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arxiv: 2606.09000 · v1 · pith:HLY3UDECnew · submitted 2026-06-08 · 🪐 quant-ph

The Transformation-Response Framework: An Operational Reformulation of Quantum Mechanics

Meng-Jun Hu This is my paper

Pith reviewed 2026-06-27 16:36 UTC · model grok-4.3

classification 🪐 quant-ph
keywords operational quantum mechanicscharacteristic functionGNS constructionBochner theoremFeynman path integralpositive-definitenessgroup automorphismsbackground independence
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The pith

Positive-definiteness of a response characteristic function alone derives the full quantum formalism including Hilbert space and path integrals.

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

The paper establishes that a quantum state is the collection of complex values χ(g) recording how a system responds to each physical transformation g drawn from its local group G. The sole assumption is that this characteristic function must be positive-definite, which prevents any superposition of transformations from producing negative probability. From that condition the standard apparatus follows: the GNS construction supplies the Hilbert space, Bochner's theorem supplies the Born rule, group automorphisms supply the Schrödinger equation, and a Trotter-product limit supplies the Feynman path integral. The reformulation is background-independent, treats time as one coordinate on a subgroup, and introduces product order positivity as a new constraint.

Core claim

A quantum state is defined as the catalog of responses χ(g) obtained from interference experiments for every transformation g in the system's local group G; the single postulate that χ is positive-definite is sufficient to recover the Hilbert space via the GNS construction, the Born rule via Bochner's theorem, the dynamics via group automorphisms, and the Feynman path integral as a Trotter limit.

What carries the argument

The characteristic function χ on the local group G, whose positive-definiteness encodes the physical consistency requirement that no superposition of transformations yields negative probability.

If this is right

  • The Hilbert space, Born rule, and Schrödinger dynamics emerge as mathematical consequences rather than independent postulates.
  • Time appears only as a coordinate along a one-parameter subgroup of G rather than a fundamental external parameter.
  • Product order positivity is identified as an extra physical constraint that may produce observable deviations from standard quantum theory.
  • The same construction applies uniformly once a local group of transformations is specified for any system.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The framework could be tested by searching for violations of product order positivity in multi-particle interference setups that remain within the domain of ordinary quantum mechanics.
  • If the local group is taken to be infinite-dimensional, the same logic might generate quantum field theory without separate postulates.
  • Interference data alone would suffice to reconstruct the state, removing any need to postulate a wave function or density operator first.

Load-bearing premise

The responses χ(g) for transformations in the local group G already contain the complete physical state with no further structure required.

What would settle it

An experiment that measures a system's response function χ and finds it violates positive-definiteness while all observable predictions still match standard quantum mechanics.

read the original abstract

We present the transformation-response framework, an operational reformulation of quantum mechanics. A quantum state is not a Hilbert space object but the catalog of a system's responses to all physical transformations: for each operation $g$ from the system's local group $G$, an interference experiment gives a complex value $\chi(g)$. The collection $\{\chi(g): g\in G \}$ is the characteristic function and defines the state. The only postulate is that $\chi$ is positive-definite, encoding the requirement that no superposition of transformations yields negative probability. From this single assumption, the entire standard formalism is derived: Hilbert space via GNS construction, Born rule via Bochner theorem, Schr\"odinger equation from group automorphisms, and especially Feynman path integral as a Trotter limit. The framework is background-independent and time-neutral: time is a coordinate along a one-parameter subgroup of $G$. It also reveals a new physical constraint, product order positivity, which may lead to testable predictions. The framework provides a unified, economical, and falsifiable foundation for quantum theory rooted in operational primitives.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript introduces the transformation-response framework as an operational reformulation of quantum mechanics. Quantum states are defined as catalogs of responses χ(g) to transformations g drawn from a system's local group G, obtained via interference experiments. The sole postulate is that χ is positive-definite. From this, the paper claims to recover the full standard formalism: Hilbert space via the GNS construction, the Born rule via Bochner's theorem, the Schrödinger equation from group automorphisms, and the Feynman path integral as a Trotter limit. The framework is presented as background-independent and time-neutral, with time as a coordinate on a one-parameter subgroup of G, and it introduces an additional constraint termed product order positivity.

