Complex Stochastic Optimal Control Foundation of Quantum Mechanics
Vasil Yordanov

Fast and Robust Expectation Propagation MIMO Detection via Preconditioned Conjugated Gradient
    Luca Schmid1, Dominik Sulz2, and Laurent Schmalen1
1Communications Engineering Lab, Karlsruhe Institute of Technology, 76187 Karlsruhe, Germany

2Mathematical Institute, University of Tübingen, 72076 Tübingen, Germany

Email: 1, 2

    April 29, 2024
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Recent studies have expanded the use of the stochastic Hamilton–Jacobi–Bellman (HJB) equation to include complex variables for deriving quantum mechanical equations. However, these studies typically assume that it’s valid to apply the HJB equation directly to complex numbers, an approach that overlooks the fundamental problem of comparing complex numbers to find optimal controls. This paper addresses how to properly apply the HJB equation in the context of complex variables. 
Our findings significantly reevaluate the stochastic movement of quantum particles, directly influenced by the Cauchy-Riemann theorem. These insights not only deepen our understanding of quantum dynamics but also enhance the mathematical rigor of the framework for applying stochastic optimal control in quantum mechanics.



§ INTRODUCTION

Recently, a number of studies <cit.> have derived non-relativistic quantum mechanical equations from the stochastic Hamilton–Jacobi–Bellman (HJB) equation. 
Although not explicitly stated in these works, the use of complex diffusion coefficients necessitates that the stochastic equations of motion incorporate complex coordinates and velocities of the particle, while the intermediate action is a complex function of the complex coordinates.

In their work, Yang et al. <cit.> introduce the concept of complex quantum mechanics, which also relies on the HJB equation. This study explicitly defines complex stochastic equations of motion and incorporates complex functions and variables within the stochastic HJB framework.

In my previous research <cit.>, I extended these ideas by deriving the Dirac equation, assuming a complex equation of motion and applying the HJB equation to complex functions and variables.

Despite the innovative approaches in these papers, all assume that the four-coordinates and the intermediate actions can be straightforwardly replaced with their complex counterparts in the stochastic HJB equation.

This paper mathematically clarifies the correctness of using complex numbers in the HJB equation, as suggested in previous studies. It demonstrates that the formal replacement of real numbers with complex ones in the HJB equation is indeed valid.
However, this formal replacement needs to be considered as a system of two HJB equations: one for the real part and one for the imaginary part of the intermediate action J.

We provide mathematical proof that the equations for finding the optimal control policy are consistent with those in the real-valued framework. However, for the complex-valued case, these equations require taking a complex derivative on the intermediate action J.
This proof depends on the crucial assumption that both the Lagrangian ℒ and the intermediate action J can be analytically continued into the complex plane.
Additionally, we establish that, due to the Cauchy-Riemann equations, the diffusion coefficient is purely imaginary, indicating that stochastic movements are confined to the imaginary components of particle coordinates.

The derivation of the complex HJB equation is based on methodologies outlined in previous research <cit.>. These works, which derive the HJB equation without considering complex numbers, establish a starting point for extending the equation to incorporate complex variables, as demonstrated in this paper.



§ DERIVATION OF THE COMPLEX STOCHASTIC HJB EQUATION

According to the first postulate of the Stochastic Optimal Control Theory of Quantum Mechanics <cit.>, we assume that the particle moves as a Brownian particle in four-dimensional spacetime, influenced by an external random spacetime force. However, this motion occurs within the complex plane of each four-coordinate component. The complex stochastic equation of motion governing this behavior is given by:

    dz_μ = w_μ ds + σ_μ dW_μ,   μ=0..3,

where w_μ represents the complex velocity and σ_μ denotes the complex-valued diffusion coefficients, with dW_μ  being the increments of a Wiener process that encapsulates the stochastic nature of the particle’s trajectory.

As assumed either implicitly or explicitly in referenced above works, it is consider a particle moving within a complex plane. The four-coordinates of this particle are defined as complex numbers, enabling the inclusion of both real and imaginary components to fully describe its position in spacetime. We express these coordinates as follows:

    z_μ = x_μ + i y_μ,   μ=0..3,

where x_μ and  y_μ represent the real and imaginary parts, respectively, and the index μ spans the four dimensions of spacetime, aligning with the notation used in relativistic mechanics.

