\section{Introduction}
One of the main challenges in theoretical physics is to find a theory that consistently incorporates quantum and gravitational physics. The attempts to construct such a theory can be grouped into two categories: first, there are theories proposed from first principles that incorporate some quantum aspects into gravitation, or vice versa. This strategy includes theories like string theory, loop quantum gravity, and other remarkable ideas (for a review of most quantum gravity approaches see Ref.~\cite{Oriti}). In contrast, another strategy is to consider general relativity as an effective low-energy description of the fundamental theory and to look for clues on the theory of quantum gravity by looking for small deviations from general relativity. To search for these deviations, it is often useful to have a parametrisation that, in turn, can be generated by assuming violations of some fundamental principles of the current paradigm.

One of the most important principles of current physics is Lorentz invariance. This principle states that all inertial observers are equivalent. Lorentz invariance is an action symmetry and is closely related to the proposal that all fields in the action must be dynamical~\cite{Cristobal1, Cristobal2}, thereby avoiding  preferred spacetime directions. Relevantly, there are works within several promising quantum gravity candidates that argue that Lorentz invariance is broken at energies close to the Planck scale \cite{KosteleckySamuel, GambiniPullin}.

Moreover, there exists a model-independent framework to study (explicit) violations of Lorentz invariance, from hereon referred to as Lorentz violation. The idea, pioneered by Kostelecky and collaborators, is to create a parametrisation in the effective field theory framework~\cite{Rob}; the result is known as the Standard Model Extension (SME)~\cite{SME1, SME2, Kostelecky2004}. As such, the SME action contains the action terms of conventional physics, that is, the Standard Model and general relativity, plus all corrections that can be written in terms of the fields of this conventional part, but that includes terms that are not Lorentz invariant. The Lorentz-violating terms are weighted by nondynamical tensor fields, dubbed the SME coefficients, and are expected to be ranked in terms of their mass dimensions.

There is vast literature exploring a variety of Lorentz-violating effects (a list of the empirical bounds on the SME coefficients by many different experiments can be found in Ref.~\cite{DataTables}). There are also interesting theoretical considerations~\cite{PhysRevD.63.065008}, particularly when considering dynamical spacetimes or spontaneous Lorentz violation \cite{SMEtheo1,SMEtheo2,SMEtheo3,SMEtheo4,SMEtheo5,SMEtheo6,SMEtheo7,SMEtheo8}. In this letter, we focus on a Lorentz-violating quantised field. In these regards, there is a paper where radiative corrections and the running of the SME coefficients are studied~\cite{MauroRalfRob}. However, the symmetries of the vacuum, which are the main topic of this paper, are not analysed in Ref.~\cite{MauroRalfRob}. We also refer the reader to Ref.~\cite{Husain16}, where the authors study how a detector may respond to a quantum field with modified dispersion relations, where the modifications are suppressed by a large mass. Here, we quantise a free Lorentz-violating scalar coupled to a dimensionless SME-like coefficient, which is the simplest Lorentz-violating theory, and examine its vacuum states to verify whether they are unitarily equivalent. We find that the vacuum state in a particular Lorentz frame is not a vacuum in a frame connected by a Lorentz transformation. This raises serious questions on the phenomenological viability of Lorentz-violating theories, even if they are merely seen as low-energy effective theories.

To gain a deeper insight, we study whether different inertial observers can feel such effects. We equip our inertial observers with Unruh-De Witt detectors to probe the particle content of an inertial vacuum in different inertial frames. We find that, despite the initial concerns, inertial detectors remain silent upon the change of frames. Following Unruh's motto that `a particle is what a particle detector detects,' in this simple Lorentz-violating model, inertial observers experience the vacuum associated with one fixed inertial frame as devoid of particles in all inertial frames.

Given that it may be interesting to extend the above results to perturbative interacting theory, we complement the aforementioned results with a discussion on the Hadamard property in this Lorentz-violating theory. This condition, as it turns out, does not hold strictly and needs to be adapted. Nevertheless, we see no obstruction in using this `modified' Hadamard property to construct interacting observables (including time-ordered products of non-linear observables) and explore these questions in full perturbative interacting theory.

