Spin Chronicles Part 39: Inside a state

 This note is a continuation of Part 38. While we have the geometric Clifford algebra Cl(V) in mind, we start with a more general case of a finite-dimensional *-algebra A with unit 1. A state on A has been defined as a normalized positive linear functional on A. Thus f is a state on A if

1. f(a*a) ≥ 0 for all a in A,

and

2. f(1) = 1.

We need to draw some simple consequences from this definition. The devil is always hiding in the details.

The devil hides in the details

The first deduction from the definition will be: every state automatically has the Hermitian property:

3. f(a*) = f(a)*.

The proof is simple, but somewhat tricky.

Let us take any two elements a,b in A, and any two complex numbers λ,μ Then λa+μb is in A, therefore we must have

f((λa+μb)*(λa+μb)) ≥ 0.

But 

(λa+μb)*(λa+μb) = |λ|² a*a + λ*μ a*b + λμ* b*a + |μ|² b*b.

Using linearity of f we get

|λ|² f(a*a) + λ*μ f(a*b) + λμ* f(b*a) + |μ|² f(b*b) ≥ 0.         (0)

This inequality holds for all complex λ,μ. In particular for λ=μ=1. Thus

f(a*a) + f(a*b) + f(b*a) + f(b*b) ≥ 0.                   (1)

Now chose λ=1, μ=i. We get

 f(a*a) + i f(a*b) - i f(b*a) + f(b*b) ≥ 0.          (2)

Now decompose f(a*b) and f(b*a) into real and imaginary parts

f(a*b) = α₁ + i β₁, (α₁,β₁ real),
f(b*a) = α₂ + i β₂, (α₂,β₂ real).

Then

f(a*b) + f(b*a) = α₁ +  α₂ +i(β₁ +β₂),
f(a*b) - f(b*a) = α₁ -  α₂ +i(β₁ - β₂).

We notice that f(a*a) and f(b*b) are real. It follows then form (1) that f(a*b)+f(b*a) must be real. The imaginary part of f(a*b)+f(b*a) must be zero. Therefore β₁ + β₂ = 0, or

β₂ = -β₁ .

. Similarly, from (2) we get that the real part of f(a*b) - f(b*a) must be zero, therefore α₁ -  α₂ = 0, or

α₂ = α₁ .

Thus we get that

f(a*b) = f(b*a)*.

In particular, for b=1, we get the desired Hermicity property of f.

But we never really used the positivity yet! We used only the fact that f(a*a) is real for every a in A. There must be more that we can deduce from positivity. The method of special choices of λ,μ can, in fact, be exploited somewhat deeper. And indeed, there is another trick, even trickier than the two simple ones that we have used. We will use this trick to prove the  Bunyakovsky-Cauchy–Schwarz inequality:

|f(a*b)|² ≤ f(a*a) f(b*b).                     (3)

First we notice that if f(a*b)=0, the inequality is evident. So, it is enough to analyze the case of f(a*b) ≠ 0. For this we set λ = t f(a*b)/|f(a*b)|, μ = 1, t real.

Substituting these values in Eq. (0) we get

t²f(a*a) + 2t |f(a*b)| + f(b*b) ≥ 0,

for all t. Notice that f(a*a) must be strictly positive, otherwise we would have a straight line that would cross the zero level, unless f(a*b)=0, which case we have excluded. Therefore we have an equation of a parabola with both arms "upward", and the parabola crosses the horizontal axis in at most one point. Thus the discriminant Δ must be Δ≤0. But

Δ = 4|f(a*b)|² - 4f(a*a)f(b*b).

And that gives us exactly the inequality (3).

So we are done, but  am not very happy. We were using "tricks". I do not think tricks are "natural". And how would Nature know in advance whether a given functional on an algebra is positive or not? Try it first on all elements a, and check if f(a*a) is positive or not? Ridiculous. So there is something unnatural with the very concept of "state". And yet physics is full of all such ridiculous assumptions. Positive definiteness is "postulated" everywhere. Measure should be (as a rule) positive. Scalar product should be "positive", state should be "positive". For practical purposes, for "shut up an calculate", this is fine. It saves time. But to believe that we "understand" something this way - that is an error, and a serious error. Geometry has been cured from this decease by admitting indefinite signature with + and - allowed. Algebra is happy with Clifford algebra Cl(p,q). But in quantum physics indefinite metric scalar product is generally treated as something that needs to be avoided or ignored. That is bad. Very very bad. Right after my Ph D Thesis that was about von Neumann algebras with cyclic and separating vectors, I wrote a long paper about "Geometry of Indefinite Metric Spaces", which was followed by another paper "Quantum Logic an Indefinite Metric Spaces". Here is the part taken from the introduction to this last paper:  

1. Introduction

The failure of all attempts to give a coherent explanation of the most fundamental facts of elementary particle physics brings some physicists to examine again and again the most profound basis of the quantum theory. Persistently new works appear on this subject. From time to time someone finds the new condition more "physical" or only different from the hitherto existing ones, leading to the necessity of the standard Hilbert space or  C*-algebra approach. It would seem that the justification of the conventional formalism is unquestionable to such an extent, that there is nothing more to do but to search for nontrivial models satisfying all the accepted axioms. Unfortunately, the nature seems to be mischievous rather than allied in all these attempts. On the other hand, none of the so proposed non-standard formalisms has been widely accepted and not because of its nonstandardness but because of the lack of proposals of a concrete and working mathematical apparatus. It is, however, well known that such a generalization of the standard formalism exists and- what is more essential -it works. We have in mind indefinite metric here. It is commonly accepted that indefinite metric offers new possibilities, removes a number of difficulties and gives concrete results (see e.g. [5]). But also, it causes the serious troubles as far as the interpretation is concerned (negative "probabilities", "non-physical" states,etc). It is for these interpretatative troubles, presumably, that indefinite metric has not been taken into account in all attempts of axiomatic formulation of the quantum theory (the only exception known to the author is a separate section in Wightman and Garding [10].

But that is, at present, a side remark. In the following post we will take a "state", as it is understood in this post and finally, finally, build a Hilbert space, a cyclic vector, and a representation of the algebra - the GNS construction.

Exercise 1. Let f be a state. Show that the set {a: f(a*a)=0} is a linear subspace of A. 

Hint: Show that {a: f(a*a)=0} = {a:f(a*b)=0 for all b}

P.S. 20-01-25 12:58 It is easy ...

P.S. 21-01-25 10:50 Recently I was usually posting three times a week. This will change. It was a little bit too much for some of my Readers, and for me too. This will change. I will still try to post regularly, but only twice a week, say Sunday and Thursday.