Probability Bracket Notation for Probability Modeling

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 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’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.

1. Introduction

The postulates of Quantum Mechanics (QM) were established in terms of Hilbert spaces and linear Hermitian operators. Values of physical observables such as energy and momentum were considered as eigenvalues, more precisely as spectral values of linear operators in a Hilbert space. Dirac’s Vector Bracket Notation (VBN) is a very powerful tool to manipulate vectors in Hilbert spaces. It has been widely used in QM and quantum field theories. The main beauty of the VBN is that many formulae can be presented in an abstract symbolic fashion, independent of state expansions or basis selections, which, when needed, can be easily performed by inserting an appropriate v-identity operator ([1] p. 96).

$$\begin{array}{ccc}\hfill \mathrm{v}\text{-}\mathrm{Bracket}:\langle {\psi }_A|{\psi }_B\rangle & \Rightarrow & \mathrm{v}\text{-}\mathrm{bra}:\langle {\psi }_A|,\mathrm{v}\text{-}\mathrm{ket}:|{\psi }_B\rangle \hfill \end{array}$$

(1)

$$\begin{array}{ccc}\hfill \mathrm{where}:\langle {\psi }_A|{\psi }_B\rangle & =& {\langle {\psi }_B|{\psi }_A\rangle }^{*},{\langle {\psi }_A|=|{\psi }_A\rangle }^{†}\hfill \end{array}$$

(2)

$$\begin{array}{ccc}\hfill \mathrm{v}\text{-}\mathrm{basis}\&\mathrm{v}\text{-}\mathrm{identity}:\widehat{H}|{\epsilon }_i\rangle & =& {\epsilon }_i|{\epsilon }_i\rangle ,\langle {\epsilon }_i|{\epsilon }_j\rangle ={\delta }_{ij},{\widehat{I}}_H=\sum _i|{\epsilon }_i\rangle \langle {\epsilon }_i|\hfill \end{array}$$

(3)

$$\begin{array}{ccc}\hfill \mathrm{Normalization}:1=\langle \Psi |\Psi \rangle & =& \langle \Psi |{\widehat{I}}_H|\Psi \rangle ={\sum }_i\langle \Psi |{\epsilon }_i\rangle \langle {\epsilon }_i|\Psi \rangle ={\sum }_i|c_i{|}^2\hfill \end{array}$$

(4)

$$\begin{array}{ccc}\hfill \mathrm{Expectation}:\langle H\rangle \equiv \overline{H}& \equiv & \langle \Psi |\widehat{H}|\Psi \rangle ={\sum }_i\langle \Psi |\widehat{H}|{\epsilon }_i\rangle \langle {\epsilon }_i|\Psi \rangle ={\sum }_i{\epsilon }_i|c_i{|}^2\hfill \end{array}$$

(5)

where † denotes the Hermitian adjoint and ∗ is the complex conjugate. However, when applying operators, one must ensure that the results are contained within a Hilbert space or subspace. One must also consider if the operator was bounded or unbounded, etc. Thus, one has to develop an entire spectral theory for Hermitian operators in a Hilbert space [2].

Inspired by the great success of the VBN in QM, we now propose the Probability Bracket Notation (PBN). The latter relies on a sample space which is less constrained than a Hilbert space. Assuming X is a random variable $(R.V)$, $\Omega$ is the set of all its outcomes and $P\left(x_i\right)$ is the probability of $x_i\in \Omega$. Then we can make an expression like Equation (4), with

$${\sum }_{i}P(x_i)={\sum }_{i}P(x_i|\Omega )={\sum }_{i}P(\Omega |x_i)P(x_i|\Omega )=P(\Omega |\left\{{\sum }_i|x_i\left)P\right(x_i|\right\}|\Omega )=P(\Omega |\Omega )=1$$

Here we have used the definition of conditional probability for $A,B\subseteq \Omega$ ([3] p. 91).

$$P(A|B)\equiv P(A\cap B)/P\left(B\right),\therefore P(x_i|\Omega )=P\left(x_i\right),P(\Omega |x_i)=1\mathrm{for}\forall x_i\in \Omega$$

(6)

Therefore, we seem to have discovered a probability “identity operator”:

$$I_X\equiv {\sum }_i|x_i\left)P\right(x_i|\Rightarrow P(\Omega |\Omega )=P(\Omega |I_X|\Omega )={\sum }_{i}P(\Omega |x_i)P(x_i|\Omega )\underset{\left(6\right)}{=}{\sum }_{i}P\left(x_i\right)=1$$

Then, following Dirac’s notation, we define the probability bra (P-bra), P-ket, P-bracket (as conditional probabilities by nature), P-basis, the system P-ket, P-identity, normalization, expectation and more, which are similar but not identical to their counterparts in Equations (1)–(5). In Section 2, for systems of one $R.V$, we show that the PBN has an advantage similar to that of the VBN: miscellaneous probability expressions [3,4,5] now can be presented in an abstract way, independent of P-basis, and can be expanded by inserting a suitable P-identity.

Next, in Section 3, we investigate the time evolution of homogeneous Markov chains (HMC) [3,4,5]. We realize that the time evolution of a continuous time HMC can be written in a symbolic abstract expression (in Section 3.2), just like the stationary Schrödinger equation in the VBN:

$$\begin{array}{c}i\hslash {\partial }_t|\Psi \left(t\right)\rangle =\widehat{H}|\Psi \left(t\right)\rangle =\left[-({\hslash }^2/2m){\partial }_x^2-V\left(x\right)\right]|\Psi \left(t\right)\rangle ,\hfill \\ |\Psi \left(t\right)\rangle =U\left(t\right)|\Psi \left(0\right)\rangle =exp(-i\widehat{H}t/\hslash )|\Psi \left(0\right)\rangle \hfill \end{array}$$

(7)

We also find that our time-dependent system P-kets can be identified with probability vectors [3]. Our system P-bra is closely related to the state function or standard bra introduced in Doi–Peliti Techniques [6,7,8]. We show that by transforming from the Schrödinger picture to the Heisenberg picture in the PBN, the time dependence of a system P-ket relocates to the $R.V$, which becomes a stochastic process; the Chapman–Kolmogorov Theorem [4,5,9] for transition probabilities can be derived by just inserting a time-dependent P-identity. Section 5 shows that a Schrödinger equation transforms to a master equation by making a special Wick rotation. Section 6 showcases the potential applications of the PBN, such as handling non-Hermitian operators and clustering text datasets. Discussion and concluding remarks are made at the end.

2. Probability Bracket Notation and Random Variable (R.V)

2.1. Discrete Random Variable

We define a probability space $(\Omega ,X,P)$ of a discrete random variable ($R.V$, or observable) X as follows: the set of all elementary events $\omega$, associated with a discrete random variable X, is the sample space $\Omega$, and

$$\mathrm{For}\forall {\omega }_i\in \Omega ,X\left({\omega }_i\right)=x_i\in \Re ,P:{\omega }_i\mapsto P\left({\omega }_i\right)=m\left({\omega }_i\right)\ge 0,{\sum }_{i}m\left({\omega }_i\right)=1$$

(8)

Definition 1.

(Probability event-bra and evidence-ket): Let $A\subseteq \Omega$ and, $B\subseteq \Omega$,

1. 

The symbol $P(A|$ represents a probability event bra, or P-bra;

2. 

The symbol $|B)$ represents a probability evidence ket, or P-ket.

Definition 2.

(Probability Event Evidence Bracket): The conditional probability (CP) of event A given evidence B in the sample space Ω can be treated as a P-bracket, and it can be split into a P-bra and a P-ket, similar to a Dirac bracket. For $A,B\subseteq \Omega$, we define:

$$\begin{array}{ccc}& & P\text{-}\mathit{bracket}:P(A|B)\Rightarrow P\text{-}\mathit{bra}:P(A|,P\text{-}\mathit{ket}:|B)\mathit{Note}:P{(A|\ne |A)}^{†}\hfill \\ & & \mathit{Here}:P(A|B)\equiv {\displaystyle \frac{P(A\cap B)}{P\left(B\right)}}(a\mathit{conditional}\mathit{probability}\mathit{by}\mathit{nature})\hfill \end{array}$$

(9)

As a CP, the P-bracket has the following properties for $A,B\subseteq \Omega$:

$$\begin{array}{ccc}\hfill P(A|B)& =& 1\mathrm{if}A\supseteq B\supset ⌀\hfill \end{array}$$

(10)

$$\begin{array}{ccc}\hfill P(A|B)& =& 0\mathrm{if}A\cap B=⌀\hfill \end{array}$$

(11)

Definition 3.