The velocity of the particle is similarly complexified to account for motions in both the real and imaginary dimensions of the spacetime:

    w_μ = u_μ + i v_μ,   μ=0..3,

where u_μ  and v_μ  denote the real and imaginary components of the four-velocity, respectively.

The real and imaginary parts of the stochastic equation of motion of the particle are represented as follows:


    dx_μ=u_μ ds+ σ^x_μ dW_μ,   μ=0..3,



    dy_μ=v_μ ds+ σ^y_μ dW_μ,   μ=0..3,


where σ^x_μ and σ^y_μ denote the diffusion coefficients associated respectively with the stochastic dynamics of the real and imaginary components of the coordinates.

According to the third postulate of the Stochastic Optimal Control Theory of Quantum Mechanics <cit.>, Nature tries to minimize the expected value for the action, in which the particle’s velocity is consider to be a control parameter of the optimization.
Formally the complex action is defined as the minimum of the expected value of stochastic action:

    S(𝐳_i, 𝐰(τ_i →τ_f))=min_𝐰(τ_i →τ_f) < ∫_τ_i^τ_f ds ℒ(𝐳(s), 𝐰(s), s) >_𝐳_i,


However, this formal definition encounters difficulties as it is not possible to directly find a minimum of a complex-valued function. To address this, we redefine the complex action to separate its real and imaginary components, each optimized independently:


    S(𝐳_i, 𝐰)=   S_R(𝐱_i, 𝐲_i, 𝐮, 𝐯) + i S_I(𝐱_i, 𝐲_i, 𝐮, 𝐯)

where the real and imaginary parts of the action are defined respectively as follows:


    S_R,I(𝐱_i, 𝐲_i, 𝐮(τ_i →τ_f), 𝐯(τ_i →τ_f)) 
    =min_𝐮(τ_i →τ_f) 
    𝐯(τ_i →τ_f)< ∫_τ_i^τ_f ds ℒ_R,I(𝐱(s), 𝐲(s), 𝐮(s),𝐯(s), s) >_𝐱_i, 𝐲_i


where ℒ_R(𝐱, 𝐲, 𝐮, 𝐯, s) and ℒ_I(𝐱, 𝐲, 𝐮, 𝐯, s) are the real and imaginary parts of the Lagrangian of the test particle.  These functions depend on the real and imaginary components of the control policies 𝐮 and 𝐯, the and four-coordinates 𝐱 and 𝐲 at proper time s.  
The subscripts 𝐱_i and 𝐲_i on the expectation value indicate that the expectation is calculated over all stochastic trajectories that originate at the complex coordinate 𝐳_i=𝐱_i + i 𝐲_i.

We define the complex Lagrangian of the particle as:

    ℒ(𝐳, 𝐰, s)=ℒ_R(𝐱, 𝐲, 𝐮,𝐯, s) + i ℒ_I(𝐱, 𝐲, 𝐮, 𝐯, s)

such that ℒ(𝐳, 𝐰, s) is the analytic continuation of the real Lagrangian of the particle.

The task of optimal control theory <cit.> is to find the controls 𝐮(s) and 𝐯(s), τ_i < s < τ_f, denoted as 𝐮(τ_i →τ_f) and  𝐯(τ_i →τ_f), that minimizes the expected value of the action S_R,I(𝐱_i, 𝐲_i, 𝐮(τ_i →τ_f),  𝐯(τ_i →τ_f)).

We introduce the optimal cost-to-go function for any intermediate proper time τ, where  τ_i < τ < τ_f:

    J_R,I(τ,  𝐱_τ,  𝐲_τ)=min_𝐮(τ_i →τ_f) 
    𝐯(τ_i →τ_f)< ∫_τ^τ_f ds ℒ_R,I(𝐱(s), 𝐲(s), 𝐮(s), s) >_𝐱_i, 𝐲_i


We define the complex cost-to-go function as:

    J(τ, 𝐳) = J_R(τ, 𝐱, 𝐲) + i J_I(τ, 𝐱, 𝐲),

assuming that it is also an analytic function in the complex plane.

