Bonjour
quand on a des opérateurs linéaires A et B agissant sur un meme espace de Hilbert on peut étudier le commutateur AB - BA.
je copie ici un passage d'un livre
Throughout this chapter (Y, ω) is a pre-symplectic space, that is, Y is a real
vector space equipped with an anti-symmetric form ω. From the point of view of
classical mechanics, Y will have the interpretation of the dual of a phase space,
or, as we will say for brevity, of a dual phase space. Note that for quantum
mechanics dual phase spaces seem more fundamental that phase spaces.
In this chapter we introduce the concept of a representation of the canonical
commutation relations (a CCR representation). According to a naive definition,
a CCR representation is a linear map
y → φ^π (y)
with values in self-adjoint operators on a certain Hilbert space satisfying
[φ^π (y1 ), φ^π (y2 )] = iy1 ·ω y2 Id
We will call (8.2) the canonical commutation relations in the Heisenberg form.
They are unfortunately problematic, because one needs to supply them with the
precise meaning of the commutator of unbounded operators on the left hand
side.
Ou se trouve la difficulté?
merci
et bonne canicule.
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