It makes sense to ask what is the distance between two continuous probability distributions. It's given by:
\int | p1(x) - p2(x) | dx
L^1 (the space of all functions for which \int |f(x)| dx < \infty) is a weird space, and does not admit concepts like "what is the angle between two vectors".
Quantum mechanics changes this to:
\int |p1(x) - p2(x)|^2 dx
Functions like this are called L^2. Once you put the square in, you can immediately derive a lot of geometry, inner products, angles between vectors, etc.
So I'd argue that QM is probability in L^2.
Far from it. As early as the first subsection, "A Less Than 0% Chance", Scott writes:
"Now, what happens if you try to come up with a theory that's like probability theory, but based on the 2-norm instead of the 1-norm? I'm going to try to convince you that quantum mechanics is what inevitably results."
So discussing that distinction is the entire point of the lecture, and not some minor point "discussed a bit towards the end".
But I don't think that's even the point. The whole article is a rationalization of basic rules of quantum mechanics; when I look back at the history of physics, rationalization of known --at the time-- physical laws (which are often replaced by "better laws" after a while when get a better we understanding of nature) is most often counter-productive.
This is a rationalization but it's useful. We have a physical theory in L^2, and we know that you can't have such a theory in L^2.1 or L^7. Our theories might be totally wrong. But we are confident they can't be only a little wrong.
The concern of "is this thing square-integrable" arises only in infinite dimensional spaces. Inner product of all $\mathbb C^N$ where N in finite is finite. There is no point in saying "L^2" when talking about spins etc. (forcefully casting a finite sum of finite terms to integration and then introducing the concept of "square integrable" isn't helpful, if not trivial).
> This is a rationalization but it's useful.
You can do that kind of rationalization when you "know" the answer ahead. It has zero prediction power.
Newton had a rationalization about the independent nature of time. Descartes also had a rationalization about how action-at-a-distance works for gravity. Aether was also a popular rationalization once.
None of those rationalizations held any value at the end.
Such "rationalizations" are better cut by Occam's razor and left for philosophers.
At least, I prefer not waste my time with such stuff as a physicist.
Your mention of Occam's razor is key. The point of this rationalization is that it shows quantum mechanics (or at least a subset of the theory) is mathematically simple, thus serving a threefold purpose:
1) Due to Occam's razor, a simpler theory is more likely to be correct than a more complicated theory with the same explanatory power. If quantum mechanics can be expressed more simply, then the estimated probability of its correctness should be increased (by only a slight amount).
2) Quantum mechanics, explained this way, is interesting from a purely mathematical standpoint. Even if we knew quantum mechanics doesn't describe reality, mathematicians (and theoretical computer scientists like Aaronson) would perhaps still investigate it. Of course, this may not interest physicists.
3) The article presents a novel way to teach quantum mechanics. Mathematical simplicity can (to some extent) replace intuition as a way for learners to grasp the theory. As Aaronson remarks, quantum mechanics is often taught by following the historical order in which the ideas discovered. Starting from the "conceptual core" (if Aaronson is correct about the conceptual core) is arguably a superior pedagogical technique.
"Simpler" theories are not ab initio more likely correct. The world is full of "simpler" theories that are wrong: the four elements, the caloric theory of heat, Newtonian dynamics and gravity, and so on.
"Simpler" is a purely human notion, and a heavily culturally laden one at that. For this reason, Occam's razor is best left in the dust-heap of philosophy. It is never useful in doing actual science, except now and then by accident.
As for the notion of simplicity itself, we have Kolmogorov complexity and Solomonoff induction, they capture the essence pretty well (though I heard that with some caveats).
The problem with all your examples is that Occams razor says "choose the simplest theory that works", not the simplest of all possible theories.
I explored this idea in more detail here: http://www.chrisstucchio.com/blog/2014/why_to_reject_complex...
