# The height of an infinite parallelotope is infinite 111To all fearless
Ukrainians defending not only their country, but the whole civilization
against putin’s rashism

A.V. Kosyak<EMAIL_ADDRESS>Institute of Mathematics, Ukrainian National
Academy of Sciences,
3 Tereshchenkivs’ka Str., Kyiv, 01601, Ukraine  London Institute for
Mathematical Sciences,
21 Albemarle St, London W1S 4BS, UK

###### Abstract

We show that
$\frac{\Gamma(f_{0},f_{1},\dots,f_{m})}{\Gamma(f_{1},\dots,f_{m})}=\infty$ for
$m+1$ vectors having the properties that no non-trivial linear combination of
them belongs to $l_{2}(\mathbb{N})$. This property is essential in the proof
of the irreducibility of unitary representations of some infinite-dimensional
groups.

###### keywords:

Gram determinant , parallelotope volume , generalized characteristic
polynomial , infinite-dimensional groups , irreducible representation ,

###### MSC:

[2020] 22E65 (5 , 15 , 26 , 40)

††journal:

###### Contents

  1. 1 One lemma about $m$ infinite vectors
  2. 2 The particular cases
    1. 2.1 Case $m=1$
    2. 2.2 Case $m=2$
  3. 3 The proof of Lemma 1.1
  4. 4 How far is a vector from a hyperplane?
    1. 4.1 The distance of a vector from a hyperplane
    2. 4.2 Cramer’s rule reformulated
    3. 4.3 Some estimates
    4. 4.4 Gram determinants and Gram matrices
    5. 4.5 The generalized characteristic polynomial and its properties
  5. 5 The explicit expression for $C^{-1}(\lambda)$ and $(C^{-1}(\lambda)a,a)$
    1. 5.1 The case where $C$ is the Gram matrix
    2. 5.2 Case $m=2$
    3. 5.3 Case $m=3$
    4. 5.4 General $m$

##  1 One lemma about $m$ infinite vectors

In the study of the irreducibility of unitary representations of infinite-
dimensional groups we need Lemma 1.1, see [4]–[7]. The particular case of this
lemma was proved for $m=1$ in [4, 5]

###### Lemma 1.1

Let $m\in\mathbb{N}$ and $(f_{r})_{r=0}^{m}$ be $m+1$ infinite real vectors
$f_{r}=(f_{rk})_{k\in\mathbb{N}},$ $0\leq r\leq m$ such that for all
$\big{(}C_{0},\dots,C_{m}\big{)}\in{\mathbb{R}}^{m+1}\setminus\\{0\\}$ holds

$\sum_{r=0}^{m}C_{r}f_{r}\not\in
l_{2}(\mathbb{N}),\quad\text{i.e.,}\quad\sum_{k\in\mathbb{N}}\Big{|}\sum_{r=0}^{m}C_{r}f_{rk}\Big{|}^{2}=\infty.$
(1.1)

Denote by $f_{r}^{(n)}=(f_{rk})_{k=1}^{n}\in\mathbb{R}^{n}$ the projections of
the vectors $f_{r}$ on the subspace $\mathbb{R}^{n}$. Then for all $s$ with
$0\leq s\leq m$

$\frac{\Gamma(f_{0},f_{1},\dots,f_{m})}{\Gamma(f_{0},\dots,\hat{f_{s}},\dots,f_{m})}:=\lim_{n\to\infty}\frac{\Gamma(f_{0}^{(n)},f_{1}^{(n)}\dots,f_{m}^{(n)})}{\Gamma(f_{0}^{(n)},\dots,\widehat{f_{s}^{(n)}},\dots,f_{m}^{(n)})}=\infty,$
(1.2)

where $\hat{f_{s}}$ means that the vector $f_{s}$ is absent and
$\Gamma(f_{0},f_{1},\dots,f_{m})$ is the Gram determinant.

###### Lemma 1.2

Let we have $m+1$ real vectors $(f_{k})_{k=0}^{m}$ such that
$\sum_{k=0}^{m}C_{k}f_{k}\not\in l_{2}(\mathbb{N})$ for any nontrivial
combination $(C_{k})_{k=0}^{m}$. Then for any $s,\,0\leq s\leq m$

$\frac{{\rm det}\big{(}I_{m+1}+\gamma(f_{0},\dots,f_{m})\big{)}}{{\rm
det}\big{(}I_{m}\\!+\\!\gamma(f_{0},\dots,\hat{f_{s}},\dots,f_{m})\big{)}}\\!=\\!\lim_{n\to\infty}\frac{{\rm
det}\big{(}I_{m+1}+\gamma(f_{0}^{(n)},\dots,f_{m}^{(n)})\big{)}}{{\rm
det}\big{(}I_{m}\\!+\\!\gamma(f_{0}^{(n)},\dots,\widehat{f_{s}^{(n)}},\dots,f_{m}^{(n)})\big{)}}\\!=\\!\infty.$
(1.3)

Here $I_{m}\\!=\\!{\rm diag}(1,\dots,1)\in{\rm Mat}(m,\mathbb{R},)$ and
$\gamma(f_{0},\dots,f_{m})$ is the Gram matrix.

###### Proof 1

The proof follows from Lemma 1.1 and (5.2).

##  2 The particular cases

###  2.1 Case $m=1$

######  Lemma 2.3 ([4, 5])

Let $f_{r}\\!=\\!(f_{rk})_{k\in{\mathbb{N}}},\,\,0\\!\leq\\!r\\!\leq\\!1$ be
two real vectors such that

$C_{0}f_{0}+C_{1}f_{1}\not\in l_{2}(\mathbb{N})\quad\quad\text{for
all}\quad(C_{0},C_{1})\in{\mathbb{R}}^{2}\setminus\\{0\\},\quad\text{then}$
(2.1)
$\frac{\Gamma(f_{0},f_{1})}{\Gamma(f_{1})}\\!=\\!\lim_{n\to\infty}\frac{\Gamma(f_{0}^{(n)},f_{1}^{(n)})}{\Gamma(f_{1}^{(n)})}\\!=\\!\infty,\quad\frac{\Gamma(f_{0},f_{1})}{\Gamma(f_{0})}\\!=\\!\infty.$
(2.2)

###### Proof 2

The initial proof can be found in [4, 5]. Here we give different proof that
can be generalized for an arbitrary $m\in\mathbb{N}$. For $t\in\mathbb{R}$ and
$f_{0},f_{1}\in H$, where $H$ is some Hilbert space, define the quadratic form

$\displaystyle
F_{1}(t)=\|tf_{1}-f_{0}\|^{2}=t^{2}(f_{1},f_{1})-2t(f_{1},f_{0})+(f_{0},f_{0})=$
$\displaystyle(A_{1}t,t)-2(t,b)+(f_{0},f_{0}),\,\,\,\text{where}\,\,\,b=(f_{1},f_{0})\in\mathbb{R}.\quad\text{We
have}$
$\displaystyle\min_{t\in\mathbb{R}}F_{1}(t)=F_{1}(t_{0})=\frac{\Gamma(f_{0},f_{1})}{\Gamma(f_{1})},\quad\text{where}\quad
t_{0}=\frac{(f_{1},f_{0})}{(f_{1},f_{1})},$ (2.3)

and $A_{1}$ is the Gram matrix (see Definition 4.1)

$A_{1}=\gamma(f_{1})=(f_{1},f_{1}).$ (2.4)

Consider the following matrix

$B_{1}:=\gamma(f_{0},f_{1})=\left(\begin{array}[]{cc}(f_{0},f_{0})&(f_{0},f_{1})\\\
(f_{1},f_{0})&(f_{1},f_{1})\end{array}\right),\quad
B^{(n)}_{1}:=\gamma(f^{(n)}_{0},f^{(n)}_{1}),$ (2.5)

then

$t_{0}=\frac{(f_{1},f_{0})}{(f_{1},f_{1})}=\frac{-A^{0}_{1}(B_{1})}{A^{0}_{0}(B_{1})}.$
(2.6)

If we replace the vectors $f_{0},f_{1}$ with $f^{(n)}_{0},f^{(n)}_{1}$, the
formulas (2.3) and (2.6) become

$\min_{t\in\mathbb{R}}F^{(n)}_{1}(t)\\!=F^{(n)}_{1}(t^{(n)}_{0})\\!=\\!\frac{\Gamma(f^{(n)}_{0},f^{(n)}_{1})}{\Gamma(f^{(n)}_{1})},\quad
t^{(n)}_{0}\\!=\\!\frac{(f^{(n)}_{1},f^{(n)}_{0})}{(f^{(n)}_{1},f^{(n)}_{1})}\\!=\\!\frac{-A^{0}_{1}(B^{(n)}_{1})}{A^{0}_{0}(B^{(n)}_{1})}.$
(2.7)

Suppose that for all $n\in\mathbb{N}$

$\frac{\Gamma(f_{0}^{(n)},f_{1}^{(n)})}{\Gamma(f_{1}^{(n)})}\leq C.$ (2.8)

Without loss of generality, we can assume that for all $n\in\mathbb{N}$

$\Gamma(f^{(n)}_{0})\leq C_{1}\Gamma(f^{(n)}_{1}),$ (2.9)

we choose some subsequence $n_{k}$, if necessary. Then it is sufficient to
verify the first part of (2.2). We prove that the seguence $t^{(n)}_{0}$
defined by (2.7) is bounded. Since all the matrices
$\gamma(f_{0}^{(n)},f_{1}^{(n)})$ are positively defined, we have

$1\geq\frac{(f_{0}^{(n)},f_{1}^{(n)})^{2}}{(f_{0}^{(n)},f_{0}^{(n)})(f_{1}^{(n)},f_{1}^{(n)})}\stackrel{{\scriptstyle\eqref{f_0<f_1.1}}}{{\geq}}\frac{(f_{0}^{(n)},f_{1}^{(n)})^{2}}{C_{1}(f_{1}^{(n)},f_{1}^{(n)})^{2}}=\frac{1}{C_{1}}\big{(}t^{(n)}_{0}\big{)}^{2}.$

Hence, the sequence $t^{(n)}_{0}$ is bounded. Therefore, there exists a
subsequence $(t^{(n_{k})}_{0})_{k\in\mathbb{N}}$ that converges to some
$t\in\mathbb{R}$. This contradics (2.8). Indeed

$\hskip 56.9055pt\lim_{n\to\infty}F^{(n)}_{1}(t)=\infty,\quad
F^{(n)}_{1}(t_{0}^{(n_{k})})\leq
C,\quad\lim_{k\to\infty}t_{0}^{(n_{k})}=t.\hskip 56.9055pt\Box$

###  2.2 Case $m=2$

###### Lemma 2.4

Let $f_{0},f_{1},f_{2}$ be three infinite real vectors
$f_{r}=(f_{rk})_{k\in\mathbb{N}},$
$0\leq r\leq 2$ such that for all
$(C_{0},C_{1},C_{2})\in{\mathbb{R}}^{3}\setminus\\{0\\}$ holds
$\sum_{r=0}^{2}C_{r}f_{r}\not\in l_{2}(\mathbb{N})$. Then for all $r,s$ with
$0\leq r<s\leq 2$ holds

$\frac{\Gamma(f_{0},f_{1},f_{2})}{\Gamma(f_{r},f_{s})}:=\lim_{n\to\infty}\frac{\Gamma(f^{(n)}_{0},f^{(n)}_{1},f^{(n)}_{2})}{\Gamma(f^{(n)}_{r},f^{(n)}_{s})}=\infty.$
(2.10)

