nLab
wave function collapse

Contents

Context

Algebraic Quantum Field Theory

algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)

Introduction

Concepts

field theory:

Lagrangian field theory

quantization

quantum mechanical system, quantum probability

free field quantization

gauge theories

interacting field quantization

renormalization

Theorems

States and observables

Operator algebra

Local QFT

Perturbative QFT

Contents

Idea

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.

Relation to conditional expectation values

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 (π’œ,βŸ¨βˆ’βŸ©)(\mathcal{A},\langle -\rangle) be a quantum probability space, hence a complex star algebra π’œ\mathcal{A} of quantum observables, and a state on a star-algebra βŸ¨βˆ’βŸ©:π’œβ†’β„‚\langle -\rangle \;\colon\; \mathcal{A} \to \mathbb{C}.

This means that for Aβˆˆπ’œA \in \mathcal{A} any observable, its expectation value in the given state is

𝔼(A)β‰”βŸ¨AβŸ©βˆˆβ„‚. \mathbb{E}(A) \;\coloneqq\; \langle A \rangle \in \mathbb{C} \,.

More generally, if Pβˆˆπ’œP \in \mathcal{A} is a real idempotent/projector

(1)P *=P,AAAPP=P P^\ast = P \,, \phantom{AAA} P P = P

thought of as an event, then for any observable Aβˆˆπ’œA \in \mathcal{A} the conditional expectation value of AA, conditioned on the observation of PP, is (e.g. Redei-Summers 06, section 7.3)

(2)𝔼(A|P)β‰”βŸ¨PAP⟩⟨P⟩. \mathbb{E}(A \vert P) \;\coloneqq\; \frac{ \left \langle P A P \right\rangle }{ \left\langle P \right\rangle } \,.

Now assume a star-representation ρ:π’œβ†’End(β„‹)\rho \;\colon\; \mathcal{A} \to End(\mathcal{H}) of the algebra of observables by linear operators on a Hilbert space β„‹\mathcal{H} is given, and that the state βŸ¨βˆ’βŸ©\langle -\rangle is a pure state, hence given by a vector Οˆβˆˆβ„‹\psi \in \mathcal{H} (β€œwave function”) via the Hilbert space inner product ⟨(βˆ’),(βˆ’)⟩:β„‹βŠ—β„‹β†’β„‚\langle (-), (-)\rangle \;\colon\; \mathcal{H} \otimes \mathcal{H} \to \mathbb{C} as

⟨A⟩ β‰”βŸ¨Οˆ|A|ψ⟩ β‰”βŸ¨Οˆ,Aψ⟩. \begin{aligned} \langle A \rangle & \coloneqq \left\langle\psi \vert A \vert \psi \right\rangle \\ & \coloneqq \left\langle\psi, A \psi \right\rangle \end{aligned} \,.

In this case the expression for the conditional expectation value (2) of an observable AA conditioned on an idempotent observable PP becomes (notationally suppressing the representation ρ\rho)

𝔼(A|P) =⟨ψ|PAP|ψ⟩⟨ψ|P|ψ⟩ =⟨Pψ|A|Pψ⟩⟨Pψ|Pψ⟩, \begin{aligned} \mathbb{E}(A\vert P) & = \frac{ \left\langle \psi \vert P A P\vert \psi \right\rangle }{ \left\langle \psi \vert P \vert \psi \right\rangle } \\ & = \frac{ \left\langle P \psi \vert A \vert P \psi \right\rangle }{ \left\langle P \psi \vert P \psi \right\rangle } \,, \end{aligned}

where in the last step we used (2).

This says that assuming that PP has been observed in the pure state |ψ⟩\vert \psi\rangle, then the corresponding conditional expectation values are the same as actual expectation values but for the new pure state |Pψ⟩\vert P \psi \rangle.

This is the statement of β€œwave function collapse”. The original wave function is Οˆβˆˆβ„‹\psi \in \mathcal{H}, and after observing PP it β€œcollapses” to PΟˆβˆˆβ„‹P \psi \in \mathcal{H} (up to normalization).

References

Discussion from the point of view of quantum probability includes

See also

Last revised on September 12, 2018 at 10:50:31. See the history of this page for a list of all contributions to it.