Algebraic approach to Physics and Applications

[Sonnet Nguyen <Sonnet.Nguyen@fuw.edu.pl>]

Dear academics,

Last time I've observed a strong and hot tendency of algebraization in
physics.  I'm not sure how many percent of my observations was correct, so
forgive me for posting my "primitive thoughts" here ...

Geometrization of physics is nothing new for theoretical physicists.  
It's shown everywhere in classical physical theories, i.e. in mechanics, 
electrodynamics, general relativity, etc..  An analogical 
tendency in quantum physics is called geometrical quantization theory,  

Does exist sth called algebraization of physics?  Is it important for 
development of physical theories? etc.

Because it's far enough to say about algebraization of physics, but
first, may be few words explaining what is algebraic approach to physics.  
In physics, there are two basical concepts: states and observables.  
Value of any measurement is an expectational value of observable acting 
on state.  This schema of thought works perfectly not only in Physics, but 
also in other sciences.   
Basical concept of the algebraic approach to physics is to reverse the
roles played by states and observables in the following sense:
one begins by constructing observables as an elements of an abstract algebra.
One then  defines states as objects which act upon observables by associating
a real (or more general: field's) numbers to each observable.  This action 
corresponds to taking expectation values in the standar approach.
The advantage of this approach is that it thereby allows one to treat all 
states - and in particular, states arising in unitarily inequivalent 
constructions - thereby enabling one to define the theory without the need 
to select a prefered constructions.


Because Many VNSA-netters are very good physicists and few of them are 
experts of quantum field theorem, then Let consider the quantum field theory 
as an example to understand why algebraic approach is required. 
(Hope specialists of QFT give me a lession, if I lie sth  here...)  
In (canonical) Quantum theory, quantum field is simply a set 
of harmonic oscillators (Let image it as a space with oscilators at 
every point).  The states are labbeled by particle occupations numbers:
in the vaccuum state all oscillators are in their ground state.  The 
n-particle state can be constructed from vaccum state by acting on it n times 
by the creation operator.  Other linear combinations of the creation operators
give other states.  Field observables are operators which acting on the state 
vectors.     
But this construction doesn't work if we need a quantum field theory 
in curved space (i.e. consistent to general relativity).  In fact, there is 
no such things as particle physics in general relativity because it's 
impossible to define anambigously what constitues a particle:  
what seems to be vacuum state in one coordinate system 
is full of particles in another !!!!!!!
How we can do particle physics in curved space??????

There is ambiguity about what we call a particle but no about what carefully 
defined particle detector will detect when following a given world
line.  What observers measure is not the state but the expectation value of 
an operator in a given state.  (In fact we try to avoid making vague 
statements of physical nature about particles and energy, where possible 
starting our conclusions in operational terms - what observer moving in such
and such way would actually measure with a particular piece of apparatus.)
More or less in this spirit, B. Kay have used algebraic approach to build 
a quantum field theory in curved space-time.  (Phys. Raport. 1991, 207).

Hope I'll be able to present a conprehensive introduction to quantum
groups theory and their (eventual) applications to Phys, in next week.

enjoys,
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|   Sonnet Nguyen,  Polish Acad. of Scie.               |
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