By definition, the action S_R,I(𝐱_i, 𝐮(τ_i →τ_f), 𝐯(τ_i →τ_f)) is equal to the cost-to-go function J_R,I(τ_i,  𝐱_τ_i, 𝐲_τ_i) at the initial proper time and spacetime coordinate:

    S_R,I(𝐱_i, 𝐲_i, 𝐮(τ_i →τ_f), 𝐯(τ_i →τ_f))=J_R,I(τ_i,  𝐱_τ_i, 𝐲_τ_i)


We can rewrite recursive formula for J_R,I(τ, 𝐱_τ, 𝐲_τ) for any intermediate time τ', where τ < τ' < τ_f:

    J_R,I(τ,  𝐱_τ, 𝐲_τ) = 
       =min_𝐮(τ_i →τ_f) 
    𝐯(τ_i →τ_f)< ∫_τ^τ' ds ℒ_R,I(s, 𝐱_s, 𝐲_s, 𝐮_s,  𝐯_s) + ∫_τ'^τ_f ds ℒ_R,I(s, 𝐱_s, 𝐲_s, 𝐮_s, 𝐯_s) >_𝐱_τ, 𝐲_τ
       =min_𝐮(τ_i →τ_f) 
    𝐯(τ_i →τ_f)< ∫_τ^τ' ds ℒ_R,I(s, 𝐱_s,  𝐲_s, 𝐮_s, 𝐯_s) + min_𝐮(τ'  →τ_f )< ∫_τ'^τ_f ds ℒ_R,I(s, 𝐱_s, 𝐲_s, 𝐮_s, 𝐯_s)  >_𝐱_τ',𝐲_τ'>_𝐱_τ, 𝐲_τ
       =min_𝐮(τ_i →τ_f) 
    𝐯(τ_i →τ_f)< ∫_τ^τ' ds ℒ_R,I(s, 𝐱_s, 𝐲_s, 𝐮_s, 𝐯_s) + J(τ',  𝐱_τ', 𝐲_τ') >_𝐱_τ, 𝐲_τ.


In above equation we split the minimization over two intervals. These are not independent, because the second minimization is conditioned on the starting value x_τ', y_τ', which depends on the outcome of the first minimization.

If τ' is a small increment of τ, τ'=τ + dτ then:


    J_R,I(τ,  x_τ, y_τ)=min_𝐮(τ_i →τ_f) 
    𝐯(τ_i →τ_f)<ℒ_R,I(τ, 𝐱_τ, 𝐲_τ, 𝐮_τ, 𝐯_τ) d τ + J_R,I(τ+dτ, 𝐱_τ + dτ, 𝐲_τ + dτ) >_𝐱_τ, 𝐲_τ



We must take a Taylor expansion of J_R and J_I in d 𝐱, d 𝐲 and dτ. 
However, since <d 𝐱^2 >=σ_x^2 dτ and <d 𝐲^2 >=σ_y^2 dτ is of order dτ, we must expand up to order d 𝐱^2 and d 𝐲^2:


    < J_R,I(τ+dτ,   𝐱_τ + dτ, 𝐲_τ + dτ) >_𝐱_τ = 
       =∫ d  𝐱_τ+dτ d  𝐲_τ+dτ𝒩 ((𝐱_τ + dτ, 𝐲_τ + dτ) |  (𝐱_τ, 𝐱_τ), σ dτ) J_R,I(τ + dτ, 𝐱_τ+dτ, 𝐲_τ+dτ) = 
       =∫ d  𝐱_τ+dτ d 𝐲_τ+dτ𝒩 ((𝐱_τ+dτ, 𝐲_τ+dτ) |  (𝐱_τ, 𝐲_τ), σ dτ) ×
       × (J_R,I(τ,  𝐱,  𝐲)+dτ∂_τ J_R,I(τ,  𝐱_τ,  𝐲_τ) + dx^μ∂_𝐱^μ J_R,I(τ,  𝐱_τ, 𝐲_τ) + dy^μ∂_𝐲^μ J_R,I(τ,  𝐱_τ, 𝐲_τ) + 
        + dx^μ dx^ν1/2∂_𝐱^μ𝐱^ν J_R,I(τ,  𝐱_τ, 𝐲_τ) + dy^μ dy^ν1/2∂_𝐲^μ𝐲^ν J_R,I(τ,  𝐱_τ,  𝐲_τ) +dx^μ dy^ν∂_𝐱^μ𝐲^ν J(τ,  𝐱_τ,  𝐲_τ) ) = 
       =J_R,I(τ, 𝐱, 𝐲)+dτ∂_τ J_R,I(τ, 𝐱_τ, 𝐲_τ) + <dx^μ> ∂_𝐱^μ J_R,I(τ, 𝐱_τ, 𝐲_τ) + <dy^μ> ∂_𝐲^μ J_R,I(τ, 𝐱_τ, 𝐲_τ) + 
        + 1/2<dx^ν dx^μ> ∂_𝐱^ν𝐱^μ J_R,I(τ, 𝐱_τ, 𝐲_τ) + 1/2<dy^ν dy^μ> ∂_𝐲^ν𝐲^μ J_R,I(τ, 𝐱_τ, 𝐲_τ)


