KCL-PH-TH/2023-01

# NEC violation: Tunnelling versus the Casimir effect

Jean Alexandre  Drew Backhouse  Theoretical Particle Physics and Cosmology,
King’s College London, WC2R 2LS, UK

###### Abstract

We show that tunnelling between two degenerate minima, as allowed in a finite
volume, leads to a non-extensive symmetric ground state. This results in Null
Energy Condition violation for sufficiently low temperatures, when a
continuous set of momenta in the box containing the field is assumed. Taking
into account discrete momenta can modify this picture and is achieved via the
addition of the Casimir energy to the tunnelling-induced ground state energy.
Focusing on zero-temperature, these non-trivial effects are found to compete,
depending on the typical length scales involved.

##  I Introduction

Spontaneous Symmetry Breaking (SSB) is strictly speaking valid for infinite
volumes only, where tunnelling between degenerate vacua is completely
suppressed. On the other hand, for a field confined in a box of finite volume,
tunnelling between degenerate vacua is allowed and we study here the energetic
consequences.

Involving tunnelling in the quantisation of a system automatically takes into
account the different vacua and is known to lead to a convex effective action
convexity . This is not the case in the situation of SSB, where the different
vacua are decoupled and quantisation over a single vacuum does not necessarily
lead to convexity. Taking into account several degenerate vacua in the
partition function comes with a remarkable energetic feature, generated
dynamically: the effective action is non-extensive, as was shown in
AlexTsapa13 ; AlexBack22  with a semi-classical approximation for the
partition function.

The latter works were done in an $O(4)$-symmetric Euclidean spacetime though,
and to account for a full description of tunnelling one needs a finite spatial
volume $V$ and an independent large Euclidean time $\beta$. The natural
context for these studies is therefore equilibrium field theory at a finite-
temperature $T=1/\beta$. The corresponding Quantum Mechanics study was done in
AlexPolo22 , involving a gas of instantons/anti-instantons which dominates the
partition function in the limit of small temperature. It is shown there that
the Null Energy Condition (NEC - see NECreview  for reviews) is violated, as a
consequence of a non-extensive effective action induced by tunnelling. The
present article extends this study to full 4-dimensional quantum fluctuations,
and we find that NEC violation occurs in any finite volume for sufficiently
low temperatures.

Our study does not however deal with high-temperature symmetry restoration, as
seen in the Kibble-Zureck mechanism KZ . We are instead interested in the low-
temperature regime, where tunnelling dominates over thermal fluctuations
providing an opportunity to violate the NEC, which the Kibble-Zurek mechanism
does not.

We first evaluate quantum corrections with continuous momentum for
fluctuations above each saddle point, to describe the fundamental dynamical
mechanism induced by tunnelling. We then take into account the modification
arising from discrete momentum in a finite volume, using results known from
studies of the Casimir effect (see Casimirreview  for a review). The latter is
known to be either attractive or repulsive, depending on the geometry of the
box containing the field, as well as the boundary conditions the field
satisfies on the walls of the box. As a consequence, as far as NEC violation
is concerned, the difference between discrete and continuous momentum can play
an important role.

In Section II we describe the semi-classical approximation in which the
partition function is derived, to take into account the different saddle
points which are relevant to tunnelling: static saddle points and the
instanton/anti-instanton dilute gas. Details of the calculations with
continuous momentum are given in Appendices A and B. Section III focuses on
the ground state of the effective action, with a non-extensive energy density
providing the origin of NEC violation. The maximum effect occurs at zero
temperature and is the regime in which we introduce corrections arising from
discrete momentum in Section IV via the Casimir energy. We find that
tunnelling and the Casimir effect compete when the typical size of the box
containing the field is of the order of the Compton wavelength of the
corresponding particle. For a larger box, the Casimir effect seems to be
dominant.

To summarise our results: in the low-temperature regime, the sum of energy
density $\rho$ and pressure $p$ can be written in the form

$\rho+p\simeq~{}A_{\text{finite-T}}~{}+~{}B_{\text{tunnelling}}~{}+~{}C_{\text{Casimir}}~{},$

where

  * •

the finite-temperature contribution $A$ is always positive (and vanishes
exponentially for $T\to 0$);

  * •

the tunnelling contribution $B$, calculated with continuous momentum, is
always negative (and vanishes exponentially for $V\to\infty$);

  * •

the discrete momentum correction $C$ has a sign which depends on the geometry
and topology of the finite box containing the field (and vanishes for
$V\to\infty$).

As expected, the NEC is satisfied at zero temperature and for infinite volume,
where $\rho+p=0$ for a homogeneous vacuum.

##  II Semi-classical approximation

###  II.1 Model

Consider a single real scalar field $\phi(t,\mathbf{x})$ in Euclidean space,
at finite-temperature $T=1/\beta$ and in a three-dimensional spacial volume
$V$, described by the Euclidean action

$\int_{0}^{\beta}dt\int_{V}d^{3}x\left(\frac{1}{2}(\partial\phi)^{2}+\frac{\lambda}{24}(\phi^{2}-v^{2})^{2}+j\phi\right)~{}.$
(1)

The finite volume is represented by a physical box containing the scalar
field, in which we assume continuous momenta to calculate quantum corrections.
Section IV discusses corrections arising from discrete momentum and the
boundary conditions the field satisfies at the walls of the box. Finite
temperature requires field configurations to have periodic boundary conditions
in Euclidean time and, as later discussed, has an impact on the saddle point
configurations which are allowed in the partition function.

