Quantum Theory From Three Reasonable Abstract Axioms

In my previous posting ( Quantum mechanics and axiomatic methods in physics ) , I argued about the need for a better axiomatization of quantum mechanics if we are to end the long debate about foundations of the subject. In the next four postings , I will put forward an axiomatization of quantum mechanics via an abstract axiomatic framework called S - formalization , where S is an arbitrary physical system. This abstract framework starts with three basic axioms ( therefore the title ) , with these basic axioms we are able to derive not just standard quantum mechanics, but also classical mechanics and the algebraic approach to quantum mechanics by gradually adding new axioms ! Which is exactly what Hilbert asked for when he wrote :

If geometry is to serve as a model for the treatment of physical axiom , 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 a more special theories.........The mathematician will have also to take into account not only 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 in his famous paper quant -ph/0101012 , Lucien Hardy proposed to axiomatize quantum mechanics via his " Five Reasonable Axioms ". Although I took the word " Reasonable " from his paper , our motives and approaches are drastically different ! I am motivated by Hilbert sixth problem on axiomatization of physics , whereas Lucien`s motivation was essentially to find an operational justification for the Hilbert space mathematical apparatus of the subject.

1 Some Mathematical Concepts

Let X1 , X2 be sets and let Map ( X1 , X2 ) be the set of all maps from X1 to X2.

Definition 1. A bridge from set X1 to X2 is a pair written ( X1 , Â( X1 , X2 ) ) such that Â( X1 , X2 ) ⊆ Map ( X1 , X2 ).

The subset Â( X1 , X2 ) is called a bridge support. When X2 ⊆ X1 then the bridge ( X1 , Â( X1 , X2 ) ) is called an auto - bridge on X1. For our purposes , X1 and X2 are either mathematical structures or contained in some mathematical structure.Obviously a category theorist can define the notion of bridge as follows : Let X1 , X2 be objects in a category C. A bridge from object X1 to X2 is a pair written ( X1 , Â( X1 , X2 ) ) such that Â( X1 , X2 ) ⊆ Hom ( X1 , X2 ). Hence one can then recover definition 1 on sets by putting C = Sets. Indeed , one can make a lot of interesting mathematical constructions with bridges , but in the next postings I will just focus on setting up the general axiomatic framework rather than mathematical constructions. Hopefully there will be a follow up by me , or by the reader given more emphasis on mathematical constructions !

Some obvious examples

1. Let H be a separable complex Hilbert space and let ad( H ) be the set of all self - adjoint operators on H. The pair ( H , ad( H ) ) is an auto - bridge on H.

2. Let M be a symplectic manifold and let C∞( M ) be the set of all real valued smooth maps on M. Now let R be the set of real numbers , then obviously the pair ( M , C∞( M ) ) is a bridge from M to R.

3. Let C be the set of complex numbers. Now let A be a unital C* - algebra and
P* ( A , C ) be the set of all positive linear functionals on A. Then the pair ( A , P* ( A , C ) ) is a bridge from A to C.

Definition1.1. Let ( X1 , Â( X1 , X2 ) ) be a bridge and let X3 be a nonempty set. An extension of ( X1 , Â( X1 , X2 ) ) over X3 is ( X1 , Â( X1 , X2 ) ∪ X3 ).

Definition 1.2 . ( X1 , Â( X1 , X2 ) ∪ X3 ) is called a proper extension if the intersection Â( X1 , X2 ) ∩ X3 is empty , otherwise it`s an improper extension.


Note : If X3 ⊆ Â( X1 , X2 ) then ( X1 , Â( X1 , X2 ) ∪ X3 ) = ( X1 , Â( X1 , X2 ) ). Hence a bridge ( X1 , Â( X1 , X2 ) ) can always be seen as an improper extension over X3 ⊆ Â( X1 , X2 ) ! This indeed , is the most economical way of recovering standard quantum mechanics within the S - formalization framework !


