Notes on Auyang’s “How is Quantum Field Theory Possible?”

Auyang moves rapidly thru QM in the first chapter. Here are some things I would have added and which may help some. You may not need the information in green.

On page 19 she speaks of the spectrum of an operator as being discrete or continuous and Self-adjoint operators having only a real spectrum. These are mathematical theorems, not physics axioms. They were established before von Neumann introduced Hilbert space to QM. The spectrum of an operator is the set of eigenvalues of that operator. The Hilbert space used in QM is over the complex numbers and an eigenvalue of an operator is a complex number. As she says, if the operator is self-adjoint then the eigenvalue is real; the spectrum of a self-adjoint operator is thus a subset of the reals. If B is an operator and if for some complex number c and non-zero vector x, Bx = cx, then c is an eigenvalue and x is a corresponding eigenvector. All operators under discussion here are ‘linear’ as defined here.

If the Hilbert space is the set of square integrable functions on the real line, then the derivative operator is a linear operator. The derivative of eax is aeax and this is true for any a. Thus every real is in the spectrum of the derivative operator. Its spectrum is continuous. No finite dimensional space has operators with a continuous spectrum. An operator in n dimensions has at most n eigenvalues.

Note that not all functions have derivatives and thus this operator does not operate on the whole space. It turns out that it operates on a linear subspace of the Hilbert space. Theorems on Hilbert space operators are well decorated with hypotheses on the domains of the operators mentioned therein.

By the bottom of page 20 I think it has not been made clear that to observe an observable, which is an operator A, A acts on the state vector x and the new state vector, after the observation, is Ax. An immediate corollary of this is that the same operator acting on a state vector leaves the state unchanged. This paragraph is obscure and incomplete. The state vector after the observation is one of the eigenvectors of the operator. Q4 says that the result of the observation is one of the eigenvalues of the observable. Another result (perhaps Q5) is that the new state vector is one of the eigenvectors that goes with that eigenvalue. If two eigenvectors have the same eigenvalue then any vector in the space spanned by those two eigenvectors is also an eigenvector with that same eigenvalue.

It is too late now but I want to record a fuzzy idea. When eigenvalues are almost equal funny thing happen which is part of the physics in Egan’s novel Teranesia (ISBN=0061059803). The state vector collapses but ambiguously.

I am uncharacteristically hung up with the introduction to this book. I read a couple of paragraphs and become distracted, such as writing this e-mail. I find it impossible to skim because it seems to say something significant. I have not found it necessary to recall what Kant thought since I took a test on the subject many years ago. I resist revisiting those obscure distinctions which I never found especially enlightening. I fear that Auyang may be right, however.

We shall see if she has found a path thru the swamp, if I can persevere. Is it really time to disinter Kant?

These are some thoughts that I should put elsewhere but so far this is the only page on my site on these issues.

We have some examples of universes which most of my friends and I think are wholly adequate to intelligent observers.

• Turing machines
• General purpose computers
• Game of LIFE (and variations)
Observers in these worlds have just the problems that Kant worried about. The range of possibilities they must contemplate is at least as wild as Kant proposed.

Let me clarify the situation of these observers as follows; there may be other possibilities. The following is a dualistic game where we (inhabitants of our real universe) get to write programs that will run in the artificial universe, called the world here, and learn the results via “printf” statements. Suppose that we have been provided a higher level language, such as C, whose semantics is incomplete. By incomplete I mean here that “*(int *)0 = 0;” is a syntactically valid statement that we can execute but with undefined consequences (semantics). We do not know what the hardware substrate is. The goal of the game is to extend the known semantics of C to be complete or at least more complete. This would seem to entail learning the instruction set architecture of the machine, or the rules of LIFE.

I claim that this game is a good model of epistemology in our own universe and also involves the same sorts of questions raised by Auyang. The difference is that we may be able to get some definite answers here, or at least clarify the questions. What are we up against in trying to do physics in our own universe?

The dualistic situation of this game is troubling and some philosophers will be unsatisfied for that reason. Others may be pleased.

Note that C never claimed to be a safe language; there are many things you can’t do in a safe language. If our real universe is implemented in a “safe language” we will be unable to break thru the abstractions below and they are in thus in a sense unreal thereby. This is why I choose an unsafe language.

A loop in C can report many artifacts of our world as follows:

`{int * j; for(j=0; j<(int*)10000; ++j) printf("%08x ", *j);}`
This is cheating, but not really. Recall, however, that we don’t know the instruction set, or even if the laws of physics are implemented in the answers we see. We might however want to change the rules of the game to disable this.

It is unclear what it means to compile a C program for the game of life. It is possible however since life has been shown to be Turing complete. It is especially unclear what the meaning of the above C program would be, there being no natural concept of numerical address in life.