🚀 go-pugleaf

Isham 3
"6.2.2  Conditional probabilities in Conventional Quantum Theory
... Let the (mixed) state of a quantum system at some time t = 0 be p(o).
In the Schrodinger representation, the state p(t) at time t is related to
the t=0 state by the unitary transformation

p(t) = U(t)p(0)U(t)^-1          (6.2.2)"

[In my quantum connection gedanken experiment motivated by a history
picture we need a nonlocal generalization

p(t,t')= U(t,t')p(0)U(t,t')^-1          (6.2.2')

for case of a pair of particles in which one particle is measured at t, the
other at t' - and we should further generalize to include spatial
locations. See Isham's eq. (6.2.8) below.]

"where U(t) = e^tH/h.  Therefore, if a measurement of an observable A is
made at time t1, the probability that the result will lie in some subset a
of the eigenvalue spectrum of the operator A is

Prob(A in a,t1;p(0)0 = tr(P(A,a,t1)p(t1))          (6.2.3)

where P(A,a,t1) is the Heisenberg-picture operator...

P(A,a,t1) = U(t1)^-1P(A,a)U(t1)                    (6.2.4)

... and P(A,a) is operator that projects onto the subset a; for example...

P(A,a) = Sum ai in a[|ai><ai|]                     (6.2.5)

If the measurement of A yields a result lying in a, any further predictions
must be made using the density matrix

p(a) = {P(A,a,t1)p(0)P(A,a,t1)}/tr[P(A,a,t1)p(o)]  (6.2.6)

and the transformation

p(t1) -> p(a)                                      (6.2.7)

is the [non-unitary! js] analog for density matrices of the familiar
reduction of the state vector.

Now let the system evolve until time t2 when a measurement of observable B
is made..... the probability of finding B in a range b, ...conditional on A
.. found to be in a at time t1, is

Prob(B in b, t2|A ina,t1;p(0)) = tr(P(B,b,t2)p(a)) (6.2.8)"