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Stochastic quantum mechanics (or the stochastic interpretation) is an interpretation of quantum mechanics. This interpretation is based on a reformulation of quantum mechancis in which the dynamics of all particles is governed by a stochastic differential equation. Thus, according to the stochastic interpretation, quantum particles follow well-defined random trajectories in space(time), similar to a Brownian motion.

The first relatively coherent stochastic theory of quantum mechanics was put forward by Hungarian physicist Imre Fényes^[1] who was able to show the Schrödinger equation could be understood as a kind of diffusion equation for a Markov process.^[2][3]

Louis de Broglie^[4] felt compelled to incorporate a stochastic process underlying quantum mechanics to make particles switch from one pilot wave to another.^[5] Perhaps the most widely known theory where quantum mechanics is assumed to describe an inherently stochastic process was put forward by Edward Nelson^[6] and is called stochastic mechanics. This was also developed by Davidson, Guerra, Ruggiero, Pavon and others.^[7]

The modern application of stochastics to quantum mechanics involves the assumption of spacetime stochasticity, the idea that the small-scale structure of spacetime is undergoing both metric and topological fluctuations (John Archibald Wheeler's "quantum foam"), and that the averaged result of these fluctuations recreates a more conventional-looking metric at larger scales that can be described using classical physics, along with an element of nonlocality that can be described using quantum mechanics. A stochastic interpretation of quantum mechanics is due to persistent vacuum fluctuation. The main idea is that vacuum or spacetime fluctuations are the reason for quantum mechanics and not a result of it as it is usually considered.

Stochastic Quantization

The postulates of Stochastic Mechanics can be summarized in a stochastic quantization condition that was formulated by Nelson^[8] and reformulated by Kuipers.^[9] For a non-relativistic theory on ${\displaystyle \mathbb {R} ^{n}}$ this condition states:

• the trajectory of a quantum particle is described by the real projection of a complex semi-martingale: ${\displaystyle X(t)={\rm {Re}}[Z(t)]}$ with ${\displaystyle Z(t)=C(t)+M(t)}$, where ${\displaystyle C(t)}$ is a continuous finite variation process and ${\displaystyle M(t)}$is a complex martingale;
• the trajectory stochastically extremizes an action ${\displaystyle S=\mathbb {E} \left[\int L\,dt\right]}$;
• the martingale ${\displaystyle M(t)}$ is a continuous process with independent increments and finite moments. Furthermore, its quadratic variation is fixed by the structure relation ${\displaystyle d[M^{i},M^{j}]={\frac {{\rm {i}}\,\hbar }{m}}\,\delta ^{ij}\,dt}$ where ${\displaystyle m}$ is the mass of the particle;
• the time reversed process exists and is subjected to the same dynamical laws.

Using the decomposition ${\displaystyle Z=X+{\rm {i}}\,Y}$, and the fact that ${\displaystyle C}$ has finite variation, one immediately finds that the quadratic variation of ${\displaystyle X}$ and ${\displaystyle Y}$ is given by

${\displaystyle {\begin{pmatrix}d[X^{i},X^{j}]&d[X^{i},Y^{j}]\\d[Y^{i},X^{j}]&d[Y^{i},Y^{j}]\end{pmatrix}}={\frac {\hbar }{2\,m}}\,\delta ^{ij}\,{\begin{pmatrix}1&1\\1&1\end{pmatrix}}\,dt}$.

Hence, by Lévy's characterization of Brownian motion, ${\displaystyle X(t)}$ and ${\displaystyle Y(t)}$ describe two maximally correlated Wiener processes with a drift described by the finite variation process ${\displaystyle C(t)}$.

