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REVIEW 4 major objections 6 minor 48 references

This paper sets out to prove that complex numbers are a convenience, not a necessity, in quantum field theory: replacing i by a real 2×2 matrix J with J² = −1 yields a real-number formulation of scalar QFT whose observables match ordinary Q

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2026-08-01 10:13 UTC pith:BQTROBAX

load-bearing objection A transparent, mostly rigorous reformulation of scalar QFT via i→J, with a solid core and a Standard Model realification whose central algebra is unverified. the 4 major comments →

arxiv 2607.20304 v1 pith:BQTROBAX submitted 2026-07-22 hep-th

Real Quantum Field Theory, J-Quantization, and Standard Model

I. Aref'eva , I. Volovich This is my paper
classification hep-th PACS 11.10.-z12.15.-y
keywords real quantum field theoryJ-quantizationreal Kähler spaceJ-Fourier transformortho-symplectic S-matrixStandard Model realificationJ-symmetry breakingMajorana neutrino mass
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper is trying to establish that the entire apparatus of complex quantum field theory—the i in commutation relations, propagators, Wightman functions, and unitarity—can be transplanted to a purely real setting. The mechanism is the substitution i → J, where J is a real 2×2 matrix with J² = −1; QFT then lives in a real Kähler Fock space. The payoff is conceptual: QFT and real symplectic geometry share a language, and complex phases become rotations generated by J. The paper further claims the Standard Model admits a real formulation with ortho-symplectic symmetry and identical physical content, and that breaking J-symmetry is a natural beyond-Standard-Model marker, notably for neutrino mass. A sympathetic reader would care because it reframes a foundational question—whether complex numbers are required—into a structural question about which real geometric data underlie the known particles.

Core claim

The central claim is that physical quantum field theory has a fully real counterpart. A field operator Φ(t,x) acting on the real Kähler analogue of bosonic Fock space satisfies the Klein–Gordon equation and the canonical commutation relation [Φ(t,x), ∂ₜΦ(t,y)] = Jδ³(x−y), with J the real 2×2 matrix squaring to −1. Every standard object is rebuilt by the substitution i → J: the J-Fourier transform, J-Wightman functions, and the J-Feynman propagator G̃_J(p) = J P(1/(p²−m²)) + πδ(p²−m²). Unitarity S†S = 1 becomes orthogonality plus symplecticity, SᵀS = 1 and SᵀJS = J. For the Standard Model, complex unitary representations are replaced by real orthogonal-symplectic representations, and the pape

What carries the argument

The carrying mechanism is the J-calculus: a systematic replacement of the complex unit i by the real matrix J together with the corresponding J-Fourier transform, J-valued distributions, and the J-Sokhotski–Plemelj formula. For gauge theories the workhorse is realification, the block map R(A+iB) = [[A, −B], [B, A]], which turns complex unitary representations into real matrices that are simultaneously orthogonal and symplectic and that commute with J. These tools convert the Feynman propagator, Cutkosky rules, the optical theorem, and the Standard Model field content into real language while preserving the underlying complex-linear physics.

Load-bearing premise

The central equivalence leans on Theorem 3 in Appendix C.2, which asserts that restricting the J-Fourier transform to the mass shell sends Schwartz functions to Schwartz functions; for m = 0 the mass-shell parametrization k ↦ (|k|, k) is not differentiable at k = 0, and the proof simply defers to the complex case, so this step is the load-bearing premise that must hold.

What would settle it

Compute the J-Fourier transform of f(t,x) = ∂ₜ(e^{−t²−|x|²}) and restrict it to the mass shell k₀ = |k|; the resulting component |k| e^{−|k|²/4} is not C∞ at k = 0, contradicting Theorem 3's assertion that the restriction is Schwartz. Repeating the massless Wightman computation (3.14)–(3.18) with this test function would determine whether (3.18) still has a valid distributional meaning.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • For scalar QFT, masses, decay widths, cross sections, and spectra come out identical in RQFT, so the real formulation is an equivalent description rather than a competing theory.
  • Unitarity is re-expressed as orthogonality plus symplecticity; the J-optical theorem and J-Cutkosky rules give the same on-shell discontinuities as the usual imaginary-part rules.
  • The realified Standard Model preserves the physical content: CKM and PMNS phases survive inside real matrices, the Higgs mechanism works, and fermions receive masses through realified Yukawa couplings.
  • Breaking J-symmetry enables gauge-invariant Majorana mass terms and the seesaw mechanism, making J-symmetry a concrete diagnostic for beyond-Standard-Model physics.
  • If the equivalence holds, complex numbers are shown to be dispensable in QFT, with the role of i played by a real complex structure acting on a Kähler space.