###### Proof 3

Suppose that for all $n\in\mathbb{N}$

$\frac{\Gamma(f_{0}^{(n)},f_{1}^{(n)},f_{2}^{(n)})}{\Gamma(f_{1}^{(n)},f_{2}^{(n)})}\leq
C.$ (2.11)

Without loss of generality, we can assume that for all $n\in\mathbb{N}$

$\Gamma(f^{(n)}_{0},f^{(n)}_{1})\leq
C_{2}\Gamma(f^{(n)}_{1},f^{(n)}_{2}),\quad\Gamma(f^{(n)}_{0},f^{(n)}_{2})\leq
C_{1}\Gamma(f^{(n)}_{1},f^{(n)}_{2}),$ (2.12)

we choose some subsequence $n_{k}$, if necessary. Then it is sufficient to
verify (2.10) for $(r,s)=(1,2)$. Define the quadratic form $F_{2}(t)$ as
follows:

$\displaystyle
F_{2}(t)=\|\sum_{r=1}^{2}t_{r}f_{r}-f_{0}\|^{2}=\sum_{k,r=1}^{2}t_{k}t_{r}(f_{k},f_{r})-2\sum_{k=1}^{2}t_{k}(f_{k},f_{0})+(f_{0},f_{0})$
$\displaystyle=(A_{2}t,t)-2(t,b)+(f_{0},f_{0}),$

where $b=(f_{k},f_{0})_{k=1}^{2}\in\mathbb{R}^{2}$ and $A_{2}$ is the Gram
matrix

$A_{2}=\gamma(f_{1},f_{2})=\big{(}(f_{k},f_{r})\big{)}_{k,r=1}^{2}.$ (2.13)

For $t_{0}$ defined by $A_{2}t_{0}=b$ we have by Lemma 4.5

${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}F_{2}(t)}=(A_{2}t,t)-2(t,b)+(f_{0},f_{0})=(A_{2}(t-t_{0}),(t-t_{0}))+\frac{\Gamma(f_{0},f_{1},f_{2})}{\Gamma(f_{1},f_{2})},$

and therefore,

${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}F_{2}(t_{0})}=\frac{\Gamma(f_{0},f_{1},f_{2})}{\Gamma(f_{1},f_{2})}.$
(2.14)

Consider the following matrix

$B_{2}:=\gamma(f_{0},f_{1},f_{2})=\left(\begin{array}[]{ccc}(f_{0},f_{0})&(f_{0},f_{1})&(f_{0},f_{2})\\\
(f_{1},f_{0})&(f_{1},f_{1})&(f_{1},f_{2})\\\
(f_{2},f_{0})&(f_{2},f_{1})&(f_{2},f_{2})\end{array}\right).$ (2.15)

By Cramer’s rule (see Lemma 4.6) we have

$t_{0}=\frac{1}{A^{0}_{0}(B_{2})}\left(\begin{array}[]{c}-A^{0}_{1}(B_{2})\\\
-A^{0}_{2}(B_{2})\end{array}\right).$ (2.16)

Define the quadratic form $F^{(n)}_{2}(t)$ fot $t\in\mathbb{R}^{2}$ as
follows:

$\displaystyle
F^{(n)}_{2}(t)=\|\sum_{r=1}^{2}t_{r}f^{(n)}_{r}-f^{(n)}_{0}\|^{2}=(A^{(n)}_{2}t,t)-2(t,b)+(f^{(n)}_{0},f^{(n)}_{0}),$

where $b=(f^{(n)}_{k},f^{(n)}_{0})_{k=1}^{2}\in\mathbb{R}^{2}$ and
$A^{(n)}_{2}$ is the Gram matrix

$A^{(n)}_{2}=\gamma(f^{(n)}_{1},f^{(n)}_{2})=\big{(}(f^{(n)}_{k},f^{(n)}_{r})\big{)}_{k,r=1}^{2}.$
(2.17)

By (3.6) we have

$F^{(n)}_{2}(t^{(n)}_{0})=\min_{t\in\mathbb{R}^{2}}F^{(n)}_{2}(t)=\frac{\Gamma(f_{0}^{(n)},f_{1}^{(n)},f_{2}^{(n)})}{\Gamma(f_{1}^{(n)},f_{2}^{(n)})}.$

We prove that the seguence $t^{(n)}_{0}$ is bounded. If we replace the vectors
$f_{0},f_{1},f_{2}$ with $f^{(n)}_{0},f^{(n)}_{1},f^{(n)}_{2}$, we will get
the following expressions:

$t^{(n)}_{0}=\frac{1}{A^{0}_{0}(B^{(n)}_{2})}\left(\begin{array}[]{c}-A^{0}_{1}(B^{(n)}_{2})\\\
-A^{0}_{2}(B^{(n)}_{2})\end{array}\right),$ (2.18)

where $B^{(n)}(2)$ is defined by

$B^{(n)}_{2}:=\\!\gamma(f^{(n)}_{0},f^{(n)}_{1},f^{(n)}_{2})\\!=\\!\left(\\!\\!\begin{array}[]{ccc}(f^{(n)}_{0},f^{(n)}_{0})&(f^{(n)}_{0},f^{(n)}_{1})&(f^{(n)}_{0},f^{(n)}_{2})\\\
(f^{(n)}_{1},f^{(n)}_{0})&(f^{(n)}_{1},f^{(n)}_{1})&(f^{(n)}_{1},f^{(n)}_{2})\\\
(f^{(n)}_{2},f^{(n)}_{0})&(f^{(n)}_{2},f^{(n)}_{1})&(f^{(n)}_{2},f^{(n)}_{2})\end{array}\\!\\!\right).$
(2.19)

Since all the matrices $B^{(n)}_{2}$ defined by (2.19) are positively defined,
the inverse matrices $\Big{(}B^{(n)}_{2}\Big{)}^{-1}$ are also positevely
defined. We have the following expression for them

$\Big{(}B^{(n)}_{2}\Big{)}^{-1}=\frac{1}{{\rm
det}B^{(n)}_{2}}\left(\begin{array}[]{ccc}A^{0}_{0}(B^{(n)}_{2})&A^{0}_{1}(B^{(n)}_{2})&A^{0}_{2}(B^{(n)}_{2})\\\
A^{1}_{0}(B^{(n)}_{2})&A^{1}_{1}(B^{(n)}_{2})&A^{1}_{2}(B^{(n)}_{2})\\\
A^{2}_{0}(B^{(n)}_{2})&A^{2}_{1}(B^{(n)}_{2})&A^{2}_{2}(B^{(n)}_{2})\end{array}\right).$
(2.20)

We prove that the sequence $t^{(n)}_{0}$ defined by (2.18)

$t^{(n)}_{0}=\frac{1}{A^{0}_{0}(B^{(n)}_{2})}\left(\begin{array}[]{c}-A^{0}_{1}(B^{(n)}_{2})\\\
-A^{0}_{2}(B^{(n)}_{2})\end{array}\right)$ (2.21)

is bounded when (2.12) holds. Set

$t^{(n)}_{rr}=A^{r}_{r}(B^{(n)}_{2}),\,\,0\leq r\leq
2,\,\,t^{(n)}_{rs}=-A^{r}_{s}(B^{(n)}_{2}),\,\,\,0\leq r\not=s\leq 2.$ (2.22)

Then
$t^{(n)}_{0}=\Big{(}\frac{t^{(n)}_{01}}{t^{(n)}_{00}},\frac{t^{(n)}_{02}}{t^{(n)}_{00}}\Big{)}$.
Since the matrix $\Big{(}B^{(n)}_{2}\Big{)}^{-1}$ is positively defined and
(2.12) holds, we have

$\displaystyle\big{(}t^{(n)}_{01}\big{)}^{2}\leq
t^{(n)}_{00}t^{(n)}_{11},\quad\big{(}t^{(n)}_{02}\big{)}^{2}\leq
t^{(n)}_{00}t^{(n)}_{22},$ (2.23) $\displaystyle t^{(n)}_{22}\leq
C_{2}t^{(n)}_{00},\quad t^{(n)}_{11}\leq C_{1}t^{(n)}_{00},$ (2.24)
$\displaystyle\|t^{(n)}_{0}\|^{2}:=\frac{|t^{(n)}_{01}|^{2}+|t^{(n)}_{02}|^{2}}{|t^{(n)}_{00}|^{2}}\stackrel{{\scriptstyle\eqref{A:>}}}{{\leq}}\frac{t^{(n)}_{11}+t^{(n)}_{22}}{t^{(n)}_{00}}$
(2.25)
$\displaystyle\stackrel{{\scriptstyle\eqref{A:(r,s)<(1,2).2}}}{{\leq}}\frac{C_{1}t^{(n)}_{00}+C_{2}t^{(n)}_{00}}{t^{(n)}_{00}}=C_{1}+C_{2}.$
(2.26)

Hence, the sequence $t^{(n)}_{0}\in\mathbb{R}^{2}$ is bounded. Therefore,
there exists a subsequence $(t^{(n_{k})}_{0})_{k\in\mathbb{N}}$ that converges
to some $t\in\mathbb{R}^{2}$. This contradics (2.11). Indeed

$\hskip 42.67912pt\lim_{n\to\infty}F^{(n)}_{2}(t)=\infty,\quad
F^{(n)}_{2}(t_{0}^{(n_{k})})\leq
C,\quad\lim_{k\to\infty}t_{0}^{(n_{k})}=t.\hskip 42.67912pt\Box$

##  3 The proof of Lemma 1.1

######  Proof 4 (Proof of the Lemma 1.1)

Suppose that for all $n\in\mathbb{N}$ we have

$\frac{\Gamma(f_{0}^{(n)},f_{1}^{(n)}\dots,f_{m}^{(n)})}{\Gamma(f_{1}^{(n)},f_{2}^{(n)}\dots,f_{m}^{(n)})}\leq
C.$ (3.1)

Without loss of generality, we can assume that for all $n\in\mathbb{N}$ and
$0\leq s\leq m-1$ hold

$\Gamma(f_{0}^{(n)},\dots,\widehat{f_{s}^{(n)}},\dots,f_{m}^{(n)})<C_{s}\Gamma(f_{1}^{(n)},f_{2}^{(n)},\dots,f_{m}^{(n)}).$
(3.2)

we choose some subsequence $n_{k}$, if necessary. Consider the following
quadratic forms for $t=(t_{r})_{r=1}^{m}\in\mathbb{R}^{m}$

$F^{(n)}_{m}(t)=\|f_{0}^{(n)}-\sum_{r=1}^{m}t_{r}f_{r}^{(n)}\|^{2},$ (3.3)

The forms $F^{(n)}_{m}(t)$ defined by (3.3) have the following properties, for
any fixed $t\in\mathbb{R}^{m}$ we have:
1) $F^{(n)}_{m}(t)\leq F^{(n+1)}_{m}(t)$,
2) $\lim_{n\to\infty}F^{(n)}_{m}(t)=\infty$.
By Lemma 4.5 there exists some $t^{(n)}_{0}\in\mathbb{R}^{m}$ such that

$F^{(n)}_{m,0}(t^{(n)}_{0})=\min_{t\in\mathbb{R}^{m}}F^{(n)}_{m,0}(t)=\frac{\Gamma(f_{0}^{(n)},f_{1}^{(n)},\dots,f_{m}^{(n)})}{\Gamma(f_{1}^{(n)},f_{2}^{(n)}\dots,f_{m}^{(n)})}.$
(3.4)

We prove that the seguence $t^{(n)}_{0}$ is bounded. To find $t^{(n)}_{0}$
explicitely in (3.4) we introduce some notations. For $t\in\mathbb{R}^{m}$ and
$f_{0},f_{1},\dots,f_{m}\in H$ we define the function