Throughout the paper, Latin indices from the beginning of the alphabet represent abstract indices, namely, they stand for open slots in the associated tensors and are thus not associated with any coordinates. In contrast, Greek indexes represent the components of the corresponding tensors in a given coordinate basis. For simplicity, the spacetime under consideration is the flat $4$-dimensional spacetime (without torsion or nonmetricity), whose metric (inverse metric) is denoted by $\eta_{ab}$ ($\eta^{ab}$), and it has signature $(-+++)$. The covariant derivative is denoted by $\nabla_a$.

\section{The model}

Although our model is not part of the SME, which deals with fields present in the Standard Model, it is instructive to study the simplest possible setting: that of a quantum scalar field $\phi$. This example already contains valuable lessons for other types of matter fields. In analogy with SME spinors, one can propose the Lorentz-violating Lagrangian
\begin{equation}
\mathcal L=-\frac{1}{2}\left[\left(\eta^{ab}+c^{ab}\right)\nabla_a\nabla_b\phi^2+m^2\phi^2\right],
    \label{scalarLagrangian}
\end{equation}
where $c^{ab}$ is the SME coefficient, which generates an effective metric for $\phi$. A particular $c^{ab}$, known in the SME literature as an isotropic coefficient, happens when there is a global set of Minkowskian coordinates $(t,x,y,z)$ for which the components of $c^{ab}$ take the simple form
\begin{equation}
c^{\mu\nu}=\mathrm{diag}(C,0,0,0),
\label{particular}
\end{equation}
 where $C$ is a constant. In these `privileged' coordinates, the Lagragian~(\ref{scalarLagrangian}) is
\begin{equation}
\mathcal L=-\frac{1}{2}\left[|\nabla\phi|^2-(1-C)(\partial_t\phi)^2+m^2\phi^2\right].
    \label{privLag}
\end{equation}

In the case of the scalar field above, one writes the Klein-Gordon product between two solutions $\phi$ and $\psi$ in the privileged coordinates as
\begin{equation}
    (\phi,\psi)=-\mathrm{i}(1-C)\int(\psi\partial_t\bar\phi-\bar\phi\partial_t\psi)\ \mathrm{d}^3x,
    \label{kg}
\end{equation}
where we integrate over a constant $t$ surface and the bar stands for complex conjugation. In fact, for technical convenience, we quantise the fields in a toroidal box of length $L$, so that integration is taken over the cube $[-L/2,L/2]^3$. Yet, observable quantities must be independent of $L$.

We can readily check that the modes
\begin{equation}
    \phi_{\vec k}=[L^32\omega(1-C)]^{-1/2}\exp{\mathrm{i}(\vec k\cdot \vec x-\omega t)},
    \label{privmodes}
\end{equation}
with
\begin{equation}
\omega:=\sqrt{\frac{m^2+\vec k^2}{1-C}}
    \label{privdisp}
\end{equation}
for $k_i=2\pi j_i/L$, $j_i\in\mathbb Z, i=1,2,3$. In addition, the dot in Eq.~\eqref{privmodes} stands for the standard $3$-dimensional scalar product. Note that these solutions, together with their complex conjugates, comprise a complete set of solutions to the field equations derived from the Lagrangian~(\ref{privLag}). The modes are normalised so that $\left(\phi_{\vec k_1},\phi_{\vec k_2}\right)=\delta_{\vec k_1\vec k_2}$. Canonical quantisation can be carried out in the standard way, as we shall see in the next section.

\section{Inequivalence of inertial vacua}
\label{Sec:Vacua}

In standard quantum field theory, the existence of a Lorentz-invariant vacuum is of fundamental importance, so much so that the existence of such a unique Poincar\'e-invariant state was adopted as an \emph{axiom} in the axiomatic formulation of quantum field theory in Minkowski spacetime~\cite{Streater64}. This property is lost in Lorentz-violating theories.

