τ keeps on slipping...

Apologies: this post is somewhat more specialized than my normal fare, and probably will be boring as hell without a some mathematical knowledge.

Electrons have a negative charge. When you think about it, this really doesn't make much sense; after all, electrons are the charge carriers for electric charge, and so we would hope to assign their charge a positive value. That electrons carry a negative charge isn't a fundamental statement about reality, though, but rather an unfortunate consequence of an arbitrary decision made early on when electricity was being studied, but when electrons were still undiscovered.

Similar cases of unfortunate arbitrary conventions can be seen in other areas of mathematics and science. Recently, for instance, Michael Hartl has argued that $\pi$ is not the right constant to use in the equations governing such things as circles, frequencies and angles. Rather, Hartl argues that $\tau = 2\pi$ is a much more natural choice. Using this convention, the circumference $c$ of a circle is $c=\tau\ r$, eliminating the awkward factor of 2 in $c=2\pi r$. It may seem that we lose something when considering the area $A = \frac12 \tau\ r^2$ of a circle in this notation, but in fact this is much more natural for expressing as an integral, as those familiar with calculus will be happy to note.

Today, I'd like to show you somewhere else in physics where changing notation makes things much more natural. Concretely, I'd like to argue that using a different $\tau$ makes quite a lot of sense, when we using $\tau = it$ as a replacement for the time $t$ in equations. In fact, I take it as a lesson of quantum mechanics that we should consider time to lie along an imaginary axis and not along the real axis. This notational trick, known as Wick rotation, simplifies many physical equations, such as Schrödinger's equation. I find that \(\frac{d}{dt} \left\vert\psi\right\rangle = i \hat{H} \left\vert\psi\right\rangle\)  makes much more sense expressed in imaginary time:
\[\frac{d}{d\tau} \left\vert\psi\right\rangle = \hat{H} \left\vert\psi\right\rangle\] Adopting this convention also makes it manifestly clear why complex conjugation is intimately related to time reversal, since $\tau^* = -\tau$.

It is, in fact, quite rare for $t$ to appear in quantum mechanics without a factor of $i$ attached. Even when describing a classical object interacting with a quantum mechanical system, such as an oscillating field introducing a time-varying term to a system's Hamiltonian (that is, the operator which describes the energy of a system--- if that makes no sense, don't worry), we write something like \[\hat H(t) = \cos(\omega t)\ \hat\sigma_x + \sin(\omega t)\ \hat\sigma_y.\] But wait!, you say! There's no $it$ in that equation! As it turns out, there actually is, but we've hidden it by using trigonometric functions where an exponential function is more natural: \[ \hat{H}(\tau) = e^{-\omega\tau\hat\sigma_z/2}\hat\sigma_x e^{\omega\tau\hat\sigma_z/2} \] This form also has the advantage of making it manifest that the oscillation of the classical field can be thought of as a coordinate rotation of a time-independent field.

Other key results of quantum mechanics become much cleaner with the imaginary-time convention. For instance, this convention along with the natural units convention that $\hbar = 1$ makes Ehrenfest's theorem much less awkward to write: \[\frac{d}{d\tau}\left\langle \hat A\right\rangle = \left\langle \frac{d\hat A}{d\tau}\right\rangle + \left\langle[\hat H, \hat A]\right\rangle\]

At the end of the day, such notational choices as the sign of an electron's charge, the choice of circle constant, or the axis which we use to represent time are all arbitrary. We can do physics quite well even when a choice lacks something in mathematical beauty. My point, then, in exploring the fun of $\tau$ is to show that even though our choice of notation is an arbitrary choice made for the convenience of the humans that work with it, by making our notational choices carefully, we can coax out and make manifest deep truths.

Much needed closure.

Closure is an important concept in mathematics, and is deceptively simple. If you have a set of things and some operation acting on those things, then the closure of your set is the smallest set that contains your original set along with everything that operation gives you.

The words get in the way, though, so let's consider an example. If you have the numbers zero and one, then their closure under addition would be all positive integers. Why? Because you can get to any positive integer by adding one to itself over and over. For instance, 2 is in the closure, since addition produces 2 from our set: 1 + 1 = 2. By the same argument, 3 is in the closure since 1 + 2 = 3, and since 2 must be in our closure.