Where Pith is reading between the lines

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

  • If RQFT is exactly equivalent to ordinary QFT, then experiments that claim to rule out real-valued quantum mechanics in finite-dimensional settings do not automatically constrain this construction; the field-theoretic equivalence turns the question from empirical prediction into mathematical reformulation.
  • The massless Wightman formula in Section 3.1.1 may still be salvageable through a distributional interpretation or a restricted test-function space, but the paper's mass-shell lemma needs repair before that formula is rigorous.
  • Promoting J to a spacetime-dependent field J(x), which the paper mentions as a future direction, would make the complex structure dynamical and could couple to real gravitational geometry—an extension the paper does not develop.
  • An odd number of real degrees of freedom after J-breaking suggests lepton-number-violating signatures; a testable extension would be to search for neutrino mass textures that cannot be written as a doubling of an even complex structure.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 6 minor

Summary. The paper proposes a purely real formulation of quantum field theory, RQFT, obtained by replacing the imaginary unit i with a real 2x2 matrix J throughout the standard complex formalism. For a real scalar field it constructs a field operator in a real Kähler Fock space, defines J-Wightman functions, J-Feynman propagators, a J-version of the Sokhotski–Plemelj formula, and an S-matrix satisfying both orthogonality and symplecticity instead of unitarity. It verifies a one-loop J-optical theorem in λΦ^4 theory. The second half claims that the Standard Model, including its Yukawa sector, can be 'realified' without loss of physical content, with ortho-symplectic gauge representations and J-symmetry breaking discussed as a beyond-Standard-Model marker.

Significance. The scalar-field part of the paper is a clear and mostly rigorous exercise in real reformulation: it gives explicit isomorphisms between real and complex Fock spaces, checks the J-Fourier calculus, and demonstrates that the two-point functions, propagator, and one-loop discontinuity are the R-images of the usual complex objects. If the SM realification were fully established, the paper would be conceptually valuable for foundations and for approaches where real geometric structures are preferred. The paper honestly states that RQFT does not produce new numerical predictions, and I credit the explicit γ-isomorphism, the J-Sokhotski derivation, and the J-optical theorem check as concrete, checkable content. However, the Standard Model claim currently rests on unverified algebraic assertions, not on a derivation that a referee can verify.

