# Noiselessly amplified thermal states and after multi-photon addition or
subtraction

Xue-feng Zhan and Xue-xiang Xu† College of Physics and Communication
Electronics, Jiangxi Normal University, Nanchang 330022, China
<EMAIL_ADDRESS>
###### Abstract

In this paper, we introduce a noiselessly amplified thermal state (ATS), by
operating the noiseless amplification operator ($g^{\hat{n}}$) on the thermal
state (TS) with corresponding mean photon number (MPN) $\bar{n}$. Actually,
the ATS is an new TS with MPN
$\bar{N}=g^{2}\bar{n}/[1-\bar{n}\left(g^{2}-1\right)]$. Furthermore, we
introduce photon-added-ATS (PAATS) and photon-subtracted-ATS (PSATS) by
operating $m$-photon addition ($\hat{a}^{{\dagger}m}$) and $m$-photon
subtraction ($\hat{a}^{m}$) on the ATS, respectively. We study photon number
distributions (PNDs), purities, and Wigner functions (WFs) for all these
states.

Keywords: noiseless amplification; photon addition; photon subtraction; Wigner
function

##  I Introduction

Quantum theory is originated from Planck’s discovery of the radiation law1 ; 2
. Often, those electromagnetic radiations emitted by a hot body with a certain
temperature were called as thermal lights3 . One of them, namely thermal state
(TS), can be expressed in terms of Fock states

$\rho_{th}\left(\bar{n}\right)=\sum_{n=0}^{\infty}\frac{\bar{n}^{n}}{(\bar{n}+1)^{n+1}}\left|n\right\rangle\left\langle
n\right|,$ (1)

where $\bar{n}$ denotes its mean photon number (MPN)4 . Usually, the TS plays
a significant role in statistical physics5 . However, this TS is an Gaussian
state, which lack some desirable properties (e.g., Wigner negativity) for
quantum supremacy in various applications6 ; 7 .

Photon addition (by operating creation operator $\hat{a}^{{\dagger}}$) and
photon subtraction (by operating annihilation operator $\hat{a}$), as the
basic tools, have been used to generate non-Gaussian quantum states8  with
some desirable properties. Recently, Guerrini et al. called these states as
photon-varied quantum states9 . Barnett et al. presented a detailed analysis
of the photon statistics for photon-subtracted and photon-added states10 .
Based on the TS, the reseachers have introduced many quantum states and done
many theoretical and experimental works11 ; 12 ; 13 . Zavatta et al.
demonstrated the bosonic communication relation in experiment14  by applying
$\hat{a}\hat{a}^{{\dagger}}-e^{i\phi}\hat{a}^{{\dagger}}\hat{a}$ on the TS.
Zavatta et al. reported the generation and the analysis of single-photon-added
or single-photon-subtracted TSs15  by applying $\hat{a}^{{\dagger}}$ or
$\hat{a}$ on the TS. Many groups have introduced and studied the multiphoton-
subtracted TSs by using $\hat{a}^{m}$ ($m$ is an integer) 16 ; 17 ; 18 ; 19 ;
20 . Chatterjee compared the nonclassicality of the added-then-subtraced and
subtracted-then-added TS by using $\hat{a}^{q}\hat{a}^{{\dagger}p}$ or
$\hat{a}^{{\dagger}p}\hat{a}^{q}$ ($p,q$ are integers)21 . Deepak and
Chatterjee reported a study related with a coherent superposition TS by using
$s\hat{a}\hat{a}^{{\dagger}}+t\hat{a}^{{\dagger}}\hat{a}$ 22 . Our group has
introduced photon annihilation-then-creation TS and photon creation-then-
annihilation TS by using $\left(\hat{a}^{{\dagger}}\hat{a}\right)^{m}$ and
$\left(\hat{a}\hat{a}^{{\dagger}}\right)^{m}$23 . Our group has also
introduced a kind of amplified TS by using
$\left(g-1\right)\hat{a}^{{\dagger}}\hat{a}+1$24 , where $g>1$ is the
amplification gain.

In recent decades, studies on the noiseless quantum amplifications have
attracted significant attention of researchers25 ; 26 ; 27 ; 28 ; 29 ; 30 . As
we know, noise is unavoidably added to the amplified signal in any
deterministic linear amplification. Observing the results in the hypothetical
noiseless amplification of coherent state (i.e.,
$\left|\alpha\right\rangle\rightarrow\left|g\alpha\right\rangle$), one can
infer that it is not a deterministic physical operation (i.e., a trace-
preserving positive map)31 . In order to overcome the limit of deterministic
amplification, one can resort to probabilistic amplification32 , which can be
approximate the conditional operation $g^{\hat{n}}$ ($g>1$ and
$\hat{n}=\hat{a}^{{\dagger}}\hat{a}$). Thus, the input coherent state
$\left|\alpha\right\rangle$ can be mapped into an amplified coherent state
$\left|g\alpha\right\rangle$, i.e.,
$g^{\hat{n}}\left|\alpha\right\rangle=e^{(g^{2}-1)\left|\alpha\right|^{2}/2}\left|g\alpha\right\rangle$
33 . However, this operation is still unphysical because no device realizing
this transformation can be implemented perfectly34 . The idea on the heralded
noiseless amplification was theoretically proposed by Ralph and Lund35 .
Afterwards, some amplifications were experimentally demonstrated by several
groups36 ; 37 ; 38 .

In a recent work, we have studied multi-photon-addition amplified coherent
states by considering the effects of $g^{\hat{n}}$ and
$\hat{a}^{{\dagger}m}$39 . However, we have never seen any reports on the
combined effect of $g^{\hat{n}}$ with $\hat{a}^{{\dagger}m}$ or with
$\hat{a}^{m}$ on the TS. So in this paper, we shall study this problem and
introduce related quantum states. The paper is structured as follows: we
introduce the ATS in Sec.2, and then introduce the PAATS and the PSATS in
Sec.3. In Sec.4-6, we study their photon number distribution (PND), purity,
Wigner function (WF), respectively. Our conclusions are given in the last
section.

##  II Introduction of amplified TS

As shown in Fig.1(a), we introduce another amplified TS (ATS) by considering
the effect of $g^{\hat{n}}$ on the TS. Making simple analysis (see Appendix
A), we prove that the operator $g^{\hat{n}}$ can map
$\rho_{th}\left(\bar{n}\right)$ into a new TS $\rho_{th}\left(\bar{N}\right)$,
i.e.

$g^{\hat{n}}\text{:
}\rho_{th}\left(\bar{n}\right)\Longrightarrow\rho_{th}\left(\bar{N}\right),$
(2)

in the condition of $g^{2}\bar{n}/(\bar{n}+1)<1$ and with new MPN $\bar{N}$.
Here, we name this new TS as noiselessly amplified TS (ATS) yielding

$\rho_{th}\left(\bar{N}\right)=\frac{1}{N_{0}}g^{\hat{n}}\rho_{th}\left(\bar{n}\right)g^{\hat{n}},$
(3)

with $N_{0}=[1-\bar{n}\left(g^{2}-1\right)]^{-1}$ and
$\bar{N}=N_{0}g^{2}\bar{n}$. Obviously, if $g=1$, leading to
$\bar{N}=\bar{n}$, then $\rho_{th}\left(\bar{N}\right)$ will reduce to
$\rho_{th}\left(\bar{n}\right)$. But if $g^{2}\bar{n}/(\bar{n}+1)>1$, it is
impossible to form a quantum state.

It is necessary to emphasize that the parameters ($g$ and $\bar{n}$) must be
chosen rationally because the operator $g^{\hat{n}}$ is unbounded. In
Fig.1(b), we plot the possible regions with $\bar{N}>0$ and $\bar{N}<0$ in the
($\bar{n}$, $g$) parameter space, where the boundary line is satisfied
$g=\sqrt{(\bar{n}+1)/\bar{n}}$. We find that: (i) Only in the condition of
$1<g<\sqrt{(\bar{n}+1)/\bar{n}}$, a physical state can be generated due to
$\bar{N}>0$. (ii) But in the condition of $g>\sqrt{(\bar{n}+1)/\bar{n}}$, it
will be unphysical because of $\bar{N}<0$.

To further demonstrate Fig.1(b), we plot Fig.1(c) and Fig.1(d) by setting
several parameters. In Fig.1(c), we plot $\bar{N}$ as a function of $g$ with
$\bar{n}=1.1$ and $\bar{n}=1.5$. There is a critical value
$g_{c}=\sqrt{(\bar{n}+1)/\bar{n}}$ for each $\bar{n}$, with $\bar{N}>0$ for
$1<g<g_{c}$ and $\bar{N}<0$ for $g>g_{c}$. In the critical value $g=g_{c}$,
$\bar{N}$ will be limited to positive infinity or negative infinity. In
Fig.1(d), we plot $\bar{N}$ as a function of $\bar{n}$ with $g=1.06$ and
$\bar{n}=1.08$. Similarly, there is a critical value
$\bar{n}_{c}=1/\left(g^{2}-1\right)$ for each $g$, with $\bar{N}>0$ for
$0<\bar{n}<\bar{n}_{c}$ and $\bar{N}<0$ for $\bar{n}>\bar{n}_{c}$. In the
critical value $\bar{n}=\bar{n}_{c}$, $\bar{N}$ will be limited to positive
infinity or negative infinity.

Figure 1: (a) Conceptual scheme of generating ATS; (b) Regional plot of
$\bar{N}>0$ and $\bar{N}<0$ in ($\bar{n},g$) space, where the boundary line is
constrained by $g=\sqrt{(\bar{n}+1)/\bar{n}}$; (c) $\bar{N}$ versus $g$ with
$\bar{n}=1.1$, $1.5$ with respective $g_{c}\doteq 1.3817$, $1.2910$; (d)
$\bar{N}$ versus $\bar{n}$ with $g=1.06$, $1.08$ with respective
$\bar{n}_{c}\doteq 8.0906$, $6.0096$.

##  III Introduction of PAATS and PSATS

Based on the ATS, we further introduce photon-added ATS (PAATS) and photon-
subtracted ATS (PSATS) by considering the effects of $\hat{a}^{{\dagger}m}$
and $\hat{a}^{m}$.

PAATS: As shown in Fig.2(a), applying $\hat{a}^{{\dagger}m}$ then
$g^{\hat{n}}$ (or $g^{\hat{n}}$ then $\hat{a}^{{\dagger}m}$) on
$\rho_{th}\left(\bar{n}\right)$ yields the photon-added amplified TS (PAATS)

$\displaystyle\rho_{m+}$ $\displaystyle=$
$\displaystyle\frac{1}{N_{m+}^{(1)}}g^{\hat{n}}\hat{a}^{{\dagger}m}\rho_{th}\left(\bar{n}\right)\hat{a}^{m}g^{\hat{n}}$
(4) $\displaystyle=$
$\displaystyle\frac{1}{N_{m+}^{(2)}}\hat{a}^{{\dagger}m}g^{\hat{n}}\rho_{th}\left(\bar{n}\right)g^{\hat{n}}\hat{a}^{m}$
$\displaystyle=$
$\displaystyle\frac{1}{N_{m+}}\hat{a}^{{\dagger}m}\rho_{th}\left(\bar{N}\right)\hat{a}^{m},$

with the normalization coefficients $N_{m+}=m!\left(\bar{N}+1\right)^{m}$,
$N_{m+}^{(1)}=g^{-2m}N_{0}N_{m+}$, and $N_{m+}^{(2)}=N_{0}N_{m+}$. In the
second step, we have used
$g^{\hat{n}}\hat{a}^{{\dagger}m}=g^{m}\hat{a}^{{\dagger}m}g^{\hat{n}}$. In
particularly, $\rho_{0+}$ is just $\rho_{th}\left(\bar{N}\right)$.

