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It is claimed here that toward the end of his life Werner Heisenberg worked on a non-linear broadening or generalization of quantum mechanics. What work was that? Was it published?

Is it listed in Cassidy's Werner Heisenberg: A Bibliography of His Writings?

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This likely refers to Heisenberg's non-linear spinor theory of elementary particles, on which he worked from 1953 to the end of his life. There was a prominent write-up in an unpublished 1958 preprint with Pauli, from which Pauli withdrew his name. Heisenberg later gave a lecture Quantum Electrodynamics in the Nonlinear Spinor Theory (1966) and published a book An Introduction to the Unified Theory of Elementary Particles (1967) based on his Munich collaborations with Durr et al. In the book Heisenberg proposes that all properties of matter should follow from a highly symmetrical field equation that plays a role of "Plato's form".

Heisenberg's publications are listed on Academictree. The latest on the spinor theory is from 1974, The unified field theory of elementary particles: Some recent advances. Considering his health problems in 1973-6 it is unlikely that he worked much on it "in the last year", see The Munich Years (1958 – 1976), the 1975-6 publications are more historical/philosophical.

Here is Todorov's description:

"Heisenberg tries to build up a theory of a (not directly observable!) spinor field, which obeys a non-linear partial differential equation, with a coupling constant of dimension length square. The proton and the neutron, the building blocks of the atomic nucleus, are not identified with the quanta of the basic field (as it is the case with electrons in quantum electrodynamics), but arise as its bound states. Heisenberg does not follow the beaten tracks (not even those that brought him success in the old days), he is looking for something really new..."

In fact, Heisenberg is conscientious,that there are questions and “details”, which are not yet clear, but he is an optimist.For less emotionally engaged theorists the remaining difficulties and open problems set doubt on the whole enterprise. Shortly before his death later the same year (1958) Pauli withdraws his name from their joint work and expresses publicly his disapproval. Heisenberg is affected and disappointed but he continues steadfastly his work (with his young collaborator H.P. Dürr). He works, like most great physicists of the past century in their later years, in virtual isolation, with the world of science ignoring his latest ideas.

With the hindsight of subsequent development we can now point to both a prophetic vision and some shortcomings in this last attempt of the great scientist at anew breakthrough in fundamental physics... The quarks’ field in what we now call the standard model can be viewed as a realization of Heisenberg’s idea. (He discusses himself such a possibility after the first success of the quark model – some 10 years after his controversial 1958 paper.) In both cases the observable (strongly interacting) hadrons are bound states of the basic field whose quanta do not correspond to free particles. The idea of a spontaneously broken symmetry, associated today with the name of Higgs, plays a prominent part inthe Heisenberg model: it originates in his treatment of ferromagnetism of the 1930’s.The concept of a fundamental length is present in the now fashionable “superstring theory”... Heisenberg is well aware of the non-renormalizability of his model but he hopes that the gauge invariant equations of quantum electrodynamics will follow from his theory at distances large compared to the fundamental length."

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    $\begingroup$ Why did Pauli disapprove of Heisenberg's non-linear spinor theory? $\endgroup$
    – Geremia
    Sep 12 '20 at 20:24
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    $\begingroup$ @Geremia He got tired of Heisenberg's metaphysical infatuations, and his self-promotion in newspapers and on the radio. As he wrote:"“What fascinates Heisenberg is also the mirror archetype... As a consequence of this fascination, he reproduces his obsessions over and over again and cannot listen to (or read) what others say". See e.g. Pauli's Ideas on Mind and Matter by Atmanspacher. $\endgroup$
    – Conifold
    Sep 12 '20 at 21:04
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As Conifold already mentioned, there were several problems with this initial approach: spinors where observable physical objects due to their dimension of $-\frac{3}{2}$, the introduction of a dimensioned coupling constant lead to non-renormalizability, etc. I have been following this topic for a while now and just recently came across this video proposing changes to this initial approach, leading to some quite interesting results. I think it is worth to watch:

A non-linear spinor theory of elementary particles

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