The stochastic quantization procedure can be generalized, such that it describes a larger class of stochastic theories.^[9] In this generalization, the structure relation is given by ${\displaystyle d[M^{i},M^{j}]={\frac {\alpha \,\hbar }{m}}\,\delta ^{ij}\,dt}$ with ${\displaystyle \alpha =|\alpha |e^{{\rm {i}}\phi }\in \mathbb {C} .}$ The covariation of ${\displaystyle X}$ and ${\displaystyle Y}$ is then given by

${\displaystyle {\begin{pmatrix}d[X^{i},X^{j}]&d[X^{i},Y^{j}]\\d[Y^{i},X^{j}]&d[Y^{i},Y^{j}]\end{pmatrix}}={\frac {|\alpha |\,\hbar }{2\,m}}\,\delta ^{ij}\,{\begin{pmatrix}1+\cos \phi &\sin \phi \\\sin \phi &1-\cos \phi \end{pmatrix}}\,dt}$.

This generalized theory describes quantum mechanics for ${\displaystyle \alpha ={\rm {i}}}$ , while, for ${\displaystyle \phi =0}$, it describes a Brownian motion with diffusion coefficient ${\displaystyle {\frac {|\alpha |\,\hbar }{2\,m}}}$.

The term stochastic quantization to describe this quantization procedure was introduced^[10] in the 1970's. Nowadays, stochastic quantization more commonly refers to a framework developed by Parisi and Wu in 1981. Consequently, the quantization procedure developed in stochastic mechanics is sometimes also referred to as Nelson's stochastic quantization or stochasticization.^[11]

Velocity of the Process

The stochastic process ${\displaystyle Z(t)}$ is almost surely nowhere differentiable, such that the velocity ${\displaystyle {\dot {Z}}(t)={\frac {dZ(t)}{dt}}}$ along the process is not well-defined. However, it turns out that there exist velocity fields, defined using conditional expectations. These are given by

${\displaystyle w_{+}(x,t)=\lim _{dt\rightarrow 0}\mathbb {E} \left[{\frac {Z(t+dt)-Z(t)}{dt}}\,{\Big |}\,X(t)=x\right],}$

${\displaystyle w_{-}(x,t)=\lim _{dt\rightarrow 0}\mathbb {E} \left[{\frac {Z(t)-Z(t-dt)}{dt}}\,{\Big |}\,X(t)=x\right],}$

and are called the forward and backward Itô velocity of the process. Since the process is not differentiable, these velocities are, in general, not equal to each other. The physical interpretation of this fact is as follows: at any time ${\displaystyle t}$ the particle is subjected to a random force that instantaneously changes its velocity from ${\displaystyle w_{-}}$ to ${\displaystyle w_{+}}$. As the two velocity fields are not equal, there is no unique notion of velocity for the process ${\displaystyle Z(t)}$. In fact, any velocity given by

${\displaystyle w_{a}=a\ w_{+}+(1-a)\ w_{-}}$ with ${\displaystyle a\in [0,1]}$ represents a valid choice for the velocity of the process ${\displaystyle Z(t)}$. This is particularly true for the special case ${\displaystyle a={\frac {1}{2}}}$ denoted by ${\displaystyle w_{\circ }={\frac {w_{+}+w_{-}}{2}}}$ , which is the Stratonovich velocity field.

Since ${\displaystyle Z(t)}$ has a non-vanishing quadratic variation, one can additionally define second order velocity fields given by

${\displaystyle w_{2,+}(x,t)=\lim _{dt\rightarrow 0}\mathbb {E} \left[{\frac {[Z(t+dt)-Z(t)]\otimes [Z(t+dt)-Z(t)]}{dt}}\ {\Big |}\ X(t)=x\right]}$,

${\displaystyle w_{2,-}(x,t)=\lim _{dt\rightarrow 0}\mathbb {E} \left[{\frac {[Z(t)-Z(t-dt)]\otimes [Z(t)-Z(t-dt)]}{dt}}\ {\Big |}\ X(t)=x\right]}$.

