous variables, while each polarization choice is a yes-no question.)

In the case we are considering, we're been lucky that the Hilbert space is directly related to two dimensions of the physical space.  That means that the rotation of axes in the Hilbert space is the same thing as a rotation of physical space.  In general, however, we are not so lucky and the Hilbert space is more abstract.  But it is still true that there are a bunch of different possible bases of the Hilbert space, that are related by rotations in the Hilbert space.  (Since the Hilbert space is complex, we are really only interested in those rotations that don't mess with the notion of "multiplying-by-$i$".  These are called unitary transformations.)

As long as I'm talking about complex numbers, I should mention that there's also such a thing as circularly polarized photons, which involve complex superpositions like $(\mathbf{X} \pm i \mathbf{Y}) / \sqrt{2}$.  But most of the bizarreness of superpositions can be illustrated without thinking about complex numbers.

Continue to the final post

A bunch of people have been asking me about the interpretation of QM.  Now, every interpretation of QM predicts (or claims to predict) the same experimental results in any experiment (or at least, any realistically feasible experiment).  Otherwise they wouldn't be rival interpretations, they would be rival theories, and we would just do an experiment to see who is right.

So before discussing what QM actually means, it's good to get the ground rules down—the ones that all physicists agree are the right ones to use to predict the results of actual experiments.

Let's suppose we're doing a physics experiment, which I am going to describe in an extremely abstract way, because that's the kind of person I am.  A (somewhat idealized) way to describe a certain class of experiments is as follows:  We start out by preparing the initial configuration of the apparatus to be in some particular configuration (or "state"), let's call it A.  For example, we start with a radioactive atom.  We can let the experimental apparatus evolve on its own, isolated from the rest of the world, until it reaches some final configuration.  Then we look inside, e.g. 10 minutes later, and check what the current state of the experiment is.  Perhaps the atom has now decayed into something else.  Let's call this final configuration B.

Several aspects of this description are clearly idealizations.  There are always some limitations in our ability to control and/or know the initial condition A; the system is never going to be completely isolated from the rest of the world no matter how hard I try.  And in some experimental setups this may be a good thing—that is, we may want to deliberately reach in to measure and/or adjust the system, part way through its "time evolution".  (Unlike biologists, we physicists use the word evolution any time anything changes!)   And, at the end of the process, we're never going to be able to measure the final outcome with perfect precision either.

But I'm a theorist so I can ignore the messiness of real life, whenever it pleases me to do so.

Now if the laws of physics were deterministic (and if we know what they are, and we know the initial state completely precisely...) then in principle we could simply solve all of the relevant equations and find out what exactly the final state would be.   So after 10 minutes, any given A will become some particular B with probability 1.

(In practice, this calculation is often impossible because of phenomena like chaos where (in some systems, not others) the final outcome depends very, very sensitively on the initial conditions.  For chaotic systems, you need to know the initial conditions to exponential precision in order to predict the future.   This is why we can't predict the weather accurately for more than about a few days out, because the number of digits accuracy you'd need to measure things to is proportional to the number of days!)

But the actual laws of physics are stranger than that.  They are not deterministic.  I think I've read that some Philosophers of Science claimed that Determinism was an important assumption underlying the possibility of doing Science at all.  Well, Determinism is false, and yet we scientists still have jobs.  I know, 20-20 hindsight, but it was still a dumb thing to say (if anyone in fact ever said it, which I haven't done the research to confirm...).

So let's try again.  Once again, we'll set up our experiment in a particular initial state A.  But now, there are several possible final states B, B', B'' etc.  Let's suppose we want to calculate the probability for some specific one: B.  So the sane, sensible way of doing this, would be to think of all the different ways that A could evolve in time to become B.  To actually do calculations, you apply the rules of probability theory:

NORMAL PROBABILITY THEORY:

• For any particular process by which A can evolve to B (a history), we survey all the events which happened in that history, and calculate the probability of each individual event (using our knowledge of the laws of physics, as worked out from experiment or theory)  Then we multiply those probabilities to calculate the probability of that particular history.
  .
• If there is more than one distinct history going from A to B, then we add up the probabilities of each history (since each of them are separate possible ways to get B), to get the total probability of observing B.
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• At the end of the day, the probabilities for all possible final outcomes should add up to 1.

