# Euclidean Quantum Mechanics and Universal Nonlinear Filtering

## Abstract

**:**

## 1. Introduction

## 2. Basic Results of Nonlinear Filtering

#### 2.1. From the DMZ SDE to the Robust DMZ PDE

#### 2.2. The Yau Equation of Continuous-Continuous Filtering

## 3. Path Integral Formula for the Fundamental solution of the YYe

## 4. YYe and Euclidean Quantum Mechanics

#### 4.1. A General Equivalence

#### 4.2. A Special Case

## 5. Nonlinear Yau Filtering System and Time-Dependent Schrödinger equation

#### 5.1. The Work of S-T. Yau and Stephen Yau

#### 5.2. Path Integral Derivation

## 6. Additional Remarks

- The general equivalence proposed in Section 4.1 is that between the fundamental solution of the FPKfe and a certain matrix element of a quantum mechanical system. Note that all previous equivalences were between fundamental solutions of FPKfe and a Schrödinger equation. While the previous equivalences were obtainable using operator methods, it is not clear how the proposed general equivalence can be derived using operator methods.
- The Feynman path integral methods used here are very different from the measure-theoretic methods used to study the nonlinear filtering problem (see, for instance, [16]). The Feynman path integral approach developed in [2,6] and this paper leads to an independent way of tackling the universal nonlinear filtering problem. In particular, unlike the standard filtering theory approaches, the DMZ equation (or its variants) is not taken as an input. On the contrary, the YYe is obtained directly as a consequence of the path integral approach. It is also noted that the Euclidean quantum physics referred to in the equivalence to nonlinear filtering developed in this paper is a quantum mechanical one, not a quantum field theoretical one. Also note that ${\hslash}_{\nu}$ here is analogous to the Planck’s constant, ℏ, in quantum physics.
- Although a formal equivalence has been developed between Euclidean quantum mechanics and universal nonlinear filtering, it is important to point out that there is a profound difference between classical and quantum probabilities. In the “real time” (as opposed to Euclidean time) quantum mechanics, the transition probability amplitude is not a probability; it may not even be real. The probability amplitude is to be multiplied with its complex conjugate to obtain a probability density. In contrast, the transition probability density in filtering theory is a classical probability density. We have shown that, in a mathematical sense, matrix elements in a Euclidean quantum mechanical system equal the transition probability density of a classical stochastic process. This equivalence is purely mathematical, not conceptual.
- The reason for some of the mathematical equivalence between nonlinear filtering and quantum physics is the semi-group property. That is, in stochastic processes the Chapman-Kolmogorov semi-group property is a fundamental property of Markov processes. The Chapman-Kolmogorov semi-group property allows the transition probability density in stochastic process to be written as follows. Let us partition the time interval $[{t}_{0},t]$ into N equi-spaced time intervals so that ${t}_{i}={t}_{0}+i\u03f5$ where $\u03f5=(t-{t}_{0})/N$. Then, from the Chapman-Kolmogorov semi-group property it follows that$$\begin{array}{cc}\hfill P(t,x|{t}_{0},{x}_{0})& =\int \{{d}^{n}x\left({t}_{1}\right)\cdots {d}^{n}x\left({t}_{N-1}\right)\}P(t,x|{t}_{N-1},x\left({t}_{N-1}\right))\cdots P({t}_{1},x\left({t}_{1}\right)|{t}_{0},{x}_{0}),\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& =\int \left(\right)open="\{"\; close="\}">\prod _{i=0}^{N}{d}^{n}x\left({t}_{i}\right)\left(\right)open="\{"\; close="\}">\prod _{i=1}^{N}P({t}_{i},x\left({t}_{i}\right)|{t}_{i-1},x\left({t}_{i-1}\right))\hfill & {\delta}^{n}(x\left(t\right)-x){\delta}^{n}(x\left({t}_{0}\right)-{x}_{0}),\end{array}\hfill \phantom{\rule{1.em}{0ex}}& ={\int}_{x\left({t}_{0}\right)={x}_{0}}^{x\left(t\right)=x}\left(\right)open="\{"\; close="\}">\prod _{i=1}^{N-1}{d}^{n}x\left({t}_{i}\right)\prod _{i=1}^{N}P({t}_{i},x\left({t}_{i}\right)|{t}_{i-1},x\left({t}_{i-1}\right)),\hfill $$
- Note that the unitarity of the time evolution operator plays a crucial role in quantum mechanics. This is because it is essential for conservation of probability. It is noted that the unitarity of the time evolution operator implies that the Hamiltonian operator is Hermitian. Note that in the standard formulation of quantum mechanics, Hermiticity of the Hamiltonian is required in order to ensure that the eigenvalues of the Hamiltonian (and hence the possible energies) are real and that the time evolution is unitary (i.e., conserves probability). This can also be ensured even if the Hamiltonian is not Hermitian provided that the Hamiltonian is space-time (or $\mathcal{PT}$) reflection symmetric. For a pedagogical discussion of non-Hermitian quantum mechanics, see [17].The evolution of the probability distribution for a continuous Markov processes is described by the FPKfe. The FPKfe operator is not a Hermitian operator. This is not inconsistent with the conservation of probability. This is because the FPKfe is a continuity equation, and so the probability is conserved (as long as the boundary terms vanish). This is yet another instance of the profound difference between classical and quantum probabilities.
- In the continuous-continuous filtering case, the YYe plays the role that the FPKfe plays in continuous-discrete filtering. However, the YYe is not a continuity equation. This is not a contradiction since the YYe is related to the unnormalized conditional probability density, whereas the FPKfe evolves (and preserves the normalization of) the normalized probability density.
- In this paper, it has been assumed that the noise is additive and the model is not explicitly time-dependent. In the more general case, a simple equivalence, as derived in this paper, is not possible. For instance, in order to obtain the YYe, it was necessary to assume that the measurement model was not explicitly time dependent (see [6]); this is not valid in the general case. Also, when the noise is multiplicative, quantization is not as straightforward. This can be traced to the well-known operator ordering ambiguity in quantum physics (since the position and momentum operators do not commute). Furthermore, standard calculus manipulations are no longer possible in the path integral. However, the path integral result is the same even for the multiplicative noise case if the diffusion matrix is proportional to the identity matrix.
- It is noted that the derivation of the proposed equivalence is not entirely satisfactory. This is because the Feynman path integral used in obtaining the equivalence is not rigorous. In particular, quantities like ${\dot{x}}_{i}\left(t\right)$ and ${\dot{x}}_{i}^{2}\left(t\right)$ in the Feynman path integral are mathematically ill-defined, especially since the trajectories contributing to the integral are not differentiable. A more satisfactory approach would be to use results in stochastic integration theory, such as the Ito-Girsanov transformation, in the mathematically well-defined Feynman-Kǎc formalism.

