Relativistic quantum heat engine from uncertainty relation standpoint

Goutam Paul; Pritam Chattopadhyay

REVIEW 4 major objections 4 minor 70 references

Relativistic quantum heat engine from uncertainty relation standpoint

T0 review · 4 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Work and efficiency of a relativistic Stirling engine are bounded by the thermal position–momentum uncertainty relation.

desk verdict The claimed uncertainty-based efficiency bounds for a relativistic quantum heat engine collapse under the paper's own approximations. read the letter →

arxiv 1908.06819 v2 pith:D572HAD4 submitted 2019-08-19 quant-ph

Pritam Chattopadhyay , Goutam Paul This is my paper · ORCID

classification quant-ph

keywords relativisticquantumheatengineStirlingcyclethermaluncertaintyrelationposition-momentuminfinitepotentialwellKlein-Gordonequationefficiencyboundsthermodynamics

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 sets out to show that a relativistic quantum Stirling engine—a particle in a one-dimensional potential well running between two heat baths—has its work output and efficiency fixed, and bounded, by the thermal uncertainty relation for the particle's position and momentum. The authors build the cycle from two isothermal processes (inserting and removing a barrier in the well) and two constant-width processes, using the Klein–Gordon spectrum of the relativistic particle. They then write the partition function, free energy, entropy, work, and efficiency in terms of the thermal variances $(\Delta X_T)^2$ and $(\Delta P_T)^2$, so that bounds on the sum uncertainty $\Delta X_T+\Delta P_T$ become bounds on efficiency. If the construction holds, an experimenter could quote the engine's efficiency range from position–momentum uncertainties alone, without measuring energy-level populations. The paper also reads the link as evidence that fundamental quantum incompatibility can directly control thermodynamic performance.

What carries the argument

The load-bearing object is the thermal sum uncertainty relation for a Klein–Gordon particle in a one-dimensional well of length $2L$: $\Delta X_T+\Delta P_T$, assembled from the thermal variances in Eqs. (11) and (12). The partition function $Z\approx \frac{1}{2}\sqrt{\pi/(\beta\alpha)}e^{-\beta mc^2}$ with $\alpha=\pi^2\hbar^2/[2m(2L)^2]$ converts the level-dependent position and momentum spreads into temperature-dependent ones, and Eq. (14) gives the lower bound $\Delta X_T+\Delta P_T\ge \hbar/2$. The reverse uncertainty inequality (20), obtained by squaring the Dunkl–Williams inequality, supplies the upper bound (22). The Stirling cycle then enters through two isothermal barrier manipulations at temperatures $T_1$ and $T_2$; the efficiency formula (30) combines the two uncertainty sums $f(\Delta X_{T_1}+\Delta P_{T_1})$ and $g(\Delta X_{T_2}+\Delta P_{T_2})$ with the partition-function logarithms $\ln(Z_B/Z_A)$ and $\ln(Z_D/Z_C)$, which is how uncertainty bounds are converted into efficiency bounds.

What would settle it

Compute the exact partition-function ratios $Z_B/Z_A$ and $Z_D/Z_C$ for the single-particle box, or keep the next order in $\alpha\beta$; if the ratios approach 1 in the regime where $U_A=U_B$ and $U_C=U_D$, then Eqs. (29) and (30) give $W=0$ and an indeterminate $\eta$, and the claimed efficiency bounds disappear. A numerical check of whether Fig. 5 can be reproduced from Eq. (30) with the stated $\beta$ values and masses would settle the same question.

Watch

Extended reading notes

Core claim

The central claim is that for a relativistic particle in a box of width $2L$, the Stirling engine's work and efficiency are determined by the thermal sum uncertainty $\Delta X_T+\Delta P_T$. Starting from the Klein–Gordon solution in the Feshbach–Villars form and the partition function $Z\approx \frac{1}{2}\sqrt{\pi/(\beta\alpha)}e^{-\beta mc^2}$, the paper defines thermal variances from Eqs. (11) and (12) and the sum relation (14). It then inverts this relation to express $Z$, the Helmholtz free energy, the entropy, the total work $W$ of Eq. (29), and the efficiency $\eta$ of Eq. (30) as functions of the uncertainty sums $f$ and $g$ at the two bath temperatures. Because $f$ and $g$ are bounded below by the sum-uncertainty inequality (17) and above by the reverse inequality (20) derived from the Dunkl–Williams inequality, the paper obtains upper and lower bounds on the efficiency. In the authors' words, they are 'able to determine the bounds (the upper and the lower bounds) of the efficiency of the heat engine through the thermal uncertainty relation.'

