# The Existence of Linear Selection and the Quotient Lifting Property

Monika ,  Fernanda Botelho and  Richard Fleming Department of Mathematical
Sciences
The University of Memphis
Memphis, TN 38152-3240, United States<EMAIL_ADDRESS><EMAIL_ADDRESS>Central Michigan University<EMAIL_ADDRESS>
###### Abstract.

Lifting properties for Banach spaces are studied. An alternate version of the
lifting property due to Lindenstrass and Tzafriri is proposed and a
characterization, up to isomorphism, is given. The quotient lifting property
for pairs of Banach spaces $(X,J)$, with $J$ proximinal in $X$, is considered
and several conditions for the property to hold are given.

###### 2000 Mathematics Subject Classification:

Primary 47L05, 46B28, 46B25
Keywords and phrases. Lifting property; Quotient lifting property; Metric
projection; Metric complement; Proximinal subspaces; Linear selection

##  1\. Introduction

Following Lindenstrass and Tzafriri [8], we say a Banach space $Y$ has the
_lifting property_ (LP) if for every bounded operator $\psi$ from a Banach
space $X$ onto a Banach space $W$ and for every $S\in\L(Y,W)$, there is
$\hat{S}\in\L(Y,X)$ such that $S=\psi\circ\hat{S}$. We note that in [8] it is
shown that $\ell_{1}$ has the LP, but that can easily be extended to show that
any space isomorphic to $\ell_{1}$ has the LP. The converse for infinite
dimensional separable spaces is given in [8]. If we put the attention on $W$
rather than $Y$, we say $W$ has the _alternate lifting property_ (ALP), if for
an operator $\psi$ from $X$ onto $W$ and $S\in\L(Y,W)$, there is
$\hat{S}\in\L(Y,X)$ such that $S=\psi\circ\hat{S}$. From the ideas in [8] we
have the following theorem.

Theorem. Let $W$ be a Banach space. Then $W$ satisfies the ALP if and only if
$W$ is isomorphic to $\ell_{1}(\Gamma)$, for a suitable index set $\Gamma$.

We consider the case where $W$ is the quotient of $X$ by a closed subspace $J$
with $\psi=\pi$, where $\pi$ is the quotient map. We are interested in the
existence of norm preserving lifts of a bounded operator $S:Y\rightarrow X/J$.
If, for every $Y$ and $S$, such a lift exists then we say that the pair
$(X,J)$ has quotient lifting property (QLP).

First, we give an alternate version of the QLP. We say that $Y$ satisfies the
alternate quotient lifting property (AQLP) if given a Banach space $X$ with
closed subspace $J$ and bounded operator $S$ from $Y$ to $X/J$, there exists
$\hat{S}\in\L(Y,X)$ such that $S=\pi\circ\hat{S}$. We obtain a similar
characterization for Banach spaces with the AQLP.

Theorem. If $Y$ is Banach space then $Y$ satisfies the AQLP if and only if $Y$
is isomorphic to $\ell_{1}(\Gamma),$ for a suitable index set $\Gamma$.

We remark that we have not required $\|\hat{S}\|=\|S\|$ in the alternate
versions as we do for the QLP.

In this paper we first study conditions on a Banach space that ensure the
existence of liftings for operators either into the space (ALP) or on the
space (AQLP), and for which a diagram commutes. We also investigate conditions
for the existence of norm preserving lifts of operators (QLP) and its
interconnections with metric projections, linear selections and proximinality.

We start by recalling some definitions to be used throughout. For a closed
subspace $J$ of $X$, we denote by $\pi$ the canonical quotient map from $X$ to
$X/J$, given by $\pi(x)=x+J$ ($\|\pi\|=1$ unless $J=X$). A subspace $J$ of a
normed linear space $X$ is called proximinal (resp. Chebyshev) if for each $x$
in $X$, the set of best approximations to $x$ in $J$,

$P_{J}(x):=\\{j\in J|\|x-j\|=dist(x,J)\\}$

is nonempty (resp. a singleton). The set valued map $P_{J}$ is called the
metric projection onto $J$. For a proximinal subspace $J$, a selection for
$P_{J}$ is a function $p:X\to J$ with $p(x)\in P_{J}(x)$, for every $x$. If
$p$ is linear we call it a linear selection. The metric complement of $J$ is
defined to be $J_{0}=\\{x\in X:\|x\|=dist(x,J)\\}.$ We also recall that an
M-ideal in a Banach space is a subspace for which the annihilator is an
L-summand of the dual space. For details we refer the reader to Chapter 1 in
[6]. It is a known fact that M-ideals are proximinal.

In section 2 we prove the theorems stated above. These theorems characterize
spaces with the ALP and spaces with the AQLP, up to isomorphism. We draw some
conclusions concerning subspaces of Banach spaces with the property.

In section 3 we show that proximinality is a necessary condition for the QLP
and the metric complement being a subspace is sufficient for QLP.

In section 4 we study the relation between properties of the metric complement
of $J$ and the QLP for $(X,J)$. It is worth to mention that the existence of a
linear selection is an invariant condition under isometric isomorphisms. From
this fact we derive that the QLP holds for several pairs of Banach spaces.
Moreover, for proximinal subspaces, we show that the QLP is equivalent to the
existence of a linear selection.

In Section 5 we consider the QLP for subspaces that are M-ideals. If $X$ is
reflexive and $J$ is an $M$-ideal in $X$, then $(X,J)$ has the QLP. The same
holds for $C([0,1]^{n})$ and $J$ an $M$-ideal in $C([0,1]^{n})$. The QLP does
not hold, in general, for $(C(\Omega),J)$, with $\Omega$ a compact Hausdorff
space and $J$ an $M$-ideal of $C(\Omega)$. Nevertheless, the QLP holds when
$\Omega$ is compact and metrizable.

##  2\. The Alternate Lifting Properties

We define the following alternate lifting properties for Banach spaces.

######  Definition 1.

Let $W$ be a Banach space. Then,

  * •

$W$ has the ALP if and only if given Banach spaces $X$ and $Y$, an operator
$\psi$ from $X$ onto $W$ and $S\in\L(Y,W)$, there is $\hat{S}\in\L(Y,X)$ such
that $S=\psi\circ\hat{S}$.

  * •

$Y$ has the AQLP if and only if for every Banach space $X$, a closed subspace
$J$ and a bounded operator $S$ from $Y$ to $X/J$, there exists
$\hat{S}\in\L(Y,X)$ such that $S=\pi\circ\hat{S}$, with $\pi$ denoting the
quotient map from $X$ onto $X/J$.

######  Theorem 2.

Let $W$ be a Banach space. Then $W$ satisfies the ALP if and only if $W$ is
isomorphic to $\ell_{1}(\Gamma)$, for some suitable index set $\Gamma$.

###### Proof.

The statement is straightforward for finite dimensional spaces. We present the
proof for the infinite dimensional case. We recall that any Banach space is
the quotient of $\ell_{1}(\Gamma)$ for some suitable index set $\Gamma$ (see
[10] page 21), and therefore is the range of a bounded operator on
$\ell_{1}(\Gamma)$. Let $W$ satisfy the ALP and in that definition, let
$X=\ell_{1}(\Gamma)$, $\psi$ a bounded operator from $X$ onto $W$, $Y=W$, and
$S$ the identity operator on $W$, denoted by $Id$. Then by the ALP, there
exists $\hat{S}$ from $W$ to $\ell_{1}(\Gamma)$ such that

$Id=S=\psi\circ\hat{S}.$

It is easy to see that $\hat{S}\circ\psi$ is a bounded projection onto a
subspace of $\ell_{1}(\Gamma)$. It follows from the Koethe extension of
Theorem 2.a.3 in [8, p. 108] that W is isomorphic to $\ell_{1}(\Gamma)$.

On the other hand, suppose $T$ is an isomorphism from $\ell_{1}(\Gamma)$ onto
$W$. Let $w_{\gamma}=Te_{\gamma}$, where $\\{e_{\gamma}\\}$ denotes the
standard family of functions from $\Gamma$ into the scalar field such that
$e_{\gamma}(\alpha)=1$ for $\gamma=\alpha,$ and $0$ otherwise. Thus if $w\in
W$, there exists an absolutely summable family $\\{\alpha_{\gamma}\\}$ of
scalars such that $w=\sum_{\gamma\in\Gamma}\alpha_{\gamma}w_{\gamma}$. There
exists $x_{\gamma}\in X$ such that $\psi(x_{\gamma})=w_{\gamma}$ and since
$\psi$ is surjective and open, the family $\\{x_{\gamma}\\}$ must be bounded.
For $y\in Y$ there is an absolutely summable family of scalars
$\\{\alpha_{\gamma}\\}$ such that
$S(y)=\sum_{\gamma\in\Gamma}\alpha_{\gamma}w_{\gamma}$. Since the series
$\sum_{\gamma\in\Gamma}\alpha_{\gamma}x_{\gamma}$ is absolutely summable, it
must converge to some $x\in X$. We define $\hat{S}(y)=x$ and it follows that
$\hat{S}$ is a bounded linear operator from $Y$ to $X$ such that
$S=\psi\circ\hat{S}$. ∎

Similar considerations apply to the AQLP (see Definition • ‣ 1) to prove the
characterization for Banach spaces with the AQLP.

