Zeno Walks Into a Bar

August 4, 2019

Zeno walks into a bar.

"I have a problem," he said.

"What is it?" said the bartender.

"Well, it has to do with the movement of physical bodies," said Zeno.

"Talk to my friend Max," said the bartender. He gestured toward a German man wearing round spectacles.

"Sir," said Zeno, "I wonder if you could help me with a problem."

"What's the problem?" said Max.

"Suppose I shoot an arrow from point $A$ to point $B$," said Zeno. "Before it reaches point $B$ it must first reach a point $C_1$ midway between points $A$ and $B$."

"Naturally," said Max.

"And before the arrow reaches point $C_1$ it must reach a point $C_2$ midway between points $C_1$ and $A$," continued Zeno.

"I see," said Max.

"And before the arrow reaches point $C_2$ it must reach a point $C_3$ midway between points $C_2$ and $A$," continued Zeno.

"Wait a minute," said Max. "How far apart are points $A$ and $B$?"

"10 meters," said Zeno.

"Then yes," said Max. "I understand your situation."

"And before the arrow reaches point $C_3$ it must reach a point $C_4$ midway between points $C_3$ and $A$," continued Zeno. "Do you see the impasse?"

"Nope," said Max, "I think we're getting somewhere. How long is the arrow?"

"One meter," said Zeno.

"The distance between points $C_3$ and $C_4$ is five eighths of a meter," said Max. "A one-meter-long arrow can be at point $C_3$ and $C_4$ at the same time."

"Let's consider the tip of the arrow then," said Zeno. "Before the tip of the arrow reaches point $C_3$ it must reach a point $C_4$ midway between points $C_3$ and $A$."

They talked deep into the night.

"And before the high-energy particle reaches point $C_{118}$ it must reach a point $C_{119}$ midway between points $C_{118}$ and $A$," continued Zeno.

"Hold on," said Max. "How far apart are points $A$ and $C_{119}$?"

"$1.5\times10^{-35}$ meters" said Zeno.

"That's shorter than $1.6\times10^{-35}$ meters," said Max. "The uncertainty in the position of a particle must always exceed $1.6\times10^{-35}$ meters, because of space-time equivalence and the quantum-mechanical velocity operator's non-commutation with position. Even theoretically, the wave function of a particle can't ever occupy a space smaller than $1.6\times10^{-35}$ meters."

"Thanks," said Zeno.

"By the way," said Max, "What brought you to this question in the first place?"

"I wanted to know how to define the momentum of a particle at an instantaneous moment of time," said Zeno.

"You could have just asked," said Max. "The probability distribution of a particle's momentum is determined by the instantaneous phase and magnitude of its wave."