major comments (4)
  1. [Appendix C.2, Theorem 3 (Eq. C.10)] The theorem asserts that for m=0 the mass-shell restriction of the J-Fourier transform of a Schwartz function is again Schwartz. This is false: the parametrization k ↦ (|k|, k) is not differentiable at k=0. A concrete counterexample is f1=f2=∂_t(e^{-t^2-|x|^2}) ∈ S(R^4); its mass-shell Fourier transform is proportional to |k| e^{-|k|^2/4}, which is not C^1 at k=0, let alone Schwartz. The proof claims the result follows from the standard case, but the massless shell is not a smooth submanifold in the relevant parametrization. This invalidates the statement as written and the comment after Eq. (2.7). The scalar construction can likely be repaired because the smeared field only needs L^2 rather than Schwartz test functions on the mass shell, and the Wightman integral (3.1) is a direct p-integral, but Theorem 3 must be corrected (e.g., replace S(R^3) by L^2(R^3)^2 or a suitable weighted Schw
  2. [§7.4, Eq. (7.42)] The realified Standard Model Yukawa Lagrangian is the load-bearing step for the abstract's claim that the Standard Model admits a consistent real formulation. Yet Eq. (7.42) is introduced with only 'after algebraic calculations' and references to unpublished Mathematica notebooks. No derivation from Eqs. (7.34)–(7.36) is given, the index conventions (r,s,t) are not defined, and the resulting gauge invariance is not checked. A sign or factor error in (7.42) would silently break the claimed equivalence. This is not a presentation issue: the central SM realification claim cannot be refereed without a complete, verifiable derivation of (7.42)–(7.44), or at least an independent check of the fermion mass matrices in (7.49).
  3. [Appendix F.3, Eq. (F.55)] The realified Weyl kinetic action is written symbolically as L_R^W = g(Ξ, JΣ^μ∂_μΞ), with the parenthetical 'with the understanding that Ξ is Grassmann odd.' This is not a well-defined action: products of Grassmann fields require a Berezin measure and an explicit statement of which variables are integrated, and the normalization relative to iχ†σ̄^μ∂_μχ is not established. Since the fermion kinetic term is essential for the realification of the Standard Model, this gap must be closed by giving the explicit Berezin integral and proving that it reproduces the complex Weyl action.
  4. [General framing] The paper's central statement that 'physical observables in RQFT coincide with those of ordinary QFT' is, for the scalar theory, guaranteed by the explicit isomorphism γ between the real Kähler Fock space and the complex Fock space, and by the R-map that sends J to i in every formula. This is not an independently derived prediction. The manuscript should state more explicitly which statements are theorems (valid for the free and perturbative scalar theory) and which are conjectural (e.g., non-perturbative existence, full SM equivalence). Otherwise the reader may attribute more independence to the coincidence claim than the construction actually provides.
minor comments (6)
  1. [§7.2, p. 24] The sentence 'More details about realification of group U(1), SU(2) and SU(2)' should read 'U(1), SU(2) and SU(3)'.
  2. [Appendix C.2] There is a typo: 'The prof is the same as in the standard Fourier transform' should be 'The proof is the same...'.
  3. [§3.1.1, Eqs. (3.17)–(3.21)] The approximate denominator in (3.17) drops a factor of 2 in the ε^2(r^2+x_0^2) term relative to the exact product (ε^2+(r-x_0)^2)(ε^2+(r+x_0)^2). The final distributional formula may still be correct, but the intermediate approximation is misleading and should be corrected or stated as leading-order only.
  4. [Eqs. (3.21), (3.25), (8.4)] Several displayed formulas contain stray '0' characters at the end (e.g., 'J sgn(x0−y0) 0' and 'J0 (x0−y0)'). These look like layout artifacts and should be removed.
  5. [Appendices E.2.1, E.3.1] The text refers to 'Mathematica: MatrixM4.nb' and 'Matrix6.nb', but these notebooks are not included with the submission. Either include them as supplementary material or remove the references, since the determinant and commutation claims should be verifiable from the displayed formulas.
  6. [General] There are duplicated equation numbers (e.g., two equations labeled (3.11)) and inconsistent notation for the conjugate Higgs doublet (\(\tilde H\) vs. \(eH\)). These should be cleaned up.

Circularity Check

0 steps flagged

No significant circularity: the claimed equivalence between RQFT and ordinary QFT is an explicit isomorphism, not a hidden prediction.

full rationale

The paper is transparent that RQFT is obtained from standard QFT by the substitution i -> J: the abstract says 'The construction is obtained from the standard complex-number formulation by replacing the imaginary unit i with a matrix J throughout all formulas,' and Section 1 states 'Quantization over the reals is constructed simply by the substitution i -> J.' The key equalities are not independent predictions but explicit equivalences: Eq. (2.11) defines the real field as R(phi)=Phi, and Eq. (4.26)-(4.27) proves the J-propagator equals R of the complex propagator by applying the same realification map. Section 5's statement that observables 'are the same in both QFT and RQFT' follows from the explicit real-complex Fock-space isomorphism in Appendix B, not from a fitted parameter or from a self-citation. The Standard Model realification is similarly framed as a reformulation: Section 1 notes 'Any complex theory can be rewritten over the reals by doubling dimensions,' and Section 7.5 says 'the purely real formulation of the Standard Model is not a new theory, but a more unified mathematical representation of the existing theory.' Self-citations [16,17] are motivational and not load-bearing; no uniqueness theorem is imported from the authors' prior work. Two correctness gaps exist — the massless Schwartz restriction claim in Appendix C.2 (Theorem 3) appears false for m=0, and the realified Yukawa sector (7.42) is stated only 'after algebraic calculations' without derivation — but these are correctness or support defects, not circular reductions. There is no fitted input presented as a prediction, no ansatz smuggled in by citation, and no derivation whose conclusion is equivalent to its input by construction in a way that the paper conceals.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 0 invented entities