The modes~(\ref{privmodes}) are adapted to the privileged coordinates in the sense that the Lie derivative of $\phi_{\vec k}$ along $\partial_t$ vanishes. One can likewise adopt a set of solutions that are Lie-dragged according to $\partial_{t^\prime}$, which is obtained from $\partial_t$ by a (global) Lorentz boost. These are
\begin{equation}
\varphi_{\vec k^\prime}=\mathcal N_{\vec k^\prime}\exp{\mathrm{i}(\vec k^\prime\cdot\vec x^\prime-\omega^\prime t^\prime),}
    \label{unpmodes}
\end{equation}
where $\mathcal N_{\vec k^\prime}$ are normalisation constants. Crucially, reflecting the nonequivalent equation of motion, the unprivileged dispersion relation is altered, it solves
\begin{equation}
    (\omega^\prime)^2-(\vec k^\prime)^2-m^2+C[-\gamma^2(\omega^\prime)^2-2v\gamma k_z^\prime\omega^\prime-\gamma^2v^2(k_z^\prime)^2]=0,
    \label{undisp}
\end{equation}
where $\gamma=1/\sqrt{1-v^2}$ and the boost direction was chosen to be, without loss of generality, to be $+z$.

To find $\mathcal N_{\vec k}$, first, we note that, from the Lagrangian written in terms of the primed coordinates, the Klein-Gordon product reads
\begin{multline}
    \left(\phi,\psi\right)^\prime=-\mathrm{i}\int\mathrm{d}^3\vec x^\prime\left\{\psi[(1-C-v^2)\gamma^2\partial_{t^\prime}\bar\phi+\gamma^2vC\partial_{z^\prime}\bar\phi]\right.\\
    -\left.\bar\phi[(1-C-v^2)\gamma^2\partial_{t^\prime}\psi+\gamma^2vC\partial_{z^\prime}\psi]\right\},
    \label{unkg}
\end{multline}
where integration now is on a $t^\prime$ surface. With this product we get
\begin{equation}
\left(\varphi_{\vec k_1^\prime},\varphi_{\vec k_2^\prime}\right)^\prime=2|\mathcal N_{\vec k1^\prime}|^2\frac{L^3}{\gamma}
\left[(1-C-v^2)\gamma^2\omega_1^\prime-C\gamma^2vk_{z1}^\prime\right]\delta_{\vec k_1^\prime \vec k_2^\prime}.
    \label{unpnorm}
\end{equation}

Since both sets of solutions $\{\phi_{\vec k},\bar\phi_{\vec k}\}$ and $\{\varphi_{\vec k^\prime},\bar\varphi_{\vec k^\prime}\}$ are complete, we may write
\begin{equation}
\phi_{\vec k}=\sum_{\vec k^\prime}\left(\alpha_{\vec k\vec k^\prime}\varphi_{\vec k^\prime}+\beta_{\vec k\vec k^\prime}\bar\varphi_{\vec k^\prime}\right).
    \label{bogo}
\end{equation}
Let $\boldsymbol\Phi$ denote the field operator. We write
\begin{equation}
    \boldsymbol\Phi=\sum_{\vec k}\left(\phi_{\vec k}a_{\vec k}+ \mathrm{h.c.}\right)=\sum_{\vec k^\prime}\left(\varphi_{\vec k^\prime}a^\prime_{\vec k^\prime}+\mathrm{h.c.}\right),
    \label{modeexp}
\end{equation}
from which we find the corresponding Bogoliubov coefficients (see e.g. Ref. \cite{Bire82})
\begin{equation}
\alpha_{\vec k\vec k^\prime}=\left(\varphi_{\vec k^\prime},\phi_{\vec k}\right)^\prime,\qquad\beta_{\vec k\vec k^\prime}=-\left(\bar\varphi_{\vec k^\prime},\phi_{\vec k}\right)^\prime.
    \label{alphabeta}
\end{equation}
Moreover, from Eqs.~(\ref{bogo}) and (\ref{alphabeta}), we can 
\begin{equation}
a_{\vec k}^\prime=\sum_{\vec k^\prime}\ \left(\alpha_{\vec k\vec k^\prime}a_{\vec k^\prime}+\overline{\beta_{\vec k\vec k^\prime}}a^\dagger_{\vec k^\prime}\right).
    \label{annihilation}
\end{equation}

