pith. sign in

arxiv: 2605.28935 · v1 · pith:EGL7AQLPnew · submitted 2026-05-27 · ✦ hep-ph · cond-mat.other· hep-th

FunKit: A computer algebra toolkit for functional approaches

Franz R. Sattler This is my paper

Pith reviewed 2026-06-29 11:04 UTC · model grok-4.3

classification ✦ hep-ph cond-mat.otherhep-th
keywords functional equationscomputer algebraMathematica packageDyson-Schwinger equationsfunctional renormalization groupmaster equationstensor tracingcode generation
0
0 comments X

The pith

FunKit supplies an expression vocabulary and rules for deriving functional equations from arbitrary master equations in any field theory.

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

The paper presents FunKit, a Mathematica package for deriving and tracing functional equations from arbitrary master equations. It supplies a vocabulary and set of rules intended to work across field theories and master equations such as Dyson-Schwinger equations, functional RG flows, nPI equations, STIs, WTIs, and Polchinski or Wegner flows. The system supports user extensions for specific cases and connects to the FORM language for tracing large tensor expressions. Derived results can be exported to C++, Julia, or Fortran code for numerical evaluation, and the tracing and export tools can operate independently. This design targets the repeated manual work required when moving between different functional approaches.

Core claim

FunKit provides an expression vocabulary and a set of rules that allow for derivations in any given field theory and master equation. It can be used in a wide range of situations, for example Dyson-Schwinger or functional RG equations, flowing reparametrisations, nPI equations, (modified) STIs and WTIs, functional Polchinski and Wegner flows, functional master equations with sources, and many others. Besides interfacing with the FORM language to trace large tensor expressions efficiently, FunKit also provides facilities to export arbitrary Mathematica expressions to C++, Julia or Fortran code.

What carries the argument

The expression vocabulary and set of rules for performing derivations from arbitrary master equations.

If this is right

  • Derivations in any field theory and master equation become possible using the supplied vocabulary and rules.
  • Large tensor expressions can be traced efficiently via the built-in interface to FORM.
  • Results of derivations can be exported directly to C++, Julia, or Fortran for numerical use.
  • Users can add custom extensions to adapt the system to more specific equation setups.
  • The tracing and code-generation features remain usable on their own or alongside other packages.

Where Pith is reading between the lines

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

  • A general toolkit of this kind could reduce the incidence of transcription errors when moving between different functional methods.
  • The export options open a direct path from symbolic derivation to high-performance numerical evaluation without intermediate manual rewriting.
  • Community use might create shared libraries of extensions for common master equations.
  • The independent tracing module could serve as a drop-in replacement for manual FORM scripting in existing workflows.

Load-bearing premise

The provided vocabulary and rules are general and correct enough to handle the listed wide range of master equations without further validation or specific implementations.

What would settle it

Applying the rules to derive a known functional equation from one of the listed master equations and finding a mismatch with an established manual result.

Figures

Figures reproduced from arXiv: 2605.28935 by Franz R. Sattler.

Figure 1
Figure 1. Figure 1: FunKit workflow: From theory specification to numerical evaluation. [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Results of the Yang–Mills example in examples/Yang-Mills/. Left: Inverse gluon dressing 1/ZA(p) compared to lattice data from [43]. Center: Ghost dressing in the scaling limit. Right: Avatars of the strong coupling. 7. Comparisons and Benchmarks In the following, we compare FunKit (v1.0.0) with the two most widely used packages for deriving functional equations, DoFun (v3.0) [27, 28] and QMeS (v1.2) [29] … view at source ↗
Figure 3
Figure 3. Figure 3: Performance comparison between FunKit (v1.0.0), DoFun (v3.0) [27, 28] and QMeS (v1.2) [29]. In all cases, we measured the total time to take the derivatives and truncate the expression. Especially for very large expressions with several thousand diagrams, e.g. the six-point and four-point flows, FunKit excels. All benchmarks were performed on a 16-core AMD Ryzen 9 7945HX machine. Two warmup-cycles were use… view at source ↗
read the original abstract

