Stochastic mechanics is a framework for describing the dynamics of particles that are subjected to an intrinsic random processes as well as various external forces. The framework provides a derivation of the diffusion equations associated to these stochastic particles. It is best known for its derivation of the Schrödinger equation as the Kolmogorov equation for a certain type of conservative (or unitary) diffusion, and for this purpose it is also referred to as stochastic quantum mechanics.
The derivation can be based on the extremization of an action in combination with a quantization prescription. This quantization prescription can be compared to canonical quantization and the path integral formulation, and is often referred to as Nelson’s stochastic quantization or stochasticization. As the theory allows for a derivation of the Schrödinger equation, it has given rise to the stochastic interpretation of quantum mechanics. This interpretation has served as the main motivation for developing the theory of stochastic mechanics.
The first relatively coherent stochastic theory of quantum mechanics was put forward by Hungarian physicist Imre Fényes. Louis de Broglie felt compelled to incorporate a stochastic process underlying quantum mechanics to make particles switch from one pilot wave to another. The theory of stochastic mechanics is ascribed to Edward Nelson, who independently discovered a derivation of the Schrödinger equation within this framework. This theory was also developed by Davidson, Guerra, Ruggiero, Pavon and others.
The stochastic interpretation interprets the paths in the path integral formulation of quantum mechanics as the sample paths of a stochastic process. It posits that quantum particles are localized on one of these paths, but observers cannot predict with certainty where the particle is localized. The only way to locate the particle is by performing a measurement. An observer can only predict probabilities for the outcomes of such a measurement based on their earlier measurements and their knowledge about the forces that are acting on the particle.
This interpretation is well-known from the context of statistical mechanics, and Brownian motion in particular. Hence, according to the stochastic interpretation, quantum mechanics should be interpreted in a way similar to Brownian motion. However, in the case of Brownian motion, the existence of a probability measure (called the Wiener measure) that defines the statistical path integral is well established, and this measure can be generated by a stochastic process called the Wiener process. On the other hand, proving the existence of a probability measure that defines the quantum mechanical path integral faces difficulties, and it is not guaranteed that such a probability measure can be generated by a stochastic process. Stochastic mechanics is the framework concerned with the construction of such stochastic processes that generate a probability measure for quantum mechanics.
For a Brownian motion, it is known that the statistical fluctuations of a Brownian particle are often induced by the interaction of the particle with a large number of microscopic particles. In this case, the description of a Brownian motion in terms of the Wiener process is only used as an approximation, which neglects the dynamics of the individual particles in the background. Instead it describes the influence of these background particles by their statistical behavior.
The stochastic interpretation of quantum mechanics is agnostic about the origin of the quantum fluctuations of a quantum particle. It introduces the quantum fluctuations as the result of new stochastic law of nature called the background hypothesis. This hypothesis can be interpreted as a strict implementation of the statement that `God plays dice’, but it leaves open the possibility that this dice game is replaced by a hidden variable theory, as in the theory of Brownian motion.
The remainder of this article deals with the definition of such a process and the derivation of the diffusion equations associated to this process. This is done in a general setting with Brownian motion and Quantum mechanics as special limits, where one obtains respectively the heat equation and the Schrödinger equation. The derivation heavily relies on tools from Lagrangian mechanics and stochastic calculus.
See main article: Stochastic quantization. The postulates of Stochastic Mechanics can be summarized in a stochastic quantization condition that was formulated by Nelson. For a non-relativistic theory on
Rn
X(t)={\rmRe}[Z(t)]
Z(t)=C(t)+M(t)
C(t)
M(t)
S=E\left[\intLdt\right]
M(t)
d[Mi,Mj]=
\alpha\hbar | |
m |
\deltaijdt
m
\hbar
\alpha=|\alpha|e{\rm\phi}\inC
\deltaij
Using the decomposition
Z=X+{\rmi}Y
C
X
Y
\begin{pmatrix} d[Xi,Xj]&d[Xi,Yj]\\ d[Yi,Xj]&d[Yi,Y
| ||||
\deltaij\begin{pmatrix} 1+\cos\phi&\sin\phi\\ \sin\phi&1-\cos\phi \end{pmatrix} dt
Hence, by Lévy's characterization of Brownian motion,
X(t)
Y(t)
C(t)
|\alpha|\in[0,infty)
\phi
\phi\in\left\{-
\pi | |
2 |
,
\pi | |
2 |
\right\}
\alpha\in{\rmi} x \R
\phi\in\{0,\pi\}
\alpha\in\R
The term stochastic quantization to describe this quantization procedure was introduced in the 1970s. 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.