We say that this set is the closure of our original set since it is the smallest set which is, well, closed. If, in our previous example, we omitted the number 2, our set wouldn't be closed any more, since addition could take us outside of the set.

As of late, however, the way that mathematicians use the word closure has started to be seen well outside of mathematics. Witness the rise of "epistemic closure" (closely related to deductive closure) as a useful term in political science. The word finds much use even outside of mathematics, as it gets to the heart of a very powerful technique in rational thinking: asking what, given some tool, one can produce. In epistemic closure, the tool is reasoning itself, while in our more pedestrian example, our tool was basic addition. In both cases, however, what remains is the use of closure as a mechanism for understanding and characterizing an operation.

In the spirit, then, of exploring closure, I'd like to bring some much needed closure forward. Specifically, I'd like to consider a kind of causal closure. If we consider some set of events which may or may not be causally related, we can for any specific event ask what events may be caused and what other events may cause it. Both of these are a kind of operation; extrapolating both directions in time to understand the causal structure of your set of events. The causal closure, then, of a set of events is the full set of events which caused the original set, along with the full set of effects caused by these events.

Luckily, we already have a term for this kind of causal closure. What we mean when we say that two events are causally related is that they lie within the same universe, so that the universe can be thought of as the set of all events which are causally related to an event representing our powers of observation. Under this realization, if A causes B, then A cannot be in a different universe than B.

All this is well and good, but why do I bring it up now? In a recent post, I asserted that religious claims were of a material nature, and thus amenable to the methods of science. That this is the case can be easily seen by invoking a principle of causal closure; if religious claims include any causal relation to the material universe, then they must, by closure, be entirely about the material universe. Of course, that alone does not mean that such claims are subject to scientific understanding, but that is an argument I have made before and don't wish to repeat here. Rather, my intent was simply to bring some much needed closure to bear on an argument that has gone on too long.

What is a gate? (Part 1)

It's been a bit since my last "what is" post, but I'd like to return to talking about science by taking a pause from my build-up to quantum states and quantum computation to instead discuss something more classical: the notion of a logic gate.

One way of modeling classical computation is as a sequence of operations performed on some data. We can then consider each operation independently. Just as we can build up complicated equations from simple arithmetic operations, these computational operations, typically called gates, can be used to build up arbitrarily complicated computations.

Take a specific example, the NOT gate, also written ¬. This gate takes a bit and produced a bit with the opposite value. Since each bit can only have one of two possible values (either 0 or 1), we can completely specify the behavior of the NOT gate by listing what it does to each of these inputs. That is, if I tell you that ¬ 0 = 1 and that ¬ 1 = 0, then in principle, I have told you everything that there is to know about the NOT gate. If this reminds you of a basis, then your intuition serves you well— we will explore that connection more in due time.

For now, though, I would like to discuss a few more examples of gates: the AND and OR gates, often written as ∧ and ∨, respectively (if these symbols seem arcane, it may think of them in terms of set unions and intersections). Each of these gates takes two bits as inputs and produces one output. AND produces 1 if and only if both its inputs are 1 (1 ∧ 1 = 1, 0 ∧ 0 = 0 ∧ 1 = 1 ∧ 0 = 0), while OR produces 1 if and only if at least one input is 1 (0 ∨ 0 = 0, 0 ∨ 1 = 1 ∨ 0 = 1 ∨ 1 = 1). Finally, the XOR gate (short for exclusive or and written ⊕) returns 1 if and only if exactly one input is 1 (0 ⊕ 0 = 1 ⊕ 1 = 0, 0 ⊕ 1 = 1 ⊕ 0 = 1).

With these four gates, we can build up any arbitrarily complicated Boolean function; that is, a function from strings of bits to a single bit. Functions returning multiple bits can in turn be built up by representing each output bit as a Boolean function. We could actually do with less kinds of gates, but that's besides the point. Rather, the point is that together, NOT, AND, OR and XOR are universal for classical computation.

It takes some effort to prove this, but an example helps to make things concrete. The full adder circuit in particular can be used to add two one-bit numbers, and is built up entirely from two XOR gates, two AND and one OR gate, as shown in this circuit diagram from Wikipedia. These full adders in turn can be combined to add arbitrarily long integers. From addition, one can get to subtraction and multiplication, demonstrating the usefulness of the gate model in capturing arithmetic.

