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{{Infobox scientist
| name = Claes Johnson
| image = Claes Johnson 2948.jpg
| alt = Claes Johnson
| birth_date = 18 November 1943
| residence = Stockholm, Sweden
| nationality = Swedish
| fields = [[Mathematics]], [[Physics]], [[Thermodynamics]], [[Climatology]]
| workplaces = Chalmers University of Technology; KTH Royal Institute of Technology, Stockholm
| known_for = Finite element methods, Computational Blackbody Radiation, Many-Minds Relativity, Real Quantum Mechanics (realQM), New Theory of Flight, criticism of greenhouse-gas climate alarmism
}}

'''Claes Göran Johnson''' (born 1943) is a Swedish applied mathematician and professor emeritus of applied mathematics at KTH Royal Institute of Technology in Stockholm. Within mainstream science he is widely known for his work on the finite element method and adaptive computational methods for differential equations, with textbooks and papers that have been cited more than twenty thousand times. Outside the mainstream he is one of the most persistent and mathematically equipped critics of what he regards as unphysical foundations in modern physics: he has published book-length challenges to the statistical interpretation of the second law of thermodynamics, to blackbody radiation theory and "back radiation" greenhouse physics, to Einstein's special relativity, and to the Copenhagen interpretation of quantum mechanics.

==Biography==
Claes G. L. Johnson was born on 18 November 1943 in Mölndal, Sweden. He took his Ph.D. in mathematics at Chalmers University of Technology in 1973 under the supervision of Vidar Thomée, and became docent in numerical analysis at Chalmers in 1978. He held a professorship in applied mathematics at Chalmers University of Technology in Göteborg before moving to KTH Royal Institute of Technology in Stockholm, where he was professor of applied mathematics in the School of Computer Science and Communication (CSC) and is now professor emeritus.

Johnson has published on the order of a hundred scientific articles and a series of books on partial differential equations, finite element methods and numerical analysis. His doctoral students include Johan Hoffman, Anders Logg, Johan Jansson, Mohammad Asadzadeh, Larisa Beilina and Rickard Bergström, several of whom went on to lead the FEniCS automated finite element software project.

At Chalmers he initiated, with Kenneth Eriksson and Donald Estep, the reform program ''Applied Mathematics: Body and Soul'', a multi-volume synthesis of mathematical analysis ("Soul"), numerical computation ("Body") and application, motivated by the view that the computer has changed what mathematics is and how it should be taught. The same computational philosophy — that a physical theory should be a computable model of the real world, not a formal apparatus requiring interpretation — underlies all of his later heterodox work.

He writes prolifically at the blog ''CJ on Mathematics and Science'' (subtitled "towards understanding by critical constructive inquiry") and at a set of companion sites devoted to individual projects.

==Scientific contributions==

===Computational foundations and General Galerkin===
Johnson's mainstream contributions centre on the finite element method: ''Numerical Solution of Partial Differential Equations by the Finite Element Method'' (1987, reprinted by Dover) is a standard introduction, and with Eriksson, Estep and Peter Hansbo he wrote ''Computational Differential Equations'' (1996). He was an early developer of the discontinuous Galerkin method for hyperbolic problems and of adaptive finite element methods with a posteriori error control for parabolic problems.

With Johan Hoffman he developed General Galerkin (G2), a computational method for turbulent flow in which no turbulence model and no wall model are introduced: the numerical method's own stabilization plays the role of the physical dissipation, and mean-value quantities such as drag and lift are computed directly from the Euler or Navier–Stokes equations. This is presented in ''Computational Turbulent Incompressible Flow'' (Springer, 2007), volume 4 of the Body and Soul series.

===Resolution of d'Alembert's paradox and a new theory of flight===
Hoffman and Johnson published "Resolution of d'Alembert's Paradox" in the ''Journal of Mathematical Fluid Mechanics'' (vol. 12, 2010, pp. 321–334), arguing that the paradox is resolved not by Prandtl's boundary layer but by an instability of potential flow: slightly viscous bluff-body flow is zero-drag potential flow modified by a three-dimensional rotational slip separation which turns it into turbulent flow with nonzero drag and lift.

