Unreal Probabilities

by Carlos C. Rodriguez

Dept. of MathematicsThe University at Albanycarlos@math.albany.edu

Real Probability

P(Whole) = 1 P(a+b) = P(a) + P(b) if ab=ú

P : A [0,1]

Real Probability

P(Whole) = 1 P(a+b) = P(a) + P(b) if ab=ú

P : A [0,1]

The Axioms

• A, Boolean algebra of propositions: a,b,c...• G, Clifford algebra: u,v,M,N...

Operations:Operations:OR : a+b AND: ab NOT: a_ 0 = FALSE 1= TRUE

Ø Defined

Hilbert Space of Ø’sHilbert Space of Ø’s

The Truth with Ø

Real TruthReal Truth

Independence

Ø makes a and b independent given c if

• symmetric• includes usual formula• generalizes to more than two propositions

Weak Law of Large Numbers

AA BB

ExampleExample

QM

x

t

xü

tü

Ú Ú»tÁ

(Û , tª)

Boolean algebra of filters: e(x,t) = “x open at time t”

“one filter on top of the other”

Independent segments

t

independent given

QM Dictionary

Amplitude Amplitude

Wave functionWave function

HamiltonianHamiltonian

i@=-1 i commutes will all the Ø’s

Schrödinger Equation

( generalized for possible non commuting Ø’s

with independent segments)

Ed, what do you think ?