We introduce FunKit, a Mathematica package for the derivation and tracing of functional equations from arbitrary master equations. FunKit provides an expression vocabulary and a set of rules that allow for derivations in any given field theory and master equation. It also allows users to add extensions for more specific equation systems. Therefore, it can be used in a wide range of situations, for example Dyson--Schwinger or functional RG equations, flowing reparametrisations, nPI equations, (modified) STIs and WTIs, functional Polchinski and Wegner flows, functional master equations with sources, and many others. Besides interfacing with the \FORM language to trace large tensor expressions efficiently, FunKit also provides facilities to export arbitrary Mathematica expressions to C++, Julia or Fortran code, including the results of derivations, which can then be evaluated numerically. Both the tracing and code generation can also be used independently and in combination with other packages.

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 paper introduces FunKit, a Mathematica package for deriving and tracing functional equations from arbitrary master equations. It supplies an expression vocabulary and rules claimed to support derivations across any field theory and master equation (Dyson-Schwinger, functional RG, nPI, STIs, WTIs, Polchinski/Wegner flows, etc.), with optional user extensions, a FORM interface for tensor tracing, and export of results to C++, Julia, or Fortran code.

Significance. If the core vocabulary and rules prove general and correct, FunKit could reduce manual effort in functional QFT calculations and enable faster numerical follow-up via code generation. The FORM interface and independent usability of tracing/export features are practical strengths for large expressions. However, the manuscript supplies no example derivations, output checks against known results, or error-validation data, so the practical significance remains unassessed.

major comments (2)
  1. [Abstract] Abstract and package description: the central claim that the provided vocabulary and rules suffice for derivations from any listed master equation (DSE, fRG, nPI, STIs, WTIs, Polchinski/Wegner flows) without further validation is unsupported; no concrete derivations, test cases, or comparisons with analytic results are shown anywhere in the manuscript.
  2. [Package description] The extensibility mechanism is described only at a high level; it is unclear whether the base rules already cover the full range of master equations cited or whether substantial additional definitions are required for each, undermining the claim of immediate applicability to arbitrary cases.
minor comments (1)
  1. Notation for the expression vocabulary and rule set should be introduced with a compact table or explicit list early in the text for readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful review and constructive comments on the manuscript. We address each major comment below and will revise the manuscript to incorporate additional material where this strengthens the presentation of the toolkit's capabilities.

read point-by-point responses

  1. Referee: [Abstract] Abstract and package description: the central claim that the provided vocabulary and rules suffice for derivations from any listed master equation (DSE, fRG, nPI, STIs, WTIs, Polchinski/Wegner flows) without further validation is unsupported; no concrete derivations, test cases, or comparisons with analytic results are shown anywhere in the manuscript.

    Authors: We agree that the manuscript would benefit from explicit demonstrations to support the generality claim. The design of the vocabulary and rules targets the common algebraic structures appearing across the listed master equations, but concrete test cases were omitted to keep the focus on the package architecture. In the revised manuscript we will add a new section containing at least one fully worked derivation (e.g., a functional RG flow equation in a simple scalar theory) together with direct comparison to the corresponding analytic result obtained by hand. revision: yes

  2. Referee: [Package description] The extensibility mechanism is described only at a high level; it is unclear whether the base rules already cover the full range of master equations cited or whether substantial additional definitions are required for each, undermining the claim of immediate applicability to arbitrary cases.

