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I would like to discuss the “next entry” in the parallel texts that Masoud was presenting in his post. On the function theory side we are talking about “real and complex variables”. A perfect book to get introduced to that is “real and complex analysis” by W. Rudin (McGraw-Hill). It is a classic and remains one of the best entrance doors to the subject. What one learns is the constant interplay between the “real variable” techniques such as the Lebesgue integral, differentiability almost everywhere, etc.. and the “complex variable” techniques. There is a saying of André Weil like “The complex world is beautiful, the real world is dirty”. One might then be tempted to ignore the “real world” and only work in the complex variable set-up where “any” function is holomorphic and hence infinitely differentiable etc… That’s fine, and one can go some distance with that, except that most of the deep results in complex analysis do rely on real analysis.

Now what about the next entry in the parallel text? It is

Complex variable…………….Operator on Hilbert space

Real variable……………………..Self-adjoint operator

where I have slightly rewritten the previous entry

functions f: X -> C ……………..operators on Hilbert space

of Masoud’s post to stress that the right column gives an ideal model for what the loose notion of a “variable” is… The set of values of the variable is the spectrum of the operator, and the number of times a value is reached is the spectral multiplicity. Continuous variables (operators with continuous spectrum) coexist happily with discrete variables precisely because of non-commutativity of operators.

The holomorphic functional calculus gives a meaning to f(T) for all holomorphic functions f on the spectrum of T, and a deep result controls the spectrum of f(T). The really amazing fact is that while for general operators T in Hilbert space the only functions f(z) that can be applied to T are the holomorphic ones (on the spectrum of T), the situation changes drastically when one deals with self-adjoint operators: for T=T* the operator f(T) makes sense for **any** function f! You can take a pencil and draw the graph of a function, it does not need to be continuous…nor even piecewise continuous, just anything you can name will do….(at the technical level the only requirement on f is that it is universally measurable but nobody can construct explicitly a function which does not fulfill this condition!)…Moreover a bounded operator is a function of T (ie is of the form f(T) ) if and only if it shares all the symmetries of T (ie if it commutes with all operators that commute with T ).

I remember that, at a very early stage of my encounter with mathematics, it is this very fact that convinced me of the power of the Hilbert space techniques in close relation with the adjoint operation T -> T*. This was enough to resist the temptation of starting directly in the “complex world” of algebraic geometry which was attracting most beginners at that time, following the aura of Grothendieck, who described so well his first encounter with that world:

“Je me rappelle encore de cette impression saisissante (toute subjective certes), comme si je quittais des steppes arides et revèches, pour me retrouver soudain dans une sorte de “pays promis” aux richesses luxuriantes, se multipliant à l’infini partout où il plait à la main de se poser, pour cueillir ou pour fouiller….”

I get confused about how far to take these analogies sometimes. Its a lot easier to work with self adjoint operators than arbitrary ones, so in that sense the analogy with complex/real valued functions is backwards. Not to menation the fact that the Borel functional calculus (the more general f(T) that AC was describing) works whenever T and T* commute. So there are “complex” variables out there whose spectral calculus is just as fexible as a “real” variable.

Its a useful metaphore, to be sure, but its not quite as clean as it looks.

Another analogy,

Real = Hermitian

Complex = non Hermitian.

Example: In the Pauli algebra, the Hermitian primitive idempotents are of the form

(1 + vec{u} cdot vec{sigma})/2

where vec{sigma} is a vector of the Pauli operators (spin matrices), and vec{u} is a unit vector also in 3 dimensions.

The non Hermitian primitive idempotents of the Pauli algebra consist of the same thing, but with u a complex vector that happens to satisfy

u_x^2 + u_y^2 + u_z^2 = 1

for u_chi complex.

It turns out that each non Hermitian primitive idempotent of the Pauli algebra can be written uniquely as a real multiple of the product of two Hermitian primitive idempotents.

The real multiple is 1/P where P is the transition probability between the two Hermitian states, (1+cos(theta))/2.

Non Hermitian stuff is of interest in elementary particles, see the most recent .

“anonymous” the point is that the class of arbitrary functions (real analysis) is that which operates on self-adjoint operators, while only the holomorphic ones operate on general operators T. The case of normal operators ([T,T*]=0) is just “two real variables” and has nothing to do with complex analysis. When the class of functions is smaller you expect more properties, but that does not mean that it is “easier” (rather the opposite) so the analogy is not backwards…

Thanks to Alain for bringing up this theme of “passing from c-numbers to q-numbers” (c for classic or complex, q for quantum; I guess the nomenclature was coined by Dirac and even used in his book). While self adjoint operators and Borel functional calculus correspond to real numbers and measurable functions on them, positive real numbers correspond to positive operators. An operator is positive if it is self adjoint and its spectrum is real and non-negative (notice that just saying

the spectrum is non-negative won’t do). An interesting theme here is the issue of operator (in particular matrix) inequalities. One can ask about the `q-analogues’ of classical inequalities for c-numbers: a simple first observation is that a product of positive operators is not necessarily positive, unless they commute. The analogy c-numbers—q-numbers is however strengthened in noncommutative geometry by including infinitesimal c-numbers and their `q-analogues’. This significant extension can be discussed later. For the moment you can take a look at Alain’s book where this is developed.

Incidentally I came across a book, “From c-Numbers to q-Numbers: The Classical Analogy in the History of Quantum Theory” (I have not read it yet) which seems to be a detailed case study of this analogy at the beginning of quantum theory. May be someone who knows this book can make a comment here.

could someone please translate Grothendieck’s words into English?

Anonymous: here is a possible translation, though Grothendieck’s text is stronger and more poetical in a way…

“I still remember this striking feeling (rather subjective of course)

as if I were leaving an arid desert to find myself suddenly in a

kind of “promised land” full of luxuriant treasures, growing

profusely at infinity, everywhere where the hand likes to settle to

pick or to search.”

(ps the words “steppes” and “revèches” have not been properly translated so one can certainly do better than this “first approximation”)

one thing about Grothendieck’s memoir “Recoltes et Semaillles”:

http://www.fermentmagazine.org/Grotproj.html

“Mainstream publishers have refused to touch it. The publisherOdile Jacob for example will only publish the first 400 pages, provided that all the real names are replaced by fictitious ones.”

no comment.