Noncommutative Mathematics for Quantum Systems

32 Noncommutative Mathematics for Quantum Systems

∆f(g 1 ,g 2 ) =f(g 1 g 2 ),

forg 1 ,g 2 ∈Gandε(f) =f(e).

Definition 1.5.3 LetBbe an involutive bialgebra. A quantum
stochastic process(jst) 0 ≤s≤tonBover some quantum probability
space (A,Φ) is called a L ́evy process, if the following four
conditions are satisfied.

(i) (Increment property) We have

jrs?jst = jrt for all 0≤r≤s≤t,

jtt = ε 1 for all 0≤t,

that is,jtt(b) =ε(b) 1 for allb∈ B, where 1 denotes the unit

ofA.

(ii) (Independence of increments) The family (jst) 0 ≤s≤t is

independent, that is, the quantum random variablesjs 1 ,t 1 ,... ,

jsntn are tensor-independent for all n ∈ N and all

0 ≤s 1 ≤t 1 ≤s 2 ≤···≤tn.

(iii) (Stationarity of increments) The distributionφst=Φ◦jstofjst

depends only on the differencet−s.

(iv) (Weak continuity) The quantum random variables jst

converge tojssin distribution fort↘s, that is,

lim

t↘s

Φ◦jst(b) =Φ◦jss(b)

for allb∈B.

Recall that an(involutive) Hopf algebra(B,∆,ε,S)is an (involutive)
bialgebra(B,∆,ε)equipped with a linear map calledantipode S :
B →Bsatisfying

S?id= 1 ◦ε=id?S. (1.5.1)

The antipode is unique, if it exists. Furthermore, it is an algebra and
coalgebra anti-homomorphism, that is, it satisfiesS(ab) =S(b)S(a)
for alla,b∈Band(S⊗S)◦∆=τ◦∆◦S, whereτ:B⊗B →B⊗B
is theflipτ(a⊗b) =b⊗a. If(B,∆,ε)is an involutive bialgebra and
S:B → Ba linear map satisfying (1.5.1), thenSsatisfies also the
relation

S◦∗◦S◦∗=id.