    Authors: The base set of rules implements the generic operations (functional differentiation, integration by parts, source handling, etc.) that recur in all cited master equations; theory- or equation-specific adjustments are handled through the documented extension interface. We acknowledge that the current description leaves the boundary between base and extension unclear. The revision will expand this section with an explicit enumeration of the core rules and a worked example of a minimal user extension for a non-standard flow equation. revision: yes

Circularity Check

0 steps flagged

No circularity: software tool description with no derivation chain

full rationale

The manuscript introduces FunKit as a Mathematica package providing an expression vocabulary and rules for functional derivations. No mathematical predictions, fitted parameters, or derivation chains are present that could reduce to inputs by construction. Claims concern software capabilities and extensibility for listed master equations; these are not self-definitional, fitted-input predictions, or dependent on load-bearing self-citations. The paper is self-contained as a tool description without invoking uniqueness theorems or ansatze from prior work in a circular manner.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

This is a software description paper; no free parameters, axioms, or invented entities are involved in any physical derivation.

pith-pipeline@v0.9.1-grok · 5683 in / 939 out tokens · 42377 ms · 2026-06-29T11:04:18.526454+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

46 extracted references · 43 canonical work pages · 24 internal anchors

  1. [1]

    A. K. Cyrol, M. Mitter, N. Strodthoff, FormTracer - a Mathematica tracing package using FORM, Comput. Phys. Commun. 219 (2017) 346–352.arXiv:1610.09331,doi:10.1016/j.cpc.2017.05.024

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/j.cpc.2017.05.024 2017

  2. [2]

    Braun, A

    J. Braun, A. Geißel, J. M. Pawlowski, F. R. Sattler, N. Wink, Juggling with tensor bases in functional approaches, Annals Phys. (3 2025).arXiv:2503.05580

    work page arXiv 2025

  3. [3]

    C. D. Roberts, A. G. Williams, Dyson-Schwinger equations and their application to hadronic physics, Prog. Part. Nucl. Phys. 33 (1994) 477–575.arXiv:hep-ph/9403224,doi:10.1016/0146-6410(94)90049-3

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/0146-6410(94)90049-3 1994

  4. [4]

    C. D. Roberts, S. M. Schmidt, Dyson-Schwinger equations: Density, temperature and continuum strong QCD, Prog. Part. Nucl. Phys. 45 (2000) S1–S103.arXiv:nucl-th/0005064, doi:10.1016/S0146-6410(00) 90011-5

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/s0146-6410(00 2000

  5. [5]

    Non-Perturbative Renormalization Flow in Quantum Field Theory and Statistical Physics

    J. Berges, N. Tetradis, C. Wetterich, Nonperturbative renormalization flow in quantum field theory and sta- tistical physics, Phys. Rept. 363 (2002) 223–386.arXiv:hep-ph/0005122, doi:10.1016/S0370-1573(01) 00098-9. 35

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/s0370-1573(01 2002

  6. [6]

    Dyson-Schwinger equations: a tool for hadron physics

    P. Maris, C. D. Roberts, Dyson-Schwinger equations: A tool for hadron physics, Int. J. Mod. Phys. E 12 (2003) 297–365.arXiv:nucl-th/0301049,doi:10.1142/S0218301303001326

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1142/s0218301303001326 2003

  7. [7]

    J. M. Pawlowski, Aspects of the functional renormalisation group, Annals Phys. 322 (2007) 2831–2915. arXiv:hep-th/0512261,doi:10.1016/j.aop.2007.01.007

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/j.aop.2007.01.007 2007

  8. [8]

    Introduction to the functional RG and applications to gauge theories

    H. Gies, Introduction to the functional RG and applications to gauge theories, Lect. Notes Phys. 852 (2012) 287–348.arXiv:hep-ph/0611146,doi:10.1007/978-3-642-27320-9_6

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/978-3-642-27320-9_6 2012

  9. [9]

    C. S. Fischer, Infrared properties of QCD from Dyson-Schwinger equations, J. Phys. G 32 (2006) R253–R291. arXiv:hep-ph/0605173,doi:10.1088/0954-3899/32/8/R02

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1088/0954-3899/32/8/r02 2006

  10. [10]