Load-bearing premise

The load-bearing assumption is that the high-temperature continuous approximation for the partition function can be used to set the internal energies equal on the two isothermal branches while the logarithmic terms $\ln(Z_B/Z_A)$ and $\ln(Z_D/Z_C)$ in Eqs. (25) and (27) remain different from zero; under the same approximation those ratios are one, so the stated work and efficiency would vanish.

Editorial extensions

If this is right

One can state the efficiency range of this relativistic engine using only $\Delta X_T+\Delta P_T$ at the two bath temperatures; no energy-level measurements or population readouts are needed.
The upper efficiency bound decreases monotonically as temperature grows, and the lower and upper bounds converge at large uncertainty, so the model predicts that larger quantum spread degrades the conversion of heat into work.
Helmholtz free energy and entropy become functions of the same uncertainty sum; entropy increases with uncertainty, giving a direct thermodynamic signature of quantum spread.
The dictionary developed for the box is meant to transfer to other cycles and working substances, including quantum phase transitions and relativistic condensed-matter-inspired engines.

Reading between the lines

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

A general recipe is latent in the paper: any working substance whose partition function can be re-expressed through variances of two noncommuting observables will produce efficiency bounds without energy measurements. Testing the recipe on a harmonic oscillator or a two-level system, where partition functions are known exactly, would show whether the box example is special.
The entropy–uncertainty link suggests a possible route toward an entanglement measure for mixed relativistic states, an open problem the paper points to; verifying it would require a separate bipartite calculation.
An experiment with a single trapped ion or ultracold atom in a tunable box potential could measure the two variances and compare the measured efficiency band with Eq. (30), giving a direct test of the uncertainty bound.

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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,

Short version: the paper's headline claim—bounds on a relativistic quantum engine's efficiency from thermal uncertainty—doesn't hold up under its own equations. The derivation mixes approximations in a way that zeroes out the work, and there's a dimension mismatch that makes the central object ill-defined.

The authors deserve credit for a real idea: connecting thermodynamic variables to position-momentum variances is a legitimate question, and they correctly note that product uncertainty can vanish for eigenstates of one observable, so they work with sum uncertainty. The relativistic particle-in-a-box setup with a KG equation is also a reasonable touch. But the execution falls apart.

Three load-bearing defects. First, Eq. (14) adds ΔX_T and ΔP_T directly. One has units of length, the other momentum. That is not a quantity you can exponentiate, log, or write a partition function in terms of, yet Eqs. (15)-(16) and everything after uses ΔX_T+ΔP_T. Second, the supposed upper bound Eq. (21) is not a bound derived from any uncertainty principle: the RHS is essentially the actual ΔX²+ΔP² plus a positive term, so the inequality is vacuously true and gives no physical information. Third, and most damaging, the work formula in Sec. III D is internally inconsistent. They set U_A=U_B using the high-temperature continuum partition function, but then keep ln(Z_B/Z_A) nonzero in Eq. (25). Under the same approximation, Z_B=Z_A, so that term is zero, Q_AB=0, and similarly Q_CD=0. Then the only heat exchanges cancel and W=0. The alternative—using the discrete sums—invalidates the U_A=U_B equality. Either way Eq. (29) does not follow.

The newness is also thinner than claimed. Ref. [7] already treats a relativistic Stirling engine in a 1D well, and Ref. [24] does the level-degeneracy version non-relativistically. The uncertainty rephrasing is an algebraic inversion of the partition function, not an independent constraint.

This is beyond repair by minor edits. The central result is unsupported and the paper should be desk-rejected. If the authors return with a dimensionally consistent, approximation-consistent treatment, the core question might be worth another look.

Referee Report

4 major / 4 minor

Summary. The paper proposes a relativistic quantum Stirling engine whose working substance is a particle in a one-dimensional infinite potential well of length 2L, described by the Klein-Gordon equation. The cycle has four stages: isothermal insertion of a central barrier at temperature T1, cooling at fixed width to T2, isothermal removal of the barrier at T2, and heating back to T1. The authors compute thermal variances of position and momentum, ΔX_T and ΔP_T, from the partition function of the relativistic particle, invert the relation to express the partition function in terms of ΔX_T + ΔP_T (Eq. (15)), and thereby rewrite the internal energy, free energy, and entropy in terms of the uncertainty sum. They then derive lower and upper bounds on the sum uncertainty (Eqs. (17)-(22)) and claim corresponding bounds on the work and efficiency of the engine (Eqs. (29)-(30)), concluding that the efficiency bounds of the relativistic heat engine follow through the thermal uncertainty relation.

