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\def\da{{\cal D}(A)}
\def\db{{\cal D}(B)}
\def\iq{\pi_{\cal Q}^*}
\def\IR{{\rm I\kern -1.6pt{\rm R}}}
\def\IP{{\rm I\kern -1.6pt{\rm P}}}
\def\ZZ{{\rm Z\kern -4.0pt{\rm Z}}}
\def\IC{\ {\rm I\kern -6.0pt{\rm C}}}
\def\O{{\Omega}}
\def\gs{\left(\matrix{0\cr1\cr}\right)}
\def\pq{\pi_{\cal Q}^{\phantom.}}
\def\Rt{R_\theta}
\def\rt{{R_{\theta(t)}}}
\def\ta{{\bf \tau}}
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\hbox to \hsize{\hfil\eightit EACEHL-Revised July-12-92}
\vglue1.5truein
\centerline{\bf OPTIMAL HYPERCONTRACTIVITY FOR FERMI FIELDS AND}
\centerline{\bf RELATED NON-COMMUTATIVE INTEGRATION
INEQUALITIES}
\bigskip
{\baselineskip = 12pt
\halign{\qquad#\hfil\qquad\qquad\hfil&#\hfil\cr
Eric A. Carlen\footnote{$^*$}{On leave from School of Math., Georgia
Institute of Technology, Atlanta, GA 30332} & Elliott H. Lieb\footnote{$^{**}$}{Work supported by U.S.
National Science Foundation grant no. PHY90--19433--A01.}\cr
Department of Mathematics & Departments of Mathematics and Physics\cr
Princeton University & Princeton University\cr
Princeton, New Jersey 08544 & Princeton, New Jersey 08544--0708\cr}}
\bigskip
\vskip .4 true in
\centerline{Dedicated to Prof. Huzihiro Araki on his 60$^{th}$ birthday}
\vskip 1 true in
{\bf Abstract:} Optimal
hypercontractivity bounds for the fermion oscillator semigroup are obtained.
These are the fermion analogs of the optimal hypercontractivity bounds for
the boson oscillator semigroup obtained by Nelson.
In the process, several results of independent interest in the theory of
non-commutative integration are established.
\vfill\eject
\def\J{{\cal J}}
\def\Q{{\cal Q}}
\def\P{{\cal P}}
\def\K{\cal K}
\def\cq{{\cal C}({\cal Q})}
\def\cqn{{\cal C}({\cal Q}_{(n-1)})}
\def\cqp{{\cal C}^p({\cal Q})}
\def\cqq{{\cal C}^q({\cal Q})}
\def\cqt{{\cal C}^2({\cal Q})}
\def\cp{{\cal C}({\cal P})}
\def\cpp{{\cal C}^p({\cal P})}
\def\ck{{\cal C}({\cal K})}
\def\ckp{{\cal C}^p({\cal K})}
\def\ckt{{\cal C}^2({\cal K})}
\def\H{{\cal H}}
\centerline{\bf I. INTRODUCTION}
\vskip .3 true in
Observables pertaining to the configuration of a quantum system
with $n$ degrees of freedom are
operators $Q_1, Q_2,\dots, Q_n$ which, depending on the system,
may or may not commute.
Our main concern is with the case in which the configuration variables are
amplitudes of certain field modes.
For boson fields, these configuration observables
do commute,
and the state space $\H$ can
be taken as the space of all complex square integrable
functions on their joint spectrum. This is
the Schr\"odinger $q$-space representation, and the fact that in it the
state space is a function space, and not just an abstract Hilbert space, is
very helpful in the analysis of such systems. As one example, it
sometimes turns out that physically interesting operators preserve the cone of
positive functions, and this opens the way to the application of the
Perron-Frobenius theorem in the study of ground states of
such systems.
For fermion fields, the configuration observables do not commute,
and this simple $q$-space representation is not available. However, the
non-commutative integration theory of Irving Segal [Se53]
permits the construction
of a suitable substitute, and in fact it was created with such a purpose in
view. This approach to the study of fermion systems has been extensively
developed by Gross [Gr72] who, among other things, proved a version of the
Perron-Frobenius theorem adapted to the setting and applied it to
prove existence and uniqueness of ground states for certain fermion
quantum field models.
The main estimate which enabled Gross to apply his Perron-Frobenius type
theorem to fermion fields was a hypercontractivity estimate for the fermion
oscillator semigroup.
Corresponding hypercontractivity estimates for boson fields had been
introduced earlier by Nelson [Ne66] who later obtained the optimal
such bound for bosons [Ne73].
Our main result, Theorem 4 below,
is the corresponding optimal hypercontractivity bound for
fermions. Before stating this theorem, we describe its mathematical and
physical contexts in some detail because it cannot even be formulated
naturally in the conventional Fock space language.
(Of course its perturbation theoretic
{\it consequences} can be expressed quite naturally in the usual
language.)
The state space for a system of $n$ fermion degrees of freedom is
conventionally realized as the
Fock space
$${\cal F} =
\oplus_{j=o}^n\bigl((\IC^n)^{\wedge j}\bigr)\eqno(1.1)$$
The basic ``free Hamiltonian'' on this space is the fermion number
operator
$$\hat H_0 = \sum_{j=1}^n c^*_jc_j\eqno(1.2)$$
where $c_j$ and $c_j^*$ are the usual fermion annihilation and creation
operators acting on ${\cal F}$. The fermion oscillator semigroup is
the semigroup of operators $\exp(-t\hat H_0)$ that it generates.