######  Theorem 3.

Let $Y$ be a Banach space. Then $Y$ satisfies the AQLP if and only if $Y$ is
isomorphic to $\ell_{1}(\Gamma)$.

###### Proof.

Suppose $Y$ has AQLP. As above, $Y$ is isometric with a quotient space
$\ell_{1}(\Gamma)/J$ for some index set $\Gamma$. Let $X=\ell_{1}(\Gamma)$ and
$J$ its subspace as given. Let $S$ be the isometry between $Y$ and
$\ell_{1}(\Gamma)$. For $\hat{S}$ guaranteed by AQLP, the operator
$\hat{S}\circ S^{-1}\circ\pi$ is a projection onto a complemented subspace of
$\ell_{1}(\Gamma)$. The remainder of the proof follows as in the proof of
Theorem 2. ∎

Therefore the class of Banach spaces with the ALP coincides with that of the
AQLP.

######  Theorem 4.

Let $W$ be a Banach space and let $W_{1}$ be a complemented subspace of $W$.
If $W$ has the AQLP then $W_{1}$ has the AQLP.

###### Proof.

Let $X$ be a Banach space and $J$ be a closed subspace of $X$. Since $W_{1}$
is complemented in $W$, there exists a projection $P$ from $W$ onto $W_{1}$.
Given $S\in L(W_{1},X/J)$ and since $W$ has the AQLP, there exists
$\widetilde{S\circ P}$, a lift of $S\circ P$, such that
$\pi\circ\widetilde{S\circ P}=S\circ P$. Then $\widetilde{S\circ P}|_{W_{1}}$
is a lift for $S$. This concludes the proof. ∎

Theorems 3 and 4 imply that every separable infinite dimensional complemented
subspace of a space with the AQLP (or ALP) is isomorphic to $\ell_{1}$. We
invoke the main theorem in [3] to conclude that $C(\Omega,E)$ (with $\Omega$
an infinite compact Hausdorff space and $E$ an infinite dimensional Banach
space) does not satisfy the AQLP. In particular, $C(\Omega,C(\Omega))$;
$C(K_{1}\times K_{2})$, where $K_{1}$ and $K_{2}$ are infinite compact
Hausdorff spaces, do not satisfy the AQLP, since each of these spaces contain
a complemented copy of $c_{0}$. See [3].

##  3\. Proximinality and the Quotient Lifting Property

In this section we start with the definition for a pair of Banach spaces to
have the QLP. The authors are grateful to T.S.S.R.K.Rao for mentioning a
property considered in [6] that lead to this definition.

######  Definition 5.

Let $X$ be a Banach space and $J$ be a closed subspace of $X$. The pair
$(X,J)$ has the QLP if and only if for every Banach space $Y$ and every
bounded operator $S:Y\to X/J$ there exists a bounded operator $T$ from $Y$ to
$X$ lifting $S$ while preserving the norm, i.e. $\|T\|=\|S\|$ and $\pi\circ
T=S$.

Given $S$, as in the Definition 5, there may exist several liftings, i.e.
bounded operators $T$ such that $S=\pi\circ T$. For each such lifting, $T$, we
have $\|S\|\leq\|T\|.$ It is easy to construct examples where the inequality
is strict.

######  Theorem 6.

Let $X$ be a Banach space and $J$ a closed subspace of $X$. Then

  1. (i)

If $(X,J)$ has the QLP then $J$ is proximinal in $X$.

  2. (ii)

If $J$ is proximinal in $X$ and $P_{J}$ has a linear selection, then $(X,J)$
has the QLP.

###### Proof.

If $J$ is not proximinal in $X$, then there must exist a norm one element
$x\in X$ such that $dist(x,J)<\|x-j\|$, for every $j\in J$. We define
$Y=span\\{x\\}$. Let $S:Y\rightarrow X/J$ be such that $S(x)=x+J$ then
$\|S\|=dist(x,J)$. Every bounded operator $T:Y\rightarrow X$ such that
$\pi\circ T=S$, satisfies $T(x)=x+j$, for some $j\in J$. Then
$\|S\|<\|x+j\|\leq\|T\|.$ This proves (i).

For (ii), suppose $P_{J}$ admits a linear selection $p:X\to J$ and let
$\psi:X/J\to X$ be defined by $x+J\mapsto x-p(x)$. To see that $\psi$ is well
defined, we suppose that $x_{1}+J=x_{2}+J$. Then $x_{2}=x_{1}+j$ for some
$j\in J$, and by the linearity of $p$, we can write

$\psi(x_{2}+J)=x_{2}-p(x_{2})=x_{1}+j-p(x_{1})-j=x_{1}-p(x_{1})=\psi(x_{1}+J).$

Thus $\psi$ is a linear isometry. For a bounded operator $S:Y\to X/J$,
$\tilde{S}=\psi\circ S$ provides the lifting as desired. ∎

######  Corollary 7.

If $J$ is an M-summand in X, then (X,J) has QLP .

###### Proof.

If $J$ is an M-summand in X, then J is proximinal and has a linear
(continuous) selection, namely, the M-projection P. Therefore the statement. ∎

David Yost in [11] proved that a Banach space is reflexive if and only if
every closed subspace is proximinal. Hence every nonreflexive space must
contain a nonproximinal closed subspace and therefore this pair does not have
the QLP.

We now give an example of a pair of spaces for which the proximinality
condition holds but the QLP does not hold. First, we recall the notion of
total subset of a Banach space. A subset $F$ of the dual space $X^{*}$ of a
Banach space $X$ is said to be total if $f(x)=0$ for each $f\in F$ implies
that $x=0$.

######  Example 8.

We consider the pair $(\ell_{\infty},c_{0})$. From [6] we know $c_{0}$ is an
$M$-ideal, hence a proximinal subspace of $\ell_{\infty}$. We show that the
pair $(\ell_{\infty},c_{0})$ does not have the QLP. To see this, we observe
that there is no injective bounded linear map from $\ell_{\infty}/c_{0}$ to
$\ell_{\infty}$. We assume the existence of an injective bounded linear map
$\phi$ from $\ell^{\infty}/c_{0}$ to $\ell_{\infty}$. Since
$\left(\ell_{\infty}\right)^{*}$ contains a countable total set,
$\\{\tau_{i}:\tau_{i}(e_{j})=\delta_{ij}\\}$, then $\phi^{*}\tau_{i}$ is a
total set for $(\ell_{\infty}/c_{0})^{*}$. Indeed, for $z\in\ell_{\infty}$
such that $\phi^{*}\tau_{i}(z)=0$ for all i, then $\phi z=0$. Since $\phi$ is
injective, then $z=0$. Arterbaum and Whitley in [1] showed that
$(\ell_{\infty}/c_{0})^{*}$ has no countable total subset. Therefore, the
identity map from $\ell_{\infty}/c_{0}$ to $\ell_{\infty}/c_{0}$ has no lift.
If such a lift existed, it would be an injective bounded linear map. This is
impossible and it shows that the pair $(\ell_{\infty},c_{0})$ does not have
the QLP.

We note that the quotient space $\ell_{\infty}/c_{0}$ does not have the ALP.
To establish this take $S$ to be the identity map from $\ell_{\infty}/c_{0}$
to $\ell_{\infty}/c_{0}$ and $\psi$ to be the quotient map from
$\ell_{\infty}$ to $\ell_{\infty}/c_{0}$. If there exists $\hat{S}$ from
$\ell_{\infty}/c_{0}$ to $\ell_{\infty}$ such that $S=\psi\circ\hat{S}$, then
$\hat{S}$ is an injective map but no such map exists. Theorem 2 implies that
$\ell_{\infty}/c_{0}$ is not isomorphic to a $\ell_{1}(\Gamma),$ for any index
set $\Gamma$.

For completeness of exposition we include the following theorems from [5] and
from [4] to be used later.

######  Theorem 9.

(See Deutsch, [5]) Let $J$ be a proximinal subspace of a normed linear space
$X$. Then the following are equivalent

  1. (1)

$P_{J}$ has a linear selection.

  2. (2)

$J_{0},$ the metric complement of $J$, contains a closed subspace $J_{1}$ such
that $X=J\oplus J_{1}$.

######  Theorem 10.

(See Cheney and Wulbert [4]) Let $J$ be a subspace of $X$. Then

  1. (1)

$J$ is a proximinal subspace if and only if $X=J+J_{0}$.

  2. (2)

$J$ is Chebyshev if and only if $X=J+J_{0}$ and the representation of each
$x\in X$ as $x=j+j_{0}$, where $j\in J$ and $j_{0}\in J_{0}$, is unique.

Thus we have the following proposition.

######  Proposition 11.

Let $J$ be a proximinal subspace of $X$. Then

  1. (i)

If $J_{0}$ is a subspace of $X$ then $(X,J)$ has the QLP.

  2. (ii)

(cf. [7]) $J_{0}$ is a subspace if and only if $J$ is Chebyshev and $P_{J}$ is
linear.

###### Proof.

From Theorem 10, we have $X=J+J_{0}$. By assumption $J_{0}$ is a subspace of
$X$, so Theorem 9 asserts that $P_{J}$ has a linear selection. So (i) follows
from Theorem (ii)-(ii).