PSATS: As shown in Fig.2(b), applying $\hat{a}^{m}$ then $g^{\hat{n}}$ (or
$g^{\hat{n}}$ then $\hat{a}^{m}$) on $\rho_{th}\left(\bar{n}\right)$ yields
the photon-subtracted amplified TS (PSATS)

$\displaystyle\rho_{m-}$ $\displaystyle=$
$\displaystyle\frac{1}{N_{m-}^{(1)}}g^{\hat{n}}\hat{a}^{m}\rho_{th}\left(\bar{n}\right)\hat{a}^{{\dagger}m}g^{\hat{n}}$
(5) $\displaystyle=$
$\displaystyle\frac{1}{N_{m-}^{(2)}}\hat{a}^{m}g^{\hat{n}}\rho_{th}\left(\bar{n}\right)g^{\hat{n}}\hat{a}^{{\dagger}m}$
$\displaystyle=$
$\displaystyle\frac{1}{N_{m-}}\hat{a}^{m}\rho_{th}\left(\bar{N}\right)\hat{a}^{{\dagger}m},$

with the normalization coefficients $N_{m-}=m!\bar{N}^{m}$,
$N_{m-}^{(1)}=g^{2m}N_{0}N_{m-}$, and $N_{m-}^{(2)}=N_{0}N_{m-}$. In the
second step, we have used
$g^{\hat{n}}\hat{a}^{m}=g^{-m}\hat{a}^{m}g^{\hat{n}}$. In particularly,
$\rho_{0-}$ is just $\rho_{th}\left(\bar{N}\right)$.

Figure 2: (a) Conceptual schemes of generating PAATSs (b) Conceptual schemes
of generating PSATSs.

##  IV Photon number distributions of considered states

After analyzing the density matrices for all above states (including TS, ATS,
PAATS and PSATS), we immediately know that their non-diagonal elements are
zero (i.e., $\rho_{kl}=0$ for $k\neq l$). So all these states can be expanded
into the following compound Poisson distribution

$\rho=\sum_{k=0}^{\infty}\rho_{kk}\left|k\right\rangle\left\langle k\right|,$
(6)

where the value of $\rho_{kk}$ denotes the probability of finding the Fock
state $\left|k\right\rangle$ in the state $\rho$. In addition, the
$\rho_{kk}$s for $\rho_{th}\left(\bar{n}\right)$,
$\rho_{th}\left(\bar{N}\right)$, $\rho_{m+}$, and $\rho_{m-}$ can be expressed
as

$\displaystyle\rho_{kk\left(\bar{n}\right)}^{\left(0\right)}$ $\displaystyle=$
$\displaystyle\frac{\bar{n}^{k}}{\left(\bar{n}+1\right)^{k+1}},$
$\displaystyle\rho_{kk\left(\bar{N}\right)}^{\left(0\right)}$ $\displaystyle=$
$\displaystyle\frac{\bar{N}^{k}}{\left(\bar{N}+1\right)^{k+1}},$
$\displaystyle\rho_{kk}^{\left(m+\right)}$ $\displaystyle=$
$\displaystyle\frac{\bar{N}^{\left(k-m\right)}k!}{\left(\bar{N}+1\right)^{k+1}m!\left(k-m\right)!},$
$\displaystyle\rho_{kk}^{\left(m-\right)}$ $\displaystyle=$
$\displaystyle\frac{\bar{N}^{k}\left(m+k\right)!}{\left(\bar{N}+1\right)^{k+m+1}m!k!},$
(7)

respectively. Therefore, we just need to explore the photon number
distributions (PNDs) for these quantum states.

Taking $\bar{n}=1.5$ with different $g=1.05$, $1.1$, and $1.2$, we plot the
PNDs for $\rho_{0+}$, $\rho_{1+}$, $\rho_{3+}$, and $\rho_{5+}$ in Fig.3 and
for $\rho_{0-}$, $\rho_{1-}$, $\rho_{3-}$, and $\rho_{5-}$ in Fig.4. As the
parameter $g$ increases, the PND of $\rho_{th}\left(\bar{N}\right)$ become
broader (see column 1 in Fig.3 and column 1 Fig.4). Obviously, photon-addition
or photon-subtraction can change the PNDs of the ATS, but not simply shift its
PNDs. Compared Fig.3 with Fig.4, we find that: (i) the components including
$\left|0\right\rangle$, $\left|1\right\rangle$, $\cdots$,
$\left|m-1\right\rangle$ are absent for $\rho_{m+}$, i.e.,
$\rho_{kk}^{(m+)}=0$ ($k=0,\cdots,m-1$); (ii) the PNDs for $\rho_{m+}$ and
$\rho_{m-}$ are the same apart from a shift, i.e.,
$\rho_{\left(k+m\right)\left(k+m\right)}^{(m+)}=\rho_{kk}^{(m-)}$. In other
word, the PNDs of $\rho_{m+}$ and $\rho_{m-}$ have the same shape, but
$\rho_{kk}^{(m-)}$ starts at $k=0$ and $\rho_{kk}^{(m+)}$ starts at $k=m$. In
Table I, we list some probabilities for $\rho_{th}\left(\bar{n}\right)$,
$\rho_{th}\left(\bar{N}\right)$, $\rho_{3+}$, and $\rho_{3-}$ and see
$\rho_{00}^{(3+)}=\rho_{11}^{(3+)}=\rho_{22}^{(3+)}=0$,
$\rho_{33}^{(3+)}=\rho_{00}^{(3-)}$, $\rho_{44}^{(3+)}=\rho_{11}^{(3-)}$,
and$\ \rho_{55}^{(3+)}=\rho_{22}^{(3-)}$.

Figure 3: PNDs for PAATSs $\rho_{0+}$ (column 1), $\rho_{1+}$ (column 2), $\rho_{3+}$ (column 3), $\rho_{5+}$ (column 4) with $g=1.05$ (row 1), $g=1.1$ (row 2), $g=1.2$ (row 3) and fixed $\bar{n}=1.5$. Figure 4: PNDs for PSATSs $\rho_{0-}$ (column 1), $\rho_{1-}$ (column 2), $\rho_{3-}$ (column 3), $\rho_{5-}$ (column 4) with $g=1.05$ (row 1), $g=1.1$ (row 2), $g=1.2$ (row 3) and fixed $\bar{n}=1.5$. Table 1: Some probability values for $\rho_{th}\left(\bar{n}\right)$, $\rho_{th}\left(\bar{N}\right)$, $\rho_{3+}$, and $\rho_{3-}$ with $\bar{n}=1.5$ and $g=1.2$. Notice: to compare the values of $\rho_{3+}$ with those of $\rho_{3-}$. PND | $\rho_{th}\left(\bar{n}\right)$ | $\rho_{th}\left(\bar{N}\right)$ | $\rho_{3+}$ | $\rho_{3-}$
---|---|---|---|---
$\rho_{00}$ | $0.4$ | $0.136$ | $0$ | $0.000342102$
$\rho_{11}$ | $0.24$ | $0.1175$ | $0$ | $0.0011823$
$\rho_{22}$ | $0.144$ | $0.1015$ | $0$ | $0.00255378$
$\rho_{33}$ | $0.0864$ | $0.08772$ | $0.000342102$ | $0.00441293$
$\rho_{44}$ | $0.05184$ | $0.07579$ | $0.0011823$ | $0.00667235$
$\rho_{55}$ | $0.031104$ | $0.06548$ | $0.00255378$ | $0.00922385$
$\vdots$ | $\vdots$ | $\vdots$ | $\vdots$ | $\vdots$

##  V Purities of considered states

The purity is defined as $P=\mathrm{Tr}\left(\rho^{2}\right)$ for a quantum
state $\rho$. Thus, we can obtain the purity

$P_{m+}=\frac{\bar{N}^{2m}}{\left(2\bar{N}+1\right)^{2m+1}}\left.{}_{2}F_{1}\right.(-m,-m;1,\frac{\left(\bar{N}+1\right)^{2}}{\bar{N}^{2}})$
(8)

for $\rho_{m+}$ and the purity

$P_{m-}=\frac{\left(\bar{N}+1\right)^{2m}}{\left(2\bar{N}+1\right)^{2m+1}}\left.{}_{2}F_{1}\right.(-m,-m;1,\frac{\bar{N}^{2}}{\left(\bar{N}+1\right)^{2}})$
(9)

for $\rho_{m-}$. Here $\left.{}_{2}F_{1}\right.(a,b;c,d)$ denotes a
hypergeometric function40 . In particularly, we have
$P_{0+}=P_{0-}=1/\left(2\bar{n}+1\right)$ for $\rho_{th}\left(\bar{n}\right)$
and $P_{0+}=P_{0-}=1/\left(2\bar{N}+1\right)$ for
$\rho_{th}\left(\bar{N}\right)$.

Figure 5 depicts the purities ($P_{m+}$/$P_{m-}$) as functions of $\bar{N}$
for different $m$. The bigger $m$ is, the smaller the purities
($P_{m+}$/$P_{m-}$) are. It was surprising to see that the identity
$P_{m+}\equiv P_{m-}$ is hold for the same $\bar{N}$. Then the question is:
why are these two purities equal? Indeed, we can link $P$ with $\rho_{kk}$
with the rule

$P=\sum_{k=0}^{\infty}\rho_{kk}^{2}.$ (10)

for our considered states. That is to say, the purity for each of these state
is the sum of squares of all its diagonal elements41 . Thus, we immediately
understand the reason that $P_{m+}$ is always equal to $P_{m-}$ for the same
$m$.

Figure 5: Purities ($P_{m+}$/$P_{m-}$) as functions of $\bar{N}$ with
$m=0,1,3,5$.

##  VI Wigner functions of considered states

Gaussian or non-Gaussian state can be judged from its WF. The Wigner
negativity (WN) is an indicator of nonclassicality of quantum state42 .
Gaussian state has Gaussian WF without WN. Non-Gaussian state has non-Gaussian
WF with or without WN. The WF for a quantum state $\rho$ is defined by

$W_{\rho}\left(\beta\right)=\mathrm{Tr}\left(\rho
O_{w}\left(\beta\right)\right)$ (11)

with
$O_{w}\left(\beta\right)=\frac{2}{\pi}:e^{-2\left(a^{{\dagger}}-\beta^{\ast}\right)\left(a-\beta\right)}:$
and $\beta=(x+iy)/\sqrt{2}$43 . Corresponding to Fig.3 and Fig.4, we plot
their WFs in Fig.6 and Fig.7, respectively.

WFs for $\rho_{0+}$ and $\rho_{0-}$ (i.e., $\rho_{th}\left(\bar{N}\right)$)
have the common Gaussian form with

$W_{\rho_{th}\left(\bar{N}\right)}\left(\beta\right)=\frac{2}{\pi\left(2\bar{N}+1\right)}e^{-\frac{2}{2\bar{N}+1}\left|\beta\right|^{2}}.$
(12)

which is positive everywhere in the phase space and peaks at the center. As
the factor $g$ increases, the distribution will become broader, which can be
shown in column 1 in Fig.6 and column 1 in Fig.7. Naturally, the distribution
of the ATS is broader than that of the TS.