For $n$ boson degrees of freedom, the state space may be realized
{\it either} as
the boson Fock space, or as $L^2(\IR^n,(2\pi)^{n/2}e^{-x^2/2}{\rm d}x)$. The
natural isomorphism between these spaces was pointed out by
Segal, and the latter may be regarded as the Schr\"odinger $q$-space
realization. The boson oscillator semigroup is the semigroup generated by the
boson number operator. Though we have just defined it in Fock space terms,
it may also be considered as an operator semigroup on
$L^2(\IR^n,(2\pi)^{n/2}e^{-x^2/2}{\rm d}x)$. In this setting, it
is {\bf hypercontractive}; i.e., for any finite $p$
greater than 2, there is $t_p$ suffciently
large, for which the semigroup is a contraction from
$L^2(\IR^n,(2\pi)^{n/2}e^{-x^2/2}{\rm d}x)$
to $L^p(\IR^n,(2\pi)^{n/2}e^{-x^2/2}{\rm d}x)$ for all $t\ge t_p$. Since
$t_p$ is independent of $n$, this result proved useful in treating
perturbation theoretic problems for boson fields. This hypercontractivity
inequality cannot be formulated naturally in the Fock space setting because
no natural notion of ``$L^p$'' can be introduced there. The
$L^p(\IR^n,(2\pi)^{n/2}e^{-x^2/2}{\rm d}x)$ setting is essential.
In order to to pass from the Fock space description
for a system of $n$ fermion degrees of freedom to a non-commutative
analog of the
Schr\"odinger $q$-space description (in which we have non-commutative
analogs of $L^p$ norms),
we introduce configuration observables
$Q_j = c_j + c_j^*$, and let $\cq$ be the algebra with unit which
they generate. This is an operator algebra on the $2^n$--dimensional Hilbert
space ${\cal F}$. As we explain in Section II, it is a Clifford algebra
naturally associated to $\IC^n$ with its usual inner product.
What follows here shall be explained in more detail in Sections II--IV,
but the key fact for our present considerations is that the $2^n$
distinct monomials in the $Q_j$ are a basis for $\cq$ as a
{\it vector space}.
Thus, any $A$ in $\cq$ can be uniquely written as
$$A = \alpha I + \sum_{j}\alpha_jQ_j + \sum_{j 0$
for all non zero $A$ in ${\cal A}$, and cyclic in the sense that
$\ta(AB) = \ta(BA)$
for all $A$ and $B$ in ${\cal A}$. Such a functional is evidently unitarily
invariant in the sense that whenever $A$ and $U$ belong to ${\cal A}$,
and $U$ is unitary, then
$\ta(U^*AU) = \ta(A)$.
Since $\ck$ is a full matrix algebra, it contains all unitaries. Hence any
trace on $\ck$ must assign the same value to all rank one projections, and
thus must be a scalar multiple of the standard trace $Tr$ on the matrix algebra.
Henceforth, $\ta$ shall denote this trace normalized by the condition that
$\ta(I) = 1$
and $Tr$ shall denote the standard unnormalized trace.
In the non-commutative integration theories of Dixmier [Di53] and Segal [Se53],
the trace functional $\ta$ is the non-commutative analog of the functional
that assigns to an integrable function its integral. When the Hilbert space
is infinite dimensional, some further regularity properties are required of
$\ta$ in order to obtain a useful analog. Since all of our estimations will
be carried out in the finite dimensional setting, we shall not go into this
here, but shall simply refer the reader to these original papers as well as
the accounts in [Gr72] and [Ne74].
Norms on $\ck$ which are the non commutative analogs of
the $L^p$ norms can now be introduced; namely for $1\le p <\infty$ we put
$$\|A\|_p = \bigl(\ta\bigl((A^*A)^{p/2}\bigr)\bigr)^{1/p}\quad,\eqno(3.1)$$
and denote the operator norm of $A$ by $\|A\|_\infty$.
$\ckp$ shall denote $\ck$ equipped with the norm $\|\cdot\|_p$; evidently
$\ckt$ is the Hilbert space of $2^n\times 2^n$ matrices equipped with the
Hilbert-Schmidt norm.
Consider the monomials
$$E_{[\alpha_1\dots,\alpha_j;\beta_1,\dots,\beta_k]} =
Q_{\alpha_1}\cdots Q_{\alpha_j}P_{\beta_1}\cdots P_{\beta_k}\eqno(3.2)$$
where
$\alpha_1>\cdots>\alpha_j$ and $\beta_1>\cdots
>\beta_k$ and $j+k>0$.
Evidently
$$E_{[\alpha_1\dots,\alpha_j;\beta_1,\dots,\beta_k]}^*
E_{[\alpha_1\dots,\alpha_j;\beta_1,\dots,\beta_k]}^{\phantom{.}} = I$$ and thus
$\|E_{[\alpha_1\dots,\alpha_j;\beta_1,\dots,\beta_k]}\|_p = 1$
for all $p$. Moreover
$$\ta(
E_{[\alpha_1\dots,\alpha_j;\beta_1,\dots,\beta_k]}) = 0\quad.\eqno(3.3)$$
To see this, first consider the case in which $j+k$ is odd. The inversion
$x\mapsto -x$ on $\K$ is orthogonal. Hence it induces an automorphism
of $\ck$, and hence there is an invertible $S$ in $\ck$ so that
$$-E_{[\alpha_1\dots,\alpha_j;\beta_1,\dots,\beta_k]} =
SE_{[\alpha_1\dots,\alpha_j;\beta_1,\dots,\beta_k]}S^{-1}\quad.$$
Then, by using cyclicity of the trace we get the desired result. Next
consider the case in which $j+k$ is even and, say, $j>0$. Then write
$E_{[\alpha_1\dots,\alpha_j;\beta_1,\dots,\beta_k]} = Q_{\alpha_1}X$,
and note that by (2.1),
$Q_{\alpha_1}X =
-XQ_{\alpha_1}$. Again the desired conclusion follows from cyclicity of the
trace.