The time-reversibility postulate imposes a relation on these two fields such that ${\displaystyle w_{2,\pm }=\pm \,w_{2}}$. Moreover, using the structure relation by which the quadratic variation is fixed, one finds that ${\displaystyle w_{2}^{ij}(x,t)={\frac {\alpha \,\hbar }{m}}\,\delta ^{ij}}$ with ${\displaystyle \alpha \in \mathbb {C} }$. It immediately follows that in the Stratonovich formulation the second order part of the velocity vanishes, i.e. ${\displaystyle w_{2,\circ }=0}$.

The real and imaginary part of the velocities are detnoted by

${\displaystyle v={\rm {Re}}(w)\qquad {\rm {and}}\qquad u={\rm {Im}}(w)}$.

Using the existence of these velocity fields, one can formally define the velocity processes ${\displaystyle W_{\pm }(t)}$ by the Itô integral ${\displaystyle \int W_{\pm }(t)\ dt=\int d_{\pm }Z(t)}$. Similarly, one can formally define a process ${\displaystyle W_{\circ }(t)}$ by the Stratonovich integral${\displaystyle \int W_{\circ }(t)\ dt=\int d_{\circ }Z(t)}$ and a second order velocity process ${\displaystyle W_{2}(t)}$ by the Stieltjes integral ${\displaystyle \int W_{2}(t)\ dt=\int d[Z,Z](t)}$. Using the structure relation, one then finds that the second order velocity process is given by ${\displaystyle W_{2}^{ij}(t)={\frac {\alpha \ \hbar }{m}}\ \delta ^{ij}}$. However, the processes ${\displaystyle W_{\pm }(t)}$ and ${\displaystyle W_{\circ }(t)}$ are not well-defined: the first moments exist and are given by ${\displaystyle \mathbb {E} [W_{\pm }(t)\ |\ X(t)]=w_{\pm }(X(t),t)}$, but the quadratic moments diverge, i.e. ${\displaystyle \mathbb {E} [||W_{\pm }(t)||^{2}\ |\ X(t)]=\infty }$. The physical interpretation of this divergence is that in the position representation the position is known precisely, but the velocity has an infinite uncertainty.

Stochastic Action

The postulates of stochastic mechanics state that the stochastic trajectory must extremize a stochastic action ${\displaystyle S=\mathbb {E} \left[\int L\,dt\right]}$, but they do not specify the stochastic Lagrangian ${\displaystyle L}$. This Lagrangian can be obtained from a classical Lagrangian using a standard procedure. Here, we consider a classical Lagrangian of the form

${\displaystyle L(x,v,t)={\frac {m}{2}}\delta _{ij}v^{i}v^{j}+q\,A_{i}(x,t)\,v^{i}-{\mathfrak {U}}(x,t)}$,

where ${\displaystyle m}$ denotes the mass of the particle, ${\displaystyle q}$ the charge under the vector potential ${\displaystyle A}$, and ${\displaystyle {\mathfrak {U}}}$ is a scalar potential.

An important property of this Lagrangian is the principle of gauge invariance. This can be made explicit by defining a new action ${\displaystyle {\tilde {S}}}$ through the addition of a total derivative term to the original action, such that

${\displaystyle {\tilde {S}}[x(t)]=S[x(t)]+\int dF(x,t)=\int \left[{\frac {m}{2}}\delta _{ij}v^{i}v^{j}+qA_{i}v^{i}-{\mathfrak {U}}+\partial _{t}F+v^{i}\partial _{i}F\right]dt=\int \left[{\frac {m}{2}}\delta _{ij}v^{i}v^{j}+q{\tilde {A}}_{i}v^{i}-{\tilde {\mathfrak {U}}}\right]dt}$,

where ${\displaystyle {\tilde {A}}_{i}=A_{i}+q^{-1}\partial _{i}F}$ and ${\displaystyle {\tilde {\mathfrak {U}}}={\mathfrak {U}}-\partial _{t}F}$. Thus, since the dynamics should not be affected by the addition of a total derivative to the action, the action is gauge invariant under the above redefinition of the potentials for an arbitrary differentiable function ${\displaystyle F}$.