Note that, since probabilities are between 0 and 1, multiplying them makes them smaller, as befits situations where multiple unlikely things need to happen to get from A to B.  On the other hand, adding them makes them bigger, as makes sense if there's more than one way for something to happen.

This is the sort of probability theory which would make sense a priori to our rational minds.  The kind from which you can prove sensible results like Bayes' theorem.  But the universe doesn't really work that way either!

One way to think about QM is that it's like a Behind the Looking Glass version of probability theory, where things almost work like how you expect them to, but not quite.  The basic weird idea of quantum mechanics that instead of assigning each path from A to B a probability (which is a real number between 0 and 1) you assign to each path an amplitude (which is a complex number whose absolute value is less than or equal to 1).

A complex number can be thought of as just a vector lying in a two-dimensional plane.  In order to specify it, you need to know how long it is (the "absolute value" of the complex number) and what direction it points in (the "phase" of the complex number).  Of course, if the absolute value is zero, then the phase is meaningless, since the complex number is just 0.

In QM, the absolute value squared of an amplitude represents the probability for an event to happen.  This is called the Born rule, and it is the necessary interface for getting actual predictions about the world out of the theory.

So let's suppose you have two different possible ways to go from A to B.  (A classic example is the double slit experiment, where a particle passes through a screen which has two holes in it, and then reaches one of several possible locations on the detector.)

If the two possible histories have the same phase, then they constructively interfere and the probability of B happening is more than you would expect, from adding up the probabilities of the two histories.  On the other hand, if the two possible histories have opposite phases, then they destructively interfere, and the final probability is less than you would expect.  In fact, if the amplitudes are equal and opposite, then the total probability of getting to B is exactly 0!

(More generally, amplitudes constructively interfere if they are at an acute angle in the complex plane, and destructively interfere if they are at an obtuse angle.  For right angles, the Pythagorean Theorem + the Born Rule tells you that you get the naive expected answer from just adding up the probabilities.)

So, to summarize, instead of doing the thing that makes sense, you do this instead:

QUANTUM PROBABILITY THEORY:

• For any particular process by which A can evolve to B (a history), we survey all the events which happened in that history, and calculate the amplitude for each individual event (using our knowledge of the laws of physics, as worked out from experiment or theory.)  Then we multiply those amplitudes to calculate the probability of that particular history.
  .
• If there is more than one history going from A to B, then we add up the amplitudes of each history (since each of them are separate possible ways to get B), to get the total probability of ending up at B.
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• The probability of observing B is given by taking the absolute value squared of the total amplitude.  Unlike amplitudes, this is always a real number between 0 and 1.  Also, the laws of physics are chosen so that, at the end of the day, the probabilities of all possible final outcomes still add up to 1.  (This requirement is called unitarity).  QM may be weird, but it's not that weird.

(You may wish to go back and compare this, point by point, with the Not-Batshit-Crazy-Probability-Theory earlier in the post.)

So, if you have a system with N different initial states (and therefore N possible final states), you can specify the time evolution over any given time $t$ by writing all of the possible transition amplitudes from each possible initial state A, A', A''... to B, B', B''... in an N x N matrix $U(t)$, with complex numbers in each slot.  If you know about the math of matrices, this matrix is required to be unitary: $UU^\dagger = U^\dagger U = I$.  That's what enforces unitarity, the rule that probabilities add to 1 no matter which state you start with.

Now, if you wanted to know which specific states are allowed, or which specific unitary matrix to use, then you need to specify a particular quantum mechanical theory, e.g. a harmonic oscillator, or Quantum Electrodynamics., or the Standard Model.  QM is a framework for constructing theories, not a specific theory.  Just like Newton's Law $F = ma$, or the rules of classical physics, are a general framework; only experiments can tell you which particular forces actually exist in Nature.

In the next post of the series, I'll spell out some of the implications of this framework, and then maybe I'll be in a position to talk about interpretation.