## Monday, 2 November 2009

### Quantum mechanics and axiomatic methods in physics

Quantum mechanics is without doubts the most challenging subject in modern physics. In one hand , its predictions have been successfully verified with great accuracy over the last 85 years. On the other hand , the foundations of the subject still for many physicists obscure and puzzling ! Now we may ask ourselves how come ? There are quite many reasons to be intrigued with quantum mechanics , but I will consider the following one :

Prior to quantum mechanics , from Newton to Einstein , a fundamental physical theory has always evolved from simple and intuitive physical principles. Take for example Newton`s principle “ An object remains at rest or uniform motion unless acted upon by an external force “ or Einstein`s principle The speed of light is the same in all inertial frames “. With these simple principles , both physicists were able to successfully derive the mathematical frameworks of their respective theories. On the other hand , standard quantum mechanics as formulated in the late twenties does not provide any physical principle that justifies the use of its abstract mathematical framework. In fact , many attempts have been made to find the physical principles ( if there is any ! ) underlying the abstract mathematical framework of the subject , but so far none of the attempts have gained a universal acceptance.

The abstract Hilbert space approach to quantum mechanics is due to von Neumann , it was definitely the starting point for applications of axiomatic methods to quantum mechanics. Few years earlier , there were two versions of quantum mechanics , namely Heisenberg`s matrix mechanics and Schrodinger wave mechanics. However von Neumann realized that the two versions had something deep in common , that deep was of course the mathematical structure of “ separable complex Hilbert space. Indeed the “ separable “ condition was initially included in the definition of Hilbert space , but later it was dropped in order to include non - separable Hilbert spaces !

Now before von Neumann’s Hilbert space approach to quantum mechanics , David Hilbert proposed his sixth problem , where he calls for the axiomatization of physics ( see David Hilbert and the Axiomatization of Physics by Leo Corry ).In his words :

To treat in the same manner, by means of axioms , the physical sciences in which mathematics plays an important part ; in the first rank are the theory of probabilities and mechanics.

Also in his own words :

If geometry is to serve as a model for the treatment of physical axioms, we shall try first by a small number of axioms to include as large a class as possible of physical phenomena, and then by adjoining new axioms to arrive gradually at the more special theories.…. The mathematician will have also to take into account not only of those theories coming near reality , but also , as in geometry , of all logically possible theories. He must be always alert to obtain a complete survey off all conclusions derivable from the system of axioms assumed.

Now since physicists are still debating about the foundations of quantum mechanics for decades now , without any general consensus , can a better axiomatization of quantum mechanics kill the debate ?

My view is yes , a better axiomatization of quantum mechanics will not only kill the debate , but will also give a new insights on Hilbert sixth problem by bringing quantum mechanics close to pure mathematics ! But , what I mean with better “ axiomatization ?

By a better axiomatization , I mean an axiomatization that starts with a small number of fundamental axioms that enable us to derive the Hilbert space approach , the algebraic approach and classical mechanics as special cases !

But why do we need to add more abstraction ? Isn`t abstraction the main problem of quantum mechanics ?

First in my view , quantum mechanics is just a very special case of a general axiomatic scheme that hasn`t been yet clarified or fully understood by physicists ! This axiomatic scheme is abstract by nature , it often lacks the so - called “ physical principles “ , but nonetheless it has an astonishing power of prediction. Second , I think abstraction is not a problem , but the only viable option to clarify and give this scheme a firm foundations. In this regard , let us take the example of classical probability theory. Before Kolmogorov , the foundations of the subject was very obscure , indeed some thought it would remain obscure forever ! However , after Kolmogorov came along with his abstract axiomatic treatment of the subject using measure theoretic methods , the nature of the subject become more clear. Similarly I think we need a kolmogorov like approach to quantum mechanics , of course this doesn`t necessarily implies employing measure theoretic methods to such axiomatization of quantum mechanics.