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:

*a*

^{†}

*a*|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*

^{†}-

*a*

^{†}

*a*acts on |0> (take the difference of the above equations).

(

*a*

*a*

^{†}-

*a*

^{†}

*a*)|0> = |0>

A stronger form of this result is the

*operator*relation

*a*

*a*

^{†}-

*a*

^{†}

*a*= 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*

^{†}-

*a*

^{†}

*a*= 1 (rather than 0). I wonder whether they saw it that way.

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