It is easy to see from this that
$$\ta(
E^*_{[\alpha_1\dots,\alpha_j;\beta_1,\dots,\beta_k]}
E^{\phantom .}_{[\gamma_1\dots,\gamma_m;\delta_1,\dots,\delta_n]}) = 0\eqno
(3.4)$$
unless the two monomials coincide. Thus,
together with the identity, the assemblage of such
monomials forms an orthonormal
basis for $\ckt$. Finally observe that since $Qe_- = e_+$
$$\langle \O,
E_{[\alpha_1\dots,\alpha_j]}\O\rangle =0\eqno(3.5)$$
whenever $j \ge 1$, and, as indicated, $k=0$. It now follows that,
restricted to $\cq$,
$$\ta(A) = \langle \O,A\O\rangle\eqno(3.6)$$
for all $A$ in $\ck$.
Formula (3.6) is very important for us. It permits us to calculate
the ``physically'' relevant quantity
$\langle \O,A\O\rangle$ in terms of the apparently mathematically simpler
quantity $\tau(A)$.
Many familiar inequalities for $L^p$ norms hold for the ${\cal C}^p$
norms as well [Di53]. This is true in particular of the H\"older inequality
$$\|AB\|_r \le \|A\|_p\|B\|_q\qquad{1\over r} = {1\over p} + {1\over
q}\quad.$$
Certain optimal inequalities expressing the uniform convexity properties of
the $L^p$ norms also hold for the
${\cal C}^p$ norms, and this fact constitutes one cornerstone of our
analysis.
The modulus of convexity $\delta_p$ of $\ckp$ is defined by
$$\delta_p(\epsilon) =
\inf\bigl\{1 - {1\over 2}\|A+B\|_p : \|A\|_p = \|B\|_p = 1\ ,\ \|A-B\|_p =
\epsilon\bigr\}\eqno(3.7)$$
for $0 < \epsilon < 2$. For $1 < p < \infty$, $\delta_p$ is always positive
which means these norms are uniformly convex. Useful geometric information
is contained in the rate at which $\delta_p(\epsilon)$ tends to zero with
$\epsilon$. It is known [TJ74] that for $2 \le p < \infty$, $\delta_p(\epsilon)
\sim \epsilon^p$, but that for $1 < p \le 2$, $\delta_p(\epsilon) \sim
\epsilon^2$.
An optimal expression of this fact is given by the following theorem which
was proved jointly with Keith Ball [BCL]:
%\vfill\eject
\vskip .3 true in
\noindent{\bf Theorem 1: (Optimal 2-uniform convexity for matrices)}.\quad
{\it For all $m\times m$ matrices $A$ and $B$ and all $p$ for $1 \le p \le
2$,
$$\biggl({Tr|A+B|^p + Tr|A-B|^p\over 2}\biggr)^{2/p} \ge
\bigl(Tr|A|^p\bigr)^{2/p} +
(p-1)\bigl(Tr|B|^p\bigr)^{2/p}\quad.\eqno(3.8)$$
For $ 1 < p < 2$, there is equality only when $B=0$}.
\vskip .3 true in
This result, which we interpret here as a statement about $\ckp$,
is proved in the appendix in the special case that both $A+B$
and $A-B$ are positive; this is the only case in which we shall use it
here, and the proof is considerably simpler
in this case. The full result is proved in
[BCL], in which other geometric inequalities for trace norms are proved as
well.
The theorem implies that
$$\delta_p(\epsilon) \ge
{(p-1)\over 2}\bigl({\epsilon\over 2}\bigr)^2
\qquad{\rm for}\qquad 1 < p \le 2 \eqno(3.9)$$
as one sees by considering
$A = (C+D)/2$, $B = (C-D)/2$, $\|C\|_p = \|D\|_p = 1$ and $\|C - D\|_p =
\epsilon$. It is easily seen that the constant $(p-1)/8$ cannot be
improved.
We make our main application of this result in Section V.
There we will also need to know that the norms on $\ckp$ are continuously
differentiable away from the origin for $1 < p < \infty$. This is known
[Gr75],
but a simple proof can be based on inequalities of the form
$\delta_p(\epsilon) \ge K_p\epsilon^{r(p)}$ such as we have found above for
$1 < p < 2$. This proof, moreover, gives the modulus of continuity of the
derivative, and is sketched in the appendix as well. Again, these estimates
are independent of the dimension and therefore apply to the case of
infinitely many degrees of freedom.
\vskip .3 true in
\centerline{\bf IV. CONDITIONAL EXPECTATIONS AND THE FERMION}
\centerline{\bf OSCILLATOR SEMIGROUP}
\vskip .3 true in
We are particularly concerned with the subalgebra $\cq$ of $\ck$, and the
conditional expectation [Di53][Um54]
with respect to it shall play a basic role in our
investigation. For any $A$ in $\ck$,
the {\bf conditional expectation} $\pq(A)$
of $A$ with respect to $\cq$ is defined to be the unique
element of $\cq$ such that
$\ta(B^*\pq(A)) = \ta(B^*A)$
for all $B$ in $\cq$. Otherwise said, $\pq$ is the orthogonal projection from
$\ckt$ onto $\cqt$. It is well known that the conditional expectation is
positivity preserving; a familiar argument shows that
$\pq(A^*A) \ge \pq(A)^*\pq(A)$.
We can use the conditional expectation to give a useful expression for the
oscillator semigroup for fermion fields.
