Histories

A homogeneous history ${\displaystyle H_{i}}$ (here ${\displaystyle i}$ labels different histories) is a sequence of Propositions ${\displaystyle P_{i,j}}$ specified at different moments of time ${\displaystyle t_{i,j}}$ (here ${\displaystyle j}$ labels the times). We write this as:

${\displaystyle H_{i}=(P_{i,1},P_{i,2},\ldots ,P_{i,n_{i}})}$

and read it as "the proposition ${\displaystyle P_{i,1}}$ is true at time ${\displaystyle t_{i,1}}$ and then the proposition ${\displaystyle P_{i,2}}$ is true at time ${\displaystyle t_{i,2}}$ and then ${\displaystyle \ldots }$". The times ${\displaystyle t_{i,1}<t_{i,2}<\ldots <t_{i,n_{i}}}$ are strictly ordered and called the temporal support of the history.

Inhomogeneous histories are multiple-time propositions which cannot be represented by a homogeneous history. An example is the logical OR of two homogeneous histories: ${\displaystyle H_{i}\lor H_{j}}$.

These propositions can correspond to any set of questions that include all possibilities. Examples might be the three propositions meaning "the electron went through the left slit", "the electron went through the right slit" and "the electron didn't go through either slit". One of the aims of the approach is to show that classical questions such as, "where are my keys?" are consistent. In this case one might use a large number of propositions each one specifying the location of the keys in some small region of space.

Each single-time proposition ${\displaystyle P_{i,j}}$ can be represented by a projection operator ${\displaystyle {\hat {P}}_{i,j}}$ acting on the system's Hilbert space (we use "hats" to denote operators). It is then useful to represent homogeneous histories by the time-ordered product of their single-time projection operators. This is the history projection operator (HPO) formalism developed by Christopher Isham and naturally encodes the logical structure of the history propositions.

Consistency

An important construction in the consistent histories approach is the class operator for a homogeneous history:

${\displaystyle {\hat {C}}_{H_{i}}:=T\prod _{j=1}^{n_{i}}{\hat {P}}_{i,j}(t_{i,j})={\hat {P}}_{i,n_{i}}\cdots {\hat {P}}_{i,2}{\hat {P}}_{i,1}}$

The symbol ${\displaystyle T}$ indicates that the factors in the product are ordered chronologically according to their values of ${\displaystyle t_{i,j}}$: the "past" operators with smaller values of ${\displaystyle t}$ appear on the right side, and the "future" operators with greater values of ${\displaystyle t}$ appear on the left side. This definition can be extended to inhomogeneous histories as well.

Central to the consistent histories is the notion of consistency. A set of histories ${\displaystyle \{H_{i}\}}$ is consistent (or strongly consistent) if

${\displaystyle \operatorname {Tr} ({\hat {C}}_{H_{i}}\rho {\hat {C}}_{H_{j}}^{\dagger })=0}$

for all ${\displaystyle i\neq j}$. Here ${\displaystyle \rho }$ represents the initial density matrix, and the operators are expressed in the Heisenberg picture.

The set of histories is weakly consistent if

${\displaystyle \operatorname {Tr} ({\hat {C}}_{H_{i}}\rho {\hat {C}}_{H_{j}}^{\dagger })\approx 0}$

for all ${\displaystyle i\neq j}$.

Probabilities

If a set of histories is consistent then probabilities can be assigned to them in a consistent way. We postulate that the probability of history ${\displaystyle H_{i}}$ is simply

${\displaystyle \operatorname {Pr} (H_{i})=\operatorname {Tr} ({\hat {C}}_{H_{i}}\rho {\hat {C}}_{H_{i}}^{\dagger })}$

which obeys the axioms of probability if the histories ${\displaystyle H_{i}}$ come from the same (strongly) consistent set.

As an example, this means the probability of "${\displaystyle H_{i}}$ OR ${\displaystyle H_{j}}$" equals the probability of "${\displaystyle H_{i}}$" plus the probability of "${\displaystyle H_{j}}$" minus the probability of "${\displaystyle H_{i}}$ AND ${\displaystyle H_{j}}$", and so forth.