Trajectory-Based Decomposition of Quantum Variances

• The paper decomposes quantum variance into a classical-like Bohmian ensemble variance and an irreducible quantum fluctuation term.
• It details how the weak actual value field links operator expectations to trajectory-based ensemble analysis via phase–amplitude coupling.
• It rigorously establishes conditions under which the variance split holds, while highlighting limits for non-position observables like spin.

A trajectory-based decomposition of quantum variances is a formalism, developed primarily within the context of Bohmian mechanics, that expresses the quantum variance of an observable as a sum of two non-negative terms: a classical-like, trajectory-based ensemble variance of a weak actual value field, and an irreducibly quantum fluctuation term associated with phase–amplitude coupling. This decomposition provides both operational and interpretive insight into the origins of quantum uncertainty, with a uniquely clear structure for observables that are functions of position and its conjugate momentum, while also delineating where such a trajectory-based picture fails, notably for intrinsic degrees of freedom such as spin (Ye, 31 Dec 2025).

1. Weak Actual Value Field and Bohmian Ensemble Variance

For a self-adjoint operator $\hat{A}$ and stationary bound-state wave function $\psi(x)$ on configuration space $\mathbb{R}^d$ (excluding the nodal set $N={x: \psi(x)=0}$), the weak actual value field $a_w(x)$ is defined as:

$$a_w(x) = \operatorname{Re} \frac{\psi^*(x) (\hat{A}\psi)(x)}{|\psi(x)|^2}$$

Under the quantum equilibrium distribution $\rho(x) = |\psi(x)|^2$, the expectation of $a_w(x)$ recovers the quantum expectation:

$$\mathbb{E}[a_w] = \int |\psi(x)|^2 a_w(x) dx = \langle \psi | \hat{A} | \psi \rangle$$

The Bohmian ensemble variance of the field is

$$\text{Var}_B[a_w] = \int |\psi(x)|^2 [a_w(x) - \mathbb{E}[a_w]]^2 dx$$

This quantity represents the statistical spread of the weak actual field over the (Bohmian) quantum-equilibrium ensemble.

2. Variance Decomposition Theorem

The central result is an identity that splits the standard quantum variance of $\hat{A}$:

$$\text{Var}_Q[\hat{A}] = \langle \psi | \hat{A}^2 | \psi \rangle - \langle \psi | \hat{A} | \psi \rangle^2 = \text{Var}_B[a_w] + Q_A$$

where the quantum fluctuation term is

$$Q_A = \int \frac{[\operatorname{Im} \left\{ \psi^*(x) (\hat{A} \psi)(x) \right\}]^2}{|\psi(x)|^2} dx$$

Both summands are non-negative. $\text{Var}_B[a_w]$ captures classical-like statistical fluctuations of the weak actual value across the ensemble, while $Q_A$ is irreducibly quantum, arising from the phase–amplitude coupling intrinsic to the wave function.

3. Formal Hypotheses and Rigorous Conditions

Establishing this decomposition requires strong regularity conditions:

• (H1) $V(x)$ real-analytic potential.
• (H2) $\psi \in H^2(\mathbb{R}^d)$, itself real-analytic, decaying exponentially.
• (H3) $\hat{A}$ is a differential operator of order $m$ with bounded $C^m$ coefficients.
• (H4) Uniform lower bound $k \geq 1$ on the order of zeros of $\psi$.
• (H5) Geometric thinness of the nodal set: $\text{Vol}(\operatorname{dist}(x,N)<r) \leq C' r$ as $r \to 0$.
• (H6) Integrability: $k - m + d/2 > 0$.

These conditions ensure all terms in the decomposition, including pointwise identities, are Lebesgue-integrable and the variance split is mathematically precise (Ye, 31 Dec 2025).

4. Momentum Variance, Quantum Potential, and Bohmian Guidance

Specialization to the momentum operator $\hat{p} = -i\hbar \nabla$ yields sharp physical insights. Writing $\psi = R e^{iS/\hbar}$:

• $a_w(x) \equiv p_w(x) = \nabla S(x)$, the guiding momentum in Bohmian mechanics.
• The quantum term:

$$Q_p = \int (\operatorname{Im}[\psi^* (\hat{p} \psi)])^2 / |\psi|^2 dx = \hbar^2 \int |\nabla R|^2 dx$$

• The Bohmian quantum potential is $Q(x) = -(\hbar^2/2m) (\nabla^2 R / R)$, whose ensemble average is

$$\langle Q \rangle = (\hbar^2/2m) \int |\nabla R|^2 dx$$

giving the transparent relation $Q_p = 2m \langle Q \rangle$.

