(a) Any textbook gives the interpretation of the density matrix in a single continuous basis |α⟩:
The diagonal elements ρ(α,α)=⟨α|ˆρ|α⟩ give the populations.
The off-diagonal elements ρ(α,α′)=⟨α|ˆρ|α′⟩ give the coherences.
(b) But what is the physical interpretation (if any) of the density matrix ρ(α,β)=⟨α|ˆρ|β⟩ for a double continuous basis |α⟩, |β⟩?
I know that when the double basis are position and momentum then ρ(p,x) is interpreted as a pseudo-probability. I may confess that I have never completely understood the concept of pseudo-probability [*], but I would like to know if this physical interpretation as pseudo-probability can be extended to arbitrary continuous basis |α⟩, |β⟩ for non-commuting operators ˆα, ˆβ and as probability for commuting ones.
[*] Specially because ρ(p,x) is bounded and cannot be 'spike'.
EDIT: To avoid further misunderstandings I am adding some background. Quantum averages can be obtained in a continuous basis |α⟩ as
⟨A⟩=∫dα⟨α|ˆρˆA|α⟩
(a) Introducing closure in the same basis |α⟩
⟨A⟩=∫dα∫dα′⟨α|ˆρ|α′⟩⟨α′|ˆA|α⟩=∫dα∫dα′ρ(α,α′)A(α′,α)
with the usual physical interpretation for the density matrix ρ(α,α′) as discussed above.
(b) Introducing closure in a second basis |β⟩, we obtain the alternative representation
⟨A⟩=∫dα∫dβ⟨α|ˆρ|β⟩⟨β|ˆA|α⟩=∫dα∫dβρ(α,β)A(β,α)
When the two basis are momentum |p⟩ and position |x⟩ the density ρ(p,x) is the well-known Wigner function whose physical interpretation is that of a pseudo-probability. My question is about the physical interpretation of ρ(α,β) in two arbitrary basis |α⟩, |β⟩.
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