    An Introduction to the Nonperturbative Renormalization Group

    B. Delamotte, An introduction to the nonperturbative renormalization group, Lect. Notes Phys. 852 (2012) 49–132.arXiv:cond-mat/0702365,doi:10.1007/978-3-642-27320-9_2

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1007/978-3-642-27320-9_2 2012

  11. [11]

    O. J. Rosten, Fundamentals of the exact renormalization group, Phys. Rept. 511 (2012) 177–272.arXiv: 1003.1366,doi:10.1016/j.physrep.2011.12.003

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/j.physrep.2011.12.003 2012

  12. [12]

    Metzner, M

    W. Metzner, M. Salmhofer, C. Honerkamp, V. Meden, K. Schönhammer, Functional renormalization group approach to correlated fermion systems, Reviews of Modern Physics 84 (1) (2012) 299–352.doi: 10.1103/revmodphys.84.299. URLhttp://dx.doi.org/10.1103/RevModPhys.84.299

    work page doi:10.1103/revmodphys.84.299 2012

  13. [13]

    Collective perspective on advances in Dyson-Schwinger Equation QCD

    A. Bashir, L. Chang, I. C. Cloet, B. El-Bennich, Y.-X. Liu, C. D. Roberts, P. C. Tandy, Collective perspective on advances in Dyson-Schwinger equation QCD, Commun. Theor. Phys. 58 (2012) 79–134. arXiv:1201.3366,doi:10.1088/0253-6102/58/1/16

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1088/0253-6102/58/1/16 2012

  14. [14]

    C. S. Fischer, QCD at finite temperature and chemical potential from Dyson–Schwinger equations, Prog. Part. Nucl. Phys. 105 (2019) 1–60.arXiv:1810.12938,doi:10.1016/j.ppnp.2019.01.002

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/j.ppnp.2019.01.002 2019

  15. [15]

    Dupuis, L

    N. Dupuis, L. Canet, A. Eichhorn, W. Metzner, J. M. Pawlowski, M. Tissier, N. Wschebor, The nonperturba- tive functional renormalization group and its applications, Phys. Rept. 910 (2021) 1–114.arXiv:2006.04853, doi:10.1016/j.physrep.2021.01.001

    work page doi:10.1016/j.physrep.2021.01.001 2021

  16. [16]

    J. M. Pawlowski, M. Reichert, Quantum gravity: A fluctuating point of view, Front. in Phys. 8 (2021) 551848.arXiv:2007.10353,doi:10.3389/fphy.2020.551848

    work page doi:10.3389/fphy.2020.551848 2021

  17. [17]

    Fu, QCD at finite temperature and density within the fRG approach: an overview, Commun

    W.-j. Fu, QCD at finite temperature and density within the fRG approach: an overview, Commun. Theor. Phys. 74 (9) (2022) 097304.arXiv:2205.00468,doi:10.1088/1572-9494/ac86be

    work page doi:10.1088/1572-9494/ac86be 2022

  18. [18]

    C. S. Fischer, J. M. Pawlowski, Phase structure and observables at high densities from first principles QCD (3 2026).arXiv:2603.11135

    work page arXiv 2026

  19. [19]

    Exact evolution equation for the effective potential

    C. Wetterich, Exact evolution equation for the effective potential, Phys. Lett. B 301 (1993) 90–94.arXiv: 1710.05815,doi:10.1016/0370-2693(93)90726-X

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/0370-2693(93)90726-x 1993

  20. [20]

    F. J. Dyson, Thes matrix in quantum electrodynamics, Phys. Rev. 75 (1949) 1736–1755.doi:10.1103/ PhysRev.75.1736. URLhttps://link.aps.org/doi/10.1103/PhysRev.75.1736

    work page doi:10.1103/physrev.75.1736 1949

  21. [21]

