Spin Chronicles Part 37: What we do with states

Hamilton discovered the algebra of quaternions on October 16, 1843. He wrote about this:

"They started into life, or light, full grown, on the 16th of October, 1843, as I was walking with Lady Hamilton to Dublin, and came up to Brougham Bridge. "

They were strange creatures at this time. Euler was very close to their discovery already 100 years before. Gauss discovered them 20 years before. But these two great mathematicians did not realize that they, quaternions, will be important. Hamilton discovered them all by himself and realized their importance. Of course not without obstacles. Lord Kelvin, the famous Scottish physicist, commented on Hamilton's discovery with a warning: "Quaternions came from Hamilton after his really good work had been done; and though beautifully ingenious, have been an unmixed evil to those who have touched them in any way."

Algebra is an unmixed evil indeed. What to do with an algebra? Multiply its elements one by another? Divide one by another? In 1858 Cayley discovered matrices and realized the abstract algebra of quaternions as a particular subalgebra of Mat(2,C). Quaternions became ready to do some real work. Matrices act on vectors creating new vectors. Cayley has found a representation of Hamilton's algebra.

Representations of abstract algebras as "algebras of operators" is today big branch of mathematics, with applications to physics, in particular to quantum physics. To the set of all (equivalence classes of) irreducible representations of a C*-algebra A has been given a name: the spectrum of A. So algebras have their spectra. Studying spectra of atoms gave birth to quantum theory. But algebras, as it seems, have their own ways of producing some kind of "light".

Here we are studying just one monadic-type of algebra - the geometric algebra of space. We should not expect more than one light ray coming from it. Perhaps two. Our algebra is isomorphic to the algebra of biquaternions (complex quaternions), thus double the size of Cayley algebra. We have already found its irreducible representations by looking for non-trivial left ideals. The left regular representation of our A can be decomposed into a direct sum of two equivalent representations (see Part 31)

A = F⊕F^⟂,

where F is the left ideal Ap, p=(1+n)/2, F^⟂=A(1-p).

As announced in my previous post we will take the same task, but using a different perspective - will use the Gelfand-Neumark construction (today known under the name GNS-construction). The GNS machine has been developed for the much more interesting infinite-dimensional case. Here  we will use it for the simplest possible case of a 4-dimensional complex algebra. In finite dimension we can disregard all talk about continuity, because all finite dimensional linear maps are automatically continuous. We can disregard all talk about one subspace being dense in another, because every finite-dimensional linear subspace is automatically closed. What remains of the Gelfand-Neumark construction is a pure algebra. However GNS construction is more "physics oriented", where by "physics" I mainly mean the lessons we have learned from quantum mechanics. And so, we will meet the important concept of "positivity". Here by positivity I shall always mean "non-negativity". Positivity is related to the fact that probabilities are usually being considered as positive, taking values in the interval [0,1]. For real numbers being positive can be defined as being a square of some other real number. For complex numbers being positive is the same as being of the form a*a, where a is another complex number, and a* denotes the complex conjugate of a. Quantum theory suggests us that this last definition also seems to work well with noncommutative *-algebras. So we define an element of our *-algebra A to be positive if it can be written as a*a, where now a* is the antilinear anti-automorphism in A. Now if b is positive, it is automatically Hermitian : b*=b (Why?). Hermitian elements in quantum theory are usually called "observables", positive observables are those that have positive eigenvalues - their spectrum is on the positive real axis. This is not evident from the definition, but it can be proved without great difficulties.

Note: It is rather intuitive, but it takes some real effort to prove it from the above definition that the sum of two positive elements is positive. But it is so.

Of course here we meet a big interpretational problem: what is the meaning of the algebra product for two noncommuting algebra elements a,b? Or even what is meaning of a+b when a and b do not commute? It is not a surprise that Feynman declared that nobody understands quantum theory. Of course many physicists and mathematicians work hard to find a way around these problems (quantum logic, Jordan algebras, noncommutative probability, "effects", etc.), but none of these many proposals has been generally accepted as "the solution". The problem still exists, and waits for a satisfactory answer. "Shut up and calculate "is not a fully satisfactory answer.

The next important ingredient of the construction is the concept of state. A state is a normalized positive linear functional on the algebra A. Linear functional means a linear map from the algebra A to complex numbers C. For every (finite-dimensional) vector space E we have its dual E' - the space of all linear functionals on E.  If v_i are the components of vector v, and fⁱ is any sequence of complex numbers, then f(v) = fⁱv_i defines an element f of E', and any element of E' is of this form (Why?). Here the fact that A is not just a vector space, but an algebra, plays no role. But we want f to be positive, which is defined as: f(v) is positive for all positive v. That is

f(a*a) ≥ 0 for all a in A.

Here the algebra structure and its star operation  play their role.

Finally state must be normalized. Here we use the fact that our algebra has a unit, which we denote simply by 1. Normalization means that we require f(1) = 1. On the left hand side 1 is the unit in the algebra, on the right-hand-side it is the number 1.

We usually interpret f(a) in a probabilistic way as an "expectation value" of a in the state f. So our requirement on f are: expectation value of a positive observable should be positive, and expectation value of an observable taking only value 1 is 1. Intuitive, but, perhaps, misleadingly simple.

So states provide numbers to algebra elements: complex numbers to general elements, real numbers to self-adjoint (a=a*) elements, positive numbers to positive elements. Numbers we understand better than abstract algebra elements. We call these numbers "expectation values" and instantly feel much better. What can we do with states? The same we do in the kitchen with the ingredients: we mix them. If f₁ and f₂ are states, and t is a real number in the interval [0,1], we can forma new state tf₁+(1-t)f₂. We can proceed with mixing adding to the mixture more and more states. By mixing states we lose information - this is known from classical probability, where we mix probability measures. Going in the reverse direction we can try to "un-mix" states. If our state can be decomposed into f₁ and f₂, we try to decompose f₁ and f₂ further, and continue until we finally arrive at states that are not mixtures of other states. These are called "pure states". They contain maximal information about the system, maximal within a given statistical model. This is common to both classical and quantum physics. The main difference between classical and quantum, in this respect, is the fact that in classical statistical mechanics the decomposition of a mixed  into pure states is unique (we say that in classical physics the statistical figure - the convex set of states - is a "simplex"), while in quantum mechanics there is no such uniqueness. This is perhaps one of the main puzzles of quantum theory. Where is this non-uniqueness coming from? And what does it mean? We will meet  this non-uniqueness on an example later on.

And then we can use states (mixed or pure) to construct representations of the algebra as algebras of operators acting on Hilbert spaces. Why do we need this? Can't we simply work with "expectation values" and be happy forever? Here comes another quantum mystery. Louis de Broglie associated waves with particles. Waves are famous for the phenomenon of "interference". Waves can "superpose". This is not the same as statistical mixing. Then came Heisenberg with his matrix quantum mechanics saying bye-bye to the wave picture, but the superposition principle was preserved: we can make superpositions of vectors in the space on which our matrices act. We can treat the superposition within the Hilbert space formalism, but they do not fit the abstract algebra framework. So, by looking for a representation of the algebra, we move from "states" to state vectors. What these state vectors represent beyond reproducing expectation values given by states - that is again a mystery.

Gelfand-Neumark construction takes a state and uses it to construct a Hilbert space and a representation of algebra as an algebra of operators in this space. It realizes this particular state used for the construction as one particular vector in a Hilbert space, and it creates a linear "envelope" of this state by acting with operators representing the algebra elements on this one distinguished vector. This is a general picture. It will be better understood when we will do it on several examples using our Clifford algebra as a toy.