System P-ket:  For any subset, the probability $P\left(E\right)$ can be written as a conditional probability, or a P-bracket:

$$P\left(E\right)=P(E|\Omega )$$

(12)

Here $|\Omega )$ is called the system P-ket. The P-bracket defined in Equation (9) now becomes:

$$P(A|B)=P(A\cap B)/P(B)=P(A\cap B|\Omega )/P(B|\Omega )$$

(13)

We have the following important property expressed in PBN:

$$\mathrm{For}\forall B\subseteq \Omega \mathrm{and}B\ne ⌀,P(\Omega |B)=1$$

(14)

The Bayes formula (see [3] Sec. (2.1)) now can be expressed as:

$$P(A|B)\equiv P(B|A)P\left(A\right)/P\left(B\right)=P(B|A\left)P\right(A|\Omega )/P(B|\Omega )$$

(15)

The set of all elementary events in $\Omega$ forms a complete mutually disjoint set:

$${\bigcup }_{{\omega }_i\in \Omega }{\omega }_i=\Omega ,{\omega }_i\cap {\omega }_j={\delta }_{ij}{\omega }_i,{\sum }_{i}m\left({\omega }_i\right)=1.$$

(16)

Definition 4.

(Discrete P-Basis and P-Identity): Using Equations (8)–(11), we have the following properties for basis elements in $(\Omega ,X,P)$:

$$X\left({\omega }_j\right)=x_j\to X|{\omega }_j)=x_j|{\omega }_j),P(\Omega |{\omega }_j)=1,P({\omega }_i|\Omega )=m\left({\omega }_i\right)$$

(17)

In view of the one-to-one correlation between $x_j$ and ${\omega }_j$, from now on, we will use $x_j$ to label basis elements, just like labeling eigenstates in Equation (3) in the VBN for QM:

$$X\left(x_j\right)=x_j\to X|x_j)=x_j|x_j),P(\Omega |x_j)=1,P(x_i|\Omega )=m\left(x_i\right)$$

(18)

Here X behaves like a right-acting operator. The complete mutually disjoint events in (16)–(18) form a probability basis (or P-basis) and a P-identity, similar to Equation (3) in QM:

$$P(x_i|x_k)={\delta }_{ik},{\sum }_{x\in \Omega }|x)P(x|={\sum }_i|x_i)P(x_i|=I_X.$$

(19)

The system P-ket, $|\Omega )$, now can be expanded from the left as:

$$|\Omega )=I_X|\Omega )=\sum _i|x_i)P(x_i|\Omega )=\sum _{i}m\left(x_i\right)|x_i)$$

(20)

While the system P-bra, has its expansion from the right as:

$$P(\Omega |=P(\Omega |I_X={\sum }_{i}P(\Omega |x_i)P(x_i|\underset{\left(14\right)}{=}{\sum }_{i}P(x_i|$$

(21)

The two expansions are quite different, and $P{(\Omega |\ne [|\Omega \left)\right]}^{†}$. But their P-bracket is consistent with the requirement of normalization, similar to Equation (4) in the VBN:

$$1=P(\Omega )\equiv P(\Omega |\Omega )={\sum }_{i,j=1}^{N}P(x_i|m\left(x_j\right)|x_j)={\sum }_{i,j=1}^{N}m\left(x_j\right){\delta }_{ij}={\sum }_{i=1}^{N}m\left(x_i\right)$$

(22)

Definition 5.

(Expectation Value): Analogous to Equation (5) in QM, the expected value of the $R.V$ or observable X in $(\Omega ,X,P)$ now can be expressed as:

$$\langle X\rangle \equiv \overline{X}\equiv E\left[X\right]=P(\Omega |X|\Omega )={\sum }_{x\in \Omega }P(\Omega |X|x)P(x|\Omega )={\sum }_{x\in \Omega }xm\left(x\right)$$

(23)

If $f\left(X\right)$ is a continuous function of observable X, then it is easy to show that:

$$\langle f\left(X\right)\rangle \equiv E\left[f\left(X\right)\right]\equiv P(\Omega |f\left(X\right)|\Omega )={\sum }_{x\in \Omega }f\left(x\right)m\left(x\right)$$

(24)

2.2. Independent Random Variables

Let $\overrightarrow{X}=\{X_1,X_2,\dots X_n\}$ be a vector of independent random variables and the sample space (i.e., the set of possible outcomes) of $X_i$ is the set ${\Omega }_i$. Then the joint probability distribution can be denoted as:

$$P(x_1,...,x_N|\Omega )=P(X_1=x_1,...,X_n=x_n|\Omega ),|\Omega )=|{\Omega }_1\otimes \dots \otimes {\Omega }_n)$$

(25)

The joint probability of independent $R.V$ are factorable, e.g.:

$$P(x_i,x_k|\Omega )=P(x_i|{\Omega }_i)P(x_k|{\Omega }_k)=P\left(x_i\right)P\left(x_k\right)$$

(26)

The factorable system have the following factorable expectation, for example:

$$P(\Omega |X_i{X}_j{X}_k|\Omega )=P({\Omega }_i|X_i|{\Omega }_i\left)P\right({\Omega }_j|X_j|{\Omega }_j\left)P\right({\Omega }_k|X_k|{\Omega }_k)$$

(27)

As an example, in Fock space, we have the following basis from the occupation numbers:

$$N_i|\overrightarrow{n})=n_i|\overrightarrow{n}),P(\overrightarrow{n}|{\overrightarrow{n}}^{\prime })={\delta }_{\overrightarrow{n},{\overrightarrow{n}}^{\prime }}={\prod }_i{\delta }_{n_i,n_i^{\prime }},I_{\overrightarrow{n}}={\sum }_{\overrightarrow{n}}|\overrightarrow{n}\left)P\right(\overrightarrow{n}|={\prod }_i{\sum }_{n_i}|n_i\left)P\right(n_i|$$

(28)

The expectation value of an occupation number now is given by:

$$\langle N_i\rangle \equiv P(\Omega |N_i|\Omega )=P({\Omega }_i|N_i|{\Omega }_i)=P(\Omega |N_i{I}_i|{\Omega }_i)={\sum }_{n_i}{n}_{i}P(n_i|{\Omega }_i)$$

(29)

Moreover, if sets A and B are mutually independent in $\Omega$, we have the following equivalence:

$$P(A|B)=P(A|\Omega )\iff P(A\cap B|\Omega )=P(A|\Omega )P(B|\Omega )$$

(30)

2.3. Continuous P-Basis and P-Identity

Equations (17)–(19) can be extended to the probability space $(\Omega ,X,P)$ of a continuous random variable X,

$$\begin{array}{ccc}\hfill X\left(x\right)=x\to X|x)& =& x|x),P(\Omega |x)=1,P:x\mapsto P(x)\equiv P(x|\Omega )\hfill \end{array}$$

(31)

$$\begin{array}{ccc}\hfill P(x|x^{\prime })& =& \delta (x-x^{\prime }),{\int }_{x\in \Omega }|x)dxP(x|=I_X\hfill \end{array}$$

(32)

We see that it is consistent with the normalization requirement:

$$P(\Omega |\Omega )=P(\Omega |I_X|\Omega )=\int P(\Omega |x)dxP(x|\Omega )={\int }_{x\in \Omega }dxP(x|\Omega )={\int }_{x\in \Omega }dxP\left(x\right)=1$$

(33)