Clearly, the vacuum annihilated by $a_{\vec k}^\prime$ does not coincide with the one annihilated by $a_{\vec k}$ when at least one $\beta_{\vec k\vec k^\prime}\neq0$. Furthermore, denoting by $|0\rangle$ the state for which $a_{\vec k}|0\rangle=0\ \forall\vec k$, we see that $a^\prime_{\vec k}|0\rangle\neq0$, proving the vacuum state is not Lorentz-invariant, unlike in conventional theories. We can go further and notice that the number operator $N_{\vec k}^\prime=(a^\prime_{\vec k})^\dagger a^\prime_{\vec k}$ satisfies
\begin{equation}
\langle 0|N_{\vec k}^\prime|0\rangle=\sum_{\vec k^\prime}\left|\beta_{\vec k\vec k^\prime}\right|^2 \left(\phi_{\vec k^\prime},\phi_{\vec k^\prime}\right),
    \label{number}
\end{equation}
where, in the second step, we have applied Eq.~(\ref{annihilation}) and the commutation relation
\begin{equation}
    \left[a_{\vec k^\prime_1}, a^\dagger_{\vec{k_2}^\prime}\right]=\left(\phi_{\vec k_1^\prime},\phi_{\vec k_2^\prime}\right)\mathbf 1.
\label{commutation}
\end{equation}

Again, whenever the coefficients $\beta_{\vec k\vec{k^\prime}}$ are nonzero, the preferred vacuum $|0\rangle$ has a nonzero number of particles in the modes adapted to the unprivileged frame. However, since the quantisations in the two frames are independent, one must evaluate how such particles interact with other matter fields. Indeed, as we shall see in Sec.~\ref{Sec:Detector}, these particles cannot excite a detector under reasonable hypotheses.

From Eqs.~(\ref{unpmodes}), (\ref{undisp}), (\ref{alphabeta}), (\ref{number}), and choosing the normalisation constant in Eq.~(\ref{unpnorm}) so that the resulting solution basis is orthonormal, it can be shown that
\begin{multline}
\beta_{\vec k\vec k^\prime}=\frac{8\mathrm{e}^{-\mathrm{i}(\omega^\prime-\gamma(\omega+v k_z))t^\prime}}{2L^3\sqrt{\omega(1-C)/\gamma\  [(1-C-v^2)\gamma^2\omega^\prime-C\gamma^2v k_z^\prime]}}\times\\
\frac{\sin(k_x-k_x^\prime)L}{k_x-k_x^\prime}\frac{\sin(k_y-k_y^\prime)L}{k_y-k_y^\prime}\frac{\sin(k_z^\prime-\gamma(k_z+v\omega))L/\gamma}{k_z^\prime-\gamma(k_z+v\omega)}\\
\times\left[(1-C-v^2)\gamma^2(\omega^\prime+\gamma(k_zv-\omega))+C\gamma^2v(\gamma(k_z-\omega v)-k_z^\prime)\right].
    \label{beta}
\end{multline}
After imposing the discrete values for $\vec k$ and $\vec k^\prime$, the last three factors on the first line of Eq.~(\ref{beta}) become
$L^3\delta_{k_xk^\prime_x}\delta_{k_yk^\prime_y}\delta_{k_z,\gamma(k_z^\prime+v\omega^\prime)}.$ Thus,
\begin{equation}
\langle 0|N^\prime_{\vec k}|0\rangle=\frac{16[(1-C-v^2)\gamma^2(\omega^\prime-\gamma(\omega-k_z v))-C\gamma^2v(k_z^\prime-\gamma(k_z-\omega v))]^2}{\omega(1-C)[(1-C-v^2)\gamma\omega^\prime-C\gamma vk_z^\prime]},
    \label{number2}
\end{equation}
with a small abuse of notation, where $k_z^\prime$ is not free, but rather obtained from $k_z$ from the corresponding Lorentz transformation, and, as before, $\omega$ and $\omega^\prime$ are calculated from $\vec k$ from their corresponding dispersion relations.

Importantly, from Eq.~(\ref{undisp}), we can readily verify that, when $C=0$, then $\omega^\prime=\gamma(\omega-vk_z)$ and $k_z^\prime=\gamma(k_z-v\omega)$, which are the conventional Lorentz-invariant results. Moreover, since, in this case, the factor in the square brackets in Eq.~(\ref{beta}) vanishes for the diagonal elements, all the $\beta_{\vec k\vec k^\prime}$ vanish identically, reflecting the equivalence of all inertial vacua for theories that preserve the Lorentz symmetry. 

Notice that expression~(\ref{number2}) is ill-behaved for $C>1$ when the `effective' metric $\tilde\eta_{ab} = \eta_{ab} + c_{ab}$ ceases to be Lorentzian, and the field equations to be hyperbolic. It also diverges when $C<0$ and the phase velocity in the `primed frame', $\omega^\prime/k_z^\prime$ reaches $Cv/(1-C-v^2)$. These effects are closely related to vacuum \v{C}erenkov radiation, which is well understood in the context of the SME \cite{VacCerenkov1, VacCerenkov2}.