Don't take his word on how quantum mechanics is thought. While we don't teach undergrads it, there is a conceptually simpler approach to quantum mechanics: Feynman's space-time approach. Not only it is conceptually simple and intuitive, it most importantly gives an elementary understanding to the principle of minimal action (sum over all possible paths in space-time) but it also offers a way of calculation that is much better suited to certain class of problems. Now, that is useful.
If you're a layman, you can read "QED: The Strange Theory of Light and Matter". If you know some physics and maths, you can read: http://www.feynmanlectures.caltech.edu/III_03.html
That is how quantum mechanics is made simple and intuitive (and that is superior pedagogically).
Not by introducing new and strange additional concepts such as negative probabilities as basic things just because you can.
The reason we don't teach path integrals in the undergrad is, we expect students to actually use it for calculations, and unfortunately the mathematics is much more involved in comparison to matrix mechanics or wave equations where you can get away with "basic" maths.
If you have a very good intuition and understanding of quantum mechanics, it is doable though, see Feynman Lectures on Physics Vol. III (not sure how many percentage of the students will actually be able to absorb the intuition along with the new information though).
While not everything in the book is perfect, it also reminds me of Penrose's Road to Reality, which painstakingly takes the question "what is a number?" to build up a sophisticated understanding of what that means and then explains how this impacts Physics in a very real way.
We've made so many advances in recent decades we sometimes forget to step back and reformulate our teaching methodologies to incorporate what we know, simplify the teaching, and make the ideas more accessible earlier, so we can put more minds to work on extending them. That's a shame.
If you look at the work of Feigenbaum - e.g. http://arxiv.org/abs/0806.1234 - you'll see how similar to Hardy deriving QM purely by continuing older lines of thought, the fact the universe has a maximum possible speed that anything can travel - the speed of light (though it's not strictly about light) can be derived from postulates as old as Galilean relativity. The speed of light cannot be infinite - it must have some value.
The "probability in L^2" cleared up a lot of confusion for me, although I still have a very poor intuition for what the Born probabilities are probabilities of. If you believe the MWI story, it seems like it's the probability you will "find yourself in the universe where this outcome happened" but even that sounds odd to me.
I'm not sure anyone knows at a deep level. Experimentally you can count how many particles go a given way and it matches the calculation but how that actually works I think remains a mystery.
The L2 norm of a vector is the square root of the sum of the entries squared. Hence a distribution would be a vector x(i) such that sum x(i)^2 is one. This allows x(i) to have both negative and even imaginary numbers.
This awesome talk by Scott
explains the difference between L1 and L2 probability using the Latke vs Hamentaschen Farsical debate. See the video from the link.
My experience (Rome University) was that first we were taught the basic mathematics (Hilbert spaces, functional analysis) for almost a year, before being introduced to the physics which followed a rather axiomatic (not dissimilar to this).
Sometimes our professor would casually drop bombs like the title of this article or things like: "you know in the end quantum mechanics is just Markov chains in imaginary time", but even those made sense in time (Wick rotation).
This line of thinking is a huge problem in science and the reason Scott has to write articles like the one discussed. "Yeah, so those computer guys are so full of themselves that they dare say they know something about how the universe works". Well, the thing is, they actually do.
<i>Two other perfect examples of "obvious-in-retrospect" theories are evolution and special relativity. Admittedly, I don't know if the ancient Greeks, sitting around in their togas, could have figured out that these theories were true. But certainly -- certainly! -- they could've figured out that they were possibly true: that they're powerful principles that would've at least been on God's whiteboard when She was brainstorming the world.</i>
To me, evolution is a perfect example of the need for practical knowledge. Darwinian and Lamarckian evolution are both absolutely reasonable theories; it's just that only one of them matches the world we live in.
This is a quite similar story: ghosts have the wrong sign for the kinetic term:
Note that he of course didn't try to introduce negative probabilities as basic things.
In both cases it was just a matter of interpretation --just with "bad" physical consequences.
> Basically, quantum mechanics is the operating system that other physical theories run on as application software (with the exception of general relativity, which hasn't yet been successfully ported to this particular OS).
