A Time-Symmetric Variational Reformulation of Nonrelativistic Quantum Mechanics

Lance H. Carter

REVIEW 3 major objections 4 minor 21 references

A Time-Symmetric Variational Reformulation of Nonrelativistic Quantum Mechanics

T0 review · 3 major / 4 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read This paper claims that the Schrödinger equation and the Born rule are consequences of a single time-symmetric variational principle acting on probability density and current.

desk verdict The variational Schrödinger derivation is correct but standard; the Born-rule 'emergence' is circular because the final-boundary density already encodes the outcome probabilities. read the letter →

arxiv 2512.22320 v6 pith:7ESUDPDC submitted 2025-12-26 quant-ph

Lance H. Carter This is my paper · ORCID

classification quant-ph

keywords variationalprincipleFisherinformationSchrödingerequationBornruletimesymmetryhydrodynamicvariablesquantumpotentialmaximumcaliber

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

The paper argues that the two dynamical principles of standard quantum mechanics—unitary evolution and collapse—can be replaced by a single time-symmetric variational principle. The state is written in terms of probability density and current, and the linear Schrödinger equation appears as the necessary condition for minimizing a Fisher-regularized action subject to conservation of probability. The Born rule for measurement outcomes follows from mass conservation in the same framework, without a separate collapse postulate. If correct, the reformulation would recast quantum randomness as effective randomness over boundary-compatible histories rather than a fundamental stochastic law.

What carries the argument

The central object is the primal–dual action A[ρ,j,S] = ∫(m|j|²/(2ρ) − Vρ − (ℏ²/8m)|∇ρ|²/ρ + S(∂_tρ + ∇·j)) dx dt, with S serving as a Lagrange multiplier enforcing continuity. The Fisher information term acts as a convex 'cost of localization' that forbids sharp trajectories, and its functional derivative generates the quantum potential Q = −(ℏ²/2m)(∇²√ρ/√ρ). The mapping ψ=√ρ e^{iS/ℏ} converts the coupled nonlinear optimality conditions into the linear Schrödinger equation, with the Born rule emerging from conservation of the probability fluid.

What would settle it

A concrete falsifier is a pair of boundary densities ρ(t0) and ρ(tf) for which the Fisher-regularized action has no finite minimizer in the admissible class, or for which the Euler–Lagrange flow fails to satisfy the continuity equation or yields a non-unique Lagrange multiplier S; either would break the claimed equivalence to the Schrödinger equation. Alternatively, an experiment showing that outcome statistics for a von Neumann pointer with fully disjoint packets deviate from |c_k|² would falsify the mass-conservation derivation of the Born rule.

Watch

Extended reading notes

Core claim

The central claim is that the linear Schrödinger equation is not an independent postulate but the Euler–Lagrange equation of a global action defined on the hydrodynamic variables (ρ, j). The action adds a Fisher information term to the classical kinetic and potential terms, and its convexity forces optimal histories to obey a quantum Hamilton–Jacobi equation and continuity equation. Defining ψ = √ρ e^{iS/ℏ} then maps those optimality conditions exactly onto the linear Schrödinger equation. Measurement corresponds to imposing a final boundary condition, and the integrated density flowing into a detector region reproduces the Born rule as a consequence of probability conservation.

Load-bearing premise

The load-bearing premise is that the Fisher-information term with coefficient ℏ²/8m is the correct physical action; if another regularizer or a different coefficient were needed to match experiments, the derivation of the linear Schrödinger equation would not follow.

Editorial extensions

If this is right

The dualism of unitary evolution and collapse is replaced by a single boundary-value problem; no collapse postulate is needed.
The Born rule for pointer measurements is shown to be a consequence of conserved hydrodynamic mass, so measurement statistics require no extra stochastic axiom.
Trajectories like the spreading Gaussian packet are recovered as action-minimizing flows between boundary conditions, suggesting trajectory selection is a global geometric necessity.
The framework interprets 1-randomness of quantum outcomes as effective randomness induced by the Fisher penalty, reframing the 'oracle' problem of hidden-variable theories.
The reformulation offers a nonrelativistic proof of concept; extension to spinors is outlined via a matrix density field leading to the Pauli equation.