WFs for $\rho_{m+}$ and $\rho_{m-}$ ($m>0$) have the respective non-Gaussian
forms with

$\displaystyle W_{\rho_{m+}}\left(\beta\right)$ $\displaystyle=$
$\displaystyle
T_{ng}^{(m+)}W_{\rho_{th}\left(\bar{N}\right)}\left(\beta\right),$
$\displaystyle W_{\rho_{m-}}\left(\beta\right)$ $\displaystyle=$
$\displaystyle
T_{ng}^{(m-)}W_{\rho_{th}\left(\bar{N}\right)}\left(\beta\right),$ (13)

with respective non-Gaussian terms $T_{ng}^{(m+)}$ and $T_{ng}^{(m-)}$.
Obviously, the PAATS and the PSATS have ring-shaped non-Gaussian
distributions. These results demonstrate that the non-Gaussian operations
($\hat{a}^{{\dagger}m}$ and $\hat{a}^{m}$) can render Gaussian states into
non-Gaussian states. From Fig.6, we see that the PAATSs are non-Gaussian
states with WNs. From Fig.7, we see that the PSATSs are non-Gaussian states
without WNs. For example, we have

$\displaystyle T_{ng}^{(1+)}$ $\displaystyle=$
$\displaystyle\frac{4\left(\bar{N}+1\right)}{\left(2\bar{N}+1\right)^{2}}\left|\beta\right|^{2}-\frac{1}{2\bar{N}+1},$
$\displaystyle T_{ng}^{(1-)}$ $\displaystyle=$
$\displaystyle\frac{4\bar{N}}{\left(2\bar{N}+1\right)^{2}}\left|\beta\right|^{2}+\frac{1}{2\bar{N}+1}.$
(14)

There exists WN for $\rho_{1+}$ if $T_{ng}^{(1+)}<0$, i.e. in the condition of
$\left|\beta\right|<\sqrt{\left(2\bar{N}+1\right)/\left(\bar{N}+1\right)}/2$.
But it is impossible to have WN for $\rho_{1-}$ because of $T_{ng}^{(1-)}>0$
for any $\beta$. Similar results are applicable to cases $m\geq 2$. In order
to see WN clearly in Fig.6, we plot $W\left(x,y=0\right)$ for $\rho_{m+}$ as a
function of $x$ in Fig.8. From Fig.8, we see that: 1) the bigger the number
$m$ is, the smaller the WN is; 2) the bigger the proper $g$ value is, the
smaller the WN is. The minimum WF values are on or below the order of
$10^{-2}$, $10^{-3}$, and $10^{-4}$ for cases $m=1,3,5$, respectively.

Figure 6: Wigner functions of PAATSs $\rho_{0+}$ (column 1), $\rho_{1+}$
(column 2), $\rho_{3+}$ (column 3), $\rho_{5+}$ (column 4) with $g=1.05$ (row
1), $g=1.1$ (row 2), $g=1.2$ (row 3) and fixed $\bar{n}=1.5$. Figure 7: Wigner
functions of PSATSs $\rho_{0-}$ (column 1), $\rho_{1-}$ (column 2),
$\rho_{3-}$ (column 3), $\rho_{5-}$ (column 4) with $g=1.05$ (row 1), $g=1.1$
(row 2), $g=1.2$ (row 3) and fixed $\bar{n}=1.5$. Figure 8: Sections for part
WFs in Fig.6, for $\rho_{m+}$ with (a) $m=1$, (b) $m=3$, (c) $m=5$. For each
figure, we take $\bar{n}=1.5$ and $g=1.05$ (black dotdashed), $1.1$ (blue
dashed), $1.2$ (red solid). The main character is to show the WNs.

##  VII Conclusion

In summary, we have introduced the ATS by acting $g^{\hat{n}}$ on the TS and
found that the ATS only exist in a certain range of relative parameters
(including $g$ and $\bar{n}$). Based on the ATS, we have further introduced
the PAATS and the PSATS by operating $\hat{a}^{{\dagger}m}$ and $\hat{a}^{m}$
on the ATS, respectively. The behaviors of their PNDs, purities, and Wigner
functions are studied by adjusting the interaction parameters ($g$, $\bar{n}$
and $m$).

Our results show that: (1) All our considered states have the compound
Poissonian PNDs. (2) The PAATS and the PSATS with same $m$ have the same
purities. (3) The TS and the ATS are Gaussian states without WNs. (4) The
PAATSs are non-Gaussian states with WNs. (5) The PSATSs are non-Gaussian
states without WNs. Our analysis in the work will be a feasible theoretical
reference for experimenters.

Appendix A: Some information of the ATS

Operating $g^{\hat{n}}$ on $\rho_{th}\left(\bar{n}\right)$ and using
$g^{\hat{n}}\left|n\right\rangle=g^{n}\left|n\right\rangle$, we obtain the
following relation

$\frac{1}{N_{0}}g^{\hat{n}}\rho_{th}\left(\bar{n}\right)g^{\hat{n}}=\frac{1}{N_{0}}\sum_{n=0}^{\infty}\frac{\left(g^{2}\bar{n}\right)^{n}}{(\bar{n}+1)^{n+1}}\left|n\right\rangle\left\langle
n\right|$ (A1)

with an undetermined coefficient $N_{0}$. In order to form a physical state,
we must ensure
$\mathrm{Tr}[\frac{1}{N_{0}}g^{\hat{n}}\rho_{th}\left(\bar{n}\right)g^{\hat{n}}]=1$
and then have

$\frac{1}{N_{0}(\bar{n}+1)}\sum_{n=0}^{\infty}(\frac{g^{2}\bar{n}}{\bar{n}+1})^{n}=1.$
(A2)

Obviously, if and only if $g^{2}\bar{n}/(\bar{n}+1)<1$, the series
$\sum_{n=0}^{\infty}(\dfrac{g^{2}\bar{n}}{\bar{n}+1})^{n}$ is convergent.
Thus, we can obtain $N_{0}=1/[1-\left(g^{2}-1\right)\bar{n}]$ and reconstruct
Eq.(A1) as

$\frac{1}{N_{0}}g^{\hat{n}}\rho_{th}\left(\bar{n}\right)g^{\hat{n}}=\frac{1}{\bar{N}+1}\sum_{n=0}^{\infty}\dfrac{\bar{N}^{n}}{(\bar{N}+1)^{n}}\left|n\right\rangle\left\langle
n\right|$ (A3)

with $\bar{N}=N_{0}g^{2}\bar{n}$. So the proof of Eq.(3) is completed.

On the other hand, it is impossible to form a quantum state if
$g^{2}\bar{n}/(\bar{n}+1)>1$ because the series
$\sum_{n=0}^{\infty}(\frac{g^{2}\bar{n}}{\bar{n}+1})^{n}$ is divergent.

Appendix B: Some information of the PAATS and the PSATS

(1) Normal ordering forms

Owe to $\hat{n}\left|n\right\rangle=n\left|n\right\rangle$, we know
$g^{\hat{n}}\left|n\right\rangle=g^{n}\left|n\right\rangle$ and
$g^{\hat{n}}\left|\alpha\right\rangle=e^{(g^{2}-1)\left|\alpha\right|^{2}/2}\left|g\alpha\right\rangle$.
According to formula
$e^{\lambda\hat{a}^{{\dagger}}\hat{a}}=:e^{(e^{\lambda}-1)\hat{a}^{{\dagger}}\hat{a}}:$,
we know $g^{\hat{n}}=e^{(\ln
g)\hat{a}^{{\dagger}}\hat{a}}=:e^{(g-1)\hat{a}^{{\dagger}}\hat{a}}:$ and then
obtain

$\displaystyle\rho_{m+}$
$\displaystyle=\frac{1}{(\bar{N}+1)N_{m+}}\partial_{\mu_{1}}^{m}\partial_{\nu_{1}}^{m}$
$\displaystyle:e^{-\frac{1}{\bar{N}+1}a^{{\dagger}}a+\mu_{1}a^{{\dagger}}+\nu_{1}a}:|_{\mu_{1}=\nu_{1}=0},$
(B1)

and

$\displaystyle\rho_{m-}$
$\displaystyle=\frac{1}{(\bar{N}+1)N_{m-}}\partial_{\mu_{2}}^{m}\partial_{\nu_{2}}^{m}e^{\mu_{2}\nu_{2}}$
$\displaystyle:e^{-\frac{1}{\bar{N}+1}a^{{\dagger}}a+\frac{\bar{N}}{\bar{N}+1}(\mu_{2}a^{{\dagger}}+\nu_{2}a)}:|_{\mu_{2}=\nu_{2}=0}.$
(B2)

in normal ordering form.

(2) Density matrix elements

For state $\rho_{m+}$, we have

$\displaystyle\rho_{kl}^{(m+)}$
$\displaystyle=\left\langle\left|l\right\rangle\left\langle
k\right|\right\rangle_{\rho_{m+}}$
$\displaystyle=\frac{1}{\left(\bar{N}+1\right)\sqrt{k!l!}N_{m+}}\partial_{\mu_{1}}^{m}\partial_{\nu_{1}}^{m}\partial_{f}^{k}\partial_{h}^{l}$
$\displaystyle
e^{f\mu_{1}+h\nu_{1}+\frac{\bar{N}}{\bar{N}+1}fh}|_{\mu_{1},\nu_{1},f,h=0}.$
(B3)

For state $\rho_{m-}$, we have

$\displaystyle\rho_{kl}^{(m-)}$
$\displaystyle=\left\langle\left|l\right\rangle\left\langle
k\right|\right\rangle_{\rho_{m-}}$
$\displaystyle=\frac{1}{\left(\bar{N}+1\right)\sqrt{k!l!}N_{m-}}\partial_{\mu_{2}}^{m}\partial_{\nu_{2}}^{m}\partial_{f}^{k}\partial_{h}^{l}$
$\displaystyle
e^{\frac{\bar{N}}{\bar{N}+1}\left(\mu_{2}\nu_{2}+\nu_{2}h+\mu_{2}f+fh\right)}|_{\mu_{2},\nu_{2},f,h=0}.$
(B4)

(3) Wigner functions

For state $\rho_{m+}$, we have

$\displaystyle W_{\rho_{m+}}\left(\beta\right)$ $\displaystyle=\left\langle
O_{w}\left(\beta\right)\right\rangle_{\rho_{m+}}$
$\displaystyle=\frac{2}{\pi\left(2\bar{N}+1\right)N_{m+}}e^{-\frac{2}{2\bar{N}+1}\left|\beta\right|^{2}}$
$\displaystyle\partial_{\mu_{1}}^{m}\partial_{\nu_{1}}^{m}$ $\displaystyle
e^{\frac{2\left(\bar{N}+1\right)}{2\bar{N}+1}(\beta^{\ast}\mu_{1}+\beta\nu_{1}-\frac{\mu_{1}\nu_{1}}{2})}|_{\mu_{1},\nu_{1}=0}.$
(B5)

For state $\rho_{m-}$, we have

$\displaystyle W_{\rho_{m-}}\left(\beta\right)$ $\displaystyle=\left\langle
O_{w}\left(\beta\right)\right\rangle_{\rho_{m-}}$
$\displaystyle=\frac{2}{\pi\left(2\bar{N}+1\right)N_{m-}}e^{-\frac{2}{2\bar{N}+1}\left|\beta\right|^{2}}$
$\displaystyle\partial_{\mu_{2}}^{m}\partial_{\nu_{2}}^{m}e^{\frac{2\bar{N}}{2\bar{N}+1}(\beta^{\ast}\mu_{2}+\beta\nu_{2}+\frac{\mu_{2}\nu_{2}}{2})}|_{\mu_{2},\nu_{2}=0}.$
(B6)

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# Is the Coleman de Luccia action minimum?: AdS/CFT approach

Naritaka Oshita1,2,3 Yutaro Shoji4 Masahide Yamaguchi5,6 1Center for
Gravitational Physics, Yukawa Institute for Theoretical Physics, Kyoto
University, Kitashirakawa Oiwakecho, Sakyo-ku, Kyoto 606-8502, Japan  2The
Hakubi Center for Advanced Research, Kyoto University, Yoshida Ushinomiyacho,
Sakyo-ku, Kyoto 606-8501, Japan  3RIKEN iTHEMS, Wako, Saitama, 351-0198, Japan
4Racah Institute of Physics, Hebrew University of Jerusalem, Jerusalem 91904,
Israel  5Cosmology, Gravity and Astroparticle Physics Group,
Center for Theoretical Physics of the Universe, Institute for Basic Science
(IBS), Daejeon, 34126, Korea  6Department of Physics, Tokyo Institute of
Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo 152-8551, Japan

###### Abstract

We use the anti-de Sitter/conformal field theory (AdS/CFT) correspondence to
find the least bounce action in an AdS false vacuum state, i.e., the most
probable decay process of the metastable AdS vacuum within the Euclidean
formalism by Callan and Coleman. It was shown that the $O(4)$ symmetric bounce
solution leads to the action minimum in the absence of gravity, but it is non-
trivial in the presence of gravity. The AdS/CFT duality is used to evade the
difficulties particular to a metastable gravitational system, such as the
problems of negative modes and unbounded action. To this end, we show that the
Fubini bounce solution in CFT, corresponding to the Coleman de Luccia bounce
in AdS, gives the least action among all finite bounce solutions in a
conformal scalar field theory. Thus, we prove that the Coleman de Luccia
action is the least action when (i) the background is AdS, (ii) the AdS radii,
$L_{+}$ and $L_{-}$, in the false and true vacua, respectively, satisfy
$L_{+}/L_{-}\simeq 1$, and (iii) a metastable potential gives a thin-wall
bounce much larger than the AdS radii.