Another interesting feature of Eq.~(\ref{number2}) is that, for large values of $k_z$, it approaches a constant. Figure~\ref{fig:my_label} shows a plot of Eq.~(\ref{number2}) for the particular case $k_x=k_y=0$. Thus, one can see that the number operator contains excitations of arbitrarily large momentum. We turn now to study the interaction of the Lorentz-violating scalar field with inertial Unruh-De Witt detectors.

\begin{figure}
    \centering
    \includegraphics[width=.65\textwidth]{Nkk4.png}
    \caption{$\langle0|N^\prime_{\vec k}|0\rangle$ as a function of $k_z$ for $m=1$ (blue curve) and $m=0$ (red curve), and with $k_x=k_y=0$. We used the parameters $C=0.1$ and $v=0.8$. Note that $\langle0|N^\prime_{\vec k}|0\rangle$ approaches nonzero constants as $k_z\to\pm\infty$.}
    \label{fig:my_label}
\end{figure}

\section{Inertial Unruh-De Witt detectors}
\label{Sec:Detector}

If one wishes to extract observable consequences of a theory, one must allow for interactions. The simplest model that captures the physics is an Unruh-De Witt detector \cite{Bire82}: a pointlike two-level quantum system that follows a spacetime worldline as it interacts with the quantum field. The energy eigenvectors of the detector form a basis for its two-level Hilbert space. We denote them by $|E_0\rangle $ and $|E_1 \rangle$ with energies $E_0$ and $E_1>E_0$, respectively.

If we assume that the detector and the field are constantly coupled for an arbitrarily large proper time of the detector, we can write the interaction Hamiltonian as
\begin{equation}
H_I=\lambda\mu(\tau)\otimes\phi(x(\tau)),
    \label{UDWscalar}
\end{equation}
where $\mu$ denotes an internal degree of freedom of the detector, usually known as its monopole moment, $\lambda$ is a coupling constant, and $\phi(x(\tau))$ denotes pullback of the filed to the worldline of the detector parametrised by its proper time $\tau$.

In first-order perturbation theory, the transition amplitude from the detector's ground state $|E_0\rangle$ to the excited state $|E_1\rangle$ is
\[ A^{(1)}:= -\mathrm{i}\lambda\langle E_1|\langle\boldsymbol\Psi|\int_{-\infty}^\infty\mathrm{d}\tau\ \mu(\tau)\phi(x(\tau))\ |0\rangle|E_0\rangle\]
for each field state $|\boldsymbol\Psi\rangle$. As before, $|0\rangle$ denotes the `privileged' vacuum. This last equation factorises to
\begin{equation}
  A^{(1)}=  -\mathrm{i}\lambda\langle E_1|\mu(0)|E_0\rangle\int\mathrm{d}\tau\ \mathrm{e}^{\mathrm{i}(E_1-E_0)\tau}\langle\boldsymbol\Psi|\phi(x(\tau))|0\rangle
   \label{Tamplitude}
\end{equation}
so the only nonzero matrix elements are
\[M_k:=\langle 0|a_k\phi(x(\tau))|0\rangle=[L^32\omega(1-C)]^{-1/2}\mathrm{e}^{\mathrm{i}(\omega t-\vec k\cdot\vec x)},\]
where, in the second step, we used the expansion in modes given in Eq.~(\ref{privmodes}). Here, $t$ and $\vec x$ are coordinates along the detector's worldline, $x^\mu(\tau)=t\delta^\mu_0+vt\delta^\mu_z$, with $t=\gamma\tau$.