Significance. If the derivations can be shown to proceed from the positive-definiteness postulate without the operational definitions of G or the interference-to-χ map already embedding Hilbert-space or probability structure, the work would supply an unusually economical and falsifiable foundation for quantum theory. The explicit use of GNS and Bochner constructions, together with the identification of a potential new constraint (product order positivity), would constitute a genuine contribution to the operational foundations literature.

major comments (2)
  1. [Abstract / definition of G] Abstract and the section defining the local group G: the claim that positive-definiteness of χ is the sole postulate requires an explicit demonstration that the multiplication table, involution, and the set of transformations counted as physical in G are fixed by operational primitives alone and do not already encode non-commutativity or the Lie-algebra structure presupposed by standard QM; otherwise the subsequent GNS and Bochner steps are not derivations from a single axiom.
  2. [Derivation of Born rule] The paragraph deriving the Born rule via Bochner's theorem: the mapping from interference-experiment outcomes to the values χ(g) must be shown to be free of any inner-product or probability interpretation; if that mapping already assumes Born-rule probabilities, the appeal to Bochner merely recovers what was inserted by hand.
minor comments (1)
  1. Notation for the characteristic function χ(g) and the group G should be introduced with an explicit table or diagram showing which operational procedures fix the group operation and which fix the numerical values of χ.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and valuable feedback on our manuscript. We address each of the major comments below, offering clarifications on the operational aspects of the framework and committing to revisions that will strengthen the presentation of the derivations.

read point-by-point responses

  1. Referee: [Abstract / definition of G] Abstract and the section defining the local group G: the claim that positive-definiteness of χ is the sole postulate requires an explicit demonstration that the multiplication table, involution, and the set of transformations counted as physical in G are fixed by operational primitives alone and do not already encode non-commutativity or the Lie-algebra structure presupposed by standard QM; otherwise the subsequent GNS and Bochner steps are not derivations from a single axiom.

    Authors: The group G is constructed purely from operational considerations: it consists of all transformations that can be physically applied to the system and whose effects can be measured through interference experiments. The multiplication operation corresponds to the sequential composition of these transformations, which is an operational primitive observed by applying one after the other and recording the net response. The involution is the inverse transformation, identified operationally by the fact that applying g followed by its inverse returns the system to the initial state, yielding χ(e) = 1. The set of included transformations is those for which such experiments are possible, without any prior assumption of algebraic properties like non-commutativity. Any such structure arises as a consequence of the positive-definiteness postulate when applying the GNS construction. We will revise the manuscript to include a more detailed operational construction of G, perhaps with an illustrative example, to make this explicit. revision: yes

  2. Referee: [Derivation of Born rule] The paragraph deriving the Born rule via Bochner's theorem: the mapping from interference-experiment outcomes to the values χ(g) must be shown to be free of any inner-product or probability interpretation; if that mapping already assumes Born-rule probabilities, the appeal to Bochner merely recovers what was inserted by hand.

    Authors: The value χ(g) is obtained directly from the outcomes of interference experiments as the normalized complex amplitude corresponding to the visibility and phase of the interference fringes when comparing the untransformed and transformed paths. This extraction relies solely on classical measurement of intensity patterns and does not invoke any probabilistic interpretation or inner product at this stage. The positive-definiteness ensures that these values can be represented as expectations in a Hilbert space via GNS, and Bochner's theorem then provides the measure that corresponds to the probability distribution in that representation. We agree that the current paragraph could be clearer on this point and will revise it to explicitly describe the operational mapping from experimental data to χ(g) without assuming Born's rule. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation applies external theorems to operationally defined inputs

full rationale

The paper posits that χ positive-definiteness on the operationally defined group G is the sole postulate, then invokes standard external theorems (GNS construction for Hilbert space, Bochner theorem for Born rule, group automorphisms for dynamics, Trotter limit for path integral). These are independent mathematical results not derived from or equivalent to the paper's inputs by construction. No self-citations are load-bearing for the central claims, no parameters are fitted then renamed as predictions, and G is presented as given by physical transformations without the paper showing that its definition already encodes the target structures. The framework is self-contained against external benchmarks and does not reduce the derived formalism to a relabeling of its premises.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 2 invented entities