In order to construct a stochastic Lagrangian corresponding to this classical Lagrangian, one must look for a minimal extension of the above Lagrangian that respects this gauge invariance.^[12] In the Stratonovich formulation of the theory, this can be done straightforwardly, since the differential operator in the Stratonovich formulation is given by

${\displaystyle \int d_{\circ }F(x,t)=\int \left(\partial _{t}F+v_{\circ }^{i}\partial _{i}F\right)dt}$.

Therefore, the Stratonovich Lagrangian can be obtained by replacing the classical velocity ${\displaystyle v}$ by the complex Stratonovich velocity ${\displaystyle w_{\circ }}$, such that

${\displaystyle L_{\circ }(x,w_{\circ },t)={\frac {m}{2}}\delta _{ij}w_{\circ }^{i}w_{\circ }^{j}+qA_{i}w_{\circ }^{i}-{\mathfrak {U}}.}$

In the Itô formulation, things are more complicated, as the total derivative is given by Itô's lemma: ${\displaystyle \int d_{\pm }F(x,t)=\int \left(\partial _{t}F+v_{\pm }^{i}\partial _{i}F\pm {\frac {1}{2}}v_{2}^{ij}\partial _{j}\partial _{i}F\right)dt}$.

Due to the presence of the second order derivative term, the gauge invariance is broken. However, this can be restored by adding a derivative of the vector potential to the Lagrangian. Hence, the stochastic Lagrangian in the Itô formulation is given by

${\displaystyle L_{\pm }(x,w_{\pm },w_{2},t)={\frac {m}{2}}\delta _{ij}w_{\pm }^{i}w_{\pm }^{j}+q\,A_{i}w_{\pm }^{i}\pm {\frac {q}{2}}\partial _{j}A_{i}w_{2}^{ij}-{\mathfrak {U}}}$.

The stochastic action can be defined using the Stratonovich Lagrangian, which is equal to the action defined by the Itô Lagrangian up to a divergent term:^[8]

${\displaystyle S=\mathbb {E} \left[\int L_{\circ }\,dt\right]=\mathbb {E} \left[\int L_{\pm }dt\right]\pm \mathbb {E} \left[\int L_{\infty }dt\right]}$.

The divergent term can be calculated^[9] and is given by

${\displaystyle \mathbb {E} \left[\int L_{\infty }dt\right]={\frac {m}{2}}\oint _{\gamma }{\frac {\delta _{ij}w_{2}^{ij}}{t}}dt=\alpha \,\hbar \,\pi \,{\rm {i}}\,\sum _{i=1}^{n}k_{i}}$,

where ${\displaystyle k_{i}\in \mathbb {Z} }$ are winding numbers that count the winding of the path ${\displaystyle \gamma (t)}$ around the pole at ${\displaystyle t=0}$.

As the divergent term is constant, it does not contribute to the equations of motion. For this reason, this term has been discarded in early works on stochastic mechanics.^[8] However, when this term is discarded, stochastic mechanics cannot account for the appearance of discrete spectra in quantum mechanics. This issue is known as Wallstrom's criticism,^[13] and can be resolved by properly taking into account the divergent term.^[9]

There also exists a Hamiltonian formulation of stochastic mechanics.^[14] It starts from the definition of canonical momenta:

${\displaystyle p_{\circ ,i}={\frac {\partial L_{\circ }}{\partial w_{\circ }^{i}}}=m\,\delta _{ij}w_{\circ }^{j}+q\,A_{i}}$,

${\displaystyle p_{\pm ,i}={\frac {\partial L_{\pm }}{\partial w_{\pm }^{i}}}=m\,\delta _{ij}w_{\pm }^{j}+q\,A_{i}}$.

