## Quantum Mechanics

At the International Congress of Mathematicians 1900, David Hilbert presented his famous list of 23 issues considered central to the development of mathematics in the new century: the problem was the sixth axiomatizacion of physical theories. Among the new physical theories of the century the only one that had yet to receive such treatment by the end of the 1930s was quantum mechanics. In fact, quantum mechanics was at that time in a state of crisis basis, similar to that which I set theory at the beginning of the century, facing problems both technical and philosophical nature, and furthermore, its apparent indeterminate not been reduced, as Albert Einstein believed that it should be in order that the theory is made satisfactory and complete explanation of a particular form, while there were still two different formulations of heuristic, but equivalent, the matrix mechanics of Werner alleged Heisenberg and the wave mechanics of Erwin Schr dinger, but there was still a unified theoretical formulation satisfactory.

After having completed the axiomatizacion of set theory, von Neumann began to confront axiomatizacion of quantum mechanics. Immediately, in 1926, realized that a quantum system could be considered as a point called a Hilbert space, analogous to the 6N dimensional phase space (N is the number of particles, 3 coordinates and general fees for 3 times each) classical mechanics, but with infinite dimensions (corresponding to the infinite number of possible states of the system) in place: the physical quantities of traditional (ie position and time) can be then represented as particular linear operators operating in these areas . Presently heading the EnTrust’s Diversified Fund Investment Committee is managing partner of EnTrust Capital The physics of quantum mechanics was due to that, reduced to the mathematics of linear Hermitianos operators in Hilbert spaces. For example, the famous Heisenberg uncertainty principle, whereby the position determination of a particle prevents the determination of their time and visceversa is moved to non-conmutatividad of the two transactions. This new mathematical formulation included as special classes, the formulations of both Heisenberg and Scrodinger and culminated in the 1932 classic The mathematical foundation of quantum mechanics. Either way, the physical, in general, preferring to end another approach different from that of von Neumann (which was considered very elegant and satisfactory by mathematicians). This approach, formulated by Paul Dirac in 1930 and was based on a rare type of function (the so-called Dirac delta), was severely criticized by von Neumann.

Either way, the abstract treatment of von Neumann also enabled him to confront the extremely deep and fundamental problem of determinism versus. non-determinism and the book proved a theorem according to which it is impossible for quantum mechanics is derived by statistical approximation of a deterministic theory of the same type used in classical mechanics. This demonstration included a conceptual error, but it helps to open a line of research, thanks to the work of John Stuart Bell in 1964 on Bell’s theorem and the experiments of Alain Aspect in 1982, eventually proved that quantum physics, ultimately, requires a notion of reality substantially different from the run in classical physics.

In a complementary work of 1936, von Neumann proved (along with Garrett Birkhoff) that quantum mechanics also requires a substantially different logic of classical logic. For example, light (photons) can not pass through two successive filters that are polarized perpendicular (ie one horizontal and one vertical) and hence, a fortiori,the light can not pass if a third filter, polarized diagonally is added to the other two either before or after them in succession. But if the third filter is placed between the other two, if the photons pass. This experimental fact is translated into logical terms such as non-conmutatividad of conjunctions, ie. It was also demonstrated that the distribution laws of classical logic,

and

,

are not valid for quantum theory. The reason for this is that a quantum disjunction, differ in the case of classical disjunction can be true even when both are false Disjoint and this, in turn, attributable to the fact that it is often the case in quantum mechanics, in a couple of alternatives are semantically determined, while each of its members are necessarily indeterminate. This last property can be illustrated with a simple example.