Reading between the lines

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

If this action principle is fundamental, it suggests that quantum systems in curved spacetime would obey a covariant version of the Fisher-regularized action; testable predictions might arise for the spread of narrow wavepackets under strong gravity.
The framework may be extended to quantum field theory by replacing the density field with a local operator, potentially giving a variational origin for interacting field dynamics.
The role of final boundary conditions suggests a natural connection to retrodictive quantum measurement and to post-selected experiments; the framework predicts that outcome statistics are consistent with the MaxCal history ensemble, which could be tested in weak-measurement scenarios.
If the Fisher penalty indeed ensures algorithmic complexity, the framework might be used to quantify the computational irreducibility of quantum measurement in terms of channel capacity.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Colleague,

The paper is a serious attempt at a time-symmetric variational reformulation, but the headline claim doesn't survive contact with the details. The Euler-Lagrange derivation of the Schrödinger equation from a Fisher-regularized hydrodynamic action is mathematically correct — and it's a known result (Reginatto 1998, Parwani 2005), which the paper acknowledges. What's presented cleanly here is the two-time boundary framing and the MaxCal ensemble, which gives the interpretive machinery a surface of plausibility.

What it does well: it's self-contained, the algebra in Appendix A checks out, and the Gaussian trajectory example is a nice illustration of how the variational principle selects a spreading packet. The discussion of Landsman's oracle problem is thoughtful: reframing the sampling of initial conditions as boundary-condition selection is a legitimate move, even if it doesn't eliminate the need for a stochastic element at the boundary.

The soft spot is the Born-rule derivation in Sec. IX.B, and it's load-bearing. The paper defines P(k) as the mass of the final boundary density in Ω_k, then uses the Schrödinger evolution of an initial state with coefficients c_k to get |c_k|². That's not a derivation; it's the Born rule read backwards. The final boundary density ρ(t_f) either is an input boundary condition (then the probabilities are put in by hand) or it's the unitary evolution of the initial density (then the coefficients already encode the probabilities). The MaxCal measure in Eq. (12) is never connected to this calculation — the paper never evaluates S_min,k or shows that the marginal over histories ending in Ω_k equals ∫ρ(t_f). The continuity equation conserves whatever mass is in the boundary data; it can't assign probabilities to outcomes unless ρ is already a probability density, which is assumed from Sec. II onward.

The Fisher coefficient is also explicitly calibrated to reproduce the Schrödinger equation, so the 'emergence' claim is a reconstruction. That's not fatal for a reformulation — many good papers are reconstructions — but it undercuts the claim that unitary dynamics and the Born rule are derived from a single principle.

Verdict: the central interpretational claim is not supported. The paper is still worth a serious referee — the math is correct, the literature engagement is honest, and the oracle discussion has value. But as a research contribution it's a well-executed synthesis of existing results with an overreach at the end. I'd send it to review, with the expectation that the Born-rule section would need major revision or removal.

For a reading group, it's a maybe — useful for seeing how far variational reconstruction can go, but not a foundational breakthrough.

Referee Report

3 major / 4 minor

Summary. The paper proposes a time-symmetric variational reformulation of nonrelativistic quantum mechanics. The state is represented by hydrodynamic fields (ρ, j) and a Fisher-information term is added to the classical action (Eq. 2). A Lagrange multiplier S enforces continuity, and the Euler–Lagrange equations yield the guidance relation j=ρ∇S/m and the quantum Hamilton–Jacobi equation; via ψ=√ρ e^{iS/ℏ} this reproduces the linear Schrödinger equation (Sec. V, Appendix A). The paper further claims that a Maximum Caliber distribution over histories (Eq. 11) selects boundary-consistent outcomes, and that the Born rule for a von Neumann pointer follows from mass conservation (Eq. 28). The Gaussian spreading example (Sec. VIII) is meant to illustrate boundary-selected flow tubes. The manuscript is positioned as a proof of concept that a single variational principle can replace unitary evolution plus collapse.

