Probability Bracket Notation for Probability Modeling

Xing M. Wang; Tony C. Scott

doi:10.20944/preprints202404.1399.v1

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preprints.org > physical sciences > quantum science and technology > doi: 10.20944/preprints202404.1399.v1

Preprint Article Version 1 Preserved in Portico This version is not peer-reviewed

Probability Bracket Notation for Probability Modeling

Xing M. Wang

and

Tony C. Scott

Version 1 : Received: 22 April 2024 / Approved: 22 April 2024 / Online: 22 April 2024 (11:16:21 CEST)

How to cite: Wang, X.M.; Scott, T.C. Probability Bracket Notation for Probability Modeling. Preprints 2024, 2024041399. https://doi.org/10.20944/preprints202404.1399.v1 Wang, X.M.; Scott, T.C. Probability Bracket Notation for Probability Modeling. Preprints 2024, 2024041399. https://doi.org/10.20944/preprints202404.1399.v1

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MDPI and ACS Style

Wang, X.M.; Scott, T.C. Probability Bracket Notation for Probability Modeling. Preprints 2024, 2024041399. https://doi.org/10.20944/preprints202404.1399.v1

APA Style

Wang, X.M., & Scott, T.C. (2024). Probability Bracket Notation for Probability Modeling. Preprints. https://doi.org/10.20944/preprints202404.1399.v1

Chicago/Turabian Style

Wang, X.M. and Tony C. Scott. 2024 "Probability Bracket Notation for Probability Modeling" Preprints. https://doi.org/10.20944/preprints202404.1399.v1

Abstract

Following Dirac’s notation in Quantum Mechanics (QM), we propose the Probability Bracket Notation (PBN), by defining a probability-bra (P-bra), P-ket, P-bracket, P-identity, etc…Using the PBN, many formulas, such as normalizations and expectations in systems of one or more random variables, can now be written in abstract basis-independent expressions, easy to expand by inserting a proper P-identity. The time evolution of homogeneous Markov processes can also be formatted in such a way. Our system P-kets are identified with probability vectors and our P-bra system is comparable with Doi’s state function or Peliti’s standard bra. In the Heisenberg picture of the PBN, a random variable becomes a stochastic process, and the Chapman-Kolmogorov equations are obtained by inserting a time-dependent P-identity. Also, some QM expressions in Dirac notation are naturally transformed to probability expressions in PBN by a special Wick rotation. Potential Applications show the usefulness of the PBN beyond the constrained domain and range of Hermitian operators on Hilbert Spaces in QM all the way to IT.

Keywords

Bra-ket notation; Chapman-Kolmogorov Theorem; Information Technology; Markov processes; Quantum Mechanics; Wick rotation

Subject

Physical Sciences, Quantum Science and Technology

Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Comments (1)

Comment 1

Received: 26 April 2024
Commenter:

Tony Scott

Commenter's Conflict of Interests: I am one of the author

Comment: The impetus behind the creation of vector calculus was to express the laws of electromagnetism, but it was found to express so much more: fluid mechanics, the laws of elasticity and continuum mechanics, etc... A good notation may play a vital role in developing scientific theories. As it is well known, Dirac’ bracket notation (VBN) has been playing a crucial role in the development of Quantum theory.

Herein, we propose a Probability Bracket Notation (PBN) to play as a similar
tool for the probability modeling. It is found useful to be useful
to build a very useful bridge between classical and quantum worlds
(through special Wick rotation) and provide the possibility to treat
non-Hermitian Hamilton operators.

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