All posts by Greg M.

Sending Your Secrets Safely with Chaos

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The theory of chaos is an extraordinarily broad mathematical topic, and we all have some intuition for what it means when a system is chaotic. The ideas of unpredictability, spontaneity, intractability, turbulence, and perhaps randomness, all come to mind. But deterministic chaos is somewhat different than our intuitions would have us believe. If you watch the shape of a flickering flame, the whitewater in a rocky river, or the price of crude oil in North America, you’re definitely seeing behavior which can’t be described without chaos. But you’re also most likely seeing the effects of any number of random influences on the system, whether a faltering breeze or some oil speculator’s whimsy. Chaos theory deals with the behavior of deterministic systems—that is, systems with no random inputs. All the intricacy and intrigue of chaotic behavior can arise in systems which might seem deceptively uncomplicated, like a pendulum hanging from another pendulum, or three stars orbiting each other.

But if you’ve ever heard of the “butterfly effect” (a term coined by a pioneer of chaos theory, Edward Lorenz), it’s likely your intuition is right about the central feature of deterministic chaos: chaotic systems have high sensitivity to initial conditions.

If chaotic systems are so unpredictable and temperamental, how can we possibly make chaos work for us? One answer is encryption. Continue reading Sending Your Secrets Safely with Chaos


Edward Lorenz’s Strange Attraction

What does it take to make a system unpredictable?

If you watch a newspaper blowing in the wind, its flopping will be pretty unpredictable. The forces that govern it are constantly changing, maybe even randomly changing: the breeze fluctuates, the pages flap, the creases spin. As onlookers we would decide that the system is governed by some really complicated rules and forces, and we’d be right. It’s easy to give a system the appearance of randomness when its inputs are (or might as well be) random. But what’s the deeper nature of chaos? In chaotic dynamics, we find that the appearance of randomness, unpredictability, and intricacy of a system’s behavior can arise from a few simple deterministic rules. The strict definition of chaos has been somewhat controversial, but a working standard is:

Chaos is non-periodic behavior in a deterministic system with high sensitivity to initial conditions.

The nature of chaos is that simple rules can give rise to unpredictable and subtle behavior.

We’ll be looking at the Lorenz system, a famous system of differential equations which comes from a 1963 paper Edward Lorenz of MIT published in the Journal of Atmospheric Sciences [1]. The publication was momentous, and the Lorenz system helped spur the development modern chaos theory. In his paper, Lorenz Continue reading Edward Lorenz’s Strange Attraction

Quantum Mechanics Fails Its Relativity Exams

When people talk about quantum mechanics, they’re usually referring to applying the principles of quantum mechanics to particles which are moving with non-relativistic energies. That’s fine, and quantum mechanics (or really, non-relativistic quantum particle mechanics) is extremely useful for huge ranges of problems, but there are also whole fields of physics for which we must take special relativity into account. Further, we know that particles are created and destroyed all the time—particle accelerators produce showers of daughter particles, which themselves decay into still other particles, and so on. So there is a definite need for some quantum framework which can account for Einstein’s special relativity, as well as for particle creation and destruction. This encompassing quantum framework is known as Quantum Field Theory.

In developing a relativistic quantum theory, it turns out that we can’t just quantize relativistic particles the way we can quantize non-relativistic particles, as I’ll detail below. Rather, we will need to construct a quantum field in order to take quantum mechanics into the relativistic regime. Here, we’ll take it for granted that a quantum field \phi(x) can be constructed which satisfies certain fundamental requirements.

In this notation, x is a 4-vector x = \langle{t}; \vec{x}\rangle  representing a point in spacetime, \vec{x} = \langle{x}_{1}, {x}_{2}, {x}_{3}\rangle is a spatial vector in \mathbf{R}^3, and I’ll use boldface for operators (like the Hamiltonian, \textbf{H}). In the end our quantum field (\phi) will be operator-valued at each spacetime point—I just won’t use boldface for the fields themselves, even though they are operators. So \phi associates each spacetime point x with an operator called \phi(x).

So, why doesn’t familiar quantum mechanics play well with relativity? Continue reading Quantum Mechanics Fails Its Relativity Exams

Seeing Without Looking: The Quantum Bomb

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Some of the most dramatic and unintuitive experimental results in quantum mechanics have come from dedicated physicists’ insistence on pushing the bounds of what should be physically possible (but not all, as Henry Becquerel’s serendipitous discovery of radioactivity reminds us). Dedication to probing the strangeness of quantum physics has been paying off for over a hundred years.
One of the more recent and striking successes of quantum theory, first posited in 1993 by Avshalom Elitzur and Lev Vaidman of Tel Aviv University, is the idea that we can actually detect an object without using photons or any other particles to look at it. What in the world does that mean? Well,

Imagine you’re tasked with determining whether a new (highly classified) special “quantum bomb” is operational or is a dud. This quantum bomb has two characteristics that make it extremely volatile:

  1. If the bomb is operational, it will explode when a single photon of light strikes it;
  2. If the bomb is a dud, it will not interact with photons in any way.

This may be an extremely strange and dangerous bomb, but you’ve found yourself before a problem which seems impossible: any light you shine on the bomb, perhaps to test it, will just cause an explosion. How can you possibly test that the bomb works without causing it to explode?

You need to perform an interaction-free measurement.

Continue reading Seeing Without Looking: The Quantum Bomb