Significance. If the central claims were correct, the paper would be significant: it would offer a single all-at-once variational principle from which linear Schrödinger dynamics and Born-rule statistics emerge without a separate collapse postulate, while addressing Landsman's critique of hidden-variable randomness. The paper is also commendably clear about its axiomatic status: it explicitly treats the Fisher term and MaxCal as axioms and candidly leaves open whether ℏ and the Fisher coefficient can be derived from deeper principles. However, the advertised novelty rests on two load-bearing steps—the 'emergence' of the Schrödinger equation and the 'derivation' of the Born rule—and both are, on inspection, either calibrated inputs or circular. The Euler–Lagrange calculation itself is standard Madelung/Reginaatto material, not a new derivation. The MaxCal measure is introduced but never used to compute an outcome probability, so the probabilistic content of the paper is not actually supplied by the variational principle.

major comments (3)

[Sec. III.A, Eq. (2); Appendix A] The claim that 'the linear Schrödinger equation emerges as the necessary condition for minimizing the Fisher-regularized action' (Sec. V) is not supported by the derivation as written. The Fisher term −(ℏ²/8m)|∇ρ|²/ρ is inserted into the action with a coefficient that, as the paper states, 'is fixed by empirical calibration to recover the standard Schrödinger equation' (Sec. III.A). With any other coefficient, or any other gradient regularizer, the resulting Euler–Lagrange equation would not be the linear Schrödinger equation. Thus the target equation is effectively an input, not an output. The paper itself acknowledges this in Sec. X.F, where it says that whether ℏ and the Fisher term are uniquely determined by information-theoretic principles remains open. This undercuts the abstract's strong statement that 'Schrödinger dynamics are not postulated.'
[Sec. IX.B, Eq. (28); Appendix A] The derivation of the Born rule is circular. In Appendix A the boundary conditions fix ρ(x,t0) and ρ(x,tf), so the final density is an input to the variational problem. Equation (28) then evaluates P(k) = ∫_{Ω_k} ρ(x,y,tf) dx dy and obtains |c_k|² only because the initial coefficients c_k are already encoded in the boundary data and evolved by the Schrödinger equation. The mass conservation equation ∂tρ+∇·j=0 only redistributes the mass specified by the boundary conditions; it cannot assign probabilities to outcomes unless ρ is already a probability density. But the paper calls ρ a 'probability density' from Sec. II onward, without deriving that status from the variational principle. Therefore the statement that the Born rule 'is a necessary consequence of mass conservation' (Sec. IX.B) is a tautology, not a derivation.
[Sec. VII.B, Eqs. (11)–(13); Sec. IX] The Maximum Caliber measure is never connected to the Born-rule calculation. Equation (12) defines P(k|M0) as a functional integral over histories weighted by exp(−S[Φ]/ℏ), and Eq. (13) gives a steepest-descent approximation in terms of S_{min,k}. But the paper never evaluates S_{min,k} for the von Neumann pointer model, never computes the prefactor C_k, and never shows that the steepest-descent result equals Eq. (14) or Eq. (28). The pointer derivation instead uses only deterministic Schrödinger evolution of the joint density and the continuity equation. Without a bridge from the MaxCal measure to the hydrodynamic mass integral, the central claim that 'probabilities arise solely from indifference over histories' (Sec. VII.A) is unsupported. The stochastic postulate is not eliminated; it is absorbed into an unevaluated functional integral.

minor comments (4)