Introducing the dimensionless variables $\tau\equiv\omega t$ and

$\omega\equiv
v\sqrt{\frac{\lambda}{6}}~{}~{},~{}~{}\varphi\equiv\sqrt{\frac{\lambda}{6}}\frac{\phi}{\omega}~{}~{},~{}~{}k\equiv\sqrt{\frac{\lambda}{6}}\frac{j}{\omega^{3}}~{},$
(2)

leads to the bare action

$\displaystyle S[\varphi]$ $\displaystyle=$ $\displaystyle\frac{\lambda
v^{4}}{12\omega}\int_{0}^{\omega\beta}d\tau\int_{V}d^{3}x\Big{(}(\varphi^{\prime})^{2}+\frac{1}{\omega^{2}}(\nabla\varphi)^{2}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\frac{1}{2}(\varphi^{2}-1)^{2}+2k\varphi\Big{)}~{},$

where a prime represents a derivative with respect to the dimensionless
Euclidean time $\tau$. As shown further in this article, the effective action
is convex and we thus focus on the true vacuum, which occurs for vanishing
source $j=0=k$. As a consequence, no bubbles of true/false vacuum can form as
they would have an infinite radius Coleman1977 . We are therefore interested
in time-dependent instantons only, beyond the static and homogeneous saddle
points. The corresponding equation of motion is then

$\varphi^{\prime\prime}-\varphi^{3}+\varphi-k=0~{},$ (4)

where the solutions to this equation, $\varphi_{i}(\tau,\mathbf{x})$, are the
saddle points of the partition function to be introduced below.

###  II.2 Static saddle points

Introducing the critical dimensionless source

$k_{c}\equiv 2/(3\sqrt{3})~{},$ (5)

allows us to distinguish two cases.

For $|k|>k_{c}$, there is only one static and homogeneous (real) solution to
the equation of motion (4), and quantisation of the theory can therefore be
based on one saddle point only, leading to the usual 1PI effective potential.

For $|k|<k_{c}$, the regime we focus on, there are two such solutions

$\displaystyle\varphi_{L}(k)$ $\displaystyle=$
$\displaystyle\frac{2}{\sqrt{3}}\cos\Big{(}\pi/3-(1/3)\arccos(k/k_{c})\Big{)}$
(6) $\displaystyle\varphi_{R}(k)$ $\displaystyle=$
$\displaystyle\frac{2}{\sqrt{3}}\cos\Big{(}\pi-(1/3)\arccos(k/k_{c})\Big{)}$
$\displaystyle=$ $\displaystyle-\varphi_{L}(-k)~{}.$

The actions for these configurations are

$\displaystyle S_{L}$ $\displaystyle\equiv$ $\displaystyle
S[\varphi_{L}(k)]=B\omega\beta\left(4k-k^{2}+{\cal O}(k^{3})\right)$ (7)
$\displaystyle S_{R}$ $\displaystyle\equiv$ $\displaystyle
S[\varphi_{R}(k)]=S[\varphi_{L}(-k)]~{},$

where

$B\equiv\frac{\lambda v^{4}V}{24\omega}~{}.$ (8)

###  II.3 Instanton/anti-instanton gas

In Euclidean time, and with the absence of a source, the motion described by
equation (4) corresponds to the motion in real-time with the upside-down
potential $V(\varphi)\equiv-(\varphi^{2}-1)/2$, for which the minimum action
$S_{\text{inst}}$ is obtained by the known solution

$\varphi_{\text{inst}}(\tau)=\pm\tanh\left(\frac{\tau-\tau_{0}}{\sqrt{2}}\right)~{},$
(9)

where $0\leq\tau_{0}\leq\omega\beta$, and

$S_{\text{inst}}\equiv S[\varphi_{\text{inst}}]=\frac{8\sqrt{2}}{3}B~{}.$ (10)

Because of finite temperature though, field configurations should be periodic
in Euclidean time, such that one needs to consider an instanton/anti-instanton
pair as the basic building block. For two distant “jumps” at $\tau_{1}$ and
$\tau_{2}$ such that $|\tau_{1}-\tau_{2}|\gg 1$, the configuration can be
approximated by AlexPolo22

$\varphi_{\text{pair}}(\tau)\simeq-\tanh\left(\frac{\tau-\tau_{1}}{\sqrt{2}}\right)\tanh\left(\frac{\tau-\tau_{2}}{\sqrt{2}}\right)~{},$
(11)

with an action exponentially close to $2S_{\text{inst}}$.

In the presence of a source, the basic building block is in principle either a
bounce or a shot (see AA  for reviews). However, since we are interested in
the limit of vanishing source and periodic boundary conditions, the
fundamental saddle point we consider behaves as the function (11). Assuming
the jumps occur over a short time in comparison to $\beta$, the
instanton/anti-instanton pair spends the same time $\beta/2$ exponentially
close to each static saddle point, resulting in an action for such a pair
$\varphi_{\text{pair}}$ of

$S_{\text{pair}}\simeq\frac{1}{2}S_{L}+\frac{1}{2}S_{R}+2S_{\text{inst}}~{}.$
(12)

Revisiting the analogy of classical mechanics in the upside-down potential
$V(\varphi)$, the other possible saddle points consist of periodic
oscillations made of $n$ instanton/anti-instanton pairs, where the value of
$n$ depends on how “exponentially close” the oscillations from a static saddle
point begins. An example of an exact saddle point is given in Fig.1(a).
Assuming the total Euclidean time $\beta$ is large enough to leave the
structure of pairs intact, the time spent close to one static saddle point is
the same as the time spent close to the other and the total action for $n$
pairs is

$S_{\text{n
pairs}}\simeq\frac{1}{2}S_{L}+\frac{1}{2}S_{R}+2nS_{\text{inst}}~{}.$ (13)

The latter “crystalline” structure, with $n$ periodic oscillations,
corresponds to an exact solution of the equation of motion. For large $\beta$,
where the average distance between instantons and anti-instantons remains
large compared to their width, a translation of each jump leaves the action
$S_{\text{inst}}$ invariant and the resulting highly degenerate “gas” of
instanton/anti-instanton pairs dominates the partition function. An example of
an approximate saddle point is given in Fig.1(b). In this “dilute gas”
approximation, the $n$ instanton/anti-instanton pair configurations spend on
average an equal time $\beta/2$ close to each static saddle point, with the
same total action (13) as for an exact $n$-pair configuration as a result of
the translational invariance of jumps.