No free parameters are fitted; the construction is a reformulation via the fixed isomorphism C≈R^2. The load-bearing assumptions are standard QFT axioms, the C≈R^2 isomorphism, and an asserted but unproved (for m=0) mass-shell restriction theorem. J is a fixed mathematical matrix, not a new physical entity; promoting J to a dynamical field is only a future suggestion.

axioms (5)
  • standard math C is isomorphic to R^2 with i mapped to the 2x2 matrix J (Appendix A, Lemma 1).
    Defines the maps R and gamma used throughout; the algebraic properties of J (J^2=-I) are the standard complex-structure identities.
  • standard math J-Sokhotski-Plemelj distributional identities (Appendix C.3, eqs. C.12-C.18).
    Used to evaluate the J-Feynman propagator and Wightman functions; these are derived from standard distribution theory, with J replacing i.
  • domain assumption Restriction of the J-Fourier transform of a Schwartz function to the mass shell k0=|k| is in S(R^3) for m=0 (Theorem 3, Appendix C.2).
    The proof is deferred to the standard complex case, but the light cone is singular at k=0 and the statement is false for general Schwartz functions. This affects the massless Wightman computation in Section 3.1.1.
  • domain assumption Standard QFT framework: LSZ reduction, optical theorem, causal perturbation theory and the adiabatic limit.
    RQFT's observables are defined as the images of standard QFT observables under the isomorphism; the S-matrix axioms in Section 6 are taken over from Bogoliubov-Shirkov.
  • domain assumption Standard Model gauge group and matter content.
    The realification in Section 7 assumes the usual SU(3)xSU(2)xU(1) representations and Yukawa structure, as supported by experiment.

pith-pipeline@v1.3.0-alltime-deepseek · 33987 in / 27751 out tokens · 228189 ms · 2026-08-01T10:13:42.227979+00:00 · methodology

0 comments
read the original abstract

We present a formulation of quantum field theory based entirely on real numbers, which we call real quantum field theory (RQFT). The construction is obtained from the standard complex-number formulation by replacing the imaginary unit i with a matrix J throughout all formulas. Here J is the real 2x2 matrix satisfying the condition J^2=-1. This extends the recently developed formulation of real quantum mechanics based on the real K\"ahler space to systems with infinitely many degrees of freedom. As the basic example, we construct the RQFT for a scalar field. We define a field operator acting on the real K\"ahler analogue of the bosonic Fock space. This operator satisfies the Klein-Gordon equation and the J-form of canonical commutation relation. To construct the RQFT we develop the corresponding J-calculus. In particular, we introduce the direct and inverse J-Fourier transforms and the associated J-valued distributions. In RQFT, the usual unitarity condition for the complex S-matrix is replaced by the statement that the real scattering operator is both orthogonal and symplectic. The physical observables in RQFT coincide with those of ordinary QFT. Thus RQFT does not change the physical predictions of scalar QFT, but provides an equivalent formulation. We explore the possibility of purely real formulations of the Standard Model of elementary particles. We show that it does admit this formulation and the resulting theory has ortho-symplectic symmetry. The choice of the Standard Model as the testbed for exploring the possibility of purely real formulations is related with the fact it provides a realistic description of all known fundamental particles. The real formulation of the Standard Model naturally suggests a possible exit beyond the Standard Model. In particular, we consider the implications of breaking the J-symmetry as a marker for new physics.

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Reference graph

Works this paper leans on

48 extracted references · 22 linked inside Pith

  1. [1]

    The logic of quantum mechanics,

    G. Birkhoff and J. von Neumann, “The logic of quantum mechanics,” Annals of Mathematics37, 823-843 (1936)

    1936

  2. [2]

    Quantum theory in real Hilbert space,

    E. C. Stueckelberg, “Quantum theory in real Hilbert space,” Helv. Phys. Acta33, 458 (1960). E. C. G. Stueckelberg. “Quantum theory in real Hilbert Space II: Addenda and errats”. Helv. Phys. Acta 34, 628 (1961). E. C. G. Stueckelberg. “Quantum theory in real Hilbert Space III: Fields of the first kind”. Helv. Phys. Acta 34, 698 (1961). E. C. G. Stueckelber...

