mathematics

Huzihiro Araki

Huzihiro Araki is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Huzihiro Araki rather than just read about it. In short: Huzihiro Araki (荒木不二洋, Araki Fujihiro; 28 July 1932 – 16 December 2022) was a Japanese mathematical physicist and mathematician who worked on the foundations of quantum field theory, on quantum statistical mechanics, and on the theory of operator algebras. Biography Araki is the son of the University of Kyoto physics professor Gentarō Araki, with whom he studied and with whom in 1954 he published his first physics…

Key takeaways

Huzihiro Araki belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Huzihiro Araki to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Huzihiro Araki from memory before moving on to harder problems.

Reference excerpt

Huzihiro Araki (荒木不二洋, Araki Fujihiro; 28 July 1932 – 16 December 2022) was a Japanese mathematical physicist and mathematician who worked on the foundations of quantum field theory, on quantum statistical mechanics, and on the theory of operator algebras.

Biography Araki is the son of the University of Kyoto physics professor Gentarō Araki, with whom he studied and with whom in 1954 he published his first physics paper. He earned his diploma under Hideki Yukawa and in 1960 he attained his doctorate at Princeton University with thesis advisors Rudolf Haag and Arthur Wightman. He was a professor at the University of Kyoto starting in 1966, and became the director of the Research Institute for Mathematical Sciences (RIMS). Araki died on 16 December 2022.

Research Araki worked on axiomatic quantum field theory, statistical mechanics, and in particular on applications of operator algebras like von Neumann algebras and C*-algebras. At the beginning of the 1960s, in Princeton, he made important contributions to local quantum physics and to the scattering theories of Haag and David Ruelle. He also supplied important contributions in the mathematical theory of operator algebras, classifying the type-III factors of von Neumann algebras. Araki originated the concept of relative entropy of states of von Neumann algebras. In the 1970s he showed the equivalence in quantum thermodynamics of, on the one hand, the KMS condition (named after Ryogo Kubo, Paul C. Martin, and Julian Schwinger) for the characterization of quantum mechanical states in thermodynamic equilibrium with, on the other hand, the variational principle for quantum mechanical spin systems on lattices. With Yanase he worked on the foundations of quantum mechanics, i.e. the Wigner-Araki-Yanase theorem, which describes restrictions that conservation laws impose upon the physical measuring process. Stated in more precise terms, they proved that an exact measurement of an operator, which additively replaces the operator with a conserved size, is impossible. However, Yanase did prove that the uncertainty of the measurement can be made arbitrarily small, provided that the measuring apparatus is sufficiently large.

Honors and awards Huzihiro Araki was an invited speaker at the International Congress of Mathematicians in 1970 in Nice and in 1978 in Helsinki. He was the second president of the International Association of Mathematical Physics, during the period 1979–1981. In 2003 he received, together with Oded Schramm and Elliott Lieb, the Henri Poincaré Prize. In 1990 he was the chief organizer of the International Congress of Mathematicians in Kyoto. He was editor of the scientific journal Communications in Mathematical Physics and founder of Reviews in Mathematical Physics. In 2012 he became a fellow of the American Mathematical Society.

Selected works

Illustrations

Worked examples

Example 1 — a first encounter with Huzihiro Araki

Start with the simplest possible case. Write down what Huzihiro Araki claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Huzihiro Araki before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Huzihiro Araki ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Huzihiro Araki

In research: Huzihiro Araki appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Huzihiro Araki in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Huzihiro Araki is common in secondary-school and first-year university syllabi. It links to neighbouring topics 1932 births, 2022 deaths, 20th-century Japanese mathematicians, so understanding it makes those chapters shorter.
In everyday life: Look for Huzihiro Araki outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Huzihiro Araki in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Huzihiro Araki means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Huzihiro Araki out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Huzihiro Araki in simple terms?

Why does Huzihiro Araki matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Huzihiro Araki?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Huzihiro Araki.