$\displaystyle F_{m}(t)$ $\displaystyle=$
$\displaystyle\|\sum_{k=1}^{m}t_{k}f_{k}-f_{0}\|^{2}=\sum_{k,r=1}^{m}t_{k}t_{r}(f_{k},f_{r})-2\sum_{k=1}^{2}t_{k}(f_{k},f_{0})+(f_{0},f_{0})$
$\displaystyle=$ $\displaystyle(A_{m}t,t)+2(t,b)+(f_{0},f_{0}),$

where $b=(f_{k},f_{0})_{k=1}^{m}\in\mathbb{R}^{m}$ and $A_{m}$ is the Gram
matrix, see Definition 4.1:

$A_{m}=\gamma(f_{1},\dots,f_{m})=\big{(}(f_{k},f_{r})\big{)}_{k,r=1}^{m}.$
(3.5)

The minimum of $F_{m}(t)$ is attained at $t_{0}$ defined by $A_{m}t_{0}=b$. By
Remark 4.1, (4.6) and (4.3) we get (for details see Section 4.1)

$F_{m}(t)=(A_{m}t,t)-2(t,b)+(f_{0},f_{0})=(A_{m}(t-t_{0}),(t-t_{0}))+\frac{\Gamma(f_{0},f_{1},\dots,f_{m})}{\Gamma(f_{1},\dots,f_{m})},$
$\text{and therefore,}\quad
F_{m}(t_{0})=\frac{\Gamma(f_{0},f_{1},\dots,f_{m})}{\Gamma(f_{1},\dots,f_{m})}.$
(3.6)

Consider the following matrix

$B_{m}:=\gamma(f_{0},f_{1},\dots,f_{m})=\left(\begin{array}[]{ccccc}(f_{0},f_{0})&(f_{0},f_{1})&(f_{0},f_{2})&\dots&(f_{0},f_{m})\\\
(f_{1},f_{0})&(f_{1},f_{1})&(f_{1},f_{2})&\dots&(f_{1},f_{m})\\\
(f_{2},f_{0})&(f_{2},f_{1})&(f_{2},f_{2})&\dots&(f_{2},f_{m})\\\ &&&\dots&\\\
(f_{m},f_{0})&(f_{m},f_{1})&(f_{m},f_{2})&\dots&(f_{m},f_{m})\end{array}\right).$
(3.7)

By Cramer’s rule (see Lemma 4.6) the solution of $A_{m}t_{0}=b$ is as follows:

$t_{0}\\!=\\!\frac{1}{A^{0}_{0}(B_{m})}\left(\\!\begin{array}[]{c}-A^{0}_{1}(B_{m})\\\
-A^{0}_{2}(B_{m})\\\ \dots\\\ -A^{0}_{m}(B_{m})\end{array}\\!\right).$ (3.8)

If we replace the vectors $f_{0},f_{1},\dots,f_{m}$ with
$f^{(n)}_{0},f^{(n)}_{1},\dots,f^{(n)}_{m}$, we will get the following
expression

$t^{(n)}_{0}\\!=\\!\frac{1}{A^{0}_{0}(B^{(n)}_{m})}\left(\\!\\!\begin{array}[]{c}-A^{0}_{1}(B^{(n)}_{m})\\\
\dots\\\ -A^{0}_{m}(B^{(n)}_{m})\end{array}\\!\\!\right),$ (3.9)

wher $B^{(n)}(m)$ is defined by

$B^{(n)}_{m}:=\\!\gamma(f^{(n)}_{0},\dots,f^{(n)}_{m})\\!=\\!\left(\\!\\!\begin{array}[]{cccc}(f^{(n)}_{0},f^{(n)}_{0})&(f^{(n)}_{0},f^{(n)}_{1})&\dots&(f^{(n)}_{0},f^{(n)}_{m})\\\
(f^{(n)}_{1},f^{(n)}_{0})&(f^{(n)}_{1},f^{(n)}_{1})&\dots&(f^{(n)}_{1},f^{(n)}_{m})\\\
&&\dots&\\\
(f^{(n)}_{m},f^{(n)}_{0})&(f^{(n)}_{m},f^{(n)}_{1})&\dots&(f^{(n)}_{m},f^{(n)}_{m})\end{array}\\!\\!\right).$
(3.10)

Since all the matrices $B^{(n)}_{m}$ defined by (3.10) are positively defined,
the inverse matrices $\Big{(}B^{(n)}_{m}\Big{)}^{-1}$ are also positevely
defined. We have the following expression for them

$\Big{(}B^{(n)}_{m}\Big{)}^{-1}\\!\\!\\!=\\!\frac{1}{{\rm
det}B^{(n)}_{m}}\left(\\!\\!\begin{array}[]{cccc}A^{0}_{0}(B^{(n)}_{m})&A^{0}_{1}(B^{(n)}_{m})&\dots&A^{0}_{m}(B^{(n)}_{m})\\\
A^{1}_{0}(B^{(n)}_{m})&A^{1}_{1}(B^{(n)}_{m})&\dots&A^{1}_{m}(B^{(n)}_{m})\\\
&&\dots&\\\
A^{m}_{0}(B^{(n)}_{m})&A^{m}_{1}(B^{(n)}_{m})&\dots&A^{m}_{m}(B^{(n)}_{m})\end{array}\\!\\!\right)\\!.$
(3.11)

We prove that the sequence

$t^{(n)}_{0}=\frac{1}{A^{0}_{0}(B^{(n)}_{m})}\left(\begin{array}[]{c}-A^{0}_{1}(B^{(n)}_{m})\\\
\dots\\\ -A^{0}_{m}(B^{(n)}_{m})\end{array}\right)$ (3.12)

is bounded when (3.2) holds. Set

$t^{(n)}_{rr}=A^{r}_{r}(B^{(n)}_{m}),\,\,0\leq r\leq
m,\,\,t^{(n)}_{rs}=-A^{r}_{s}(B^{(n)}_{m}),\,\,\,0\leq r\not=s\leq m.$ (3.13)

Then
$t^{(n)}_{0}=\Big{(}\frac{t^{(n)}_{01}}{t^{(n)}_{00}},\frac{t^{(n)}_{02}}{t^{(n)}_{00}},\dots\frac{t^{(n)}_{0m}}{t^{(n)}_{00}}\Big{)}$.
Since the matrix $\Big{(}B^{(n)}_{m}\Big{)}^{-1}$ is positively defined and
(2.12) holds, we have

$\displaystyle\big{(}t^{(n)}_{rs}\big{)}^{2}\leq
t^{(n)}_{rr}t^{(n)}_{ss},\quad\text{for all}\quad 0\leq r<s\leq m,$ (3.14)
$\displaystyle t^{(n)}_{ss}\leq C_{s}t^{(n)}_{00},\quad\text{for all}\quad
0\leq s\leq m-1,$ (3.15)
$\displaystyle\|t^{(n)}_{0}\|^{2}:=\frac{\sum_{s=1}^{m}|t^{(n)}_{0s}|^{2}}{|t^{(n)}_{00}|^{2}}\stackrel{{\scriptstyle\eqref{A:>.m}}}{{\leq}}\frac{\sum_{s=1}^{m}t^{(n)}_{ss}}{t^{(n)}_{00}}$
(3.16)
$\displaystyle\stackrel{{\scriptstyle\eqref{A:(r,s)<(1,2).m}}}{{\leq}}\frac{\sum_{s=1}^{m}C_{s}t^{(n)}_{00}}{t^{(n)}_{00}}=\sum_{s=1}^{m}C_{s}.$
(3.17)

Hence, the sequence $t^{(n)}_{0}\in\mathbb{R}^{m}$ is bounded. Therefore,
there exists a subsequence $(t^{(n_{k})}_{0})_{k\in\mathbb{N}}$ that converges
to some $t\in\mathbb{R}^{m}$. This contradics (3.1). Indeed

$\hskip 42.67912pt\lim_{n\to\infty}F^{(n)}_{m,0}(t)=\infty,\quad
F^{(n)}_{m,0}(t_{0}^{(n_{k})})\leq
C,\quad\lim_{k\to\infty}t_{0}^{(n_{k})}=t.\hskip 42.67912pt\Box$

##  4 How far is a vector from a hyperplane?

###  4.1 The distance of a vector from a hyperplane

In this section we follow [6, Section 1.3]. We start with a classical result,
see, e.g. [3]. Consider the hyperplan $V_{n}$ generated by $n$ arbitrary
vectors $f_{1},\dots,f_{n}$ in some Hilbert space $H$.

###### Lemma 4.5

The square of the distance $d(f_{0},V_{n})$ of a vector $f_{0}$ from the
hyperplaen $V_{n}$ is given by the ratio of two Gram determinants (see
Definition 4.1)

$d^{2}(f_{0},V_{n})=\frac{\Gamma(f_{0},f_{1},f_{2},\dots,f_{n},)}{\Gamma(f_{1},f_{2},\dots,f_{n})}.$
(4.1)

###### Proof 5

We follow closely the book by Axiezer and Glazman [1]. Set
$f=\sum_{k=1}^{n}t_{k}f_{k}\in V_{n}$ and $h=f-f_{0}$. Since $h$ should be
orthogonal to $V_{n}$ we conclude that $f_{r}\perp h$, i.e., $(f_{r},h)=0$ for
all $r$, or

$\sum_{k=1}^{n}t_{k}(f_{r},f_{k})=(f_{r},f_{0}),\quad 1\leq r\leq n.$ (4.2)

Set $A=\gamma(f_{1},f_{2},\dots,f_{n})$ and
$b=(f_{k},f_{0})_{k=1}^{n}\in\mathbb{R}^{n}$. By definition we have

$d^{2}=\min_{f\in V_{n}}\|f-f_{0}\|^{2}=(At,t)-2(t,b)+(f_{0},f_{0}).$ (4.3)

Since $d^{2}=(h,h)=(f_{0},h)$ we conclude that
$d^{2}=\sum_{k=1}^{n}t_{k}(f_{0},f_{k})-(f_{0},f_{0})$ or

$\sum_{k=1}^{n}t_{k}(f_{0},f_{k})=(f_{0},f_{0})-d^{2}.$ (4.4)

So we have the system of equations:

$\left\\{\begin{array}[]{ccc}t_{1}(f_{1},f_{1})+t_{2}(f_{1},f_{2})+\dots+t_{n}(f_{1},f_{n})&=&(f_{1},f_{0})\\\
t_{1}(f_{2},f_{1})+t_{2}(f_{2},f_{2})+\dots+t_{n}(f_{2},f_{n})&=&(f_{2},f_{0})\\\
\dots&&\\\
t_{1}(f_{n},f_{1})+t_{2}(f_{n},f_{2})+\dots+t_{n}(f_{n},f_{n})&=&(f_{n},f_{0})\\\
t_{1}(f_{0},f_{1})+t_{2}(f_{0},f_{2})+\dots+t_{n}(f_{0},f_{n})&=&(f_{0},f_{0})-d^{2}\end{array}\right..$
(4.5)

Excluding $t_{k}$ from the system we get
$d^{2}=\frac{\Gamma(f_{0},f_{1},f_{2},\dots,f_{n},)}{\Gamma(f_{1},f_{2},\dots,f_{n})}$.

###### Remark 4.1

From the system (4.5) we conclude that $At=b$, where
$b=(f_{k},f_{0})_{k=1}^{n}\in\mathbb{R}^{n}$, hence $t=A^{-1}b$. By (4.3) we
get

$d^{2}=(f_{0},f_{0})-(A^{-1}b,b)=\frac{\Gamma(f_{0},f_{1},f_{2},\dots,f_{n})}{\Gamma(f_{1},f_{2},\dots,f_{n})}.$
(4.6)

See also [5, Chap. 4.3, Lemma 4.3.2].