Significance. If the derivation were correct, connecting the work and efficiency of a relativistic quantum Stirling engine to position-momentum uncertainty would be a useful contribution to quantum thermodynamics, and the topic is timely. The physical setup and the use of the Feshbach-Villars formalism for the relativistic box are reasonable, and the reference list covers the relevant engine literature. These strengths do not outweigh the technical problems. The cycle calculation is internally inconsistent (Sec. III D), the thermal variance formulas are incorrect as written (Eqs. (11)-(12)), the central equations add quantities of different physical dimension (Eqs. (14)-(16), (21)-(22)), and the claimed upper bound in Eq. (21) reduces to the trivial variance-versus-second-moment inequality. The analytic results are not supported by an independent numerical check, and the figures do not give sufficient parameters for reproduction. The central claim is therefore not supported.

major comments (4)

[Sec. III D (Eqs. (24)-(30))] The central result is invalidated by an internal consistency error in the treatment of the high-temperature approximation. Equation (24) sets U_A = U_B = 1/(2β1) + mc^2 using the continuum approximation Z ≈ (1/2)√(π/(βα)) e^{−βmc^2} of Eq. (9). The same approximation applied to the partitioned well gives Z_B = 2·(1/2)√(π/(β·4α)) e^{−βmc^2} = Z_A, and identically Z_D = Z_C, because inserting the barrier replaces α by 4α and adds the factor 2 for the two chambers. Hence ln(Z_B/Z_A) = ln(Z_D/Z_C) = 0, so Q_AB and Q_CD vanish in Eqs. (25) and (27), and the remaining exchanges Q_BC = U_C − U_B and Q_DA = U_A − U_D sum to zero; the total work in Eq. (29) is W = 0 and the efficiency in Eq. (30) is η = 0. If instead the exact discrete sums are retained, then U_A ≠ U_B and Eq. (24) is invalid, so Eq. (29) does not follow from the preceding derivation. Either way, the claimed bounds on efficiency from the thermal uncertainty relation are not supported by the calculation as written.
[Eqs. (13)-(14) and (17)-(22)] The central quantities are dimensionally inconsistent. Since ℏ and c are kept explicit throughout, ΔX_T has the dimension of length while ΔP_T has the dimension of momentum, yet they are added directly in Eq. (14) and their squares are added in Eqs. (18) and (22); the same problem affects ΔX + ΔP in Eqs. (17)-(21) and the combination ΔX_T + ΔP_T + C_T in Eqs. (15)-(16), where C_T itself mixes a length with √(2mc). No natural-unit convention is stated that would make these sums meaningful. In addition, the right-hand side of the claimed upper bound in Eq. (21) is exactly ⟨X^2⟩ + ⟨P^2⟩ for the n-th eigenstate as listed in Eq. (8), so the bound is the trivial inequality ΔX^2 + ΔP^2 ≤ ⟨X^2⟩ + ⟨P^2⟩; the Dunkl-Williams chain in Eqs. (19)-(20) cannot produce this bound because Δ(X − P) is not dimensionally defined for position and momentum operators.
[Eqs. (11)-(12)] The thermal variance in Eq. (11) is incorrect by definition. In a canonical ensemble, (ΔX)^2_T = ⟨X^2⟩_T − (⟨X⟩_T)^2 with ⟨X⟩_T = Z^{−1}Σ_n ⟨ψ_n|X|ψ_n⟩e^{−βE_n}; the second line of Eq. (11) instead subtracts the unsquared thermal average Z^{−1}Σ_n ⟨ψ_n|X|ψ_n⟩e^{−βE_n} from ⟨X^2⟩_T, which is neither the variance nor dimensionally consistent. Equation (12) has a related problem: it replaces ⟨n^2⟩_T by n̄^2 and writes +2mc^2 where the per-level momentum expectation in Eq. (8) has +2m^2c^2, so (ΔP)^2_T is dimensionally wrong as well. Since Eqs. (13)-(15), (22), and (29)-(30) are all constructed from these variances, the error propagates into the central results.
[Secs. III B and III D (Eqs. (15), (29)-(30))] The claim that the efficiency bounds follow from the uncertainty principle is largely an algebraic rearrangement. The variances ΔX_T and ΔP_T are computed from the same eigenstates and partition function Z that define the heat exchanges; Eq. (15) then inverts this dependence to express Z in terms of ΔX_T + ΔP_T + C_T, and the functions f and g in Eqs. (29)-(30) are the same combination evaluated at T1 and T2. Both sides are functions of the same parameters (β, L, m), so the resulting bounds on W and η restate the temperature dependence of the partition-function ratios rather than constituting an independent restriction from position-momentum uncertainty. Correspondingly, the abstract's assertion that the efficiency is bounded through the thermal uncertainty relation without performing measurements overstates what is shown, since computing ΔX_T and ΔP_T requires the same spectral and thermal information as computing Z.

