Stochastic quantum mechanics

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 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 ${\displaystyle {\frac {|\alpha |\,\hbar }{2\,m}}}$. In addition, the generalized theory is symmetric under the operation ${\displaystyle (t,\phi )\leftrightarrow (-t,\phi +\pi )}$.

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]

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

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

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

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

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

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

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 ${\displaystyle w_{2}^{ij}(X(t),t)={\frac {\alpha \ \hbar }{m}}\,\delta ^{ij}}$ with ${\displaystyle \alpha \in \mathbb {C} }$. It immediately follows that in the Stratonovich formulation the second order part of the velocity vanishes, i.e. ${\displaystyle w_{2,\circ }=0}$.

Using the existence of the velocity fields ${\displaystyle w(X(t),t)}$, 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.

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]

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

Stochastic quantum mechanics can be applied to the field of electrodynamics and is called stochastic electrodynamics (SED).[16] SED differs profoundly from quantum electrodynamics (QED) but is nevertheless able to account for some vacuum-electrodynamical effects within a fully classical framework.[17] In classical electrodynamics it is assumed there are no fields in the absence of any sources, while SED assumes that there is always a constantly fluctuating classical field due to zero-point energy. As long as the field satisfies the Maxwell equations there is no a priori inconsistency with this assumption.[18] Since Trevor W. Marshall[19] originally proposed the idea it has been of considerable interest to a small but active group of researchers.[20]