(a) An exact saddle point configuration with 3 instanton/anti-instanton pairs
and action $S_{\text{3 pairs}}$: the oscillations are periodic.

(b) An approximate saddle point configuration with 3 instanton/anti-instanton
pairs: the jumps are randomly distributed, but the average distance between
them is larger than their width, such that they keep their shape and the
action of the configuration is also $S_{\text{3 pairs}}$.

Figure 1: Examples of exact and approximate saddle points. In the dilute gas
approximation, the difference between the corresponding actions is of order
$B\omega\beta\exp(-\omega\beta)\ll 1$, and the partition function is dominated
by the whole set of approximate saddle points.

###  II.4 Partition function

The partition function is evaluated in the semi-classical approximation via a
sum over the two static saddle points, $\varphi_{L}$ and $\varphi_{R}$, and
the dilute gas of $n$ instanton/anti-instanton pairs for all possible values
of $n$. Together, with the corresponding one-loop fluctuation factors
$F_{L,R}$ and $F_{n}$, the semi-classical approximation of the partition
function reads

$\displaystyle Z[k]$ $\displaystyle\simeq$ $\displaystyle
F_{L}(\beta)\exp(-S_{L})+F_{R}(\beta)\exp(-S_{R})$ $\displaystyle+$
$\displaystyle\sum_{n=1}^{\infty}\left(\prod_{i=1}^{2n}\int_{\tau_{i-1}}^{\omega_{R}\beta}\tau_{i}\right)F_{n}\exp(-S_{\text{n
pairs}})~{}.$

In the latter expression, the product of integrals over the times $\tau_{i}$
where the jumps occur corresponds to the zero-mode of the fluctuation factor
for the saddle point made of $n$ instanton/anti-instanton pairs. Indeed, the
translational invariance of the action means that the $i_{\text{th}}$ jump can
happen at any time $\tau_{i}\in[\tau_{i-1},\beta]$. For finite temperature
though, there is a maximum number of instanton/anti-instanton pairs, however,
the error made in the summation for $n\to\infty$ is negligible since each term
is suppressed by $\exp(-nS_{\text{inst}})$. With the fluctuation factors
derived in Appendix A and Appendix B, we can write

$\displaystyle Z[k]$ $\displaystyle=$
$\displaystyle\exp\Big{(}-\Sigma_{L}(\beta)\Big{)}+\exp\Big{(}-\Sigma_{R}(\beta)\Big{)}$
(15) $\displaystyle+\exp\Big{(}-\Sigma_{\text{gas}}(\beta)\Big{)}~{},$

where $\Sigma_{L},\Sigma_{R}$ and $\Sigma_{\text{gas}}$ are the connected
graphs generating functionals for the static saddle points and the gas of
instanton/anti-instanton pairs respectively. We note that an instanton or
anti-instanton does not lead to any imaginary part in the partition function,
unlike a bounce, since the former are monotonous functions of the Euclidean
time, such that the fluctuation operator does not have negative eigenvalues
Kleinert .

####  II.4.1 Static saddle points

One-loop quantum corrections can be split into two contributions: the zero-
temperature corrections, containing all the divergences, and the divergence
free finite-temperature dependent corrections. The zero-temperature
contribution is calculated in AlexBack22  and is expressed in terms of the
renormalised parameters. It is mentioned here that, in the case of several
saddle points and in order to avoid confusion between loop orders,
renormalisation should be done at the level of the individual connected graphs
generating functionals before performing the Legendre transform. The finite-
temperature contribution can be calculated using the Schwinger proper time
representation \- see Appendix A \- and the overall contribution is

$\displaystyle\Sigma_{L,R}(\beta)$ $\displaystyle=$ $\displaystyle
B_{r}\omega_{r}\beta\Bigg{(}(\varphi_{L,R}^{2}-1)^{2}+4k\varphi_{L,R}$
$\displaystyle+\frac{\lambda_{r}}{96\pi^{2}}(3\varphi_{L,R}^{2}-1)^{2}\ln\left(\frac{3}{2}\varphi_{L,R}^{2}-\frac{1}{2}\right)$
$\displaystyle-\frac{\lambda_{r}(3\varphi_{L,R}^{2}-1)}{3\pi^{2}}\sum_{l=1}^{\infty}\frac{K_{2}\left(l\omega_{r}\beta\sqrt{3\varphi_{L,R}^{2}-1}\right)}{(l\omega_{r}\beta)^{2}}\Bigg{)}~{}.$

In the previous expression, the renormalised parameters are

$\displaystyle\lambda_{r}$ $\displaystyle\equiv$
$\displaystyle\lambda-\frac{3\lambda^{2}}{32\pi^{2}}\log\left(\frac{\Lambda^{2}}{\lambda
v^{2}}\right)~{},$ (17) $\displaystyle v_{r}^{2}$ $\displaystyle\equiv$
$\displaystyle v^{2}-\frac{3\Lambda^{2}}{16\pi^{2}}+\frac{\lambda
v^{2}}{16\pi^{2}}\log\left(\frac{\Lambda^{2}}{\lambda v^{2}}\right)~{},$
$\displaystyle B_{r}$ $\displaystyle\equiv$
$\displaystyle\frac{\lambda_{r}v_{r}^{4}V}{24\omega_{r}}~{},$
$\displaystyle\omega_{r}$ $\displaystyle\equiv$ $\displaystyle
v_{r}\sqrt{\frac{\lambda_{r}}{6}}~{},$

and $K_{2}(z)$ is a modified Bessel function of the second kind with
asymptotic behaviour

$K_{2}(z\to\infty)\simeq e^{-z}\sqrt{\frac{\pi}{2z}}~{}.$ (18)

We note that $l$ does not correspond to Matsubara modes. Also, the
temperature-independent part of the expression (II.4.1) reproduces the zero-
temperature result derived in AlexBack22 .