The expected value $E\left(X\right)$ can be easily extended from (23):

$$\langle X\rangle \equiv \overline{X}\equiv E\left[X\right]=P(\Omega |X|\Omega )={\int }_{x\in \Omega }P(\Omega |X|x)dxP(x|\Omega )={\int }_{x\in \Omega }dxxP\left(x\right)$$

(34)

The basis-independent expressions in the PBN are similar to those in the Dirac VBN, and among them are the expectation and normalization formulae in the two notations:

$$\begin{array}{ccc}\hfill PBN:\langle f\left(X\right)\rangle & \equiv & E\left[f\right(X\left)\right]=P(\Omega |f\left(X\right)|\Omega ),P(\Omega |\Omega )=1,\hfill \end{array}$$

(35)

$$\begin{array}{ccc}\hfill VBN:\langle f\left(\widehat{X}\right)\rangle & \equiv & E\left[f\left(\widehat{X}\right)\right]=\langle \psi |f\left(\widehat{X}\right)|\psi \rangle ,\langle \psi |\psi \rangle =1.\hfill \end{array}$$

(36)

Of course, there exist differences between the PBN and Dirac VBN. For example, the expansions of bra, ket and normalization have their own expressions:

$$\begin{array}{ccc}\hfill PBN:|\Omega )& =& \int dx|x)P(x|\Omega ),P(\Omega |=\int dxP(x|,P(\Omega |\Omega )=\int dxP(x|\Omega )=1,\hfill \end{array}$$

(37)

$$\begin{array}{ccc}\hfill VBN:|\psi \rangle & =& \int dx|x\rangle \langle x|\psi \rangle ,\langle \psi |=\int dx\langle \psi |x\rangle \langle x|,\langle \psi |\psi \rangle =\int dx|\langle \psi |x\rangle {|}^2=1.\hfill \end{array}$$

(38)

2.4. Conditional Probability and Expectation

The conditional expectation of X given $H\subset \Omega$ in the continuous basis (31) can be expressed as ([5] p. 61):

$$E[X|H]\equiv P(\Omega |X|H)=\int P(\Omega |X|x)dxP(x|H)=\int xdxP(x|H)$$

(39)

$$\mathrm{where}:P(x|H)\underset{\left(9\right)}{=}{\displaystyle \frac{P(x\cap H|\Omega )}{P(H|\Omega )}}(\mathrm{conditional}\mathrm{probability})$$

(40)

To become familiar with the PBN, let us see two simple examples.

2.4.1. Example: Rolling a Die (Ref. [4] Examples 2.6–2.8)

A die is rolled once. We let X denote the outcome of this experiment. Then the sample space for this experiment is the six-element set $\Omega =\{1,2,3,4,5,6\}$. We assumed that the die was fair, and we chose the distribution function defined by $m\left(i\right)=1/6$, for $i=1,\dots 6$. Using the PBN, we have the P-identity for this sample space and the probability of each outcome:

$$\begin{array}{ccc}\hfill \sum _{i=1}^6|i)P(i|& =& I_D,\hfill \end{array}$$

(41)

$$\begin{array}{ccc}\hfill 1=P(\Omega |\Omega )& =& \sum _{i=1}^{6}P(\Omega |i\left)P\right(i|\Omega )=\sum _{i=1}^{6}P(i|\Omega )=6p.\hfill \end{array}$$

(42)

Hence, the probability for each outcome has the same value:

$$P\left(i\right)\equiv P(i|\Omega )=p=1/6.$$

(43)

Its expectation value can be readily calculated:

$$P(\Omega |X|\Omega )=\sum _{i=1}^{6}P(\Omega |X|i)P(i|\Omega )=\sum _{i=1}^{6}P(\Omega |i|i)P(i|\Omega )=\sum _{i=1}^{6}i/6=21/6=7/2,$$

(44)

and the variance can be calculated as:

$${\sigma }^2=\langle X^2\rangle -{\overline{X}}^2={\sum }_{i=1}^6{i}^2/6-49/4=91/6-49/4=45/4$$

(45)

2.4.2. Example: Rolling a Die (Examples 2.1 Continued, [4] 2.8 Continued)

If E is the event that the result of the roll is an even number, then $E=\{2,4,6\}$ and $P\left(E\right)=m\left(2\right)+m\left(4\right)+m\left(6\right)=1/6+1/6+1/6=\frac{1}{2}$. Using the PBN, the probability of event E can be easily calculated as:

$$P\left(E\right)\equiv P(E|\Omega )=P(E|\widehat{I}|\Omega )={\sum }_{i\in E}P(E|i)P(i|\Omega )={\sum }_{i\in E}P(i|\Omega )=3p=3/6=1/2$$

(46)

Applying Equation (40), we can calculate the conditional probabilities $P(i|E)$ as follows:

$$P(i|E)={\displaystyle \frac{P(E\cap i|\Omega )}{P(E|\Omega )}}=\left\{\begin{array}{c}P(i|\Omega )/(1/2)=(1/6)/(1/2)=1/3,(ieven)\\ 0,\left(iodd\right)\end{array}\right.$$

(47)

Our discussions above can be easily extended to systems of multiple $R.V.$ For example, we can introduce the following system of three independent discrete $R.V$:

$$\begin{array}{ccc}\hfill X|x,y,z)& =& x|x,y,z),Y|x,y,z)=y|x,y,z),Z|x,y,z)=z|x,y,z)\hfill \\ \hfill P(x,y,z)& =& P(x,y,z|\Omega );P(\Omega |\Omega )=1;P(\Omega |H)=1\mathrm{if}H\subseteq \Omega \to P(\Omega |x,y,z)=1\hfill \end{array}$$

(48)

$$\begin{array}{ccc}\hfill \mathrm{Orthonormality}:& & P(x,y,z|x^{\prime },y^{\prime },z^{\prime })={\delta }_{x{x}^{\prime }}{\delta }_{y{y}^{\prime }}{\delta }_{z{z}^{\prime }}\hfill \end{array}$$

(49)

$$\begin{array}{ccc}\hfill \mathrm{Completeness}:& & {\sum }_{x,y,z}|x,y,z)P(x,y,z|=I_{X,Y,Z}\hfill \end{array}$$

(50)

3. Probability Vectors and Homogeneous Markov Chains (HMCs)

For simplicity, from now on, we will only discuss the time evolution for one $R.V$. We assume our probability space $\Omega$ has the following stationary discrete P-basis from a random variable $\widehat{N}$ (which is possible for state labeling or occupation number counting):

$$\widehat{N}|i)=i|i),P(i|j)={{\delta }_i}_j,{\sum }_{i=1}^N|i)P(i|=I$$

(51)

3.1. Discrete-Time HMC

The transition matrix element $p_{ij}$ is defined as ([3] p. 407):

$${p_i}_j\equiv P(N_{t+1}=j|N_t=i)\equiv P(j,t+1|i,t),{\sum }_{j=1}^N{p}_{ij}=1$$

(52)

In matrix form, if we define a probability row vector (PRV) at $t=0$ as $\langle u^{\left(0\right)}|$, then matrix P acting on it from the right for k times gives the PRV at $time=k$ ([3] theorem 11.2):

$$\langle u^{\left(k\right)}|=\langle u^{\left(0\right)}|P^k,\mathrm{or}:\langle u^{\left(k\right)}|i\rangle ={u^{\left(k\right)}}_i={u^{\left(0\right)}}_j{P^k}_{ji}$$

(53)

Definition 6.

Time-Dependent System P-ket: we use the following system P-ket, to represent a probability column vector in probability space $(\Omega ,X,P)$:

$$|{\Omega }_t)={\sum }_i^N|i)P(i|{\Omega }_t)={\sum }_i^{N}m(i,t)|i),P(\Omega |{\Omega }_t)={\sum }_i^{N}m(i,t)=1$$

(54)

The time evolution Equation (53) now can be written in a basis-independent way:

$$|{\Omega }_t)={\left(P^T\right)}^t|{\Omega }_0)\equiv \widehat{U}\left(t\right)|{\Omega }_0),t\in \mathbb{N};|{\Omega }_0)\equiv [{\langle u^{\left(0\right)}|]}^T$$

(55)

Definition 7.

