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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.

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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