###  4.2 Cramer’s rule reformulated

Consider two Gram matrices, (see Definition 4.1):

$A_{m}=\gamma(f_{1},\dots,f_{m})=\big{(}(f_{k},f_{r})\big{)}_{k,r=1}^{m},\quad
B_{m}=\gamma(f_{0},f_{1},\dots,f_{m}),$ (4.7)

and a vector $b=(f_{k},f_{0})_{k=1}^{m}\in\mathbb{R}^{m}$. The solution of the
equation $A_{m}t=b$ is as follows.

###### Lemma 4.6

We have

$t=A_{m}^{-1}b=\frac{1}{A^{0}_{0}(B_{m})}\left(\\!\begin{array}[]{c}-A^{0}_{1}(B_{m})\\\
-A^{0}_{2}(B_{m})\\\ \dots\\\ -A^{0}_{m}(B_{m})\end{array}\\!\right).$ (4.8)

###### Proof 6

By Cramer’s rule if we have a system of linear equations

$At=b$ (4.9)

where $A\in{\rm Mat}(m,\mathbb{C})$ with ${\rm det}\,A\not=0$ and
$t,b\in\mathbb{C}^{m}$, than the soultions are given by the following
formulas:

$t_{k}=\frac{{\rm det}\,(A_{k})}{{\rm det}\,(A)},\quad 1\leq k\leq m,$ (4.10)

where $A_{k}$ is the matrix formed by replacing the $i$-th column of $A$ by
the column vector $b$. Consider the matrix $B_{m}$ defined by (4.7). We have

${\rm det}\,(A)=A^{0}_{0}(B_{m}),\quad{\rm
det}\,(A_{k})=-A^{0}_{k}(B_{m}),\quad 1\leq k\leq m,$ (4.11)

thus implying (4.8). Recall that $A^{r}_{s}(C)$ denote cofactors of the matrix
$C$, for details, see Section 4.5.

###  4.3 Some estimates

We use some material from [5], Section 1.4.1, pp. 24-25.

###### Lemma 4.7

For a strictly positive operator $A$ (i.e., $(Af,f)>0,\,\,f\not=0$) acting in
${\mathbb{R}}^{n}$ and a vector $b\in{\mathbb{R}}^{n}\backslash\\{0\\}$ we
have

$\min_{x\in{\mathbb{R}}^{n}}\Big{(}(Ax,x)\mid(x,b)=1\Big{)}=(A^{-1}b,b)^{-1}.$
(4.12)

The minimum is assumed for $x=A^{-1}b\left((A^{-1}b,b)\right)^{-1}$.

Lemma 4.7 is a direct generalization of the well known result (see, for
example, [2], Chap. I, §52), for $a_{k}>0,\,\,1\leq k\leq n$ we have

$\min_{x\in{\mathbb{R}}^{n}}\Big{(}\sum_{k=1}^{n}a_{k}x_{k}^{2}\mid\sum_{k=1}^{n}x_{k}=1\Big{)}=\Big{(}\sum_{k=1}^{n}\frac{1}{a_{k}}\Big{)}^{-1}.$
(4.13)

We will also use the same result in a slightly different form:

$\min_{x\in{\mathbb{R}}^{n}}\Big{(}\sum_{k=1}^{n}a_{k}x_{k}^{2}\mid\sum_{k=1}^{n}x_{k}b_{k}=1\Big{)}=\Big{(}\sum_{k=1}^{n}\frac{b_{k}^{2}}{a_{k}}\Big{)}^{-1}.$
(4.14)

The minimum is assumed for
$x_{k}=\frac{b_{k}}{a_{k}}\Big{(}\sum_{k=1}^{n}\frac{b_{k}^{2}}{a_{k}}\Big{)}^{-1}$.

###  4.4 Gram determinants and Gram matrices

###### Definition 4.1

Let us recall the definition of the Gram determinant and the Gram matrix (see
[3], Chap IX, §5). Given the vectors $x_{1},x_{2},...,x_{m}$ in some Hilbert
space $H$ the Gram matrix $\gamma(x_{1},x_{2},...,x_{m})$ is defined by the
formula

$\gamma(x_{1},x_{2},...,x_{m})=\big{(}(x_{k},x_{n})\big{)}_{k,n=1}^{m}.$

The determinant of this matrix is called the Gram determinant for the vectors
$x_{1},x_{2},...,x_{m}$ and is denoted by $\Gamma(x_{1},x_{2},...,x_{m})$.
Thus,

$\Gamma(x_{1},x_{2},\dots,x_{m}):={\rm det}\,\gamma(x_{1},x_{2},\dots,x_{m}).$
(4.15)

###  4.5 The generalized characteristic polynomial and its properties

Notations. For a matrix $C\in{\rm Mat}(n,\mathbb{R})$ and $1\leq
i_{1}<i_{2}<\dots i_{r}\leq n,$ $1\leq j_{1}<j_{2}<\dots j_{r}\leq n,\,\,r\leq
n$ denote by

$M^{i_{1}i_{2}\dots i_{r}}_{j_{1}j_{2}\dots j_{r}}(C)\quad\text{ and}\quad
A^{i_{1}i_{2}\dots i_{r}}_{j_{1}j_{2}\dots j_{r}}(C)$

the corresponding minors and cofactors of the matrix $C$.

###### Definition 4.2

([5, Ch.1.4.3]) For the matrix $C\in{\rm Mat}(m,{\mathbb{C}})$ and
$\lambda=(\lambda_{1},\dots,\lambda_{m})\in{\mathbb{C}}^{m}$ define the
generalization of the characteristic polynomial, $p_{C}(t)={\rm
det}\,(tI-C),\,t\in{\mathbb{C}}$ as follows:

$P_{C}(\lambda)={\rm det}\,C(\lambda),\quad\text{where}\quad C(\lambda)={\rm
diag}(\lambda_{1},\dots,\lambda_{m})+C.$ (4.16)

###### Lemma 4.8

([5, Ch.1.4.3]) For the generalized characteristic polynomial $P_{C}(\lambda)$
of $C\\!\in\\!{\rm Mat}(m,{\mathbb{C}})$ and
$\lambda=(\lambda_{1},\lambda_{2},...,\lambda_{m})\in{\mathbb{C}}^{m}$ we have

$P_{C}(\lambda)={\rm det}\,C+\sum_{r=1}^{m}\sum_{1\leq
i_{1}<i_{2}<...<i_{r}\leq
m}\lambda_{i_{1}}\lambda_{i_{2}}...\lambda_{i_{r}}A^{i_{1}i_{2}...i_{r}}_{i_{1}i_{2}...i_{r}}(C).$
(4.17)

###### Remark 4.2

If we set $\lambda_{\alpha}=\lambda_{i_{1}}\lambda_{i_{2}}...\lambda_{i_{r}}$,
where $\alpha=(i_{1},i_{2},...,i_{r})$ and
$A^{\alpha}_{\alpha}(C)=A^{i_{1}i_{2}...i_{r}}_{i_{1}i_{2}...i_{r}}(C),\,\,M^{\alpha}_{\alpha}(C)=M^{i_{1}i_{2}...i_{r}}_{i_{1}i_{2}...i_{r}}(C),\,\,\lambda_{\emptyset}=1,\,\,A^{\emptyset}_{\emptyset}(C)={\rm
det}\,C$ we may write (4.17) as follows:

$P_{C}(\lambda)={\rm
det}\,C(\lambda)=\sum_{\emptyset\subseteq\alpha\subseteq\\{1,2,...,m\\}}\lambda_{\alpha}A^{\alpha}_{\alpha}(C),$
(4.18) $P_{C}(\lambda)={\rm
det}\,C(\lambda)=\left(\prod_{k=1}^{n}\lambda_{k}\right)\sum_{\emptyset\subseteq\alpha\subseteq\\{1,2,...,m\\}}\frac{M^{\alpha}_{\alpha}(C)}{\lambda_{\alpha}},$
(4.19)

Let

$X=X_{mn}=\left(\begin{array}[]{cccc}x_{11}&x_{12}&...&x_{1n}\\\
x_{21}&x_{22}&...&x_{2n}\\\ ...&...&...&...\\\
x_{m1}&x_{m2}&...&x_{mn}\end{array}\right).$ (4.20)

Setting

$x_{k}=(x_{1k},x_{2k},...,x_{mk})\in{\mathbb{R}}^{m},\quad
y_{r}=(x_{r1},x_{r2},...,x_{rn})\in{\mathbb{R}}^{n},$ (4.21)

we get

$X^{*}X=\left(\begin{array}[]{cccc}(x_{1},x_{1})&(x_{1},x_{2})&...&(x_{1},x_{n})\\\
(x_{2},x_{1})&(x_{2},x_{2})&...&(x_{2},x_{n})\\\ ...&...&...&...\\\
(x_{n},x_{1})&(x_{n},x_{2})&...&(x_{n},x_{n})\end{array}\right)=\gamma(x_{1},x_{2},...,x_{n}),$
(4.22)
$XX^{*}=\left(\begin{array}[]{cccc}(y_{1},y_{1})&(y_{1},y_{2})&...&(y_{1},y_{m})\\\
(y_{2},y_{1})&(y_{2},y_{2})&...&(y_{2},y_{m})\\\ ...&...&...&...\\\
(y_{m},y_{1})&(y_{m},y_{2})&...&(y_{m},y_{m})\end{array}\right)=\gamma(y_{1},y_{2},...,y_{m}),$
(4.23)

therefore, we obtain

$\ \Gamma(x_{1},x_{2},...,x_{n})={\rm det}(X^{*}X)={\rm
det}(XX^{*})=\Gamma(y_{1},y_{2},...,y_{m}).$ (4.24)

##  5 The explicit expression for $C^{-1}(\lambda)$ and $(C^{-1}(\lambda)a,a)$

In this section we follow [6, Section 2]. Fix $C\in{\rm Mat}(n,\mathbb{R}),$
$a\in\mathbb{R}^{n}$ and $\lambda\in\mathbb{C}^{n}$. Our aim is to find the
explicit formulas for $C^{-1}(\lambda)$ and $(C^{-1}(\lambda)a,a)$, where
$C(\lambda)$ is defined by (4.16). Set $M(i_{1}i_{2}\dots i_{r})(C)$
$=M^{i_{1}i_{2}\dots i_{r}}_{i_{1}i_{2}\dots i_{r}}(C)$ and
$a_{i_{1}i_{2}\dots i_{r}}=(a_{i_{1}},a_{i_{2}},\dots,a_{i_{r}})$. Let also
$C_{i_{1}i_{2}\dots i_{r}}$ be the corresponding submatrix of the matrix $C$.
The elements of this matrix are on the intersection of
$i_{1},i_{2},\dots,i_{r}$ rows and column of the matrix $C$. Denote by
$A(C_{i_{1}i_{2}\dots i_{r}})$ the matrix of the cofactors of the first order
of the matrix $C_{i_{1}i_{2}\dots i_{r}}$, i.e.