Let $\Rt$ be the orthogonal
transformation of ${\cal K}$ given by
$$\Rt(q_j) = (\cos\theta)q_j + (\sin\theta)p_j\eqno(4.1)$$
for each $j$. Of course $\Rt$ gives the evolution at time $\theta$ on phase
space ${\cal K}$ generated by the classical oscillator Hamiltonian
$H({\bf p},{\bf q}) = \sum_{j=1}^n{1\over 2}(p_j^2 + q_j^2)$.
Let $\Rt$ denote
the automophism of $\ck$ generated by the orthogonal transformation $\Rt$
as in the first section.
For each $t\ge 0$, define $\theta(t) = \arccos(e^{-t})$ and define the
operator $P_t$ on $\ckt$ by
$$P_tA = \pq\circ\rt\circ\iq A\quad.\eqno(4.2)$$
Note that $\iq$ is the {\bf natural imbedding} of $\cq$ into $\ck$,
and regarded as such, it is a $*$-automorphism.
Formula (4.2) is the analog of the familiar expression for the
boson oscillator semigroup on $L^2(\Q,(2\pi)^{-n/2}e^{-q^2/2}{\rm d}^nq)$, i.e.
the Mehler semigroup
$$P_t^{(boson)}A({\bf q}) = \int_\Q A\biggl(e^{-t}{\bf q} + (1-e^{-2t})^{1/2}
{\bf p}\biggr)
(2\pi)^{-n/2}e^{-p^2/2}{\rm d}^np\quad.$$
Note that since all of the operators on the right in (4.2) are positivity
preserving, so is $P_t$. Also, since the first two operations on the right
preserve the ${\cal C}^p$-norms, and since the conditional
expectation is readily seen to
be a contraction from $\ckp$ to $\cqp$ for each $p$, it is readily seen that
$P_t$ possesses this property as well.
To obtain a more familiar expression for $P_t$, note that
$$R_{\theta(t)}\circ\iq\bigl(E_{[\alpha_1,\dots,\alpha_k]}\bigr)
= e^{-kt}E_{[\alpha_1,\dots,\alpha_k]} + \bigl({\rm terms\ annihilated\ by}\
\pq\bigr)\quad.$$ Hence
$$P_t\bigl(E_{[\alpha_1,\dots,\alpha_k]}\bigr)
= e^{-kt}E_{[\alpha_1,\dots,\alpha_k]}.\eqno(4.3)$$
Evidently $\{P_t : t\ge0\}$ is generated by $H_0$ where
$H_0\bigl(E_{[\alpha_1,\dots,\alpha_k]}\bigr) =
kE_{[\alpha_1,\dots,\alpha_k]}$.
It is easy to see that under the unitary equivalence between $\cqt$
and fermion Fock space ${\cal F}$ described by Segal [Se56], $H_0$
is equivalent to the usual number operator, or in other words, oscillator
Hamiltonian on ${\cal F}$.
Our primary goal is to prove optimal hypercontractivity bounds for $P_t$.
That is,
given $1 < p < q <\infty$ we want to show that for some finite $t$, $P_t$ is a
contraction from $\cqp$ to $\cqq$, and to find the smallest such $t$. Let
$$\|P_t\|_{p\rightarrow q}^{\phantom .} =
\sup\{\|P_tA\|_q\ :\ \|A\|_p = 1\ \}\quad.
\eqno(4.4)$$
As a first reduction, we shall show that the supremum on the right in (4.4)
can be restricted to the positive operators $A$ with $\|A\|_p = 1$.
In the boson case this follows immediately from the fact that, in
ordinary probability theory, the absolute value of a conditional expectation
is no greater than the conditional expectation of the absolute value.
In general, matters concerning the absolute value in the non-commutative
setting are more troublesome than in the commutative setting. An
example is provided by the Araki-Yamagami inequality [ArYa] which,
specialized to our context, asserts
that the map $A\mapsto |A|$ is Lipschitz continuous on $\ckt$ with constant
$\sqrt 2$ instead of the constant 1 which we would have in the commutative
setting.
Thus while the conditional expectation in an
operator algebra has many properties
analogous to those of the conditional expectation in ordinary probability
theory [Um54], it is not in general true that $|\pq(A)|$ will be a
smaller operator than $\pq(|A|)$. The following theorem expresses a useful
property in this direction which does hold, and after proving it we shall
show by example
that stronger properties do not hold. The theorem and its proof are easily
extended to a more general von Neumann algebra setting by the methods in
[Ru72].
\vskip .3 true in
\noindent{\bf Theorem 2: (A Schwarz inequality for conditional expectations)}.
\quad {\it For all $A$ in $\ck$ and all $p$ with} $1\le
p\le\infty$,
$$\|\pq(A)\|_p \le \|\pq(|A|)\|_p^{1/2}
\|\pq(|A^*|)\|_p^{1/2}\quad.\eqno(4.5)$$
\vskip .3 true in
\noindent{\bf Remark:}\ If we let $F(A)$ denote $\|\pi_{\cal Q}A\|_p$,
then the same argument which we shall use to prove
Theorem 2 also establishes that
$$F(A^*B) \le F(A^*A)^{1/2}F(B^*B)^{1/2}\quad.\eqno(4.6)$$
In this form, the term ``Schwarz inequality'', by which we referred to
(4.5), is more evidently appropriate. Moreover, inequalities of the
type (4.6) are well known in matrix analysis for many familiar functions;
for example when F(A) is the determinant of $A$ or the spectral radius of
$A$. Further examples can be found in [MeDS].