(Time-Dependent Expectation): The expectation value of a continuous function f of occupation number $\widehat{N}$ in $(\Omega ,N,P)$ can be expressed as:

$$\langle f\left(\widehat{N}\right)\rangle =P(\Omega |f\left(\widehat{N}\right)|{\Omega }_t)={\sum }_{i}P(\Omega |f\left(i\right)|i)P(i|{\Omega }_t)={\sum }_{i}f(i)m(i,t)$$

(56)

We can map the P-bra and P-ket into the Hilbert space by using Dirac’s notation:

$$P(\Omega |={\sum }_{i}P(i|\leftrightarrow \langle \Psi |={\sum }_i\langle i|,|{\Omega }_t)\leftrightarrow |{\Psi }_t\rangle ={\sum }_i|i\rangle \langle i|{\Psi }_t\rangle ={\sum }_{i}c(i,t)|i\rangle$$

(57)

Then the expectation Equation (56) can be rewritten in Dirac’s notation as:

$$\begin{array}{ccc}\hfill \langle f\left(\widehat{N}\right)\rangle & =& \langle {\Psi }_t|f\left(\widehat{N}\right)|{\Psi }_t\rangle ={\sum }_i\langle {\Psi }_t|f\left(i\right)|i\rangle \langle i|{\Psi }_t\rangle ={\sum }_{i}f(i)|c(i,t){|}^2\hfill \end{array}$$

(58)

$$\begin{array}{ccc}& \therefore & P(i|{\Omega }_t)\equiv m(i,t)=|c(i,t){|}^2\equiv |\langle i|{\Psi }_t\rangle {|}^2,t\in \mathbb{N}\hfill \end{array}$$

(59)

One can see in Equation (59) that the local phases of quantum states are not contained in the probability distribution, and they have no contribution to the distribution:

$$VBN:|{\Psi }_t\rangle ={\sum }_i{c}_i\left(t\right)|{\psi }_i\rangle \underset{\text{one to many}}{\overset{\text{many to one}}{\rightleftarrows }}PBN:|{\Omega }_t)={\sum }_{i}P(i,t)|i);p(i,t)={\left|c_i(t)\right|}^2$$

This means a certain loss of quantum information, and it is well known that these local phase factors are crucial in explaining experiments like the quantum interference [10] and the Aharonov–Bohm effect [11].

3.2. Continuous-Time HMC

The time evolution equation of a continuous-time HMC with a discrete basis (see Equation $\left(85\right)$ or Ref. [5] p. 221) can be written as:

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\partial }{\partial t}}{p}_j\left(t\right)={\sum }_k{p}_k\left(t\right)q_{kj}& \Rightarrow & {\displaystyle \frac{\partial }{\partial t}}P(j|{\Omega }_t)={\sum }_k{\widehat{Q}}_{jk}^{T}P(k|{\Omega }_t)\hfill \end{array}$$

(60)

$$\begin{array}{ccc}\hfill ={\sum }_{k}P(j|{\widehat{Q}}^T|k)P(k|{\Omega }_t)& =& P(j|{\widehat{Q}}^{T}I|{\Omega }_t)=P(j|{\widehat{Q}}^T|{\Omega }_t)\equiv P(j|\widehat{L}|{\Omega }_t)\hfill \end{array}$$

(61)

Equations (60)–(61) lead to a basis-independent master equation (see [8] Eq. (2.12)):

$${\displaystyle \frac{\partial }{\partial t}}|{\Omega }_t)=\widehat{L}|{\Omega }_t),|{\Omega }_t)=\widehat{U}\left(t\right)|{\Omega }_0)=e^{\widehat{L}t}|{\Omega }_0),t\ge 0$$

(62)

It looks just like Schrödinger’s Equation (7) of a conserved quantum system in Dirac’s notation. With the discrete P-basis of Fock space in Equation (28), Equations (56)–(59) now can be written as:

$$|{\Omega }_t)={\sum }_{\overrightarrow{n}}m(\overrightarrow{n},t)|\overrightarrow{n}),P(\Omega |={\sum }_{\overrightarrow{n}}(\overrightarrow{n}|,\therefore \langle f\left(\overrightarrow{n}\right)\rangle =P(\Omega |f\left(\overrightarrow{n}\right)|{\Omega }_t)$$

(63)

Doi’s definition of the state function and state vector [8,9] correspond to our system P-bra and P-ket respectively:

$$\begin{array}{ccc}\hfill P(\Omega |={\sum }_{\overrightarrow{n}}P(\overrightarrow{n}|& \leftrightarrow & \langle s|\equiv {\sum }_{\overrightarrow{n}}\langle \overrightarrow{n}|\hfill \\ \hfill |F\left(t\right)\rangle \equiv {\sum }_{\overrightarrow{n}}P(\overrightarrow{n},t)|\overrightarrow{n}\rangle & \leftrightarrow & |{\Omega }_t)={\sum }_{\overrightarrow{n}}|\overrightarrow{n}\left)P\right(\overrightarrow{n}|{\Omega }_t)\hfill \\ \hfill \therefore \langle \widehat{B}\left(\overrightarrow{n}\right)\rangle =\langle s|\widehat{B}\left(\overrightarrow{n}\right)|F\left(t\right)\rangle & =& P(\Omega |\widehat{B}\left(\overrightarrow{n}\right)|{\Omega }_t)\hfill \end{array}$$

(64)

Note that the vector-basis here corresponds to the P-basis in Equation (28):

$${\widehat{n}}_i|\overrightarrow{n}\rangle =n_i|\overrightarrow{n}\rangle ,{\sum }_{\overrightarrow{n}}|\overrightarrow{n}\rangle \langle \overrightarrow{n}|=I_{\overrightarrow{n}},\langle \overrightarrow{n}|{\overrightarrow{n}}^{\prime }\rangle ={\delta }_{\overrightarrow{n},{\overrightarrow{n}}^{\prime }}=\prod _{i=1}{\delta }_{n_i,{n_i}^{\prime }}$$

(65)

In Peliti’s formalism ([8] p. 1472), the vector-basis (from population operator n) is normalized in a special way, so the expansion of the system P-bra is also changed:

$${\sum }_n|n\rangle {\displaystyle \frac{1}{n!}}\langle n|=I_n,\langle m|n\rangle =n!{\delta }_{m,n}$$

(66)

$$P(\Omega |=P(\Omega |I_n=\sum _n^{\infty }P(\Omega |n){\displaystyle \frac{1}{n!}}P(n|\underset{\left(2.3\right)}{=}\sum _n^{\infty }P(n|{\displaystyle \frac{1}{n!}}$$

(67)

Equation (67) can be identified with the standard bra, introduced in [8] Eq. (2.27–28):

$$\begin{array}{ccc}\hfill P(\Omega |& =& {\sum }_n{\displaystyle \frac{1}{n!}}P(n|\leftrightarrow \langle |\equiv {\sum }_n{\displaystyle \frac{1}{n!}}\langle n|\hfill \\ \hfill |\Omega )& =& {\sum }_n|n){\displaystyle \frac{1}{n!}}P(n|\Omega )\leftrightarrow |\varphi \rangle ={\sum }_m|m\rangle {\displaystyle \frac{1}{m!}}\langle m|\varphi \rangle \hfill \\ \hfill \therefore E\left[\widehat{A}\right]& \equiv & \langle \widehat{A}\rangle =\langle |\widehat{A}|\varphi \rangle =P(\Omega |\widehat{A}|\Omega )\hfill \end{array}$$

(68)

3.3. The Heisenberg Picture

We call Equations (55) and (62) the evolution equations in the Schrödinger picture. Now we introduce the Heisenberg picture of a $R.V$ (or observable), similar to what is used in QM ([1] p. 541):

$$|{\Omega }_t)=\widehat{U}\left(t\right)|{\Omega }_0)\Rightarrow \widehat{X}\left(t\right)={\widehat{U}}^{-1}\left(t\right)\widehat{X}\widehat{U}\left(t\right)$$

(69)

Based on $\widehat{U}\left(t\right)$, we can introduce following the time-dependent P-basis:

$$\begin{array}{cc}& |x,t)={\widehat{U}}^{-1}\left(t\right)|x),P(x|\widehat{U}\left(t\right)=P(x,t|,P(x,t|x^{\prime },t)=P(x|x^{\prime }),P(\Omega |x,t)=1\end{array}$$

(70)

$$\begin{array}{cc}& P(x^{\prime },t|\widehat{X}\left(t\right)|x,t)=P(x^{\prime }|\widehat{U}\left(t\right){\widehat{U}}^{-1}\left(t\right)\widehat{X}\widehat{U}\left(t\right){\widehat{U}}^{-1}\left(t\right)|x)=P(x^{\prime }|\widehat{X}|x)=xP(x^{\prime }|x)\end{array}$$

(71)

The probability density now can be interpreted in the two pictures:

$$P(x,t)\equiv P(x|{\Omega }_t)=P(x|\widehat{U}\left(t\right)|{\Omega }_0)=P(x,t|{\Omega }_0)=P(x,t|\Omega )$$

(72)

In the last step, we have used the fact that in the Heisenberg picture, $|{\Omega }_0)=|\Omega )$.