The framework rests on the assumption that every quantum state is fully captured by a positive-definite function on a transformation group and that standard theorems (GNS, Bochner) can be applied directly without additional physical input.

axioms (1)
  • domain assumption The characteristic function χ is positive-definite, encoding that no superposition of transformations yields negative probability.
    Stated as the only postulate in the abstract.
invented entities (2)
  • Characteristic function χ(g) no independent evidence
    purpose: Catalog of a system's responses to all transformations g in the local group G that defines the quantum state.
    Introduced as the fundamental object replacing the Hilbert-space vector.
  • Local group G no independent evidence
    purpose: The set of all physical transformations applicable to the system.
    Assumed to exist and to carry the structure needed for GNS and Bochner constructions.

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discussion (0)

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Works this paper leans on

43 extracted references · 1 linked inside Pith

  1. [1]

    This axiomatic structure has enjoyed extraordinary empi rical success

    INTRODUCTION Standard quantum mechanics, as formalized by von Neumann in 1932 [1], rests upon a set of logically independent postulates: a quantum state is a ray i n a Hilbert space, observables are self- adjoint operators, the Born rule assigns measurement proba bilities, and the Schr¨ odinger equation governs time evolution [2]. This axiomatic structure...

    1932

  2. [2]

    state itself

    THE OPERA TIONAL ORIGIN OF QUANTUM STA TES: MOTIV A TION ANDPOSTULATE In this section, we formulate the physical motivation for th e TRF and state its core postulate. The guiding principle is that a quantum state is not an entitythat we inspect directly. It is something defined through the operations we perform and the responses w e register. Our task is t...

  3. [3]

    The mathematical engine is the Gelfand–Naimark–Segal construc- tion [ 15]

    RECONSTRUCTING THE HILBERT SPACE: THE GNS CONSTRUCTION Now we demonstrate how the entire Hilbert space structure of quantum mechanics emerges from the TRF postulate alone. The mathematical engine is the Gelfand–Naimark–Segal construc- tion [ 15]. The physical result is that every admissible response cat alogχautomatically generates a Hilbert space Hχ, a u...

  4. [4]

    zero superposition

    Thus, the positive-definiteness postulate guarantees th at ⟨·, ·⟩χis a genuine pre-inner product on A. A pre-inner product may admit non-zero vectors with vanish ing norm. These null vectors constitute a subspace Nχ= {a ∈ A|⟨a, a⟩χ= 0}. Physically, a superposition a with ⟨a, a⟩χ= 0 is one for which the operation superposition produces no detec table respon...

  5. [5]

    Why the probability should be given by the modulus squared of a complex amplitude, rather than any other function, is left unexplained

    PROBABILITY WITHOUT POSTULATES: THE BORN RULE FROM BOCHN ER THEOREM In standard quantum mechanics, the Born rule is an independe nt postulate: given a state vec- tor |ψ⟩ and a measurement associated with a self-adjoint operator ˆA with spectral decomposition ˆA = ∑ k ak|ak⟩⟨ak|, the probability of obtaining outcome ak is P(ak) = ⟨ψ|ak⟩⟨ak|ψ⟩ = |⟨ak|ψ⟩|2. ...

  6. [6]

    The time parameter t and the Planck constant ℏ both appear as fun- damental ingredients whose origins are left unexplained

    DYNAMICS WITHOUT EXTERNEL TIME: THE SCH ¨ODINGER EQUA TION AS GROUP AUTOMORPHISM In standard quantum mechanics, the Schr¨ odinger equationiℏd|ψ⟩/ dt = ˆH|ψ⟩ is postulated as an independent dynamical law. The time parameter t and the Planck constant ℏ both appear as fun- damental ingredients whose origins are left unexplained. W hen spacetime itself become...