[Sec. VIII.B, Eq. (20)] The derivation of the Gaussian flow field assumes the ansatz ρ(x,t) ∝ exp(−x²/2σ(t)²) and then verifies the HJ equation only for the x² terms, explicitly ignoring the spatially constant phase factor. This is acceptable for illustrating the envelope equation, but the paper should state more clearly that full consistency requires a time-dependent phase factor that is not derived.
[Sec. VI.C] The statement that the Fisher regularization 'increases the effective channel capacity of the trajectory' is metaphorical; no definition of channel capacity is given, and the paper immediately concedes that it has not proved that the resulting histories are 1-random. This section would benefit from being framed as a conjecture.
[Sec. VII.D] The double-slit example claims that dark fringes are suppressed because the quantum potential Q becomes singular as ρ→0. This is true for exact nodes, but the argument does not connect to the MaxCal action cost S; the exponential suppression is asserted rather than shown. Please clarify how the divergence of Q enters the action for a history ending at a dark fringe.
[General] There are several typographical and notation issues, e.g., 'IF isher' in Sec. VI.A, inconsistent use of 'Schr¨odinger'/'Schrödinger' in the header, and the phrase 'The boundary conditions do not generate this randomness from ignorance; they harness the substrate's fundamental complexity' is vague. A careful proofread is needed.

Circularity Check

3 steps flagged · score 8.0 of 10

Central 'emergence' claims reduce to constructed inputs: the Fisher action is calibrated to reproduce Schrödinger, and the Born-rule calculation integrates an initially assumed probability density.

fitted input called prediction [Sec. III.A, Sec. V (Eqs. 2, 7-8), Appendix A]
"We select the Fisher information term |∇ρ|^2/ρ because it is the unique gradient functional that preserves additivity ... While its numerical value is fixed by empirical calibration to recover the standard Schrödinger equation ... Thus, the linear Schrödinger equation emerges as the necessary condition for minimizing the Fisher-regularized action subject to continuity."

The primal action (Eq. 2) contains the Fisher term -ℏ^2/8m |∇ρ|^2/ρ with the coefficient explicitly calibrated so that the Euler-Lagrange equations coincide with the Madelung hydrodynamic system. After the substitution ψ=√ρ e^{iS/ℏ}, this system is algebraically equivalent to the linear Schrödinger equation. The 'emergence' is therefore an equivalence with a functional chosen for the purpose; the linearity and the coefficient are inputs, not outputs. The variational derivation is internally valid, but it cannot support the paper's claim that Schrödinger dynamics are not postulated but arise uniquely.
self definitional [Sec. II, Sec. IX.B (Eq. 28), Appendix A]
"When the packets are disjoint ... the total probability mass flowing into region Ω_k is: P(k)=∫...≈|c_k|^2. Thus, the Born rule P(k)=|c_k|^2 is not an independent postulate. It is a necessary consequence of mass conservation."

The initial state in the von Neumann model is ψ0(x)=Σ c_k |a_k>, and ρ is called a probability density from the outset (Sec. II). Thus the initial mass in each outcome channel is |c_k|^2 by assumption. Continuity and Schrödinger evolution merely transport that prescribed probability to the final disjoint packets; integrating the transported density returns the same input weights. Appendix A fixes ρ(t_f) as boundary data, so the mass in Ω_k is an input constraint, not a derived probability. The derivation assumes the Born rule in the initial probability assignment and then re-derives it from mass conservation.

1 more flagged steps

other [Sec. VII.B (Eqs. 11-13) vs. Sec. IX.B (Eq. 28)]
"The 'MaxCal' weight of the history ending in Ω_k is exactly the integrated density of the flow tube leading to that region."

