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Optimal hypercontractivity for Fermi fields and related non-commutative integration inequalities. (English) Zbl 0796.46054
Let \({\mathcal F}\) be the Fock space, with the vacuum vector \(\Omega\), for a system of \(n\) fermion degrees of freedom, let \(c_ j\) and\(c_ j^*\), \(j=1,\dots,n\), be the fermion annihilation and creation operators, repsectively, and let \(H=\sum_{j=1}^ n c_ j^* c_ j\) be the number operator generating the oscillator semigroup \(\text{exp}(-tH)\). Put \(Q_ j= c_ j+ c_ j^*\), and denote by \({\mathfrak U}\) the algebra with unit generated by the \(Q_ j\)’s. Any vector \(v\) in \({\mathcal F}\) can be written as \(v=A\Omega\) for some unique \(A\) in \({\mathfrak U}\), and thus \(\text{exp}(-tH)v= B\Omega\) for some unique \(B\) in \({\mathfrak U}\). This gives rise to maps \(P_ t: A\to B\) which together with the algebra \({\mathfrak U}\) can replace the usual Fock space and the oscillator semigroup in various problems concerning Fermi fields. On \({\mathfrak U}\) (non- commutative) \(L^ p\)-norms can be introduced for \(1\leq p<\infty\) by the formula \[ \| A\|_ p= [\langle \Omega,(A^*A)^{p/2} \Omega\rangle]^{1/p}, \] and the main result of the paper lies in establishing the following optimal fermion hypercontractivity inequality \[ \| P_ t A\|_ q\leq \| A\|_ q \qquad \text{when} \qquad e^{-2t}\leq (p-1)/(q-1), \] for all \(1<p\leq q<\infty\) and all \(A\) in \({\mathfrak U}\). Moreover, optimal fermion logarithmic Sobolev inequality is also obtained.

MSC:
46N50 Applications of functional analysis in quantum physics
46L51 Noncommutative measure and integration
46L53 Noncommutative probability and statistics
46L54 Free probability and free operator algebras
81T10 Model quantum field theories
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References:
[1] [A1HK] Albeverio, S., Høegh-Krohn, R.: Dirichlet forms and Markov semigroups onC *-algebras. Commun. Math. Phys.56, 173–187 (1977) · Zbl 0384.46043 · doi:10.1007/BF01611502
[2] [ArYa] Araki, H., Yamagami, S.: An inequality for the Hilbert-Schmidt norm, Commun. Math. Phys.81, 89–96 (1981) · Zbl 0468.47013 · doi:10.1007/BF01941801
[3] [BCL] Ball, K., Carlen, E.A., Lieb, E. H.: preprint 1992
[4] [BrWe] Brauer, R., Weyl, H.: Spinors inn dimensions. Am. J. Math.57, 425–449 (1935) · Zbl 0011.24401 · doi:10.2307/2371218
[5] [CL91] Carlen, E.A., Loss, M.: Extremalsof functionals with competing symmetries. J. Func. Anal.88, 437–456 (1991) · Zbl 0705.46016 · doi:10.1016/0022-1236(90)90114-Z
[6] [Da76] Davies, E.B.: Quantum Theory of Open Systems. New York: Academic Press, 1976
[7] [Di53] Dixmier, J.: Formes linéaires sur un anneau d’opérateurs. Bull.Soc. Math. France81, 222–245 (1953)
[8] [Far] Faris, W.: Product spaces and Nelson’s inequality, Helv. Phys. Acta48, 721–730 (1975)
[9] [Fe69] Federbush, P.: A partially alternate derivation of a result of Nelson. J. Math. Phys.10, 50–52 (1969) · Zbl 0165.58301 · doi:10.1063/1.1664760
[10] [Gr72] Gross, L.: Existence an uniqueness of physical ground states. J. Funct. Anal.10, 52–109 (1972) · Zbl 0237.47012 · doi:10.1016/0022-1236(72)90057-2
[11] [Gr75] Gross, L.: Hypercontractivity and logarithmic Sobolev inequalities for the Clifford-Dirichlet form. Duke Math. J.43, 383–396 (1975) · Zbl 0359.46038 · doi:10.1215/S0012-7094-75-04237-4
[12] [Gr89] Gross, L.: Logarithmic Sobolev inequalities for the heat kernel on a Lie group and a bibliography on logarithmic Sobolev inequalities and hypercontractivity. In: White Noise Analysis, Mathematics and Applications. Hida et al.(eds.) Singapore: World Scientific, 1990, pp. 108–130
