On 9/23/2015 12:19 PM, Bruno Marchal wrote:
On 23 Sep 2015, at 06:51, Brent Meeker wrote:
On 9/22/2015 9:26 PM, Bruno Marchal wrote:
On 22 Sep 2015, at 19:27, Brent Meeker wrote:
On 9/22/2015 5:17 AM, Bruno Marchal wrote:
On 22 Sep 2015, at 00:29, Brent Meeker wrote:
A fascinating application of computability theory to physics:
Undecidability of the Spectral Gap
Toby Cubitt, David Perez-Garcia, and Michael M. Wolf
The spectral gap—the difference in energy between the ground
state and the first excited state—is one of the most important prop-
erties of a quantum many-body system. Quantum phase transitions
occur when the spectral gap vanishes and the system becomes
critical. Much of physicsis concerned with understanding the
phase diagrams of quantum systems, and some of the most challenging
and long-standing open problems in theoretical physics concern
the spectral gap, 1–3 such as the Haldane conjecture 4 that the
Heisen-
berg chain is gapped for integer spin, proving existence of a
gapped topological spin liquid phase, 2,3 or the Yang-Mills gap
conjecture 5
(one of the Millennium Prize problems). These problems are all
particular cases of the general spectral gap problem: Given a quan-
tum many-body Hamiltonian, is the system it describes gapped or
gapless? Here we show that this problem is undecidable, in the
same sense as the Halting Problem was proven to be undecidable by
Turing.
I guess he means unsolvable.
"undecidable" is relative to a theory. Unsolvable or uncomputable
is absolute and does not depend on any theory. It means that there
is no alogorithm to do some task, like computing some function or
deciding some set.
Yes, and it would be quite surprising to find there is no algorithm
to compute whether or not a Hamiltonian system has a mass gap - since
it is presumably a fact of nature whether it does or not.
Not sure that this entails the existence of an algorithm. In the
arithmetical reality, many facts exists with provably no algorithm to
decide them.
But many scientists implicitly assume that reality is computable, that
there is an algorithm for deciding how the state of the universe
evolves. And so they reject the idea that reality is isomorphic to
arithmetic. If the mass gap is not computable that's very surprising.
But I'm afraid it depends on systems being potentially infinite, which
is dubious.
Brent
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