The Hamiltonian in the Stratonovich formulation can then be obtained by the first order Legendre transform:

${\displaystyle H_{\circ }(x,p_{\circ },t)=p_{\circ ,i}v_{\circ }^{i}-L(x,v_{\circ },t)}$.

In the Itô formulation, on the other hand, the Hamiltonian is obtained through a second order Legendre transform:^[15]

${\displaystyle H_{\pm }(x,p^{\pm },\partial p^{\pm },t)=p_{i}^{\pm }w_{\pm }^{i}\pm {\frac {1}{2}}w_{2}^{ij}\partial p_{ij}^{\pm }-L(x,w_{\pm },w_{2},t)}$.

Euler-Lagrange Equations

The stochastic action can be extremized,^[16] which leads to a stochastic version of the Euler-Lagrange equations. In the Stratonovich formulation, these are given by

${\displaystyle \int d_{\circ }\left({\frac {\partial L_{\circ }}{\partial w_{\circ }^{i}}}\right)=\int \left({\frac {\partial L_{\circ }}{\partial x^{i}}}\right)dt}$.

For the Lagrangian, discussed in previous section, this leads to the following second order stochastic differential equation in the sense of Stratonovich:

${\displaystyle m\,\delta _{ij}\,d_{\circ }^{2}Z^{j}(t)=q\,F_{ij}(X(t),t)\,d_{\circ }Z^{j}(t)\,dt-q\,\partial _{t}A_{i}(X(t),t)\,dt^{2}-\partial _{i}{\mathfrak {U}}(X(t),t)\,dt^{2}}$,

where, the field strength is given by ${\displaystyle F_{ij}:=\partial _{i}A_{j}-\partial _{j}A_{i}}$.

In the Itô formulation, the stochastic Euler-Lagrange equations are given by

${\displaystyle \int d_{\pm }\left({\frac {\partial L_{\pm }}{\partial w_{\pm }^{i}}}\right)=\int \left({\frac {\partial L_{\pm }}{\partial x^{i}}}\right)dt}$.

This leads to a second order stochastic differential equation in the sense of Itô, given by

${\displaystyle m\,\delta _{ij}\,d_{\pm }^{2}Z^{j}(t)=q\,F_{ij}(X(t),t)\,d_{\pm }Z^{j}(t)\,dt\pm {\frac {\alpha \,\hbar \,q}{2\,m}}\,\delta ^{jk}\,\partial _{k}F_{ij}(X(t),t)\,dt^{2}-q\,\partial _{t}A_{i}(X(t),t)\,dt^{2}-\partial _{i}{\mathfrak {U}}(X(t),t)\,dt^{2}}$.

Hamilton-Jacobi Equations

The equations of motion can also be obtained in a stochastic generalization of the Hamilton-Jacobi formulation of classical mechanics. In this case, one starts by defining Hamilton's principal function. For the Lagrangian ${\displaystyle L_{+}}$, this function is defined as

${\displaystyle S_{+}(x,t;x_{f},t_{f}):=-\mathbb {E} \left[\int _{t}^{t_{f}}L_{+}(X(s),W_{+}(s),W_{2}(s),s)\,ds\,{\Big |}\,X(t)=x,X(t_{f})=x_{f}\right]}$,

where it is assumed that the process ${\displaystyle \{X(s):s\in [t,t_{f}]\}}$ obeys the stochastic Euler-Lagrange equations. Similarly, for the Lagrangian ${\displaystyle L_{-}}$, Hamilton's principal function is defined as

${\displaystyle S_{-}(x,t;x_{0},t_{0}):=\mathbb {E} \left[\int _{t_{0}}^{t}L_{-}(X(s),W_{-}(s),W_{2}(s),s)\,ds\,{\Big |}\,X(t)=x,X(t_{0})=x_{0}\right]}$,

where it is assumed that the process ${\displaystyle \{X(s):s\in [t_{0},t]\}}$ obeys the stochastic Euler-Lagrange equations. Due to the divergent part of the action, these principal functions are subjected to the equivalence relation

${\displaystyle {\tilde {S}}_{\pm }\equiv S_{\pm }\;{\rm {if}}\;\exists k\in \mathbb {Z} ^{n},\;{\rm {such\,that}}\;{\tilde {S}}_{\pm }=S_{\pm }\pm \alpha \,\pi \,{\rm {i}}\,\hbar \,\sum _{i=1}^{n}k_{i}}$.

