The Wizard strode into class, but, instead of starting his lecture as usual, he sat down among the Acolytes at their tables. Everyone wondered what was going on, when, all of a sudden, a masked man in a black robe walked in and stood in front of the blackboard.
"Who are you?" asked Toby the Acolyte.
"I am the wizard JB." the masked man replied.
This was most confusing, since all the Acolytes knew that their own Wizard went by the initials "JB" when he was trying to downplay his wizardliness. Toby quickly glanced to make sure that the Wizard was still at his desk, not playing a clever trick of some sort. Yes, the Wizard was still there -- and he was smiling, almost chuckling!
"Meet my alter ego and collaborator from across the seas!" said the Wizard. "You could almost call him my twin. His name is indeed JB, and he is indeed a wizard. In fact, he is an expert on quantum gravity, spin networks, categories -- everything I am an expert on.".
"Wow!" said Oz "Do you look alike, too?".
"That you will never know." said the masked man "And now, I pose to you a challenge! I hear your wizard has been teaching you diagrammatic methods for doing tensor algebra. If you've mastered these, you should be able to solve this problem:
Find me linear operators
| |
| |
V v v V
\ /
\_______/
and
_______
/ \
/ \
V v v V
| |
| |
satisfying the following equations,
devised by the famous and powerful wizard Penrose,
in the days when the earth was younger and magic still filled the air:
| | |
| | |
| | |
V v _______ | _______ V v
| / \ = | = / \ |
| / \ | / \ |
| V v | V v | V v |
\ / | | | \ /
\_______/ | | | \______/
V v | V v
| | |
| | |
| | |
and
_______
/ \
/ \
| |
| |
V v v V = -2
| |
| |
\ /
\_______/
and best of all, the so-called "binor identity":
| | | | | |
| | | | | |
V v v V V v v V | |
\ / \ / | |
\ / \_______/ | |
\ / | |
/ = + V v v V
/ \ _______ | |
/ \ / \ | |
/ \ / \ | |
V v v V V v v V | |
| | | | | |
| | | | | |
There is not a unique solution to this challenge,
but all solutions are isomorphic.".
Oz groaned inwardly -- whenever a wizard used the word "isomorphic", things seemed to get tricky.
"Any takers?" asked the masked man.
Oz thought hard but wisely kept his mouth shut. The required operators reminded him of the cap and cup, but they were different somehow ... . Ah, yes, there were no arrows going up! Also, the equation
_______
/ \
/ \
| |
| |
V v v V = -2
| |
| |
\ /
\_______/
reminded him of an equation the Wizard had taught them last week:
_______
/ \
/ \
| |
| |
V ^ v V = dim(V)
| |
| |
\ /
\_______/
But of course the dimension of a vector space could never be negative!
Or could it? Wizards were full of baffling ideas like that.
It was all very strange.
Oz was not alone in his puzzlement. All the Acolytes sat flummoxed as the masked man stood there waiting for a reply.
"No takers?" said the Wizard "Well then, this will be homework. Okay, JB, go ahead and tell them what you can do with this stuff.".
And the masked man went ahead and told them of many strange and marvelous things that made Oz's head hurt, leading up to a theory of 4D quantum gravity which the masked man had devised, but, unfortunately, none of the Acolytes were fully prepared to understand these things.
"Well," the masked man asked confidentially afterwards "do you think they got much out of it?".
"I'm not sure." the Wizard replied "I'm sure it was good for them, but I'll have to go over it all again.".
© 2000 John Baez