Here 𝒩 ((𝐱_τ + dτ, 𝐲_τ + dτ )| (𝐱_τ, 𝐲_τ), σ dτ) is the conditional probability starting from state (𝐱_τ, 𝐲_τ) to end up in state (𝐱_τ+dτ, 𝐲_τ+dτ). The integration is over the entire spacetime for 𝐱 and 𝐲.

We can calculate the expected values of dx^μ using Equation (<ref>):

    < dx^μ> = ∫ d 𝐱_τ + dτ d 𝐲_τ + dτ𝒩 ((𝐱_τ + dτ, 𝐲_τ + dτ) |  (𝐱_τ, 𝐲_τ), σ^μ dτ) dx^μ
       =  ∫ d 𝐱_τ + dτ d 𝐲_τ + dτ𝒩 ((𝐱_τ + dτ, 𝐲_τ + dτ) |  (𝐱_τ, 𝐲_τ), σ^μ dτ) (𝐮^μ dτ + σ^μ d W^μ) 
       =𝐮^μ dτ∫ d 𝐱_τ + dτ d 𝐲_τ + dτ𝒩 ((𝐱_τ + dτ, 𝐲_τ + dτ) |  (𝐱_τ, 𝐲_τ), σ^μ dτ) + 
        +   ∫ d 𝐱_τ + dτ d 𝐲_τ + dτ𝒩 ((𝐱_τ + dτ, 𝐲_τ + dτ) |  (𝐱_τ, 𝐲_τ), σ^μ dτ) σ^μ d W^μ


From this, we derive that:

    <dx^μ> = u^μ dτ


Similarly, we can find the expected value for dy^μ as follows:

    <dy^μ> = v^μ dτ


Similarly, the calculation of  <dx^ν dx^μ>  is performed using:

    < dx^ν dx^μ>  =  ∫ d 𝐱_τ + dτ d 𝐲_τ + dτ𝒩 ((𝐱_τ + dτ, 𝐲_τ + dτ) |  (𝐱_τ, 𝐲_τ), σ^μ dτ)  dx^μ dx^ν
       =  ∫ d 𝐱_τ + dτ d 𝐲_τ + dτ𝒩 ((𝐱_τ + dτ, 𝐲_τ + dτ) |  (𝐱_τ, 𝐲_τ), σ^μ dτ)  (u^μ dτ + σ^μ d W^μ) (u^ν dτ + σ^ν d W^ν) 
       =  ∫ d 𝐱_τ + dτ d 𝐲_τ + dτ𝒩 ((𝐱_τ + dτ, 𝐲_τ + dτ) |  (𝐱_τ, 𝐲_τ), σ^μ dτ)  ×
       × (u^μ u^ν d^2 τ + u^μσ^ν dτ d W^ν + σ^μ d W^μ u^ν dτ+  σ^μ d W^μσ^ν d W^ν )


From which we derive:

    < dx^ν dx^μ> = 0, μ≠ν,  < (dx^μ)^2 > = σ_x^μσ_x^μ d τ


Respectively, for the imaginary components:

    < dy^ν dy^μ> = 0, μ≠ν,  < (dy^μ)^2 > = σ_y^μσ_y^μ d τ


After substituting the above equations into equation (<ref>) , we derive the stochastic HJB equation for the real part of our system:

    -∂_τ J_R(τ, 𝐱)=min_𝐮, 𝐯( ℒ_R(τ, 𝐱, 𝐲, 𝐮, 𝐯) + u^μ∂_𝐱 ^μ J_R(τ, 𝐱, 𝐲) + v^μ∂_𝐲 ^μ J_R(τ, 𝐱, 𝐲) ) +  
        + 1/2∑_μ=0^3σ_x^μσ_x^μ∂_𝐱^μ𝐱^μ J_R(τ, 𝐱_τ, 𝐲_τ) + 1/2∑_μ=0^3σ_y^μσ_y^μ∂_𝐲^μ𝐲^μ J_R(τ, 𝐱_τ, 𝐲_τ)


In a similar manner, the equation for the imaginary part is derived:

    -∂_τ J_I(τ, 𝐱)=min_𝐮, 𝐯( ℒ_I(τ, 𝐱, 𝐲, 𝐮, 𝐯) + u^μ∂_𝐱 ^μ J_I(τ, 𝐱, 𝐲) + v^μ∂_𝐲 ^μ J_I(τ, 𝐱, 𝐲) ) +  
        + 1/2∑_μ=0^3σ_x^μσ_x^μ∂_𝐱^μ𝐱^μ J_I(τ, 𝐱_τ, 𝐲_τ) + 1/2∑_μ=0^3σ_y^μσ_y^μ∂_𝐲^μ𝐲^μ J_I(τ, 𝐱_τ, 𝐲_τ)


The optimal control policies 𝐮 and 𝐯, which minimize the expected cost, can be determined from the following conditions:

    ∂_u_μRe(L_z(w_μ,z)) + ∂_x^μ J_R (τ, 𝐱, 𝐲) = 0 
    ∂_v_μRe(L_z(w_μ,z)) + ∂_y^μ J_R (τ, 𝐱, 𝐲) = 0


For the imaginary part of the action, the minimization conditions are obtained from:

    ∂_u_μIm(L_z(w_μ,z)) + ∂_x^μ J_I (τ, 𝐱, 𝐲) = 0 
    ∂_v_μIm(L_z(w_μ,z)) + ∂_y^μ J_I (τ, 𝐱, 𝐲) = 0


These conditions ensure that the stochastic Hamilton-Jacobi-Bellman (HJB) equations for both the real and imaginary actions are satisfied, leading to the minimum expected value of the action across all possible trajectories of the system.
Later, we will prove that the equations derived for the optimal control policies, specifically equations (<ref>) and (<ref>), are equivalent. 

By multiplying equation (<ref>) by the imaginary unit i and adding it to
equation (<ref>), we obtain the combined formal form of the stochastic Hamilton-Jacobi-Bellman (HJB) equation for the complex action.

    -∂_τ J(τ, 𝐱)=min_𝐮, 𝐯 ( ℒ(τ, 𝐱, 𝐲, 𝐮, 𝐯) + u^μ∂_𝐱 ^μ J_R(τ, 𝐱, 𝐲) + v^μ∂_𝐲 ^μ J_R(τ, 𝐱, 𝐲) + 
       + i u^μ∂_𝐱 ^μ J_I(τ, 𝐱, 𝐲) + i v^μ∂_𝐲 ^μ J_I(τ, 𝐱, 𝐲) ) +  
        + 1/2∑_μ=0^3σ_x^μσ_x^μ∂_𝐱^μ𝐱^μ J_R(τ, 𝐱_τ, 𝐲_τ) + 1/2∑_μ=0^3σ_y^μσ_y^μ∂_𝐲^μ𝐲^μ J_R(τ, 𝐱_τ, 𝐲_τ) + 
        + i 1/2∑_μ=0^3σ_x^μσ_x^μ∂_𝐱^μ𝐱^μ J_I(τ, 𝐱_τ, 𝐲_τ) + i 1/2∑_μ=0^3σ_y^μσ_y^μ∂_𝐲^μ𝐲^μ J_I(τ, 𝐱_τ, 𝐲_τ)

 
Since the complex intermediate action J(τ, 𝐳) is analytic, the following equations are satisfied:

    ∂_𝐳^μ J(τ, 𝐳) = ∂_𝐱^μ J_R(τ, 𝐱,  𝐲) + i ∂_𝐱^μ J_I(τ, 𝐱, 𝐲) = ∂_𝐲^μ J_I(τ, 𝐱, 𝐱) - i ∂_𝐲^μ J_R(τ, 𝐱, 𝐲)


The Cauchy-Riemann equations are:

    ∂_𝐱^μ J_R(τ, 𝐱)= ∂_𝐲^μ J_I (τ, 𝐲) ,   ∂_𝐱^μ J_I(τ, 𝐲) = - ∂_𝐲^μ J_R(τ, 𝐱)


The second derivative of the intermediate action is:

    ∂_𝐳^μ𝐳^μ J(τ, 𝐳) =    ∂_𝐱^μ𝐱^μ J_R(τ, 𝐱,  𝐲) + i ∂_𝐱^μ𝐱^μ J_I(τ, 𝐱, 𝐲) = 
    
    =     - ∂_𝐲^μ𝐲^μ J_R(τ, 𝐱, 𝐲) - i ∂_𝐲^μ𝐲^μ J_I(τ, 𝐱, 𝐲)