####  II.4.2 Gas of instanton/anti-instanton pairs

The evaluation of $\Sigma_{gas}$ involves the fluctuation factor above each
jump and includes a summation over the allowed jump positions in the interval
$\tau\in[0,\beta]$ Kleinert . The additional contribution of quantum
fluctuations arises from the “flat” parts of the instanton/anti-instanton
configurations, which are exponentially close to each static saddle point for
the approximate average time of $\beta/2$ when neglecting the width of each
jump compared to $\beta$. Performing the resummation over instantons/anti-
instantons, we show in Appendix B that the corresponding connected graphs
generating functional is then

$\displaystyle\Sigma_{\text{gas}}(\beta)$ $\displaystyle\simeq$
$\displaystyle\Sigma_{L}(\beta/2)+\Sigma_{R}(\beta/2)$
$\displaystyle~{}-\ln\Big{(}\cosh(\bar{N})-1\Big{)}~{},$

where

$\bar{N}\equiv\omega_{r}\beta\sqrt{\frac{6}{\pi}S_{\text{inst}}}~{}e^{-S_{\text{inst}}}~{},$
(20)

corresponding to the average number of instanton/anti-instanton pairs at
temperature $T=1/\beta$. In this article we are interested in the limit
$\omega_{r}\beta\gg 1$ for a fixed volume - and thus fixed action
$S_{\text{inst}}$ \- such that we consider the situation where $\bar{N}\gg 1$,
corresponding to the full tunnelling regime. In the situation where $\beta$ is
fixed and $V$ becomes large we have $\bar{N}\ll 1$, where tunnelling is
suppressed and the system is better approximated by SSB AlexPolo22 .

##  III Non-extensive ground sate

###  III.1 One-particle-irreducible effective action

From the partition function evaluated for a constant source $j$, the classical
field is obtained as

$\phi_{c}\equiv-\frac{1}{Z}\frac{\delta Z}{\delta j}~{}\to~{}-\frac{1}{V\beta
Z}\frac{\partial Z}{\partial j}~{},$ (21)

which, in terms of the dimensionless quantities previously introduced, can be
written as

$\varphi_{c}=-\frac{1}{4B_{r}\omega_{r}\beta Z}\frac{\partial Z[k]}{\partial
k}~{}.$ (22)

From the expression (15) for the partition function, together with the
expressions (II.4.1) and (II.4.2), the classical field is expanded in powers
of the source $k$

$\varphi_{c}=\left(-f_{0}+\frac{\lambda_{r}}{128\pi^{2}}f_{1}\right)k+\mathcal{O}(k^{3})~{},$
(23)

where

$\displaystyle f_{0}$ $\displaystyle\equiv$
$\displaystyle\frac{1+16B_{r}\omega_{r}\beta+\cosh(\bar{N})}{2\Big{(}1+\cosh(\bar{N})\Big{)}}$
(24) $\displaystyle f_{1}$ $\displaystyle\equiv$
$\displaystyle\frac{7+32B_{r}\omega_{r}\beta+7\cosh(\bar{N})}{1+\cosh(\bar{N})}~{}.$

Consistently with the symmetry of the bare potential, the classical field
$\phi_{c}$ is an odd function of $k$: the even powers of $k$ cancel out in the
expression for $\varphi_{c}$ after adding the contribution of the different
saddle points, leading to the mapping $k=0\Leftrightarrow\varphi_{c}=0$.

We then perform the Legendre transform, after expressing the source as a
function of the classical field

$k(\varphi_{c})=-\frac{1}{2}\left(g_{0}+\frac{\lambda}{16\pi^{2}}g_{1}\right)\varphi_{c}+\mathcal{O}(\varphi_{c}^{3})~{},$
(25)

where

$\displaystyle g_{0}$ $\displaystyle\equiv$
$\displaystyle\frac{4\Big{(}1+\cosh(\bar{N})\Big{)}}{1+16B_{r}\omega_{r}\beta+\cosh(\bar{N})}$
(26) $\displaystyle g_{1}$ $\displaystyle\equiv$
$\displaystyle\frac{\Big{(}1+\cosh(\bar{N})\Big{)}\Big{(}7+32B_{r}\omega_{r}\beta+7\cosh(\bar{N})\Big{)}}{\Big{(}1+16B_{r}\omega_{r}\beta+\cosh(\bar{N})\Big{)}^{2}}~{}.$

The effective action for a constant configuration is finally

$\displaystyle\Gamma(\varphi_{c})$ $\displaystyle=$ $\displaystyle-\ln
Z\Big{(}k(\varphi_{c})\Big{)}-4B_{r}\omega_{r}\beta\int
k(\varphi_{c})~{}d\varphi_{c}$ $\displaystyle=$
$\displaystyle\Gamma(0)+B_{r}\omega_{r}\beta\left(g_{0}+\frac{\lambda}{16\pi^{2}}g_{1}\right)\varphi_{c}^{2}+\mathcal{O}(\varphi_{c}^{4})~{},$

where

$\displaystyle\Gamma(0)$ $\displaystyle=$ $\displaystyle-\ln Z(0)$
$\displaystyle=$
$\displaystyle-\ln\left(2e^{-\Sigma_{0}(\beta)}+e^{-2\Sigma_{0}(\beta/2)}\Big{(}\cosh(\bar{N})-1\Big{)}\right)~{},$

and $\Sigma_{0}\equiv\Sigma_{L}|_{k=0}=\Sigma_{R}|_{k=0}$. The effective
potential $U_{\text{eff}}$ is finally given by

$\Gamma(\phi_{c})=V\beta U_{\text{eff}}(\phi_{c})~{},$ (29)

and, as expected, it satisfies the following properties:

  * •

it is a convex function of $\phi_{c}$, since the mass term is positive;

  * •

the ground state is at $\varphi_{c}=0$, or equivalently $k=0$;

  * •

it has a non-trivial volume-dependence and is thus non-extensive.