The stochastic process
Z(t)
Z |
(t)=
dZ(t) | |
dt |
w+(x,t)=\limdt → 0E\left[
Z(t+dt)-Z(t) | |
dt |
|X(t)=x\right],
w-(x,t)=\limdtE\left[
Z(t)-Z(t-dt) | |
dt |
|X(t)=x\right],
and can be associated to the Itô integral along the process
Z(t)
t
w-
w+
Z(t)
wa=a w++(1-a) w-
a\in[0,1]
Z(t)
a=
1 | |
2 |
w\circ=
w++w- | |
2 |
Z(t)
Since
Z(t)
w2,+(x,t)=\limdtE\left[
[Z(t+dt)-Z(t)] ⊗ [Z(t+dt)-Z(t)] | |
dt |
| X(t)=x\right],
w2,-(x,t)=\limdtE\left[
[Z(t)-Z(t-dt)] ⊗ [Z(t)-Z(t-dt)] | |
dt |
| X(t)=x\right].
The time-reversibility postulate imposes a relation on these two fields such that
w2,\pm=\pmw2
ij | |
w | |
2 |
(x,t)=
\alpha\hbar | |
m |
\deltaij
w2,\circ=0
The real and imaginary part of the velocities are detnoted by
v={\rmRe}(w) {\rmand} u={\rmIm}(w).
Using the existence of these velocity fields, one can formally define the velocity processes
W\pm(t)
\intW\pm(t) dt=\intd\pmZ(t)
W\circ(t)
\intW\circ(t) dt=\intd\circZ(t)
W2(t)
\intW2(t) dt=\intd[Z,Z](t)
ij | |
W | |
2 |
(t)=
\alpha \hbar | |
m |
\deltaij
W\pm(t)
W\circ(t)
E[W\pm(t) | X(t)]=w\pm(X(t),t)
2 | |
E[||W | |
\pm(t)|| |
| X(t)]=infty
The stochastic quantization condition states that the stochastic trajectory must extremize a stochastic action
S=E\left[\intLdt\right]
L
L(x,v,t)=
m | |
2 |
\deltaijvivj+qAi(x,t)vi-ak{U}(x,t).
Here,
(x,v)
\deltaij
Rn
m
q
A
ak{U}
An important property of this Lagrangian is the principle of gauge invariance. This can be made explicit by defining a new action
\tilde{S}
\tilde{S}[x(t)]=S[x(t)]+\intdF(x,t)=\int\left[
m | |
2 |
\deltaijvivj+qAivi-ak{U}+\partialtF+vi\partialiF\right]dt =\int\left[
m | |
2 |
\deltaijvivj+q\tilde{A}ivi-\tilde{ak{U}}\right]dt,
where
\tilde{A}i=Ai+q-1\partialiF
\tilde{ak{U}}=ak{U}-\partialtF
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. In the Stratonovich formulation of the theory, this can be done straightforwardly, since the differential operator in the Stratonovich formulation is given by
\intd\circF(x,t)=\int\left(\partialtF+
i | |
v | |
\circ |
\partialiF\right)dt.
Therefore, the Stratonovich Lagrangian can be obtained by replacing the classical velocity
v
w\circ
L\circ(x,w\circ,t)=
m | |
2 |
\deltaij
i | |
w | |
\circ |
j | |
w | |
\circ |
+qAi
i | |
w | |
\circ |
-ak{U}.
In the Itô formulation, things are more complicated, as the total derivative is given by Itô's lemma:
\intd\pmF(x,t)=\int\left(\partialtF+
i | |
v | |
\pm |
\partialiF\pm
1 | |
2 |
ij | |
v | |
2 |
\partialj\partialiF\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 is given by a Lagrangian of the form
L\pm(x,w\pm,w2,t)=
m | |
2 |
\deltaij
i | |
w | |
\pm |
j | |
w | |
\pm |
+qAi
i | |
w | |
\pm |
\pm
q | |
2 |
\partialjAi
ij | |
w | |
2 |
-ak{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:
S=E\left[\intL\circdt\right]=E\left[\intL\pmdt\right]\pmE\left[\intLinftydt\right].
The divergent term can be calculated and is given by
E\left[\intLinftydt\right]=
m | |
2 |
\oint\gamma
| |||||||||
t |
dt =\alpha\hbar\pi{\rmi}
n | |
\sum | |
i=1 |
ki,
where
ki\inZ
\gamma(t)
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. 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, and can be resolved by properly taking into account the divergent term.