    1960

  3. [3]

    Geometry of quantum theory

    V. Varadarajan, “Geometry of quantum theory” (Springer, New York, 1985)

    1985

  4. [4]

    Quantum theory based on real numbers can be experimentally falsified,

    M. O. Renou, D. Trillo, M. Weilenmann, T. P. Le, A. Tavakoli, N. Gisin, A. Acin and M. Navascues, “Quantum theory based on real numbers can be experimentally falsified,” Nature600, 625-629 (2021) [arXiv:2101.10873 [quant-ph]]

    Pith/arXiv arXiv 2021

  5. [5]

    Experimental Refutation of Real-Valued Quantum Mechanics under Strict Locality Conditions,

    D. Wu, Y. F. Jiang, X. M. Gu, L. Huang, B. Bai, Q. C. Sun, S. Q. Gong, Y. Mao, H. S. Zhong and M. C. Chen,et al.“Experimental Refutation of Real-Valued Quantum Mechanics under Strict Locality Conditions,” Phys. Rev. Lett.129, no.14, 140401 (2022) [arXiv:2201.04177 [quant-ph]]

    Pith/arXiv arXiv 2022

  6. [6]

    Ruling Out Real-Valued Standard Formalism of Quantum Theory,

    M. C. Chen, C. Wang, F. M. Liu, J. W. Wang, C. Ying, Z. X. Shang, Y. Wu, M. Gong, H. Deng and F. Liang,et al.“Ruling Out Real-Valued Standard Formalism of Quantum Theory,” Phys. Rev. Lett.128, no.4, 040403 (2022) [arXiv:2103.08123 [quant-ph]]

    Pith/arXiv arXiv 2022

  7. [7]

    Finkelstein, [arXiv:2103.12740 [quant-ph]]

    J. Finkelstein, [arXiv:2103.12740 [quant-ph]]

    Pith/arXiv arXiv

  8. [8]

    Positive Maps and Entanglement in Real Hilbert Spaces,

    G. Chiribella, K. R. Davidson, V. I. Paulsen and M. Rahaman, “Positive Maps and Entanglement in Real Hilbert Spaces,” Annales Henri Poincare24, no.12, 4139-4168 (2023) [arXiv:2207.02510 [quant-ph]]

    Pith/arXiv arXiv 2023

  9. [9]

    Hiding and masking quantum information in complex and real quantum mechanics,

    H. Zhu, “Hiding and masking quantum information in complex and real quantum mechanics,” Phys. Rev. Res.3, no.3, 033176 (2021) [arXiv:2010.07843 [quant-ph]]

    Pith/arXiv arXiv 2021

  10. [10]

    Quantum mechanics? It’s all fun and games until someone loses ani,

    C. A. Fuchs, M. Olshanii and M. B. Weiss, “Quantum mechanics? It’s all fun and games until someone loses ani,” [arXiv:2206.15343 [quant-ph]]

    Pith/arXiv arXiv

  11. [11]

    Quantum Mechanics with Real Numbers: Entanglement, Superselection Rules and Gauges,

    V. Vedral, “Quantum Mechanics with Real Numbers: Entanglement, Superselection Rules and Gauges,” Quanta12, 164-170 (2023) [arXiv:2308.05473 [quant-ph]]

    Pith/arXiv arXiv 2023

  12. [12]

    Gap between quantum theory based on real and complex numbers is arbitrarily large,

    S. Sarkar, D. Trillo, M. O. Renou and R. Augusiak, “Gap between quantum theory based on real and complex numbers is arbitrarily large,” Rep. Prog. Phys. 89 070503 [arXiv:2503.09724 [quant-ph]]

    Pith/arXiv arXiv

  13. [13]

    Quantum mechanics based on real numbers: A consistent description,

    P. B. Hita, A. Trushechkin, H. Kampermann, M. Epping and D. Bruß, “Quantum mechanics based on real numbers: A consistent description,” Phys. Rev. Lett.136, no.24, 240202 (2026), [arXiv:2503.17307 [quant-ph]]. – 61 –

    Pith/arXiv arXiv 2026

  14. [14]

    Quantum theory does not need complex numbers,

    T. Hoffreumon and M. P. Woods, “Quantum theory does not need complex numbers,” [arXiv:2504.02808 [quant-ph]]

    arXiv

  15. [15]

    Locality Implies Complex Numbers in Quantum Mechanics,

    T. Feng, C. Ren and V. Vedral, “Locality Implies Complex Numbers in Quantum Mechanics,” [arXiv:2504.07808 [quant-ph]]

    Pith/arXiv arXiv

  16. [16]