[13] [Gr92] Gross, L.: Logarithmic Sobolev inequalities and contractivity properties of semigroups. 1992 Varenna summer school lecture notes (preprint)
[14] [HuPa] Hudson, R., Parthasarathy, K.R.: Quantum Itô’s formula and stochastic evolutions. Commun. Math. Phys.93, 301–323 (1984) · Zbl 0546.60058 · doi:10.1007/BF01258530
[15] [JoKl] Jordan, P., Klein, O.: Zum Mehrkörperproblem der Quantentheorie. Zeits. für Phys.45, 751–765 (1927) · JFM 53.0857.02 · doi:10.1007/BF01329553
[16] [JoWi] Jordan, P., Wigner, E.P.: Über das Paulische Äquivalenzverbot. Zeits. für Phys.47, 631–651 (1928) · JFM 54.0983.03 · doi:10.1007/BF01331938
[17] [Li76] Lieb, E.H.: Inequalities for some operator and matrix functions. Adv. Math.20, 174–178 (1976) · Zbl 0324.15013 · doi:10.1016/0001-8708(76)90185-7
[18] [Li90] Lieb, E.H.: Gaussian kernels have only Gaussian maximizers. Invent. Math.102, 179–208 (1990) · Zbl 0726.42005 · doi:10.1007/BF01233426
[19] [Lin] Lindsay, M.: Gaussian hypercontractivity revisited. J. Funct. Anal.92, 313–324 (1990) · Zbl 0713.46050 · doi:10.1016/0022-1236(90)90053-N
[20] [LiMe] Lindsay, M., Meyer, P.A.: preprint, 1991
[21] [MeDS] Merris, R., Dias da Silva, J. A.: Generalized Schur functions. J. Lin. Algebra35, 442–448 (1975) · Zbl 0309.15006 · doi:10.1016/0021-8693(75)90057-5
[22] [Me85] Meyer, P.A.: Eléments de probabilités quantiques, exposés I–V. In: Sem. de Prob. XX, Lecture notes in Math.1204,New York: Springer, 1985 pp. 186–312
[23] [Me86] Meyer, P.A.: Elements de probabilités quantiques, exposés VI–VIII. In: Sem. de Prob. XXI, Lecture notes in Math.1247, New York: Springer, 1986 pp. 27–80
[24] [Ne66] Nelson, E.: A quartic interaction in two dimensions. In: Mathematical Theory of Elementary Particles, R. Goodman, I. Segal (eds.) Cambridge, MA, MIT Press, 1966
[25] [Ne73] Nelson, E.: The free Markov field. J. Funct. Anal.12, 211–227 (1973) · Zbl 0273.60079 · doi:10.1016/0022-1236(73)90025-6
[26] [Ne74] Nelson, E.: Notes on non-commutative integration. J. Funct. Anal.15, 103–116 (1974) · Zbl 0292.46030 · doi:10.1016/0022-1236(74)90014-7
[27] [Nev] Neveu, J.: Sur l’esperance conditionelle par rapport à un mouvement Brownien. Ann. Inst. H. Poincaré Sect. B. (N.S.)12, 105–109 (1976) · Zbl 0356.60028
[28] [Ru72] Ruskai, M.B.: Inequalities for traces on Von Neumann algebras. Commun. Math. Phys.26, 280–289 (1972) · Zbl 0257.46101 · doi:10.1007/BF01645523
[29] [SML] Schultz, T.D., Mattis, D.C., Lieb, E.H.: Two dimensional Ising model as a soluble problem of many fermions. Rev. Mod. Phys.36, 856–871 (1964) · doi:10.1103/RevModPhys.36.856
[30] [Se53] Segal, I.E.: A non-commutative extension of abstract integration. Ann. Math.57, 401–457 (1953) · Zbl 0051.34201 · doi:10.2307/1969729
[31] [Se56] Segal, I.E.: Tensor algebras over Hilbert spaces II. Ann. Math.63, 160–175 (1956) · Zbl 0073.09403 · doi:10.2307/1969994
[32] [Se70] Segal, I.E.: Construction of non-linear local quantum processes: I. Ann. Math.92, 462–481 (1970) · Zbl 0213.40904 · doi:10.2307/1970628
[33] [TJ74] Tomczak-Jaegermann, N.: The moduli of smoothness and convexity and Rademacher averages of trace classesS p(1<. Studia Mathematica50, 163–182 (1974) · Zbl 0282.46016
[34] [Um54] Umegaki, H.: Conditional expectation in operator algebras I. Tohoku Math. J.6, 177–181 (1954) · Zbl 0058.10503 · doi:10.2748/tmj/1178245177
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