In [Li76] it is shown that for a function $F$
that satisfies (4.6), and which is monotone
increasing; i.e. satisfies $F(B) \ge F(A)$ for all $B\ge A\ge 0$,
the following inequalities hold:
$$F(\sum_{j=1}^mA_j^*B_j) \le F(\sum_{j=1}^m(A_j^*A_j)^{1/2}
F(\sum_{j=1}^m(B_j^*B_j)^{1/2}$$
and
$$F(\sum_{j=1}^mA_j) \le F(\sum_{j=1}^m(|A_j|)^{1/2}
F(\sum_{j=1}^m(|A^*_j|)^{1/2}\quad.$$
In particular, these inequalities hold for $F(A) =
\|\pi_{\cal Q}A\|_p$.
Specializing the last inequality to the case $m=1$ then yields (4.5).
In our present case however, the proof of the (4.6) is essentially
the same as the direct proof of (4.5). Nonetheless,
it should not be considered novel that by taking $|A^*|$
into consideration as well as $|A|$, we can obtain a suitable bound on
$\|\pi_{\cal Q}A\|_p$.
\vskip .3 true in
\noindent{\bf Proof:}\
Let $A = U|A|$ be the polar decomposition of $A$.
Then $\|\pq(A)\|_p = \ta(CU|A|)$ for some $C$ in
$\cq$ with $\|C\|_{p'} = 1$. Let $C = V|C|$ be the polar decomposition of
$C$. Both $V$ and $|C|$ belong to $\cq$ as well. Thus
$$\|\pq(A)\|_p = \ta\bigl(CU|A|^{1/2}|A|^{1/2}\bigr)
= \ta\bigl(|C|^{1/2}U|A|^{1/2}|A|^{1/2}V|C|^{1/2}\biggr)$$
$$\le \ta\bigl(|C|^{1/2}U|A|U^*|C|^{1/2}\bigr)^{1/2}
\ta\bigl(|C|^{1/2}V^*|A|V|C|^{1/2}\bigr)^{1/2}$$
$$=\ta\bigl(|C|(U|A|U^*)\bigr)^{1/2}
\ta\bigl((V|C|V^*)|A|\bigr)^{1/2}
\le\|\pq(U|A|U^*)\|_p^{1/2}
\|\pq(|A|)\|_p^{1/2}\quad.$$
Finally, we note that $U|A|U^* = |A^*|$.
\quad$\square$
\vskip .4 true in
\noindent{\bf Example}\quad Let $A$ be the matrix $A = \left[\matrix{0&1\cr
0&1\cr}\right]$. Then
$$|A| = \sqrt2\left[\matrix{0&0\cr0&1}\right]\qquad{\rm and}\qquad
|A^*| = {1\over \sqrt2}\left[\matrix{1&1\cr1&1}\right]\quad.$$
Note that $|A^*|$ is in $\cq$, but $A$ and $|A|$ are not. One easily finds
$$\pq(A) =
{1\over 2}\left[\matrix{1&1\cr1&1}\right]\quad,\quad \pq(|A|) =
{1\over \sqrt2}\left[\matrix{1&0\cr0&1}\right]\quad,\quad
\pq(|A^*|) =
{1\over \sqrt2}\left[\matrix{1&1\cr1&1}\right]\ .$$
$$\|\pq(A)\|_\infty = 1\qquad,\qquad \|\pq(|A|)\|_\infty = {1\over
\sqrt2}\qquad{\rm and}\qquad
\|\pq(|A^*|)\|_\infty = \sqrt2\quad.$$
\vskip .4 true in
\noindent{\bf Theorem 3: ($P_t$ has positive maximizers)}.\quad
{\it The norm of $P_t$ from $\cqp$ to $\cqq$ is achieved on the positive
operators; i.e.}
$$\|P_t\|_{p\rightarrow q} =
\sup\{\|P_tA\|_q\ :\ A\ge 0\quad{\rm and}\quad\|A\|_p = 1\ \}\quad.\eqno(4.7)$$
\vskip .4 true in
\noindent{\bf Proof:}\quad Since $R_\theta$ and $\iq$ are both
$*$-automorphisms, $|\rt\circ\iq A| = \rt\circ\iq|A|$ and
$|(\rt\circ\iq A)^*| = \rt\circ\iq |A^*|$. Thus
$\|P_tA\|_q \le \|P_t|A|\|_q^{1/2}\|P_t|A^*|\|_q^{1/2}$,
and of course $\||A^*|\|_p = \||A|\|_p = \|A\|_p$.
\quad $\square$
\vskip .4 true in
\centerline{\bf V. HYPERCONTRACTIVITY FOR FERMIONS}
\vskip .3 true in
Our main result is the following theorem which is established in this section.
\vskip .3 true in
\noindent{\bf Theorem 4: (Optimal fermion hypercontractivity)}.\quad {\it For all $1 < p \le q < \infty$, $\|P_t\|^{\phantom .}_{p\rightarrow
q} = 1$ exactly when $$e^{-2t} \le {p-1\over q-1}\quad.$$}
\vskip .3 true in
The heart of the matter is the following lemma:
\vskip .3 true in
\noindent{\bf Lemma:}\quad {\it For all $1 < p \le 2$, \ $\|P_t\|_{p\rightarrow
2} = 1$ exactly when $e^{-2t} \le (p-1)$.}
\vskip .3 true in
\noindent{\bf Proof:}\quad Fix a positive element A of $\cq$.
Pick a basis $\{q_1,\dots,q_n\}$ of $\Q$, and let
$\cqn$ denote the Clifford algebra associated with the span of the first
$n-1$ of these basis elements. It is evident from the form of our standard
basis of $\cq$ that $A$ can be uniquely decomposed as
$A = B + CQ_n$
where $B$ and $C$ belong to $\cqn$. Then using the Jordan-Wigner transform
we can write
$A = B + CV_n\hat Q_n$.
