What happens to higher-order differential terms like $(dx)^2$ when extending classical mechanics to a stochastic space-time trajectory?

Question

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1. What the question is asking – in plain language

In ordinary (deterministic) calculus we write

[ dx = v\,dt ,\qquad (dx)^2 = v^{2}\,dt^{2}\;, ]

and because (dt\to 0) the term ((dx)^2) is always thrown away – it is “higher‑order”.

If the trajectory of a particle is not a smooth curve but a wildly fluctuating one (the kind of path that appears in Brownian motion or in stochastic formulations of quantum mechanics), the increment (dx) behaves like the increment of a Wiener process (dW).
For a Wiener process

[ dW\sim\mathcal{N}(0,dt)\qquad\Longrightarrow\qquad dW=O(\sqrt{dt}), ]

so that

[ (dW)^2 = O(dt)\neq 0 . ]

The student therefore asks:
1. Mathematically – how does stochastic calculus (Itô calculus) treat the product ((dx)^2) when the path is non‑smooth?
2. Physically – what does the presence of a non‑vanishing ((dx)^2) imply for the notion of “velocity” of a quantum particle?

The answer will explain the Itô rule ((dW)^2 = dt), show where it comes from, and interpret the result for quantum‑stochastic models (e.g., Nelson’s stochastic mechanics).

2. Detailed solution

2.1 Classical deterministic case (review)

If the trajectory (x(t)) is differentiable, we can write a Taylor expansion for a smooth function (f(x,t)):

[ df = \frac{\partial f}{\partial t}\,dt+\frac{\partial f}{\partial x}\,dx +\frac12\frac{\partial^{2}f}{\partial x^{2}}\,(dx)^{2}+ \cdots . ]

Because (dx = \dot x\,dt) and (\dot x) is finite, ((dx)^2 = \dot x^{\,2}dt^{2}) is of order (dt^{2}) and vanishes faster than the linear term when we divide by (dt) and let (dt\to0). Hence we drop it and obtain the ordinary chain rule

[ \frac{df}{dt}= \frac{\partial f}{\partial t}+ \frac{\partial f}{\partial x}\,\dot x . ]

2.2 Brownian (Wiener) motion – why ((dx)^2) is not negligible

A standard Wiener process (W(t)) satisfies

(W(0)=0);
increments are independent;
(W(t+dt)-W(t)\sim\mathcal N(0,dt)).

Define the infinitesimal increment

[ dW \equiv W(t+dt)-W(t). ]

From the Gaussian distribution we have

[ \mathbb{E}[dW]=0,\qquad \mathbb{E}[(dW)^{2}]=dt . ]

Thus, in the mean-square sense

[ (dW)^{2}=dt\quad\text{(to leading order)}. ]

This is the core of Itô’s calculus: the product of two infinitesimal Wiener increments is first order in (dt), not second order.

The rigorous statement is

[ \boxed{(dW)^{2}=dt,\qquad dW\,dt=0,\qquad (dt)^{2}=0 } . ]

The equalities are to be understood inside stochastic integrals; they are derived from the limit

[ \sum_{k=0}^{n-1}(W_{t_{k+1}}-W_{t_{k}})^{2} \xrightarrow[n\to\infty]{} \int_{0}^{t} ds = t , ]

which is the quadratic variation of Brownian motion.

2.3 Itô’s Lemma – the stochastic analogue of the chain rule

Let (X(t)) satisfy an Itô stochastic differential equation (SDE)

[ dX = a(X,t)\,dt + b(X,t)\,dW . ]

For a twice‑differentiable scalar function (f(x,t)) define (Y(t)=f\bigl(X(t),t\bigr)).
Itô’s Lemma (proved by applying the Taylor expansion and using the rules above) gives

[ \boxed{ \begin{aligned} dY &=\frac{\partial f}{\partial t}\,dt +\frac{\partial f}{\partial x}\,dX +\frac12\frac{\partial^{2}f}{\partial x^{2}}\, (dX)^{2} \[4pt] &=\Bigl[\,\frac{\partial f}{\partial t} +a\,\frac{\partial f}{\partial x} +\frac12 b^{2}\,\frac{\partial^{2}f}{\partial x^{2}} \Bigr]dt +b\,\frac{\partial f}{\partial x}\,dW . \end{aligned}} ]

The term (\frac12 b^{2}\partial^{2}_{x}f\,dt) originates solely from the replacement ((dW)^{2}=dt).
If the trajectory were smooth ((b\equiv0)) this term would disappear and we would recover the ordinary chain rule.