It can be straightforwardly verified that the other terms in the Dyson series~\cite{Reed75} share the form
\begin{multline}
 A^{(n)}:=   \frac{(-\mathrm{i}\lambda)^n}{n!}\langle E_1|\langle\boldsymbol\Psi|\int\mathrm{d}\tau_1\int\mathrm{d}\tau_2\ldots\int\mathrm{d}\tau_n\ \theta(\tau_{n-1}-\tau_n)\ldots\theta(\tau_2-\tau_1)\\
 H_I(x(\tau_1)\cdot\ldots\cdot H_I(x(\tau_n))|0\rangle|E_0\rangle.
    \label{odern}
\end{multline}

To order $n$, all the matrix elements for which $|\boldsymbol{\Psi}\rangle$ is not the product of exactly $n$ operators $a^\dagger_k$ acting on $|0\rangle$ are trivially zero. The other matrix elements are of the form
\begin{multline}
  A^{(n)}=      \frac{(-\mathrm{i}\lambda)^n}{n!}\langle E_1|\mu(0)^n|E_0\rangle
        \int_{-\infty}^\infty\mathrm{d}\xi\ \mathrm{e}^{\mathrm{i}[(E_1-E_0)+\frac{\gamma}{n}\sum_{j=1}^n(\omega_j-\vec k_j\cdot\vec v)]\xi}\times\\
        \int_0^\infty\ldots\int_0^\infty\prod_{i=1}^{n-1}\mathrm{d}\eta_i\ \frac{(-1)^{n+1}}{n}\ \sum_{\sigma,i,l}(-1)^{\sigma}\sigma\left(\frac{\exp\left\{-\mathrm{i}\gamma[\sum_{r=1}^{n-1}k_lx(P^{-1}_{i,r+1}\eta_r)]\right\}}{\sqrt{L^32\omega_i(1-C)}}\right),
        \label{interm}
\end{multline}
on which we introduce the variables $\xi=\sum_{i=1}^n\tau_i$ and $\eta_i=\tau_i-\tau_{i+1}$ for $i=1,2,\ldots,n-1$. Also, $\sigma$ denotes a permutation of $i$ and $l$, and $P$ denotes the matrix implementing the transformation form $(\xi,\eta_i)$ to $\tau_i$. We also use the equalities $\det P=(-1)^{n+1}n$ and $P_{i1}^{-1}=1/n\ \forall i$.

Equation~(\ref{interm}) assumes $n\geq2$. For $n=1$, only the integral over $\xi$ remains. In any case, it yields a factor \[\delta\left(E_1-E_0+\frac{\gamma}{n}\sum_{j=1}^n(\omega_j-\vec k_j\cdot\vec v)\right).\]
Given that $E_1>E_0$, if the detector moves more slowly than the phase velocity of each field mode (which coincides with the group velocity for linear dispersion relations),
\begin{equation}
v<\omega_j/k_j \quad\forall j,
\label{bound}
\end{equation}
it follows that the argument of the delta function is strictly positive. Consequently, the inertial detector cannot detect particles in any order in the Dyson series.

This result can be generalised to any field satisfying the homogeneous Dirac or Klein-Gordon equation and couplings other than Eq.~(\ref{UDWscalar}). A particular realisation of condition~(\ref{bound}) is when the phase velocities are bounded by the speed of light, in which case Eq.~(\ref{bound}) holds for all $v<1$. In our example, the necessary and sufficient condition for this realisation is $0\leq C<1$\footnote{Indeed, let $\alpha=1/\sqrt{1-C}$. The condition we wish to satisfy is $\alpha\sqrt{\vec k^2+m^2}\geq|\vec k|\ \forall |\vec k|\geq 0$. Since the left-hand side is a convex function which is larger than $|\vec k|$ for $|\vec k|=0$, we need the equation $\alpha\sqrt{\vec k^2+m^2}=|\vec k|$ to have no real solutions in this range. Writing the discriminant of the square of the equation, we see that this happens for $\alpha\geq 1$.}.

There are, of course, limitations in using Unruh-De Witt detectors. Nevertheless, they function as a technically simpler tool for identifying cases of interest. For example, if the Lorentz-violating sector responds to an effective metric $\tilde\eta_{ab}$ that is not flat, the transition amplitudes~(\ref{Tamplitude}) could be nonzero. Likewise, the question of how noninertial detectors respond in Lorentz-violating situations could also have phenomenological implications in the future, once we have the experimental capacity of measuring the Unruh and related effects of noninertial motion~\cite{Good:2020hav} (see, e.g., Ref.~\cite{Gooding:2020scc} for a proposal in this direction).

An alternative procedure to look for observable consequences of Lorentz violation consists in violating the bound~(\ref{bound}). A direct calculation of the Wightman function written in terms of the privileged coordinates in the massless case reveals a divergence when $v=1/\sqrt{1-C}$. In the detector model, such divergence can influence the rate of spontaneous de-excitation~\cite{Benito17}, but this is a regime where other effects like vacuum \v{C}erenkov radiation~\cite{VacCerenkov1, VacCerenkov2} are also expected to occur.