Now write
$$\H = \H_{(n-1)}\otimes\IC^2\eqno(5.1)$$
so that $B$, $C$ and $V_nC$ can be
considered as operators on the first factor ${\cal H}_{(n-1)}$.
Let $u_\pm = (e_+ \pm
e_-)/\sqrt 2$, so that $Qu_\pm = \pm u_\pm$. Then if $v$ is any
vector in $\H_{(n-1)}$,
$$\langle (v\otimes u_\pm),A(v\otimes u_\pm)\rangle_{\H}^{\phantom .} =
\langle v,\bigl(B \pm CV_n\bigr)v\rangle_{\H_{(n-1)}}^{\phantom .}
\quad.\eqno(5.2)$$
We see from this that since $A\ge0$, so are both $B + CV_n\ge 0$ and
$B - CV_n\ge 0$. Now let $Tr_1$ and $Tr_2$ denote the partial traces
over the first and second factors in (5.1), so that with $Tr$ still denoting
the full trace,
we have $Tr = Tr_1Tr_2$. Now applying Theorem 1 in the special case
which is proved in the appendix
$$\|A\|_p^2 = \biggl({1\over 2^n}Tr|B + CV_n\hat Q_n|^p\biggr)^{2/p}$$
$$= \biggl({1\over 2^{(n-1)}}\biggr)^{2/p}
\biggl({Tr_1|B + CV_n|^p + Tr_1|B - CV_n|^p\over 2}\biggr)^{2/p}$$
$$\ge \biggl({1\over 2^{(n-1)}}\biggr)^{2/p}
\biggl((Tr_1|B|^p)^{2/p} + (p-1)(Tr_1|C|^p)^{2/p}\biggr)$$
since $V_n$ is unitary.
Thus
$\|A\|_p^2 \ge \|B\|_p^2 + (p-1)\|C\|_p^2$
where the norms on the right are all norms on $\cqn$.
Now we make the inductive assumption that the lemma has been established
for $\cqn$. This is clearly the case when $n=1$. We then have
$$\|A\|_p^2 \ge \|P_tB\|_2^2 + (p-1)\|P_tC\|_2^2$$
where $e^{-2t} = (p-1)$. But clearly from (4.3)
$\|P_tCQ_n\|_2^2 = (p-1)\|P_tC\|_2^2$
where the norms are once again norms on $\cq$. Moreover by (3.4) $P_tB$ and
$P_tCQ_n$ are orthogonal. Thus
$\|P_tB\|_2^2 + (p-1)\|P_tC\|_2^2 =
\|P_t(B + CQ_n)\|_2^2 = \|P_tA\|_2^2$.\quad
$\square$
\vskip .3 true in
\noindent{\bf Theorem 5: (Optimal fermion logarithmic Sobolev inequality)}.\
{\it For all} $A \in \cq$,
$$\ta\bigl(|A|^2\ln|A|^2\bigr) -
\bigl(\|A\|_2^2\ln\|A\|_2^2\bigr) \le 2\langle A,H_0A\rangle\quad.\eqno(5.3)$$
\vskip .3 true in
\noindent{\bf Proof:}\ By the lemma, $\|P_tA\|_2^2 \le \|A\|^2_{(1+e^{-2t})}$
and there is equality at $t=0$. Both sides are continuously differentiable,
and comparing derivatives at $t=0$ we obtain the result. Indeed,
$${{\rm d}\over {\rm d}p}\|A\|_p = {1\over p}\|A\|_p^{1-p}\biggl(
\ta(|A|^p\ln|A|) - \|A\|_p^p\ln\|A\|_p\biggr) \eqno(5.4)$$
and of course
$${{\rm d}\over {\rm d}t}\|P_tA\|_2^2 =
{{\rm d}\over {\rm d}t}\langle A,P_{2t}A\rangle =
-2\langle A,H_0A\rangle\quad.\ \square$$
\vskip .5 true in
Gross refers to the quadratic form on the right side of (5.3) as the
Clifford Dirichlet form since it shares many properties of Dirichlet forms
in the ordinary commutative setting. An approach to the development of a
theory of Dirichlet forms in the non-commutative setting can be found in
[AlHK].
\vskip .5 true in
\noindent{\bf Proof of Theorem 4:}\quad By a deep result of Gross, when $A\ge0$
and $1 < p < \infty$,
$$\langle A^{p/2},H_0A^{p/2}\rangle \le
{(p/2)^2\over p-1}\langle A,H_0A^{p-1}\rangle \quad.$$
Replacing $A$ in (5.3) by $A^{p/2}$ and using the inequality just quoted
we obtain, following Gross's ideas [Gr75],
$$\ta(A^p\ln A) - \|A\|_p^p\ln\|A\|_p \le {p/2\over p-1}\langle
A^{p-1},H_0A\rangle\quad.$$
By combining this with (5.4) a differential inequality is obtained which implies
that
$\|P_tA\|_{q(t)}$
is a decreasing function of $t$
when $q(t) = 1 + e^{2t}(p-1)$.
This establishes the result for $A\ge0$,
and by Theorem 3 it is established in general.
By (4.3), $P_tI = I$, and therefore $\|P_t\|_{p\rightarrow q}$ is always
at least 1 for all $p$ and $q$.
That the inequality is best
possible follows from a direct computation with one degree of freedom.
To be precise, $\|P_t(I+Q_1)\|_q = \|I+e^{-t}Q_1\|_q$ is easily computed
and compared with $\|I+Q_1\|_p$ [Gr72]. The first quantity is greater than
the second if $e^{-2t} > (p-1)/(q-1)$.
