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What Bohmian mechanic says about arrival times of 1D vacuum squeezed states

T0 review · 3 major / 5 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read This paper derives a closed-form, analytic arrival-time distribution for Bohmian particles guided by a one-dimensional vacuum squeezed state, and shows it differs from the standard quantum prediction.

desk verdict The paper has a clean trajectory derivation and a correct φ=0 result, but the central arccos inversion misidentifies first arrival times for initial conditions above L when φ≠0, leaving the main distribution unsupported. read the letter →

arxiv 2502.05734 v2 pith:WLPTDD5O submitted 2025-02-09 quant-ph physics.optics

Angel Garcia-Chung , Humberto G. Laguna This is my paper · ORCID
classification quant-phphysics.optics PACS 03.65.Ta42.50.Dv
keywords Bohmianmechanicstimeofarrivalsqueezedvacuumstatearrival-timeprobabilitydistributionsymplecticgroupunitaryrepresentationSchrödingerquantumtrajectoriesharmonicoscillator
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper derives, in closed form, the time of arrival for a Bohmian particle guided by a one-dimensional vacuum squeezed state, and the probability distribution of those arrival times. The pilot wave is written in the Schrödinger representation with the help of the unitary representation of the symplectic group; the resulting Bohmian trajectories are bounded and oscillate with period $\pi/\omega$, so a particle can reach a detector at $L$ only if its initial position lies in a finite interval $I_{\mathrm{BSS}}$. Inverting the trajectory with the principal branch of arccos gives an analytic arrival-time function, and substituting it into the standard Bohmian TOA distribution yields a closed-form density $\Pi_\xi(\tau)$ and a mean arrival time. The paper's central claim is that this is the first analytic TOA distribution for this state, and that its predictions differ from those of standard quantum mechanics, giving an experimentally distinguishable counting statistic.

What carries the argument

The central object is the time-evolved pilot wave $\Psi_\xi(x,t)$ (Eq. 25), a Gaussian obtained by acting on the vacuum with the unitary representation of the symplectic group element $M(t,\xi)=M_H M(\xi)$. The mechanism that carries the argument is the exact solution of the Bohm equation (Eq. 30), whose boundedness defines the interval of initial conditions that can ever be detected at $L$. Inversion of that trajectory uses the principal branch of arccos (Eq. 42), and the arrival-time distribution (Eq. 47) is obtained by integrating $\delta(t(q_0)-\tau)$ against the initial-position density $|\Psi_\xi(q_0)|^2$ over that interval. The same trajectory inversion also yields the mean arrival time (Eq. 49).

What would settle it

Numerically integrate the Bohm equation (Eq. 29) for a concrete case with nonzero squeezing phase, e.g. $\phi=\pi/2$, $r=0.5$, $q_0=1.2$, $L=1$, and record the first time $q(t)$ crosses $L$; if that time is smaller than the value returned by Eq. (42), the arccos-branch assumption fails and $\Pi_\xi(\tau)$ in Eq. (47) does not describe all initial conditions in $I_{\mathrm{BSS}}$.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is that the Bohmian arrival-time problem for the vacuum squeezed state is exactly solvable. The time-evolved pilot wave is a Gaussian whose coefficients are built from the symplectic matrix $M(t,\xi)=M_H M(\xi)$, and the Bohm equation reduces to $\dot q=\omega\tanh(2r)\sin(2\omega t-\phi)/[1-\tanh(2r)\cos(2\omega t-\phi)]\,q(t)$, with solution $q(t)=q_0\sqrt{[1-\tanh(2r)\cos(2\omega t-\phi)]/[1-\tanh(2r)\cos\phi]}$. Inverting this with the principal branch of arccos gives $t_{\mathrm{oa}}(q_0)$ as in Eq. (42), valid for initial positions in $I_{\mathrm{BSS}}=[q^{\min}_0,q^{\max}_0]$. Using the delta-function definition of the TOA distribution and weighting initial positions by $|\Psi_\xi(q_0)|^2$ yields the closed form $\Pi_\xi(\tau)$ in Eq. (47). The paper claims this is the first analytic arrival-time distribution for a one-dimensional vacuum squeezed state in the Bohmian formalism, and it also derives the corresponding mean arrival time and shows it is bounded above and below.

