##### volume **II:** Synopsis

#### part III: Modern Physics

### page 19: 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. Their objection was based on confusion of 'Platonic' mathematical existence and real existence. Hilbert pioneered the 'formalist' approach to mathematics which sees mathematics as a game played with symbols whose only constraint is consistency. In Hilbert's view, the only requirement for mathematical existence is self consistency, and the transfinite numbers are self-consistent. Cohen, David Hilbert - Wikipedia

Hilbert showed that Cantor's new infinities fitted easily into mathematics and described new class of infinite spaces known as function spaces. John von Neumann used the function space now called 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. John von Neumann - Wikipedia, Hilbert space - Wikipedia, von Neumann

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 represent this mapping by the expression *y = f(x). *Some functions can be represented very simply as an algebraic expression, eg * y = x ^{2}*. Then if

*x*= 1,

*y*= 1;

*x*= 2,

*y*= 4;

*x*= 3,

*y*= 9; and so on.

This may be 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 *ℵ _{0}* natural numbers, there are

*ℵ*functions whose range and domain are the natural numbers. Each of these functions is represented by a single point in a suitable function space. The information carried by a point in a space is the same as the complexity or entropy of the space.

_{0}! = ℵ_{1}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 of the particles observed. Dirac

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. Tensor product - Wikipedia

It is this exponential 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.

(revised 23 May 2013)