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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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