For the following studies of NEC violation we focus on the ground state
$\varphi_{c}=0$.

###  III.2 NEC violation

The ground state density $\rho$ and pressure $p$ are obtained from the free
energy

$F=\frac{1}{\beta}\Gamma(0)=-\frac{1}{\beta}\ln Z(0)~{},$ (30)

and their sum can be written as AlexPolo22

$\displaystyle\rho+p$ $\displaystyle=$
$\displaystyle\frac{1}{V}\left(F-T\frac{\partial F}{\partial
T}\right)-\frac{\partial F}{\partial V}$ (31) $\displaystyle=$
$\displaystyle-T\frac{\partial U_{\text{eff}}(0)}{\partial T}-V\frac{\partial
U_{\text{eff}}(0)}{\partial V}~{}.$

From the expression (III.1), we obtain for $\omega_{r}\beta\gg 1$

$\displaystyle\rho+p$ $\displaystyle\simeq$
$\displaystyle\frac{4\omega_{R}^{5/2}}{(\sqrt{2}\pi\beta)^{3/2}}~{}e^{-\omega_{R}\beta/\sqrt{2}}$
$\displaystyle-\frac{\omega_{R}}{V}\left(S_{\text{inst}}+\frac{1}{2}\right)\sqrt{\frac{6}{\pi}S_{\text{inst}}}~{}e^{-S_{\text{inst}}}~{}.$

On the right-hand side, the first term corresponds to thermal fluctuations and
the second term corresponds to tunnelling. These terms compete for the overall
sign of $\rho+p$ leading to the following cases:

  * •

Infinite volume: $\rho+p\geq 0$
In the limit of infinite volume tunnelling is suppressed, as seen via the
vanishing of the average number (20) of instanton/anti-instanton pairs for any
fixed temperature: $\lim_{V\to\infty}\bar{N}=0$ for fixed $\beta$. Hence only
thermal fluctuations contribute and

$\rho+p=\frac{4\omega_{R}^{5/2}}{(\sqrt{2}\pi\beta)^{3/2}}~{}e^{-\omega_{R}\beta/\sqrt{2}}~{},$
(33)

with $\rho+p\to 0$ as the temperature goes to 0 or equivalently
$\beta\to\infty$. This result is not surprising: the limit of infinite volume
corresponds to SSB and, as expected, the NEC is satisfied;

  * •

Finite volume and zero temperature: $\rho+p<0$
In this situation, only the tunnelling term contributes and

$\rho+p=-\frac{\omega_{R}}{V}\left(S_{\text{inst}}+\frac{1}{2}\right)\sqrt{\frac{6}{\pi}S_{\text{inst}}}~{}e^{-S_{\text{inst}}}~{}.$
(34)

The NEC is violated as a consequence of the explicit volume-dependence of the
effective potential;

  * •

Boundary $\rho+p=0$
We sketch in Fig.2 the boundary $V(T)$ between the region where the NEC is
satisfied and the region where the NEC is violated.

Figure 2: The boundary between the regions where the NEC is satisfied and
where it is violated due to the competition of tunnelling and thermal
fluctuations. The plot shows the curve $V(T)$ in terms of the dimensionless
variables used in this article.

Finally, we note that NEC violation is suppressed exponentially with the
volume, unlike the power law suppression which is found with $O(4)$-symmetric
Euclidean spacetime coordinates AlexCloughPolo ; AlexBack22 .

##  IV Discrete momentum corrections

We focus here on the ground state obtained for $k=0$ in the case of zero
temperature, where NEC violation arising from tunnelling is maximum.

The previous sections ignore quantisation of momentum when calculating the
connected graphs generating functional for each static saddle point in a
finite volume. As we explain below, the evaluation of $\Sigma_{L,R}$ with
discrete momentum consists of taking into account the relevant Casimir energy.
There is no such contribution from the jumps in the instantons/anti-instantons
since the corresponding one-loop corrections do not depend on momentum.

###  IV.1 Vacuum energy

The Casimir contribution to the connected graphs generating functional is
defined as

$\Sigma_{\text{Cas}}\equiv\Sigma_{L,R}|_{\text{discrete}}-\Sigma_{L,R}|_{\text{continuum}}~{},$
(35)

where the ultraviolet divergences cancel out since they are identical in the
discrete and continuum cases. For zero temperature and vanishing source, the
expression (II.4.1) gives

$\Sigma_{L,R}(k=0,T=0)|_{\text{continuum}}=\lim_{\beta\to\infty}\Sigma_{0}(\beta)=0~{},$
(36)

such that, instead of eq.(III.1), one-loop corrections obtained with discrete
momentum lead to

$\displaystyle\Gamma(0)$ $\displaystyle=$
$\displaystyle-\ln\left(2e^{-\Sigma_{\text{Cas}}}+e^{-\Sigma_{\text{Cas}}}\Big{(}\cosh(\bar{N})-1\Big{)}\right)$
(37) $\displaystyle=$
$\displaystyle~{}\Sigma_{\text{Cas}}-\ln\Big{(}\cosh(\bar{N})+1\Big{)}~{}.$