Despite not being that popular a view among scientists generally, it is however a fairly popular view among quantum physicists, many of whom aren't that willing to commit to the physical reality of a good deal of the mathematical apparatus of QM.
But a regularity or pattern is nothing more or less than a method for compressing the data: if a particular pattern shows up in many places in a data set, then we can create a compressed version of the data by describing the pattern only once, and then specifying the different places that the pattern shows up. The most compressed version of the data is in some sense the ultimate scientific description.
There is a sense in which the goal of all science is finding theories that provide ever more concise descriptions of data."
Linear Algebra and Analysis can be either applied or pure and of many different levels of sophistication.
* The Schrodinger Equation --> Differential Equations. Wave equation.
* Wavefunctions and Normalisation --> Probability Density. Differential Equations and Function continuity
* Bohr Model --> Algebra
* Bracket Notation --> Matrices, vectors, vector spaces, linear independence
* Operators --> Transformations, matrices, unitary matrices
* Commutators --> see operators! maybe a little algebraic field theory for motivation on why AB - BA is not necessarily 0.
* Eigenstates --> these are special wavefunctions. Eigenvalues and Eigenvectors
I recommend Paul Dirac's original Principles of Quantum Mechanics for a baptism by fire. He's the primary source for a lot of this stuff. He simplified a lot of the methods in QM with his Bracket notation.
Sorry if I missed something.
Not sure if it's my learning style, but most math-related wikipedia pages read as a reference and seem to assume prior knowledge (especially with notation).
This is by design, which is why it is called an encyclopedia. The fact that it assumes prior knowledge just means that you need to refer to other entries when in doubt.
The same goes for cryptography. Most cryptography courses spend at least the first hour talking about historical irrelevance like substitution ciphers etc. Crypto I  (Dan Boneh) follows the latter approach, i.e. starting from modern theoretical principles, defining security properties in terms of computational complexity and games.
I quite liked the Quantum Computing course  (Anuj Dawar) from the Cambridge CST, which also followed that approach, though it didn't present this stuff as a "generalisation of probability". No-cloning theorem in 3rd or 4th lecture, IIRC.
edit: After reading this article fully, I think it would have made for a good "lecture 0" in the above course, bridging the gap between more elementary maths and it.
The squaring rule is actually a special case of multiplying a number by its complex conjugate, which the article doesn't mention, unfortunately.
That is to say, if we have a number z = x + iy, we can obtain its norm from sqrt(xx + yy). But another way to express this is simply sqrt(z z). The product z z is just (x + iy)(x - iy). That of course is just x^2 - (iy)^2 which goes to x^2 - (-1y^2) -> x^2 + y^2.
Geometrically, the conjugate of a complex number has the opposite angle. If z is 20 degrees from the real axis, z* is -20 degrees. Since multiplication of complex numbers is additions of their arguments (i.e. angle components), the two cancel out and the result is on the real number line.
Obviously they can't, which is why the math shows that they don't.
After the first rotation, you are the state (|0>+|1>)/sqrt(2). The physical interpretation of this this state is that it represents a 50% chance of being in |0>, and a 50% chance of being in |1>. If you apply this rotation to either of those two possibilities, you arrive back at (|0>+|1>)/sqrt(2), which still has a 50% chance of being in the state |0>. The two paths leading to this are when the intermediate state is |0> or |1>.
When you actually do the math (in which "rotation" is just a name we give to multiplying by a unitary matrix, U". You find that you end up in the state (.5-.5)|0>+(.5+.5)|1> = .5|0>-.5|0>+.5|1>+.5|1> = |1>.
Here, we can again see the semblance of 2 paths leading to zero (the two |0> terms), however they have opposite signs, so cancel out.
If you start at |0> and do a 45 degree rotation you end up at (|0> + |1>)/sqrt(2). If you start at |1> and do a 45 degree rotation you end up on (-|0> + |1>)/sqrt(2) (note that the coefficient on the |0> is negative now).