††preprint: YITP-23-99, RIKEN-iTHEMS-Report-23

##  I Introduction

The vacuum decay process can be important both in the early and later
Universe. In the early universe, vacuum decay may lead to the graceful exit of
the open inflation Gott (1982); Gott and Statler (1984); Sasaki et al. (1993).
In the later Universe, the possible Higgs metastability Degrassi et al.
(2012); Buttazzo et al. (2013), predicted in the particle physics, would
eventually lead to the nucleation of a negative-energy vacuum bubble and
destroy the structure of the present Universe. Also, the string theory
predicts the existence of many vacuum states with various values of
cosmological constants, which is known as the string landscape Susskind
(2003). In the picture of landscape, a universe could have various
cosmological constants by experiencing vacuum decay.

To quantify the decay rate $\Gamma$, we consider the Euclidean path integral
under the semi-classical approximation and obtain $\Gamma=Ae^{-B}$ from the
bounce solution Coleman (1977); Callan and Coleman (1977), where $A$ is a pre-
factor and $B$ is the on-shell Eucldiean action of the bounce. The pre-factor
$A$ can be estimated by the energy scale of a metastable system and the
exponent $B$ governs the order of magnitude of the decay rate. Therefore,
determining the factor $B$ is rather important to estimate the probability of
a vacuum decay. Finding the most probable process among all possible processes
is equivalent to finding the least Euclidean action among all possible bounce
solutions in the Euclidean formalism. In the absence of gravity, Coleman,
Glaser, and Martin (CGM) has proven Coleman et al. (1978) that the
$O(4)$-symmetric vacuum bubble leads to the least action under some
conditions. However, with gravity, there exist serious issues, e.g., the
negative mode problem Lavrelashvili et al. (1985); Tanaka and Sasaki (1992)
and unboundedness problem Gibbons et al. (1978), and it is non-trivial if the
maximally symmetric non-trivial solution, i.e., an $O(4)$ bounce, leads to the
minimum action in the existence of gravity. For the vacuum decay processes
that we are interested in, gravity can be strong and one cannot get rid of the
degrees of freedom of gravity from the system. In this sense, we could say
that the theory of vacuum decay has been facing the aforementioned serious
issues.

We consider how the anti-de Sitter/conformal field theory (AdS/CFT)
correspondence Maldacena (1998); Gubser et al. (1998); Witten (1998) can shed
light on the issues. We assume that the correspondence holds for a metastable
AdS and CFT, that is, there exists a one-to-one correspondence between the
partition functions of a bounce solution in AdS and CFT sides. We then find
the least action in the CFT side where gravity is absent, which infers what is
the least action in AdS side by virtue of the AdS/CFT correspondence (see
Figure 1). As mentioned, finding the least action among possible bounce
solutions in the presence of gravity is challenging, but we can use the
AdS/CFT correspondence to evade the complicated issues caused by gravity. We
will then argue that the CdL bounce would correspond to the Fubini bounce
under certain conditions. We then prove that the CdL bounce in the CFT side is
always spherically symmetric and hence it is given by the Fubini bounce.
Knowing that the spherically symmetric thin-wall bounce in the AdS side gives
the same action as the Fubini bounce in the CFT side, we conclude that the
spherically symmetric bounce gives the least action in the AdS side under
certain conditions.

This paper is organized as follow. In Sec. II, we set up the condition under
which our strategy works. We consider a metastable scalar field theory in
AdSD+1 and review a way of how to determine the corresponding CFTD with the
correct coupling constant based on Ref. Barbon and Rabinovici (2010). In Sec.
III, we prove that the Fubini bounce solution is the least action among
possible finite non-trivial solutions to a metastable conformal scalar field
theory. We then provide our conclusions in Sec. IV. Throughout the paper, we
use the natural units with $c=\hbar=1$ and $G=1$.

Figure 1: A schematic picture showing the role of the AdS/CFT correspondence
in our strategy to find the least bounce action in the presence of gravity.

##  II Correspondence between a metastable AdSD+1 and CFTD

In this section, we consider the correspondence between a metastable AdSD+1
and a metastable CFTD:

$\displaystyle S_{{\rm AdS}_{D+1}}$ $\displaystyle=\int
d^{D+1}x\sqrt{-g}\left(\frac{1}{16\pi}{\cal
R}-\frac{1}{2}g^{\mu\nu}\partial_{\mu}\psi\partial_{\nu}\psi-U(\psi)\right),$
(1) $\displaystyle S_{\text{CFT}_{D}}$ $\displaystyle=\int
d^{D}y\sqrt{-g_{\text{bdy}}}\left(-\frac{1}{2}g_{\text{bdy}}^{ij}\partial_{i}\phi\partial_{j}\phi-\frac{1}{2}\xi_{D}{\cal
R}_{\text{bdy}}\phi^{2}-\lambda\phi^{2D/(D-2)}\right),$ (2)

where $\xi_{D}\equiv\frac{D-2}{4(D-1)}$ (see e.g. Ref. Jacobson (2003)),
$U(\psi)$ is a metastable potential, and the coupling constant $\lambda$ will
be determined later but is negative in order for the CFT to be a metastable
system. Following the AdS/CFT correspondence, we assume that there is a one-
to-one correspondence between bounce solutions $\psi=\bar{\psi}$ in AdSD+1 and
$\phi=\bar{\phi}$ in CFTD such that

$Z_{{\rm AdS}_{D+1}}(\bar{\psi})=Z_{{\rm CFT}_{D}}(\bar{\phi}),$ (3)

where the left (right) hand side is the partition function of a bounce
solution nucleated in the bulk (on the boundary). As the partition functions
of the bounce and the initial false vacuum determine the transition amplitude
in the Euclidean path integral formalism, we may expect that the corresponding
bubbles in AdS and CFT sides would be nucleated with the same transition
amplitude. Such a one-to-one correspondence means that the transition
amplitude of the most probable decay process in the metastable AdS is
equivalent to that in the metastable CFT side. Our goal in this paper is to
confirm that, with this assumption, the CdL nucleation process is the most
probable process at least in the AdS background.

Given the metastable potential in AdS, $U(\psi)$, how can we determine the
coupling constant $\lambda$ in CFT? We can demonstrate the determination of
$\lambda$ from the AdS side when the $U(\psi)$ satisfies the conditions shown
below. We here consider a metastable potential $U(\psi)$ for which the
$O(D+1)$ bounce solution, i.e., the CdL solution, has a large wall with the
exterior and the interior AdS radii, $L_{+}$ and $L_{-}$, respectively, and a
potential barrier of the tension $\sigma$ such that

$0<q/\sigma-1\ll 1\ \text{and}\ L_{+}/L_{-}\simeq 1,$ (4)

where

$q\equiv\frac{(L_{+}^{2}/L_{-}^{2}-1)-L_{+}^{2}\Sigma^{2}}{16\pi L_{+}/(D-1)}\
\text{and}\ \Sigma\equiv 8\pi\sigma/(D-1).$ (5)

Here the tension of the wall is given by $\sigma\sim\sqrt{V_{\rm
top}}\Delta\phi$ where $\Delta\phi$ is the separation of the true and false
vacuum states in the field space and $V_{\rm top}$ is the height of the
potential barrier $V_{\rm top}$. The two quantity $q$ and $\sigma$ are
associated with the bulk and surface energy, respectively, and the balance
between them determines the size of the CdL bubble. In the condition (5), one
of the possible bounce solutions, the CdL solution, has the bubble radius of
$R_{0}$ which is much larger than the false AdS radius as

$R_{0}\equiv\frac{\sigma}{\sqrt{q^{2}-\sigma^{2}}}L_{+}\gg L_{+},$ (6)

where the explicit form of $R_{0}$ is derived below. For the CdL solution, all
degrees of freedom in the bulk, i.e., the thin wall or a probe brane, lives in
the vicinity of the AdS boundary, and its dynamics can be translated into that
of CFT Barbon and Rabinovici (2010). Then one can read the unknown coupling
constant $\lambda$ from the bulk side by sending the probe brane to the
vicinity of the AdS boundary at $r\gg L_{+}$111The coordinate $r$ is the
radial coordinate in the static AdS patch. and obtaining the effective action
of the probe brane in the canonical form Barbon and Rabinovici (2010). In the
following, we review the procedure of Ref. Barbon and Rabinovici (2010).

The dynamics of a thin-wall spherical bubble can be described by the Israel
junction conditions Israel (1966). It means that the effective degrees of
freedom of the bulk reduces to a scalar quantity, i.e., the radius of the
bubble. As we consider a spherical probe brane, the first Israel junction
condition is trivially satisfied and the second Israel junction condition
reduces to

$\displaystyle\frac{\sqrt{f_{+}+(dR/d\tau)^{2}}}{R}-\frac{\sqrt{f_{-}+(dR/d\tau)^{2}}}{R}=-\Sigma,$
(7) $\displaystyle f_{\pm}\equiv 1+R^{2}/L_{\pm}^{2},$ (8)

where $r=R(\tau)$ denotes the radius of the brane and $\tau$ is its proper
time. The junction condition (7) reduces to

$\displaystyle\left(\frac{dR}{d\tau}\right)^{2}+1-\frac{q^{2}-\sigma^{2}}{\sigma^{2}L_{+}^{2}}R^{2}=0.$
(9)

Note that $q>0$ should hold for the positivity of the exterior and interior
extrinsic curvatures. From (9), we find the radius at the moment of the bubble
nucleation is $R=R_{0}$, at which the potential term in (9) becomes zero, and
$R_{0}\to\infty$ for $\sigma\to q$. Using the asymptotic time, $t$, (9) can be
rewritten as

$\left(\frac{dR}{dt}\right)^{2}+f_{+}^{2}\left(\frac{f_{+}\sigma^{2}L_{+}^{2}}{q^{2}R^{2}}-1\right)=0.$
(10)

The action leading to the integrated equation of motion (10) is given by

$S_{\text{AdS}}=\int dtL=-\sigma\Omega_{D-1}\int
dtR^{D-1}\sqrt{f_{+}-\frac{\dot{R}^{2}}{f_{+}}}+\frac{q}{L_{+}}\Omega_{D-1}\int
dtR^{D},$ (11)

where $\Omega_{D-1}$ denotes the area of the $(D-1)$-dimensional unit sphere
and a dot denotes the derivative with respect to $t$. One can show that the
action (11) indeed derives the integrated equation of motion (10) by computing
the Hamiltonian (total energy) of the bubble $E$ as