Load-bearing premise

The load-bearing assumption is that the principal branch of arccos in Eq. (42) gives the first time the trajectory reaches $L$, which holds only when the crossing occurs on the rising branch; for initial positions $q_0>L$ (which occur whenever the squeezing phase is nonzero) the first crossing is on the falling branch and happens earlier than Eq. (42) predicts.

Editorial extensions

If this is right

  • Only initial positions in the finite interval $[q^{\min}_0,q^{\max}_0]$ can ever be detected at $L$; particles launched outside that interval never arrive, so the arrival-time distribution is supported entirely on this interval.
  • Every arrival at $L$ happens within a bounded time window: the maximum arrival time is $(\phi+\pi)/(2\omega)$ for $\phi<\pi$, independent of the squeezing parameter $r$.
  • The mean arrival time approaches $\phi/(2\omega)$ as $r$ grows and $(\phi+\pi)/(2\omega)$ as $L/l$ grows, so the mean is bracketed between two parameter-independent bounds.
  • The Bohmian click count for a detector at $L$, Eq. (54), differs from the standard quantum click count, Eq. (52), so the two frameworks make distinguishable predictions for the same squeezed state.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A testable extension is to use the same symplectic-representation route for thermal or multimode squeezed states, where trajectory boundedness may still permit a closed-form arrival-time distribution.
  • The paper itself notes the detector is modeled only at zero order; if detector back-reaction matters, the counting-rate comparison with standard quantum mechanics is provisional.
  • The forbidden phase-space region bounded by $\dot q=\pm 2\omega\sinh(2r)\,q$ points toward a possible relativistic signature if squeezed modes are treated as a quantum field, though the paper does not pursue this.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper uses the unitary representation of Sp(2,R) to write the one-dimensional vacuum squeezed state in the Schrödinger representation, solves the Bohmian equation of motion in closed form, and uses the resulting trajectory to define the time of arrival at a fixed detector position L. From the interval of initial conditions that can reach L, the authors derive a closed-form arrival-time distribution (Eq. 47), its mean value (Eq. 49), and compare the Bohmian prediction with a standard-quantum-mechanics counting formula. The advertised result is the first analytic Bohmian TOA distribution for a 1D vacuum squeezed state, depending on the squeezing parameter, the detector position, and the squeezing phase.

Significance. The derivation up to Eq. (30) is a genuine strength: the symplectic-group route produces an explicit Gaussian pilot wave and an exact Bohmian trajectory with no fitted parameters, and the algebraic inversion is economical. If the first-arrival identification were correct for general phase, Eq. (47) would be a valuable analytic benchmark for Bohmian arrival-time studies and a concrete point of contrast with standard TOA prescriptions. However, the TOA formulas are valid only for the special case φ = 0 as presented; the general-phase claim made in the abstract and in Section 4 is unsupported. The paper therefore cannot be accepted as a general result, although the φ = 0 special case and the trajectory derivation remain useful contributions.

