algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)
quantum mechanical system, quantum probability
interacting field quantization
In probability theory, the concept of noncommutative probability space or quantum probability space is the generalization of that of probability space as the concept of “space” is generalized to non-commutative geometry.
The basic idea is to encode a would-be probability space dually in its algebra of functions $\mathcal{A}$, typically regarded as a star algebra, and encode the probability measure as a state on this star algebra
Hence this primarily axiomatizes the concept of expectation values $\langle A\rangle$ (Segal 65, Whittle 92) while leaving the nature of the underlying probability measure implicit (in contrast to the classical formalization of probability theory by Andrey Kolmogorov).
Often $\mathcal{A}$ is assumed/required to be a von Neumann algebra (e.g. Kuperberg 05, section 1.8). Often $\mathcal{A}$ is taken to be the full algebra of bounded operators on some Hilbert space (e.g. Attal, def. 7.1).
In quantum physics, $\mathcal{A}$ is an algebra of observables (or a local net thereof) and $\langle (-)\rangle$ is a particular quantum state, for instance a vacuum state.
The formulation of non-perturbative quantum field theory from the algebraic perspective of quantum probability is known as algebraic quantum field theory (AQFT).
The formulation of perturbative quantum field theory from the algebraic perspective of quantum probability is known as perturbative algebraic quantum field theory (pAQFT).
The sentiment that quantum physics is quantum probability theory is also referred to as the Bayesian interpretation of quantum mechanics (“QBism”).
The idea that
quantum probability is “just as” classical probability theory but generalized to non-commutative probability spaces, hence, for quantum physics, to quantized phase spaces
may be made precise and fully manifest by understanding quantum probability theory as being classical probability theory internal to the Bohr topos of the given quantum mechanical system.
For details see at Bohr topos the section Kinematics in a Bohr topos.
For going deeper, see at order-theoretic structure in quantum mechanics.
Quantum probability theory shows that “wave function collapse” is just part of the formula for conditional expectation values in quantum probability theory (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, see also Fröhlich-Schubnel 15, (5.49), Fröhlich 19 (45))
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 an vector $\psi \in \mathcal{H}$ (“wave function”) 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 (1).
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 “wave function collapse”:
The original wave function is $\psi \in \mathcal{H}$, and after observing $P$ it “collapses” to $P \psi \in \mathcal{H}$ (up to normalization).
The axiomatization of probability theory in terms of the concept of expectation values (instead of probability measures) is amplified in
Irving Segal, Algebraic integration theory, Bull. Amer. Math. Soc. Volume 71, Number 3, Part 1 (1965), 419-489 (eculid:1183526903)
Peter Whittle, Probability via expectation, Springer 1992 (doi:10.1007/978-1-4612-0509-8)
Introductions to quantum probability theory
Greg Kuperberg, A concise introduction to quantum probability, quantum mechanics, and quantum computation, 2005 (pdf)
Miklos Redei, Stephen Summers, Quantum Probability Theory, Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics Volume 38, Issue 2, June 2007, Pages 390-417 (arXiv:quant-ph/0601158, doi:10.1016/j.shpsb.2006.05.006)
S. Attal, Quantum probability (pdf)
Jonathan Gleason, The $C^*$-algebraic formalism of quantum mechanics, 2009 (pdf, pdf)
Qiaochu Yuan, Finite noncommutative probability, the Born rule, and wave function collapse, 2012
A. Ibort, V.I. Manko, G. Marmo, A. Simoni, F. Ventriglia, A pedagogical presentation of a $C^\ast$-algebraic approach to quantum tomography, Phys. Scr., 84 (2011) 065006 (arXiv:1204.5231)
Jürg Fröhlich, B. Schubnel, Quantum Probability Theory and the Foundations of Quantum Mechanics. In: Blanchard P., Fröhlich J. (eds.) The Message of Quantum Science. Lecture Notes in Physics, vol 899. Springer 2015 (arXiv:1310.1484, doi:10.1007/978-3-662-46422-9_7)
Jürg Fröhlich, The structure of quantum theory, Chapter 6 in The quest for laws and structure, EMS 2016 (doi, doi:10.4171/164-1/8)
Jan Swart, Introduction to Quantum Probability, 2017 (pdf, pdf)
Klaas Landsman, Foundations of quantum theory – From classical concepts to Operator algebras, Springer Open 2017 (doi:10.1007/978-3-319-51777-3, pdf)
Further discussion of quantum probability:
Last revised on February 18, 2020 at 00:35:12. See the history of this page for a list of all contributions to it.