Even more compellingly, we can efficiently simulate Turing machines with these few gates, meaning that NOT, AND, OR and XOR are at least as expressive as Turing machines. Thinking about gates, then, is a powerful way of thinking about classical computation. As we shall see, this power carries very nicely to the quantum case as well.

What is a matrix? (Part 2)

Now we have a new kind of mathematical toy to play with, the matrix. As I said in the previous post, the easiest way to get a sense of what a matrices do is to use them for a while. In this post, then, I just want to go over a couple useful examples.
Suppose you wish to make all vectors in ℝ² longer or shorter by some factor s ≠ 0. You can represent this by a function f(v) = sv. With a moment's work, we can verify that this is a linear function because of the distributive law. Thus, we can represent f by a matrix. To do so, remember that we calculate f for each element of a basis. For simplicity, we will use the elementary basis {x, y}. Then, f(x) = sx and f(y) = sy. By using coordinates, we can write this as f([1; 0]) = [s; 0] and f([0; 1]) = [0; s]. The matrix representation of f then becomes:

Note that if s = 1, the function f doesn't do anything. Representing f(v) = v as a matrix, we get the very special matrix called the identity matrix, written as I, 𝟙 or 𝕀:

The identity matrix has the property that for any matrix M, M𝟙 = 𝟙M = M, much like the number 1 acts.

Of course, there's no requirement that we stretch x and y by the same amount. The matrix [a 0; 0 b], for instance, stretches x by a and y by b. If one or both of a and b is negative, then we flip the direction of x or y, respectively, since -v is the vector of the same length as v but pointing in the opposite direction.

A more complicated example shows how matrices can "mix up" the different parts of a vector by rotating one into the other. Consider, for instance, a rotation of the 2D plane by some angle θ (counterclockwise, of course). This is more difficult to write down as a function, and so a picture may be useful:

By referencing this picture, we see that f(x) = cos θ x + sin θ y, while f(y) = - sin θ x + cos θ y. Thus, we can obtain the famous rotation matrix:

As a sanity check, note that if θ = 0, then R_θ = 𝟙, as we would expect for a matrix that "does nothing."
One very important note that needs to be made about matrices is that multiplication of matrices is not always (or even often) commutative. To see this we let the matrix S swap the roles of x and y; that is, S = [0 1; 1 0]. Then, consider A = SR_θ and B = S. Since applying S twice does nothing (that is, S² = 𝟙), we have that BA = R_θ. On the other hand, if we calculate AB = SR_θS, we find that AB = R_-θ:

(Sorry for the formatting problems with that equation.)We conclude that AB ≠ BA unless sin θ = 0, neatly demonstrating that not all the typical rules of multiplication carry over to matrices.

I'll leave it here for now, but hopefully seeing a few useful matrices makes them seem less mysterious. Until next time!

What is a matrix? (Part 1)

Functions are an important tool in mathematics, and are used to represent many different kinds of processes in nature. Like so many mathematical objects, however, functions can be difficult to use without making some simplifying assumptions. One particularly nice assumption that we will often make is that a function is linear in its arguments:

One can think of a linear function as one that leaves addition and scalar multiplication alone. To see where the name comes from, let's look at a few properties of a linear function f:

This implies that f(0) = 0 for any linear function. Next, suppose that f(x) = 1 for some x. Then:

This means that if f represents a line passing through 0 having slope m = 1 / x.

So what does all this have to do with matrices? Suppose we have a linear function which takes vectors as inputs. (To avoid formatting problems, I'll write vectors as lowercase letters that are italicized and underlined when they appear in text, such as v.) In particular, let's consider a vector v in ℝ². If we use the {x, y} basis discussed last time, then we can write v = ax + by. Now, suppose we have a linear function f : ℝ² → ℝ² (that means that takes ℝ² vectors as inputs and produces ℝ² vectors as output). We can use the linear property to specify how f acts on any arbitrary vector by just specifying a few values:

This makes it plain that f(x) and f(y) contain all of the necessary information to describe f. Since each of these may itself be written in the {x, y} basis, we may as well just keep the coefficients of f(x) and f(y) in that basis:

We call the object F made up of the coefficients of f(x) and f(y) a matrix, and say that it has four elements. The element in the ith row and jth column is often written F_ij. Application of the function f to a vector v can now be written as the matrix F multiplied by the column vector representation of v:

We can take this as defining how a matrix gets multiplied by a vector, in fact. This approach gives us a lot of power. For instance, if we have a second linear function g : ℝ² → ℝ², then we can write out the composition (g ∘ f)(v) = g(f(v)) in the same way:

That means that we can find a matrix for g ∘ f from the matrices for g and f. The process for doing so is what we call matrix multiplication. Concretely, if we want to find (AB)_ij, the element in the ith row and jth column of the product AB, we take the dot product of the ith row of A and the jth column of B, where the dot product of two lists of numbers is the sum of their products:

To find the dot product of any two vectors, we write them each out in the same basis and use this formula. It can be shown that which basis you use doesn't change the answer.

If this all seems arcane, then try reading through it a few times, but rest assured, it makes a lot of sense with some more practice. Next time, we'll look at some particular matrices that have some very useful applications.

Looking back at what is.

Over the past few weeks, I've tried to tunnel through the potential barriers to actual science blogging, with mixed success. One of my bigger oversights thus far has been to omit any sort of a "big picture" from my posts, leaving them as little islands in a vast sea of scientific ideas. Today, I'd like to correct that.

My most immediate goal in science blogging has been to explain how physical states function in the beautiful formalism of quantum mechanics. The language of quantum mechanics, however, is one of probabilities, of complex numbers and of linear algebra. Arguably the most fundamental part of the language of quantum mechanics, linear algebra may be roughly thought of as the study of vectors, and how they transform. As I shall discuss in a future post, by using the idea of a basis, we can represent a special kind of vector transformation by an object called a matrix (more generally, an operator). Thus, what we have discussed thus far is not a set of disparate islands so much as a set of stepping stones. If you prefer a more concrete metaphor, we have poured a foundation for future discussions, including a discussion of the quantum state itself.

Once we have the idea of a quantum state, the horizon opens wide for exploration. The quantum state gives us a language in which we can understand seemingly arcane consequences of a world described by quantum mechanics, such as entanglement or superposition. With the formal tools of mathematics at our disposal, we can overcome the limitations of our intuition, so that we can understand even such tricky concepts as these.

One downside to my stepping-stone approach, however, is to seemingly put the concept of a quantum state on a pedestal, inaccessible without a high degree of mathematical maturity. Little could be further from the case. Indeed, the mathematics with which we understand quantum states are not so difficult as they are esoteric. It is my own opinion that these areas of math need not be esoteric, save for that it has been arbitrarily decided upon (at least in my home, the United States) that Math Is Hard, and that concepts such as those discussed here Should Be Left To the Professionals. Bollocks. We live in a probabilistic world, and one in which statistics guide nearly every aspect of society, so why should understanding probability be so inaccessible? While complex numbers are not so manifestly real, even to the point that i is called the imaginary unit, it takes but a small amount of study to see that the complex numbers form an integral part of how we describe reality. Similarly, the concept of a vector may seem too far removed from reality for the layman to pursue, but in many ways, vectors formalize and encode much of our intuition about geometry, and are just as accessible as the sort of geometry that is taught in many grade schools.

No, quantum states are there for those who want them. My goal is to bring the concepts just a little bit closer, and to let the mathematical beauty underlying them shine through just a little bit brighter. In doing so, I won't always go from point to point in most straightforward way, but I ask your patience, for I am going somewhere. With a bit of looking back at what is, I hope you'll agree that we're going somewhere interesting.

What is a basis?

Consider a vector. Just to make things concrete, consider a vector on the 2-D plane. In fact, let's consider this one (call it v⃑):

It's a vector, to be sure, but it's hardly clear how one is supposed to work with it. It doesn't make sense to pull out a ruler and pencil every time we want to add our vector to something; mathematics is supposed to be a model of the world, and thus we should be able to understand things about that model without recourse to physical measurements. To solve this problem for vectors on the plane, we can introduce two new vectors, x̂ and ŷ, then use vector addition to write v⃑ as a sum:

Now we can write v⃑ = ax̂ + bŷ̂, which doesn't at first seem to buy us much. Note, however, that we can write any vector on the 2D plane as a sum of these two new vectors in various linear combinations. Mathematically, we write this as ℝ² = span {x̂, ŷ}. Whenever a space can be written this way for some set of vectors B, we say that B is a basis for the space.