  7. [7]

    NONCOMMUTA TIVITY FROM GROUP STRUCTURE In the TRF, noncommutativity follows directly from the stru cture of the symmetry group G. When the group of physical operations is non-abelian, the St one generators of its one-parameter subgroups necessarily fail to commute, and their commutation relations are completely determined by the Lie algebra of G. The deri...

  8. [8]

    THE PA TH INTEGRAL AS TROTTER LIMITS OF THE CHARACTERISTIC FUNCTION In standard quantum mechanics, the Feynman path integral is introduced as an independent formulation of the theory [ 6]. In the TRF, however, the Feynman path integral can be deriv ed as the continuum limit of the characteristic function itself, via the Trotter product formula applied to ...

  9. [9]

    If the framework did only this, it would be an eleg ant reformulation

    PRODUCT ORDER POSITIVITY: A POSSIBLE NEW PHYSICAL CONSTR AINT The TRF has so far derived the entire standard apparatus of qu antum mechanics from a single postulate. If the framework did only this, it would be an eleg ant reformulation. However, the operational logic of the framework naturally supports a structure that standard quantum mechanics does not ...

  10. [10]

    DISCUSSION AND CONCLUSION The TRF is not actually limited to quantum mechanics. It is a g eneral operational logic for physical theories, grounded in a single principle that phys ical state is defined as response cata- 28 log to all physical transformation operations. The mathema tical machinery, the GNS construc- tion, Bochner theorem, the commutation the...

  11. [11]

    von Neumann, Mathematical Foundations of Quantum Mec hanics (Princeton University Press, 1955)

    J. von Neumann, Mathematical Foundations of Quantum Mec hanics (Princeton University Press, 1955)

    1955

  12. [12]

    P . A. M. Dirac, The Principles of Quantum Mechanics (Clar endon Press, 1930)

    1930

  13. [13]

    B. S. DeWitt, Quantum theory of gravity. I. The canonical theory, Phys. Rev. 160, 1113 (1967)

    1967

  14. [14]

    J. B. Hartle and S. W. Hawking, Wave function of the univer se, Phys. Rev. D 28, 2960 (1983)

    1983

  15. [15]

    Rovelli, Quantum Gravity (Cambridge University Pres s, 2004)

    C. Rovelli, Quantum Gravity (Cambridge University Pres s, 2004)

    2004

  16. [16]

    R. P . Feynman, Space–time approach to non-relativistic quantum mechanics, Rev. Mod. Phys. 20, 367 (1948). 30

    1948

  17. [17]

    G. W. Mackey, The Mathematical Foundations of Quantum Me chanics (W. A. Benjamin, 1963)

    1963

  18. [18]

    Ludwig, Foundations of Quantum Mechanics I (Springer , 1985)

    G. Ludwig, Foundations of Quantum Mechanics I (Springer , 1985)

    1985

  19. [19]

    Hardy, Quantum theory from five reasonable axioms, arX iv:quant-ph/0101012 (2001)

    L. Hardy, Quantum theory from five reasonable axioms, arX iv:quant-ph/0101012 (2001)

    Pith/arXiv arXiv 2001

  20. [20]

    Chiribella, G

    G. Chiribella, G. M. D’Ariano, and P . Perinotti, Inform ational derivation of quantum theory, Phys. Rev. A 84, 012311 (2011)

    2011

  21. [21]

    Rovelli, Relational quantum mechanics, Int

    C. Rovelli, Relational quantum mechanics, Int. J. Theo r. Phys. 35, 1637 (1996)

    1996

  22. [22]

    P . W. Bridgman, The Logic of Modern Physics (Macmillan, 1927)

    1927

  23. [23]

    M. A. Nielsen and I. L. Chuang, Quantum Computation and Q uantum Information: 10th Anniversary Edition (Cambridge University Press, 2010)

    2010

  24. [24]

    Birkho ffand J

    G. Birkho ffand J. von Neumann, The logic of quantum mechanics, Ann. Math . 37, 823 (1936)

    1936

  25. [25]