Since $J$ is proximinal in $X$ and $J_{0}$ is a subspace, Theorem 10-(1)
implies that $X=J+J_{0}$. It is clear that $J\cap J_{0}=\\{0\\}$, then
$X=J\oplus J_{0}$. The existence of a linear selection of the metric
projection $P_{J}$ follows from Theorem 9. The uniqueness of the
representation of any element in $X$ implies that $J$ is Chebyshev by Theorem
10-(2). Conversely, since $J_{0}=\\{x-P_{J}(x):\,x\in X\\}$, the linearity of
the metric projection implies $J_{0}$ is a subspace of $X$. ∎

Next example shows that $J_{0}$ being a subspace is not a necessary condition
for the QLP.

######  Example 12.

(See example 2.7 of [5]) Let $X=\ell_{\infty}^{(2)}(\mathbb{R})$ and
$J=span\\{e_{1}\\}$. It is easy to see that $J_{0}$ is not a subspace. Just
consider $(0,2)$ and $(1,-2)$, both in $J_{0}$ with sum $(1,0)\notin J_{0}$.
We observe that $p:X\rightarrow J$ given by $p(x,y)=(x,0)$ is a linear
selection for the metric projection, Theorem 2-(ii) implies that
$(\ell_{\infty}^{(2)}({\mathbb{R}}),J)$ has the QLP.

##  4\. On metric complement of subspaces for low dimensional Banach spaces

In this section we start with a characterization of the metric complements of
subspaces of $\ell_{\infty}^{(2)}(\mathbb{R})$. First, we notice that for
$J=\\{0\\}$ or $J=X$ then $J_{0}=X$ or $J_{0}=\\{0\\}$ respectively.

######  Proposition 13.

Let $X=\ell^{(2)}_{\infty}(\mathbb{R})$ and let $J$ be a non-trivial subspace
of $X$. Then $J_{0}$ is a subspace of $X$ if and only if $J$ is generated by
$(u,v)$ with $uv\neq 0$.

###### Proof.

We claim that $J_{0}$ is homogeneous, i.e. given $(a,b)\in J_{0}$ and
$\lambda$ a nonzero scalar, then $(\lambda a,\lambda b)\in J_{0}$. Towards
this claim we just observe that

$\inf_{j\in J}\|(\lambda a,\lambda b)-j\|=|\lambda|\inf_{j\in
J}\|(a,b)-\lambda^{-1}j\|=|\lambda|\|(a,b)\|=\|(\lambda a,\lambda b)\|.$

Let $J$ be a non-trivial subspace of $\ell^{(2)}_{\infty}(\mathbb{R})$
generated by $(u,v)$. If $uv=0$, WLOG assume that $u\neq 0$ and $v=0$, then
$J=span(1,0)$. If $J_{0}$ is a subspace we have $w_{0}=(0,1)$ and
$w_{1}=(1,1)$ belong to $J_{0}$ but $w_{1}-w_{0}=(1,0)$ does not belong to
$J_{0}$. This shows that $J_{0}$ is not a subspace. If $uv\neq 0$, let
$a=\frac{v}{u}$ then $J$ is generated by $(1,a)$. For $x_{1}=(1,c)$

$dist(x_{1},J)=\inf_{t\in\mathbb{R}}\\{\max\\{|1-t|,|c-at|\\}\\}$

The graphs of $|1-t|$ and $|c-at|$ plotted in the figure above illustrate the
cases to be considered in the computation of $dist(x_{1},J)$. In each case,
the distance is equal to the second coordinate of $P$. We now proceed with the
computations.
Case I: $a>0$
If $c>a$, $dist(x_{1},J)=\frac{c-a}{1+a}$ which is not equal to
$\|x_{1}\|_{\infty}$ for any choice of $c$. If $c<a$,
$dist(x_{1},J)=\frac{a-c}{1+a}$ which is equal to $\|x_{1}\|_{\infty}$ only if
$c=-1$. So $J_{0}$ is generated by $(1,-1)$ and is a subspace of $X$.
Case II: $a<0$
If $c>a$, $dist(x_{1},J)=\frac{c-a}{1-a}$ which is equal to
$\|x_{1}\|_{\infty}$ only if $c=1$. If $c<a$, $dist(x_{1},J)=\frac{a-c}{1-a}$
which is not equal to $\|x_{1}\|_{\infty}$ for any choice of $c$. So $J_{0}$
is generated by $(1,1)$ and is a subspace of $X$.

∎

######  Remark 14.

  1. (1)

Let $X$ be a Banach space and $J$ a proximinal subspace of codimension one.
Then $(X,J)$ has the QLP. This follows from Corollary 2.8 in [5].

  2. (2)

If $J$ is a closed subspace of a Hilbert space, $\mathcal{H}$, then the pair
$(\mathcal{H},J)$ has the QLP since the metric projection on a subspace of a
Hilbert space is linear.

The next result gives a sufficient condition for the existence of a linear
selection of the metric projection onto a proximinal subspace.

######  Proposition 15.

Let $X$ be a Banach space and let $J$ be a closed subspace of $X$. If $(X,J)$
has the QLP then the metric projection $P_{J}$ has a linear selection.

###### Proof.

Since $(X,J)$ has the quotient lifting property then there exists a bounded
operator $\tilde{Id}:X/J\rightarrow X$ such that $Id=\pi\circ\tilde{Id}$ and
$\|\tilde{Id}\|=\|Id\|=1$. It is easy to check that $\tilde{Id}\circ\pi$ is a
projection on $X$. For simplicity we denote this composition by $P$. Further,
$X=Range(P)\oplus Ker(P)$ and $ker(P)=J$. It is clear that $J\subset ker(P)$,
if $x\in Ker(P)$ then $\tilde{Id}(x+J)=0$ and
$x+J=\pi\circ\tilde{Id}(x+J)=\pi(0)=J$. Hence $x\in J$. We claim that
$Range(P)\subset J_{0}$, the metric complement of $J$. Towards this claim, we
observe that, for $x\in Range(P)$, $P(x)=\tilde{Id}(x+J)=x$ and
$\|x\|\leq\|x+J\|=dist(x,J)\leq\|x\|.$ Therefore $x\in J_{0}$. An application
of the Theorem 9 implies that the metric projection onto $J$ has a linear
selection. ∎

######  Remark 16.

Given $J$, a proximinal subspace of a Banach space $X$, and let $P_{J}$ denote
the metric projection onto $J$. Then the theorems 6 and 15 imply the
equivalence of the following statements:

  * •

$(X,J)$ has the QLP.

  * •

$P_{J}$ has a linear selection.

######  Example 17.

Let $\Omega$ be a compact Hausdorff space and let $J$ be the subspace of all
constant functions in $C(\Omega)$. The pair $(C(\Omega),J)$ has the QLP if and
only if the cardinality of $\Omega$ is less or equal to 2. It is clear that
$J$ is Chebyshev.

If $\Omega$ has 3 distinct points, $a,b,c$, we consider three pairwise
disjoint open neighborhoods, $U_{a}$, $U_{b}$ and $U_{c}$ of $a$, $b$ and $c$,
respectively. Then there exist continuous functions $f_{a}$ and $f_{b}$ on
$\Omega$ satisfying the conditions: $f_{a}:\Omega\rightarrow[0,1]$,
$f_{a}(a)=1$ and $f_{a}(x)=0,$ for all $x\notin U_{a}$;
$f_{b}:\Omega\rightarrow[-1,0]$, $f_{b}(b)=-1$ and $f_{b}(x)=0$, for all
$x\notin U_{b}$. We set $f=f_{a}+f_{b}$. Similarly we define $g=g_{b}+g_{c}$,
with $g_{b}:\Omega\rightarrow[-1,0]$, $g_{b}(b)=-1$ and $g_{b}(x)=0$, for all
$x\notin U_{b}$; $g_{c}:\Omega\rightarrow[0,1]$, $g_{c}(c)=1$ and
$g_{c}(x)=0,$ for all $x\notin U_{c}$. The constant function equal to $0$ is
closest to $f$ and $g$ but $-1/2$ is closest to $f+g$. This implies that
$P_{J}$ is not linear and then $(C(\Omega),J)$ does not have the QLP by Remark
16. The other implication follows from Proposition 13 .

This example shows that $X/J$ having the ALP does not imply that the metric
projection onto $J$ has a linear selection. The lift of the identity operator
on $X/J$ may not have norm 1, in which case the argument given in the proof
for the Proposition 15 does not hold.

We add a few remarks that follow straightforwardly from previous results and
theorems in [5].

  * •

If $\Omega$ contains n isolated points, there is an n-dimensional subspace $J$
of $C_{0}(\Omega)$ for which $P_{J}$ admits a linear selection. Therefore
$(C_{0}(\Omega),J)$ has the QLP.

  * •

If $J=span\\{f\\}$, then $(C_{0}(\Omega),J)$ has the QLP iff the support of
$f$ contains at most 2 points.

  * •

If $f$ is in $L^{1}(\Omega,\Sigma,\mu)=X,$ then for $J=span\\{f\\}$, $(X,J)$
has the QLP if support of $f$ is purely atomic and contains at most two atoms.
A similar statement holds for $L^{p}$, with $1<p<\infty$ and $p\neq 2$.