There also exists a Hamiltonian formulation of stochastic mechanics. It starts from the definition of canonical momenta:
p\circ,i=
\partialL\circ | ||||||||
|
=m\deltaij
j | |
w | |
\circ |
+qAi,
p\pm,i=
\partialL\pm | ||||||||
|
=m\deltaij
j | |
w | |
\pm |
+qAi.
The Hamiltonian in the Stratonovich formulation can then be obtained by the first order Legendre transform:
H\circ(x,p\circ,t)=p\circ,i
i | |
v | |
\circ |
-L(x,v\circ,t).
In the Itô formulation, on the other hand, the Hamiltonian is obtained through a second order Legendre transform:
H\pm(x,p\pm,\partialp\pm,t)=
\pm | |
p | |
i |
i | |
w | |
\pm |
\pm
1 | |
2 |
ij | |
w | |
2 |
\partial
\pm | |
p | |
ij |
-L(x,w\pm,w2,t).
The stochastic action can be extremized, which leads to a stochastic version of the Euler-Lagrange equations. In the Stratonovich formulation, these are given by
\intd\circ\left(
\partialL\circ | ||||||||
|
\right)=\int\left(
\partialL\circ | |
\partialxi |
\right)dt.
For the Lagrangian, discussed in previous section, this leads to the following second order stochastic differential equation in the sense of Stratonovich:
m\deltaij
2 | |
d | |
\circ |
Zj(t)=qFij(X(t),t)d\circZj(t)dt-q\partialtAi(X(t),t)dt2-\partialiak{U}(X(t),t)dt2,
where, the field strength is given by
Fij:=\partialiAj-\partialjAi
In the Itô formulation, the stochastic Euler-Lagrange equations are given by
\intd\pm\left(
\partialL\pm | ||||||||
|
\right)=\int\left(
\partialL\pm | |
\partialxi |
\right)dt.
This leads to a second order stochastic differential equation in the sense of Itô, given by a stochastic version of Newton's second law in the form
m\deltaij
2 | |
d | |
\pm |
Zj(t)=qFij(X(t),t)d\pmZj(t)dt\pm
\alpha\hbarq | |
2m |
\deltajk\partialkFij(X(t),t)dt2 -q\partialtAi(X(t),t)dt2-\partialiak{U}(X(t),t)dt2.
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
L+
S+(x,t;xf,tf):=-E\left[
tf | |
\int | |
t |
L+(X(s),W+(s),W2(s),s)ds|X(t)=x,X(tf)=xf \right],
where it is assumed that the process
\{X(s):s\in[t,tf]\}
L-
S-(x,t;x0,t0):=E\left[
t | |
\int | |
t0 |
L-(X(s),W-(s),W2(s),s)ds|X(t)=x,X(t0)=x0 \right],
where it is assumed that the process
\{X(s):s\in[t0,t]\}
\tilde{S}\pm\equivS\pm {\rmif} \existsk\inZn, {\rmsuchthat} \tilde{S}\pm=S\pm\pm\alpha\pi{\rmi}\hbar
n | |
\sum | |
i=1 |
ki.
By varying the principal functions with respect to the point
(x,t)
\begin{cases} | \partial |
\partialxi |
S\pm(x,t)&=
\pm | |
p | |
i |
(x,t),\\
\partial | |
\partialt |
S\pm(x,t)&=-H\pm(x,p\pm(x,t),\partialp\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
\oint
\pm | |
\left(p | |
i |
i | |
v | |
\pm |
\pm
1 | |
2 |
ij | |
v | |
2 |
\partialipj\right)dt=\pm\alpha\hbar\pi{\rmi}
n | |
\sum | |
i=1 |
ki.
For the given Lagrangian the first two Hamilton-Jacobi equations yield
\begin{cases} \partialiS&=m\deltaij
j | |
w | |
\pm |
+qAi,\\ \partialtS&=-
m | |
2 |
\deltaij
i | |
w | |
\pm |
j | |
w | |
\pm |
\mp
m | |
2 |
\deltaij
ik | |
w | |
2 |
\partialk
j | |
w | |
\pm |
-ak{U}. \end{cases}
These two equations can be combined, yielding
\left[m\deltaij\left(\partialt+
k | |
w | |
\pm |
\partialk\pm
1 | |
2 |
kl | |
w | |
2 |
\partiall\partialk\right)-qFij\right]
j = | |
w | |
\pm |
\pm
q | |
2 |
jk | |
w | |
2 |
\partialkFij-q\partialtAi-\partialiak{U}.