Now the trick is that rotations are a linear transformation so rotate(a(x+y)) = a(rotate(x) + rotate(y)). In our case, when we rotate |0> twice, we first end up at (|0> + |1>)/sqrt(2), then we can use linearity to split that into a |0> and a |1> component, rotate them individually and then add up the the results. When we rotate the |0> component we get a +|0> and when we rotate the |1> we get a -|0> and those cancel out (destructive interference).
> Then you compute the outer product of the vector with itself
I'm not sure what he means by outer product here. Isn't the outer product of a vector by itself always nul?
I think that if someone could explain this, they probably wouldn't need the explanation. You can't ask "why don't you understand?" and expect to get a meaningful answer.
(They might be able to point at a specific bit and say "you lost me here", but that's a what, not a how, and they might not be able to.)
Quantum mechanics is what you would inevitably come up with if you started from probability theory, and then said, let's try to generalize it so that the numbers we used to call "probabilities" can be negative numbers.
That's it. So in QM you can model events whose probabilities 'interfere' with each other by canceling each other out (eg independent events A and B have 'probabilities' -20% and +20% respectively, but you want the 'probability' of either A or B occurring to be 0%), and do all other sorts of weird stuff.
Now why would you want to do such a thing is a whole different matter. But at least this should get you started.
It seems to me reasonable to say that the physical sciences can, for example, lend very little or no credence to the claim that Jesus of Nazareth was the long expected Messiah of the Jews, i.e. because a judgment on the matter seems rather outside their scope.
But let's consider a different claim: that the existence of God, the origin and end of all things, can be known with certainty by the natural light of reason.
To develop the idea we might reasonably consider whether any of the loftier domains of the physical sciences, e.g. physical cosmology, can tell us anything about God's existence. Fr. Robert Spitzer, S.J. (among others), has been writing and speaking articulately on the subject for a number of years, and a fair bit of his material is freely available online.
Yes, certainly. Einstein was called out on his frequent allusions to god in his public talks and writings, and under some pressure he finally described how he saw god and religion.
Einstein said that his references to god were in fact with respect to Spinoza's god, an abstract god who played no part in human affairs and that bore no resemblance to the god religious believers picture. In other words, nature -- not a judge, but a morally neutral environment.
> But let's consider a different claim: that the existence of God, the origin and end of all things, can be known with certainty by the natural light of reason.
But that's not possible without evidence. Let me explain the difference between a scientist's attitude toward issues of fact, and a religious believer's attitude.
To a religious believer, a claim is assumed to be true until evidence proves it false. To a scientist, a claim is assumed to be false until evidence proves it true -- the exact opposite.
Why do scientists take this position, formally known as the null hypothesis? Because it's the only rational way to address issues of evidence. Let's take Bigfoot as an example -- to a nonscientist, Bigfoot exists until his nonexistence is proven. But disproving Bigfoot's existence requires proof of a negative, which is an impossible evidentiary burden.
Bigfoot could be hiding under some rock on a distant planet, therefore proving his nonexistence is not possible, therefore Bigfoot exists. Therefore everything exists -- UFOs, fairies, a teapot orbiting out in space in Bertrand Russell's famous argument on this issue (http://en.wikipedia.org/wiki/Russell's_teapot), and god -- all without a shred of evidence.
Imagine if law adopted a religious outlook -- people would be guilty of any crimes they were unable to prove they didn't commit. But law (at least in modern times) adopts an approximately scientific attitude toward evidence, usually codified as "innocent until proven guilty."
This is the real meaning of the chasm between religion and science, and it's not a trivial one.
> To develop the idea we might reasonably consider whether any of the loftier domains of the physical sciences, e.g. physical cosmology, can tell us anything about God's existence.
Very easy to answer -- without evidence, no such claim can be sustained. Full stop.
(Sort of like Freud's psychoanalysis. Everything is a penis, or your mother.)
This has probably nothing to do with the article though.