The next result addresses the problem of whether the existence of a linear
selection for a pair of spaces transfers to other pairs of spaces defined from
the given one. We consider sequence spaces, spaces of vector valued continuous
functions on a compact Hausdorff space $\Omega$, Lebesgue-Bochner integrable
function spaces ($L_{1}(\mu,X)$) and space of $\mu$-measurable Pettis
integrable functions ($\hat{P_{1}}(\mu,X)$). For the definitions and
properties of these spaces we refer the reader to [10]. Given a Banach space
$X$ we denote by $\mathcal{S}(X)$ any one of the following sequence spaces:
$\ell_{p}(X)$ ($1\leq p\leq\infty$), $c_{0}(X)$ or $c(X)$.

We denote by $\mathcal{F}(X)$ any one of the following function spaces:
$C(\Omega,X)$, with a compact Hausdorff space, or $Lip(\Omega,X)$ , with
$\Omega$ a compact metric space endowed with one of the standard norms. We
denote by $\mathcal{I}(X)$ any one of the following function spaces:
$L_{1}(\mu,X)$ or $\hat{P_{1}}(\mu,X)$. Further, $L(Y,X)$ denotes the space of
bounded operators from $Y$ into $X$.

######  Proposition 18.

Let $X$ and $Y$ be Banach spaces and let $J$be a proximinal subspace of $X$
such that the metric projection $P_{J}$ has a linear selection $p:X\to J$.
Then the map $f\mapsto p\circ f$ from $\mathcal{S}(X)$ ($\mathcal{F}(X)$,
$\mathcal{I}(X)$ or $L(Y,X)$) onto $\mathcal{S}(J)$ (resp. $\mathcal{F}(J)$,
$\mathcal{I}(J)$, $\mathcal{L}(Y,J)$) is a linear selection.

###### Proof.

The proof follows easily from $\|(p(x_{n}))\|\leq 2\|(x_{n})\|.$ ∎

Let $J$ be a proximinal subspace of $X$ and let $p$ be a linear selection. We
observe that if $f:Y\rightarrow X$ is an isometric isomorphism, then
$f^{-1}(J)$ is proximinal in $Y$, and $f^{-1}\circ p\circ f$ defines a linear
selection from $Y$ to $f^{-1}(J).$

We denote by $X\hat{\otimes}_{\pi}Y$ the tensor product space of $X$ and $Y$
endowed with the projective norm and by $X\hat{\otimes}_{\epsilon}Y$ the
tensor product space of $X$ and $Y$ endowed with the injective norm. We refer
the readers to [10] for more details. If $X$ and $Y$ are isometrically
isomorphic we denote this by $X\cong Y$. The following identifications of
tensor products are known: $\ell_{1}\hat{\otimes}_{\pi}X\cong\ell_{1}(X)$;
$\ell_{1}\hat{\otimes}_{\epsilon}X\cong\ell_{1}[X]$, the space of
unconditionally summable sequences in $X$;
$c_{0}\hat{\otimes}_{\epsilon}X\cong c_{0}(X)$;
$C(\Omega)\hat{\otimes}_{\epsilon}X\cong C(\Omega,X)$, the space of continuous
functions from $\Omega$ to $X$; $L_{1}(\mu)\hat{\otimes}_{\pi}X\cong
L_{1}(\mu,X)$ and
$L_{1}(\mu)\hat{\otimes}_{\epsilon}X\cong\hat{P_{1}}(\mu,X)$. These facts,
together with the above observation and Proposition 18, establish the next
corollary.

######  Corollary 19.

If $X$ is a Banach space, $J$ is a closed proximinal subspace of $X$, and
$p:X\to J$ is a linear selection, then there is a linear selection from
$\ell_{1}\hat{\otimes}_{\pi}X$ onto $\ell_{1}\hat{\otimes}_{\pi}J$,
$\ell_{1}\hat{\otimes}_{\epsilon}X$ onto $\ell_{1}\hat{\otimes}_{\epsilon}J$,
$c_{0}\hat{\otimes}_{\epsilon}X$ onto $c_{0}\hat{\otimes}_{\epsilon}J$,
$C(\Omega)\hat{\otimes}_{\epsilon}X$ onto
$C(\Omega)\hat{\otimes}_{\epsilon}J$, $L_{1}(\mu)\hat{\otimes}_{\pi}X$ onto
$L_{1}(\mu)\hat{\otimes}_{\pi}J$, and $L_{1}(\mu)\hat{\otimes}_{\epsilon}X$
onto $L_{1}(\mu)\hat{\otimes}_{\epsilon}J.$

######  Remark 20.

Using Theorem (ii) we conclude that all the pairs listed in Proposition 18
have the QLP.

##  5\. M-ideals and the quotient lifting property

In this section we explore the QLP for pairs where the subspace is an M-ideal.
Towards this we establish the existence of a linear selection for the metric
projection.

In [6] (see p.59) it is mentioned that Ando, Choi and Effros proved that if
$J$ is an $L_{1}$-predual M-ideal in Banach space $X$ and $Y$ is a separable
Banach space, any bounded linear operator $T\in L(Y,X/J)$ has a bounded linear
lifting $\tilde{T}\in L(Y,X)$ such that $\|T\|=\|\tilde{T}\|$ and
$T=\pi\circ\tilde{T}$. Our next result shows that this statement holds for
$X=C(\Omega)$, where $\Omega$ is compact metrizable, any M-ideal $J$ and all
Banach spaces $Y$. Therefore the pair $(C(\Omega),J)$ has the QLP. Let us
recall a definition from [2].

######  Definition 21.

A Banach space $X$ is called a $\pi$-space if there is a sequence ${F_{n}}$ of
finite dimensional subspaces such that $F_{1}\subset F_{2}\subset F_{3}\dots$
with $\overline{\cup_{n=1}^{\infty}F_{n}}=X$ and each $F_{n}$ is the range of
a projection of norm one.

######  Remark 22.

Separable $L_{p}(1\leq p<\infty)$ and $C(\Omega)$ on compact metrizable
$\Omega$ are $\pi$-spaces (See [2].)

######  Proposition 23.

The pair $(C(\Omega),J)$, where $\Omega$ is a compact metrizable space and $J$
is an M-ideal of $C(\Omega)$, has the QLP.

###### Proof.

Since $J$ is an M-ideal of $C(\Omega)$, then $J=\\{f\in C(\Omega):f|_{D}\equiv
0\\}$ for some closed subset $D$ of $\Omega$. By Remark 22, $C(\Omega)$ is a
$\pi$-space and $C(D)$ is isometrically isomorphic to $C(\Omega)/J$. Therefore
$C(\Omega)/J$ is a $\pi$\- space. By Theorem 5 of [2] there exists a norm one
linear map $\phi$ from $C(\Omega)/J$ to $C(\Omega)$ such that
$\pi\circ\phi=Id$. For a Banach space $Y$ and bounded linear map $S:Y\to
C(\Omega)/J$, $\tilde{S}:Y\to C(\Omega)$ defined by $\tilde{S}=\phi\circ S$ is
a lifting such that $S=\pi\circ\tilde{S}$ and $\|S\|=\|\tilde{S}\|$. This
concludes the proof. ∎

Our next result shows on how to construct a linear selection onto an arbitrary
M-ideal of the space of continuous functions on a particular class of
Euclidean domains.

######  Proposition 24.

Let $J$ be an M-ideal of $C([0,1]^{n})$, where $n$ is a positive integer. Then
the metric projection $P_{J}$ has a linear selection.

###### Proof.

Since $J$ is an M-ideal of $C([0,1]^{n})$ there exists a closed subset $D$ of
$[0,1]^{n}$ such that $J=\\{f\in C([0,1]^{n}):f|_{D}\equiv 0\\}$ (See [6]
p.4.) We observe that $g\in J_{0}$ if and only if
$\|g\|_{\infty}=dist(g,J)=\|g|_{D}\|_{\infty}.$ We present the proof for the
cases: $n=1$ and $n=2$; other cases follow similarly. For $n=1$, we first
consider $D=[t_{1},t_{2}]\cup[t_{3},t_{4}].$ Let $f\in C[0,1]$, then we define
$f_{1}$ as

$(\star)\,\,f_{1}(x)=\begin{cases}f(x)-f(t_{1}\chi_{(-\infty,t_{1})}+t_{4}\chi_{(t_{4},\infty)})&\text{for
}x\notin[t_{2},t_{3}]\\\ 0&\text{for }x\in D\\\
f(x)-f(t_{2})+\dfrac{x-t_{2}}{t_{3}-t_{2}}\left[f(t_{2})-f(t_{3})\right]&\text{for
}x\in[t_{2},t_{3}]\end{cases}$

The map $p:C([0,1])\to J$ defined by $f\mapsto f_{1}$ is a linear selection
and $f-f_{1}\in J_{0}$. For an arbitrary closed subset of $[0,1]$ we assign
the value $0$ on $D$ and extend over the open intervals in $[0,1]\setminus D$
as in $(\star)$.