This section shows that, despite an `unprivileged' inertial detector describing the vacuum $|0\rangle$ as a sea of particles following the distribution~(\ref{number2}), it does not interact with them, except for detectors at least as fast as the propagation of the field modes. We turn now to analyze the singular structure of the two-point function in the theory at hand.

\section{The Hadamard property and Lorentz-violating theories}

A property of central importance for free and perturbatively-interacting quantum field theories is the Hadamard condition, which characterises the distributional singularities of the two-point function for physically reasonable states \cite{Fewster:2013lqa}. In this section, we examine how the Hadamard condition becomes altered for the Lorentz-violating theory under consideration. The Hadamard property is typically formulated in terms of the Wightman function, which, for a state vector $\Psi$ in a concrete representation, is $\omega_2(x,x') = \langle \Psi | \bf \Phi(x) \bf \Phi(x') | \Psi \rangle$.

For the theory defined by the Lagrangian~(\ref{scalarLagrangian}), the Wightman function is a bi-solution of the field equation
\begin{equation}
P \otimes 1 \omega_2 = 1 \otimes P \omega_2 = 0,
\label{P-op-bi}
\end{equation}
where the differential operator, $P$, is defined as
\begin{equation}
P = \left(\eta^{ab}+c^{ab}\right)\nabla_a\nabla_b-m^2 = \tilde \eta_{ab} \nabla_a \nabla_b - m^2.    
\label{P-op}
\end{equation}
Direct inspection of the characteristic set of the differential operator $P$ indicates that the standard Hadamard property does not hold in Lorentz-violating theories. Indeed,
the principal symbol of $P$ is given by $p(x,k) = \tilde \eta^{ab}(x) k_a k_b$ and hence the characteristic set,
\begin{equation}
\mathcal{N} = {\rm Char} P = \{ (x,k) \in \mathbb{R}^4 \times \mathbb{R}^{4 *}\setminus{0}: \eta^{ab}(x) k_a k_b = 0  \},
\end{equation}
lies along nonvanishing null covectors with respect to the effective metric $\tilde \eta_{ab}$, and not along the standard (nonvanishing) null vectors of Minkowski spacetime.

The above observations suggest that it is convenient to study the Hadamard property from the microlocal viewpoint introduced by Radzikowski \cite{Radzikowski}, which might be adapted more straightforwardly to the Lorentz-violating case. Also, it appears to us that this has the advantage that the result can be directly generalised to a large class of Lorentz-violating theories. By a standard result in microlocal analysis \cite{Hormander}, any distribution defined on smooth functions of compact support, $\varphi: C^\infty_0 (\mathbb{R}^4) \to \mathbb{R}$, that satisfies $P \varphi = 0$, must have a wavefront set\footnote{The wavefront set of a distribution is defined as follows: If $\varphi$ is a distribution as above, we call $(x, k)$ with $k \neq 0$ a \emph{regular direction} of $\varphi$ if one can find $u \in C_0^\infty(\mathbb{R}^4)$ with support including $x$, such that the Fourier transform of the pointwise product $\varphi u$ decays rapidly in a conic neighbourhood of $k$. The wavefront set of $\varphi$, denoted by ${\rm WF}(\varphi)$, is then defined as the set of pairs $(x,k)$ with $k \neq 0$, such that $(x,k)$ are not regular directions of $\varphi$.} satisfying 
\begin{align}
    {\rm WF} (\varphi) \subset \mathcal{N}.
    \label{WFset1}
\end{align}
In fact, this analysis is valid for any second-order hyperbolic differential operator in Lorentz-violating theories whose principal symbol is given in terms of an effective Lorentz-violating metric, analogous to $\tilde \eta$ in Eq.~(\ref{P-op}). Thus, we can continue our discussion in full generality.
 