minor comments (4)

[Figs. 1-2] The vertical axis of Figs. 1 and 2 is labeled 'Sum uncertainty (X^2+P^2)', but Eq. (14) defines ΔX_T + ΔP_T and the text refers to the sum of uncertainties; the label, the plotted quantity, and Eq. (22) should be reconciled, and the values of m and φ+(p) used in the plots should be stated.
[Fig. 5] The efficiency plot in Fig. 5 carries an unexplained vertical offset '+1.637e3', and the particle mass and the bath temperatures T1 and T2 used for the plot are not given, so the displayed bounds cannot be reproduced from the manuscript.
[Sec. III C] The statement that the product uncertainty for the system 'results to zero' is unexplained and appears incorrect: for every eigenstate of a particle in a box, ΔxΔp is nonzero and satisfies the Heisenberg bound, so the motivation for switching from the product to the sum uncertainty should be clarified.
[Secs. II A and III D (Eqs. (6), (23)-(30))] The 'relativistic' engine is built on the spectrum E_n ≈ mc^2 + n^2π^2ℏ^2/(2m(2L)^2), which is the non-relativistic spectrum plus a constant rest-energy shift; the constant cancels in the partition-function ratios and internal-energy differences, so the work and efficiency derived in Sec. III D are identical to those of the non-relativistic engine under the same approximation, and the specifically relativistic content of the engine results should be stated explicitly.

Circularity Check

1 steps flagged · score 8.0 of 10

The central work/efficiency formulas are a repackaging of the partition function: Eq. (15) defines Z from the same thermal variances used to define those variances, so Eqs. (29)-(30) reduce to the original Stirling expression by construction.

self definitional [Sec. III A, Eqs. (11)-(12); Sec. III B, Eq. (15); Sec. III D, Eqs. (29)-(30)]
"The partition function [38] of the system, Z, in terms of the variance by using Eq. (14) for replacing n in Eq. (9) is expressed as Z = π/2 e^{−βmc²}[16c√(2mc)/(π³ℏ²)(ΔX_T + ΔP_T + C_T)]^{1/2} ... W ≡ Q_AB + Q_BC + Q_CD + Q_DA = 8L²α/(ℏ²π²)[ f ln(Z_B/Z_A) + g ln(Z_D/Z_C)] , where f = [16c√(2mc)/(π³ℏ²)(ΔX_T1 + ΔP_T1 + C_T1)]"

The thermal uncertainties ΔX_T and ΔP_T are not independent inputs: they are computed in Eqs. (11)-(12) from the same Boltzmann-weighted eigenstates and partition function Z used for the thermodynamics. Eq. (15) is the algebraic inverse of that construction, making Z a one-to-one function of ΔX_T+ΔP_T+C_T. Substituting this identity into the standard Stirling heat formula Q_AB = k_B T1 ln(Z_B/Z_A), Q_CD = k_B T2 ln(Z_D/Z_C) gives Eq. (29), with f and g equal to the same bracket; the log ratios are untouched. The bounds on f and g from Eqs. (17) and (22) are therefore bounds on a repackaged form of Z, so the claimed determination of W and η through the thermal uncertainty relation is equivalent by construction to the original partition-function calculation, not an independent derivation.

full rationale

The paper's headline claim is that efficiency bounds follow from the thermal uncertainty relation, but the thermal uncertainty relation itself is derived from the equilibrium partition function of the same potential-well model. Eq. (15) then solves Z in terms of ΔX_T+ΔP_T, and Eqs. (29)-(30) insert that expression (through f and g) into the standard Stirling work formula. This is an exact change of variables, not a derivation from an independent principle: every 'uncertainty' quantity is a thermal expectation value over the same E_n and Z that define the heat engine. The external inequalities used for bounds (Eqs. (17), (20), (22)) do constrain the repackaged variable, but the underlying freedom/choice of the engine is still entirely parametrized by Z. Separately, the derivation has a serious internal inconsistency that is not itself circularity: Eq. (24) sets U_A=U_B and U_C=U_D using the same continuum approximation that also gives Z_B=Z_A and Z_D=Z_C, which would make the logarithm terms in Eqs. (25) and (27) vanish and W=0. That flaw makes the central result numerically unsupported, but the circularity score above is based on the definitional relabeling in Eqs. (15) and (29).