Definition 8.

Time-Dependent P-Identity: Equations (69)–(71) also provide us with a time-dependent P-identity in the Heisenberg picture:

$$\begin{array}{ccc}\hfill \mathit{Discrete}:I_X\left(t\right)& =& {\widehat{U}}^{-1}\left(t\right)I_X\widehat{U}\left(t\right)={\widehat{U}}^{-1}\left(t\right){\sum }_i|x_i\left)P\right(x_i|\widehat{U}\left(t\right)\underset{\left(70\right)}{=}{\sum }_i|x_i,t\left)P\right(x_i,t|\hfill \\ \hfill \mathit{Continuous}:I_X\left(t\right)& =& {\widehat{U}}^{-1}\left(t\right)I_X\widehat{U}\left(t\right)={\widehat{U}}^{-1}\left(t\right)\int |x)dxP(x|\widehat{U}\left(t\right)=\int |x,t)dxP(x,t|\hfill \end{array}$$

(73)

Now the expectation value of the stochastic process $X\left(t\right)$ can be expressed as:

$$\begin{array}{ccc}\hfill P(\Omega |\widehat{X}\left(t\right)|\Omega )& =& P(\Omega |\widehat{X}\left(t\right)I_X\left(t\right)|\Omega )=\int dxP(\Omega |\widehat{X}\left(t\right)|x,t)P(x,t|\Omega )\hfill \\ & =& \int dxxP(x,t|\Omega )=\int dxxP(x|{\Omega }_t)=P(\Omega |X|{\Omega }_t)\hfill \end{array}$$

(74)

This suggests that a Markov stochastic process can be thought of as an operator in the Heisenberg picture, and its expectation value can be found from its Schrödinger picture. Additionally, if a Markov process $X\left(t\right)\equiv X_t$ is homogeneous, with independent and stationary increments ([4] p. 15), we can always set $X_0=0$, and obtain the following useful property:

$$\begin{array}{ccc}\hfill P(X_{t+s}-X_s=x)& =& P(X_t-X_0=x|\Omega )\hfill \\ & =& P(X_t=x|\Omega )=P(x,t|\Omega )=P(x|{\Omega }_t)\equiv P(x,t)\hfill \end{array}$$

(75)

Moreover, there is a relation between their transition probability and probability density:

$$P(x_2,t_2|x_1,t_1)=P(x_2-x_1,t_2-t_1|\Omega )\equiv P(x_2-x_1,t_2-t_1),(t_1<t_2)$$

(76)

3.4. Chapman–Kolmogorov Equations of Transition Probability [5,9]

These can now be easily obtained by inserting our time-dependent P-identity in Equation (73). For the HMC of discrete $R.V$ ([5] p. 174):

$$\begin{array}{ccc}\hfill {p^{m+n}}_{ij}& \equiv & P(j,m+n|i,0)=P(j,m+n|\widehat{I}\left(m\right)|i,0)=\sum _{k}P(j,m+n|k,m\left)P\right(k,m|i,0)\hfill \\ & \underset{\mathrm{HMC}}{=}& {\sum }_{k}P(j,n|k,0)P(k,m|i,0)\underset{52}{=}{\sum }_k{p^m}_{ik}{p^n}_{kj}(\mathrm{Discrete}\mathrm{time})\hfill \end{array}$$

(77)

$$\begin{array}{ccc}\hfill p_{ij}(t+s)& \equiv & P(j,t+s|i,0)=P(j,t+s|\widehat{I}\left(s\right)|i,0)={\sum }_{k}P(j,t+s|k,s\left)P\right(k,s|0,i)\hfill \\ & \underset{\mathrm{HMC}}{=}& {\sum }_{k}P(j,t|k,0\left)P\right(k,s|0,i)={\sum }_k{p_i}_k\left(s\right){p_k}_j\left(t\right)(\mathrm{Continuous}\mathrm{time})\hfill \end{array}$$

(78)

For a Markov process of both continuous $R.V$ and time [9] (Eq. (3.9), p. 31):

$$P(x,t|y,s)=P(x,t|\widehat{I}\left(\tau \right)|y,s)=\int P(x,t|z,\tau \left)dzP\right(z,\tau |y,s)\mathrm{where}t>\tau >s$$

(79)

Absolute probability distribution (APD)

Likewise, we can find the time evolution of APD for a HMC simply by inserting P-identity $I(t=0)=I\left(0\right)$. For the HMC with discrete states ([5] pp. 174, 214):

$$\begin{array}{ccc}\hfill \mathrm{Discrete}\mathrm{time}:P(i,m)& =& P(i,m|\Omega )=P_i^{\left(m\right)},P(i,0)\equiv p_i^{\left(0\right)}\hfill \\ \hfill \therefore P_i^{\left(m\right)}=P(i,m|I\left(0\right)|\Omega )& =& {\sum }_{k}P(i,m|k,0)P(k,0|\Omega )={\sum }_k{p}_k^{\left(0\right)}{p^m}_{ki}\hfill \end{array}$$

(80)

Note that Equations (80) are identical to Equation (53):

$$\begin{array}{ccc}\hfill \mathrm{Continuous}\mathrm{time}:P(x_i,t)& =& P(x_i,t|\Omega )=P(x_i|{\Omega }_t),P(x_i,0)\equiv p_i\left(0\right)\hfill \\ \hfill \therefore P(x_i,t)=P(x_i,t|I\left(0\right)|\Omega )& =& {\sum }_{k}P(x_i,t|x_k,0)P(x_k,0|\Omega )={\sum }_k{p}_k\left(0\right)p_{ki}\left(t\right)\hfill \end{array}$$

(81)

For a Markov process of both continuous $R.V$ and time ([9] p. 31):

$$\begin{array}{ccc}\hfill P(x,t)& =& P(x,t|\Omega )=P(x|{\Omega }_t),P(x,0)\equiv p^{\left(0\right)}\left(x\right)\hfill \\ \hfill \therefore P(x,t)& =& P(x,t|I\left(0\right)|\Omega )=\int dyP(x,t|y,0)P(y,0|\Omega )=\int dyP(x,t|y,0)p^{\left(0\right)}\left(y\right)\hfill \end{array}$$

(82)

3.5. Kolmogorov Forward and Backward Equations

If the Markov chains are stochastically continuous, then for infinitesimal h, the transition probability has the Taylor expansions ([4] Sec. (6.8); [5] p. 217):

$$p_{ij}\left(h\right)=p_{ij}\left(0\right)+{p_{ij}}^{\prime }\left(0\right)h+o\left(h^2\right)={\delta }_{ij}+q_{ij}h+o\left(h^2\right)$$

(83)

Then, using the Chapman–Kolmogorov Equation (78), we have:

$$\begin{array}{ccc}\hfill {p_i}_j(t+h)=\sum _k{p_i}_k\left(t\right){p_k}_j\left(h\right)& =& \sum _k{p}_k\left(t\right)({\delta }_{kj}+q_{kj}h+o\left(h^2\right))\hfill \\ \hfill & =& {p_i}_j\left(t\right)+\sum _k{p_i}_k\left(t\right)(q_{kj}h+o\left(h^2\right))\hfill \end{array}$$

(84)

Therefore, we obtain the following Kolmogorov Forward equations:

$${p^{\prime }}_{ij}\left(t\right)=\underset{h\to 0}{lim}[({p_i}_j(t+h)-{p_i}_j\left(t\right))/h]=\sum _k{p_i}_k\left(t\right){q_k}_j$$

(85)

Similarly, we can derive the Kolmogorov Backward equations:

$${p_i}_j(h+t)=\sum _k{p_i}_k\left(h\right){p_k}_j\left(t\right)\Rightarrow {p^{\prime }}_{ij}\left(t\right)=\sum _k{q_i}_k{p_k}_j\left(t\right)$$

(86)

Using Equations (81) and (83) one can extend Equation (62) to a continuous-time HMC.