The final form for the momentum variance reads

$$\text{Var}_Q[\hat{p}] = \text{Var}_B[p_w] + 2m \langle Q \rangle$$

Here, $\text{Var}_B[p_w]$ quantifies the spread of Bohmian guiding momenta, while $2m \langle Q \rangle$ is the irreducibly quantum part, directly tied to the amplitude inhomogeneity.

Table: Explicit Decomposition Components for $\hat{A}=\hat{p}$

Term	Expression	Interpretation
$\text{Var}_B[p_w]$	$$\int |\psi|^2 (\nabla S - \langle \nabla S \rangle)^2 dx$$	Ensemble spread of local momentum field
$2m \langle Q \rangle$	$2m (\hbar^2/2m) \int |\nabla R|^2 dx$	Ensemble mean quantum potential, amplitude fluctuation

5. Interpretational Boundaries: Failure for Spin and Primacy of Position

The decomposition, when formally extended to spin-½ observables (e.g., $S_z$ on two-component spinors), yields:

• $$a_w^z(x) = (\hbar/2)[|\psi_1|^2 - |\psi_2|^2]/(|\psi_1|^2 + |\psi_2|^2)$$
• $Q_{S_z} = 0$. Hence, $\text{Var}_Q[S_z] = \text{Var}_B[a_w^z]$.

However, Bohmian mechanics recognizes only position as an ontological beable; spin is not an additional local variable transported along trajectories. The vanishing of $Q_{S_z}$, and absence of any genuine "spin-field fluctuation" corresponding to quantum spin variance, illustrates the interpretive limit: for spin, the decomposition is a tautology devoid of physical content, reflecting the contextual emergence of spin measurement outcomes from entangled wave function structure.

6. Connections to Weak Measurement, Quantum Fluctuations, and Related Frameworks

Operationally, in weak measurement experiments (such as trajectory reconstructions via p_w(x) [Kocsis et al., Science 332, 1170 (2011)]), the ensemble variance of p_w and the residual quantum piece match the two terms in the trajectory-based variance decomposition (Ye, 31 Dec 2025). More generally, this formalism offers a rigorous connection between variance-level quantum fluctuations and the spatial distribution of quantum potential, providing a tangible tool for analyzing and interpreting quantum noise and fluctuation mechanisms.

Moreover, links to other formalisms (e.g., the ensemble and conditional variances studied via Wigner functions (Feyereisen, 2015), operator-weak values (Lee et al., 2020), and the theory of weak-value variances in pre- and post-selected systems (Ogawa et al., 2021)) clarify that the Bohmian decomposition uniquely separates ensemble-trajectory spread from irreducible quantum amplitude-phase effects. For momentum, these approaches coalesce: the total quantum variance splits into an ensemble variance of local weak values and an additional term linked to quantum potential and phase-amplitude coupling.

7. Representative Examples and Practical Implications

• Harmonic oscillator stationary states: For real eigenfunctions $S_n(x)=0$, so $\text{Var}_B[p_w] = 0$, and quantum variance is fully due to the quantum potential part.
• Energy in stationary states: $a_w^H(x) = E$ (constant), $Q_H=0$, so $\text{Var}_Q[H] = 0$.
• Operational measurement: In weak measurement protocols, empirical fluctuations of the reconstructed p_w(x) ensemble sum to the two decomposition terms.

In conclusion, the trajectory-based decomposition gives a mathematically rigorous, physically transparent split of quantum variance into classical-like ensemble fluctuations and an essentially quantum remainder, solid under explicit regularity conditions. For position and momentum observables, this underscores the interplay of deterministic pilot-wave guidance and genuinely quantum amplitude effects. For intrinsic properties such as spin, it highlights the boundaries of any trajectory-centric interpretation and re-emphasizes the singular foundational status of position in the Bohmian ontology (Ye, 31 Dec 2025).