Figure 1. $D$ is a closed subset of $[0,1]^{2}$ and
$D^{\prime}=D\cap\overline{OP}$

Let $n=2$ and $f\in C([0,1]^{2})$. For $x\in[0,1]^{2}$ we will define
$f_{1}=f_{1_{m}}$ along the line of slope $m$ passing through $0$ and $x$. Let
$D^{\prime}$ be the intersection of $D$ with the line passing through $0$ and
$x$, see Figure 1. For simplicity of notation, we assume that $D^{\prime}$ has
two connected components, i.e. $D^{\prime}=[t_{1},\,t_{2}]\cup[t_{3},t_{4}].$
Then,

$f_{1_{m}}(x)=\begin{cases}f_{1}(x)&\text{for
}\|x\|\notin[\|t_{2}\|,\|t_{3}\|]\\\
f(x)-f(t_{2})+\frac{\|x-t_{2}\|}{\|t_{3}-t_{2}\|}[f(t_{2})-f(t_{3})]&\text{for
}\|x\|\in[\|t_{2}\|,\|t_{3}\|]\end{cases}$

The map $p:C([0,1]^{2})\to J$ defined by $f\mapsto f_{1}$ is a linear
selection. ∎

The same techniques are unsuitable for general Hausdorff spaces.

It is well known that M-ideals are proximinal (see [6], p.50). It is not the
case that for every Banach space $X$ and an M-ideal $J$ in $X$, the pair
$(X,J)$ has the QLP, as we have seen with the pair $(\ell_{\infty},c_{0})$.
However the conclusion is different for reflexive Banach spaces.

Given a Banach space $B$, let $J_{B}$ denote the canonical isometric embedding
of $B$ into its double dual, $B^{**}$.

######  Proposition 25.

If $X$ is reflexive and $J$ is an M-ideal of $X$ then $(X,J)$ has the QLP.

###### Proof.

Let $Y$ be a Banach space and $S:Y\rightarrow X/J$, a bounded operator. We
denote by $\pi:X\rightarrow X/J$, the quotient map, we assume that $J\neq X$.
Since the dual of $X/J$ is identified with the annihilator of $J$,
$J^{\perp}$, we have $S^{*}:J^{\perp}\rightarrow Y^{*}$. Let $P$ denote the
projection $X^{*}=J^{\perp}\oplus_{1}J^{*}\rightarrow J^{\perp}$. The
composition $S^{*}\circ P:X^{*}\rightarrow Y^{*}$, is such that $\|S^{*}\circ
P\|=\|S\|$, and $(S^{*}\circ P)^{*}:Y^{**}\rightarrow X^{**}=X$ also has the
same norm as $S$. We now show that $\pi\circ(J^{-1}_{X}\circ(S^{*}\circ
P)^{*}\circ J_{Y})=S$. Let $y\in Y$, $(S^{*}\circ P)^{*}(y^{**})=J_{X}(x)$,
and $\pi(x)=x+J$. Given $y\in Y$ and $\eta\in X^{*}$ then

$(S^{*}\circ P)^{*}\circ J_{Y}(y)(\eta)=P(\eta)(S(y)).$

On the other hand, $S(y)=x+J$, identified with its image in $(X/J)^{**}$,
$(x+J)^{**}(P(\eta))=P(\eta)(x+J)=P(\eta)(S(y))$. This shows that
$\pi\circ(J^{-1}_{X}\circ(S^{*}\circ P)^{*}\circ J_{Y})=S$ and completes the
proof. ∎

## References

  * [1] D. Arterburn, R. Whitley, Projections in the space of bounded linear operators, Pacific Journal of Mathematics, 15:3 (1965), 739‐746.
  * [2] T. Ando, Closed range theorem for convex sets and linear liftings, Pacific Journal of Mathematics, Vol. 44, No. 2, (1973).
  * [3] P. Cembranos, $C(K,E)$ contains a complemented copy of $c_{0}$, Proceedings of the American Mathematical Society, Vol. 91, No. 4 (Aug., 1984), pp.556-558
  * [4] E.W. Cheney and D.E. Wulbert, The existence and unicity of best approximations Math. Scand. 24 (1969), 113-140.
  * [5] F. Deutsch, Linear selections for the metric projection, J. Functional Analysis 49 (1982), 269-292.
  * [6] P. Harmand, D. Werner and W. Werner, M-ideals in Banach spaces and Banach algebras, Lecture Notes in Mathematics, 1547 (1993) Springer-Verlag, Berlin.
  * [7] R. B. Holmes and B. R. Kripke, Smoothness of approximation, Michigan Math. J. 15 (1968), 225-248.
  * [8] J. Lindenstrauss and L. Tzafriri, Classical Banach spaces I and II, Springer Classical Mathematics. Springer-Verlag London, Ltd., London (1996).
  * [9] F. J. Murray, On complementary manifolds and projections in spaces $L_{p}$ and $\ell_{p}$, Trans. Amer. Math. Soc. 41:1 (1937), 138–152.
  * [10] R. Ryan, Introduction to tensor products of Banach spaces, Springer Monographs in Mathematics. Springer-Verlag London, Ltd., London (2002), 225 pp.
  * [11] D. Yost, Best approximation operators in functional analysis. Miniconference on nonlinear analysis, (1984) 249-270, Canberra AUS, 1984.


# Quantum Electrodynamics of Non-Hermitian Dirac Fermions

Sk Asrap Murshed  Department of Physics, Lehigh University, Bethlehem,
Pennsylvania, 18015, USA  Bitan Roy  Department of Physics, Lehigh University,
Bethlehem, Pennsylvania, 18015, USA

###### Abstract

We develop an effective quantum electrodynamics for non-Hermitian (NH) Dirac
materials interacting with photons. These systems are described by Lorentz
invariant NH Dirac operators, featuring two velocity parameters $v_{{}_{\rm
H}}$ and $v_{{}_{\rm NH}}$ associated with the standard Hermitian and a
masslike anti-Hermitian Dirac operators, respectively. They display linear
energy-momentum relation, however, in terms of an effective Fermi velocity
$v_{{}_{\rm F}}=\sqrt{v^{2}_{{}_{\rm H}}-v^{2}_{{}_{\rm NH}}}$ of NH Dirac
fermions. Interaction with the fluctuating electromagnetic radiation then
gives birth to an emergent Lorentz symmetry in this family of NH Dirac
materials in the deep infrared regime, where the system possesses a unique
terminal velocity $v_{{}_{\rm F}}=c$, with $c$ being the speed of light. While
in two dimensions such a terminal velocity is set by the speed of light in the
free space, dynamic screening in three spatial dimensions permits its
nonuniversal values. Manifestations of such an emergent spacetime symmetry on
the scale dependence of various physical observables in correlated NH Dirac
materials are discussed.

_Introduction_. Quantum electrodynamics (QED) is one of the most successful
theoretical description of the light-matter interaction [1]. Among its many
triumphs, perhaps the most prominent one is the highly accurate prediction of
the radiative correction for the $g$ factor of electrons due the interaction
with photons [2]. Even the leading-order (one-loop) correction agrees
remarkably well with experiments. What plays an important role in the
formulation of QED and the subsequent development of the electroweak and
strong forces, ultimately leading to the Standard model of fundamental
particles, is the spacetime Lorentz symmetry. It ensures a single velocity for
all the participating degrees of freedom, the speed of light ($c$), realized
at sufficiently high energies [3, 4]. Dynamics of the underlying fermionic
degrees of freedom in the Standard model is governed by the Dirac equation
[5], manifesting a linear energy-momentum relation, and thus a Lorentz
invariance.

Often quantum materials also foster linearly dispersing emergent fermionic
quasiparticles at low energies around a few isolated (and special) points in
the Brillouin zone [6, 7, 8]. The Fermi velocity ($v_{{}_{\rm F}}$), typically
much smaller than $c$, of such low energy quasiparticles in this family of
materials plays the role of the speed of light, with $v_{{}_{\rm F}}=c/300$ in
graphene, for example. On the other hand, when Dirac quasiparticles interact
with photons, ultimately the system arrives at an infrared stable
renormalzation group (RG) fixed point where $v_{{}_{\rm F}}=c$, thereby
showcasing an emergent spacetime Lorentz symmetry in the crystalline universe,
although it is generically absent at the lattice or ultraviolet scale [9, 10,
11, 12].

While these outcomes are extremely robust and well appreciated in isolated or
closed Dirac crystals, the fate of QED and the possible emergence of the
Lorentz symmetry in the presence of system-to-environment interactions in non-
Hermitian (NH) Dirac materials remain unanswered so far. Throughout we assume
that the effective Fermi velocity of a collection of linearly dispersing NH
Dirac fermions ($v_{{}_{\rm F}}$) and the speed of light ($c$) are different
at the microscopic level, and $v_{{}_{\rm F}}<c$. Then, from the leading-order
perturbative analysis of the NH Dirac fermions interacting with photons [see
Fig. 1], here we show that both in two and three spatial dimensions
$v_{{}_{\rm F}}$ always approaches the speed of light in the deep infrared
regime through radiative quantum corrections. See Figs. 2 and 3. In two
spatial dimensions ($d=2$), while the speed to light does not renormalize, in
$d=3$ it gets reduced under coarse grain due to the many-body dynamic
screening. As a consequence, the unique terminal velocity in NH Dirac
materials is _nonuniversal_ in $d=3$. Even though a one-to-one correspondence
between the nature of the system-to-environment interaction and the resulting
NH operator is missing, we arrive at these conclusions by considering symmetry
protected Lorentz invariant NH Dirac operators in $d=2$ and $d=3$. These
outcomes may thus suggest a possible generic emergence of the Lorentz symmetry
in open NH Dirac crystals. We also outline the impact of such an emergent
spacetime symmetry on the scaling behavior of various experimentally
measurable physical observables.