3.6. Transition Probability and Path Integrals

From Equations (62) and (70) the transition probability of a Markov process can be expressed as follows (assuming $t_a<t_0<t_b$):

$$P(x_b,t_b|x_a,t_b)=P(x_b|U(t_b,t_0)U^{-1}(t_a,t_0)|x_a)=P(x_b|U(t_b,t_a)|x_a)=P(x_b|e^{{\int }_{t_a}^{t_b}dtL}|x_a)$$

(87)

We can divide the time interval into small pieces and insert the P-identity N times:

$$\begin{array}{ccc}\hfill \mathsf{\Delta }t=t_{n+1}-t_n=(t_b-t_a)/(N+1)>0,& & t_N=t_b,t_0=t_a;x_N=x_b,x_0=x_a,\hfill \end{array}$$

(88)

$$\begin{array}{ccc}\hfill P(x_b,t_b|x_a,t_a)=P(x_b,t_b|I\left(t_N\right)..I\left(t_1\right)|x_a,t_a)& =& {\prod }_{n=1}^N{\int }_{-\infty }^{\infty }d{x}_{n}P(x_n,t_n|x_{n-1},t_{n-1})\hfill \\ \hfill ={\prod }_{n=1}^N{\int }_{-\infty }^{\infty }d{x}_{n}P(x_n|\widehat{U}(t_n,t_{n-1})|x_{n-1})& =& {\prod }_{n=1}^N{\int }_{-\infty }^{\infty }d{x}_{n}P(x_n|e^{L\mathsf{\Delta }t}|x_{n-1})\hfill \end{array}$$

(89)

It perfectly matches the starting equation of Feynman’s path integral for the transition amplitude in QM ([12] Eq. (2.4), p. 90):

$$\langle x_b,t_b|x_a,t_a\rangle ={\prod }_{n=1}^N{\int }_{-\infty }^{\infty }d{x}_n\langle x_n|\widehat{U}(t_n,t_{n-1})|x_{n-1}\rangle ={\prod }_{n=1}^N{\int }_{-\infty }^{\infty }d{x}_n\langle x_n|e^{-{\displaystyle \frac{i}{\hslash }}\widehat{H}\mathsf{\Delta }t}|x_{n-1}\rangle$$

(90)

Since the special Wick rotation can directly transform Equations (89) and (90) and vice versa, it might imply a possibly new connection between the Hilbert space and the probability space for certain quantum systems:

$$\langle x_b,t_b|x_a,t_a\rangle \underset{it\leftarrow t}{\overset{it\to t}{\rightleftarrows }}P(t_b,t_b|x_a,t_a)$$

However, at the moment this is just a potential application, and we need investigate this further. For a free particle, $V\left(x\right)=0$, the Schrödinger equation and resulting transition amplitude from Feynman’s path integral are (see [12] Eq. (2.125)):

$$i\hslash {\displaystyle \frac{\partial }{\partial t}}\psi (x,t)=\widehat{H}\psi (x,t)=-{\displaystyle \frac{{\hslash }^2}{2m}}{\displaystyle \frac{{\partial }^2}{\partial x^2}}\psi (x,t),\psi (x,t)=\langle x|\Psi \left(t\right)\rangle$$

(91)

$$\langle x_b,t_b|x_a,t_a\rangle =\sqrt{{\displaystyle \frac{m}{2\pi i\hslash (t_b-t_a)}}}exp\left[{\displaystyle \frac{im{(x_b-x_a)}^2}{2\hslash (t_b-t_a)}}\right]$$

(92)

4. Examples of Homogeneous Markov Processes

Now let us observe some important examples of the HMC using the PBN.

Poisson Process ([4] p. 250, [5] p. 161)

This is a counting process, $N\left(t\right)$, with the following properties:

1.

$\left\{N\right(t),t\ge 0\}$ is a non-negative process with independent increments and $N\left(0\right)=0$;

2.

It is homogeneous and its probability distribution is given by:

$$\begin{array}{ccc}\hfill m(k,t)& \equiv & P([N(t+s)-N\left(s\right)=k]|\Omega )\underset{\begin{array}{c}independent\\ increment\end{array}}{=}P([N\left(t\right)-N\left(0\right)=k]|\Omega )\hfill \\ & \underset{N\left(0\right)=0}{=}& P([N\left(t\right)=k]|\Omega )\equiv P(k|\Omega \left(t\right))\underset{\begin{array}{c}Poisson\\ Distribution\end{array}}{=}{\displaystyle \frac{{\left(\lambda t\right)}^k}{k!}}{e}^{-\lambda t}\hfill \end{array}$$

(93)

It can be shown ([5] p. 161) that:

$$\mu \left(t\right)\equiv \overline{N}\left(t\right)=\sum _{k}km(k,t)=\lambda t;{\sigma }^2\left(t\right)\equiv P(\Omega |{[N\left(t\right)-\overline{N}\left(t\right)]}^2|\Omega )=\lambda t$$

(94)

It can be shown (see [5] p. 215; [13] Theorem 1.5, p. 6) that the Poisson Process has a Markov property, and its transition probability is:

$$\begin{array}{ccc}& & P([N(t+s)=j]|N\left(t\right)=i)=P([N(t+s)-N\left(t\right)=j-i]|\Omega )={\displaystyle \frac{{\left(\lambda t\right)}^{j-i}}{(j-i)!}}{e}^{-\lambda t},\mathrm{if}j\ge i\hfill \\ & & p_{ij}\left(t\right)=0,\mathrm{if}j<i\hfill \end{array}$$

(95)

4.1. Wiener–Levy Process (see [5] p. 159; [9] sec. (3.6), p. 32)

This is a homogeneous process with independent and stationary increments and $W\left(0\right)=0$. Its probability density is a normal (Gaussian) distribution $N(0,t{\sigma }^2)$:

$$P(x,t)\equiv P([W(t+s)-W\left(s\right)=x]|\Omega )\underset{homogeneous}{=}P([W\left(t\right)-W\left(0\right)=x]|\Omega )$$

(96)

Its stationary increment is defined as:

$$P(x_2,t_2|x_1,t_1)={\displaystyle \frac{1}{\sqrt{2\pi (t_2-t_1)}\sigma }}exp\left[-{\displaystyle \frac{{(y_2-y_1)}^2}{2(t_2-t_1){\sigma }^2}}\right],(t_1<t_2)$$

(97)

We see that the Wiener–Levy process satisfies Equation (76).

$$P(x_2,t_2|x_1,t_1)=P(x_2-x_1,t_2-t_1),(t_1<t_2)$$

(98)

It can be verified ([5] p. 161) that it is an $N(0,t{\sigma }^2)$:

$$\tilde{\mu }\left(t\right)\equiv P(\Omega |W\left(t\right)|\Omega )=0,{\tilde{\sigma }}^2\left(t\right)\equiv P(\Omega |W{\left(t\right)}^2|\Omega )=t{\sigma }^2$$

(99)

