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Interesting attempt to axiomatize quantum mechanics
Posted:
Sep 1, 2011 8:34 PM
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I have been reading two very interesting papers by the same authors,
G. Chiribella, G. M. D'Ariano, and P. Perinotti,
"Probabilistic theories with purification",
Phys. Rev. A 81, 062348 (2010), arXiv:9008.1583, and
"Informational derivation of quantum theory",
Phys. Rev. A 84, 012311 (2011), arXiv:1011.6451.
These attempt an axiomatization of finite dimensional quantum mechanics
which avoids purely mathematical axioms without physical meaning, such
as an assumption that "pure states" of a quantum system are represented
by rays in a Hilbert space.
I am hoping to get in touch with others who may be interested
in discussing the ideas of these authors. To that end, I will attempt
a partial review indicating what intrigues me and what disturbs me about
these papers. I will refer to the first paper as CDP10 and the second
as CDP11.
These are basically papers in pure mathematics which are motivated
by fundamental physics. They are both long and intricately complicated.
CDP10 is 40 pages with 64 definitions, 34 lemmas, 30 theorems, and
50 corollaries. CDP11 is 39 pages with 12 definitions, 78 lemmas,
20 theorems, and 51 corollaries. I have read only a fraction of these
papers in detail, but enough that I think I have a sense of their results,
methods, and probable correctness. I am impressed with them and think that
they will be important even if should turn out that some of their results
are in error.
There is some overlap between the two papers, but not a great deal.
I think that most readers will need to read CDP10 in order to understand
CDP11, which constantly refers to CDP10 for needed results.
Both papers are well written, but in unusual notations invented by the
authors, and the notations are different for the two papers. I thought
the CDP10 notation was quite successful, but the CDP11 notation less so.
For example, CDP11 uses a thickened horizontal line to denote equality
instead of the usual "=", without explicitly informing the readers of this.
I found this really puzzling even after I had guessed its meaning. What's
wrong with "=", which everybody understands, and why make the reader guess
the meaning of unfamiliar symbols?
As a mathematician, I was sometimes disturbed by the mathematical
vagueness of some of the definitions, but I think most of them could
probably be reformulated in a rigorous way. This is ground-breaking work,
which may be akin to early 19'th century mathematics
before notions like "continuity" were fully understood on a rigorous basis.
Although the papers are basically pure mathematics,
the definitions are neither standard mathematical ones
nor presented as is customary in pure mathematics.
Mathematicians will probably need some familiarity with standard
quantum mechanics to follow the paper. Reading it may be similar
to the experience a bright high school student might have trying to
read a text on abstract linear algebra on his own. The abstractions
are not likely to be meaningful to someone completely unfamiliar with
what is being abstracted.
I think it is unfortunate that the authors did not publish this
in a more mathematically oriented journal. The Physical Review journals
are not known for careful refereeing, and they do publish a lot of badly
incorrect material. Experience has taught me to view
with initial skepticism just about everything in these journals,
including the papers under review.
A purely mathematical work of this complexity
would be accepted by the mathematical community
only after years of careful study by experts;
publication in a seriously refereed journal would only be a first step.
In this case, even the first step has not been accomplished.
I have identified a few points in these papers which seem to me
questionable, but I am mindful that I could be mistaken. Even if they are
incorrect, it is not clear that they could not be repaired. Even if
they cannot be repaired it is not clear that the main arguments
of the papers would be affected. In a work of this complexity,
it would be surprising not to find any substantial errors.
I have considered writing the authors about the possible errors,
but hesitate because of my still-limited understanding of this intricate
work. I am hoping to find someone else who may have read enough of it
to either put me straight about my misunderstandings or to give me
confidence that the problems I see may indeed be essential.
I will be happy to discuss this with anyone, but "discussion" is
a reciprocal process. I would expect that anyone who wants to discuss it
would *first* have read it to at least a depth to understand the main
definitions and to evaluate technical arguments in detail. In the past
when I have sent out such requests, I have gotten responses from
anonymous addresses, sometimes signed only with a first name, of the order
of:
"Tell me what you think is wrong with it, and I will read it tonight
and let you know."
Stephen Parrott
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