    Gelfand and M

    I. Gelfand and M. Naimark, On the imbedding of normed rin gs into the ring of operators in Hilbert space, Mat. Sb. 12, 197 (1943)

    1943

  26. [26]

    W. F. Stinespring, Positive functions on C*-algebras, Proc. Am. Math. Soc. 6, 211 (1955)

    1955

  27. [27]

    Bochner, V orlesungen ¨ uber Fouriersche Integrale (Akademische V erlagsgesellschaft, 1932)

    S. Bochner, V orlesungen ¨ uber Fouriersche Integrale (Akademische V erlagsgesellschaft, 1932)

    1932

  28. [28]

    M. H. Stone, On one-parameter unitary groups in Hilbert Space, Ann. Math. 33, 643 (1932)

    1932

  29. [29]

    Wigner, On unitary representations of the inhomogen eous Lorentz group, Ann

    E. Wigner, On unitary representations of the inhomogen eous Lorentz group, Ann. Math. 40, 149 (1939)

    1939

  30. [30]

    D. N. Page and W. K. Wootters, Evolution without evoluti on: Dynamics described by stationary ob- servables, Phys. Rev. D 27, 2885 (1983)

    1983

  31. [31]

    Gell-Mann and Y

    M. Gell-Mann and Y . Ne’eman, The Eightfold Way (W. A. Ben jamin, 1964)

    1964

  32. [32]

    H. F. Trotter, On the product of semi-groups of operator s, Proc. Am. Math. Soc. 10, 545 (1959)

    1959

  33. [33]

    Haag, Local Quantum Physics: Fields, Particles, Alg ebras (Springer, 1992)

    R. Haag, Local Quantum Physics: Fields, Particles, Alg ebras (Springer, 1992)

    1992

  34. [34]

    Oreshkov, F

    O. Oreshkov, F. Costa, and ˇC. Brukner, Quantum correlations with no causal order, Nat. C ommun. 3, 1092 (2012)

    2012

  35. [35]

    Brukner, Quantum causality, Nat

    ˇC. Brukner, Quantum causality, Nat. Phys. 10, 259 (2014)

    2014

  36. [36]

    Chiribella, Tsirelson bounds for quantum c orrelations with indefinite causal order, Nat

    Z, Liu and G. Chiribella, Tsirelson bounds for quantum c orrelations with indefinite causal order, Nat. Commun. 16, 3314 (2025)

    2025

  37. [37]

    Chiribella, G

    G. Chiribella, G. M. D’Ariano, P . Perinotti, and B. V ali ron, Quantum computations without definite causal structure, Phys. Rev. A 88, 022318 (2013)

    2013

  38. [38]

    L. M. Procopio et al., Experimental superposition of orders of quantum gates, Na t. Commun. 6, 7913 (2015). 31

    2015

  39. [39]

    Rubino, et al., Experimental verification of an indefinite causal order, Sc i

    G. Rubino, et al., Experimental verification of an indefinite causal order, Sc i. Adv. 3, e1602589 (2017)

    2017

  40. [40]

    L. A. Rozema, T. Str¨ omberg, H. Cao, Y . Guo, B. H. Liu, and P . Walther, Experimental aspects of indefinite causal order in quantum mechanics, Nat. Rev. Phys . 6, 483-499 (2024)

    2024

  41. [41]

    Deng et al

    Y . Deng et al. , Generalized Indefinite Causal Orders in an Integrated Quan tum Switch, Phys. Rev. Lett. 135, 160202 (2025)

    2025

  42. [42]

    Y . Guo, Y . Chen, G. Chen, X. M. Hu, Y . F. Huang, C. F. Li, G. C . Guo, and B. H. Liu, Surpassing the Global Heisenberg Limit Using a High-e ffciency Quantum Switch, arXiv:2505.03290[quant-ph] (2025)

    arXiv 2025

  43. [43]

    C. M.D. Richter, M. Antesberger, H. Cao , P . Walther, and L. A. Rozema1, Toward an Experimental Device-Independent V erification of Indefinite Causal Order , PRX Quantum 7, 010354 (2026)

    2026