$A(C_{i_{1}i_{2}\dots i_{r}})=(A^{i}_{j}(C_{i_{1}i_{2}\dots i_{r}}))_{1\leq
i,j\leq r}$ (5.1)

Let $n=3$, then $A(C_{123})=A(C)$ is the following matrix:

$A(C)=A(C_{123})=\left(\begin{array}[]{ccc}A^{1}_{1}&A^{1}_{2}&A^{1}_{3}\\\
A^{2}_{1}&A^{2}_{2}&A^{2}_{3}\\\
A^{3}_{1}&A^{3}_{2}&A^{3}_{3}\end{array}\right)=\left(\begin{array}[]{ccc}M^{23}_{23}&-M^{23}_{13}&M^{23}_{12}\\\
-M^{13}_{23}&M^{13}_{13}&-M^{13}_{12}\\\
M^{12}_{23}&-M^{12}_{13}&M^{12}_{12}\end{array}\right),$ (5.2)

where we write $M^{ij}_{rs}$ instead of $M^{ij}_{rs}(C)$ and $A^{i}_{j}$
instead of $A^{i}_{j}(C)$.

###### Remark 5.1

Let $A^{T}$ be the transposed matrix of $A$. Then

$A^{T}(C_{i_{1}i_{2}\dots i_{r}})={\rm det}\,C^{-1}_{i_{1}i_{2}\dots
i_{r}}\Big{(}C^{-1}_{i_{1}i_{2}\dots i_{r}}\Big{)},$ (5.3)

In what follows we will consider the submatrix $C_{i_{1}i_{2}\dots i_{r}}$ of
the matrix $C\in{\rm Mat}(n,\mathbb{R})$ as an appropriate element of ${\rm
Mat}(n,\mathbb{R})$.

###### Theorem 5.9

For the matrix $C(\lambda)$ defined by (4.16) $a\in\mathbb{R}^{n}$ and
$\lambda\in\mathbb{C}^{n}$ we have

$P_{C}(\lambda)=\Big{(}\prod_{k=1}^{n}\lambda_{k}\Big{)}\sum_{r=1}^{n}\sum_{1\leq
i_{1}<i_{2}<\dots<i_{r}\leq n}\frac{M(i_{1}i_{2}\dots
i_{r})}{\lambda_{i_{1}}\lambda_{i_{2}}\dots\lambda_{i_{r}}},$ (5.4)
$C^{-1}(\lambda)=\frac{1}{P_{C}(\lambda)}\Big{(}\prod_{k=1}^{n}\lambda_{k}\Big{)}\sum_{r=1}^{n}\sum_{1\leq
i_{1}<i_{2}<\dots i_{r}\leq n}\frac{A(C_{i_{1}i_{2}\dots
i_{r}})}{\lambda_{i_{1}}\lambda_{i_{2}}\dots\lambda_{i_{r}}},$ (5.5)
$(C^{-1}(\lambda)a,a)=\frac{1}{P_{C}(\lambda)}\Big{(}\prod_{k=1}^{n}\lambda_{k}\Big{)}\sum_{r=1}^{n}\sum_{1\leq
i_{1}<i_{2}<\dots i_{r}\leq n}\frac{(A(C_{i_{1}i_{2}\dots
i_{r}})a_{i_{1}i_{2}\dots i_{r}},a_{i_{1}i_{2}\dots
i_{r}})}{\lambda_{i_{1}}\lambda_{i_{2}}\dots\lambda_{i_{r}}}.$ (5.6)

###  5.1 The case where $C$ is the Gram matrix

Fix the matrix $X_{mn}$ defined by (4.20). Denote by $C$ the Gram matrix
$\gamma(x_{1},x_{2},...,x_{n})$, i.e.,

$C=\gamma(x_{1},x_{2},...,x_{n}),$ (5.7)

where $(x_{1},x_{2},...,x_{n})$ are defined by (4.21) and
$\gamma(x_{1},x_{2},...,x_{n})$ by (4.22). In what follows we consider the
operator $C(\lambda)$ defined by (4.16).

###### Remark 5.2

In this case we have

$\displaystyle P_{C}(\lambda)$ $\displaystyle=$ $\displaystyle{\rm
det}\Big{(}\sum_{k=1}^{n}\lambda_{k}E_{kk}+\gamma(x_{1},x_{2},...,x_{n})\Big{)}$
$\displaystyle=$
$\displaystyle\prod_{k=1}^{n}\lambda_{k}\Big{(}1+\sum_{r=1}^{n}\sum_{1\leq
i_{1}<i_{2}<...<i_{r}\leq
m}\Big{(}\lambda_{i_{1}}\lambda_{i_{2}}...\lambda_{i_{r}}\Big{)}^{-1}\Gamma(x_{i_{1}},x_{i_{2}},...,x_{i_{r}})\Big{)}$
$\displaystyle=$
$\displaystyle\prod_{k=1}^{n}\lambda_{k}\Big{(}1+\sum_{r=1}^{n}\sum_{\begin{subarray}{c}1\leq
i_{1}<i_{2}<...<i_{r}\leq n;\\\ 1\leq j_{1}<j_{2}<...<j_{r}\leq
n\end{subarray}}\Big{(}\lambda_{i_{1}}\lambda_{i_{2}}...\lambda_{i_{r}}\Big{)}^{-1}\Big{(}M^{i_{1}i_{2}...i_{r}}_{j_{1}j_{2}...j_{r}}(X)\Big{)}^{2}\Big{)},$

where we have used the following formula (see [3], Chap IX, §5 formula (25)):

$\Gamma(x_{i_{1}},x_{i_{2}},...,x_{i_{r}})=\sum_{1\leq
j_{1}<j_{2}<...<j_{r}\leq
m}\left(M^{i_{1}i_{2}...i_{r}}_{j_{1}j_{2}...j_{r}}(X)\right)^{2}.$ (5.9)

###  5.2 Case $m=2$

Fix two natural numbers $n,m\in\mathbb{N}$ with $m\leq n$, two matrices
$A_{mn}$ and $X_{mn}$, vectors $g_{k}\in\mathbb{R}^{m-1},\,\,1\leq k\leq n$
and $a\in\mathbb{R}^{n}$ as follows

$A_{mn}\\!=\\!\left(\begin{array}[]{cccc}a_{11}&a_{12}&...&a_{1n}\\\
a_{21}&a_{22}&...&a_{2n}\\\ &&...&\\\
a_{m1}&a_{m2}&...&a_{mn}\end{array}\right),\,\,g_{k}\\!=\\!\left(\begin{array}[]{c}a_{2k}\\\
a_{3k}\\\ ...\\\
a_{mk}\end{array}\right)\in\mathbb{R}^{m-1},\,\,\,a\\!=\\!(a_{1k})_{k=1}^{n}\in\mathbb{R}^{n}.$
(5.10)

Set $C=\gamma(g_{1},g_{2},\dots,g_{n})$. We calculate $\Delta_{n}(\lambda,C)$
and $(C^{-1}(\lambda)a,a)$ for an arbitrary $n$. Consider the matrix (4.20)

$X_{mn}\\!=\\!\left(\begin{array}[]{cccc}x_{11}&x_{12}&...&x_{1n}\\\
x_{21}&x_{22}&...&x_{2n}\\\ &&...&\\\
x_{m1}&x_{m2}&...&x_{mn}\end{array}\right),\quad\text{where}\quad
x_{kr}\\!=\\!\frac{a_{1k}}{\sqrt{\lambda}_{k}},\quad
y_{r}\\!=\\!(x_{rk})_{k=1}^{n}\in\mathbb{R}^{n}.$ (5.11)

######  Lemma 5.10 ([6], Lemma 2.2)

For $m=2$ we have

$(C^{-1}(\lambda)a,a)=\Delta(y_{1},y_{2}):=\frac{\Gamma(y_{1})+\Gamma(y_{1},y_{2})}{1+\Gamma(y_{2})},$
(5.12)

where $y_{1}$ and $y_{2}$ are defined as follows

$y_{1}=\left(\frac{a_{1k}}{\sqrt{\lambda_{k}}}\right)_{k=1}^{n}\quad
y_{2}=\left(\frac{a_{2k}}{\sqrt{\lambda_{k}}}\right)_{k=1}^{n}.$ (5.13)

###  5.3 Case $m=3$

Define

$\Delta(y_{1},y_{2},y_{3}):=\frac{\Gamma(y_{1})+\Gamma(y_{1},y_{2})+\Gamma(y_{1},y_{3})+\Gamma(y_{1},y_{2},y_{3})}{1+\Gamma(y_{2})+\Gamma(y_{3})+\Gamma(y_{2},y_{3})}.$

######  Lemma 5.11 ([6], Lemma 2.3)

For $m=3$ we have

$(C^{-1}(\lambda)a,a)=\Delta(y_{1},y_{2},y_{3}),$ (5.14)

where the $y_{r}$ are defined as follows:

$y_{r}=\left(\frac{a_{rk}}{\sqrt{\lambda_{k}}}\right)_{k=1}^{n}\in\mathbb{R}^{n},\quad
1\leq r\leq 3.$ (5.15)

###  5.4 General $m$

We note by (5.2) that

$\displaystyle\Delta(y_{1},y_{2})\\!=\\!\frac{{\rm
det}\big{(}I_{2}\\!+\\!\gamma(y_{1},y_{2})\big{)}}{{\rm
det}\big{(}I_{1}+\gamma(y_{2})\big{)}}\\!-\\!1,\,\,\Delta(y_{1},y_{2},y_{3})\\!=\\!\frac{{\rm
det}\big{(}I_{3}\\!+\\!\gamma(y_{1},y_{2},y_{3})\big{)}}{{\rm
det}\big{(}I_{2}+\gamma(y_{2},y_{3})\big{)}}\\!-\\!1.$

Similarly, define

$\Delta(y_{1},y_{2},\dots,y_{m}):=\frac{{\rm
det}\big{(}I_{m}+\gamma(y_{1},\dots,y_{m})\big{)}}{{\rm
det}\big{(}I_{m-1}+\gamma(y_{2},\dots,y_{m})\big{)}}-1.$ (5.16)

######  Lemma 5.12 ([6])

For the general $m$ we have

$(C^{-1}(\lambda)a,a)=\Delta(y_{1},y_{2},\dots,y_{m}),$ (5.17)

where the $y_{r}$ are defined as follows:

$y_{r}=\left(\frac{a_{rk}}{\sqrt{\lambda_{k}}}\right)_{k=1}^{n}\in\mathbb{R}^{n},\quad
1\leq r\leq m.$ (5.18)

Acknowledgement. The author is very grateful to Prof. K.-H. Neeb, Prof. M.
Smirnov and Dr Moree for their personal efforts to make academic stays
possible at their respective institutes. A. Kosyak visited: MPIM from March to
April 2022 and from January to April 2023, University of Augsburg from June to
July 2022, and University of Erlangen-Nuremberg from August to December 2022,
all during the Russian invasion in Ukraine. Also, Prof. R. Kashaev kindly
invited A. Kosyak to Geneva.

Further, he would like to pay his respect to Prof. P. Teichner at MPIM, for
his immediate efforts started to help mathematicians in Ukraine after the
Russian invasion. Also Danil Radchenko helped the author to understand better
the intial problem.

Since the spring of 2023 A. Kosyak is an Arnold Fellow at the London Institute
for Mathematical Sciences, and he would like to express his gratitude to Mrs
Myers Cornaby to Miss Ker Mercer and to Miss Madeleine Hall and especially to
the Director of LIMS Dr T. Fink and Prof. Y.-H. He.