Equation (12) defines P(k) as a MaxCal partition integral over histories weighted by exp(-S[Φ]/ℏ), and Eq. (13) approximates it by exp(-S_min,k/ℏ). But Eq. (28) simply takes P(k) to be ∫_{Ω_k} ρ(t_f), and no calculation shows that the MaxCal sum equals this transported boundary density. The identification is asserted, not derived; since the boundary density itself carries the |c_k|^2 weights, this defines the MaxCal weight to be the Born-rule weight rather than deriving the Born rule from the MaxCal measure.

full rationale

The paper's mathematical manipulations are largely self-contained: the Euler-Lagrange equations of the Fisher-regularized action are correctly shown to be the Madelung equations and hence equivalent to the linear Schrödinger equation, and the Gaussian trajectory computation is a consistent hydrodynamic exercise. However, the paper's advertised conclusions go beyond this equivalence. The Schrödinger equation is not an unforced emergence: the action explicitly includes the Fisher term |∇ρ|^2/ρ, and its coefficient is fixed by empirical calibration to recover the Schrödinger equation, so the target equation is built into the variational principle. Similarly, the Born-rule derivation is not a derivation: ρ is assumed to be a probability density, the initial channel weights are |c_k|^2, and Eq. (28) integrates the unitary evolution of that assumed density over disjoint final regions. Mass conservation conserves the probability that was already inserted; it does not generate Born-rule statistics. The MaxCal expressions (11)-(13) are never connected to Eq. (28), so they provide no independent route to P(k)=|c_k|^2. These are not mere technical gaps; they are places where the claimed predictions coincide with the inputs by construction. I do not see a load-bearing self-citation chain: the cited Reginatto/Parwani results are external and not used to forbid alternatives. The circularity is definitional/calibrational, not a self-citation loop. Overall score 8.

Assumptions & free parameters 2 free parameters · 5 assumptions · 1 invented entities

The derivation rests on an engineered action: the Fisher term and its coefficient are calibrated to reproduce the Schrödinger equation, and ρ is assumed to be a probability density from the outset. The MaxCal ensemble is an additional postulate whose connection to ∫|ψ|² is not demonstrated.

free parameters (2)

Fisher regularization coefficient = ℏ²/8m
Coefficient of |∇ρ|²/ρ is fixed by empirical calibration to recover the Schrödinger equation (Sec. III.A). No derivation of this value is given.
MaxCal inverse temperature β = 1/ℏ
History weight is set to exp(−S/ℏ) (Eq. 11); β≡1/ℏ is chosen by hand to match the quantum action, not derived from MaxCal.

assumptions (5)

domain assumption ρ is a normalized probability density and (ρ,j) obey continuity.
Sec. II; this installs the probability interpretation before any probabilities are derived.
ad hoc to paper Fisher-information term −(ℏ²/8m)|∇ρ|²/ρ is part of the action.
Sec. III.A; coefficient calibrated to reproduce the Schrödinger equation; this term generates the quantum potential Q.
ad hoc to paper Maximum Caliber distribution P[Φ]∝exp(−S[Φ]/ℏ) over histories.
Sec. VII.B Eq. (11); real Boltzmann weight is posited, not connected to the Feynman path-integral measure.
domain assumption Madelung transformation ψ=√ρ e^{iS/ℏ} is single-valued and smooth enough.
Sec. V and Appendix A; no treatment of vorticity, nodal surfaces, or singular Q.
domain assumption Final boundary constraint ρ(tf) can be imposed or selected from the MaxCal ensemble.
Sec. VII.B; what fixes the final outcome is asserted rather than derived.

invented entities (1)

Objective selected flow tube (block-universe history)
purpose: Provides a definite outcome without collapse by selecting one hydrodynamic flow from the ensemble.
Sec. VIII.C and Fig. 1; no falsifiable handle beyond standard quantum predictions.

