The Status of State Reduction by Steven Johnston The Status of State Reduction by Steven Johnston

Single-time Quantum Logic Physical System Hilbert space Orthocomplemented lattice of propositions Orthocomplemented lattice of projection operators

Propositions All propositions can be broken down to statements of the form: “The observable has a value in the range ” Lattice Operations: MEET(AND) ORTHOCOMPLEMENT (NOT) JOIN(OR) ORDERING(IMPLICATION)

Projectors

Lattice Operations (for commuting projectors): MEET ORTHOCOMPLEMENT JOIN ORDERING

Probabilities We require: Gleason’s Theorem: If the Hilbert space has dimension greater than two, the only such is given by:

Special Cases Definite proposition Indefinite proposition Result 1:

State Reduction A non-unitary time-evolution of the system’s state. Postulated to occur in a measurement. Result 2: No state reduction occurs if a proposition is definite in the initial state: true: false:

Histories Single-time quantum logic is extended using the “…and then…” sequential conjunction. A homogeneous history is of the form: An inhomogeneous history can’t be written in this form. e.g.

Multiple-time Quantum Logic Physical System (Considered over time) Hilbert space ? Orthocomplemented lattice of propositions Orthocomplemented lattice of projection operators ? What’s the correct way to represent history propositions?

History Projection Operator Formalism (HPO) Example (Negation): History propositions are represented by a projection operators on a tensor product Hilbert space. Homogeneous histories are of the form:

States in the HPO formalism Homogeneous history: System in initial state: State at first time: The history is true if each proposition is true at each time. If the history is true then the first proposition must be true: But no state reduction occurs for true propositions so the state at the second time is just: Repeat for the other times…

States in the HPO formalism In general is true if and only if: More compactly: for Extending to inhomogeneous histories as well, we get: Result 3: A general history is true if and only if: A general history is false if and only if:

The Status of State Reduction The key to understanding state reduction is the condition for a history to be false in an initial state: A general history is false if and only if: If is homogeneous then this means for some time we have: Propositions at other times can be indefinite. The appropriate state to evaluate each proposition at each time is just the time-evolved state. No state reduction occurs! Even for indefinite propositions!

Probabilities for Histories Probabilities are usually assigned using decoherence functionals. Their form is motivated by state reduction. Without state reduction, what’s the appropriate probability assignment? Use Gleason’s theorem: for and homogeneous history we get:

Conclusions Don’t postulate state reduction. A new probability assignment for history propositions is given by: where Thanks for listening!