## References

  * [1] N. I. Akhiezer, I. M. Glazman, Theory of Linear Operators in Hilbert Space (Dover Books on Mathematics) 1966. M. Nauka. (in Russian).
  * [2] E. F. Beckenbach and R. Bellmann, Inequalities, Springer, Berlin, Göttingen, Heidelberg, 1961.
  * [3] R. F. Gantmacher, Matrizenrechnung. Teil 1. Veb Deutscher Verlag der Wissenschaften, Berlin, 1958.
  * [4] A. V. Kosyak, Criteria of irreducibility of the Koopman representations for the group ${\rm GL}_{0}(2\infty,{\mathbb{R}})$, J. Funct. Anal. 276 (1) (2019) 78–126.
  * [5] A. V. Kosyak, Regular, Quasi-regular and Induced Representations of Infinite-dimensional Groups, EMS Tracts in Mathematics 29, 587 p. 2018.
  * [6] A. V. Kosyak, Establishing irreduciblity of representations via Gram determinants, 105 p. (in progress).
  * [7] A. Kosyak and P. Moree, Irreducibility of the Koopman representations for the group ${\rm GL}_{0}(2\infty,{\mathbb{R}})$ acting on three infinite rows, arXiv:2307.11198 [math.RT] (submitted to Journ. Funct. Anal.).


# Born-Infeld Gravity: Constraints from Light-by-Light Scattering and an
Effective Field Theory Perspective

Jose Beltrán Jiménez<EMAIL_ADDRESS>Departamento de Física Fundamental
and IUFFyM, Universidad de Salamanca, E-37008 Salamanca, Spain.  Adrià Delhom
<EMAIL_ADDRESS>Departamento de Física Teórica and IFIC, Centro Mixto
Universidad de Valencia - CSIC. Universidad de Valencia, Burjassot-46100,
Valencia, Spain  Gonzalo J. Olmo<EMAIL_ADDRESS>Departamento de Física
Teórica and IFIC, Centro Mixto Universidad de Valencia - CSIC. Universidad de
Valencia, Burjassot-46100, Valencia, Spain  Departamento de Física,
Universidade Federal da Paraíba, 58051-900 João Pessoa, Paraíba, Brazil
Emanuele Orazi<EMAIL_ADDRESS>International Institute of Physics,
Federal University of Rio Grande do Norte, Campus Universitário-Lagoa Nova,
Natal-RN 59078-970, Brazil  Escola de Ciencia e Tecnologia, Universidade
Federal do Rio Grande do Norte, Caixa Postal 1524, Natal-RN 59078-970, Brazil

###### Abstract

By using a novel technique that establishes a correspondence between general
relativity and metric-affine theories based on the Ricci tensor, we are able
to set stringent constraints on the free parameter of Born-Infeld gravity from
the ones recently obtained for Born-Infeld electrodynamics by using light-by-
light scattering data from ATLAS. We also discuss how these gravity theories
plus matter fit within an effective field theory framework.

After Dirac’s proposal of a relativistic theory for the electron Dirac (1928),
Halpern and Heisenberg Halpern (1933); Heisenberg (1934) realized that quantum
effects would induce light-by-light scattering (via 1-loop processes involving
charged particles), which was forbidden in the classical theory. First
calculations of this effect were performed by Euler and Kockel in the low
frequency limit Euler and Kockel (1935), and later on, a complete calculation
of light-by-light scattering within QED was published by Euler and Heisenberg
and by Karplus and Neumann Heisenberg and Euler (1936); Karplus and Neuman
(1951, 1950). Several attempts to observe this phenomenon have been performed
since then, focusing on different photon energy ranges Jarlskog  _et al._
(1973); Moulin  _et al._ (1996); Bernard  _et al._ (2000); Ishikawa  _et al._
(2012); Inada  _et al._ (2014); Yamaji  _et al._ (2016) (see also Fouché  _et
al._ (2016) for a compilation of different set-ups that constrain non-linear
electrodynamics). It was not until 2017 that the ATLAS Collaboration presented
strong evidence for this process in ultra-peripherical heavy ion collisions at
LHC Aaboud  _et al._ (2017). A recent account of the most up-to-date light-by-
light scattering physics can be found in Schoeffel  _et al._ (2020). The
results obtained in the ATLAS experiment are consistent with QED predictions,
which allows to set bounds on theories of nonlinear electrodynamics (NED). In
this direction, Ellis et. al. recently used ATLAS data to strongly constrain
the mass scale of Born-Infeld (BI) electrodynamics to be $M_{\rm BI}\geqslant
100$ GeV Ellis  _et al._ (2017), improving previous constraints. One of the
aims of this paper is to use those data to place constraints on a Born-Infeld-
like theory of gravity.

The origin of Born-Infeld electrodynamics Born and Infeld (1934) can be traced
back to some early attempts to regularize the electric field of a point charge
at the charge’s location and hence solve the problem of the divergent self-
energy of point charges. Born first attempted to tackle this issue by invoking
a sort of finiteness principle and, borrowing ideas from special relativity,
he suggested to non-linearly promote the Maxwell Lagrangian following a
prescription similar to the passage from Galilean to special relativity. In
collaboration with Infeld, they then embraced the framework of General
Relativity and wrote down what is today known as Born-Infeld eletromagnetism
described by the action

$\displaystyle\mathcal{S}_{\rm BI}$ $\displaystyle=$ $\displaystyle M_{\rm
BI}^{4}\int\text{d}^{4}x\left[\sqrt{-g}-\sqrt{\left|\det\left(g_{\mu\nu}+\frac{1}{M_{\rm
BI}^{2}}F_{\mu\nu}\right)\right|}\;\right]$ $\displaystyle=$ $\displaystyle
M_{\rm
BI}^{4}\int\text{d}^{4}x\sqrt{-g}\left(1-\sqrt{1+\frac{F^{\mu\nu}F_{\mu\nu}}{2M_{\rm
BI}^{4}}-\frac{(F^{\mu\nu}\tilde{F}_{\mu\nu})^{2}}{16M_{\rm
BI}^{8}}}\right)\;,$

where $M_{\rm BI}$ denotes the energy scale where modifications to QED become
non-perturbative or, equivalently, at electromagnetic fields
$|\vec{E}|,|\vec{B}|\sim\mathcal{O}(M_{\rm BI}^{2})$ deviations from Maxwell’s
electromagnetism become non-perturbatively relevant. The interest of the
community in BI actions was boosted by Fradkin and Tseytlin Fradkin and
Tseytlin (1985), who showed that actions of the Born-Infeld type arise in
different scenarios related to M-theory.

The many appealing properties found in BI electromagnetism (see e.g. Ketov
(2001) and references therein) motivated to seek for similar extensions in the
gravitational sector. A first incursion on this path was carried out by Deser
and Gibbons Deser and Gibbons (1998), although the seemingly unavoidable
presence of ghosts in these extensions detracted substantially from their
phenomenological interest. It was later realised that the Palatini formulation
of Born-Infeld-like gravity theories seemed less pathological Vollick (2004)
and these efforts eventually crystalised in a combination of the old Eddington
purely affine gravity with Born-Infeld ideas that is now widely known as
Eddington-inspired-Born-Infeld (EiBI) and whose action reads Banados and
Ferreira (2010)

$\mathcal{S}_{\rm
EiBI}=\pm\Lambda_{Q}^{4}\int\text{d}^{4}x\left(\sqrt{-\left|g_{\mu\nu}\pm\frac{\mathcal{R}_{(\mu\nu)}(\Gamma)}{\Lambda_{\rm
EiBI}^{2}}\right|}-\sqrt{-|g_{\mu\nu}|}\right)\;,$ (1)

where $\Lambda_{Q}$ is a high energy scale which can be factorized as
$\Lambda_{Q}\equiv\sqrt{M_{\rm Pl}\Lambda_{\rm EiBI}}$ so that GR with the
standard value of the Newton’s constant is recovered in the weak curvature
regime. Then the model has only one free parameter which can be represented by
either $\Lambda_{\rm EiBI}$ or $\Lambda_{Q}$. Although $\Lambda_{\rm EiBI}$ is
the scale that suppresses the higher order curvature corrections,
$\Lambda_{Q}$ might be endowed with a more physically transparent meaning
since this is the scale that suppresses the associated interactions in the
matter sector. In this respect, we can make an educated guess for a constraint
on this scale from the absence of anomalous scattering amplitudes at LHC so we
can impose $\Lambda_{Q}$ to be larger than the scales probed at LHC, i.e.,
$\Lambda_{Q}\gtrsim 1-10$ TeV. Of course, the precise constraint depends on
the particular process we look at, but the quoted limit is a natural lower
bound to expect. EiBI gravity has been extensively studied in recent years in
cosmology, astrophysics, and in compact objects scenarios, where it exhibits a
remarkable ability to yield nonsingular solutions and interesting
phenomenology. For a review on the different approaches to Born-Infeld gravity
and their phenomenological applications we refer to Beltran Jimenez  _et al._
(2018).

The dependence of the Ricci tensor on the connection
${\Gamma^{\alpha}}_{\mu\nu}$ has been written explicitly to emphasize that it
is not assumed to be a priori related to the metric, but rather that metric
and connection are naturally regarded as equally fundamental and independent
geometrical objects (the so called metric-affine formulation). The fact that
only the symmetric part of the Ricci tensor appears in the action guarantees
that the field equations for the connection are algebraic and, furthermore, it
can be solved as the Levi-Civita connection of a metric that can be expressed
in terms of $g_{\mu\nu}$ and the matter fields. In other words, the connection
plays the role of an auxiliary field that can be integrated out and whose
effect is the modification of the interactions in the matter sector. As a
matter of fact, a field redefinition of the metric allows to build an Einstein
frame representation of the theory in which the matter fields are minimally
coupled to the redefined (or Einstein-frame) metric but the interactions are
modified. We will discuss below how the same procedure applies to a much
larger class of theories, so this property is not specific of the EiBI
Lagrangian. It is crucial to mention that going to Einstein frame within these
theories does not require the introduction of additional degrees of freedom
because we only need to solve for the connection that is an auxiliary field.
This is radically different from other theories like metric $f(R)$ where the
space of propagating fields is enlarged with respect to GR. This is the reason
why in the EiBI theory the whole effect of going to the Einstein frame simply
consists in the generation of interactions in the matter sector for the fields
that were already present in that sector in the original frame.

By construction, the matter Lagrangian in the Einstein frame retains the
physical symmetries of the original theory so that the same conserved Noether
charges and degrees of freedom remain. The construction of the Einstein frame
requires solving an equation for $g_{\mu\nu}$ in terms of the energy-momentum
tensor of the matter sector so, as long as the energy-momentum tensor
preserves the symmetries, these will remain. This is generally expected to be
the case for internal symmetries that do not involve spacetime
transformations. A potential problem with gauge symmetries could arise for
spin-2 fields because, as it is well-known, no gauge (diffs) invariant and
Lorentz covariant energy-momentum tensor exists for a massless spin-2 field.
In this case, integrating out the connection could lead to pathologies as the
appearance of (ghostly) new degrees of freedom. Of course, these degrees of
freedom were already latent in the theory before integrating out the
connection. In any case, since having more than two massless spin-2 fields is
already pathological, this should not be a cause of problems. On the other
hand, it may occur that some properties like e.g. dualities are lost or might
be appropriately re-formulated. Remarkably, in the particular case of EiBI
gravity coupled to Maxwell electrodynamics, it was recently shown that the
Einstein-frame matter action exactly coincides with BI NED111This is strictly
true if we pick the negative sign in the gravitational Lagrangian (1),
otherwise the sign of the $\mathcal{O}(\beta^{-2})$ term in (Born-Infeld
Gravity: Constraints from Light-by-Light Scattering and an Effective Field
Theory Perspective) would be negative. If the EiBI Lagrangian is written as in
Delhom  _et al._ (2019), this choice corresponds to $\epsilon<0$., provided
that one identifies $\Lambda_{Q}=\sqrt[4]{2}M_{BI}$ Delhom  _et al._ (2019).
This correspondence is precisely the one that we will exploit in the following
to set constraints on EiBI gravity from the observed bounds on BI
electromagnetism.