major comments (3)
  1. [§4, Eq. (42)] The principal branch of the arccos in Eq. (42) gives the crossing on the rising branch of the trajectory, not the first crossing for every q0 in IBSS. For 0 < φ < π and q0 > L, which is a nonempty subset of IBSS whenever φ > 0, the trajectory starts above L and descends immediately, so the first arrival occurs on the falling branch at t = (φ − arccos c)/(2ω), with c = (1/tanh(2r))[1 − (1 − tanh(2r) cos φ)L²/q0²]. For example, with r = 0.5, φ = π/2, L = 1, and q0 = 1.2, Eq. (42) gives roughly 1.364/ω, whereas the actual first arrival is about 0.207/ω. At q0 = L the true first arrival is t = 0, while Eq. (42) gives φ/ω for 0 < φ < π. The correct first-arrival time is the minimum over the branches t = (φ ± arccos c + 2πn)/(2ω), n ∈ Z, subject to t ≥ 0.
  2. [§4.1, Eq. (47)] Because Eq. (47) is obtained by substituting Eq. (42) into the delta-function expression (45), it is the distribution of rising-branch crossing times, not of first arrival times, for φ ≠ 0. The omitted falling-branch crossings correspond to a finite-measure set of initial conditions and to an interval of τ values, so the error is not a measure-zero artifact. The figures use φ = 0, where qmax0 = L and no initial condition exceeds L, which is why the branch problem is invisible there; this does not validate the general-phase claim made in the abstract and in Section 4.
  3. [§4, Eq. (43)] The endpoint evaluation in Eq. (43) is also not the first arrival for φ > π. For instance, at φ = 3π/2 the trajectory with q0 = qmin0 reaches its maximum L at t = (φ − π)/(2ω) = π/(4ω), whereas Eq. (42) gives (φ + π)/(2ω) = 5π/(4ω), which is one full period later. Thus the text's statement that the minimum time is always toa = 0 is inconsistent with Eq. (42) itself at q0 = L for general φ. The branch error therefore affects the claimed set of arrival times and their extrema, not only the interior of IBSS.
minor comments (5)
  1. [§3, after Eq. (30)] The sentence 'the period of the trajectories doubles the period of the classical harmonic oscillator' should read 'is half', since q(t) has period π/ω while the harmonic-oscillator period is 2π/ω.
  2. [§4, Eq. (45)] Eq. (45) integrates over suppΨξ, but the time function t(q0) is defined only on IBSS and is set to infinity outside via Eq. (44); the integration domain and the handling of the piecewise extension should be stated explicitly.
  3. [§4.1, Eq. (47)] The change of variables leading from Eq. (45) to Eq. (47) should display the Jacobian and the branch of t(q0) that is being used; as written, the substitution is only formal and hides the branch issue discussed above.
  4. [§2 and §3] There are numerous typographical errors, including 'squezeed', 'most be', 'recaling', 'a fixed value ot t', and 'close expression', which should be corrected.
  5. [Figures 5 and 6] The horizontal axis is labeled ωτ − φ even though Eq. (47) and the text present Πξ as a function of τ; please clarify whether the plotted variable is ωτ − φ or ωτ.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the arrival-time distribution is derived from the standard symplectic representation and an explicit inversion of the Bohmian trajectory, with no fitted parameters or self-referential steps.

full rationale

The paper's central claim—the TOA distribution in Eq. (47)—is obtained by a transparent chain: (i) constructing the squeezed-state wave function via the standard unitary representation of Sp(2,R); (ii) solving the Bohm equation to obtain q(t) in Eq. (30); (iii) inverting q(t)=L to define toa(q0) in Eq. (42); and (iv) performing the delta-function change of variables in Eq. (45) to get the distribution. None of these steps presupposes the final distribution. The only self-citation, [32], is used for a standard Lie algebra isomorphism between quadratic operators and sp(2,R); this result is independently grounded and is also supported by the other cited references [28-31], so it is not a load-bearing self-citation. No parameters are fitted to any subset of data, and no quantity is defined in terms of the claimed prediction. The evident mathematical concern—that the principal branch of arccos in Eq. (42) may not give the first crossing time for initial conditions q0>L when the squeezing phase is nonzero—is a correctness or branch-selection issue, not a circularity: it does not make the derivation equivalent to its inputs. Given the Bohmian postulates and standard symplectic representation theory, the derivation is self-contained.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The paper introduces no new entities or fitted parameters; the state parameters (r, φ, ω, l) are physical inputs. The central hidden assumption is the branch choice in the arccos inversion, which is invalid over part of the stated domain.

assumptions (3)
  • domain assumption Bohmian guidance equation (Eq. 28) and quantum equilibrium postulate (initial positions distributed as |Ψ0|^2)
    The entire TOA calculation is conditional on adopting the Bohmian interpretation of quantum mechanics, as stated in Section 3.
  • standard math The metaplectic (unitary) representation of Sp(2,R) correctly produces the position-space squeezed state (Eqs. 14-18)
    Standard linear canonical transform theory cited via refs. [28-32].
  • ad hoc to paper The first arrival at L corresponds to the principal branch of the arccos inversion of q(t) (Eq. 42)
    This is the load-bearing assumption that fails for q0 > L (e.g., φ = π/2, q0 = 1.2, L = 1), where the principal branch gives the second crossing. The paper does not justify this branch choice.

