Sunday, August 24, 2008

Dropping the Baton

There seem to have been rather a lot of dropped batons in the relay races at the Olympics (e.g. see here).

It set me thinking about where I might have seen this sort of thing happening elsewhere, and I realised that dropping the baton is like annihilating the vacuum state.

How so?

The simplest possible algebra that one can use to model the process of baton-passing goes like this:

a increments (by 1) the number of hands holding the baton
a decrements (by 1) the number of hands holding the baton

|0> is the "vacuum" state where the baton has one hand holding it
a|0> is the state where the baton has two hands holding it

a|0> = 0 is the annihilation of the vacuum where the baton has zero hands holding it, i.e. a state from which there is no way to recover

Note that it is important to define the vacuum state as corresponding to one (rather than zero) hand holding the baton, otherwise the algebra (i.e. annihilation of the vacuum) doesn't correctly model the dropping of the baton. Thus the counting of hands holding the baton is really a measure of how many excess hands are holding the baton, because the case of one hand is actually the ground (or vacuum) state in a relay race.

Most of the time the state is |0>, and during a successful handover of the baton it passes through the transition state a|0>, after which it returns to the state |0>. However, during an unsuccessful handover of the baton it goes to the state a|0> which is 0, where the vacuum has been annihilated.

Successful handover: a a|0> = |0>
Unsuccessful handover: aa|0> = 0

The order in which the a and a operations are applied is important, and is neatly summarised by how their commutator a a - aa acts on |0> (take the difference of the above equations).

(a a - aa)|0> = |0>

A stronger form of this result is the operator relation

a a - aa = 1

This relation takes note of the fact that there are n ways of applying a to the state (a)n |0> (i.e. choose from 1 of n excess hands to decrement by 1 the number of excess hands holding the baton), but there is only 1 way of applying a to the state (a)n |0>. The case n=0 is when the vacuum gets annihilated by application of a.

The Olympic athletes who dropped the baton were the victims of a a - aa = 1 (rather than 0). I wonder whether they saw it that way.

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