    Schwinger, On the Green’s functions of quantized fields

    J. Schwinger, On the Green’s functions of quantized fields. I, Proceedings of the National Academy of Sciences 37 (7) (1951) 452–455. arXiv:https://www.pnas.org/doi/pdf/10.1073/pnas.37.7.452, doi:10.1073/pnas.37.7.452. URLhttps://www.pnas.org/doi/abs/10.1073/pnas.37.7.452

    work page doi:10.1073/pnas.37.7.452 1951

  22. [22]

    Schwinger, On the Green’s functions of quantized fields

    J. Schwinger, On the Green’s functions of quantized fields. II, Proceedings of the National Academy of Sciences 37 (7) (1951) 455–459. arXiv:https://www.pnas.org/doi/pdf/10.1073/pnas.37.7.455, doi:10.1073/pnas.37.7.455. URLhttps://www.pnas.org/doi/abs/10.1073/pnas.37.7.455

    work page doi:10.1073/pnas.37.7.455 1951

  23. [23]

    Braun, et al., Renormalised spectral flows, SciPost Phys

    J. Braun, et al., Renormalised spectral flows, SciPost Phys. Core 6 (2023) 061. arXiv:2206.10232, doi:10.21468/SciPostPhysCore.6.3.061

    work page doi:10.21468/scipostphyscore.6.3.061 2023

  24. [24]

    Ihssen, J

    F. Ihssen, J. M. Pawlowski, Flowing fields and optimal RG-flows (5 2023).arXiv:2305.00816

    work page arXiv 2023

  25. [25]

    Ihssen, J

    F. Ihssen, J. M. Pawlowski, Physics-informed renormalisation group flows, Annals Phys. 481 (2025) 170177. arXiv:2409.13679,doi:10.1016/j.aop.2025.170177. 36

    work page doi:10.1016/j.aop.2025.170177 2025

  26. [26]

    Algorithmic derivation of Dyson-Schwinger Equations

    R. Alkofer, M. Q. Huber, K. Schwenzer, Algorithmic derivation of Dyson-Schwinger equations, Comput. Phys. Commun. 180 (2009) 965–976.arXiv:0808.2939,doi:10.1016/j.cpc.2008.12.009

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/j.cpc.2008.12.009 2009

  27. [27]

    M. Q. Huber, J. Braun, Algorithmic derivation of functional renormalization group equations and Dyson- Schwinger equations, Comput. Phys. Commun. 183 (2012) 1290–1320.arXiv:1102.5307, doi:10.1016/j. cpc.2012.01.014

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/j 2012

  28. [28]

    M. Q. Huber, A. K. Cyrol, J. M. Pawlowski, DoFun 3.0: Functional equations in Mathematica, Comput. Phys. Commun. 248 (2020) 107058.arXiv:1908.02760,doi:10.1016/j.cpc.2019.107058

    work page doi:10.1016/j.cpc.2019.107058 2020

  29. [29]

    J. M. Pawlowski, C. S. Schneider, N. Wink, QMeS-Derivation: Mathematica package for the symbolic derivation of functional equations, Comput. Phys. Commun. 287 (2023) 108711. arXiv:2102.01410, doi:10.1016/j.cpc.2023.108711

    work page doi:10.1016/j.cpc.2023.108711 2023

  30. [30]

    J. A. M. Vermaseren, New features of FORM (10 2000).arXiv:math-ph/0010025

    work page internal anchor Pith review Pith/arXiv arXiv 2000

  31. [31]

    FORM version 4.2

    B. Ruijl, T. Ueda, J. Vermaseren, FORM version 4.2 (7 2017).arXiv:1707.06453

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  32. [32]

    M. Q. Huber, A beginner’s guide to functional methods in particle physics (10 2025).arXiv:2510.18960

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  33. [33]

    F. R. Sattler, FunKit package,https://github.com/satfra/FunKit(2026)

    2026

  34. [34]

    J. M. Martín-García, xAct (2025). URLhttps://www.xact.es

    2025

  35. [35]