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Pith. "Pith review of What Bohmian mechanic says about arrival times of 1D vacuum squeezed states." pith.science (2026). https://pith.science/paper/WLPTDD5O

@misc{pith2026250205734,
  author       = {Pith},
  title        = {Pith review of: What Bohmian mechanic says about arrival times of 1D vacuum squeezed states},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WLPTDD5O}},
  note         = {Machine review of arXiv:2502.05734}
}
abstract

We calculate the time of arrival probability distribution of a quantum particle using the Bohmian formalism. The pilot-wave is given by the wave function of the one dimensional vacuum squeezed state but written in the Schr\"odinger representation. We made use of the unitary representation of the symplectic group in the Hilbert space $L^2(\mathbb{R})$. The solution to the Bohmian equations are analytical function thus allowing for a closed expression of the time of arrival distribution which differs from the counterparts in the standard quantum mechanics formulation.

Figures

Figures reproduced from arXiv: 2502.05734 by the authors.

Figure 1
Figure 1. Heat map of the function |Ψξ(x, t)| 2 . In (1a) r = 0.5 and r = 1 in (1b), illustrating the effect of the squeezing parameter r, both cases with ϕ = 60◦ . 3 Bohmian analysis In this section we are going to introduce the analysis of the particles’ trajectories using the Bohmian equations [7, 8, 9] and then we are going to analyze a scenario in which a position detection is performed. Let us recall that in Bohm’s desc… view at source ↗
Figure 2
Figure 2. Bohmian trajectories of the squeezed state in Eqn (25) using the squared absolute value of the wave function in Eqn (16) for the probability distribution of the initial conditions. In (a) the phase ϕ = 0 and in (b) the phase ϕ = 2π/3. In both cases we plotted n = 200 trajectories with ℏ = m = 1 and ω = 0.5 and a squeezed parameter value of r = 0.5. then q(t) = 0 for any t ≥ 0 and viceversa, having q(t) = 0 for some … view at source ↗
Figure 3
Figure 3. Bohmian trajectories of the squeezed state in Eqn (25) using the squared absolute value of the wave function in Eqn (16) for the probability distribution of the initial conditions. In (a) the phase ϕ = 0 and in (b) the phase ϕ = 2π/3. In the second case we plotted n = 200 trajectories and in both we have: ℏ = m = 1, ω = 0.5 and a squeezed parameter value of r = 0.5. The purpose of (a) is to show the shape of individ… view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: Maximum initial condition (dashed line) and minimum initial conditions (solid line) as functions of the phase ϕ, see Eqn (36) and Eqn (37). In (a) the squeezing parameter r = 0.5 and in (b) r = 1 and in both cases L = 2. This plot shows that all the trajectories detect…
Figure 5
Figure 5. Figure 5: Probability distribution of Time of Arrival Πξ as a function of ωτ . In (5a) we varied the squeezing parameter r and consider L = 1 fixed. In (5b) we fixed r = 1 and L changes. In both cases, ℏ = m = 1, ω = 0.5 and ϕ = 0. r and for different values of the ratio L/l. (i…
Figure 6
Figure 6. Figure 6: Plot of 2ω⟨toa⟩ − ϕ as a function of the squeezing parameter r (6a) and the ratio L/l (6b). In (6a) for different values of the ratio L/l and in (6b) for different values of the squeezing parameter r. 5 Discussion and conclusions Let us examine more closely the implica…

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