In order to obtain the bound $M_{\rm BI}\gtrsim 100$ GeV, it was implicitly
assumed in Ellis  _et al._ (2017) that the gravitational sector was described
by GR, with the matter sector represented by the BI nonlinear modification of
Maxwell’s electrodynamics. An apparently different approach would be to take
the gravity side as described by the EiBI action (1) but keeping in the matter
sector the standard Maxwell theory. However, as explained above, given that a
field redefinition transforms one theory into the other, the bound found in
Ellis  _et al._ (2017) for the matter sector can be reinterpreted so as to set
a bound on the EiBI gravitational sector. Accordingly, and using the results
of Ellis  _et al._ (2017), we find that light-by-light scattering data from
ultraperipherical Pb-Pb collisions in ATLAS sets a conservative constraint to
the EiBI gravity free parameter of

$\displaystyle\Lambda_{Q}\gtrsim 120\text{ GeV}$ (2)

This constraint is similar to the most stringent constraints to EiBI up to
date, although it is obtained in a more precise way222Notice that, although
the constraint is only strictly valid for the negative sign in (1),
corrections to QED of the same order of magnitude are expected for the
positive sign ($\epsilon>0$ in Delhom  _et al._ (2019)), thus a similar bound
would be obtained in this case. Let us mention however that the sign of this
parameter could be further constrained from positivity bounds so as to embed
the theory into a local Lorentz invariant UV theory as nicely discussed in
e.g. Adams  _et al._ (2006). In this respect, it is interesting to note that
the UV string theory embedding of BI electromagnetism can be related to the
absence of superluminal propagation around any background Gibbons and Herdeiro
(2001).. Concretely it is of the same order of magnitude as the one obtained
in Delhom  _et al._ (2020) from Compton scattering data, and one order of
magnitude lower than that obtained in Latorre  _et al._ (2018) from Bhabha
scattering data. However, notice that both constraints were obtained by using
a perturbative version of the mapping in powers of $1/\Lambda_{Q}$, whereas
here we are using the full non-perturbative relation between the EiBi and
Einstein frames of the theory. To compare with other existing constraints to
EiBI in the literature, it is useful to use the parameter $|\kappa|\equiv
2c^{7}\hbar^{3}\Lambda_{Q}^{-4}$ so the bound (2) translates as

$|\kappa|\lesssim 4\times 10^{-24}\text{m}^{5}\text{kg}^{-1}\text{s}^{-2}.$
(3)

Other constraints to EiBI gravity from nuclear physics, astrophysics, or
gravitational wave data are several orders of magnitude weaker Avelino (2012a,
b); Jana  _et al._ (2018); Avelino (2019). As well, we note that this
constraint gives an order of magnitude estimate of the constraints that would
be found for generic RBG theories coupled to Maxwell electromagnetism, see
e.g. Latorre  _et al._ (2018); Delhom  _et al._ (2020).

EiBI as a Ricci-Based Gravity theory

Let us now turn our attention to the implications that the correspondence
between theories that we used above may have for other theories of gravity.
The EiBI theory is a member of a larger family of theories known as Ricci-
Based Gravity theories (RBGs), which are characterised by an action of the
form Beltran Jimenez  _et al._ (2018); Afonso  _et al._ (2017)

$\mathcal{S}_{\rm
RBG}=\int\text{d}^{4}x\sqrt{-|g|}F(g^{\mu\nu},\mathcal{R}_{(\mu\nu)}(\Gamma))+\mathcal{S}_{\rm
m}(g_{\mu\nu},\Psi_{m})\ ,$ (4)

where $F$ is any analytic function of the metric and the symmetrized Ricci
tensor, and $\mathcal{S}_{\rm m}$ represents the matter action, in which the
generic matter fields $\Psi_{m}$ are minimally coupled to the metric
$g_{\mu\nu}$.333The assumption of minimally coupled fields is not fundamental
for our argument here and could be readily included as in e.g. Afonso  _et
al._ (2017). We prefer to stick to the minimally coupled case for clarity of
the exposition. In RBGs only the symmetric part of the Ricci tensor is
considered because the antisymmetric part is not invariant under projective
transformations,
${\Gamma^{\alpha}}_{\mu\nu}\to{\Gamma^{\alpha}}_{\mu\nu}+\xi_{\mu}\delta^{\alpha}{}_{\nu}$,
and an explicit breaking of projective symmetry leads to the propagation of
unstable degrees of freedom in arbitrary backgrounds Beltrán Jiménez and
Delhom (2019, 2020).

As in the EiBI theory, it is possible to show that the connection enters as an
auxiliary field in all RBGs and they also admit an Einstein-frame
representation where the gravitational sector is described by standard GR for
the redefined metric $q_{\mu\nu}$ and integrating the connection out results
in additional interactions in the matter sector. In this frame, matter is
minimally coupled to the Einstein-frame metric $q_{\mu\nu}$ (see Beltran
Jimenez  _et al._ (2018); Afonso  _et al._ (2018a, 2019a, b); Delhom  _et al._
(2019) for further details). This redefined metric is associated to $\partial
F/\partial\mathcal{R}_{(\mu\nu)}$, and its relation with $g_{\mu\nu}$ can be
encoded into a deformation matrix $\hat{\Omega}$ defined via the relation

$q_{\mu\alpha}(\Omega^{-1})^{\alpha}{}_{\nu}=g_{\mu\nu}.$ (5)

When the field equations are satisfied, $(\Omega^{-1})^{\alpha}{}_{\nu}$ can
be written as a (model-dependent) algebraic function of the matter fields and
one of the metrics Beltran Jimenez  _et al._ (2018); Delhom  _et al._ (2019),
and it can be expanded in a series of the form444Here we refer to the RBGs
that have GR as a low energy limit and which are characterised by a single
energy scale $\Lambda_{Q}$ at which deviations from GR become non-
perturbative. Theories characterized by more than one energy scale could be
included in a straightforward manner, although they would unnecessarily
obscure the argument.

$(\Omega^{-1})^{\mu}{}_{\nu}=\delta^{\mu}{}_{\nu}+\sum_{n=1}^{\infty}\sum_{i_{n}=0}^{3}\frac{\beta_{(n,i_{n})}}{\Lambda_{Q}^{4n}}(T^{(n,i_{n})})^{\mu}{}_{\nu},$
(6)

where each $(T^{(n,i_{n})})^{\mu}{}_{\nu}$ stands for a tensor structure that
can be built with $n$ powers of the stress energy tensor. At each order $n$,
the sum over $i_{n}$ runs up to three because the Cayley-Hamilton theorem
guarantees that a given matrix is always a root of its characteristic
polynomial. In other words, the vector space of $D$-dimensional matrices that
are analytical functions of some given matrix has, at most, dimension $D$ and,
in our case, we can take the first $(D-1)$ powers of $T^{\mu}{}_{\nu}$ as a
basis of this space.555Notice that the vector space spanned by the first
$(D-1)$ powers of $T^{\mu}{}_{\nu}$ does not need to have dimension $D$, but
its dimension will be determined by the number of linearly independent powers.
In other words, the space of matrices that are analytical functions of
$T^{\mu}{}_{\nu}$ has dimension $\text{rank}(\\{\hat{T}^{i}\\}|_{i=0,\cdots
D-1}$. Strictly speaking, the relation (6) holds for the (likely more
physical) branch of solutions that reduce to GR in the low energy limit. This
can be important because the deformation matrix satisfies a non-linear
equation in terms of the energy-momentum tensor. Thus, besides the solution
(6) that could be obtained by imposing Lorentz covariance, there could be
other branches with spontaneous symmetry breaking as discussed in Beltrán
Jiménez  _et al._ (2020). However, these non-standard branches are seemingly
pathological and, since we want to recover GR at low energies, (6) is the
relevant series expansion for the solution.

The mapping into GR for an arbitrary RBG (minimally) coupled to scalar fields,
anisotropic fluids, and massless spin-1 fields has been developed in Afonso
_et al._ (2018a, 2019a, b); Delhom  _et al._ (2019), and also a few examples
in which the full non-perturbative solution of the mapping equations can be
obtained have been worked out Delhom  _et al._ (2019); Afonso  _et al._
(2019b); Guerrero  _et al._ (2020); Shao  _et al._ (2021); Olmo  _et al._
(2020). For fermions only a perturbative solution of the mapping is available
so far Latorre  _et al._ (2018); Delhom  _et al._ (2020) though, in principle,
nothing prevents the existence of an Einstein frame with an additional four-
fermion interaction as the one appearing in the Einstein-Cartan-Sciamma-Kibble
theory Kibble (1961); Sciama (1964); Hehl  _et al._ (1976). From the EFT
viewpoint, the perturbative series is in fact the relevant regime.

It is important to note that extensions of the RBG family (4) involving the
Riemann tensor and its derivatives can also be considered, leading to a
broader spectrum of theories. Such extensions generically propagate a massless
spin-2 mode encoded in some $q_{\mu\nu}$ metric plus (probably) other degrees
of freedom that are not present in GR. Our focus here on the RBG sector is
justified because a crucial part of the action of such theories will be of the
RBG type and may be constrained well before those new degrees of freedom may
become observable. Furthermore, the most general framework involving arbitrary
powers of the Riemann tensor has been shown to be generically plagued by
pathologies Beltrán Jiménez and Delhom (2019, 2020).

Effective matter sector in RBGs

The equivalence between RBG’s and GR is particularly relevant when the matter
sector is interpreted within the realm of an effective field theory (EFT)
because constraints on nonlinearities of the matter sector can be used to
place constraints on RBGs, in much the same way as we did above with the EiBI
model. In the EFT framework (see e.g.Pich (1998)) one intends to give a
description of the phenomena occurring below some energy scale $\Lambda$,
which could take into account small corrections with origin in the UV 666Here
UV/low-energy refers to energies greater/lower than the EFT scale $\Lambda$.
By construction, the EFT will only be physically meaningful at scales below
$\Lambda$, typically breaking unitarity above this scale. regime without
actually knowing the details of the underlying UV theory. This is done by 1)
identifying the set of fields that describe the spectrum at low energies and
choosing a set of symmetries that should be satisfied in the UV, and 2) using
those fields to build the most general Lagrangian consistent with those
symmetries.

Let us sketch how to build such Lagrangian. Assume that we have a given set of
asymptotic states described by the matter fields $\Psi_{i}$ whose dynamics
satisfies certain symmetries. This matter sector defines an EFT Lagrangian by
$\mathcal{L}_{\rm{eff}}=\mathcal{L}_{0}+\mathcal{L}_{\rm eff}^{\rm d>4}$ where
$\mathcal{L}_{0}$ contains the relevant and marginal operators and
$\mathcal{L}_{\rm eff}^{\rm d>4}$ is an infinite tower of all the possible
irrelevant operators that can be built upon the $\Psi_{i}$’ s and which
respect the given symmetries. As is well known, the set of dimension-$d$
operators of a given quantum field theory forms a vector space
$\mathcal{A}_{d}$. Furthermore two operators are said to be equivalent, in the
sense that they contribute equally to physical observables, if they differ (up
to a total derivative) by an on-shell constraint777This is an operator
proportional to the lower order field equations, which by definition vanish
on-shell see e.g. Weinberg (2005, 2013). $\mathcal{A}_{d}$ can be split into
the equivalence classes defined by this relation, and it suffices to consider
one operator of each equivalent class to have a basis of $\mathcal{A}_{d}$
(see e.g. Einhorn and Wudka (2013)). Thus, without loss of generality we can
write

$\mathcal{L}_{\rm eff}^{\rm
d>4}=\sum_{n=5}^{\infty}\sum_{i_{n}}\frac{\alpha_{(n,i_{n})}}{\Lambda^{n}}\mathcal{O}_{(n,i_{n})},$
(7)

where the set $\\{\mathcal{O}_{(n,i_{n})}\\}_{i_{n}}$ is a basis of
$\mathcal{A}_{n}$ and $i_{n}$ runs from 1 to dim$\\{\mathcal{A}_{n}\\}$ for
each $n$ in the above expression. The dimensionless constant $\alpha_{(p,q)}$
is the Wilson coefficient of the operator $\mathcal{O}_{(p,q)}$. Assuming
naturalness, the Wilson coefficients will be of $\mathcal{O}(1)$ and the EFT
defined by (7) will be generically valid at energies below $\Lambda$. In this
respect, it is interesting to notice that the violation of tree-level
unitarity does not imply the necessity of including new physics at that
precise scale as shown for instance with the self-healing mechanism discussed
in Aydemir  _et al._ (2012).