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central derivation rests on the KG-in-a-box energy spectrum, the continuous partition-function approximation, and two uncertainty inequalities that are used in nonstandard forms. The paper provides no numerical fitting and introduces no new entities, so the free-parameter and invented-entity lists are empty; the main burden is carried by unverified domain assumptions and the internally inconsistent use of the partition function.

assumptions (5)

domain assumption The Klein-Gordon equation with Feshbach-Villars formalism and a position-dependent mass m(x) correctly describes a relativistic particle in a 1D infinite well without Klein paradox.
Sec. II A, Eqs. (3)-(6): the energy spectrum and all variance formulas used later depend on this prescription.
domain assumption The partition function can be replaced by the continuous integral Z ≈ (1/2)√(π/(βα)) e^{-βmc²}.
Sec. III A, Eq. (9): requires αβ << 1, but this is not verified for the lengths (0.1-0.5 Å) and temperatures (40-320 K) used in the figures.
ad hoc to paper The sum uncertainty lower bound of Eq. (17) is a valid theorem applicable to X and P.
Sec. III C, Eq. (17): the relation as written is not a recognized sum-uncertainty theorem and is asserted without derivation; all subsequent lower bounds rely on it.
ad hoc to paper The Dunkl-Williams inequality can be applied in the form of Eq. (19) to derive the reverse uncertainty bound Eq. (20).
Sec. III C, Eqs. (19)-(20): the algebraic and dimensional validity of the applied form is not established, and the resulting Eq. (21) is not actually a bound on ΔX²+ΔP².
domain assumption Quasi-static insertion and removal of an infinite barrier gives a double-well spectrum of doubly degenerate E_{2n} levels and a reversible Stirling cycle.
Sec. III D, Eq. (23): the work and efficiency calculation depends on this spectral assignment and reversibility.

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Pith. "Pith review of Relativistic quantum heat engine from uncertainty relation standpoint." pith.science (2026). https://pith.science/paper/D572HAD4

@misc{pith2026190806819,
  author       = {Pith},
  title        = {Pith review of: Relativistic quantum heat engine from uncertainty relation standpoint},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/D572HAD4}},
  note         = {Machine review of arXiv:1908.06819}
}

read the original abstract

Established heat engines in quantum regime can be modeled with various quantum systems as working substances. For example, in the non-relativistic case, we can model the heat engine using infinite potential well as a working substance to evaluate the efficiency and work done of the engine. Here, we propose quantum heat engine with a relativistic particle confined in the one-dimensional potential well as working substance. The cycle comprises of two isothermal processes and two potential well processes of equal width, which forms the quantum counterpart of the known isochoric process in classical nature. For a concrete interpretation about the relation between the quantum observables with the physically measurable parameters (like the efficiency and work done), we develop a link between the thermodynamic variables and the uncertainty relation. We have used this model to explore the work extraction and the efficiency of the heat engine for a relativistic case from the standpoint of uncertainty relation, where the incompatible observables are the position and the momentum operators. We are able to determine the bounds (the upper and the lower bounds) of the efficiency of the heat engine through the thermal uncertainty relation.

Figures

Figures reproduced from arXiv: 1908.06819 by the authors.

**Figure 2.** FIG. 2. This shows the variation of sum uncertainty relation for dif [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** FIG. 3. The variation of entropy from Eq. (16) for different temper [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: FIG. 4. The figure constitutes of four stages of the Stirling cycle of [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

**Figure 5.** Figure 5: FIG. 5. The efficiency bound for a relativistic model of heat engine. [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]

Discussion (0). Continue with ORCID to comment.

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(13) Along with the product uncertainty relation, we also evaluate the thermal sum uncertainty relation of the position and the momentum operator for the potential well problem. Here, we have calculated the sum uncertainty as we are concerned about the fact that the product uncertainty relation is unable to cap- ture the uncertainty of the incompatible ob...

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(14) 4 Fig.1 describes the variation of uncertainty with respect to 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 Length L in A 1.67090 1.67092 1.67094 1.67096 1.67098 1.67100 1.67102 1.67104Sum uncertainty ( X2 + P2) Sum uncertainty relation for different temperature temp = 80K temp = 320K FIG. 1. The variation of sum uncertainty relation for different te...

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