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Cite this review

Pith. "Pith review of A Time-Symmetric Variational Reformulation of Nonrelativistic Quantum Mechanics." pith.science (2026). https://pith.science/paper/7ESUDPDC

@misc{pith2026251222320,
  author       = {Pith},
  title        = {Pith review of: A Time-Symmetric Variational Reformulation of Nonrelativistic Quantum Mechanics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7ESUDPDC}},
  note         = {Machine review of arXiv:2512.22320}
}

abstract

Standard quantum mechanics relies on two distinct dynamical principles: unitary evolution and collapse. A mathematically self-contained variational framework is presented that replaces this dualism with a single principle, in which nonrelativistic Schr\"odinger dynamics are not postulated but emerge as an admissible optimality condition of a primal-dual boundary-value problem. By expressing the state in terms of hydrodynamic variables $(\rho,\mathbf{j})$ subject to a continuity constraint, it is shown that Fisher-information regularization yields the linear Schr\"odinger equation within the admissible single-valued variational class. Rather than evolving an initial state forward in time, the dynamics arise from minimizing a global action that connects the initial and final boundary constraints, with the selected solution corresponding to a specific hydrodynamic flow within an ensemble of admissible histories. A von Neumann pointer model illustrates how Born-rule statistics for recorded outcomes arise without introducing a separate collapse law. Within this formulation, quantum uncertainty is interpreted as effective randomness over boundary-compatible histories rather than as a fundamental stochastic postulate. The resulting framework provides a nonrelativistic proof of concept for how a single time-symmetric variational reformulation can recover key features of quantum theory.

Figures

Figures reproduced from arXiv: 2512.22320 by the authors.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

21 extracted references · 8 linked inside Pith

[1]

Setup of the Augmented F unctional We begin with the primal action functional defined over the hydrodynamic variablesρ(x, t) and j(x, t): Aprimal[ρ,j] = Z tf t0 Z Ω m|j|2 2ρ −V(x)ρ− ℏ2 8m |∇ρ|2 ρ d3x dt.(A1) The continuity constraint∂ tρ+∇ ·j= 0 is enforced via the dual variable (Lagrange multiplier) S(x, t). The augmented functional is: ˜A[ρ,j, S] =Aprim...
[2]

We requireδ S ˜A= 0: Z tf t0 Z Ω δS(∂ tρ+∇ ·j)d3x dt= 0.(A3) SinceδSis arbitrary, this enforces theContinuity Equation: ∂tρ+∇ ·j= 0.(A4)

Primal F easibility (V ariation w.r.t Dual) Varying with respect to the dual variableSrecovers the constraint. We requireδ S ˜A= 0: Z tf t0 Z Ω δS(∂ tρ+∇ ·j)d3x dt= 0.(A3) SinceδSis arbitrary, this enforces theContinuity Equation: ∂tρ+∇ ·j= 0.(A4)
[3]

Dual Optimality Condition I (V ariation w.r.t Current) We compute the variation with respect to the primal currentj. The relevant terms in ˜Aare the kinetic energy and the constraint coupling: δj ˜A= Z tf t0 Z Ω δj m|j|2 2ρ +S∇ ·(δj) d3x dt.(A5) 20 Using the identityδ(|j| 2) = 2j·δjand integrating the second term by parts (moving∇ontoS): δj ˜A= Z tf t0 Z ...
[4]

Dual Optimality Condition II (V ariation w.r.t Density) We vary with respect to the primal densityρ. This variation has four contributions: δρ ˜A=δ ρT+δ ρV+δ ρIF isher+δ ρC.(A8) 1.Kinetic Term (T):Treatingjas independent (fixed duringρ-variation): δρ m|j|2 2ρ =− m|j|2 2ρ2 δρ.(A9) Substituting the result from Eq. (A7) (|j| 2/ρ2 =|∇S| 2/m2): − m|j|2 2ρ2 =− ...
[5]

We compute the time evolution ofψusing the optimality conditions (A4) and (A19)

Equivalence to Schr¨ odinger Equation To verify equivalence, we define the complex primal variableψ= √ρeiS/ℏ. We compute the time evolution ofψusing the optimality conditions (A4) and (A19). Expanding the Schr¨ odinger operator: iℏ∂tψ=iℏ ∂tρ 2√ρ + i ℏ √ρ∂tS eiS/ℏ.(A20) Substituting∂ tρ=−∇ ·(ρ∇S/m) and∂ tS=−H cl −Q: iℏ∂tψ= −iℏ ∇ ·(ρ∇S/m) 2ρ + |∇S|2 2m +V+Q...
[6]

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Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.