<sub>2 The Mathematical Frameworks of Classical and Quantum mechanics

The general standard classical and quantum mechanical frameworks for describing a physical system S consist in the following schemes :

Classical mechanics ( Hamilton aproach ). We first indentify a symplectic manifold M ( phase space ) , then we indentify the states of our system S with points in M and the observables with real valued smooth maps on M. The dynamics of the system is modeled with the flow generated by the Hamiltonian vector field on M.

Quantum mechanics ( Hilbert space approach ). We first indentify a separable complex Hilbert space H , then we identify the states of our system S with the unit vectors in H and the observables with self - adjoint operators on H. Now in respect to the dynamics , there are two approaches :

1. Schrodinger picture. The dynamics is modeled with the so - called Schrodinger equation that we`ll meet up later. More precisely the state vectors obey the Schrodinger equation.

2. Heisenberg picture. The dynamics is modeled with the so - called Heisenberg equations.Here the operators representing the observables obey the Heisenberg equations.

Another approach to quantum mechanics , is the algebraic approach. Here we start with a unital C* - algebra A , then we identify the states of our system with positive linear functionals on A and the observables with self - adjoint elements of A.

2.1 The relationship between classical and quantum mathematical frameworks

As we can observe from above , the language of classical mechanics is symplectic geometry whereas the language of quantum mechanics is linear algebra ( with sophisticated functional analysis ). However according to a physicists convention , if ψ ∈ H represents some state of our system S and β ∈ C is non zero , then βψ represents the same state as ψ ! If we accept this convention , then the quantum mechanical state space is actually the projective Hilbert space PH , .ie. the space of non zero vectors with equivalence relation</sub> ψ ~
ψ` iff there exists a non zero β ∈ C such that ψ = β ψ`.
Now PH has obviously interesting geometrical properties , for instance it can be into a Kahler manifold. In fact many attempts have been to geometrize quantum mechanics via PH ( see reference 7 for example ). Now the attempts to geometrize quantum mechanics , are undermined by the lack of the so - called superposition principle, which is a key conceptual tool for very important practical applications of the subject.Indeed , for most practical applications of quantum mechanics , the linear algebraic structure of the Hilbert space is more predominant than the geometric structure induced by Hilbert space itself.
Given the differences between the classical and quantum mathematical frameworks , an attempt known by quantization was introduced , I will not get into much details but roughly speaking , the goal of quantization is to find a meningful general procedure that produces a quantum mechanical description of a system S from its classical description ! This obviously means that , such a general proceduce must :

1. Produce a meaningful Hilbert space H from the symplectic manifold M underlying the classical description of system S.

2. Produce a meningful self - adjoint operator f^ on H for every real valued smooth functions f ∈ C∞( M ) representing an observable of system S.

3. Produce a meaningful commutation relation iℏ [ f^ , g^ ] from Poisson bracket { f^ , g^ } = ω ( Xf , Xg ) , where ω is the the symplectic form on M.
Now it turns out that , finding a meaningful quantization of a system S from its classical description is physically and mathematically a very hard task ! In fact , there are the so - called " No go theorems "
showing the impossibility of meaningful general quantization over a vast class of symplectic manifolds ( phase spaces ) including the standard one. Geometric quantization is one of the popular approaches to quantization amongst mathematical physicists and mathematicians.
From foundational point of view , the goal in the next three postings is to develop a general axiomatic framework that will serve as a commom ground for both classical and quantum mathematical frameworks !

Note : This is the first of four planned postings , the next one will follow as soon as possible. In meantime , please leave your comments or corrections via the comment box or via baldeserifo@gmail.com. Thanks for reading !

Reading suggestions ( I feel lazy to writte the full details of the suggestions below , please google or search in arxiv.org for respective suggestions if you wish more details ! )

1. David Hilbert and the Axiomatization of Physics , Leo Corry.

2. Foundations of Mechanics , Ralph Abraham and Jerrold E. Marsden.

3. Mathematical Foundations of Quantum Mechanics , J. von Neumann.

4. Principles of Quantum Mechanics , P.M. Dirac.

5. Geometric Quantization , N. Woodhouse.

6. Quantization , Poisson brackets and beyond , Theodore Voronov.

7. Geometrical Formulation of Quantum Mechanics , Abhay Ashtekar and Troy A. Schilling.

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.