_NH Dirac operator_. The Lorentz invariant NH Dirac operator in $d$ spatial
dimensions is given by [13]

$~{}H_{\rm NH}(\bm{k})=v_{{}_{\rm
H}}\left(\sum^{d}_{j=1}\Gamma_{j}k_{j}\right)+v_{{}_{\rm
NH}}\;\left(M\sum^{d}_{j=1}\Gamma_{j}k_{j}\right),$ (1)

where $\bm{k}=(k_{1},\cdots,k_{d})$ is the momentum, and $v_{{}_{\rm H}}$ and
$v_{{}_{\rm NH}}$ bear the dimension of the Fermi velocity. The internal
structure of the associated Dirac spinor ($\Psi$) depends on the microscopic
details, which we do not delve into here. For concreteness, we work with the
minimal four component Dirac system. The Hermitian matrices satisfy the
anticommuting Clifford algebra $\\{\Gamma_{j},\Gamma_{k}\\}=2\delta_{jk}$ and
$\\{\Gamma_{j},M\\}=0$ for $j,k=1,\cdots,d$ with $M^{2}=1$. The first term in
$H_{\rm NH}(\bm{k})$ corresponds to the standard Dirac Hamiltonian yielding
the conventional linear real energy-momentum relation $E_{\rm H}=\pm
v_{{}_{\rm H}}|\bm{k}|$. The second term in $H_{\rm NH}(\bm{k})$, on the other
hand, is anti-Hermitian by construction as
$(M\Gamma_{j})^{\dagger}=-M\Gamma_{j}$ for any $j$. These two terms
anticommute with each other. Therefore, the anti-Hermitian operator acts as a
momentum-dependent masslike term to regular Dirac fermions. As a consequence,
the eigenvalue spectrum of $H_{\rm NH}$ is given by $E_{\rm
NH}=\pm\sqrt{v^{2}_{{}_{\rm H}}-v^{2}_{{}_{\rm NH}}}|\bm{k}|\equiv v_{{}_{\rm
F}}|\bm{k}|$, where $v_{{}_{\rm F}}=\sqrt{v^{2}_{{}_{\rm H}}-v^{2}_{{}_{\rm
NH}}}$ is the effective Fermi velocity of NH Dirac fermions. The eigenvalues
of $H_{\rm NH}(\bm{k})$ are, therefore, purely real and purely imaginary for
$v_{{}_{\rm H}}>v_{{}_{\rm NH}}$ and $v_{{}_{\rm H}}<v_{{}_{\rm NH}}$,
respectively. Throughout, we assume the former situation. The mass matrix $M$
fully anticommutes with the standard Dirac Hamiltonian and its NH counterpart.
A uniform expectation value of such an operator in the ground state, i.e. when
$\langle\Psi^{\dagger}M\Psi\rangle\neq 0$, gives rise to an isotropic gapped
state. However, in NH Dirac systems condensation of any mass order must always
break at least one of its four _nonspatial_ , namely time-reversal, anti-
unitary and unitary particle-hole, and pseudo-Hermiticity symmetries [14].
Thus in any dimensions, the nodal NH Dirac quasiparticles are symmetry
protected and cannot be gapped unless at least one of these four nonspatial
symmetries is broken [13]. Next, we consider interaction between such gapless
system and photons, leading to QED for NH Dirac materials.

Figure 1: Leading-order (one-loop) self-energy diagrams for (a) non-Hermitian
Dirac fermions (solid lines) and (b) photons (wavy lines). The explicit form
of their interaction vertex (indexed by $\rho$ and $\lambda$) is shown in Eq.
(7).

_Quantum electrodynamic_. The imaginary time ($\tau$) action describing the
interaction between fluctuating electromagnetic fields and NH Dirac fermions
takes the form

$S^{\rm QED}_{\rm NH}=\int d\tau d^{D}\bm{x}\;\left(L_{\rm F}+L_{\rm
EM}\right),$ (2)

where $D=d-\epsilon$ about which more in a moment,

$L_{\rm F}=\Psi^{\dagger}\left[\left(\partial_{0}-ieA_{0}\right)+H_{\rm
NH}\left(\bm{k}\to-i{\bm{\nabla}}-\frac{e}{c}\bm{A}\right)\right]\Psi,$ (3)

and $e$ is the electronic charge. The Maxwellian Lagrangian in three spatial
dimensions is [15]

$L_{\rm EM}=\frac{1}{4}F_{\rho\lambda}F_{\rho\lambda},$ (4)

where $\rho,\lambda=0,\cdots,d$ and
$F_{\rho\lambda}=\partial_{\rho}A_{\lambda}-\partial_{\lambda}A_{\rho}$ is the
electromagnetic field strength tensor. Here,
$\Psi^{\dagger}\equiv\Psi^{\dagger}(\tau,\bm{x})$ and
$\Psi\equiv\Psi(\tau,\bm{x})$ are two independent Grassmann variables, and
$A_{\rho}\equiv A_{\rho}(\tau,\bm{x})$. The dynamics of NH Dirac fermions is
set by the fermionic Green’s function

$G_{\rm F}(i\omega,\bm{k})=\frac{i\omega-H_{\rm
NH}(\bm{k})}{\omega^{2}+v^{2}_{{}_{\rm F}}|\bm{k}|^{2}}.$ (5)

The gauge field propagator in $d$ dimensions is given by

$D_{\rho\lambda}(i\omega,\bm{k},d)=\delta_{\rho\lambda}\;\bigg{(}\varepsilon\left[\omega^{2}+c^{2}|\bm{k}|^{2}\right]^{\frac{d-1}{2}}\bigg{)}^{-1},$
(6)

where the speed of light in the medium is $c=1/\sqrt{\varepsilon\mu}$, and
$\varepsilon$ and $\mu$ are the permittivity and permeability of the medium,
respectively. Here, $\omega$ is the Matsubara frequency. The non-analytic
structure of the gauge field propagator in $d=2$ is obtained upon integrating
the analytic one in $d=3$ (stemming from the Maxwellian Lagrangian $L_{\rm
EM}$) over the momentum in the $z$ direction ($k_{z}$). Therefore, the speed
of light does not renormalize in $d=2$, while it receives perturbative
corrections due to many-body dynamic screening in $d=3$. The interaction
vertex between NH Dirac fermions and photons is given by $-ie\gamma_{\rho}$,
where $\rho=0,\cdots,d$ and

$~{}\gamma_{\rho}=\left(1,-\frac{i}{c}\left[v_{{}_{\rm
H}}\Gamma_{j}+v_{{}_{\rm NH}}M\Gamma_{j}\right]\right).$ (7)

In what follows we compute the perturbative corrections to the velocity
parameters $v_{{}_{\rm H}}$, $v_{{}_{\rm NH}}$ and $c$, arising from such QED
interactions. To this end we compute the self-energy diagrams for the NH Dirac
fermions and photons, shown in Fig. 1, in $D$ spatial dimensions within the
framework of an $\epsilon$ expansion, where $\epsilon=d-D$.

_Feynman diagrams_. The contribution of the fermionic self-energy diagram,
shown in Fig. 1(a), reads as

$\displaystyle\Sigma(i\nu,\bm{k})$ $\displaystyle=$
$\displaystyle(-ie)^{2}\int\frac{d^{D}\bm{p}}{(2\pi)^{D}}\int^{\infty}_{-\infty}\frac{d\omega}{2\pi}\gamma_{\rho}G_{\rm
F}(i\omega+i\nu,\bm{p}+\bm{k})$ (8) $\displaystyle\times$
$\displaystyle\gamma_{\lambda}D_{\rho\lambda}(i\omega,\bm{p},d).$

The self-energy correction of the photon field (polarization bubble), shown in
Fig. 1(b), yields

$\displaystyle\Pi_{\rho\lambda}(i\nu,\bm{k})$ $\displaystyle=$
$\displaystyle-(-ie)^{2}\;{\rm
Tr}\bigg{[}\int\frac{d^{D}\bm{p}}{(2\pi)^{D}}\int^{\infty}_{-\infty}\frac{d\omega}{2\pi}\gamma_{\rho}G_{\rm
F}(i\omega,\bm{p})$ (9) $\displaystyle\times$
$\displaystyle\gamma_{\lambda}G_{\rm F}(i\omega+i\nu,\bm{p}+\bm{k})\bigg{]}.$

Due to the distinct structure of the photon propagators in $d=2$ and $d=3$, we
discuss these two cases separately. We compute the above integrals using the
standard Feynman parametrization and identify their logarithmic divergence
from $1/\epsilon$ poles [1].