Let us now couple the EFT described by $\mathcal{L}_{\rm eff}$ to a
gravitational sector given by a particular RBG theory. This might appear as an
arbitrary construction because the EFT philosophy should also be employed in
the construction of the gravity sector. However, there is nothing a priori
inconsistent with considering an RBG gravity sector and our interest here is
precisely to discuss how these gravity theories can be fit into an EFT
framework. Since the theory can be mapped into GR coupled to a matter sector
with the same degrees of freedom and symmetries as the original one, the
mapped EFT retains its structure, i.e., the basis of operators of the original
EFT is still a basis of the mapped EFT. Thus, the ‘new’ operators that appear
after the mapping were already present in the original matter Lagrangian.

Before proceeding further, it may be in order to digress a bit here again on
the preservation of the symmetries when going to the Einstein frame. In the
EFT matter sector there will be gauge symmetries that have to do with massless
particles and are fundamental for the correct number of degrees of freedom. We
do not expect the Einstein frame formulation of the theory to change this
because the energy-momentum tensor of the gauge fields will also be gauge
invariant (with the exception of massless spin-2 particles that we have
already addressed above). Regarding global symmetries, these typically arise
as accidental symmetries of the low energy theory but can be broken by higher
dimension operators. Thus, when going to the Einstein frame in the RBGs no new
operators will be generated.

In view of the above discussion and since the original EFT matter Lagrangian
was built with an arbitrary linear combination (defined by the
$\alpha_{(n,i_{n})}$) of the $\\{\mathcal{O}_{(n,i_{n})}\\}_{i_{n}}$, which
provide a full basis of the space of operators of the theory, it should be
clear that the effect of the non-linearities introduced in the matter sector
after the mapping can be reabsorbed by a redefinition of the Wilson
coefficients888Note that by the structure of (6), the new operators that will
enter the Lagrangian after the mapping are of mass dimension $4n$, and
therefore only the Wilson coefficients corresponding to $4n$-dimensional
operators will be non-trivially changed.
$\alpha_{(n,i_{n})}\mapsto\tilde{\alpha}_{(n,i_{n})}$ which schematically
looks like

$\tilde{\alpha}_{(4n,i_{n})}=\alpha_{(4n,i_{n})}+\sum_{i_{k}}\beta_{(n-4,i_{k})}\left(\frac{\Lambda}{\Lambda_{Q}}\right)^{n-4}.$
(8)

where the $\beta_{(n,i_{n})}$ are (up to a constant factor) those appearing in
(6) and $\Lambda_{Q}$ is the high-energy scale that parametrises departures
from GR within the given RBG model. Since by construction the Wilson
coefficients are arbitrary in an EFT999Of course, they can be constrained by
experiments, thus ruling out regions of parameter space. But regarding the
theoretical construction of the EFT, these are arbitrary coefficients, this
redefinition is not relevant, and the matter sector is described by the same
EFT before and after the mapping. There is a subtlety that can arise here: If
the scale $\Lambda_{Q}$ is much below the EFT scale, then the redefinitions in
(8) imply that either the $\alpha_{(4n,i_{n})}$ or the
$\tilde{\alpha}_{(4n,i_{n})}$ will typically be far from unity by an amount of
the same order, which could restrict the range of validity of the EFT to
energies below $\Lambda_{Q}$.

Discussion and conclusions.

As a consequence of the above discussions we arrive at the following
conclusion: the predictions of a given RBG coupled to a given matter sector
will differ, in general, from those of GR coupled to the same matter sector
(when the connection is integrated out), but if the matter sector coupled to
the RBG is an EFT its predictions will be indistinguishable from those of GR
coupled to an EFT. This has been illustrated in the first part of this paper
by the coupling of EiBI gravity to Maxwell, which is equivalent to the
coupling of GR to Born-Infeld electromagnetism rather than of GR to Maxwell.
In the EFT framework, this is stated by saying that the operators pertaining
to the RBG sub-class of metric-affine actions are redundant when one considers
all the allowed operators in the matter sector. Although other metric-affine
models can also be shown to be equivalent to GR, it is still an open question
whether there is any other disjoint class of metric-affine operators which are
redundant in the sense described above.

If the EFT approach is extended to the gravitational interactions, it is
straightforward to see that the EFT for a general metric-affine sector cannot
be reduced to an RBG by imposing any symmetries involving the metric and/or
affine connection, even at the lowest order. In fact, RBGs only propagate a
massless spin-2 mode, and the most general diffeomorphism and projective
invariant action propagates extra degrees of freedom already at quadratic
order, some of which are likely to be unstable unless fine-tuned models are
considered Beltrán Jiménez and Delhom (2019, 2020); Percacci and Sezgin
(2020). Furthermore, it is easy to see that by imposing extra symmetries to
this action one would forbid operators that lie within the RBG class. Hence,
it is difficult to envision how a general metric-affine EFT could be reduced
to an RBG theory by means of symmetries even at the lowest order. Nonetheless,
this does not preclude RBGs from being regarded as physically sound effective
theories for perturbative phenomena at energy scales $E<\Lambda_{Q}$ but then
these effects would already be there from the irrelevant operators of the
matter sector.

From a non-perturbative perspective, there are RBGs able to remove
cosmological and black hole singularities at the classical level, restoring
geodesic completeness via the emergence of wormholes or cosmological bounces.
However, quantum corrections in the matter sector of those theories are likely
to strongly backreact onto those backgrounds, potentially rendering them as
unphysical Beltran Jimenez  _et al._ (2017). In this regard, it is important
to recall that there are effective theories with irrelevant operators that
satisfy non-renormalization theorems. This property is important at a
phenomenological level because it allows to have non-perturbative classical
effects from non-renormalisable operators while maintaining the quantum
corrections under control. The paradigmatic example is of course General
Relativity where the Planck mass in front of the Einstein-Hilbert term does
not get renormalised by graviton loops.101010There is a simple argument that
explains why this is the case. If we consider the so-called $\Gamma\Gamma$
Einstein form for the GR action (i.e. the Einstein-Hilbert Lagrangian deprived
of the total derivative term), then diffeomorphisms are only realised up to a
boundary term. Since Feynman diagrams realise the symmetries in an exact form,
all loops can only generate quantum corrections to the higher order terms. It
receives quantum corrections from matter loops, but these are typically
$\mathcal{O}(m^{2}/M_{\rm pl}^{2})\lesssim 10^{-30}$ for the standard model
particles so GR is actually an excellent quantum EFT (see e.g. Donoghue
(1995); Dobado  _et al._ (1997); Burgess (2004))111111We cannot resist
referring to Weinberg’s words on the topic in Weinberg (2009).. Of course,
there is the problem of the cosmological constant, but this is a naturalness
problem rather than a breakdown of the EFT.

Similarly, theories like non-linear electrodynamics, $K$-essence, or Galileon
theories exhibit analogous properties for their quantum corrections. Thus, if
we take RBG theories and the matter sector is composed by e.g. a massless
scalar field solely, the Einstein frame version of the theory after
integrating out the connection will give rise to a $K$-essence model with a
Lagrangian of the form
$\mathcal{L}=\Lambda^{4}_{\phi}K(\partial_{\mu}\phi\partial^{\mu}\phi/M^{4})$,
with $\Lambda$ and $M$ some mass scales. The non-perturbative regime of the
original RBG theory can then be mapped into a phenomenological effect arising
from non-renormalisable operators in $K$ where $M$ corresponds to the scale of
non-linearities and $\Lambda$ is parameterically given by
$\Lambda\simeq\sqrt{M_{\rm pl}M}$. The structure of these scalar theories, in
particular their shift symmetry, guarantees that the quantum corrections will
come in with derivatives of the Lagrangian $\sim\partial K$ so it is plausible
to have a regime where $\partial_{\mu}\phi\partial^{\mu}\phi/M^{4}\gtrsim 1$
while quantum corrections are kept under control and the irrelevant operators
in the classical action are technically natural (see e.g. de Rham and Ribeiro
(2014); Brax and Valageas (2016) for explicit derivations of these
statements).Thus, the non-perturbative classical solutions have a chance of
surviving and we can trust their predictions. Similarly, the quantum
corrections in non-linear electrodynamics theories can be kept small even in
regions where the electromagnetic fields are
$|F_{\mu\nu}|\gtrsim\Lambda^{2}_{\rm non-linear}$, being Born-Infeld
electromagnetism a paradigmatic example.

An important concern with RBG theories is that the irrelevant operators
generated after integrating out the non-dynamical connection are somewhat
universal in the sense that they will involve all fields in the matter sector
and, at the very least, those of the Standard Model. This universal nature of
the RBG theories will typically lead to an EFT in the Einstein frame without
any underlying symmetry or structure guaranteeing any non-renormalisation
result or naturalness of the resulting interactions like in the case of pure
K-essence or non-linear electromagnetisms. Quite the opposite, the very
presence of the Higgs field and its potential already points towards the
impossibility of having non-perturbative classical solutions based on
irrelevant operators without going beyond the regime of validity of the would-
be EFT. Leaving aside these quantum corrections issues, one should not forget
that the interest on these theories stems from their non-perturbative
properties as classical field theories, as they accommodate a plethora of
exact solutions with interesting properties which can expand our dictionary of
viable spacetimes. For instance, they are furnished with singularity free
cosmological and spherically symmetric backgrounds Barragan  _et al._ (2009);
Banados and Ferreira (2010); Olmo  _et al._ (2014); Masó-Ferrando  _et al._
(2021); Benisty  _et al._ (2021), as well as wormholes and other compact
objects which behave in interesting and unexpected ways (see e.g. Lobo  _et
al._ (2013, 2014a, 2014b, 2014c, 2020)) which are worth to be explored.

Acknowledgments.

JBJ acknowledges support from the “Atracción del Talento Científico” en
Salamanca programme and from project PGC2018-096038-B-I00 by Spanish
Ministerio de Ciencia, Innovación y Universidades. AD is supported by a PhD
contract with reference FPU15/05406 (Spanish Ministry ofEconomy and
Competitiveness). GJO acknowledges support from the Spanish Agencia Estatal de
Investigación (FIS2017-84440-C2-1-P), by the Generalitat Valenciana
(PROMETEO/2020/079), and by the Spanish Research Council (i-COOPB20462, CSIC).
This work has further been supported by the European Union’s Horizon 2020
Research and Innovation (RISE) programme H2020-MSCA-RISE-2017 Grant No.
FunFiCO-777740.

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