Once we have a basis picked out, we can work with the coefficients (a and b in our example) instead of the vector itself, as they completely characterize the vector. For example, adding vectors becomes a matter of adding their respective coefficients.

In spaces other than the 2-D plane, we can also apply the same idea to find bases for representing vectors. Consider, for instance, the space of column vectors such as [a; b] (pretend they're stacked in a column, OK?). Then, a perfectly fine basis would be the set:

It's easy to see that we can write any other 2-dimensional column vector as a sum of the form a[1; 0] + b[0; 1] = [a; 0] + [b; 0] = [a; b].

A point that can get lost in this kind of discussion, however, is that there's absolutely nothing special about the bases I've given here as examples. We could just as well used [1; 1] and [1; -1] as a basis for column vectors, or just as well used a different pair of vectors in the plane:

Put differently, a basis is a largely arbitrary choice that you make when working with vectors. The relevant operations work regardless of what basis you use, since each of the vectors in a basis can itself be expanded. For example, [1; 0] = ½([1;1] + [1; -1]) and [0; 1] = ½([1; 1] - [1; -1]), so that we have a way of converting from a representation in the {[1; 0], [0; 1]} basis to the {[1; 1], [1; -1]} basis.

While there is much, much more to be said on the topic of bases for vectorspaces, I'm happy to say a few words about bases. As we shall see when we get into discussing linear operations, the existence of bases for vectorspaces is a large part of what gives us so much power in linear algebra. We shall need this power in the quantum realm, as linear algebra may well be said to be the language of quantum mechanics. Hopefully I'll get a few more words in on the subject before my vacation!

What are vectors?

As I've said before, science is social-- oops. Wrong mantra. What I meant to say is that vectors are an abstract way of describing a pattern. Specifically, the vectorspace axioms formally describe a kind of mathematical object, the vector, that encapsulates the geometric and algebraic properties of a large class of seemingly disparate objects. By using the vectorspace axioms, we will be able see that lists of numbers such as are vectors, as are arrows on the 2D plane.

Rather than describe how to do so myself, though, I will try something different. Vectors are important in much of physics, and so lots of people have already written much about them. Thus, for the bulk of the work in describing vectors, I will defer to these other writings. A very physics-oriented approach can be found over at Dot Physics, starting with a trig-based introduction to vectors, followed by a discussion of how to represent vectors. An alternate physics-motivated discussion of vectors can be found at HyperPhysics.

For the more mathematically motivated amongst us, Wikipedia has a good page describing a very special family of vector spaces called ℝⁿ that is used to describe points in Euclidean space. MathWorld has a few good articles on vectors, including a technical definition and listing of properties and a more concise listing of the vectorspace axioms. Finally, the Unapologetic Mathematician derives vectorspaces from a more general construction called a module (warning: not for the feint of math).

To understand why we care about vectors in quantum information and computation, however, takes one more observation. A quantum state can be written as a linear combination of some set of basis states. For example, an arbitrary qubit state can be written as . This important property means that quantum states are a kind of vector in what we call a Hilbert space. This has some profound implications for how we think of and manipulate quantum states, as we shall explore in forthcoming posts.

Intuition, mathematics and the conspicuous absence of "vs."

I am now at the point in my academic career where some people see fit to ask me for advice. As shocking as this change is to me, I have found that when people ask me for advice, they mean it sincerely and take my advice seriously. It is thus incumbent upon me to be just as serious and sincere in what advice I give. Tonight, I'd like to ponder one particular kind of advice I have found myself giving in recent weeks.

In physics, we use mathematics. A lot. Math is the language of physics and the method by which our science is performed. How, then, can physics ever advance in directions unanticipated by math? Perhaps more to the point, since mathematical thinking is not inherent but a skill, how can we check ourselves against mistakes in our mathematics? In physics, we often find that intuition is the answer to these problems. A good physicist will have developed a keen intuition that guides them to new and novel discoveries, checks them against mistakes and that helps with the interpretation of very abstract concepts.