4.2. Brownian Motion ([4] Sec. (10.1), p. 524; [9] p. 6, 42)

The stochastic process $X\left(t\right)$ has $X\left(0\right)=0$, has stationary and independent increments for $t\ge 0$, and its density function for $t>0$ is a normal distribution $N(t\mu ,t{\sigma }^2)$, or:

$$P(x,t)\equiv P(x|\Omega \left(t\right))={\displaystyle \frac{1}{\sqrt{2\pi t}\sigma }}exp\left[-{\displaystyle \frac{{(x-\mu t)}^2}{2t{\sigma }^2}}\right]$$

(100)

Brownian motions have stationary and independent increments ([4] Sec. (10), pp. 524, 529), so they are a HMC and satisfy Equation (76). They are the solution of the following master equation (Einstein’s diffusion equation, see [9] p. 6) with a drift speed $\mu$:

$${\displaystyle \frac{\partial }{\partial t}}P(x,t)=D{\displaystyle \frac{{\partial }^2}{\partial x^2}}P(x,t),D={\displaystyle \frac{\sigma }{\sqrt{2}}}.$$

(101)

Here the constant D is called the diffusion coefficient. Equation (101) can be interpreted as a special HMC case of Equation (62), given in the x-basis:

$${\displaystyle \frac{\partial }{\partial t}}P(x|{\Omega }_t)=\int d{x}^{\prime }P(x|\widehat{L}|x^{\prime })P(x^{\prime }|{\Omega }_t),P(x|\widehat{L}|x^{\prime })=D{\displaystyle \frac{{\partial }^2}{\partial x^2}}\delta (x-x^{\prime })$$

(102)

Equation (102) closely resembles Schrödinger’s Equation (7) for a free particle in the x-basis:

$$i\hslash {\displaystyle \frac{\partial }{\partial t}}\langle x|\Psi \left(t\right)\rangle =\int d{x}^{\prime }\langle x|\widehat{H}|x^{\prime }\rangle \langle x^{\prime }|\Psi \left(t\right)\rangle ,P(x|\widehat{H}|x^{\prime })={\displaystyle \frac{-{\hslash }^2}{2m}}{\displaystyle \frac{{\partial }^2}{\partial x^2}}\delta (x-x^{\prime })$$

(103)

5. Special Wick Rotation, Time Evolution and Induced Diffusions

The induced microscopic diffusion is defined by the following equation:

$${\displaystyle \frac{\partial }{\partial t}}P(x,t)=\widehat{G}\left(x\right)P(x,t),\widehat{G}={\displaystyle \frac{1}{2{\mu }_{\hslash }}}{\displaystyle \frac{{\partial }^2}{\partial x^2}}-{\mu }_{\hslash }u\left(x\right),{\mu }_{\hslash }\equiv {\displaystyle \frac{m}{\hslash }}$$

(104)

The Special Wick Rotation (SWR), caused by the imaginary time rotation Wick rotation $(it\to t)$ [14], is defined by:

$$\mathrm{SWR}:it\to t,|\psi \left(t\right)\rangle \to |{\Omega }_t),\langle x_b,t_b|x_a,t_a\rangle \to P(x_b,t_b|x_a,t_a)$$

(105)

Under a SWR, Schrödinger Equation (7) is shifted to the induced micro diffusion (104):

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\partial }{i\partial t}}|\psi \left(t\right)\rangle & =& {\displaystyle \frac{-1}{\hslash }}\widehat{H}|\psi \left(t\right)\rangle \to {\displaystyle \frac{\partial }{\partial t}}|{\Omega }_t)=G|{\Omega }_t)\hfill \end{array}$$

(106)

$$\begin{array}{ccc}\hfill -{\displaystyle \frac{1}{\hslash }}\widehat{H}={\displaystyle \frac{\hslash {{\partial }_x}^2}{2m}}-{\displaystyle \frac{m}{\hslash }}{\displaystyle \frac{V\left(x\right)}{m}}& \equiv & {\displaystyle \frac{{{\partial }_x}^2}{2{\mu }_{\hslash }}}-{\mu }_{\hslash }u\left(x\right)\to \widehat{G}={\displaystyle \frac{1}{2{\mu }_{\hslash }}}{\displaystyle \frac{{\partial }^2}{\partial x^2}}-{\mu }_{\hslash }u\left(x\right)\hfill \end{array}$$

(107)

The simplest example is the Induced Micro-Einstein–Brown motion when $u\left(x\right)=0$ in Equation (104). Applying (105) to the transition amplitude in Equation (92), we obtain the transition probability:

$$P(x_b,t_b|x_a,t_a)=\sqrt{{\displaystyle \frac{1}{4\pi D_{\hslash }(t_b-t_a)}}}exp\left[-{\displaystyle \frac{{(x_b-x_a)}^2}{4D_{\hslash }(t_b-t_a)}}\right]$$

(108)

Here we have introduced the induced micro-diffusion coefficient,

$$D_{\hslash }\equiv 1/\left(2{\mu }_{\hslash }\right)=\hslash /\left(2m\right)$$

(109)

Similarly, applying (105) to the Schrödinger for free particle, Equation (103), we obtain Einstein’s diffusion Equation (102), with $D_{\hslash }\to D$.

A Special Non-Hermitian Case

We can apply Dirac notation and the PBN together to solve the following special quantum system with a non-Hermitian Hamiltonian:

$$\widehat{H}={\widehat{H}}_1-i{\widehat{H}}_2;{\widehat{H}}_1={{\widehat{H}}_1}^{†};{\widehat{H}}_2={{\widehat{H}}_2}^{†};[{\widehat{H}}_1,{\widehat{H}}_2]=0;{\partial }_t\widehat{H}=0$$

(110)

Then the time evolution equation can be written as:

$$i\hslash {\partial }_t\left[|{\Psi }_t\rangle |{\Omega }_t)\right]=\left[{\widehat{H}}_1|{\Psi }_t\rangle \right]|{\Omega }_t)+|{\Psi }_t\rangle \left[-i{\widehat{H}}_2|{\Omega }_t)\right]$$

(111)

It leads to two equations:

$${\partial }_t|{\Psi }_t\rangle =-{\displaystyle \frac{i}{\hslash }}{\widehat{H}}_1|{\Psi }_t\rangle ,{\partial }_t|{\Omega }_t)={\displaystyle \frac{-1}{\hslash }}{\widehat{H}}_2|{\Omega }_t)\underset{\left(106\right)}{=}\widehat{G}|{\Omega }_t\left)\right)$$

(112)

The first is an ordinary Schrödinger equation while the second is a master equation for an induced micro-diffusion. The product of their solutions is the solution of Equation (111). Suppose we have ${\widehat{H}}_1\left(x\right)$ and ${\widehat{H}}_2\left(y\right)$; then the product of Equations (89) and (90) gives the path-integral expression of the system:

$$[\langle x_b,t_b|P(y_b,t_b|]e^{-{\int }_{t_a}^{t_b}dt{\displaystyle \frac{i}{\hslash }}[{\widehat{H}}_1\left(x\right)-i{\widehat{H}}_2\left(y\right)]}|[x_a,t_a\rangle |y_a,t_a\left)\right]$$

$$={\prod }_{m=1}^N{\int }_{-\infty }^{\infty }d{x}_m\langle x_m|e^{-{\displaystyle \frac{i}{\hslash }}{\widehat{H}}_1\left(x\right)\mathsf{\Delta }t}|x_{m-1}\rangle {\prod }_{n=1}^N{\int }_{-\infty }^{\infty }d{y}_{n}P(y_n|e^{\widehat{G}\left(y\right)\mathsf{\Delta }t}|y_{n-1})$$

(113)

Moreover, since both ${\widehat{H}}_1$ and ${\widehat{H}}_2$ are Hermitian, we assume that they have complete sets of discrete orthonormal eigenstates like:

$${\widehat{H}}_1|{\psi }_k\rangle ={\epsilon }_k|{\psi }_k\rangle ,{\widehat{H}}_2|{\varphi }_{\mu })={\lambda }_{\mu }|{\varphi }_{\mu })$$

(114)

