mathematics

Paul Chernoff

Paul Chernoff 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 Paul Chernoff rather than just read about it. In short: Paul Robert Chernoff (21 June 1942, Philadelphia – 17 January 2017) was an American mathematician, specializing in functional analysis and the mathematical foundations of quantum mechanics. He is known for Chernoff's Theorem, a mathematical result in the Feynman path integral formulation of quantum mechanics.

Key takeaways

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

Reference excerpt

Paul Robert Chernoff (21 June 1942, Philadelphia – 17 January 2017) was an American mathematician, specializing in functional analysis and the mathematical foundations of quantum mechanics. He is known for Chernoff's Theorem, a mathematical result in the Feynman path integral formulation of quantum mechanics. He was also the author of limericks.

Education and career Chernoff graduated from Central High School in Philadelphia. He matriculated at Harvard University, where he received bachelor's degree, summa cum laude, in 1963, master's degree in 1965, and Ph.D. in 1968 under George Mackey with thesis Semigroup Product Formulas and Addition of Unbounded Operators. At the University of California, Berkeley, he became in 1969 a lecturer, in 1971 an assistant professor, and in 1980 a full professor. U. C. Berkeley awarded him multiple Distinguished Teaching Awards and the Lili Fabilli and Eric Hoffer Essay Prize. In 1986 he was a visiting professor at the University of Pennsylvania. Chernoff was elected in 1984 a Fellow of the American Association for the Advancement of Science and in 2012 a Fellow of the American Mathematical Society. He gave in 1981 a simplified proof of the Groenewold-Van Hove theorem, which is a no-go theorem that relates classical mechanics to quantum mechanics.

Selected publications Chernoff Paul (1968). "Note on product formulas for operator semigroups". J. Funct. Analysis. 2 (2): 238–242. doi:10.1016/0022-1236(68)90020-7. Chernoff, Paul; Rasala, Richard; Waterhouse, William (1968). "The Stone-Weierstrass theorem for valuable fields" (PDF). Pacific Journal of Mathematics. 27 (2): 233–240. doi:10.2140/pjm.1968.27.233. Chernoff Paul (1972). "Some remarks on quasi-analytic vectors". Trans. Amer. Math. Soc. 167: 105–113. doi:10.1090/S0002-9947-1972-0295125-5. Chernoff Paul (1973). "Representations, automorphisms, and derivations of some operator algebras". J. Funct. Analysis. 12 (3): 275–289. doi:10.1016/0022-1236(73)90080-3. Chernoff Paul (1973). "Essential self-adjointness of powers of generators of hyperbolic equations". J. Funct. Analysis. 12 (4): 401–414. doi:10.1016/0022-1236(73)90003-7. Paul Chernoff Product formulas, nonlinear semigroups, and addition of unbounded operators, American Mathematical Society 1974. Paul Chernoff; Jerrold Marsden: Properties of infinite dimensional Hamiltonian systems, Springer 1974 Chernoff Paul (1976). "Understanding mathematical proofs: Conceptual barriers". Science. 193 (4250): 276. doi:10.1126/science.193.4250.276.a. PMID 17745711. Chernoff Paul (1977). "The quantum n-body problem and a theorem of Littlewood" (PDF). Pacific J. Math. 70: 117–123. doi:10.2140/pjm.1977.70.117. Chernoff Paul (1995). "Irreducible representations of infinite-dimensional transformation groups and Lie algebras, I.". J. Funct. Anal. 130 (2): 255–282. doi:10.1006/jfan.1995.1069. Chernoff Paul, Hughes Rhonda (1993). "A new class of point interactions in one dimension". Journal of Functional Analysis. 111 (1): 97–117. doi:10.1006/jfan.1993.1006. Chernoff Paul, Hughes R (1996). "Some examples related to Kato's conjecture". J. Austral. Math. Soc. Ser. A. 60 (2): 274–286. doi:10.1017/S1446788700037666. Chernoff Paul (2000). "Quantization and irreducible representations of infinite-dimensional transformation groups and Lie algebras" (PDF). Eletron. J. Differ. Equ. Conf. 4. Proceedings of the Symposium on Mathematical Physics and Quantum Field Theory (Berkeley, CA, 1999): 17–22. Chernoff Paul (2000). "A pseudo zeta function and the distribution of primes". Proc. Natl. Acad. Sci. USA. 97 (14): 7697–7699. doi:10.1073/pnas.97.14.7697. PMC 16606. PMID 10884402. (There is a typographical error: "One can show that C(s) may be analytically continued at least into the half-plane Re s > 0 except for an isolated singularity (presumably a simple pole) at s = 0." This should be "at s = 1" according to the mathematical argument given.)

References

Illustrations

Worked examples

Example 1 — a first encounter with Paul Chernoff

Start with the simplest possible case. Write down what Paul Chernoff 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 Paul Chernoff 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 Paul Chernoff 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 Paul Chernoff

In research: Paul Chernoff 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 Paul Chernoff 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: Paul Chernoff is common in secondary-school and first-year university syllabi. It links to neighbouring topics 1942 births, 2017 deaths, 20th-century American mathematicians, so understanding it makes those chapters shorter.
In everyday life: Look for Paul Chernoff 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 Paul Chernoff in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Paul Chernoff 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 Paul Chernoff out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Paul Chernoff in simple terms?

Why does Paul Chernoff 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 Paul Chernoff?

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 Paul Chernoff.