\quad $\square$
\vskip .3 true in
\noindent{\bf VI. HYPERCONTRACTIVITY FOR BOSONS AND FERMIONS TOGETHER}
\vskip .3 true in
As a result of the present work and of earlier work on bosons, we know
that for $t$ given by $e^{-2t} = (p-1)/(q-1)$, both the fermion and the
boson oscillator semigroups are contractive from the appropriate
$p$--spaces to the appropriate $q$--spaces, and that this value of
$t$ is optimal for each case {\it separately}.
It is natural to expect that the same condition governs hypercontractivity
in a situation in which we have bosons and fermions {\it together}. This
is indeed the case, as we now show using Minkowski's inequality in an
argument based on Segal's method for showing that the optimal conditions
for hypercontractivity with $m$ boson degrees of freedom are the same as
for one degree of freedom.
Let $\mu({\rm d}x) = (2\pi)^{-m/2}e^{-x^2/2}{\rm d}x$ be the unit Gauss measure
on $\IR^m$. Then in our mixed setting, with $m$ boson degrees of freedom
and $n$ fermion degrees of freedom, the relevant $p$--space is
$${\cal B}^p = L^p(\IR^m,\mu)\otimes{\cal C}^p({\cal Q}_{(n)}),$$
which may be regarded as consisting of
${\cal C}^p({\cal Q}_{(n)})$ valued measurable functions $x\mapsto A(x)$
such that
$$|\!|\!|A|\!|\!|_p^p = \int_{\IR^m}\|A(x)\|_p^p\mu({\rm d}x)$$
is finite. This equation defines the norm on ${\cal B}^p$.
For $p=2$, ${\cal B}^p$ is naturally isomorphic to the tensor product of
the symmetric tensor algebra over $\IC^m$ and the antisymmetric tensor
algebra over $\IC^n$ as shown by Segal. On the latter space we have the
mixed oscillator semigroup generated by the sum of the boson and fermion
number operators
$$\exp\biggl\{-t\biggl[\sum_{j=1}^m a^*_ja_j+\sum_{j=1}^n c^*_jc_j
\biggr]\biggr\}.$$
Considered as operators on ${\cal B}^p$, the operators
${\cal P}_t$ which constitute this semigroup are given by
$${\cal P}_tA(x) =
\int_{R^M}M_t(x,x')P_t\bigl[A(x')\bigr]\mu({\rm d}x')\eqno(6.1)$$ where
$M_t(x,x')$ is the Mehler kernel; i.e., the positive integral kernel for the
boson oscillator semigroup $P^{(boson)}_t$ discussed in Section IV.
Of course $P_t$ denotes the fermion oscillator semigroup studied throughout
this paper.
Now successively applying Minkowski's inequality, our theorem on optimal
fermion hypercontractivity, and Nelson's theorem on optimal boson
hypercontractivity, we have for $e^{-2t} \le (p-1)/(q-1)$:
$$\eqalign{|\!|\!|{\cal P}_tA|\!|\!|_q^q&= \int_{\IR^m}\bigg\|\int_{\IR^m}M_t(x,x')P_t\bigl[A(x')\bigr]
\mu({\rm d}x')
\bigg\|_q^q\mu({\rm d}x)\cr
&\le \int_{\IR^m}\biggl(\int_{\IR^m}M_t(x,x')\|P_tA(x')\|_q\mu({\rm d}x')\biggr)
^q\mu({\rm d}x)\cr
&\le \int_{\IR^m}\biggl(\int_{\IR^m}M_t(x,x')\|A(x')\|_p\mu({\rm d}x')\biggr)^q\mu({\rm d}x)\cr
&\le \bigl(\int_{\IR^m}\|A(x)\|_p^p\mu({\rm d}x)\bigr)^{q/p} =
|\!|\!|A|\!|\!|_p^q \quad.\cr}$$
\vskip .3 true in
\centerline{\bf APPENDIX}
\vskip .3 true in
\noindent{\bf Proof of Theorem 1 when $A\pm B\ge0$:}\quad
Let $Z$ and $W$ be the $2m\times 2m$ matrices given by
$$Z = \left[\matrix{A&0\cr 0&A\cr}\right]\quad,\quad W =
\left[\matrix{B&0\cr 0&-B\cr}\right]\quad.$$
Our goal is to establish that for all $r$ with $0\le r \le 1$,
$$\biggl({Tr(A+rB)^p + Tr(A-rB)^p\over 2}\biggr)^{2/p} \ge (Tr(A)^p)^{2/p} +
r^2(p-1)(Tr|B|^p)^{2/p}\quad,$$
or what is the same,
$$Tr(Z+rW)^{2/p} \ge (Tr(Z)^p)^{2/p} + r^2(p-1)(Tr|W|^p)^{2/p}
\quad.\eqno(A.1)$$
First, note that the null space of $Z+rW$ is exactly the null space of $Z$ for
$0 \le r < 1$. Thus by carrying out all of the following computations on
the orthogonal complement of this fixed null space, we may freely assume
that $Z+rW > 0$ for all $0 \le r < 1$.
Next, both sides of (A.1) agree at $r=0$,
and the first derivatives in $r$ of both
sides vanish there as well. We define $\psi(r)$ to be $Tr(Z+rW)^p$. Then the
second derivative in $r$ of the left side of (A.1) satisfies
$${{\rm d}^2\over {\rm d}r^2}\bigl(\psi(r)\bigr)^{2/p} \ge
{2\over p}\psi(r)^{(2-p)/p}{{\rm d}^2\over {\rm d}r^2}\psi(r)\quad.$$
The second derivative on the right side is just
$$2(p-1)(Tr|W|^p)^{2/p}\quad,$$
and we are left with showing that
$${1\over p}\psi(r)^{(2-p)/p}
{{\rm d}^2\over {\rm d}r^2}\psi(r) \ge (p-1)(Tr|W|^p)^{2/p} \eqno(A.2)$$
for all $0 < r < 1$. By redefining $Z$ to be $Z+rW$, it suffices to
establish (A.2) at $r=0$.