Then we obtain the following solutions of Equation (112):

$$|{\psi }_i\left(t\right)\rangle =e^{-i{\epsilon }_{k}t/\hslash }|{\psi }_k\rangle ,|{\varphi }_{\mu }\left(t\right))=e^{-{\lambda }_{\mu }t/\hslash }|{\varphi }_{\mu })$$

(115)

It leads to the expectation value of H in the mixed product state $|{\psi }_i\left(t\right)\rangle |{\varphi }_{\mu }\left(t\right))$ as:

$$E_{k,\mu }\left[\widehat{H}\right]=\langle {\psi }_k\left(t\right)|{\widehat{H}}_1|{\psi }_k\left(t\right)\rangle -iE[H_2|{\phi }_{\mu ,t}]={\epsilon }_k-i{\lambda }_{\mu }$$

(116)

Here we have used the conditional prediction of ${\widehat{H}}_2$ given ${\varphi }_{\mu }\left(t\right)$, defined from Equation (40):

$$E[H_2|{\phi }_{\mu ,t}]={\sum }_{\alpha =\mu }{\lambda }_{\alpha }P\left({\phi }_{\alpha ,t}\right)/P\left({\phi }_{\mu ,t}\right)={\lambda }_{\mu }$$

The imaginary part in Equation (116) looks strange, but it would appear in expectation values of similar complex Hamiltonians, as in [15] p. 290 and [16] p. 3, because $\widehat{H}$ is not Hermitian. However, we know that ${\widehat{H}}^{†}\widehat{H}$ is Hermitian. Therefore, it is reasonable to use the following expression:

$${\left\{E_{k,\mu }\left[{\widehat{H}}^{†}\widehat{H}\right]\right\}}^{1/2}={\left\{E_{k,\mu }[{\left({\widehat{H}}_1\right)}^2+{\left({\widehat{H}}_2\right)}^2]\right\}}^{1/2}={\left\{{{\epsilon }_k}^2+{{\lambda }_{\mu }}^2\right\}}^{1/2}$$

(117)

It is interesting to note that if the initial condition of $|{\Omega }_t)$ is a linear combination of a set of P-kets $|{\phi }_{\mu })$, then, because each P-ket has its probability proportional to $exp[-{\lambda }_{\mu }t/\hslash ]$, the P-ket with the lowest eigenvalue ${\lambda }_{min}$ will eventually dominate, and the expectation value of energy $H_2$ will approach ${\lambda }_{min}$. Therefore, our master Equation (104) describes the processes that will ultimately jump to its lowest energy level, included in its initial condition! Of course, further research on this is required.

6. Potential Applications

Hermitian operators are sufficient for “pure” eigenvalue states for closed systems where the energies are conserved and real valued. However, for mixed states and in a number of physical circumstances, non-Hermitian operators have had to be considered. It is well known that any non-Hermitian linear operator $\widehat{C}$ can be expressed in the form:

$$\widehat{C}=\widehat{A}+i\widehat{B}$$

(118)

where $\widehat{A}$ and $\widehat{B}$ are Hermitian by letting $\widehat{A}=(\widehat{C}+{\widehat{C}}^{†})/2$ and $\widehat{B}=(\widehat{C}-{\widehat{C}}^{†})/\left(2i\right)$. Note that $\widehat{B}\to i\widehat{B}$ can be treated as a Wick rotation in the path-integral application of the PBN. Section 5 dovetails into the matter of general linear operators where there have been a number of applications. We mention:

• The method of “complex scaling” applied to quantum mechanical Hamiltonians was a “hot” area in the area of atomic and molecular physics during the 1970s and 1980s and involved non-Hermitian linear operators. We cite an application by the mathematician Barry Simon on complex scaling to non-relativistic Hamiltonians for molecules [17].
• Another application requiring such an operator was performed by Botten et al. [18]. They showed how to solve a practical problem involving wave scattering using a bi-orthogonal basis, where there is a VPN bra basis and a ket basis consisting of different functions. In a unitary problem, these VPN bra and ket basis functions would be the same. Here, the Helmholtz equation Laplacian ${\nabla }^{2}f+k^{2}f=0$ has a wave number k which is complex. The imaginary part of k indicates loss or gain depending on its sign.
• K. G. Zloshchastiev has performed many applications on the general density operator approach with non-Hermitian Hamiltonians of the form Equation (118) applied to, e.g., open dissipative systems, which automatically deal with mixed states (see [19,20,21]) and von Neumann quantum entropy [22]. For non-Hermitian, open/dissipative systems, there is also the work of Refs. [23,24].
• Consider a rectangular real data matrix $\widehat{Q}$. Its similarity matrix (or adjacency matrix) $\widehat{S}$ and corresponding row stochastic (Markov) matrix $\widehat{R}$ are defined by:

  $$\widehat{S}=\widehat{Q}\ast {\widehat{Q}}^T{R}_{i,j}=S_{ij}/\sum _k{S}_{i,k}$$

  (119)

The symmetric matrix $\widehat{S}$ has real eigenvalues, and as does $\widehat{R}$, even though the latter is non-symmetric. $\widehat{R}$ is a transition matrix in the language of QM but also the key operator in the Meila–Shi algorithm of spectral clustering, as well as quantum clustering with IT applications in science, engineering, (unstructured) text using a “bag-of-words” model [25,26,27,28] and even medicine [29]. Probably, the PBN may provide us with some new approaches to quantum clustering.

If the real data matrix $\widehat{Q}$ in Equation (119) is huge but very sparse (as is often the case in a “bag-of-words” model [27]), we may use the PBN to find other alternative algorithms for text document clustering. Suppose that the dataset has a vocabulary of N labeled keywords, which serve as the P-basis in Equation (28), and $q_{\mu ,k}$ represents the frequency of $k^{th}$ keyword in document $Q_{\mu }$. Since $q_{\mu ,k}$ can be greater than 1, they behave like the occupation numbers of a Boson system in Quantum Field Theory. The conditional probability of finding doc $Q_{\mu }$ given doc $Q_{\nu }$ is:

$$P(Q_{\mu }|Q_{\nu })={\sum }_k^{N}P(Q_{\mu }|k)(k|Q_{\nu })\underset{\left(14\right)}{=}{\sum }_{k\in Q_{\mu }}P(k|Q_{\nu });P(k|Q_{\mu })=q_{\mu ,k}/{\sum }_{k\in Q_{\mu }}{q}_{\mu ,k}$$

(120)

Now, for example, we can define the relevance of two docs and obtain its expression as:

$$R_{\mu ,\nu }\equiv {\displaystyle \frac{1}{2}}[P(Q_{\mu }|Q_{\nu })+P(Q_{\nu }|Q_{\mu })]={\displaystyle \frac{1}{2}}\left[{\sum }_{k\subset Q_{\mu }}P(k|Q_{\nu })+{\sum }_{k\subset Q_v}P(k|Q_{\mu })\right]$$

(121)

This algorithm may be very effective for huge datasets in numerous applications.

7. Summary and Discussion

Inspired by Dirac’s notation used in quantum mechanics (QM), we proposed the Probability Bracket Notation (PBN). We demonstrated that the PBN could be a very useful tool for symbolic representation and manipulation in probability modeling, like the various normalization and expectation formulae for systems of single or multiple random variables, as well as the master equations of homogeneous Markov processes. We also show that a stationary Schrödinger Equation (103) naturally becomes a master equation under the special Wick rotation (see Equations (106) and (107)).

We have shown the similarities between many QM expressions in VBN and related probabilistic expressions in PBN, which might provide us with a beneficial bridge connecting the quantum world with the classical one. We also showed how the PBN could be applied to problems which require non-Hermitian operators. We make no pretense that the PBN creates new physics. Rather it provides a notational interdisciplinary “umbrella” to various statistical and physical processes and thereby opens up the possibility of their synthesis and integration.

8. Conclusions

Of course, more investigations need to be undertaken to verify the consistency (or correctness), usefulness and limitations of our proposal of the PBN, but it is intriguing in its possibility of expressing formulations from classical statistics and QM and handling non-Hermitian operators.