Using the above equations, we can simplify the HJB equation:

    -∂_τ J(τ, 𝐳)=min_𝐰 ( ℒ(τ, 𝐳, 𝐰) + w^μ∂_𝐱 ^μ J_R(τ, 𝐱, 𝐲) + i w^μ∂_𝐱 ^μ J_I(τ, 𝐱, 𝐲) ) +  
        + 1/2∑_μ=0^3σ_x^μσ_x^μ∂_𝐳^μ𝐳^μ J(τ, 𝐳_τ) - i 1/2∑_μ=0^3σ_y^μσ_y^μ∂_𝐳^μ𝐳^μ J(τ, 𝐳_τ)= 
        = min_𝐰 ( ℒ(τ, 𝐳, 𝐰) + w^μ∂_𝐳 ^μ J(τ, 𝐳) ) + 1/2∑_μ=0^3σ^μσ^μ∂_𝐳^μ𝐳^μ J(τ, 𝐳_τ)


Finally, we can formally write the complex HJB equation:

    -i ∂_τ J_z(τ, 𝐳)=min_𝐰(ℒ_z(τ, 𝐳, 𝐰) + w^μ∂_z^μ J(τ, 𝐱) + 1/2∑_μ=0^3σ^μσ^μ∂_𝐳^μ𝐳^μ J_z(τ, 𝐳_τ) ),

where the complex diffusion coefficient satisfies

    σ^μσ^μ = σ_x^μσ_x^μ - i σ_y^μσ_y^μ


It is crucial to emphasize again that this formal form of the HJB equation is conceptually meaningful only when considering the distinct equations for its real  (<ref>) and imaginary  (<ref>) parts.

It is clear from its definition that the boundary condition for J_z(τ, 𝐳_τ) is:

    J_z(τ_f, 𝐳_τ_f) = 0


Sinces L_z(𝐰_μ, 𝐳) is an analytic function, the derivative operator and the operator for taking the real part commute. 
Consequently, equation (<ref>) can be expressed as:

    Re(∂_u_μ  L_z(𝐰_μ, 𝐳)) + ∂_x^μ J_R (τ, 𝐱, 𝐲) = 0 
    Re(∂_v_μ L_z(𝐰_μ, 𝐳)) + ∂_y^μ J_R (τ, 𝐱, 𝐲) = 0


Similarly, from equation (<ref>), we can derive the equations for the optimal control that result from minimizing the imaginary part of the intermediate action:

    Im(∂_u_μ  L_z(𝐰_μ, 𝐳)) + ∂_x^μ J_I (τ, 𝐱, 𝐲) = 0 
    Im(∂_v_μ L_z(𝐰_μ, 𝐳)) + ∂_𝐲^μ J_I (τ, 𝐱, 𝐲) = 0


We will prove that Equations (<ref>) and (<ref>) are equivalent. To do this, we find the derivatives:

    ∂_u_μ  L_z(w_μ,z) = ∂_w_μ  L_z(𝐰, 𝐳) ∂ w_μ/∂ u_μ=  ∂_w_μ  L_z(𝐰, 𝐳) 
    ∂_v_μ L_z(w_μ,z)  = ∂_w_μ  L_z(𝐰, 𝐳) ∂ w_μ/∂ v_μ= i ∂_w_μ  L_z(𝐰, 𝐳)


If we substitute the above derivatives into Equation (<ref>), multiply the second equation by the imaginary unit, and add it to the first equation, we obtain:

    Re(∂_w_μ  L_z(𝐰_μ, 𝐳)) + i Re(i ∂_w_μ L_z(𝐰_μ, 𝐳)) + ∂_x^μ J_R (τ, 𝐱, 𝐲) + i ∂_y^μ J_R (τ, 𝐱, 𝐲) = 0


Similarly, if we substitute the above derivatives into Equation (<ref>), multiply the second equation by the imaginary unit, and subtract it from the first equation, we obtain

    Im(∂_w_μ  L_z(𝐰_μ, 𝐳)) - i Im(i ∂_w_μ L_z(𝐰_μ, 𝐳)) + ∂_x^μ J_I (τ, 𝐱, 𝐲) - i ∂_y^μ J_I (τ, 𝐱, 𝐲) = 0


From the Cauchy-Riemann equation (<ref>), the definition of the complex derivative (<ref>), and the identities Re(Z)+i Re (i Z)=Z and -Im (i Z) + i Im(Z)=Z, where Z is any complex number, we prove that both equations can be written as:

    ∂_w_μ  L_z(𝐰_μ, 𝐳) + ∂_z^μ J_R (τ, 𝐱, 𝐲) = 0


The equation for the optimal control policy, 𝐰 = 𝐮 + i 𝐯, maintains the same form as that of the real-valued HJB equation. However, it requires taking a complex derivative of the intermediate action, as can be observed.