2.4 What does this mean for the “velocity” of a quantum particle?

No ordinary derivative.
For a Brownian‑type path the limit

[ \lim_{dt\to0}\frac{x(t+dt)-x(t)}{dt} ]

does not exist (it diverges like (1/\sqrt{dt})). Hence a classical velocity (v(t)=\dot x(t)) is undefined.
Forward and backward drifts.
In stochastic mechanics (Nelson, 1966) one introduces mean forward and mean backward derivatives:

[ D_{+}x(t)=\lim_{dt\downarrow0}\mathbb{E}!\left[\frac{x(t+dt)-x(t)}{dt}\,\bigg|\,\mathcal F_t\right], \qquad D_{-}x(t)=\lim_{dt\uparrow0}\mathbb{E}!\left[\frac{x(t)-x(t-dt)}{dt}\,\bigg|\,\mathcal F_t\right]. ]

These are finite and play the role of “velocity fields” (called current velocity (v) and osmotic velocity (u)). They are not instantaneous microscopic speeds but statistical averages over the random fluctuations.
Physical interpretation of ((dx)^{2}=dt).
The non‑vanishing quadratic variation is the mathematical expression of the fact that the particle’s path is nowhere differentiable. In quantum‑stochastic models the diffusion coefficient is fixed by Planck’s constant:

[ b = \sqrt{\frac{\hbar}{m}} \quad\Longrightarrow\quad (dx)^{2}=b^{2}\,dt = \frac{\hbar}{m}\,dt . ]

Thus the stochastic term carries the quantum of action (\hbar); it is the source of the uncertainty that replaces a sharp velocity.
Consequences for the Schrödinger equation.
Starting from the SDE

[ dX = v\,dt + \sqrt{\frac{\hbar}{m}}\,dW , ]

and applying Itô’s Lemma to the wave‑function amplitude (\psi(X,t)), one recovers the Schrödinger equation (Nelson’s derivation). The extra term (\frac12 b^{2}\partial_{x}^{2}\psi\,dt) coming from ((dW)^{2}=dt) is precisely the kinetic term (-(\hbar^{2}/2m)\partial_{x}^{2}\psi).

2.5 Summary of the mathematical answer

Situation	Increment size	Quadratic variation	Rule for ((dx)^{2})
Smooth deterministic path (x(t))	(dx = v\,dt = O(dt))	(\displaystyle\lim_{t\to0}\sum(dx)^{2}=0)	((dx)^{2}=0) (higher‑order, discarded)
Wiener‑type stochastic path (W(t))	(dW = O(\sqrt{dt}))	(\displaystyle\lim_{n\to\infty}\sum_{k}(dW_{k})^{2}=t)	((dW)^{2}=dt) (kept)
General Itô process (dX = a\,dt + b\,dW)	(dX = O(dt) + O(\sqrt{dt}))	((dX)^{2}=b^{2}dt)	((dX)^{2}=b^{2}dt)

Thus, in stochastic calculus ((dx)^{2}) is not discarded; it is replaced by a term proportional to (dt), and that term is crucial for the dynamics.

3. Final answer – concise statement

In ordinary calculus the square of an infinitesimal displacement is of order (dt^{2}) and is omitted.
For a trajectory that is a Wiener (Brownian) process, the increment scales as (\sqrt{dt}); consequently

[ (d x)^{2}= (b\,dW)^{2}=b^{2}\,dt , ]

where (b) is the diffusion coefficient (e.g., (b=\sqrt{\hbar/m}) in Nelson’s stochastic mechanics).

This rule, ((dW)^{2}=dt), is the cornerstone of Itô calculus. It appears in Itô’s Lemma as an extra drift term (\frac12 b^{2}\partial_{x}^{2}f\,dt).
Physically, the non‑zero quadratic variation means the particle’s path is nowhere differentiable, so a classical instantaneous velocity does not exist. One can instead define mean forward/backward drifts (statistical velocities). The extra ((dx)^{2}) term encodes the quantum diffusion that ultimately yields the Schrödinger equation.

4. Common mistakes when tackling this type of problem

Mistake	Why it is wrong	How to avoid it
Treating (dx) as a regular number and writing ((dx)^{2}=0) even for stochastic paths.	In Itô calculus (dx) denotes a random increment; its square is of order (dt), not (dt^{2}).	Remember the scaling: (dW=O(\sqrt{dt})) → ((dW)^{2}=O(dt)).
Confusing Itô and Stratonovich calculus and using the wrong rule for ((dx)^{2}).	Stratonovich integrals obey the ordinary chain rule, but the conversion adds a term (\frac12 b^{2}\partial_{x}^{2}f\,dt).	State clearly which interpretation you are using; for most quantum‑stochastic derivations Itô is standard.
Assuming a well‑defined instantaneous velocity for a Brownian path.	The limit (\lim_{dt\to0}(x(t+dt)-x(t))/dt) diverges; only mean drifts are finite.	Introduce forward/backward derivatives or the concept of drift instead of classical velocity.
Dropping the ((dx)^{2}) term in the Taylor expansion of a stochastic differential because “it is higher order”.	The term is first order in (dt) for a stochastic increment, so it contributes to the dynamics.	Perform the expansion including the quadratic variation, then replace ((dW)^{2}) by (dt).
Neglecting the physical meaning of the diffusion coefficient (e.g., setting (b=1) without justification).	In stochastic mechanics (b) is fixed by (\hbar) and the particle mass; arbitrary choice changes the physics.	Relate (b) to the quantum constants when interpreting the result.

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