[1] [1]

Setup of the Augmented F unctional We begin with the primal action functional defined over the hydrodynamic variablesρ(x, t) and j(x, t): Aprimal[ρ,j] = Z tf t0 Z Ω m|j|2 2ρ −V(x)ρ− ℏ2 8m |∇ρ|2 ρ d3x dt.(A1) The continuity constraint∂ tρ+∇ ·j= 0 is enforced via the dual variable (Lagrange multiplier) S(x, t). The augmented functional is: ˜A[ρ,j, S] =Aprim...

[2] [2]

We requireδ S ˜A= 0: Z tf t0 Z Ω δS(∂ tρ+∇ ·j)d3x dt= 0.(A3) SinceδSis arbitrary, this enforces theContinuity Equation: ∂tρ+∇ ·j= 0.(A4)

Primal F easibility (V ariation w.r.t Dual) Varying with respect to the dual variableSrecovers the constraint. We requireδ S ˜A= 0: Z tf t0 Z Ω δS(∂ tρ+∇ ·j)d3x dt= 0.(A3) SinceδSis arbitrary, this enforces theContinuity Equation: ∂tρ+∇ ·j= 0.(A4)

[3] [3]

Dual Optimality Condition I (V ariation w.r.t Current) We compute the variation with respect to the primal currentj. The relevant terms in ˜Aare the kinetic energy and the constraint coupling: δj ˜A= Z tf t0 Z Ω δj m|j|2 2ρ +S∇ ·(δj) d3x dt.(A5) 20 Using the identityδ(|j| 2) = 2j·δjand integrating the second term by parts (moving∇ontoS): δj ˜A= Z tf t0 Z ...

[4] [4]

Dual Optimality Condition II (V ariation w.r.t Density) We vary with respect to the primal densityρ. This variation has four contributions: δρ ˜A=δ ρT+δ ρV+δ ρIF isher+δ ρC.(A8) 1.Kinetic Term (T):Treatingjas independent (fixed duringρ-variation): δρ m|j|2 2ρ =− m|j|2 2ρ2 δρ.(A9) Substituting the result from Eq. (A7) (|j| 2/ρ2 =|∇S| 2/m2): − m|j|2 2ρ2 =− ...

[5] [5]

We compute the time evolution ofψusing the optimality conditions (A4) and (A19)

Equivalence to Schr¨ odinger Equation To verify equivalence, we define the complex primal variableψ= √ρeiS/ℏ. We compute the time evolution ofψusing the optimality conditions (A4) and (A19). Expanding the Schr¨ odinger operator: iℏ∂tψ=iℏ ∂tρ 2√ρ + i ℏ √ρ∂tS eiS/ℏ.(A20) Substituting∂ tρ=−∇ ·(ρ∇S/m) and∂ tS=−H cl −Q: iℏ∂tψ= −iℏ ∇ ·(ρ∇S/m) 2ρ + |∇S|2 2m +V+Q...

[6] [6]

Bohm, ”A Suggested Interpretation of the Quantum Theory in Terms of ’Hidden’ Variables,” Phys

D. Bohm, ”A Suggested Interpretation of the Quantum Theory in Terms of ’Hidden’ Variables,” Phys. Rev.85, 166 (1952)

1952

[7] [7]

Hydrodynamical formulation of quantum mechanics, K¨ ahler structure, and Fisher in- formation,

M. Reginatto, “Hydrodynamical formulation of quantum mechanics, K¨ ahler structure, and Fisher in- formation,” arXiv:quant-ph/9909065 (1998)

arXiv 1998

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An Information-Theoretic Link Between Spacetime Symmetries and Quantum Linear- ity,

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