Why algebra? Well, there is a natural construction that associates algebras with spaces. So, we should ask "why space?", and "why why?". There is a certain magic in algebras, and this magic attracts me. That is how I started to be interested in physics. Through the beauty and usefulness of mathematics involved in our handling of natural phenomena. Numbers and symbols - they seem to have something to do with how we can comprehend the mysteries of the world around us.
In this series we are playing with particular algebras - geometric
algebras related to the geometry of space and, perhaps, "time" as well.
We play with them like girls play with their dolls, and boys with their
toy soldiers (bad boys, I know, I was a bad boy too).
Probably the first algebra we ever meet is the algebra of sets - a
model of logic. It has "sum" (union of sets) and product (intersection
of sets) - that model logical operations "or" and "and". This is nicely
embedded into the algebra of real-valued functions through the use of
characteristic functions. Algebras of real or complex-valued functions
are commutative. But then came the discovery of quaternions, matrix
algebras, and then quantum mechanics in its Heisenberg's version, with
non-commuting complementary variables. John von Neumann developed it
into a whole big theory of "Algebras of operators". The new era begun.
Here we are playing with a toy: the geometric Clifford algebra of
space. It is 8-dimensional real by nature, but it is naturally equipped
with a complex structure. Then it starts to resemble the algebra of spin
1/2 with its Pauli matrices. Pure coincidence? Or there is something
deeper here? To ralate geometric algebra to quantum-mechanical spinors
we have followed the standard route of algebraists - we analyzed left
ideals, and we were able to re-discover Pauli matrices. But followers of
von-Naumann have found another way of constructing representations of
*-algebras, closer related to their use in quantum field theory. As a
result we have today "algebraic quantum field theory" that enables us to
handle certain difficulties that have been found with divergent
expressions. Today we understand that there is more than one way to provide a
Hilbert space than by constructing what is called "the Fock space". The quantum "vacuum" may be, perhaps, not so empty (In Fock space the ground state, the vacuum, is with zero particles).
So far we have just one spin, so there is no real need to use these advanced tools. But, perhaps, we can learn something new by using these tools to our toy? I want to play with what is called GNS-construction. Wikipedia has an article devoted to this subject: "Gelfand–Naimark–Segal construction". In the History section of this article we find:
"Gelfand and Naimark's paper on the Gelfand–Naimark theorem was published in 1943.[3] Segal recognized the construction that was implicit in this work and presented it in sharpened form.[4]."
But I want to start with something deeper, something that will give
us a real taste of the problem. So below is the full quote from the
opening part of Ref. [4] of the Wikipedia article.
Notes on the Gelfand-Neumark Theorem
RICHARD V. KADISON
Dedicated to Irving Kaplansky and Irving Segal with gratitude and respect.
ABSTRACT. The Gelfand-Neumark Theorem, the GNS construction and some of their consequences over the past fifty years are studied.
1. Introduction
In 1943, a paper [G-N], written by I. M. Gelfand and M. Neumark, "On the imbedding of normed rings into the ring of operators in Hilbert space," appeared (in English) in Mat. Sbornik (see previous paper). From the vantage point of a fifty year history, it is safe to say that that paper changed the face of modern analysis. Together with the monumental "Rings of operators" series [M-vN I, II, III, IV], authored by F. J. Murray and J. von Neumann, it introduced "non-commutative analysis," the vast area of mathematics that provides the mathematical model for quantum physics.
The founders of the theory underlying
quantum mechanics (Schrodinger and Heisenberg, primarily) were groping
their way toward this mathematics ("wave" and "matrix" mechanics). With
his magnificent volume [D], P. A. M. Dirac all but invents the operator
algebra and uses Hilbert space techniques to produce powerful
conclusions in physics. Of course, simultaneously with his introduction
of "rings of operators," von Neumann's book [vN2] appeared, providing a
model for "quantum measurement" and some of the fundamentals of quantum
statistical mechanics.
Extremely knowledgeable and vitally interested in quantum physics, I. E.
Segal, who had been developing commutative and non-commutative harmonic
analysis in the Hilbert space context, recognized the construction
buried in the Gelfand-Neumark paper- a construction that is basic and
crucial for the subject of operator algebras. Just after publication of
his "Postulates for quantum mechanics" [Sl], Segal published his
groundbreaking "Irreducible operator algebras" [S2] in which that
construction is sharpened and made explicit and then used in one of the
earliest general studies of (infinite-dimensional) unitary
representations of (non-commutative) locally compact groups.
A statement of the Gelfand-Neumark theorem follows.
THEOREM (GELFAND- NEUMARK 1943). If A is an algebra over the complex numbers C with unit I, with a norm A → ||A|| relative to which it is a Banach space for which
||AB|| ≤ ||A|| ||B||,
and ||I|| = 1 (A is a Banach algebra),
and with a mapping (involution) A → A* such that
i) (aA + B)* = a* A* + B* (a* is the complex conjugate of a),
ii) (AB)* = B*A*,
iii) (A*)* =A,
iv) ||A* A||= ||A*|| ||A||,
v) A*A + I has an inverse (in A) for each A in A,
vi) ||A*|| = ||A||,
then there is an isomorphism φ of A with a norm-closed subalgebra B of the algebra B(H) of all bounded operators on a Hilbert space H such that φ(A*) = φ(A)*, where φ(A)* is the adjoint (in B(H)) of φ(A). Moreover, ||φ(A)|| = ||A|| for all A in A.
Gelfand and Neumark conjecture, in their paper, that conditions ( v) and (vi) are superfluous, that is, derivable from the others. They were proved right ten years later on (v) and seventeen years later on (vi).
The Gelfand-Neumark construction allows us to construct
representations (both reducible and irreducible) starting from the
concept of "state", as it is understood in quantum mechanics. We will do
it in the next post. Our geometric algebra A has all the required properties.