    Real Quantum Mechanics in a K¨ ahler Space,

    I. Volovich, “Real Quantum Mechanics in a K¨ ahler Space,” [arXiv:2504.16838 [quant-ph]]

    Pith/arXiv arXiv

  17. [17]

    Notes on Real Quantum Mechanics in a K¨ ahler Space,

    I. Aref’eva and I. Volovich, “Notes on Real Quantum Mechanics in a K¨ ahler Space,” Theor. and Math.Phys, (2026)228186-196 [arXiv:2506.07632 [quant-ph]]

    Pith/arXiv arXiv 2026

  18. [18]

    On whether quantum theory needs complex numbers: the foil theories perspective,

    Y. Ying, M. C. Ala˜ n´ on, D. Centeno, J. Surace, M. M. Ansanelli, R. Liu, D. Schmid and R. W. Spekkens, “On whether quantum theory needs complex numbers: the foil theories perspective,” [arXiv:2506.08091 [quant-ph]]

    Pith/arXiv arXiv

  19. [19]

    Quantum mechanics over real numbers fully reproduces standard quantum theory,

    A. C. Maioli, E. M. F. Curado and J. P. Gazeau, “Quantum mechanics over real numbers fully reproduces standard quantum theory,” [arXiv:2604.19482 [quant-ph]]

    Pith/arXiv arXiv

  20. [20]

    Quantum dynamics in real Hilbert space: Algebraic isomorphism and symplectic geometry of the Schr¨ odinger equation,

    P. Singh, “Quantum dynamics in real Hilbert space: Algebraic isomorphism and symplectic geometry of the Schr¨ odinger equation,” Annals Phys.487, 170368 (2026)

    2026

  21. [21]

    Comment on

    F. M. Kalarde, X. Xu and M. O. Renou, “Comment on ”Quantum theory based on real numbers cannot be experimentally falsified”: On the compatibility of physical principles with information theory for fermions,” [arXiv:2604.07425 [quant-ph]]

    Pith/arXiv arXiv

  22. [22]

    Hidden Complex Structure in Quotient-Space Real Quantum Mechanics,

    J. Bang, K. Cho and K. Baek, “Hidden Complex Structure in Quotient-Space Real Quantum Mechanics,” [arXiv:2607.05865 [quant-ph]]

    Pith/arXiv arXiv

  23. [23]

    Nielsen and Isaac L

    Michael A. Nielsen and Isaac L. Chuang. Quantum computation and quantum information. Cambridge university press, 2010

    2010

  24. [24]

    Mathematical foundations of quantum information and computation and its applications to nano-and bio-systems

    Masanori Ohya and Igor Volovich. Mathematical foundations of quantum information and computation and its applications to nano-and bio-systems. Springer Science&Business Media, 2011

    2011

  25. [25]

    Complete integrability of quantum and classical dynamical systems

    Igor V. Volovich, “Complete integrability of quantum and classical dynamical systems”, P-Adic Numbers Ultrametric Anal. Appl., 11:4 (2019), 328–334, 1911.01335 [math-ph]; I. V. Volovich, “Integrability of dynamical systems and monoidal categories” (2019), https://www.mathnet.ru/php/presentation.phtml?presentid=26205&option_lang=eng\\ https://www.mathnet.r...

    Pith/arXiv arXiv 2019

  26. [26]

    Quantum vs. Symplectic Computers,

    I. Volovich, “Quantum vs. Symplectic Computers,” [arXiv:2407.12755 [quant-ph]]

    Pith/arXiv arXiv

  27. [27]

    Symplectic perspective to quantum computing for Hamiltonian systems,

    E. Koukoutsis, K. Hizanidis, L. I. Inigo Gamiz, O. Amaro, C. Tsironis, A. K. Ram and G. Vahala, “Symplectic perspective to quantum computing for Hamiltonian systems,” [arXiv:2604.10794 [quant-ph]]

    Pith/arXiv arXiv

  28. [28]

    Differential calculus

    V. S. Vladimirov, I. V. Volovich, “Differential calculus”, TMP (1984), v. 59, pp.317–335

    1984

  29. [29]

    p-adic string

    I. V. Volovich, “p-adic string”, Classical Quantum Gravity, 4:4 (1987), L83–L87 – 62 –

    1987

  30. [30]

