Noncommutative Mathematics for Quantum Systems

58 Noncommutative Mathematics for Quantum Systems

then X+Y is essentially self-adjoint and the distribution w.r.t.Ωof its
closure is equal to the additive free convolution of the distributions of X
and Y w.r.t. toΩ, i.e.

L(X+Y,Ω) =L(X,Ω)
L(Y,Ω).

One can show that the reciprocal Cauchy transform Fμ of a
probability measureμ ∈ M 1 (R)is invertible in an appropriate
domain, and define the so-calledVoiculescu transformofμas

φμ(z) =F〈−^1 〉(z)−z,

whereF〈−^1 〉denotes the inverse ofF. This transform linearizes the
free convolution, we have

φμ
ν=φμ+φν.

as can be checked using the ‘parallelogram equation’ in Theorem
1.7.3.

Example 1.7.5 Consider the Bernoulli distribution

μ=

1

2

(δ+ 1 +δ− 1 ).

Then we compute

Gμ(z) =

z

z^2 − 1

and Fμ(z) =z−

1

z

.

Therefore,

Fμ〈−^1 〉(z) =

1

2

(

z+

√

z^2 + 4

)

and

φμ
μ(z) = 2 φμ(z) = 2

(

Fμ〈−^1 〉(z)−z)

)

=

√

z^2 + 4 −z

forzwith Im(z)>1. We get

Fμ
μ(z) =

√

z^2 − 4

and see that the free convolution of two Bernoulli distributions is
an arcsine distribution,

μ
μ=

1

π

1

√

4 −x^2

(^1) ]−2,2[(x)dx.