## 7. Conclusions and Future Work

- The equivalence to Euclidean quantum mechanics immediately leads to many filtering problems that can be solved exactly, as will be explored in future papers. The remarkable fact is that many exactly solvable Euclidean quantum mechanical problems correspond to filtering problems that are not finite-dimensional (i.e., Lie algebra of ${\mathcal{L}}_{0}$ and ${h}_{i}\left(x\right),i=1,\cdots ,m$ is not finite-dimensional, see [6]). Thus, simplicity in filtering theory does not imply finite dimensionality.
- The path integral formulation naturally leads to a perturbative solution of the nonlinear filtering problem. Such a perturbative solution which is analogous to extended Kalman filtering is called extended Yau filtering (EYF). Thus, it is possible to perturb about the Yau filter (a generalization of the linear Kalman filter), rather than the linear Kalman filter. Clearly, since the EKF is a special case of the EYF, such a formulation will be superior to the EKF.
- However, it is noted that the fundamental solution is defined nonperturbatively. This is important since sometimes the perturbative approaches, like EKF, fail.
- Perhaps the most important advantage of the path integral lies in numerical methods of computation [19]. In fact, path integrals are the only known way to carry out nonperturbative computation in quantum chromodynamics (QCD). Note that QCD, the gauge theory of strong interactions, is a quantum field theory, not merely quantum mechanical and the QCD Lagrangian is highly nonlinear. The excellent numerical results in QCD suggest that currently known path integral methods should give very good performance for the simpler case of (large dimensional) universal nonlinear filtering. It is sufficient to note that the Dirac-Feynman approximation, the crudest approximation of the path integral, already yields excellent results (see [2]), and is adequate for smaller dimensional problems.

## Acknowledgments

## A Verification of the Path Integral Formula

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Balaji, B.
Euclidean Quantum Mechanics and Universal Nonlinear Filtering. *Entropy* **2009**, *11*, 42-58.
https://doi.org/10.3390/e11010042

**AMA Style**

Balaji B.
Euclidean Quantum Mechanics and Universal Nonlinear Filtering. *Entropy*. 2009; 11(1):42-58.
https://doi.org/10.3390/e11010042

**Chicago/Turabian Style**

Balaji, Bhashyam.
2009. "Euclidean Quantum Mechanics and Universal Nonlinear Filtering" *Entropy* 11, no. 1: 42-58.
https://doi.org/10.3390/e11010042