Example
Let φ be an operator valued distribution and the (Klein-Gordon) PDE be
- .
This is a bosonic field. Let's call the distribution given by the Peierls bracket Δ.
Then,
where here, φ is a classical field and {,} is the Peierls bracket.
Then, the canonical commutation relation relation is
- .
Note that Δ is a distribution over two arguments, and so, can be smeared as well.
Equivalently, we could have insisted that
where is the time ordering operator and that if the supports of f and g are spacelike separated,
- .
Read more about this topic: Free Field
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