$\displaystyle E$ $\displaystyle=\dot{R}\frac{\partial
L}{\partial\dot{R}}-L=\frac{\sigma\Omega_{D-1}R^{D-1}f_{+}}{\sqrt{f_{+}-\dot{R}^{2}/f_{+}}}-\frac{q}{L_{+}}\Omega_{D-1}R^{D},$
(12)

and setting $E=0$ as the total energy of the nucleated bubble is zero, one
finds that (12) reduces to (10). In the following, we obtain the translation
of $R(t)\to\phi(t)$, by which the action $(\ref{ads_system})$ reduces to the
canonical form of

$L=L_{+}^{D-1}\Omega_{D-1}\left(\frac{1}{2}\dot{\phi}^{2}-V_{\text{AdS}}(\phi)+{\cal
O}(\dot{\phi}^{4})\right),$ (13)

in the non-relativistic situation $\dot{R}\ll 1$. This is the case when the
bubble is nucleated with a small velocity222Indeed, the bubble is nucleated
with $\dot{R}=0$ in the CdL formalism.. In this procedure, we can read the
$\lambda$ in CFT from the bulk side. Expanding $L(\dot{R},R)$ in (11) with
respect to $\dot{R}$ and comparing it with (13), one can read

$\phi=L_{+}^{\frac{4-D}{2}}\frac{2\sqrt{\sigma}}{D-2}R^{(D-2)/2}(1+{\cal
O}(L_{+}^{2}/R^{2})),$ (14)

and substituting this relation and $\dot{R}=0$ in (12), one finds

$\displaystyle
V_{\text{AdS}}(\phi)=\sigma\,(R(\phi)/L_{+})^{D-1}\sqrt{f_{+}(R(\phi))}-q\,(R(\phi)/L_{+})^{D}\simeq\frac{(D-2)^{2}}{8}(\phi/L_{+})^{2}+\lambda_{\text{AdS}}\phi^{2D/(D-2)},$
(15)

for $R\gg L_{+}$, where

$\lambda_{\text{AdS}}\equiv-\left(\frac{D-2}{2}\right)^{\frac{2D}{D-2}}\frac{1}{\sigma^{\frac{2}{D-2}}}\frac{q-\sigma}{\sigma}\frac{1}{L_{+}^{2D/(D-2)}}.$
(16)

The potential term (15) reduces to

$V_{\text{AdS}}(\phi)\simeq\frac{1}{2}\xi_{\rm D}{\cal
R}_{\text{bdy}}\phi^{2}+\lambda_{\text{AdS}}\phi^{2D/(D-2)},$ (17)

as the Ricci scalar is ${\cal R}_{\text{bdy}}=(D-1)(D-2)/L_{+}^{2}$ on the AdS
boundary whose topology is ${\bf R}\times{\bf S}^{D-1}$. Identifying $\lambda$
in (2) with $\lambda_{\rm AdS}$, we obtain the CFT action corresponding to the
metastable AdS satisfying the condition of (4). Let us consider the
correspondence in the Wick rotated space ($t\to-it_{\rm E}$) and perform the
conformal transformation leading to ${\bf R}\times{\bf S}^{D-1}\to{\bf
R}^{D}$. The latter procedure is possible as

$dt_{\rm E}^{2}+d\Omega_{D-1}=\frac{1}{u^{2}}(du^{2}+u^{2}d\Omega_{D-1}^{2}),$
(18)

where $u\equiv\exp(t_{\rm E})$ and the factor $1/u^{2}$ is the conformal
factor of the tranformation. Performing the conformal transformation, ${\cal
R}_{\rm bdy}$ vanishes and (17) becomes

$V_{\text{AdS}}(\phi)\simeq V_{\rm
CFT}(\phi)\equiv\lambda_{\text{AdS}}\phi^{2D/(D-2)}.$ (19)

Then, the $D$-dimensional Fubini bounce Fubini (1976) becomes a solution to
the equations of motion for (2) with ${\cal R}_{\rm bdy}=0$. Here, the Fubini
bounce is given by

$\phi=\left(\frac{2}{|\lambda|}\right)^{1/2}\left(\frac{\Delta
b}{x^{2}+b^{2}}\right)^{\Delta},$ (20)

where $\Delta=(D-2)/2$ is the mass dimension of a scalar field $\phi$ and $b$
is an arbitrary constant determining the size of the bounce. As the conformal
transformation does not affect the action, the original CFT of (2) also admits
the on-shell Fubini action. Remarkably, the Fubini action with
$\lambda=\lambda_{\rm AdS}$

$S_{\rm Fubini}=\frac{c}{\lambda_{\rm AdS}^{\Delta}},\
c\equiv\frac{\Omega_{D-1}}{2^{\Delta}}\Delta^{D}B\left(\frac{3}{2},\frac{D-2}{2}\right),$
(21)

is equivalent to the Coleman de Luccia action in the limit of $L_{+}/L_{-}\to
1$ as it has the form of

$S_{\rm AdS}=\frac{L_{+}}{L_{-}}S_{\rm Fubini},$ (22)

for $q/\sigma\to 1$. The exterior and interior AdS radii, $L_{+}$ and $L_{-}$,
respectively, should satisfies $L_{+}/L_{-}-1\sim 1/N\ll 1$ for the AdS/CFT
correspondence to be valid, where $N$ is a large integer. In the context of
AdS/CFT, the $N$ is the number of branes, and for $N\gg 1$, the spacetime near
the branes is approximated with AdS spacetime. Also, a nucleated bubble can be
regarded as bundled $n$ branes with $n\ll N$ Barbon and Rabinovici (2010).

In the following section, we show that the Fubini bounce leads to the least
bounce action among all finite bounce solutions by extending the theorem on
the minimum action proven by Coleman, Glaser, and Martin Coleman et al. (1978)
to cover a scalar CFT (see Sec. III). Based on the relation of (22), we argue
that the CdL bounce action gives the most probable transition amplitude among
all possible processes, at least, in our setup.

##  III Most probable decay process in the metastable CFT

In this section, we will prove that the Fubini bounce gives the least
Euclidean action of the metastable CFT. To this end, we extend the theorem
proven by Coleman, Glaser, and Martin Coleman et al. (1978) (hereinafter, we
refer it as the CGM theorem). In the former part of this section, we will
briefly review the CGM theorem, and in the latter part, we will extend the CGM
theorem so that it applies to the Fubini bounce.

###  III.1 CGM theorem

CGM has shown that there exists at least one non-trivial solution to the
differential equation

$\nabla^{2}\phi-V^{\prime}(\phi)=0,$ (23)

and the solution leading to the lowest action is spherically symmetric and
monotone if $V(\phi)$ is admissible. Here, $\nabla^{2}$ is the Laplacian in
the $D$ dimensional Euclidean space $\\{x_{1},x_{2},...,x_{D}\\}$, and
$V(\phi)$ is said to be admissible if i) $V$ is continuously differentiable
for all $\phi$, ii) $V(0)=V^{\prime}(0)=0$, iii) $V$ is somewhere negative,
and iv) there exist positive numbers $a$, $b$, $\alpha$, and $\beta$ such that

$\alpha<\beta<2D/(D-2),$ (24)

with

$V-a|\phi|^{\alpha}+b|\phi|^{\beta}\geq 0.$ (25)

Notice that this is not the case for the potential of (19) as we need
$\beta=2D/(D-2)$ and $a=0$ to satisfy the inequality.

The main theorem proven by CGM is:

######  The CGM Theorem.

In $D$-dimensional Euclidean space with $D>2$, for any admissible $V$, the
equation of motion (23) has at least one monotone spherical solution vanishing
at infinity, other than the trivial solution of $\phi=0$. Furthermore, this
solution has Euclidean action,

$S=\int d^{D}x\left[\frac{1}{2}(\partial\phi)^{2}+V(\phi)\right],$ (26)

less than or equal to that of any other solution vanishing at infinity. If the
other solution is not both spherical and monotone, the action is strictly less
than that of the other solution.

Before proving the theorem, CGM defined the reduced problem as follows.

######  Definition.

“The reduced problem” is the problem of finding a function vanishing at
infinity which minimizes $T$ for some fixed negative $W$, where

$T[\phi]\equiv\int d^{D}x\frac{1}{2}(\partial\phi)^{2},\ W[\phi]\equiv\int
d^{D}xV(\phi).$ (27)

It is equivalently stated as the problem to minimize the scale-invariant
ratio,

$X[\phi]=-\frac{(T[\phi])^{D/(D-2)}}{W[\phi]},$ (28)

with negative $W$.

The CGM theorem is proven by showing that the following theorems hold.

######  Theorem A.

If a solution of the reduced problem exists, then, for an appropriate value of
$W$, it is a solution of (23) that has an action less than or equal to that of
any non-trivial solution of (23).

######  Theorem B.

There exists at least one solution to the reduced problem. All solutions to
the reduced problem are spherically symmetric and monotone.

The proof of Theorem B is composed of a sequence of statements with short
proofs. CGM start from an infinite minimizing sequence, $\\{\phi_{n}\\}$,
$n\in{\mathbb{Z}}_{+}$, such that

$\lim_{n\to\infty}T[\phi_{n}]=\inf_{\phi}T[\phi],$ (29)

with a fixed negative $W$. The sequence is chosen so that $\phi_{n}$ is
differentiable and has compact support, and $T[\phi_{n}]$ is finite. Notice
that such a choice of the sequence is always possible. Then, CGM has proven
the following statements.

  1. (a).

[CGM Statement 4] There exists a sequence of spherical and monotone functions,
$\\{\phi_{n}^{\rm sph}\\}$, such that

$X[\phi_{n}^{\rm sph}]\leq X[\phi_{n}],$ (30)

for all $n\in I$. Here, $I$ is an infinite subset of ${\mathbb{Z}}_{+}$.

  2. (b).

[CGM pp.220-221] There is no non-spherical or non-monotone function that has
the same $R$ as the spherical monotone rearrangement of the original function.

  3. (c).

[CGM Statement 6] There exist an infinite subsequence, $\\{\Phi_{n}\\}$, of
$\\{\phi^{\rm sph}_{n}\\}$ and a bounded continuous function, $\Phi(r)$, such
that

$\lim_{n\to\infty}\Phi_{n}(r)=\Phi(r),$ (31)

pointwise for all $r\in(0,\infty)$ and uniformly on any finite closed interval
in $(0,\infty)$.

  4. (d).

[CGM Statement 6] $\Phi$ satisfies

$\lim_{r\to\infty}\Phi(r)=0.$ (32)

  5. (e).

[CGM Statement 8] $\Phi$ satisfies

$W[\Phi]<0.$ (33)

  6. (f).

[CGM Statement 10] $\Phi$ satisfies

$X[\Phi]=\lim_{n\to\infty}X[\Phi_{n}].$ (34)

Here, (a) and (b) show that the solution to the reduced problem is always
spherically symmetric and monotone, and (c)-(f) show the sequence converges to
the actual minimum of $X$ satisfying $\lim_{r\to\infty}\Phi(r)=0$ and
$W[\Phi]<0$. Hence, these statements prove Theorem B.

The other statements of CGM are used to prove Statements 4, 6, 8 and 10 shown
above. The dependencies of the statements are summarized in Appendix A.

###  III.2 Extension of the CGM Theorem

We consider non-trivial solutions to the differential equation (23) with the
potential333Addition of dimensionful terms, such as a mass term in $D=4$,
potentially results in non-existence of the bounce as discussed in Frishman
and Yankielowicz (1979); Affleck (1981); Nielsen and Nielsen (2000); Di Luzio
et al. (2016). of

$V=\lambda\phi^{\gamma},$ (35)

where $\gamma=2D/(D-2)$ and $\lambda$ is a negative constant. The theorem we
here prove is described below.