The above expression takes advantage of the proportinality between
$\Sigma_{0}$ and $\beta$ in the limit of vanishing temperature, such that

$2\Sigma_{0}(\beta/2)\to\Sigma_{0}(\beta),$ (38)

as $\beta\to\infty$. In the situation of one saddle point, and therefore no
tunnelling, $\Gamma(0)=\Sigma_{\text{Cas}}=\beta E_{\text{Cas}}$ where
$E_{\text{Cas}}$ is the Casimir energy corresponding to quantum fluctuations
about a single vacua $\pm v$ (where one has approximately quadratic
fluctuations with mass $m=\sqrt{2}\omega_{r}$). Hence

$U_{\text{eff}}(0)=\frac{E_{\text{Cas}}}{V}-\frac{1}{V\beta}\ln\Big{(}\cosh(\bar{N})+1\Big{)}~{},$
(39)

and we see the additive nature of the Casimir effect and tunnelling
contributions, similarly to the finite-temperature contribution. The sum of
density and pressure reads finally

$\displaystyle\rho+p$ $\displaystyle=$
$\displaystyle\frac{E_{\text{Cas}}}{V}-\frac{\partial E_{\text{Cas}}}{\partial
V}$
$\displaystyle-\frac{\omega_{R}}{V}\left(S_{\text{inst}}+\frac{1}{2}\right)\sqrt{\frac{6S_{\text{inst}}}{\pi}}~{}e^{-S_{\text{inst}}}~{}.$

###  IV.2 Casimir contribution to the NEC

The Casimir energy is highly sensitive to the geometry of the box containing
the field, as well as the boundary conditions used on the corresponding
surfaces Casimirreview . For a scalar field $\varphi(t,x)$ in the interval
$x\in[0,L]$ for example, the possible choices of boundary conditions are
defined as follows

Dirichlet: $\displaystyle\varphi(t,0)=\varphi(t,L)=0$ (41) Neumann:
$\displaystyle\partial_{x}\varphi(t,0)=\partial_{x}\varphi(t,L)=0$ Periodic:
$\displaystyle\varphi(t,0)=\varphi(t,L)~{}.$ (42)

For the cases we consider, the asymptotic form of the Casimir effect is
identical for both Dirichlet and Neumann boundary conditions. We thus consider
mixed boundary conditions, where different subsets of the boundary can possess
either Dirichlet or Neumann conditions. For the case of mixed boundary
conditions, the Casimir energy is dependent on the size/curvature of the
material boundaries, and for the case of periodic boundary conditions, it is
dependent on the period length/curvature of the non-trivial spacetime. A
‘general rule’ states that flat geometries lead to exponential suppression of
the Casimir energy for $mL\gg 1$, where $L$ is the length scale of the
relevant boundaries, and that curved geometries lead to power-law suppression
of the Casimir energy for $mR\gg 1$, where $R$ is the radius of curvature of
the relevant surfaces. There are exceptions to this general rule though, which
are highlighted in the following examples.

$\bullet$ Dirichlet boundary conditions, flat boundaries
The original Casimir configuration consists of a scalar field constrained
between two parallel, flat mirrors with surface area $A$ and separation $a$,
with the scalar field satisfying Dirichlet conditions on the boundaries. The
corresponding Casimir energy is Casimirreview

$E_{\text{Cas}}\simeq\left\\{\begin{array}[]{c}-\frac{A\pi^{2}}{1440a^{3}}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\mbox{for}~{}~{}~{}~{}am\ll
1\\\ -\frac{A}{8\sqrt{2}}\left(\frac{m}{\pi
a}\right)^{3/2}~{}e^{-2ma}~{}~{}~{}~{}\mbox{for}~{}~{}~{}~{}am\gg
1\end{array}\right.$ (43)

and is always negative.

$\bullet$ Dirichlet boundary conditions, curved boundaries
For dimensional reasons, the Casimir energy for a scalar field confined within
the curved boundary of a 2-sphere of radius $R$ with Dirichlet boundary
conditions is given in terms of the dimensionless function

$E_{\text{Cas}}=\frac{1}{R}f(mR)~{},$ (44)

and is found to obey power law suppression in $mR$, for $mR\gg 1$ KK .

$\bullet$ Periodic boundary conditions, flat spacetime
For a scalar field confined to the surface of a 3-torus (a rectangular box
with periodic boundary conditions), the sign of the Casimir energy depends on
the ratio of the lengths of the box and we have piston

$E_{\text{Cas}}\simeq-\frac{(mL)^{3/2}}{L}\exp(-mL)~{}~{}~{}~{}\mbox{for}~{}~{}~{}~{}mL\gg
1~{},$ (45)

where $L$ is the typical size of the period length.

$\bullet$ Periodic boundary conditions, curved spacetime
For a scalar field confined to the surface of a 3-sphere with radius $R$, we
would expect the asymptotic form to be a power law in R. However, this special
case is an exception to the general rule as a consequence of the accidental
vanishing of the heat-kernel coefficients (see Sec. 3 of Casimirreview  for
details). The resulting Casimir energy has instead an exponential asymptotic
form, as in the case of flat geometries MMS

$E_{\text{Cas}}\simeq+\frac{(mR)^{5/2}}{R}\exp(-2\pi
mR)~{}~{}~{}~{}\mbox{for}~{}~{}~{}~{}mR\gg 1~{}.$ (46)

The above examples display how the Casimir effect for a massive scalar field
is at most suppressed by the exponential $e^{-mL}$, where $L$ is a typical
size of the boundary containing the field. On the other hand, the tunnelling
contribution to the NEC, calculated with continuous momentum, is proportional
to

$e^{-S_{\text{inst}}}\sim\exp\left(-\frac{(mL)^{3}}{\lambda}\right)~{},$ (47)

and is therefore negligible compared to the Casimir contribution in the regime
$mL\gg\sqrt{\lambda}$. For $mL\sim\sqrt{\lambda}$ though, tunnelling competes
with the Casimir effect and can change the sign of $\rho+p$ in the situation
where the Casimir energy is positive. As an example, we sketch in Fig.3 the
boundary $R(\lambda)$ between the region where the NEC is satisfied and the
region where it is violated, due to the competition between tunnelling and the
Casimir effect on a 3-sphere.

