Abstract
We discuss a natural generalization of the uncertainty inequality holding in the case of non-hermitian operators. The inequality is employed to derive useful constraints on the behaviour of quantum fluctuations in problems with continuous group symmetries. Applications to Bose superfluids, antiferromagnets and crystals at zero temperature are discussed. We provide, in particular, a simple and direct proof of the absence of long range order at zero temperature in the1D case. A new inequality involving the spectral function at finite temperature is finally derived.
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References
N. N. Bogoliubov,Phys. Abh. Su 6, 1 (1962).
H. Wagner,Z. Physik 195, 273 (1966).
G. Baym, inMathematical Methods of Solid State and Superfluid Theory, R. C. Clark and G. H. Derrick, eds. (Oliver and Boyd, Edinburgh, 1969) p. 121.
P. C. Hohenberg,Phys. Rev. 158, 383 (1967).
N. D. Mermin and H. Wagner,Phys. Rev. Lett. 17, 1133 (1966).
N. D. Mermin,Phys. Rev. 176, 250 (1968).
L. D. Landau and E. M. Lifshitz,Quantum Mechanics (Pergamon Press, Oxford, 1977).
N. N. Bogoliubov,J. Phys. USSR 11, 23 (1947).
D. Pines and Ph. Nozieres,Phys. Rev. 109, 741 (1958).
J. Gavoret and Ph. Nozieres,Ann. Phys. (N.Y.)28, 349 (1964).
G. V. Chester and L. Reatto,Phys. Lett. 22, 276 (1966).
S. Coleman,Commun. Math. Phys. 31, 259 (1973).
L. P. Pitaevskii,JETP Lett. 45, 185 (1987).
L. L. Foldy,Phys. Rev. 124, 649 (1961).
P. W. Anderson,Phys. Rev. 86, 694 (1952).
I. Affleck,J. Phys.: Condens. Matter 1, 3047 (1989); I. Affleck, inFields, Strings and Critical Phenomena, Les Houches 1988, E. Brezin and J. Zinn-Justin, eds. (Elsevier Science Publishers 1989).
L. B. Ioffe and A. I. Larkin,Int. J. of Mod. Phys. B 2, 203 (1988).
E. Rastelli and A. Tassi,Phys. Rev. B 40, 5282 (1989).
P. Bruesch,Phonons: Theory and Experiments II (Springer Series in Solid State Sciences) (Springer, Berlin, 1986), Vol. 65.
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Pitaevskii, L., Stringari, S. Uncertainty principle, quantum fluctuations, and broken symmetries. J Low Temp Phys 85, 377–388 (1991). https://doi.org/10.1007/BF00682193
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DOI: https://doi.org/10.1007/BF00682193