Using that
ij | |
w | |
2 |
=
\alpha\hbar | |
m |
\deltaij
w+(x,t0)=w0(x)
w-(x,tf)=wf(x)
w\pm(x,t)
\begin{cases} d\pmZi(t)&=
i(x,t) | |
w | |
\pm |
dt+dMi(t),\\ d[Mi,Mj](t)&=
\alpha\hbar | |
m |
\deltaijdt, \end{cases}
which can be solved for the process
\{Z(t):t\in[t0,tf]\}
X(t0)=x0
+
X(tf)=xf
-
\{X(t):t\in[t0,tf]\}
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
\begin{cases} | \partial |
\partialxi |
S\pm(x,t)&=
\pm | ||
p | (x,t)\\ | |
i |
\partial | |
\partialt |
S\pm(x,t)&=-H\pm(x,p\pm(x,t),\partialp\pm(x,t),t) \end{cases}
are combined, such that one obtains the equation
2m\left(\partialtS\pm+ak{U}\right) +\deltaij\left(\partialiS\pm\partialjS\pm \pm\alpha\hbar\partialj\partialiS\pm -2qAi\partialjS\pm \mp\alpha\hbarq\partialjAi +q2AiAj \right) =0.
Subsequently, one defines the wave function
\Psi\pm(x,t)=\exp\left(\pm
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
\tilde{\Psi}+\equiv\Psi+ {\rmif} \tilde{\Psi}+=\pm\Psi+ {\rmand} \tilde{\Psi}-\equiv\Psi- {\rmif} \tilde{\Psi}-=\pm\Psi-.
Furthermore, the wave functions are subjected to the complex diffusion equations
-\alpha\hbar
\partial | |
\partialt |
\Psi+=\left[
\deltaij | |
2m |
\left(\alpha\hbar
\partial | |
\partialxi |
+qAi\right) \left(\alpha\hbar
\partial | |
\partialxj |
+qAj\right) +ak{U} \right]\Psi+,
\alpha\hbar
\partial | |
\partialt |
\Psi-=\left[
\deltaij | |
2m |
\left(\alpha\hbar
\partial | |
\partialxi |
+qAi\right) \left(\alpha\hbar
\partial | |
\partialxj |
+qAj\right) +ak{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
\{X(t):t\in[t0,tf]\}
Finally, one can construct a probability density
\rho\pm(x,t):=
|\Psi\pm(x,t)|2 | |
\int|\Psi\pm(y,t)|2dny |
,
which describes transition probabilities for the process
\{X(t):t\in[t0,tf]\}
\rho+
(x,t)
(xf,tf)
\Psi+
\{X(t):t\in[t0,tf]\}
\rho-
(x,t)
(x0,t0)
\Psi-
\{X(t):t\in[t0,tf]\}
\rho-
(x,t)
(x0,t0)
\Psi-
\{X(t):t\in[t0,tf]\}
The theory contains various special limits:
\alpha=0
X
Y
\alpha\in(0,infty)
X
Y
\alpha\in{\rmi} x (0,infty)
X
Y
\alpha\in(-infty,0)
X
Y
\alpha\in{\rmi} x (-infty,0)
X
Y
u(x,t0)=0
u(x,tf)=0
u(x,t)=0 \forallt
X
Y
Y
X
\alpha ≠ 0
Y
X
The theory is symmetric under the time reversal operation
(t,\alpha,q)\leftrightarrow(-t,-\alpha,-q)
In the Brownian limits, the theory is maximally dissipative, whereas the quantum limits are unitary, such that
d | |
dt |
\int|\Psi\pm(y,t)|2dny|\alpha x \R}=0.
The diffusion equation can be rewritten as
\mp\alpha\hbar
\partial | |
\partialt |
\Psi\pm=\hat{H}(\hat{x},\hat{p}\pm,t)\Psi\pm,
where
\hat{H}
\hat{x}i=xi {\rmand}
\pm | |
\hat{p} | |
i |
=\pm\alpha\hbar
\partial | |
\partialxi |
,
such that the Hamiltonian has its familiar shape
H(\hat{x},\hat{p},t)=
\deltaij | |
2m |
[\hat{p}i-qAi(\hat{x},t)][\hat{p}j-qAj(\hat{x},t)] +ak{U}(\hat{x},t).
These operators obey the canonical commutation relation
[\hat{x}i,
\pm | |
\hat{p} | |
j |
]=\mp\alpha\hbar
i | |
\delta | |
j |
.