By varying the principal functions with respect to the point ${\displaystyle (x,t)}$ one finds the Hamilton-Jacobi equations. These are given by

${\displaystyle {\begin{cases}{\frac {\partial }{\partial x^{i}}}S_{\pm }(x,t)&=p_{i}^{\pm }(x,t)\,,\\{\frac {\partial }{\partial t}}S_{\pm }(x,t)&=-H_{\pm }(x,p_{\pm }(x,t),\partial p_{\pm }(x,t),t)\,.\end{cases}}}$

Note that these look the same as in the classical case. However, the Hamiltonian, in the second Hamilton-Jacobi equation is now obtained using a second order Legendre transform. Moreover, due to the divergent part of the action, there is a third Hamilton-Jacobi equation, which takes the form of the non-trivial integral constraint

${\displaystyle \oint \left(p_{i}^{\pm }\,v_{\pm }^{i}\pm {\frac {1}{2}}v_{2}^{ij}\,\partial _{i}p_{j}\right)dt=\pm \alpha \,\hbar \,\pi \,{\rm {i}}\,\sum _{i=1}^{n}k_{i}}$.

For the given Lagrangian the first two Hamilton-Jacobi equations yield

${\displaystyle {\begin{cases}\partial _{i}S&=m\,\delta _{ij}w_{\pm }^{j}+q\,A_{i}\,,\\\partial _{t}S&=-{\frac {m}{2}}\,\delta _{ij}w_{\pm }^{i}w_{\pm }^{j}\mp {\frac {m}{2}}\,\delta _{ij}w_{2}^{ik}\partial _{k}w_{\pm }^{j}-{\mathfrak {U}}\,.\end{cases}}}$

These two equations can be combined, yielding

${\displaystyle \left[m\,\delta _{ij}\left(\partial _{t}+w_{\pm }^{k}\partial _{k}\pm {\frac {1}{2}}\,w_{2}^{kl}\partial _{l}\partial _{k}\right)-q\,F_{ij}\right]w_{\pm }^{j}=\pm {\frac {q}{2}}\,w_{2}^{jk}\partial _{k}F_{ij}-q\,\partial _{t}A_{i}-\partial _{i}{\mathfrak {U}}}$.

Using that ${\displaystyle w_{2}^{ij}={\frac {\alpha \,\hbar }{m}}\,\delta ^{ij}}$, this equation, subjected to the integral condition and the initial condition ${\displaystyle w_{+}(x,t_{0})=w_{0}(x)}$ or terminal condition ${\displaystyle w_{-}(x,t_{f})=w_{f}(x)}$, can be solved for ${\displaystyle w_{\pm }(x,t)}$. The solution can then be plugged into the Itô equation

${\displaystyle {\begin{cases}d_{\pm }Z^{i}(t)&=w_{\pm }^{i}(x,t)\,dt+dM^{i}(t)\,,\\d[M^{i},M^{j}](t)&={\frac {\alpha \,\hbar }{m}}\delta ^{ij}\,dt\,,\end{cases}}}$

which can be solved for the process ${\displaystyle \{Z(t):t\in [t_{0},t_{f}]\}}$. Thus, when an initial condition ${\displaystyle X(t_{0})=x_{0}}$ (for the future directed equation labeled with ${\displaystyle +}$) or terminal condition ${\displaystyle X(t_{f})=x_{f}}$ (for the past directed equation labeled with ${\displaystyle -}$) is specified, one finds a unique stochastic process ${\displaystyle \{X(t):t\in [t_{0},t_{f}]\}}$ that describes the trajectory of the particle.

