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I assume that normal order of a sum of operators is a sum of normal orders. It follows

from it that for bosonic creation and anihilation operators holds:

[tex]

a^\dagger a= ~: [a,a^\dagger ]:~ =~ :[a^\dagger,a]+1:~ = ~ :[a^\dagger,a]: +:1:=

a^\dagger a + :1:

[/tex]

So we have [tex] :1: =0[/tex]. Is it OK?