######  The Main Theorem.

In $D$-dimensional Euclidean space with $D>2$, the equation of motion (23)
with the potential of (35) has at least one monotone spherical solution
vanishing at infinity, other than the trivial solution of $\phi=0$.
Furthermore, the solution has the Euclidean action (26), which is less than or
equal to that of any other solution vanishing at infinity. If the other
solution is not both spherical and monotone, the action is strictly less than
that of the other solution.

To prove the Main Theorem, we show that Theorem A and B holds in our setup.
Theorem A has been proven without the condition of (25) and thus our main
focus is on Theorem B. As we have mentioned, Theorem B has been proven by
showing (b) and Statements 4, 6, 8 and 10. As summarized in Appendix A, (b)
and Statements 4 and 6 hold independently of (25), and Statement 10 follows
from Statement 8. However, the proof of Statement 8 by CGM depends on (25) and
does not apply to our case. This can be understood in the following way. Since
$V(\phi)<0$ for any $\phi\neq 0$, there is a possibility that a sequence of
$\Phi_{n}$ having a fixed negative $W$ converges to $\Phi$ that is zero almost
everywhere. If such a sequence exists, we obtain $W[\Phi]=0$ although
$W[\Phi_{n}]<0$ for all $n$, which contradicts Statement 8. In fact, we can
construct such a sequence utilizing the scale invariance of the theory. (One
can see that any value of $b$ in (20) gives the same bounce action, which is
the consequence of the scale invariance.) For any sequence that converges to
the Fubini bounce, we can execute the scale transformation at each step of $n$
so that the new sequence converges to the Fubini bounce with $b\to 0$ or
$b\to\infty$, which is zero almost everywhere.

Let us move on to the proof of the Main Theorem. Since the proof of Statement
4 by CGM applies to our setup, there exists a minimizing sequence,
$\\{\phi_{n}\\}$, such that $\phi_{n}$ is spherically symmetric and monotone
for all $n$. Hereafter, we write the functions in terms of the radius from the
center, $r=e^{y}$.

We prove the following propositions.

######  Proposition 1.

There exists a minimizing sequence of spherically symmetric monotone
functions, $\\{\Phi_{n}\\}$, such that (i)
$f_{n}(y)=\Phi_{n}(e^{y})e^{\frac{D-2}{2}y}$ is symmetric under $y\to-y$ and
monotone for $y>0$ for all $n$, and (ii) there exists a bounded continuous
function, $f(y)$, such that

$\lim_{n\to\infty}f_{n}(y)=f(y),$ (36)

pointwise for all $y$ and uniformly on any finite interval.

######  Proposition 2 (Statement 8’).

For the minimizing sequence of the preceeding proposition,

$W[\Phi]=W[\Phi_{n}].$ (37)

Since the scale transformation corresponds to the translation in $y$ space,
Proposition 1 excludes the sequences that converge to the Fubini bounce with
$b\to 0$ or $b\to\infty$. Then, Proposition 2 replaces the Statement 8 of CGM.
Once Proposition 2 is proven, Statement 10 of CGM immediately follows from
Proposition 2 and Statement 9 of CGM, which completes the proof of Theorem B
and the Main Theorem.

######  Definition (Spherical rearrangement).

Let $F(x)$ be a non-negative measurable function on ${\mathbb{R}}^{d}$ $(d\geq
1)$ that vanishes at infinity. A spherically symmetric monotone function,
$F^{\rm sph}(r)$, is obtained by symmetrizing $F(x)$ around
$r\equiv\sqrt{x_{1}^{2}+x_{2}^{2}+...+x_{d}^{2}}=0$ keeping

${\cal A}\left\\{x|F^{\rm sph}(r)\geq M\right\\}={\cal A}\left\\{x|F(x)\geq
M\right\\},$ (38)

for any positive value $M$. Here, ${\cal A}$ is the Lebesgue measure. Then,
$F^{\rm sph}(r)$ is said to be a spherical rearrangement of $F(x)$. (See
Figure 2.)

The spherical rearrangement has the following properties.

$\displaystyle||F^{\rm sph}||_{L^{p}}$ $\displaystyle=||F||_{L^{p}},$ (39)
$\displaystyle||\nabla F^{\rm sph}||_{L^{p}}$ $\displaystyle\leq||\nabla
F||_{L^{p}},$ (40)

where $||*||_{L^{p}}$ is the $L^{p}$-norm with $1\leq p<\infty$.

Figure 2: A schematic picture of the spherical rearrangement $F^{\rm sph}(x)$.
The area of the level set of $F=M_{i}$ ($i=1,2,3$) is equal to that of $F^{\rm
sph}=M_{i}$.

###### proof of Prop 1.

With $\tilde{f}_{n}=\phi_{n}(e^{y})e^{\frac{D-2}{2}y}$, $W$ and $T$ can be
rewritten as

$\displaystyle W[\tilde{f}_{n}]$
$\displaystyle=\Omega_{D-1}\int_{-\infty}^{\infty}dy\lambda\tilde{f}_{n}^{\gamma},$
(41) $\displaystyle T[\tilde{f}_{n}]$
$\displaystyle=\Omega_{D-1}\int_{-\infty}^{\infty}dy\left[\frac{1}{2}\tilde{f}^{\prime
2}_{n}+\frac{(D-2)^{2}}{8}\tilde{f}_{n}^{2}\right].$ (42)

Let $\tilde{f}_{n}^{\rm sph}$ be the spherical rearrangement of
$\tilde{f}_{n}$ in $y$ space. Then, from (39) and (40),

$\displaystyle W[\tilde{f}_{n}^{\rm sph}]=W[\tilde{f}_{n}],$ (43)
$\displaystyle T[\tilde{f}_{n}^{\rm sph}]\leq T[\tilde{f}_{n}].$ (44)

Thus, there exists a subsequence of $\\{\tilde{f}_{n}^{\rm sph}\\}$ that is a
minimizing sequence satisfying property (i). Then, the sequence with property
(ii) is obtained by applying Statement 6 of CGM to this sequence. ∎

###### proof of Prop 2.

Statement 5 (B) of CGM holds in our setup: there exists $M>0$ such that

$\int_{0}^{\infty}dyf_{n}^{2}(y)\leq M.$ (45)

Let us take an arbitrary $y_{1}>0$. Since $|f_{n}(y)|$ is a non-negative
monotonically decreasing function for $y>0$, we have

$\frac{|W|}{2\Omega_{D-1}|\lambda|}\geq\int_{0}^{y_{1}}dy|f_{n}(y)|^{\gamma}\geq|f_{n}(y_{1})|^{\gamma}y_{1},$
(46)

Since $W[f_{n}]$ is independent of $n$, $|f_{n}(y_{1})|$ is uniformly bounded
as

$|f_{n}(y_{1})|\leq\left(\frac{|W|}{2\Omega_{D-1}|\lambda|y_{1}}\right)^{1/\gamma}.$
(47)

From (45) and (47), it follows that

$\displaystyle\int_{y_{1}}^{\infty}dy|f_{n}(y)|^{\gamma}$
$\displaystyle\leq|f_{n}(y_{1})|^{\gamma-2}\int_{y_{1}}^{\infty}dyf_{n}^{2}(y)$
$\displaystyle\leq\left(\frac{|W|}{2\Omega_{D-1}|\lambda|y_{1}}\right)^{2/D}M.$
(48)

Thus, this integral converges to zero uniformly as $y_{1}\to\infty$.

Since $\lim_{n\to\infty}f_{n}$ converges to $f$ uniformly on $y\in[0,y_{1}]$,
we have

$\lim_{n\to\infty}\int_{0}^{y_{1}}dyf_{n}^{\gamma}(y)=\int_{0}^{y_{1}}dyf^{\gamma}(y).$
(49)

From (48) and (49), it follows that

$W[\Phi]=W[\Phi_{n}].$ (50)

∎

We now know that the Fubini bounce solution, which is the spherical monotone
bounce solution of the conformal scalar field theory, gives the least action
among all finite non-trivial solutions.

##  IV Discussion and conclusion

We here showed that the Fubini bounce is a solution leading to a minimum of
the Euclidean action among all finite solutions to a conformal scalar field
theory. Our proof is based on the proof in Ref. Coleman et al. (1978), which
did not cover the conformal scalar field theory.

The action of a metastable conformal scalar field theory, corresponding to a
metastable AdS, is uniquely determined except for the value of the coupling
constant $\lambda$. Assuming the one-to-one correspondence between the
partition function of a metastable AdS bulk and that of the corresponding
metastable CFT, we consider a case where the coupling constant of the CFT can
be determined with $\lambda=\lambda_{\rm AdS}$ from the dynamics of a large
bubble in the AdS. Such a special situation is realized when a metastable
effective potential of the bulk theory admits the nucleation of the CdL bubble
in the vicinity of the AdS boundary.

It was shown Barbon and Rabinovici (2010) that the Fubini solution with the
coupling constant $\lambda_{\rm AdS}$ has the Euclidean action equivalent to
the CdL action when $L_{+}/L_{-}\simeq 1$. We then conclude that the CdL
action is the action minima among all possible finite solutions under the
conditions we have discussed, provided that there exists the one-to-one
correspondence between the metastable AdS and CFT.

The situation we investigated is restrictive and many open issues are
remained: i) How does our procedure work for a case where a small CdL bubble
is nucleated? ii) Is the CdL action minimum when the difference between the
exterior and interior cosmological constants is not negligible? (i.e.,
$L_{+}/L_{-}-1\gtrsim 1$) iii) Is it possible to generalize our procedure to
the case of Minkowski or de Sitter background? Beyond the Euclidean path
integral technique, it is non-trivial that finding a solution leading to the
least Euclidean action corresponds to finding the most probable process in
vacuum decay. In Ref. Oshita et al. (2023); Shoji (2022), the authors indeed
considered another scheme, the polychronic tunneling (or mixed tunneling),
where the Euclidean and Lorentzian evolution coexist during a phase
transition. The authors then found that the polychronic tunneling is more
probable than the CdL process at least when the false vacuum is almost
Minkowskian.

The Euclidean path integral in the existence of gravity has many open issues
that are challenging to address, e.g., unbounded-ness of the action, negative
mode problems, and the choice of vacuum state. As such, it is totally non-
trivial if the CdL action is one of the action minima among all possible
bounce solutions. Our strategy to attack this problem is based on the one-to-
one correspondence between the theory including gravity (AdS) and one without
gravity (CFT). As mentioned, this is the first step towards solving the
problem. As the vacuum decay process with strong gravity plays an important
role in cosmology, the aforementioned open issues are very important. We would
also expect that the AdS/CFT correspondence and other possible dualities
(e.g., dS/CFT correspondence Strominger (2001)) could be helpful to understand
vacuum decay.

##  Appendix A The Structure of CGM Statements

The CGM statements 4, 6, 8 and 10 are proven by other statements of CGM, and
(b) [CGM pp.220-221] is an independent statement proven without (25). Here, we
summarize the dependencies of the CGM statements.

  * •

Statement 4

    * –

Statement 3

  * •

Statement 6

    * –

Statement 5 (C)

    * –

Statement 5 (D)

  * •

Statement 8 $\clubsuit$

    * –

Statement 7 $\clubsuit$

      * *

Statement 5 (D)

      * *

Statement 5 (F) $\clubsuit$

      * *

Statement 6

  * •

Statement 10 $\clubsuit$

    * –

Statement 8 $\clubsuit$

    * –

Statement 9

The parent bullet depends on the child bullets, and $\clubsuit$ indicates that
the statement requires (25). For the details of each statement, see Coleman et
al. (1978).

Statement 5 (C), (D) and (F) further depend on other statements as follows.