Diffusion Equation

The key result of stochastic mechanics is that it derives the Schrödinger equation from the postulated stochastic process. In this derivation, the Hamilton-Jacobi equations

${\displaystyle {\begin{cases}{\frac {\partial }{\partial x^{i}}}S_{\pm }(x,t)&=p_{i}^{\pm }(x,t)\\{\frac {\partial }{\partial t}}S_{\pm }(x,t)&=-H_{\pm }(x,p_{\pm }(x,t),\partial p_{\pm }(x,t),t)\end{cases}}}$

are combined, such that one obtains the equation

${\displaystyle 2\,m\left(\partial _{t}S_{\pm }+{\mathfrak {U}}\right)+\delta ^{ij}\left(\partial _{i}S_{\pm }\partial _{j}S_{\pm }\pm \alpha \,\hbar \,\partial _{j}\partial _{i}S_{\pm }-2\,q\,A_{i}\partial _{j}S_{\pm }\mp \alpha \,\hbar \,q\,\partial _{j}A_{i}+q^{2}\,A_{i}A_{j}\right)=0}$.

Subsequently, one defines the wave function

${\displaystyle \Psi _{\pm }(x,t)=\exp \left(\pm {\frac {S_{\pm }(x,t)}{\alpha \,\hbar }}\right)}$.

Since Hamilton's principal functions are multivalued, one finds that the wave functions are subjected to the equivalence relations

${\displaystyle {\tilde {\Psi }}_{+}\equiv \Psi _{+}\quad {\rm {if}}\quad {\tilde {\Psi }}_{+}=\pm \Psi _{+}\qquad {\rm {and}}\qquad {\tilde {\Psi }}_{-}\equiv \Psi _{-}\quad {\rm {if}}\quad {\tilde {\Psi }}_{-}=\pm \Psi _{-}}$.

Furthermore, the wave functions are subjected to the complex diffusion equations

${\displaystyle -\alpha \,\hbar \,{\frac {\partial }{\partial t}}\Psi _{+}=\left[{\frac {\delta ^{ij}}{2\,m}}\left(\alpha \,\hbar \,{\frac {\partial }{\partial x^{i}}}+q\,A_{i}\right)\left(\alpha \,\hbar \,{\frac {\partial }{\partial x^{j}}}+q\,A_{j}\right)+{\mathfrak {U}}\right]\Psi _{+}}$,

${\displaystyle \alpha \,\hbar \,{\frac {\partial }{\partial t}}\Psi _{-}=\left[{\frac {\delta ^{ij}}{2\,m}}\left(\alpha \,\hbar \,{\frac {\partial }{\partial x^{i}}}+q\,A_{i}\right)\left(\alpha \,\hbar \,{\frac {\partial }{\partial x^{j}}}+q\,A_{j}\right)+{\mathfrak {U}}\right]\Psi _{-}}$.

Thus, for any for any process that solves the postulates of stochastic mechanics, one can construct a wave function that obeys these diffusion equations. Due to the equivalence relations on Hamilton's principal function, the opposite statement is also true: for any solution of these complex diffusion equations, one can construct a stochastic process ${\displaystyle \{X(t):t\in [t_{0},t_{f}]\}}$ that is a solution of the postulates of stochastic mechanics. A similar result has been established by the Feynman-Kac theorem.

