Waiting and Watching

Kosmas L. Tsakmakidis, Ortwin G. Hess

Frank Verstraete

John H. Schwarz, John H. Schwarz

Hands-on exercise: wave and particle properties The purpose of this exercise is for you to observe some basic wave and particle properties. To complete this exercise you will need the following: Tub of water, or access to a quiet pond, lake or swimming pool. Things to float on the surface of the water. Pen or pencil and some drawing paper. Small projectile such as a stone. Disturb the middle of the tub just until you are able to make a visible wave on the surface. Watch how the wave propagates. Wait until the surface of the water returns to being flat and make another wave. Keep doing this as many times as necessary to be able to draw what you see on the paper and answer the following questions: Does the wave have a definite location at any one moment in time? Does the wave have a definite direction as it propagates? Approximately how far does the wave travel in 1 s? Describe the motion of the water in which the wave moves. Throw your small projectile in the air at various angles, letting it drop down (not in the water). Keep doing this as many times as necessary to be able to draw what you see and answer the following questions: Does the object have a definite location at any one moment in time? […]

Yoothana Suansook

This paper tells the story of a mathematical object — the dagger — and its quiet, persistent presence in two of the most consequential theories of the twentieth and twenty-first centuries. The dagger, or conjugate transpose, is the defining operation of quantum mechanics: it turns a state into a dual, an operator into an observable, and a symmetry into a conserved quantity. In 2017, when Vaswani and colleagues introduced the Transformer [1], they wrote down an equation whose core operation —the scaled dot-product — contained the dagger in disguise, hidden behind the innocent transpose of a real-valued matrix. This paper is a narrative exploration of that coincidence, and an argument that it is not a coincidence at all. We trace the dagger from Born and Jordan’s matrix mechanics in 1926 [11] to the attention mechanism of modern artificial intelligence; we watch it split into real and imaginary parts, we follow its phase as it survives or fails to survive the softmax bottleneck, and we witness its reunion with unitarity and the Born rule [5]. The story is not merely one of analogy. It is a story of structural identity — a single mathematical operation applied to two different substrates, revealing a unity that was there all along, waiting for the field to become complex enough to see it.

Ron Cottam, Willy Ranson