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SUMMARY:Undecidability of the spectral gap - Toby Cubitt
DTSTART:20150429T131500Z
DTEND:20150429T141500Z
UID:TALK58173@talks.cam.ac.uk
CONTACT:Mahdi Godazgar
DESCRIPTION:The spectral gap -- the difference in energy between the groun
 d\nstate and the first excited state -- is of central importance to quantu
 m\nmany-body physics. Some of the most challenging and long-standing open\
 nproblems in theoretical physics concern the spectral gap\, such as the\nf
 amous Haldane conjecture\, or the infamous Yang-Mills gap conjecture (one\
 nof the Millennium Prize problems). These problems -- and many others --\n
 are all particular cases of the general spectral gap problem: Given a\nqua
 ntum many-body Hamiltonian\, is the system it describes gapped or\ngapless
 ?\n\nWe prove that this problem is undecidable (in exactly the same sense 
 as\nthe Halting Problem was proven to be undecidable by Turing). This also
 \nimplies that the spectral gap of certain quantum many-body Hamiltonians\
 nis not determined by the axioms of mathematics (in much the same sense as
 \nGoedel's incompleteness theorem implies that certain theorems are\nmathe
 matically unprovable). The results also extend to many other\nimportant lo
 w-temperature properties of quantum many-body systems\, such\ncorrelation 
 functions.\n\nThe proof is complex and draws on a wide variety of techniqu
 es\, ranging\nfrom mathematical physics to theoretical computer science\, 
 from\nHamiltonian complexity theory\, quantum algorithms and quantum compu
 ting\nto fractal tilings. I will explain the result\, sketch the technique
 s\ninvolved in the proof at an accessible level\, and discuss the striking
 \nimplications this may have both for theoretical physics\, and for physic
 s\nmore generally (which\, after all\, happens in the laboratory not in\nH
 ilbert space!).
LOCATION:MR2\, Centre for Mathematical Sciences
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