This answer, as correct as it may be, is nonetheless incomplete, however. Our intuitions are ill-equipped to deal with phenomena outside the "middle world" in which nothing is too big, too small or too fast. Thus, when we first encounter quantum mechanics, for instance, our intuitions often betray us, leading us to reject such beautiful facets of the theory as entanglement. As we learn, we must fight our intuitions as much as we utilize them.

It is common to paint this quandary as being about intuition versus mathematical truth, but herein lies my perhaps not-so-humble advice: trust your intuition, but trust the mathematics more. Our intuitions are wonderful tools for doing science, but at the end of the day, it is mathematics upon which our theories must be founded. When we encounter something unintuitive, it does indeed behoove us to exercise extra caution and skepticism, but these must ultimately give way to experiment and to theory.

As we learn to put our trust (not faith) in the explanatory power of mathematics and the veracity of our experiments, then we can build up a new intuition that serves us more faithfully. In short, by trusting in mathematics, we turn that most admirable of human qualities to our intuitions: the capacity for self-improvement. Rather than thinking of intuition and mathematical truth as being at odds, we can see conflicts between these as opportunities to learn and to improve. The two tools, when used to their fullest, work together in a virtuous cycle that results in the expansion of knowledge.

All our relations laid bare.

Update: Thank you to Diandra for the kind words about this post!

The Internet is abuzz recently with a depressing bit of news about the state of math education in the United States: the vast majority of adults in the US do not understand what the equals sign means. For a particularly good take on this, please see the excellent article on Cocktail Party Physics. As for my part, I'd like to take this as an opportunity to expand on a point that the author, Diandra Leslie-Pelecky, made in the Cocktail Party post. Specifically, I want to elaborate on the use of the equals sign to indicate a relation.

Fundamental to mathematics is the idea of a relation, which is a formal way of stating that two objects are related in some specific way. For instance, the object "2 + 3" is related to the object "5" by the equality. This notion, however, can hide that something very important has occurred. We have taken a conceptual process, addition, and restated it in terms of a statement about static relations. No matter what I do, I cannot break the relation "2 + 3 = 5." By contrast, if I constrain myself to thinking about the addition process, then it is harder to separate that statement about Platonic ideals from the perhaps imperfect implementation of the addition process. The equality relation, then, tells us about what is.

To take a tangent for a moment, mathematics can be thought of as the process of identifying and abstracting patterns. The concept of addition, for instance, is an abstract way of discussing and modeling a very common pattern in the natural world. We need not specify whether we are adding apples or planets; the pattern is the same. Thus, taking the relational view is a natural step in this methodology of abstraction, as relations tell us about the patterns that we can identify in other patterns. We recognize that the pattern "three objects" is indistinguishable from the pattern "one object and two objects added together," and so we say that these two patterns are equal to each other. In doing so, we make no statement about which pattern precedes which in a process, meaning that we can represent the "three objects" pattern as the "1 + 2 objects" pattern should we find the latter more convenient.

Of course, processes exist, and so mathematics would be much less useful were it not able to describe them. One may well point out, for instance, that the concept of a function has a very clean intuitive description as a mathematical formalization of a process. That is, the expression "f(x)" means "take x and do f to it." Notice, however, that we ultimately rely on the idea of a relation to make sense of functions. We say things like "y = f(x)," meaning that the result of f acting upon x is related to y such that the two objects are indistinguishable. We can thus remove any notion of dynamics from our description, focusing on the pattern that our process introduces. One can even go as far as to think of a function f as a kind of relation between other objects, so that "x f y" means that the objects x and y are related by the action of f.

This shift to relational thinking is very powerful, and underlies not only much of mathematics, but also much of our language. When I say "I am Chris," there is no naming process implied, but only the statement that the concepts "I" and "Chris" are related by the verb "to be." That is, in announcing my name, I am relating the concepts of self and name. Recognizing this has been a boon to the Semantic Web, and is used to express concepts in terms of abstract relations, such as is done in Notation3. The entire concept of the World Wide Web comes back to the concept of a very special relation known as the hyperlink.

Understanding the notion of a relation can be seen to be critical to understanding the means by which we understand and model the world around us. A key part of being human is our capacity to identity patterns and relationships between objects around us, and it is precisely this capacity that we bring to bear in mathematics.

PS: This is why ":=" is often preferred over "=" for indicating "set equal to" by mathematicians.