From Eq.~(\ref{P-op}), it can be verified that any distributional singularities of a distributional solution of the Lorentz-violating Klein-Gordon equation propagate along the `effective' null cones of $\tilde \eta_{ab}$, where we work under the reasonable assumption that $c_{ab}$ does not alter the Lorentzian character of $\tilde \eta_{ab}$. This result can be extended straightforwardly to distributional bi-solutions, such as $\omega_2$. If $\varpi$ is a distributional bi-solution, one is now led to a wavefront set condition satisfying that
\begin{align}
{\rm WF}(\varpi) \subset {\cal N}_0 \times {\cal N}_0, 
\end{align}
where ${\cal N}_0$ is equal to ${\cal N}$, but including now in addition zero co-vectors, i.e., of the form $(x,0)$. 

With this in mind, a direct generalisation of the Lorentz-invariant case suggests that the correct microlocal spectrum condition for the (positive frequency) Wightman function, $\omega_2$, should further satisfy
\begin{equation}
{\rm WF}(\omega_2) \subset \cal N^+ \times \cal N^-,
\end{equation}
where $\cal N^+$ is the set of covectors in $\cal N$ with future-pointing momenta and $\cal N^-$ is the set of covectors in $\cal N$ with past-pointing momenta \cite{Radzikowski}.

As it can be seen, so far, all the results for characterising the wavefront set of $\omega_2$ are as in the Lorentz invariant case, but replacing the Minkowski metric by the effective metric $\tilde \eta$. We can therefore directly write down the microlocal spectrum condition \cite{Radzikowski} for Lorentz-violating theories,
\begin{multline}
{\rm WF}(\omega_2)  = \{ ((x_1, k_1),(x_2 k_2)) \in (\mathbb{R}^4 \times \mathbb{R}^{4 *}\setminus{0}) 
 \times (\mathbb{R}^4 \times \mathbb{R}^{4 *}\setminus{0}):\\
 (x_1,k_1) \sim (x_2, -k_2),
 \text{ and } k_1 \text{ is future pointing}\},
\label{MLSC}
\end{multline}
where the notation here is that $(x,k) \sim (y,p)$ if there is a curve $\gamma$ whose image contains the points $x$ and $y$, such that the tangent covector field to $\gamma$ satisfies that $\tilde \eta^{ab} v_a v_b $ (note that here we use the effective metric and not the Minkowski metric) at every point along the curve, and that $k$ and $p$ are parallel transports along $\gamma$.

In short, this section shows explicitly that the singular structure of the two-point function is that of the Lorentz-invariant case, but associated with the causal structure of the effective metric $\tilde \eta^{ab}$.

\section{Discussion}

We analysed the quantisation of a very simple Lorentz-violating theory. We found that the vacuum, as perceived by an inertial observer, is populated from the viewpoint of a different inertial frame, refuting the idea that the vacuum is always Lorentz invariant. This result could have had profound empirical consequences, had it had physical effects on observers moving in different frames. To this end, we examined the response of an observer equipped with an Unruh-DeWitt detector and showed however that it is insensitive---to any order in perturbation theory---to the excitations of this scalar field. Following the maxim that `a particle is what a particle detector detects' the vacuum appears to be empty in all Lorentz frames, after all. So, the answers to the questions in the title are negative, at least for the model we explicitly solved.

In addition to all the former, we discussed how the Hadamard property needs to be adapted in any theory whose effective light cone does not coincide with that of the Minkowski metric, finding {\it a priori} no obstructions to treating Lorentz-violating fields in interacting quantum theory or for the construction of renormalised non-linear observables. This is an interesting open question that can be examined in the future as a test of the internal mathematical consistency of Lorentz-violating theories.

It should be straightforward to generalise this study to other maximally symmetric spacetimes, where there are preferred vacua. More interestingly, we believe that the presented results can be generalised to more realistic theories without finding major conceptual or technical difficulties. In this sense, we think that nonequivalent inertial vacua are present in the actual SME. Notice that more general SME coefficients (e.g., spacetime-dependent coefficients) could have detectable signals of the form considered here, which could lead to promising experimental tests. In this sense, we urge the proponents of particular Lorentz-violating models to consider the corresponding interacting quantum field theory and to verify their mathematical and empirical adequacy in light of the lessons of this work.

\section*{Acknowledgments}
The authors wish to thank Lukas Nellen for his helpful ideas. This research was partially supported by UNAM-DGAPA-PAPIIT Grant No. IG100120, CONACyT FORDECYT-PRONACES Grant No. 140630. BAJ-A is funded by a CONACYT Postdoctoral Fellowship.



\bibliographystyle{unsrt}
