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CATEGORIES:Wednesday HEP-GR Colloquium
SUMMARY:Undecidability of the spectral gap - Toby Cubitt
DTSTART;TZID=Europe/London:20150429T141500
DTEND;TZID=Europe/London:20150429T151500
UID:TALK58173AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/58173
DESCRIPTION:The spectral gap -- the difference in energy betwe
en the ground\nstate and the first excited state -
- is of central importance to quantum\nmany-body p
hysics. Some of the most challenging and long-stan
ding open\nproblems in theoretical physics concern
the spectral gap\, such as the\nfamous Haldane co
njecture\, or the infamous Yang-Mills gap conjectu
re (one\nof the Millennium Prize problems). These
problems -- and many others --\nare all particular
cases of the general spectral gap problem: Given
a\nquantum many-body Hamiltonian\, is the system i
t describes gapped or\ngapless?\n\nWe prove that t
his problem is undecidable (in exactly the same se
nse as\nthe Halting Problem was proven to be undec
idable by Turing). This also\nimplies that the spe
ctral gap of certain quantum many-body Hamiltonian
s\nis not determined by the axioms of mathematics
(in much the same sense as\nGoedel's incompletenes
s theorem implies that certain theorems are\nmathe
matically unprovable). The results also extend to
many other\nimportant low-temperature properties o
f quantum many-body systems\, such\ncorrelation fu
nctions.\n\nThe proof is complex and draws on a wi
de variety of techniques\, ranging\nfrom mathemati
cal physics to theoretical computer science\, from
\nHamiltonian complexity theory\, quantum algorith
ms and quantum computing\nto fractal tilings. I wi
ll explain the result\, sketch the techniques\ninv
olved in the proof at an accessible level\, and di
scuss the striking\nimplications this may have bot
h for theoretical physics\, and for physics\nmore
generally (which\, after all\, happens in the labo
ratory not in\nHilbert space!).
LOCATION:MR2\, Centre for Mathematical Sciences
CONTACT:Mahdi Godazgar
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