Figure 3: The boundary between the regions where the NEC is satisfied and
where it is violated, due to the competition of the Casimir energy and
tunnelling at zero temperature on a 3-sphere. The plot shows the curve
$R(\lambda)$ in terms of the dimensionless variables used in this article.

We note however two important points regarding the Casimir examples cited
here: (i) they are valid for ideal surfaces only, and a realistic confining
mechanism for the scalar field would lead to a modification of the Casimir
vacuum energies, especially if the field is confined by an external potential
instead of a physical box GrahamOlum ; (ii) they assume free scalar fields and
ignore its self-interactions. On the other hand, the tunnelling mechanism
described here: (i) necessitates the field to be self-interacting; (ii) is not
sensitive to the geometry/topology of the box containing the field. Hence the
conclusions regarding which effect dominates could be modified by a more
thorough study, depending on the situation which is considered.

Finally, the Average Null Energy Condition is not violated by the present
mechanism. Indeed, if we take into account the energy necessary to maintain
the confining mechanism the overall ground state of the system does not
violate the NEC ANEC , consistently with what is expected from causality
causality .

##  V Conclusions

Tunnelling between degenerate vacua is exponentially suppressed with the
volume of the box containing the field, but nevertheless allows the
possibility of NEC violation at low temperatures. Taking into account discrete
momentum of fluctuations in a finite volume implies this effect is mainly
relevant for situations where the typical size of the box is not too large
compared to the Compton wave length of the particle, and where tunnelling can
lead to an overall NEC violation. A potential application lies in axion
physics, where the de Broglie wavelength can be of order 1 kpc Doddy  with the
confinement provided by a gravitational well.

Exponential suppression in the volume could potentially be avoided by a
consideration of non-degenerate vacua, where other saddle points with a
volume-independent action become relevant, as in the original study of false
vacuum decay Coleman1977 . The resulting effective action would be non-
extensive in a certain regime of the classical field, but more studies need to
be done for the status of NEC violation in the corresponding vacuum.

Finally, NEC violation could play an important role in Early Universe
Cosmology, where tunnelling could provide a dynamical mechanism for a
cosmological bounce, as explained in AlexCloughPolo : as the Universe
contracts, tunnelling switches on and violates the NEC, which induces a bounce
after which tunnelling is suppressed as the Universe expands. This scenario
necessitates the study of tunnelling in a Friedman-Lemaitre-Robertson-Walker
background though, and is left for future work.

## Acknowledgements

The authors would like to thank Klaus Kirsten for valuable correspondence
regarding the Casimir effect, and JA would like to thank Janos Polonyi for
enlightening discussions. This work is supported by the Leverhulme Trust
(grant RPG-2021-299) and the Science and Technology Facilities Council (grant
STFC-ST/T000759/1). For the purpose of Open Access, the authors have applied a
CC BY public copyright licence to any Author Accepted Manuscript version
arising from this submission.

##  Appendix A Fluctuation factor for a static saddle point

The fluctuation factors for the static saddle points are calculated with
continuous 3-dimensional momenta, introducing the cut-off $\Lambda$ in the
Schwinger proper time representation of the propagator. Introducing the
dimensionless Matsubara frequency $\nu_{n}\equiv 2\pi n/\omega\beta$, we have

$\displaystyle\mbox{Tr}\left\\{\ln\left(\delta^{2}S[\varphi_{i}]\right)\right\\}$
$\displaystyle=$ $\displaystyle
V\int\frac{d^{3}p}{(2\pi)^{3}}\sum_{n=-\infty}^{\infty}\int_{1/\Lambda^{2}}^{\infty}\frac{\mathrm{d}s}{s}\mathrm{e}^{-4B\omega\beta
s(p^{2}/\omega^{2}+\nu_{n}^{2}+3\varphi_{i}-1)}$ $\displaystyle=$
$\displaystyle\frac{V\omega^{3}}{2\pi^{2}}\sum_{n=-\infty}^{\infty}\int_{1/X^{2}}^{\infty}\frac{\mathrm{d}x}{x}\int_{0}^{\infty}\mathrm{d}q~{}q^{2}\mathrm{e}^{-x(q^{2}+\nu_{n}^{2}+3\varphi_{i}-1)}$
$\displaystyle=$
$\displaystyle\frac{V\omega^{3}}{8\pi^{3/2}}\sum_{n=-\infty}^{\infty}\int_{1/X^{2}}^{\infty}\frac{\mathrm{d}x}{x^{5/2}}\mathrm{e}^{-x(\nu_{n}^{2}+3\varphi_{i}-1)}$
$\displaystyle=$
$\displaystyle\frac{V\omega^{3}}{8\pi^{3/2}}\int_{1/X^{2}}^{\infty}\frac{\mathrm{d}x}{x^{5/2}}\mathrm{e}^{-x(3\varphi_{i}-1)}\vartheta_{0}\left(\frac{4\pi
x}{\omega^{2}\beta^{2}}\right)~{},$

where the dimensionless variables are

$q\equiv\frac{p}{\omega}\quad,\quad x\equiv 4B\omega\beta s\quad,\quad
X^{2}\equiv\frac{\Lambda^{2}}{4B\omega\beta}~{},$ (49)

and $\vartheta_{0}(y)$ is the Jacobi function

$\vartheta_{0}(y)\equiv\sum_{n=-\infty}^{\infty}\mathrm{e}^{-\pi yn^{2}}~{}.$
(50)