  * •

Statement 5 (C)

    * –

Statement 5 (A)

  * •

Statement 5 (D)

    * –

Statement 5 (A)

    * –

Statement 5 (B)

  * •

Statement 5 (F) $\clubsuit$

    * –

Statement 2 $\clubsuit$

      * *

Statement 1 $\clubsuit$

    * –

Statement 5 (E)

      * *

Statement 5 (B)

      * *

Statement 5 (D)

###### Acknowledgements.

N.O. is supported by Grant-in-Aid for Scientific Research (KAKENHI) project
for FY2023 (23K13111). Y.S. is supported by the US-Israeli Binational Science
Foundation (grant No. 2020220) and the Israel Science Foundation (grant No.
1818/22). M.Y. is supported by IBS under the project code, IBS-R018-D3, and by
JSPS Grant-in-Aid for Scientific Research Number JP21H01080.

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# Wild quantum dilogarithm identities

Markus Reineke

###### Abstract.

We exhibit and discuss “wild” analogues of the five-term quantum dilogarithm
identity. We derive these from the representation theory of quivers, using
motivic wall-crossing, the geometricity of motivic Donaldson–Thomas
invariants, and special properties of Kronecker moduli.

##  1\. Introduction

The quantum dilogarithm is a $q$-series with many remarkable properties [16],
including the famous five-term identity [2]. Cluster algebra theory and wall-
crossing of motivic invariants of quivers have led to vast generalizations of
such dilogarithm identities [4].
In this note, we explore the outer limits of this circle of ideas, by
investigating “wild” dilogarithm identities, those arising from wild quivers.
For this we use again motivic wall-crossing, interpret the resulting series in
terms of motivic Donaldson–Thomas invariants, and use the geometric
interpretation of the latter in terms of intersection homology of quiver
moduli spaces [7] to establish very strong positivity properties. In the rank
two case, originating from generalized Kronecker quivers, the well-explored
and very special symmetries of Kronecker moduli yield many additional explicit
properties of the individual terms of our highly infinite dilogarithm
identities; see Theorem 2.6.
To derive this identity, we collect in Section 3 the available material on
wall-crossing of motivic invariants (see [8] for an introduction), and adapt
it to the present notation and special setting in Section 4. Although similar
approaches are used, for example, in the context of the tropical vertex [3,
12], it it desirable to state the nature of such wild identities as explicitly
as possible.
Acknowldegments: The author would like to thank Daping Weng for several
discussions on these wild identities, Bernhard Keller for the opportunity to
present them in his seminar, Vladimir Fock as well as Sergey Mozgovoy for
valuable comments, and Timm Peerenboom for carefully reading a draft of this
text and suggesting several improvements.

##  2\. Quantum dilogarithm identities

We first define the quantum dilogarithm, and state the classical five-term
identity. The coefficient ring $\mathbb{Q}(q^{1/2})[[x]]$, and in particular
the twist by half-powers of $q$, will become natural in the context of motivic
Donaldson–Thomas invariants.

######  Definition 2.1.

We define the quantum dilogarithm $\Phi(x)\in\mathbb{Q}(q^{1/2})[[x]]$ as

$\Phi(x)=\sum_{n\geq 0}\frac{q^{n/2}x^{n}}{(1-q)\cdot\ldots\cdot(1-q^{n})}=$
$=\exp\left(\sum_{n\geq
1}\frac{x^{n}}{n\cdot(q^{-n/2}-q^{n/2})}\right)=\prod_{n\geq
0}\frac{1}{1-q^{n+1/2}\cdot x}.$

Remark: The translation from the present definition to the ones in the
literature is straightforward. For example, [2, 14] use the definition
$(x;q)_{\infty}=\Phi(q^{-1/2}x)^{-1}$, and [4] uses $\mathbb{E}(x)=\Phi(-qx)$.
The classical Euler dilogarithm [16] arises as the following limit:

$(q^{-1/2}-q^{1/2})\log\Phi(x)=\sum_{n\geq
1}\frac{x^{n}\cdot(q^{-1/2}-q^{1/2})}{n\cdot(q^{-n/2}-q^{n/2})}\stackrel{{\scriptstyle
q\rightarrow 1}}{{\longrightarrow}}\sum_{n\geq 1}\frac{x^{n}}{n^{2}}={\rm
Li}_{2}(x)$

######  Definition 2.2.

For a positive integer $m$, we define $\mathbb{Q}(q^{1/2})_{q^{m}}[[x,y]]$ as
the skew formal power series ring with skew commutativity relation
$xy=q^{m}yx$.

We can now formulate the five-term quantum dilogarithm identity:

######  Theorem 2.3 (Schützenberger, Fadeev, Kashaev, Volkov).

In $\mathbb{Q}(q^{1/2})_{q}[[x,y]]$, we have

$\Phi(x)\Phi(y)=\Phi(y)\Phi(-q^{-1/2}xy)\Phi(x)$

For $m=2$, we will obtain the following identity as a special case of Theorem
2.6.

######  Theorem 2.4.

In $\mathbb{Q}(q^{1/2})_{q^{2}}[[x,y]]$, we have

$\Phi(x)\Phi(y)=\Phi(y)\Phi(q^{-2}xy^{2})\Phi(q^{-6}x^{2}y^{3})\Phi(q^{-12}x^{3}y^{4})\ldots\cdot$
$\cdot\Phi(q^{-1/2}xy)^{-1}\Phi(q^{-3/2}xy)^{-1}\cdot$
$\cdot\ldots\Phi(q^{-12}x^{4}y^{3})\Phi(q^{-6}x^{3}y^{2})\Phi(q^{-2}x^{2}y)\Phi(x).$

To make such identities more readable, we will now introduce a shorthand
notation, which again will be motivated later by the framework of
Donaldson–Thomas invariants:

######  Definition 2.5.

For $a,b\geq 0$, define

$\Phi_{(a,b)}=\Phi({(-1)}^{mab}{q}^{(a^{2}+b^{2}-2mab-1)/2}x^{a}y^{b})^{{(-1)}^{a^{2}+b^{2}-mab-1}}.$

More generally, for a Laurent polynomial
$P(q)=\sum_{k}c_{k}(-q^{1/2})^{k}\in\mathbb{Q}[q^{\pm 1/2}]$, define

$\Phi_{(a,b)}^{{\circ
P}}={\prod_{k}}\Phi((-1)^{mab}q^{(a^{2}+b^{2}-2mab-1{+k})/2}x^{a}y^{b})^{(-1)^{a^{2}+b^{2}-mab-1{+k}}\cdot{c_{k}}}.$

Then the previous identities simplify to

$\Phi_{(1,0)}\Phi_{(0,1)}=\Phi_{(0,1)}\Phi_{(1,1)}\Phi_{(1,0)}\mbox{ for
}m=1,$
$\Phi_{(1,0)}\Phi_{(0,1)}=\Phi_{(0,1)}\Phi_{(1,2)}\Phi_{(2,3)}\ldots\cdot\Phi_{(1,1)}^{\circ(q+1)}\cdot\ldots\Phi_{(3,2)}\Phi_{(2,1)}\Phi_{(1,0)}$

for $m=2$.
In the case $m\geq 3$, we can no longer give explicit identities, but we will
obtain a rather complete qualitative description.

To formulate this main result, we need two more definitions. We denote by
$\sigma$ the operator on $\mathbb{Z}^{2}$ given by $\sigma(a,b)=(b,mb-a)$,
which generates an infinite dihedral group together with the involution
$(a,b)\mapsto(b,a)$. We also define $\mu_{\pm}={(m\pm\sqrt{m^{2}-4})}/{2}$,
the two roots of the quadratic equation $x^{2}-mx+1=0$.

######  Theorem 2.6.

In $\mathbb{Q}(q^{1/2})_{q^{m}}[[x,y]]$ for $m\geq 3$, we have

$\Phi_{(1,0)}\Phi_{(0,1)}=\Phi_{(0,1)}\Phi_{\sigma(0,1)}\Phi_{\sigma^{2}(0,1)}\Phi_{\sigma^{3}(0,1)}\cdot\ldots\cdot$
$\cdot\prod^{\rightarrow}_{{\mu_{-}\leq a/b\leq\mu_{+}}\atop{\mbox{\tiny
increasing}}}\Phi_{(a,b)}^{\circ P_{(a,b)}}\cdot$
$\cdot\ldots\Phi_{\sigma^{-3}(1,0)}\Phi_{\sigma^{-2}(1,0)}\Phi_{\sigma^{-1}(1,0)}\Phi_{(1,0)},$

where the $P_{(a,b)}$ satisfy the following properties:

  1. (1)

“Wildness”/Completeness: We have the non-vanishing property

$P_{(a,b)}\not=0\mbox{ for all }\mu_{-}\leq a/b\leq\mu_{+}.$

  2. (2)

Dihedral symmetry: We have the symmetries

$P_{(b,a)}=P_{(a,b)},\;\;\;P_{\sigma(a,b)}=P_{(a,b)}.$

  3. (3)

Positivity: We have

$P_{(a,b)}\in\mathbb{N}[q],\mbox{ of degree }d=mab-a^{2}-b^{2}+1>0.$

  4. (4)

Unimodality: The polynomial $P_{(a,b)}=\sum_{k}c_{k}q^{k}$ is palindromic and
unimodal:

$c_{d-k}=c_{k}\mbox{ and }1=c_{0}\leq c_{1}\leq\ldots\geq c_{d-1}\geq
c_{d}=1.$

  5. (5)

Lowest order terms:

$P_{(1,k)}=\left[{m\atop k}\right]_{q}.$

  6. (6)

Special value:

$P_{(k,k)}(1)=\frac{1}{(m-2)k^{2}}\sum_{d|k}\mu(\frac{k}{d})(-1)^{md+1}{{(m-1)^{2}d-1}\choose{d}}.$

As a consequence of the dihedral symmetry property, we see that all
$P_{(a,b)}$ are determined by those for $a\leq b\leq\frac{m}{2}a$. All
properties will follow from interpreting the $P_{(a,b)}$ as the Poincaré
polynomials in intersection homology of Kronecker moduli [1], a class of
projective varieties parametrizing certain tuples of matrices up to base
change.

##  3\. Quiver setup

In this section, we recall the necessary terminology on quiver representations
and their moduli spaces, and we formulate the main ingredients for general
quiver dilogarithm identities, namely the motivic wall-crossing formula, and
the definition and geometricity of motivic Donaldson–Thomas invariants. The
reader is referred to [8] for a detailed introduction into motivic wall-
crossing for quivers, and to [11] for a short summary.
Let $Q$ be a finite acyclic quiver. We order the set of vertices
$Q_{0}=\\{i_{1},\ldots,i_{n}\\}$ such that $i_{k}\rightarrow i_{l}$ implies
$k>l$. The Euler form of $Q$ is the (in general non-symmetric) bilinear form
on $\mathbb{Z}Q_{0}$ given by $\langle\mathbf{d},\mathbf{e}\rangle=\sum_{i\in
Q_{0}}d_{i}e_{i}-\sum_{\alpha:i\rightarrow j}d_{i}e_{j}$ for
$\mathbf{d},\mathbf{e}\in\mathbb{Z}Q_{0}$.
We define the formal quantum affine space $\mathbb{Q}(q^{1/2})_{q}[[Q_{0}]]$
as the skew formal power series ring with topological basis $t^{\mathbf{d}}$
for $\mathbf{d}\in\mathbb{N}Q_{0}$ and multiplication twisted by the
antisymmetrized Euler form

$t^{\bf d}\cdot t^{\bf e}=(-q^{1/2})^{\langle{\bf d},{\bf
e}\rangle-\langle{\bf e},{\bf d}\rangle}t^{{\bf d}+{\bf e}}.$