On this basis they proposed a "New Theory of Flight" (''Journal of Mathematical Fluid Mechanics'', 2015), a mathematical account of subsonic flight that is fundamentally different from the century-old Prandtl–Kutta–Zhukovsky boundary-layer/circulation theory. In their account the large lift generated at the leading edge by potential flow is preserved through the separation mechanism at the cost of only small drag, giving lift-to-drag ratios of 15–20 at the Reynolds numbers of real aircraft — and, crucially, this is obtained by computation from the equations rather than by fitted models. Johnson maintains a dedicated site, ''The Secret of Flight'', for this work, and has argued that the textbook explanation of lift is not merely incomplete but wrong.

===The second law of thermodynamics===
Johnson rejects the standard reduction of the second law to statistics and molecular disorder. In ''Computational Thermodynamics'' (Body and Soul vol. 5) he proposes instead a deterministic account in which irreversibility arises from finite precision computation: a physical process, like a numerical solution, cannot resolve arbitrarily fine scales, and the turbulent dissipation that results from this finite resolution is what produces the arrow of time. Entropy on this view is not a measure of ignorance or of microscopic disorder but an effect of the impossibility of exact resolution — a physical, computable quantity rather than a statistical bookkeeping device. This is the position he presented in his 2024 CNPS talk "Resolution of the Mystery of the 2nd Law of Thermodynamics".

===Computational Blackbody Radiation===
In ''Mathematical Physics of BlackBody Radiation'' and the associated Computational Blackbody Radiation project, Johnson derives Planck's radiation law from a deterministic wave model with finite precision computation, without quanta and without statistics of light particles. He argues that Planck's original statistical derivation was, as Planck himself described it, an act of despair, and that the quantization it introduced — and with it much of the foundational mystery of quantum mechanics — was unnecessary.

A central consequence is his treatment of two-way radiative exchange. Johnson argues that a body cannot absorb and thermalize radiation from a colder body: such radiation resonates with the warmer body without being converted into heat, so there is no circulatory flow of heat energy, because such a flow would be unstable. Radiative heat transfer between two bodies is therefore one-way, from hot to cold, in direct correspondence with the second law, and the "back radiation" that appears in standard greenhouse accounting is on his analysis a violation of the second law rather than a mechanism of warming.

===Climate and greenhouse physics===
Johnson is a prominent Swedish critic of greenhouse-gas climate alarmism. Applying his blackbody analysis to the atmosphere, he argues that the radiative forcing attributed to a doubling of atmospheric CO<sub>2</sub> is much smaller than the consensus estimates, and that atmospheric temperature structure is governed principally by thermodynamics — convection, gravitation and the lapse rate — rather than by radiative trapping. He published "Basic Thermodynamics of the Atmosphere" (2010) and lectured in 2011 on "CO2 Climate Alarmism Debunked by Mathematics", in which he also defended the scientific blogosphere as an uncensored arena for open criticism at a time when, in his view, the journals were closed to it. He has been sharply critical of what he sees as the handling of data in climate science, and in 2017 he was among the signatories of a petition organized by Richard Lindzen urging the United States to withdraw from international climate agreements.

===Many-Minds Relativity===
In ''Many-Minds Relativity'' (2011) Johnson argues that special relativity as usually taught is a formal mathematical theory without a coherent physical interpretation, defined by the Lorentz transformation connecting the measurements of observers in relative motion in place of the Galilean transformation of Newtonian mechanics. In his reading Einstein made a free mix of physics about reality with mathematics not about reality, and the contradictions that follow — the twin paradox and its relatives — are not deep truths about nature but symptoms of that confusion. His "many-minds" alternative assigns each observer a coordinate system with its own local clock and rods, so that what relativity describes is a relation between observers' descriptions rather than a property of space and time themselves. He develops the historical and philosophical side of this critique in ''Dr Faustus of Modern Physics'' (2011), where he presents the founders of modern physics as having traded physical understanding for formal power.