Making use of the following property

$\vartheta_{0}(y)=y^{-1/2}\vartheta_{0}(1/y)~{},$ (51)

the above becomes

$\displaystyle\mbox{Tr}\left\\{\ln\left(\delta^{2}S[\varphi_{i}]\right)\right\\}$
$\displaystyle=$
$\displaystyle\frac{V\omega^{4}\beta}{16\pi^{2}}\int_{1/X^{2}}^{\infty}\frac{\mathrm{d}x}{x^{3}}\mathrm{e}^{-x(3\varphi_{i}-1)}\vartheta_{0}\left(\frac{\omega\beta}{4\pi
x}\right)$ $\displaystyle=$
$\displaystyle\frac{V\omega^{4}\beta}{16\pi^{2}}\int_{1/X^{2}}^{\infty}\frac{\mathrm{d}x}{x^{3}}\mathrm{e}^{-x(3\varphi_{i}-1)}\sum_{n=-\infty}^{\infty}e^{-\omega^{2}\beta^{2}n^{2}/4x}$
$\displaystyle=$
$\displaystyle\lambda\frac{B\omega\beta}{24\pi^{2}}\Big{(}I_{\Lambda}(\varphi_{i})+I_{T}(\varphi_{i})\Big{)}~{},$

where

$\displaystyle I_{\Lambda}(\varphi_{i})$ $\displaystyle\equiv$
$\displaystyle\int_{1/\Lambda^{2}}^{\infty}\frac{\mathrm{d}x}{x^{3}}~{}\mathrm{e}^{-x(3\varphi_{i}-1)}$
(53) $\displaystyle I_{T}(\varphi_{i})$ $\displaystyle\equiv$ $\displaystyle
2\sum_{n=1}^{\infty}\int_{0}^{\infty}\frac{\mathrm{d}x}{x^{3}}~{}\mathrm{e}^{-x(3\varphi_{i}-1)-\omega^{2}\beta^{2}n^{2}/4x}~{}.$

The first integral $I_{\Lambda}$ is the temperature-independent divergent
integral which, after renormalisation, produces the same results as in the
zero-temperature case AlexBack22 . The second integral $I_{T}$ is the
temperature-dependent contribution corresponding to the finite-temperature
corrections. It is finite, which is why the cut-off is taken to infinity in
this specific term. This temperature-dependent integral can be written in
terms of the modified Bessel functions of the second kind $K_{2}(z)$ as

$I_{T}(\phi_{i})=\sum_{n=1}^{\infty}\frac{16(3\varphi_{i}-1)}{(n\omega\beta)^{2}}K_{2}(n\omega\beta\sqrt{3\varphi_{i}-1})~{}.$
(54)

Together with the integral $I_{\Lambda}$, the connected graphs generating
functional for homogeneous saddle points is given by eq.(II.4.1).

##  Appendix B Fluctuation factor for the instantons/anti-instantons gas

We calculate here the contribution $\exp(-\Sigma_{\text{gas}})$ to the
partition function (15), following the known approach in studies of tunnelling
effects Kleinert .

The invariance of the action for $n$ instanton/anti-instanton pairs under the
translation of the jumps leads to the degeneracy factor in the partition
function

$\displaystyle\left(\prod_{i=1}^{2n}\int_{\tau_{i-1}}^{\omega\beta}\tau_{i}\right)=\frac{(\omega\beta)^{2n}}{(2n)!}~{},$
(55)

where $\tau_{i}\in[\tau_{i-1},\omega\beta]$ and $\tau_{0}=0$, since successive
instanton jumps can only occur after previous ones. Each jump has an
associated fluctuation factor $\sqrt{6S_{\text{inst}}/\pi}$ and thus the total
fluctuation factor is given by the product of the contributions of the “flat”
parts of the $n$-pairs of instanton/anti-instantons and the $n$ pairs of
jumps. On average, each configuration of $n$ instanton/anti-instanton pairs
spends the same time $\simeq\beta/2$ close to each static saddle point, such
that the expression for $F_{n}$ is finally

$F_{n}=F_{L}(\beta/2)F_{R}(\beta/2)\left(\frac{6S_{int}}{\pi}\right)^{n}~{}.$
(56)

Substituting the above results into the partition function (II.4), along with
the total action (13) for $n$ pairs, yields the total contribution to the
partition function due to instanton/anti-instanton pairs

$\displaystyle\exp(-\Sigma_{\text{gas}})$ $\displaystyle=$ $\displaystyle
e^{-\Sigma_{L}[\beta/2]}e^{-\Sigma_{R}[\beta/2]}\sum_{n=1}^{\infty}\frac{(\omega\beta)^{2n}}{(2n)!}\left(\frac{6S_{\text{int}}}{\pi}\right)^{n}e^{-2nS_{\text{int}}}$
$\displaystyle=$
$\displaystyle\exp\Big{(}-\Sigma_{L}[\beta/2]-\Sigma_{R}[\beta/2]\Big{)}$
$\displaystyle\times\left(\cosh\left(\omega\beta\sqrt{\frac{6S_{\text{int}}}{\pi}}~{}e^{-S_{\text{inst}}}\right)-1\right)~{},$

This leads to the expression (II.4.2), where the parameters can be replaced by
their renormalised version, since the overall expression is already at one-
loop.

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