    Horvát, MaTeX, software package (Mar

    S. Horvát, MaTeX, software package (Mar. 2024).doi:10.5281/zenodo.10828124. URLhttps://doi.org/10.5281/zenodo.10828124

    work page doi:10.5281/zenodo.10828124 2024

  36. [36]

    A. K. Cyrol, M. Mitter, J. M. Pawlowski, N. Strodthoff, Nonperturbative quark, gluon, and meson correlators of unquenched QCD, Phys. Rev. D 97 (5) (2018) 054006.arXiv:1706.06326, doi:10.1103/PhysRevD.97. 054006

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.97 2018

  37. [37]

    Ihssen, J

    F. Ihssen, J. M. Pawlowski, F. R. Sattler, N. Wink, Towards quantitative precision in functional QCD I (8 2024).arXiv:2408.08413

    work page arXiv 2024

  38. [38]

    The three-gluon vertex in Landau gauge

    G. Eichmann, R. Williams, R. Alkofer, M. Vujinovic, Three-gluon vertex in Landau gauge, Phys. Rev. D 89 (10) (2014) 105014.arXiv:1402.1365,doi:10.1103/PhysRevD.89.105014

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.89.105014 2014

  39. [39]

    Four-point functions and the permutation group S4

    G. Eichmann, C. S. Fischer, W. Heupel, Four-point functions and the permutation group S4, Phys. Rev. D 92 (5) (2015) 056006.arXiv:1505.06336,doi:10.1103/PhysRevD.92.056006

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.92.056006 2015

  40. [40]

    Eichmann, R

    G. Eichmann, R. D. Torres, Five-point functions and the permutation group S5 (2 2025).arXiv:2502.17225

    work page arXiv 2025

  41. [41]

    F. R. Sattler, J. M. Pawlowski, DiFfRG: A Discretisation Framework for functional Renormalisation Group flows (12 2024).arXiv:2412.13043

    work page arXiv 2024

  42. [42]

    A. K. Cyrol, L. Fister, M. Mitter, J. M. Pawlowski, N. Strodthoff, Landau gauge Yang-Mills correlation functions, Phys. Rev. D 94 (5) (2016) 054005.arXiv:1605.01856,doi:10.1103/PhysRevD.94.054005

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevd.94.054005 2016

  43. [43]

    Lattice study of the infrared behavior of QCD Green's functions in Landau gauge

    A. Sternbeck, E. M. Ilgenfritz, M. Müller-Preussker, A. Schiller, I. L. Bogolubsky, Lattice study of the infrared behavior of QCD Green’s functions in Landau gauge, PoS LAT2006 (2006) 076.arXiv:hep-lat/0610053, doi:10.22323/1.032.0076

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.22323/1.032.0076 2006

  44. [44]

    M. Q. Huber, M. Mitter, CrasyDSE: A Framework for solving Dyson-Schwinger equations, Comput. Phys. Commun. 183 (2012) 2441–2457.arXiv:1112.5622,doi:10.1016/j.cpc.2012.05.019

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1016/j.cpc.2012.05.019 2012

  45. [45]

    Braun, Y.-r

    J. Braun, Y.-r. Chen, W.-j. Fu, F. Gao, C. Huang, F. Ihssen, K. Kockler, Y. Lu, J. M. Pawlowski, F. Rennecke, F. R. Sattler, J. Stoll, Y.-y. Tan, Z.-n. Wang, R. Wen, J. Wessely, S. Yin, H.-w. Zheng, N. Zorbach, fQCD collaboration,https://fqcd-collaboration.github.io/(2026)

    2026

  46. [46]

    S. P. Klevansky, The Nambu-Jona-Lasinio model of quantum chromodynamics, Rev. Mod. Phys. 64 (1992) 649–708.doi:10.1103/RevModPhys.64.649. 37

    work page doi:10.1103/revmodphys.64.649 1992