Figure 2: Renormalization group (RG) flows of two velocity components of non-
Hermitian Dirac fermions $v_{{}_{\rm H}}$ and $v_{{}_{\rm NH}}$ [Eq. (1)], and
the effective Fermi velocity $v_{{}_{\rm F}}=\sqrt{v^{2}_{{}_{\rm
H}}-v^{2}_{{}_{\rm NH}}}$ due to their interaction with photons in two
dimensions for (a) $v^{0}_{{}_{\rm H}}=0.5$ and $v^{0}_{{}_{\rm NH}}=0.31$,
such that $v^{0}_{{}_{\rm NH}}<v^{0}_{{}_{\rm H}}<c$, (b) $v^{0}_{{}_{\rm
H}}=1.1$ and $v^{0}_{{}_{\rm NH}}=0.9$, such that $v^{0}_{{}_{\rm
NH}}<c<v^{0}_{{}_{\rm H}}$, and (c) $v^{0}_{{}_{\rm H}}=1.25$ and
$v^{0}_{{}_{\rm NH}}=1.05$, such that $c<v^{0}_{{}_{\rm NH}}<v^{0}_{{}_{\rm
H}}$. Notice that $v^{0}_{{}_{\rm F}}<c$ always. The results are obtained by
numerically solving Eq. (12). Throughout we measure all the velocities in
units of $c$ (unrenormalized), which we set to be unity. Here, $b$ is the RG
time. The quantities with the superscript ‘0’ correspond to their bare values.
For convenience, we set the fine structure constant $\alpha^{\rm 2D}_{{}_{\rm
F}}=1$.

_Two dimensions_. In two spatial dimensions the contribution from the
polarization bubble is _finite_ , and thus the field renormalization condition
for the gauge field ($A_{\rho}$) is trivial $Z_{A}=1$. By contrast, the
fermionic self-energy diagram is logarithmically divergent, yielding the
fermionic field renormalization [16]

$Z_{\Psi}=1+2\;\alpha^{\rm 2D}_{{}_{\rm
F}}\;\frac{1-2x^{2}}{1-x^{2}}\;\left[x-\frac{x^{2}\;{\rm{cos^{-1}}}(x)}{(1-x^{2})^{1/2}}\right]\frac{1}{\epsilon},$
(10)

where $x=v_{{}_{\rm F}}/c$ and $\alpha^{\rm 2D}_{{}_{\rm
F}}=e^{2}/(4\pi^{2}\varepsilon v_{{}_{\rm F}}c)$ is the fine structure
constant in two dimensions. The renormalization conditions for two velocity
parameters of NH Dirac fermions are [16]

$Z_{Q}=Z^{-1}_{\Psi}\left\\{1+\frac{\alpha^{\rm 2D}_{{}_{\rm
F}}}{1-x^{2}}\bigg{[}x-\frac{{\rm{cos^{-1}}}(x)}{(1-x^{2})^{1/2}}\bigg{]}\frac{1}{\epsilon}\right\\}$
(11)

for $Q=v_{{}_{\rm H}}$ and $v_{{}_{\rm NH}}$. Fascinatingly, both the velocity
components receive identical perturbative correction, which is expressed
solely in terms of the effective Fermi velocity ($v_{{}_{\rm F}}$) for NH
Dirac fermions. Consequently, their RG flow equations are given by [16]

$~{}\beta_{Q}=\alpha^{\rm 2D}_{{}_{\rm
F}}\left[\frac{x-4x^{3}}{1-x^{2}}+\frac{1-2x^{2}+4x^{4}}{(1-x^{2})^{3/2}}\;{\rm{cos^{-1}}}(x)\right]Q$
(12)

for $Q=v_{{}_{\rm H}}$ and $v_{{}_{\rm NH}}$, where $\beta_{Q}\equiv dQ/d\ln
b$ and $b$ is the RG time. Remarkably, in Hermitian Dirac systems with
$v_{{}_{\rm NH}}=0$ from the outset, the RG flow equation for the Fermi
velocity $v_{{}_{\rm H}}=v_{{}_{\rm F}}$ takes the identical form [10]. Such a
similarity stems from the Lorentz invariant form of the NH Dirac operator [Eq.
(1)]. The RG fixed point of the above flow equation for any arbitrary initial
condition or bare value of $v_{{}_{\rm H}}$ and $v_{{}_{\rm NH}}$ is placed at
$x=1$ or equivalently at $v_{{}_{\rm F}}=c$. As the speed of light does not
renormalize in $d=2$, the terminal effective Fermi velocity for two-
dimensional NH Dirac fermions in the deep infrared regime is always the speed
of light in vacuum, with both the velocity components $v_{{}_{\rm H}}$ and
$v_{{}_{\rm NH}}$ being nonzero, irrespective of their bare values. We also
confirm this outcome by numerically solving the RG flow equation from Eq.
(12). The results are shown in Fig. 2. Notice that in NH Dirac materials, the
bare or renormalized values of $v_{{}_{\rm H}}$ and $v_{{}_{\rm NH}}$ can in
principle be larger than $c$, while maintaining $v_{{}_{\rm F}}\leq c$ always.
These scenarios are also shown in Fig. 2. Therefore, two-dimensional massless
NH Dirac quasiparticles always feature an emergent spacetime Lorentz symmetry
with their terminal effective Fermi velocity being equal to the speed of light
in the free space.

_Three dimensions_. In three dimensional NH Dirac systems the speed of light
gets renormalized due to the interaction between the fermionic degrees of
freedom and photons, causing dynamic many-body screening. Consequently, the
polarization bubble [Fig. 1(b)] shows a logarithmic or an $1/\epsilon$
divergence, resulting in the renormalization conditions for the permittivity
and permeability of the medium, respectively give by [16]

$Z_{\varepsilon}=1+N_{f}\frac{4}{3}\frac{\alpha^{\rm 3D}_{{}_{\rm
F}}}{\epsilon}\;\;\text{and}\;\;Z_{\mu}=1-N_{f}\frac{4}{3}\frac{\alpha^{\rm
3D}_{{}_{\rm F}}}{\epsilon}x^{2},$ (13)

where $\alpha^{\rm 3D}_{{}_{\rm F}}=e^{2}/(8\pi^{2}\varepsilon v_{{}_{\rm
F}}c^{2})$ is the fine structure constant in three dimensions and $N_{f}$ is
the number of four-component fermionic flavor. As the speed of light is
related to these two quantities according to $c=1/\sqrt{\varepsilon\mu}$, we
then arrive at the following RG flow equation [16]

$\displaystyle~{}\beta_{c}=-\frac{2}{3}N_{f}\alpha^{\rm 3D}_{{}_{\rm
F}}\left(1-x^{2}\right)c.$ (14)

The fine structure constant decreases monotonically under coarse grain
according to

$~{}\beta_{\alpha^{\rm 3D}_{{}_{\rm F}}}=-\frac{4}{3}N_{f}\left(\alpha^{\rm
3D}_{{}_{\rm F}}\right)^{2}.$ (15)

The RG flow equation for $c$ in a conventional Dirac system with $v_{{}_{\rm
NH}}=0$ is identical to the above one, but in terms of the ratio $v_{{}_{\rm
H}}/c$, as $v_{{}_{\rm F}}=v_{{}_{\rm H}}$ therein.

Figure 3: Renormalization group (RG) flows of two velocity components of non-
Hermitian Dirac fermions $v_{{}_{\rm H}}$ and $v_{{}_{\rm NH}}$ [Eq. (1)], as
well as the effective Fermi velocity $v_{{}_{\rm F}}=\sqrt{v^{2}_{{}_{\rm
H}}-v^{2}_{{}_{\rm NH}}}$ along with the speed of light in the medium ($c$)
due to the gauge-fermion interaction in three dimensions for (a)
$v^{0}_{{}_{\rm H}}=0.5$ and $v^{0}_{{}_{\rm NH}}=0.4$, such that
$v^{0}_{{}_{\rm NH}}<v^{0}_{{}_{\rm H}}<c^{0}$, (b) $v^{0}_{{}_{\rm H}}=1.1$
and $v^{0}_{{}_{\rm NH}}=0.9$, such that $v^{0}_{{}_{\rm
NH}}<c^{0}<v^{0}_{{}_{\rm H}}$, and (c) $v^{0}_{{}_{\rm H}}=1.15$ and
$v^{0}_{{}_{\rm NH}}=1.05$, such that $c^{0}<v^{0}_{{}_{\rm
NH}}<v^{0}_{{}_{\rm H}}$. Notice that $v^{0}_{{}_{\rm F}}<c$ always. The
results are obtained by numerically solving Eqs. (14), (15) and (18). Here,
$b$ is the RG time and the quantities with the superscript ‘0’ correspond to
their bare values. Throughout we measure all velocities in units of $c^{0}$,
which we set to be unity. For convenience, we set the bare fine structure
constant $\alpha^{\rm 3D,0}_{{}_{\rm F}}=1$ and $N_{f}=1$.

