{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:20:54Z","timestamp":1760145654349,"version":"build-2065373602"},"reference-count":29,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2024,8,20]],"date-time":"2024-08-20T00:00:00Z","timestamp":1724112000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Following Dirac\u2019s 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 formulae, such as normalizations and expectations in systems of one or more random variables, can now be written in abstract basis-independent expressions, which are 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\u2019s state function or Peliti\u2019s standard bra. In the Heisenberg picture of the PBN, a random variable becomes a stochastic process, and the Chapman\u2013Kolmogorov 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.<\/jats:p>","DOI":"10.3390\/axioms13080564","type":"journal-article","created":{"date-parts":[[2024,8,20]],"date-time":"2024-08-20T01:38:45Z","timestamp":1724117925000},"page":"564","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Probability Bracket Notation for Probability Modeling"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8673-925X","authenticated-orcid":false,"given":"Xing M.","family":"Wang","sequence":"first","affiliation":[{"name":"Sherman Visual Lab, Sunnyvale, CA 94085, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0228-1673","authenticated-orcid":false,"given":"Tony C.","family":"Scott","sequence":"additional","affiliation":[{"name":"Institut f\u00fcr Physikalische Chemie, RWTH Aachen University, 52056 Aachen, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,8,20]]},"reference":[{"key":"ref_1","unstructured":"Liboff, R.L. (1992). Introductory Quantum Mechanics, Addison-Wesley Publishing Company."},{"key":"ref_2","unstructured":"Rudin, W. (1973). Functional Analysis, McGraw-Hill."},{"key":"ref_3","unstructured":"Grinstead, C.M., and Snell, J.L. (1997). Introduction to Probability, American Mathematical Society."},{"key":"ref_4","unstructured":"Ross, S.M. (2006). Introduction to Probability Models, ISE, Introduction to Probability Models, Academic Press. [9th ed.]."},{"key":"ref_5","unstructured":"Ye, E., and Zhang, D. (2005). Probability Theory and Stochastic Processes, Science Publications."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1465","DOI":"10.1088\/0305-4470\/9\/9\/008","article-title":"Second quantization representation for classical many-particle system","volume":"9","author":"Doi","year":"1976","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"51121","DOI":"10.1103\/PhysRevE.74.051121","article-title":"Master equation and two heat reservoirs","volume":"74","author":"Trimper","year":"2006","journal-title":"Phys. Rev. E"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1469","DOI":"10.1051\/jphys:019850046090146900","article-title":"Path integral approach to birth-death processes on a lattice","volume":"46","author":"Peliti","year":"1985","journal-title":"J. Phys. France"},{"key":"ref_9","unstructured":"Garcia-Palacios, J.L. (2007). Introduction to the Theory of Stochastic Processes and Brownian motion problems. arXiv."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"033018","DOI":"10.1088\/1367-2630\/15\/3\/033018","article-title":"Controlled double-slit electron diffraction","volume":"15","author":"Bach","year":"2013","journal-title":"New J. Physics"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"485","DOI":"10.1103\/PhysRev.115.485","article-title":"Significance of Electromagnetic Potentials in the Quantum Theory","volume":"115","author":"Aharonov","year":"1959","journal-title":"Phys. Rev."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Kleinert, H. (2009). Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, World Scientific. [5th ed.]. Available online: https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/7305.","DOI":"10.1142\/9789814273572"},{"key":"ref_13","unstructured":"Berestycki, N., and Sousi, P. (2024, August 10). Applied Probability-Online Lecture Notes. Available online: http:\/\/www.statslab.cam.ac.uk\/~ps422\/notes-new.pdf."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"633","DOI":"10.1119\/1.18168","article-title":"Introduction to the diffusion Monte Carlo method","volume":"64","author":"Kosztin","year":"1996","journal-title":"Am. J. Phys."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"287","DOI":"10.1007\/s12043-009-0120-x","article-title":"Classical and quantum mechanics of complex hamiltonian systems: An extended complex phase space approach","volume":"73","author":"Kaushal","year":"2009","journal-title":"Pramana"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Rajeev, S.G. (2007). Dissipative Mechanics Using Complex-Valued Hamiltonians. arXiv.","DOI":"10.1016\/j.aop.2007.02.004"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"L167","DOI":"10.1088\/0022-3700\/14\/5\/002","article-title":"The calculation of molecular resonances by complex scaling","volume":"14","author":"Morgan","year":"1981","journal-title":"J. Phys. B At. Mol. Opt. Phys."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1087","DOI":"10.1080\/713820680","article-title":"The Finitely Conducting Lamellar Diffraction Grating","volume":"28","author":"Botten","year":"1981","journal-title":"Opt. Acta Int. J. Opt."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Zloshchastiev, K.G. (2016). Quantum-statistical approach to electromagnetic wave propagation and dissipation inside dielectric media and nanophotonic and plasmonic waveguides. Phys. Rev. B, 94.","DOI":"10.1103\/PhysRevB.94.115136"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Zloshchastiev, K.G. (2024). Generalization of the Schr\u00f6dinger Equation for Open Systems Based on the Quantum-Statistical Approach. Universe, 10.","DOI":"10.3390\/universe10010036"},{"key":"ref_21","unstructured":"Zloshchastiev, K. (2024, August 10). PROJECT Density Operator Approach for non-Hermitian Hamiltonians. Available online: https:\/\/www.researchgate.net\/publication\/369022872_PROJECT_Density_Operator_Approach_for_non-Hermitian_Hamiltonians."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"033102","DOI":"10.1088\/1742-5468\/2016\/03\/033102","article-title":"Quantum entropy of systems described by non-Hermitian Hamiltonians","volume":"2016","author":"Sergi","year":"2016","journal-title":"J. Stat. Mech. Theory Exp."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"486","DOI":"10.1038\/nature09801","article-title":"An open-system quantum simulator with trapped ions","volume":"470","author":"Barreiro","year":"2011","journal-title":"Nature"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Zheng, C. (2021). Universal quantum simulation of single-qubit nonunitary operators using duality quantum algorithm. Sci. Rep., 11.","DOI":"10.1038\/s41598-021-83521-5"},{"key":"ref_25","unstructured":"Fertik, M.B., Scott, T., and Dignan, T. (2014). Identifying Information Related to a Particular Entity from Electronic Sources, Using Dimensional Reduction and Quantum Clustering. (No. 8,744,197), U.S. Patent."},{"key":"ref_26","unstructured":"Wang, S., and Dignan, T.G. (2014). Thematic Clustering. (No. 888,665,1 B1), U.S. Patent."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Scott, T.C., Therani, M., and Wang, X.M. (2017). Data Clustering with Quantum Mechanics. Mathematics, 5.","DOI":"10.3390\/math5010005"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"33","DOI":"10.5121\/ijdkp.2021.11103","article-title":"A Comprehensive Analysis of Quantum Clustering: Finding All the Potential Minima","volume":"11","author":"Maignan","year":"2021","journal-title":"Int. J. Data Min. Knowl. Manag. Process"},{"key":"ref_29","first-page":"2308","article-title":"The Gender-Oriented Perspective in the Development of Type 2 Diabetes Mellitus Complications; SGOP Study","volume":"30","author":"Kumar","year":"2023","journal-title":"JPTCP"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/8\/564\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T15:39:18Z","timestamp":1760110758000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/13\/8\/564"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,8,20]]},"references-count":29,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2024,8]]}},"alternative-id":["axioms13080564"],"URL":"https:\/\/doi.org\/10.3390\/axioms13080564","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2024,8,20]]}}}