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Mathematical Physics

Applied Mathematics and Mathematical Physics:

From Dirac Notation to Probability Bracket Notation

Author: Dr. Xing M (Sherman) Wang

Dirac notation (or Bra-ket notation) is a very powerful and indispensable tool for modern physicists. Unfortunately it is only taught in Quantum Mechanics . I believe it would be great to introduce it in Applied Mathematics (like Linear Algebra ).

On the other hand, while studying probability theories, I felt that it would be very helpful if we had a similar notation to represent or derive probabilistic formulas .

That was why I posted following articles online, in which Dirac Notation was insroduced to IR, and Probability Bracket Notation was proposed and applied do IR, too.

No mater you agree or disagree with my work, I welcome and appreciate your opinions.

How to email to the author?
Subject line: About the articles on your web site
Email address: swang (at) shermanlab (dot) com,
or from if you are a member

1: Probability Bracket Notation, Probability Vectors, Markov Chains and Stochestic Processes

PDF: Current (11/14/2009)   


Dirac notation has been widely used for vectors in Hilbert spaces of Quantum Theories. It now has also been introduced to Information Retrieval. In this paper, we propose a new set of symbols, the Probability Bracket Notation (PBN), for probability theories. We define new symbols like probability bra (p-bra), p-ket, p-bracket, sample base, unit operator, state ket and more as their counterparts in Dirac notation, which we refer as Vector Bracket Notation (VBN). By applying PBN to represent fundamental definitions and theorems for discrete and continuous random variables, we show that PBN could play the same role in probability sample space as Dirac notation in Hilbert space. We also find that there is a close relation between our probability state kets and probability vectors in Markov chains, which are involved in data clustering like Diffusion Maps .We summarize the similarities and differences between PBN and VBN in the two tables of Appendix A.

2: Probability Bracket Notation: Probability Space, Conditional Expectation and Introductory Martingales

PDF: Current (07/16/2007)  


In this paper, we continue to explore the consistence and usability of Probability Bracket Notation (PBN) proposed in our previous articles. After a brief review of PBN with dimensional analysis, we investigate probability spaces in terms of PBN by introducing probability spaces associated with random variables (R.V) or associated with stochastic processes (S.P). Next, we express several important properties of conditional expectation (CE) and some their proofs in PBN. Then, we introduce martingales based on sequence of R.V or based on filtration in PBN. In the process, we see PBN can be used to investigate some probability problems, which otherwise might need explicit usage of Measure theory. Whenever applicable, we use dimensional analysis to validate our formulas and use graphs for visualization of concepts in PBN. We hope this study shows that PBN, stimulated by and adapted from Dirac notation in Quantum Mechanics (QM), may have the potential to be a useful tool in probability modeling, at least for those who are already familiar with Dirac notation in QM.

3: Probability Bracket Notation and Probability Modeling (a short version of the first article)

PDF: Current (11/07/2009)


Inspired by the Dirac notation, a new set of symbols, the Probability Bracket Notation (PBN) is proposed for probability modeling. By applying PBN to discrete and continuous random variables, we show that PBN could play a similar role in probability spaces as the Dirac notation in Hilbert vector spaces. The time evolution of homogeneous Markov chains with discrete-time and continuous-time are discussed in PBN. Our system state p-kets are identified with the probability vectors, while our system state p-bra can be identified with Doi's state function and Peliti's standard bra. We also suggest that, by transforming from the Schrodinger picture to the Heisenberg picture, the time-dependence of a system p-ket of a homogeneous MC can be shifted to the observable as a stochastic process.

  Errata (09/06/2012)  

4: From Dirac Notation to Probability Bracket Notation: Time Evolution and Path Integral under Wick Rotations

PDF: Current (01/29/2009)  


In this article, we continue to investigate the application of Probability Bracket Notation (PBN). We show that, under Special Wick Rotation (caused by imaginary-time rotation), the Schrodinger equation of a conservative system and its path integral in Dirac rotation are simultaneously shifted to the master equation and its Euclidean path integral of an induced micro diffusion in PBN. Moreover, by extending to General Wick Rotation and using the anti-Hermitian wave-number operator, we execute the path integral in Dirac notation side-by-side with the Euclidean path integral in PBN, and derive the Euclidean Lagrangian of induced diffusions and Smoluchowski equation.

5: Probability Bracket Notation: the Unified Expressions of Conditional Expectation and Conditional Probability in Quantum Modeling

PDF: Current (11/07/2009)


After a brief introduction to Probability Bracket Notation (PBN), indicator operator and conditional density operator (CDO), we investigate probability spaces associated with various quantum systems: system with one observable (discrete or continuous), system with two commutative observables (independent or dependent) and a system of indistinguishable non-interacting many-particles. In each case, we derive unified expressions of conditional expectation (CE), conditional probability (CP), and absolute probability (AP): they have the same format for discrete or continuous spectrum; they are defined in both Hilbert space (using Dirac notation) and probability space (using PBN); and they may be useful to deal with CE of non-commutative observables.

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