Finally, one can construct a probability density

${\displaystyle \rho _{\pm }(x,t):={\frac {|\Psi _{\pm }(x,t)|^{2}}{\int |\Psi _{\pm }(y,t)|^{2}d^{n}y}}}$,

which describes transition probabilities for the process ${\displaystyle \{X(t):t\in [t_{0},t_{f}]\}}$. More precisely, ${\displaystyle \rho _{+}}$ describes the probability of being in the state ${\displaystyle (x,t)}$ given that the system ends up in the state ${\displaystyle (x_{f},t_{f})}$. Therefore, the diffusion equation for ${\displaystyle \Psi _{+}}$ can be interpreted as the Kolmogorov backward equation of the process ${\displaystyle \{X(t):t\in [t_{0},t_{f}]\}}$. Similarly, ${\displaystyle \rho _{-}}$ describes the probability of being in the state ${\displaystyle (x,t)}$ given that the system ends up in the state ${\displaystyle (x_{0},t_{0})}$, when it is evolved backward in time. Therefore, the diffusion equation for ${\displaystyle \Psi _{-}}$ can be interpreted as the Kolmogorov backward equation of the process ${\displaystyle \{X(t):t\in [t_{0},t_{f}]\}}$ when it is evolved towards the past. By inverting the time direction, one finds that ${\displaystyle \rho _{-}}$ describes the probability of being in the state ${\displaystyle (x,t)}$ given that the system starts in the state ${\displaystyle (x_{0},t_{0})}$, when it is evolved forward in time. Thus, the diffusion equation for ${\displaystyle \Psi _{-}}$ can also be interpreted as the Kolmogorov Forward equation of the process ${\displaystyle \{X(t):t\in [t_{0},t_{f}]\}}$ when it is evolved towards the future.

The theory contains various special limits:

• The classical limit with ${\displaystyle \alpha =0}$. In this case, the process ${\displaystyle X}$ and auxiliary process ${\displaystyle Y}$ describes two decoupled deterministic trajectories.
• The Brownian limit with ${\displaystyle \alpha \in (0,\infty )}$. In this case, the process ${\displaystyle X}$ describes a Wiener process (a.k.a. Brownian Motion) for which the above result is established by the Feynman-Kac theorem, whereas the auxiliary process ${\displaystyle Y}$ describes a deterministic process.
• The quantum limit with ${\displaystyle \alpha \in {\rm {i}}\times (0,\infty )}$. In this case, the process ${\displaystyle X}$ and auxiliary process ${\displaystyle Y}$ describe two positively correlated Wiener processes.
• The time-reversed Brownian limit with ${\displaystyle \alpha \in (-\infty ,0)}$. In this case, the process ${\displaystyle X}$ describes a deterministic process, whereas the auxiliary process ${\displaystyle Y}$ describes a Wiener process.
• The time-reversed quantum limit with ${\displaystyle \alpha \in {\rm {i}}\times (-\infty ,0)}$. In this case, the process ${\displaystyle X}$ and auxiliary process ${\displaystyle Y}$ describe two negatively correlated Wiener processes.

Furthermore, the theory is symmetric under the time reversal operation ${\displaystyle (t,\alpha ,q)\leftrightarrow (-t,-\alpha ,-q)}$.

In the Brownian limit with initial condition ${\displaystyle u(x_{0},t_{0})=0}$ or terminal condition ${\displaystyle u(x_{f},t_{f})=0}$, the processes ${\displaystyle X}$ and ${\displaystyle Y}$ are decoupled, such that the dynamics of the auxiliary process ${\displaystyle Y}$ can be discarded, and ${\displaystyle X}$ can simply be described as a real Wiener process. In all other cases with ${\displaystyle \alpha \neq 0}$, the processes are coupled to each other, such that the auxiliary process ${\displaystyle Y}$ must be taken into account in deriving the dynamics of ${\displaystyle X}$.

In the Brownian limits, the theory is maximally dissipative, whereas the quantum limits are unitary, such that

${\displaystyle {\frac {d}{dt}}\int |\Psi _{\pm }(y,t)|^{2}d^{n}y{\Big |}_{\alpha \in {\rm {i}}\times \mathbb {R} }=0}$.

See also

References