In the next section, we will illustrate this concept with a specific example, applying a concrete Lagrangian for a relativistic particle in an electromagnetic field.



§ ANALYTIC CONTINUATION OF THE COVARIANT RELATIVISTIC LAGRANGIAN

The relativistic Lagrangian for a particle in an electromagnetic field is given by:

    Λ= σ̃ m c √(σ̃ u_μ u^μ) + q A_μ u^μ,

where q represents the charge of the particle and A_μ denotes the 4-vector potential. The symbol σ̃ indicates the sign convention for the metric tensor: it takes the value of +1 for the metric with diagonal elements (1,-1,-1,-1) and -1 for the metric (-1,1,1,1), as elaborated in <cit.>.

The components of the four-velocity of the particle are related to the speed of light by the equation:


    u_μ u^μ = σ̃ c^2.


This relation, referred to as the “weak equation” by Dirac, allows us to treat u^μ as unconstrained quantities until all differentiation operations have been carried out, at which point we impose the condition of equation (<ref>) (see <cit.> Chapter 7.10).
This will be the approach we employ as we seek to minimize the expected value of the stochastic action.

The Lagrangian in equation (<ref>) is a real-valued function of real arguments – the coordinates and velocity of the particle. In complex stochastic optimal control, we assume that this Lagrangian is the analytic continuation of the real-valued Lagrangian referenced in equation (<ref>).

    L_z(𝐳, w_μ)=σ̃ m c √(σ̃ w_μ w^μ) + q A_μ(τ,  𝐳) w^μ,


The “weak equation” should be also analytically continued:

    w_μ w^μ = σ̃ c^2.


The derivative of the complex Lagrangian can be calculated using the “weak equation” (<ref>):

    ∂_w_μ L_z(w_μ,z) =∂_w_μ  (σ̃ m c √(σ̃ w_μ w^μ) + q A_μ(τ,  𝐳) w^μ) =1/2 √(σ̃ w_μ w^μ) 2 w_μ + q A_μ(τ,  𝐳),


Finally the complex velocity is:

    w_μ = -1/m (∂_z^μ J (τ, 𝐳) + q A_μ(τ,  𝐳))




§ STOCHASTIC EQUATION OF MOTION

In my previous work <cit.>, I demonstrated that to linearize the HJB equation, the diffusion coefficient must be purely imaginary:

    g_μμσ^μσ^μ= - 2 i ϵ_r ħ/m


From equation (<ref>), it is evident that the real stochastic coefficient is equal to zero:

    g_μμσ_x^μσ_x^μ=0


Conversely, the imaginary stochastic coefficient is represented by:

    g_μμσ_y^μσ_y^μ= 2 ϵ_r ħ/m


This leads us to the stochastic equations of motion:

    d x_μ=u_μ ds



    d y_μ=v_μ ds + ϵ_r g_μμ√(2 ħ/m) dW_μ,   μ=0..3


This result is particularly interesting because it reveals that only the imaginary part of the particle's motion is governed by a random process, while the real part of the coordinate remains deterministic.



§ CONCLUSION

In this paper, we demonstrate that it is possible to formally substitute real variables such as the four-coordinates, control policy, and cost-to-go function in the HJB equation with complex-valued ones. However, such substitution can only be performed if we define the complex-valued Lagrangian as an analytical continuation of the real-valued Lagrangian of the system.

In consequence of Cochy-Rieman theorem we prove that the diffusion coefficient in the complex stochastic HJB equation is: σ^μσ^μ = σ_x^μσ_x^μ - i σ_y^μσ_y^μ.

The last result is quite interesting because in all previous works <cit.> that derive quantum mechanical equations it is proved that in order to linerize the HJB equation it is required the square of the diffusion coefficient to be a pure imaginary number. 
This result tell us that the equation only for the imaginary coordinates in the quantum mechnical equations is stochastic. 


unsrt