===Real Quantum Mechanics (realQM)===
Johnson's most sustained recent project is realQM, a reformulation of atomic physics as classical continuum mechanics in ordinary three-dimensional space. Instead of a wave function on a 3N-dimensional configuration space with a statistical Born interpretation, realQM uses a system of non-linear Schrödinger equations for N electron densities in three space dimensions with local, essentially disjoint supports, meeting at free boundaries. The model has, he argues, two decisive advantages over standard quantum mechanics: it has a direct physical meaning, since everything in it lives in real space, and it is computable without the approximations that standard quantum mechanics requires for anything beyond the simplest atoms. He regards this as a return to Schrödinger's original physical intention, against the Copenhagen interpretation that he holds Bohr and Born imposed over Schrödinger's and Einstein's objections.

An extension, RealNucleus, proposes that nuclear binding can be accounted for by Coulomb interaction between protons and electrons alone, without a strong or weak force, and Johnson argues against the Standard Model on the ground that it cannot predict nuclear binding energies from first principles. In related work he argues that magnetic phenomena can be derived from moving charge densities in ordinary space without invoking relativity.

==CNPS talks==
He has presented in the CNPS online seminar series:
* [https://www.youtube.com/watch?v=rT0dkhrE4rc "Resolution of the Mystery of the 2nd Law of Thermodynamics with Claes Johnson"] (9 November 2024)
* [https://www.youtube.com/watch?v=Pv7naqxiak0 "Where Critical Thinking Challenges Theory with Claes Johnson"] (3 February 2024)

==Works==

===Books===
* ''Numerical Solution of Partial Differential Equations by the Finite Element Method'' (Cambridge University Press, 1987; Dover reprint, 2009)
* ''Computational Differential Equations'' (with K. Eriksson, D. Estep and P. Hansbo; Cambridge University Press, 1996)
* ''Applied Mathematics: Body and Soul'', vols. 1–3 (with K. Eriksson and D. Estep; Springer, 2003–2004)
* ''Computational Turbulent Incompressible Flow'' — Applied Mathematics: Body and Soul vol. 4 (with J. Hoffman; Springer, 2007)
* ''Computational Thermodynamics'' — Applied Mathematics: Body and Soul vol. 5
* ''Dreams of Calculus: Perspectives on Mathematics Education'' (with J. Hoffman and J. Jansson)
* ''Many-Minds Relativity'' (2011)
* ''Dr Faustus of Modern Physics'' (2011)
* ''Mathematical Physics of BlackBody Radiation''
* ''The Secret of Flight''
* ''The Secret of Sailing''
* ''The Clock and the Arrow: A Brief Theory of Time''
* ''Real Quantum Mechanics''

===Selected papers===
* J. Hoffman and C. Johnson, "Resolution of d'Alembert's Paradox", ''Journal of Mathematical Fluid Mechanics'' 12 (2010), 321–334.
* J. Hoffman, J. Jansson and C. Johnson, "New Theory of Flight", ''Journal of Mathematical Fluid Mechanics'' (2015).
* C. Johnson, "Basic Thermodynamics of the Atmosphere" (2010).
* C. Johnson, "Computational Blackbody Radiation".
* C. Johnson, "Radiated Energy and the Second Law of Thermodynamics".
* C. Johnson, "An Analysis of the Discontinuous Galerkin Method for a Scalar Hyperbolic Equation" (1986).
* K. Eriksson and C. Johnson, "Adaptive Finite Element Methods for Parabolic Problems I" (1991).

==External links==
* [https://claesjohnson.blogspot.com/ CJ on Mathematics and Science] — his main blog
* [https://claesjohnsonmathscience.wordpress.com/ The World As Computation] — companion blog
* [https://physicalquantummechanics.wordpress.com/ Real Quantum Mechanics] — realQM project site
* [https://computationalblackbody.wordpress.com/ Computational Blackbody Radiation] — project site
* [https://secretofflight.wordpress.com/ The Secret of Flight] — project site
* [https://scholar.google.com/citations?user=40UhzoMAAAAJ Google Scholar profile]
* [https://www.researchgate.net/profile/Claes-Johnson-2 ResearchGate profile]
* [https://kth.academia.edu/ClaesJohnson Academia.edu profile]

[[Category:Scientist|Johnson Claes]]
[[Category:Worldwide List of Dissident Scientists]]
[[Category:Relativity|Johnson Claes]]
[[Category:Thermodynamics|Johnson Claes]]
[[Category:Mathematics|Johnson Claes]]
[[Category:Climatology|Johnson Claes]]

[[Category:Quantum Theory]]