vol 2: Synopsis
part III: Modern physics
page 19: John von Neumann
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John von Neumann
(1903-1957)
When George Cantor first announced the transfinite numbers, some theologians objected on the ground that the only actual infinity in existence is God. David Hilbert showed that these new infinities fitted easily into mathematics and described new class of infinite spaces known as function spaces. John von Neumann used Hilbert space to resolve the apparent conflict between the particle and wave (discrete and continuous) descriptions of the world, opening the way for the consistent development of quantum theory.
Hilbert space is a function space. This means that each point in the space represents a function. A function is a mapping between a set of elements called the domain of the function (say x ) to a set called the range of the function (say y ). We write this y = f(x). Some functions can be represented very simply as an arithmetic expression, eg y = x 2. Then if x = 1, y = 1; x = 2, y = 4; x = 3, y = 9; and so on.
This is a function whose range and domain are the natural numbers. When no law exists that enables the succinct expression of a function, we must represent it by a table of values. We note here that there are n! ways of mapping n things onto themselves. Since there are aleph(0) natural numbers, there are aleph(0)! functions whose range and domain are the natural numbers.
In Hilbert space, functions are expressed as ordered lists of values called vectors. For a given application, the dimension of the appropriate Hilbert space must be equal to complexity of the state represented, ranging from 2 for the spin states of an electron to a countable infinity for the energy states of a hydrogen atom. In this respect, Hilbert space is a natural extension of ordinary three dimensional space.
Quantum mechanics represents physical motion, that is changes of state, by operators which transform state vectors into one another. Since a change of state in the quantum world is generally accompanied by the emission or absorption of observable particles, quantum mechanics uses operators to represent observables. Different operators acting on state vectors yield the energy, momentum and angular momentum on the particles involved.
When we come to consider two or more particles, the Hilbert space we need is the tensor product of the Hilbert spaces for the original particles. The size of the resulting Hilbert space grows exponentially with the number of particles represented, in the same way as the size of the number represented by an 'Arabic' numeral grows exponentially with the length of the numeral.
It is this growth that leads us to suspect that the 'state vectors of the universe' are so large that they require the formal Cantor universe for their adequate representation.
Books
| Hilbert, David, and Leon Unger (translator, from the tenth German edition). Revised and Enlarged by Paul Bernays, Foundations of Geometry (Grundlagen der Geometrie), Open Court 1999 Jacket: 'Along with the writings of Hilbert's friend and correspondent Frege, Hilbert's Grundlagen der Geometrie is the major prop that set the stage for Russell and Whitehead's Principa Mathematica. Hilbert presents a new axiomatization of geometry, the reduction of geometry to algebra, and introduces the distinction between mathematics and metamathematics, with a new theory of proof. This edition is translated from the tenth German edition, including all the improvements which Hilbert derived from his own reflections and the contributions of other writers. Amazon back |
| Pais, Abraham, 'Subtle is the Lord...': The Science and Life of Albert Einstein, Oxford UP 1982 Jacket: In this ... major work Abraham Pais, himself an eminent physicist who worked alongside Einstein in the post-war years, traces the development of Einstein's entire ouvre. ... Running through the book is a completely non-scientific biography ... including many letters which appear in English for the first time, as well as other information not published before.' Amazon back |
| Reid, Constance, Hilbert-Courant, Springer Verlag 1986 Jacket: '[Hilbert] is woven out of three distinct themes. It presents a sensitive portrait of a great human being. It describes accurately and intelligibly on a non-technical level the world of mathematical ideas in which Hilbert created his masterpieces. And it illuminates the background of German social history against which the drama of Hilbert's life was played. ... Beyond this, it is a poem in praise of mathematics.' Science Amazon back |
| von Neumann, John, and Robert T Beyer (translator), Mathematical Foundations of Quantum Mechanics, Princeton University Press 1983 Jacket: '... a revolutionary book that caused a sea change in theoretical physics. ... JvN begins by presenting the theory of Hermitean operators and Hilbert spaces. These provide the framework for transformation theory, which vN regards as the definitive form of quantum mechanics. ... Regarded as a tour de force at the time of its publication, this book is still indispensible for those interested in the fundamental issues of quantum mechanics.' Amazon back |
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