Quantum Probability 1982-2017 Hans Maassen, Universities of Nijmegen and Amsterdam Nijmegen, June 23, 2017. 1582: Pope Gregory XIII audaciously replaces the old-fashioned Julian calender by the new Gregorian calender, which still prevails. 1982: A bunch of mathematicians and physicists replaces the old-fashioned Kolmogorovian probability axioms by new ones: quantum probability, which still prevail. Two historic events 1982: A bunch of mathematicians and physicists replaces the old-fashioned Kolmogorovian probability axioms by new ones: quantum probability, which still prevail. Two historic events 1582: Pope Gregory XIII audaciously replaces the old-fashioned Julian calender by the new Gregorian calender, which still prevails. Two historic events 1582: Pope Gregory XIII audaciously replaces the old-fashioned Julian calender by the new Gregorian calender, which still prevails. 1982: A bunch of mathematicians and physicists replaces the old-fashioned Kolmogorovian probability axioms by new ones: quantum probability, which still prevail. Two historic events 1582: Pope Gregory XIII audaciously replaces the old-fashioned Julian calender by the new Gregorian calender, which still prevails. 1982: A bunch of mathematicians and physicists replaces the old-fashioned Kolmogorovian probability axioms by new ones: quantum probability, which still prevail. ∗ 400 years time difference; ∗ They happened in the same hall, in Villa Mondragone, 30 km south-west of Rome, in the Frascati hills. Villa Mondragone First conference in quantum probability, 1982 How are these two events related? ∗ They happened in the same hall, in Villa Mondragone, 30 km south-west of Rome, in the Frascati hills. Villa Mondragone First conference in quantum probability, 1982 How are these two events related? ∗ 400 years time difference; Villa Mondragone First conference in quantum probability, 1982 How are these two events related? ∗ 400 years time difference; ∗ They happened in the same hall, in Villa Mondragone, 30 km south-west of Rome, in the Frascati hills. Villa Mondragone First conference in quantum probability, 1982 How are these two events related? ∗ 400 years time difference; ∗ They happened in the same hall, in Villa Mondragone, 30 km south-west of Rome, in the Frascati hills. How are these two events related? ∗ 400 years time difference; ∗ They happened in the same hall, in Villa Mondragone, 30 km south-west of Rome, in the Frascati hills. Villa Mondragone First conference in quantum probability, 1982 (A;') to replace(Ω ; Σ; µ) Quantum probability space A: C*-algebra containing 1l (of operators on a Hilbert space). ': state (positive complex functional on A with '(1l) = 1). Interpretation: p 2 A projection ! question, proposition; 1l − p ! not-p ; pq = qp ! p and q are compatible propositions; pq ! p and q (if compatible); p + q − pq ! p or q (if compatible); p1 + ::: + pn = 1l ! exactly one of p1;::: pn holds : The new scheme to replace(Ω ; Σ; µ) Quantum probability space A: C*-algebra containing 1l (of operators on a Hilbert space). ': state (positive complex functional on A with '(1l) = 1). Interpretation: p 2 A projection ! question, proposition; 1l − p ! not-p ; pq = qp ! p and q are compatible propositions; pq ! p and q (if compatible); p + q − pq ! p or q (if compatible); p1 + ::: + pn = 1l ! exactly one of p1;::: pn holds : The new scheme (A;') Quantum probability space A: C*-algebra containing 1l (of operators on a Hilbert space). ': state (positive complex functional on A with '(1l) = 1). Interpretation: p 2 A projection ! question, proposition; 1l − p ! not-p ; pq = qp ! p and q are compatible propositions; pq ! p and q (if compatible); p + q − pq ! p or q (if compatible); p1 + ::: + pn = 1l ! exactly one of p1;::: pn holds : The new scheme (A;') to replace(Ω ; Σ; µ) A: C*-algebra containing 1l (of operators on a Hilbert space). ': state (positive complex functional on A with '(1l) = 1). Interpretation: p 2 A projection ! question, proposition; 1l − p ! not-p ; pq = qp ! p and q are compatible propositions; pq ! p and q (if compatible); p + q − pq ! p or q (if compatible); p1 + ::: + pn = 1l ! exactly one of p1;::: pn holds : The new scheme (A;') to replace(Ω ; Σ; µ) Quantum probability space ': state (positive complex functional on A with '(1l) = 1). Interpretation: p 2 A projection ! question, proposition; 1l − p ! not-p ; pq = qp ! p and q are compatible propositions; pq ! p and q (if compatible); p + q − pq ! p or q (if compatible); p1 + ::: + pn = 1l ! exactly one of p1;::: pn holds : The new scheme (A;') to replace(Ω ; Σ; µ) Quantum probability space A: C*-algebra containing 1l (of operators on a Hilbert space). Interpretation: p 2 A projection ! question, proposition; 1l − p ! not-p ; pq = qp ! p and q are compatible propositions; pq ! p and q (if compatible); p + q − pq ! p or q (if compatible); p1 + ::: + pn = 1l ! exactly one of p1;::: pn holds : The new scheme (A;') to replace(Ω ; Σ; µ) Quantum probability space A: C*-algebra containing 1l (of operators on a Hilbert space). ': state (positive complex