Now
${{\rm d}\over {\rm d}r}\psi(r) =
p\bigl(Tr(Z+rW)^{(p-1)}W\bigr)$,
since $A\pm B\ge 0$, $Z+rW\ge0$ for small $r$, and
we can use the integral representation
$$\bigl(Z+rW\bigr)^{(p-1)} = c_p\int_0^\infty t^{(p-1)}\biggl[{1\over t} -
{1\over t + (Z+rW)}\biggr]dt$$ to conclude that
$${{\rm d}^2\over {\rm d}r^2}\psi(0) =
pc_p\int_0^\infty t^{(p-1)}Tr\biggl[
{1\over t + Z}W
{1\over t + Z}W\biggr]dt\quad. \eqno(A.3)$$
Consider the right side as a function, $f(Z)$, of $Z$ for fixed $W$. It is easy to
see that $f$ is convex in $Z$. (Simply replace $Z$ by $Z+tX$, with $X$
self--adjoint, and then differentiate twice with respect to $t$; the
positivity follows from the Schwarz inequality for traces.) Also,
$f(UZU^*)=f(Z)$
provided $U$ is unitary and $U$ commutes with $W$.
In a basis in which $W$ is diagonal, we form the set $\cal{U}$ consisting
of the $2^{2m}$ distinct
diagonal unitary
matrices, each with $+1$ or $-1$ in each diagonal entry. Each of these
clearly commutes with $W$. Then
$$f(Z)=2^{-2m}\sum_{U\in\cal{U}}f(UZU^*)\ge f(2^{-2m}\sum_{U\in\cal{U}}UZU^*)
=f(Z_{{\rm diag}}),$$
where
$Z_{{\rm diag}}$ is the matrix that is diagonal in the basis diagonalizing
$W$, and whose diagonal entries are those of $Z$ in this
basis.
Replacing $Z$ by
$Z_{{\rm diag}}$ in (A.3), the integration can be carried out, and we
obtain
$${{\rm d}^2\over {\rm d}r^2}\psi(0) \ge p(p-1)
\biggl(\sum_{j=1}^{2m}z_j^{(p-2)}w_j^2\biggr)$$
where $z_j$ and $w_j$, respectively, denote
the $j$th diagonal entries of $Z$ and $W$ in a
basis diagonalizing $W$.
Now consider $\psi(0) = Tr(Z^p)$ as a function of $Z$. It is clearly
convex,
and thus by the averaging method just employed, we obtain
$$\psi(0) \ge
\biggl(\sum_{j=1}^{2m}z_j^p\biggr)\quad.$$
To establish (A.2), we are only left with showing that
$$\biggl(\sum_{j=1}^{2m}z_j^p\biggr)^{(2-p)/p}
\biggl(\sum_{j=1}^{2m}z_j^{(p-2)}w_j^2\biggr) \ge
\biggl(\sum_{j=1}^{2m}|w_j|^p\biggr)^{2/p}\quad,\eqno(A.4)$$
but this follows immediately from H\"older's inequality.
To complete the proof, observe that equality in (A.1) for $r=1$ and $1 < p
< 2$ implies equality in
(A.4) for almost every $r$ in $[0,1]$. Here, recall that $z_j$ in (A.4)
really denotes the $j$th diagonal element of $Z+rW$; these are the numbers
$z_j+rw_j$, where $z_j$ denotes the $j$th diagonal element of $Z$.
Let us assume that $w_j \neq 0$ for some $j$. Then
equality in H\"older's inequality (A.4)
requires that the vector with positive components
$z_j + rw_j$ be proportional to the vector with components $|w_j|$. Thus,
for almost every $r$ in $[0,1]$ we require
$$z_j + rw_j = c(r)|w_j|$$
for some number $c(r)$ that depends on $r$ but not on $j$. The left side
above is a linear function, and thus $c(r) = a+rb$ for some numbers $a$ and
$b$.
But then clearly $b = w_j/|w_j|$, and all
non-zero
eigenvalues of $W$ would necessarily
have the same sign. This is impossible since $TrW
= 0$.
\quad $\square$
\vskip .3 true in
We now give an application of the uniform convexity implied by this theorem
to the differentiability of the $\ckp$ norms. First we recall that for
$2 \le p < \infty$, the modulus of convexity is given by an analog of an
inequality of Clarkson for integrals which Dixmier [Di53]
established for traces. Specializing to $\ckp$, this inequality reads
$$\biggl\|{A+B\over 2}\biggr\|_p^p +
\biggl\|{A-B\over 2}\biggr\|_p^p
\le {1\over 2}\bigl[\|A\|^p_p + \|B\|_p^p\bigr]\qquad
2\le p < \infty\quad,$$
which implies that in this range ({\it cf}. (3.7))
$$\delta_p(\epsilon) \ge {1\over p}\bigl({\epsilon\over 2}\bigr)^p
\quad.\eqno(A.5)$$
For any non-zero $A$ in $\ckp$, define $\da$ by
$$\da = \|A\|_p^{(1-p)}|A|^{(p-1)}U^*$$
where $A = U|A|$ is the polar decomposition of $A$. Let $p'$ be defined by
$1/p + 1/p' = 1$. Then for $1