algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)
quantum mechanical system, quantum probability
interacting field quantization
In the context of quantum mechanics, the collapse of the wave function, also known as the reduction of the wave packet, is said to occur after observation or measurement, when a wave function expressed as the sum of eigenfunctions of the observable is projected randomly onto one of them. Different interpretations of quantum mechanics understand this process differently.
The perspective associated with the Bayesian interpretation of quantum mechanics observes (see below) that the apparent collapse is just the mathematical reflection of the formula for conditional expectation values in quantum probability theory.
There is a close relation between wave function collapse and conditional expectation values in quantum probability (e.g. Kuperberg 05, section 1.2, Yuan 12):
Let $(\mathcal{A},\langle -\rangle)$ be a quantum probability space, hence a complex star algebra $\mathcal{A}$ of quantum observables, and a state on a star-algebra $\langle -\rangle \;\colon\; \mathcal{A} \to \mathbb{C}$.
This means that for $A \in \mathcal{A}$ any observable, its expectation value in the given state is
More generally, if $P \in \mathcal{A}$ is a real idempotent/projector
thought of as an event, then for any observable $A \in \mathcal{A}$ the conditional expectation value of $A$, conditioned on the observation of $P$, is (e.g. Redei-Summers 06, section 7.3)
Now assume a star-representation $\rho \;\colon\; \mathcal{A} \to End(\mathcal{H})$ of the algebra of observables by linear operators on a Hilbert space $\mathcal{H}$ is given, and that the state $\langle -\rangle$ is a pure state, hence given by a vector $\psi \in \mathcal{H}$ (βwave functionβ) via the Hilbert space inner product $\langle (-), (-)\rangle \;\colon\; \mathcal{H} \otimes \mathcal{H} \to \mathbb{C}$ as
In this case the expression for the conditional expectation value (2) of an observable $A$ conditioned on an idempotent observable $P$ becomes (notationally suppressing the representation $\rho$)
where in the last step we used (2).
This says that assuming that $P$ has been observed in the pure state $\vert \psi\rangle$, then the corresponding conditional expectation values are the same as actual expectation values but for the new pure state $\vert P \psi \rangle$.
This is the statement of βwave function collapseβ. The original wave function is $\psi \in \mathcal{H}$, and after observing $P$ it βcollapsesβ to $P \psi \in \mathcal{H}$ (up to normalization).
Discussion from the point of view of quantum probability includes
Greg Kuperberg, A concise introduction to quantum probability, quantum mechanics, and quantum computation, 2005 (pdf)
Miklos Redei, Stephen Summers, Quantum Probability Theory (arXiv:quant-ph/0601158)
Qiaochu Yuan, Finite noncommutative probability, the Born rule, and wave function collapse, 2012
See also
Wikipedia, Wave function collapse
Wikipedia, Born rule
Last revised on September 12, 2018 at 10:50:31. See the history of this page for a list of all contributions to it.