    Introduction to Quantum Field Theory

    N. N. Bogolyubov and D. V. Shirkov, “Introduction to Quantum Field Theory” (Nauka, 1973)

    1973

  31. [31]

    Wave function of the universe and p-adic gravity,

    I. Y. Arefeva, B. Dragovich, P. H. Frampton and I. V. Volovich, “Wave function of the universe and p-adic gravity,” Int. J. Mod. Phys. A6, 4341-4358 (1991)

    1991

  32. [32]

    Dragovich, A

    B. Dragovich, A. Y. Khrennikov, S. V. Kozyrev, I. V. Volovich, and E. I. Zelenov, p-adic mathematical physics: The first 30 years. p-adic Numbers, Ultrametric Analysis and Applications 9, 87–121 (2017)

    2017

  33. [33]

    I. M. Gel’fand and G. E. Shilov,Generalized Functions, Vol. 2: Spaces of Fundamental and Generalized Functions, Academic Press, 1968

    1968

  34. [34]

    Equations of mathematical physics

    V.S.Vladimirov, (1976). Equations of mathematical physics. Moscow, Nauka

    1976

  35. [35]

    H¨ ormander,The Analysis of Linear Partial Differential Operators I, Springer-Verlag, 2nd ed., 1990

    L. H¨ ormander,The Analysis of Linear Partial Differential Operators I, Springer-Verlag, 2nd ed., 1990

    1990

  36. [36]

    M. Reed, B. Simon, (1975). II: Fourier analysis, self-adjointness (Vol. 2). Elsevier

    1975

  37. [37]

    Perturbation theory of Wightman functions

    Steinmann O. Perturbation theory of Wightman functions. Commun.Math. Phys. 152, 627–645 (1993)

    1993

  38. [38]

    Itzykson and J.-B

    C. Itzykson and J.-B. Zuber,Quantum Field Theory, McGraw-Hill, 1980

    1980

  39. [39]

    Schwartz, Quantum Field Theory and the Standard Model (Cambridge University Press, Cambridge, UK, 2014) pp

    Matthew D. Schwartz, Quantum Field Theory and the Standard Model (Cambridge University Press, Cambridge, UK, 2014) pp. 456–459

    2014

  40. [40]

    One Loop Corrections for e+ e- Annihilation Into mu+ mu- in the Weinberg Model,

    G. Passarino and M. J. G. Veltman, “One Loop Corrections for e+ e- Annihilation Into mu+ mu- in the Weinberg Model,” Nucl. Phys. B160, 151-207 (1979)

    1979

  41. [41]

    Electroweak Radiative Corrections to Higgs Production via Vector Boson Fusion using SCET: Numerical Results,

    F. Siringo and G. Buccheri, “Electroweak Radiative Corrections to Higgs Production via Vector Boson Fusion using SCET: Numerical Results,” Phys. Rev. D86, 053013 (2012) [arXiv:1207.1906 [hep-ph]]

    Pith/arXiv arXiv 2012

  42. [42]

    R. E. Cutkosky, Singularities and discontinuities of Feynman amplitudes, J. Math. Phys. (N.Y.) 1, 429 (1960)

    1960

  43. [43]

    Abramowitz and I

    M. Abramowitz and I. A. Stegun,Handbook of Mathematical Functions, Dover Publications, New York (1964)

    1964

  44. [44]

    The Standard Modelhttps

    D. Tong, "The Standard Modelhttps", (2026) https://davidtong.org/pdfs/teaching/standard-model/standardmodel.pdf

    2026

  45. [45]

    Kozlov, O.G

    V.V. Kozlov, O.G. Smolyanov. ”Hamiltonian approach to secondary quantization”, Doklady Mathematics, (2018) 98:3, 571-574

    2018

  46. [46]

    Generalized imaginary units in quantum mechanics,

    S. Giardino, “Generalized imaginary units in quantum mechanics,” [arXiv:2311.14162 [quant-ph]]

    arXiv

  47. [47]

    V. S. Vladimirov, I. V. Volovich, E. I. Zelenov, p-adic analysis and mathematical physics, World Scientific Publishing Co. Inc., River Edge, NJ, 1994,

    1994

  48. [48]

    Structure and applications of realC∗-algebras,

    J. Rosenberg, “Structure and applications of realC∗-algebras,” [arXiv:1505.04091 [math.OA]]. – 63 –

    Pith/arXiv arXiv