functional on A with '(1l) = 1). p 2 A projection ! question, proposition; 1l − p ! not-p ; pq = qp ! p and q are compatible propositions; pq ! p and q (if compatible); p + q − pq ! p or q (if compatible); p1 + ::: + pn = 1l ! exactly one of p1;::: pn holds : The new scheme (A;') to replace(Ω ; Σ; µ) Quantum probability space A: C*-algebra containing 1l (of operators on a Hilbert space). ': state (positive complex functional on A with '(1l) = 1). Interpretation: 1l − p ! not-p ; pq = qp ! p and q are compatible propositions; pq ! p and q (if compatible); p + q − pq ! p or q (if compatible); p1 + ::: + pn = 1l ! exactly one of p1;::: pn holds : The new scheme (A;') to replace(Ω ; Σ; µ) Quantum probability space A: C*-algebra containing 1l (of operators on a Hilbert space). ': state (positive complex functional on A with '(1l) = 1). Interpretation: p 2 A projection ! question, proposition; pq = qp ! p and q are compatible propositions; pq ! p and q (if compatible); p + q − pq ! p or q (if compatible); p1 + ::: + pn = 1l ! exactly one of p1;::: pn holds : The new scheme (A;') to replace(Ω ; Σ; µ) Quantum probability space A: C*-algebra containing 1l (of operators on a Hilbert space). ': state (positive complex functional on A with '(1l) = 1). Interpretation: p 2 A projection ! question, proposition; 1l − p ! not-p ; pq ! p and q (if compatible); p + q − pq ! p or q (if compatible); p1 + ::: + pn = 1l ! exactly one of p1;::: pn holds : The new scheme (A;') to replace(Ω ; Σ; µ) Quantum probability space A: C*-algebra containing 1l (of operators on a Hilbert space). ': state (positive complex functional on A with '(1l) = 1). Interpretation: p 2 A projection ! question, proposition; 1l − p ! not-p ; pq = qp ! p and q are compatible propositions; p + q − pq ! p or q (if compatible); p1 + ::: + pn = 1l ! exactly one of p1;::: pn holds : The new scheme (A;') to replace(Ω ; Σ; µ) Quantum probability space A: C*-algebra containing 1l (of operators on a Hilbert space). ': state (positive complex functional on A with '(1l) = 1). Interpretation: p 2 A projection ! question, proposition; 1l − p ! not-p ; pq = qp ! p and q are compatible propositions; pq ! p and q (if compatible); p1 + ::: + pn = 1l ! exactly one of p1;::: pn holds : The new scheme (A;') to replace(Ω ; Σ; µ) Quantum probability space A: C*-algebra containing 1l (of operators on a Hilbert space). ': state (positive complex functional on A with '(1l) = 1). Interpretation: p 2 A projection ! question, proposition; 1l − p ! not-p ; pq = qp ! p and q are compatible propositions; pq ! p and q (if compatible); p + q − pq ! p or q (if compatible); The new scheme (A;') to replace(Ω ; Σ; µ) Quantum probability space A: C*-algebra containing 1l (of operators on a Hilbert space). ': state (positive complex functional on A with '(1l) = 1). Interpretation: p 2 A projection ! question, proposition; 1l − p ! not-p ; pq = qp ! p and q are compatible propositions; pq ! p and q (if compatible); p + q − pq ! p or q (if compatible); p1 + ::: + pn = 1l ! exactly one of p1;::: pn holds : From probability: ∗ `We understand that random variables form an algebra, ∗ but what is this funny non-commutativity? ∗ Is not Kolmogorov good enough?' From quantum physics: ∗ `We understand that observables do not commute, ∗ but what is the role of this funny algebra? ∗ Is not Hilbert space good enough?' In general: ∗ `This is nothing new, just operator algebras in quantum mechanics. ∗ Where is the innovation?' Reactions to the new framework ∗ `We understand that random variables form an algebra, ∗ but what is this funny non-commutativity? ∗ Is not Kolmogorov good enough?' From quantum physics: ∗ `We understand that observables do not commute, ∗ but what is the role of this funny algebra? ∗ Is not Hilbert space good enough?' In general: ∗ `This is nothing new, just operator algebras in quantum mechanics. ∗ Where is the innovation?' Reactions to the new framework From probability: ∗ but what is this funny non-commutativity? ∗ Is not Kolmogorov good enough?' From quantum physics: ∗ `We understand that observables do not commute, ∗ but what is the role of this funny algebra? ∗ Is not Hilbert space good enough?' In general: ∗ `This is nothing new, just operator algebras in quantum mechanics. ∗ Where is the innovation?' Reactions to the new framework From probability: ∗ `We understand that random variables form an algebra, ∗ Is not Kolmogorov good enough?' From quantum physics: ∗ `We understand that observables do not commute, ∗ but what is the role of this funny algebra? ∗ Is not Hilbert space good enough?' In general: ∗ `This is nothing new, just operator algebras in quantum mechanics. ∗ Where is the innovation?' Reactions to the new framework From probability: ∗ `We understand that random variables form an algebra, ∗ but what is this funny non-commutativity? From quantum physics: ∗ `We understand that observables do not commute, ∗ but what is the role of this funny algebra? ∗ Is not Hilbert space good enough?' In general: ∗ `This is nothing new, just operator algebras in quantum mechanics.