We fix linear functions $\Theta,\kappa\in(\mathbb{Z}Q_{0})^{*}$ on $Q$ such
that $\kappa({\bf d})>0$ for ${\bf d}\in\mathbb{N}Q_{0}\setminus 0$, and
consider the associated slope function $\mu({\bf d})=\frac{\Theta({\bf
d})}{\kappa({\bf d})}$ for $\mathbf{d}\in\mathbb{N}Q_{0}\setminus 0$. We
denote by $\Lambda_{a}$ the set of all ${\bf d}\in\mathbb{N}Q_{0}\setminus 0$
of slope $a\in\mathbb{Q}$.
We define the Grothendieck ring of varieties $K_{0}({\rm Var}_{\mathbb{C}})$
as the free abelian group in isomorphism classes of complex algebraic
varieties $X$ modulo the “cut-and-paste” relation ${}[X]=[A]+[U]\mbox{ if
}A\subset X\mbox{ closed, }U=X\setminus A,$ with product given by
$[X]\cdot[Y]=[X\times Y]$. We abbreviate the Lefschetz motive
$[\mathbb{A}^{1}]=:q$.
We consider the localization $R=K_{0}({\rm Var}_{\mathbb{C}})[q^{\pm
1/2},(1-q^{n})^{-1}:n\geq 1]$, and define the formal motivic affine space
$R_{q}[[Q_{0}]]$ as above, with $R$ replacing the coefficient ring
$\mathbb{Q}(q^{1/2})$. In fact, all our computations will happen in the
smaller coefficient ring of motives which are rational functions in $q$. Note
that the existence of motivic measures such as the virtual Hodge polynomial
shows that this subring is isomorphic to a subring of $\mathbb{Q}(q^{1/2})$.
Given a dimension vector ${\bf d}\in\mathbb{N}Q_{0}$, we fix
$\mathbb{C}$-vector spaces $V_{i}$ of dimension $d_{i}$ for $i\in Q_{0}$. We
consider the base change action

$\prod_{i\in Q_{0}}{\rm GL}(V_{i})=G_{\bf d}\curvearrowright R_{\bf
d}(Q)=\bigoplus_{\alpha:i\rightarrow j}{\rm Hom}(V_{i},V_{j})$

given by

$(g_{i})_{i}\cdot(f_{\alpha})_{\alpha}=(g_{j}f_{\alpha}g_{i}^{-1})_{\alpha:i\rightarrow
j},$

whose orbits, by definition, correspond bijectively to the isomorphism classes
of complex representations of $Q$ of dimension vector $\mathbf{d}$.
We denote by $R_{\bf d}^{\mu-{\rm sst}}(Q)\subset R_{\bf d}(Q)$ the open
subset of $\mu$-semistable points. Using this notation, we can formulate the
motivic wall-crossing formula, which is formally equivalent to the existence
of the Harder–Narasimhan filtration [9]:

######  Theorem 3.1 (Motivic wall-crossing formula).

In $R_{q}[[Q_{0}]]$, we have the identity

$\Phi(t_{1})\cdot\ldots\cdot\Phi(t_{n})=\sum_{{\bf d}}(-q^{1/2})^{\langle{\bf
d},{\bf d}\rangle}\frac{[R_{\bf d}(Q)]}{[G_{\bf d}]}t^{\bf d}=$
$=\prod^{\rightarrow}_{a\in\mathbb{Q}\mbox{ \tiny
decreasing}}\left(1+\sum_{{\bf d}\in\Lambda_{a}}(-q^{1/2})^{\langle{\bf
d},{\bf d}\rangle}\frac{[R_{\bf d}^{\mu-{\rm sst}}(Q)]}{[G_{\bf d}]}t^{\bf
d}\right).$

Next, we can define motivic Donaldson–Thomas invariants. We generalize the
shorthand notation of the previous section to

$\Phi(x)^{\circ P(q)}=\prod_{k}\Phi(q^{k/2}x)^{(-1)^{k}c_{k}}$

for $P(q)=\sum_{k}c_{k}(-q^{1/2})^{k}$.

######  Definition 3.2.

[6] Assume that $\langle\\_,\\_\rangle$ is symmetric on $\Lambda_{a}$. Define
${\rm DT}^{\mu}_{\bf d}(q)\in\mathbb{Q}[q^{\pm 1/2}]$ by factorization in
$R_{q}[[Q_{0}]]$:

$1+\sum_{{\bf d}\in\Lambda_{a}}(-q^{1/2})^{\langle{\bf d},{\bf
d}\rangle}\frac{[R_{\bf d}^{\mu-{\rm sst}}(Q)]}{[G_{\bf d}]}t^{\bf
d}=\prod_{{\bf d}\in\Lambda_{a}}\Phi(t^{\bf d})^{\circ{\rm DT}^{\mu}_{\bf
d}(q)}.$

The ${\rm DT}^{\mu}_{\bf d}(q)$ are called the motivic Donaldson–Thomas
invariants of the quiver $Q$ with stability $\mu$.

We remark that the more common definition in terms of the plethystic
exponential ${\rm Exp}$ is equivalent to this one since, by definition,
$\Phi(x)={\rm Exp}(\frac{x}{q^{-1/2}-q^{1/2}})$.
The motivic Donaldson–Thomas invariants admit a geometric interpretation in
terms of intersection homology of moduli spaces. Namely, we consider the GIT
quotient

$R_{\bf d}^{\mu-{\rm sst}}(Q)//G_{\bf d}=M_{\bf d}^{\mu-{\rm sst}}(Q),$

the moduli space of $\mu$-semistable representations of $Q$ of dimension
vector ${\bf d}$.
It is an irreducible projective normal (typically singular) complex algebraic
variety. If ${\bf d}$ is $\mu$-stable, that is, if there exists a $\mu$-stable
representation of dimension vector ${\bf d}$, then $\dim M_{\bf d}^{\mu-{\rm
sst}}(Q)=1-\langle{\bf d},{\bf d}\rangle$. In terms of this moduli space, we
have the following geometric interpretation of the motivic Donaldson–Thomas
invariants [7]:

######  Theorem 3.3 (Geometricity of DT invariants).

We have

${\rm DT}_{\bf d}^{\mu}(q)=(-q^{1/2})^{\langle{\bf d},{\bf
d}\rangle-1}\sum_{k\geq 0}\dim{{\rm IH}}^{k}(M_{\bf d}^{\mu-{\rm
sst}}(Q),\mathbb{Q})(-q^{1/2})^{k}$

if ${\bf d}$ is $\mu$-stable, and ${\rm DT}_{\bf d}^{\mu}(q)=0$ otherwise.

##  4\. Quantum dilogarithm identity for quivers

To combine the methods prepared in the previous section, we consider quivers
and stabilities such that the restriction of $\langle\\_,\\_\rangle$ is
symmetric on all $\Lambda_{a}$. In particular, this holds if the
antisymmetrized Euler form of $Q$ is determined by the stability function, in
the sense that

$\langle{\bf d},{\bf e}\rangle-\langle{\bf e},{\bf d}\rangle=\kappa({\bf
d})\Theta({\bf e})-\kappa({\bf e})\Theta({\bf d})\mbox{ for all }{\bf d},{\bf
e}\in\mathbb{Z}Q_{0},$

compare [11, Proposition 5.2]. Besides generalized Kronecker quiver, this
property holds, for example, for complete bipartite quivers.

######  Theorem 4.1.

Assume that $\langle\\_,\\_\rangle$ is symmetric on all $\Lambda_{a}$. Then we
have a factorization

$\Phi(t_{1})\cdot\ldots\cdot\Phi(t_{n})=\prod^{\rightarrow}_{\mu({\bf
d})\mbox{ \tiny decreasing}}\Phi(t^{\bf d})^{\circ(-q^{1/2})^{\langle{\bf
d},{\bf d}\rangle-1}\cdot P_{\bf d}(q)}$

in $R_{q}[[Q_{0}]]$, for polynomials $P_{\bf d}(q)$ with the following
properties:

  1. (1)

Non-vanishing: We have $P_{\bf d}(q)\not=0$ if and only if ${\bf d}$ is
$\mu$-stable,

  2. (2)

Positivity: We have $P_{\bf d}\in\mathbb{N}[q]$, of degree $\deg P_{\bf
d}(q)=1-\langle{\bf d},{\bf d}\rangle$,

  3. (3)

Unimodality: $P_{\bf d}(q)$ is palindromic and unimodal,

  4. (4)

Simplicity: We have $P_{\bf d}(q)=1$ if, additionally, $\langle{\bf d},{\bf
d}\rangle=1$.

This is now readily proved: combining Theorem 3.1, Definition 3.2 and Theorem
3.3, we see that $P_{\bf d}(q)$ is precisely the Poincaré polynomial in
intersection homology of $M_{\bf d}^{\mu-{\rm sst}}(Q)$ in case ${\bf d}$ is
$\mu$-stable, proving the non-vanishing property. Positivity and unimodality
are proven in [7, Corollary 1.2]. The degree statement follows from the
dimension formula for the moduli space, and consequently the simplicity
statement follows.
To derive Theorem 2.6, we consider the $m$-arrow Kronecker quiver
$K_{m}=1\stackrel{{\scriptstyle(m)}}{{\leftarrow}}2$, with stability function
given by $\Theta({\bf d})=m\cdot(d_{2}-d_{1})$ and $\kappa({\bf
d})=d_{1}+d_{2}$.
We identify the variables $t_{1}=x$ and $t_{2}=y$, leading to an
identification of $\mathbb{Q}(q^{1/2})_{q}[[(K_{m})_{0}]]$ with
$\mathbb{Q}(q^{1/2})_{q^{m}}[[x,y]]$, such that $t^{\bf
d}=(-q^{1/2})^{md_{1}d_{2}}x^{d_{1}}y^{d_{2}}$ and

$\Phi(t^{{\bf d}})^{\circ(-q^{1/2})^{\langle{\bf d},{\bf d}\rangle-1}\cdot
P(q)}=\Phi_{(a,b)}^{\circ P(q)}$

for ${\bf d}=(a,b)$.
Theorem quiverdilog then provides a factorization in
$\mathbb{Q}(q^{1/2})_{q^{m}}[[x,y]]$ of the form

$\Phi_{(1,0)}\Phi_{(0,1)}=\prod^{\rightarrow}_{a/b\mbox{ \tiny
increasing}}\Phi_{(a,b)}^{\circ P_{(a,b)}},$

such that

$P_{(a,b)}(q)=\sum_{k\geq 0}\dim{\rm IH}^{2k}(K_{(a,b)},\mathbb{Q})q^{k},$

where

$K_{(a,b)}=M_{a\times b}(\mathbb{C})^{m}_{\rm sst}//{\rm
GL}_{a}(\mathbb{C})\times{\rm GL}_{b}(\mathbb{C})$

are the Kronecker moduli of [1].
All remaining properties in Theorem 2.6 now follow from properties of these
Kronecker moduli spaces. Using the analysis of [13, Section 4], we see that a
dimension vector $(a,b)$ is $\mu$-stable if and only if
$\langle(a,b),(a,b)\rangle\leq 1$. In case $\langle(a,b),(a,b)\rangle=1$, the
dimension vector is a real root, and the moduli space $K_{(a,b)}$ reduces to a
point, thus $P_{(a,b)}=1$. Otherwise, we have $\langle(a,b),(a,b)\rangle\leq
0$, which translates to $\mu_{-}\leq a/b\leq\mu_{+}$. This proves the
completeness statement of the theorem. By linear duality, we have
$K_{(a,b)}\simeq K_{(b,a)}$. Moreover, reflection functors induce isomorphisms
$K_{(a,b)}\simeq K_{(b,mb-a)}$ by [15, Proposition 4.3]. This establishes the
dihedral symmetry. The Kronecker moduli $K_{(1,k)}$ identify with the
Grassmannians ${\rm Gr}_{k}^{m}$, determining the lowest order terms in the
factorization. Finally, the special value $P_{(k,k)}(1)$ is computed in [10,
Theorem 5.2].

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