The fermionic field renormalization in $d=3$ is [16]

$Z_{\Psi}=1-2\;\alpha^{\rm 3D}_{{}_{\rm
F}}\;\frac{x(1-3x^{2})}{(1+x)^{2}}\;\frac{1}{\epsilon}.$ (16)

The renormalization conditions for two velocity parameters for three-
dimensional NH Dirac fermions are [16]

$\displaystyle Z_{Q}=Z^{-1}_{\Psi}\left\\{1-\frac{2}{3}\alpha^{\rm
3D}_{{}_{\rm
F}}\;\frac{(1+x^{2})(2+x)}{(1+x)^{2}}\;\frac{1}{\epsilon}\right\\},$ (17)

for $Q=v_{{}_{\rm H}}$ and $v_{{}_{\rm NH}}$. Similar to the situation in
$d=2$, the perturbative corrections for both the velocity components are
identical in $d=3$ as well. Therefore, their RG flow equations take identical
form, given by [16]

$~{}\beta_{Q}=\frac{4}{3}\frac{\alpha^{\rm 3D}_{{}_{\rm
F}}}{(1+x)^{2}}\left[1+2x+x^{2}-4x^{3}\right]\;Q,$ (18)

for $Q=v_{{}_{\rm H}}$ and $v_{{}_{\rm NH}}$. Notice that the pertubative
correction in the above flow equations for $v_{{}_{\rm H}}$ and $v_{{}_{\rm
NH}}$ is _identical_ to that for $v_{{}_{\rm H}}$ in conventional Dirac system
[11, 12]. However, in such systems $x=v_{{}_{\rm H}}/c$, as $v_{{}_{\rm
H}}=v_{{}_{\rm F}}$ therein.

The coupled RG flow equations from Eqs. (14) and (18) support a single stable
fixed point at $x=1$ or equivalently at $v_{{}_{\rm F}}=c$. However, in $d=3$,
while the effective Fermi velocity for NH Dirac fermions increases under
coarse grain, the speed of light in Dirac crystals decreases as we approach
the deep infrared regime. As a result, even though the system ultimately
always features a unique terminal velocity for all the participating degrees
of freedom, thereby yielding a robust spacetime Lorentz symmetry, its
magnitude is _nonuniversal_ and depends on the bare values of $v_{{}_{\rm H}}$
and $v_{{}_{\rm NH}}$. These outcomes are depicted in Fig. 3. At the same
time, we also note that the fine structure constant ($\alpha^{\rm 3D}_{{}_{\rm
F}}$) decreases monotonically (but logarithmically slowly) as we approach the
infrared regime, following Eq. (15).

_Summary & discussions_. To summarize, we develop an effective description of
QED for NH Dirac materials, when the underlying fermoinic degrees of freedom,
dispersing linearly with an effective Fermi velocity ($v_{{}_{\rm F}}$),
interact with photons. Even though the collection of NH Dirac fermions is
described by a Lorentz invariant NH operator [Eq. (1)], in the presence of
fluctuating electromagnetic radiation the system generically lacks the Lorentz
symmetry as $v_{{}_{\rm F}}$ and the speed of light $c$ are not equal at the
ultraviolet or lattice scale. Nevertheless, we show that when such an
interacting system approaches the deep infrared regime, quantum corrections
[Fig. 1] always restore the spacetime Lorentz symmetry therein, where all the
participating degrees of freedom possess a unique terminal velocity. See Figs
2 and 3. As a two-dimensional Dirac system floats like a ‘brane’ in a three-
dimensional world, hosting photons, the speed of light does not renormalize
due to the absence of any fermion mediated dynamic screening in $d=2$. Thus
the terminal velocity in $d=2$ is always the speed of light in vacuum. By
contrast, in $d=3$ both $v_{{}_{\rm F}}$ and $c$ flow under coarse grain and
ultimately they reach a common, but nonuniversal terminal velocity.

We realize that the renormalized velocity components of NH Dirac fermions,
namely $v_{{}_{\rm H}}$ and $v_{{}_{\rm NH}}$, related to its effective Fermi
velocity according to $v_{{}_{\rm F}}=\sqrt{v^{2}_{{}_{\rm H}}-v^{2}_{{}_{\rm
NH}}}$, can become larger than the speed of light, while always maintaining
$v_{{}_{\rm F}}\leq c$. This observation tempted us to consider a situation
when the bare values of $v_{{}_{\rm H}}$ and $v_{{}_{\rm NH}}$ are larger than
$c$, while that for $v_{{}_{\rm F}}<c$. In all these circumstances, we find a
robust and generic restoration of the spacetime Lorentz symmetry. This
observation may therefore suggest a possible generic emergence of the Lorentz
symmetry in the NH universe of Dirac fermions, encompassing system-to-
environment interactions.

The RG flow of the effective Fermi velocity of NH Dirac fermions leaves its
signature on thermodynamic and transport observables. For example, the density
of states and the specific heat in a $d$-dimensional NH Dirac system scale as
$\varrho(E)\sim|E|^{d-1}/v^{d}_{{}_{\rm F}}$ and $C_{v}\sim
T^{d}/v^{d}_{{}_{\rm F}}$, respectively. Here $T$ is the temperature.
Therefore, increasing $v_{{}_{\rm F}}$ under coarse grain should manifest via
decreasing magnitudes of these two thermodynamic quantities, while preserving
their leading power-law scaling, resulting from the dynamic scaling exponent,
measuring the relative scaling between energy and momentum, being locked at
$z=1$. The frequency ($\omega$) dependent zero temperature longitudinal
optical conductivity in two and three dimensional NH Dirac systems in units of
$e^{2}/h$ are respectively $\sigma^{\rm 2D}_{\rm NH}=N_{f}\pi/4$ and
$\sigma^{\rm 3D}_{\rm NH}=N_{f}\omega/(6v_{{}_{\rm F}})$, where $h$ is the
Planck’s constant [13]. So, $\sigma^{\rm 2D}_{\rm NH}$ remains unaffected by
the renormalization of $v_{{}_{\rm F}}$. But, $\sigma^{\rm 3D}_{\rm NH}$
decreases at lower frequency while maintaining its $\omega$-linear scaling.

In crystalline materials the non-relativistic (NR) limit of our findings can
be experimentally more pertinent, as typically $v_{{}_{\rm H}}$, $v_{{}_{\rm
NH}}$, $v_{{}_{\rm F}}\ll c$ at the bare level, and the effect of the
retardation and the restoration of the Lorentz symmetry develops
logarithmically slowly. In the NR limit, the electron-photon interaction
vertex is composed of only the instantaneous Coulomb part (density-density
interaction) and is given by $-ie\gamma_{\rho}$ with
$\gamma_{\rho}=(1,{\bm{0}})$. The soft photon propagator in $d$ dimensions
takes the form
$D_{\rho\lambda}(i\omega,\bm{k},d)=\delta_{\rho\lambda}/|\bm{k}|^{d-1}$. The
pertutbative corrections are obtained by taking the limit $x=v_{{}_{\rm
F}}/c\to 0$ therein. Then in $d=2$, the RG flow equation reads as

$\beta_{Q}=\frac{\pi}{2}\>\alpha^{\rm 2D}_{\rm F,NR}\>Q$ (19)

for $Q=v_{{}_{\rm H}}$ and $v_{{}_{\rm NH}}$, where $\alpha^{\rm 2D}_{\rm
F,NR}=e^{2}/(16\pi\varepsilon v_{{}_{\rm F}})$ is the fine structure constant
for two-dimensional NR systems. Therefore, both the velocity components and
the effective Fermi velocity increase logarithmically without any upper bound.
Hence, the retardation effects in the form of the current-current interaction
must be included in the formalism from the outset to showcase the emergence of
a Lorentz symmetry. We arrive at similar conclusions in $d=3$, which follow
from the RG flow equations

$\beta_{\alpha^{\rm 3D}_{\rm F,NR}}=-\frac{4}{3}N_{f}\;\left(\alpha^{\rm
3D}_{\rm F,NR}\right)^{2},\>\text{and}\>\beta_{Q}=\frac{4}{3}\>\alpha^{\rm
3D}_{\rm F,NR}\>Q,$ (20)

for $Q=v_{{}_{\rm H}}$ and $v_{{}_{\rm NH}}$. The fine structure constant for
three-dimensional NR systems is $\alpha^{\rm 3D}_{\rm
F,NR}=e^{2}/(8\pi^{2}\varepsilon v_{{}_{\rm F}})$. These outcomes closely
mimic the ones in Hermitian Dirac systems, however, in terms of only
$v_{{}_{\rm H}}$ [10, 17, 18].

As far as the material realization of NH Dirac semimetals is concerned,
designer electronic systems constitute the most promising ground, where a
prominent two-dimensional Dirac system, graphene, has been engineered [19]. In
graphene, conventional Dirac fermions stem from the nearest-neighbor hopping.
Graphene accommodates a plethora of mass orders, represented by mass matrices
$M$ [20, 21]. The simplest of them is the charge-density-wave, featuring a
density imbalance between two sublattices of the underlying honeycomb lattice
[22]. Then the anti-Hermitian component of the NH Dirac operator in Eq. (1)
also corresponds to nearest-neighbor hopping, however, with different
amplitudes in the opposite directions, yielding non-Hermiticity in the system.
Such an hopping imbalance can in principle be generated by placing electronic
valves along the nearest-neighbor bonds, but at different operational voltages
in the opposite directions. Similar engineering can also be executed in three-
dimensional Dirac and Weyl materials, given the overall simplicity of the
construction of the NH Dirac operator [13] and the existence of mass matrices
therein [23]. The present discussion should stimulate future experiments in
this direction, unfolding novel QED and emergent Lorentz symmetry in NH Dirac
materials.

_Acknowledgments_. This work was supported by NSF CAREER Grant No. DMR-
2238679 of B.R. and Dr. Hyo Sang Lee Graduate Fellowship from Lehigh
University (S.A.M.). We are thankful to Vladimir Juričić for useful
discussions and critical reading of the manuscript.

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