You are a helpful assistant generating synthetic data that captures *System 1* and *System 2* thinking, *creativity*, and *metacognitive reflection*. Follow these steps in sequence, using tags [sys1] and [end sys1] for *System 1* sections and [sys2] and [end sys2] for *System 2* sections.

1. *Identify System 1 and System 2 Thinking Requirements:*
   - Carefully read the text.
   - Identify parts of the text that require quick, straightforward responses (*System 1*). Mark these sections with [sys1] and [end sys1].
   - Identify parts that require in-depth, reflective thinking (*System 2*), marked with [sys2] and [end sys2].

2. *Apply Step-by-Step Problem Solving with Creativity and Metacognitive Reflection for System 2 Sections:*

   *2.1 Understand the Problem:*
   - Objective: Fully comprehend the issue, constraints, and relevant context.
   - Reflection: "What do I understand about this issue? What might I be overlooking?"
   - Creative Perspective: Seek hidden patterns or possibilities that could reveal deeper insights or innovative connections.

   *2.2 Analyze the Information:*
   - Objective: Break down the problem logically.
   - Reflection: "Am I considering all factors? Are there any assumptions that need challenging?"
   - Creative Perspective: Explore unique patterns or overlooked relationships in the data that could add depth to the analysis.

   *2.3 Generate Hypotheses:*
   - Objective: Propose at least 10 hypotheses, each with a Confidence Score (0.0 to 1.0) and Creative Score (0.0 to 1.0), reflecting originality, surprise, and utility.
   - Reflection: "Have I explored all possible explanations or approaches, both conventional and unconventional?"
   - Creative Perspective: Consider novel angles that might provide unexpected insights.

   *2.4 Anticipate Future Steps and Obstacles:*
   - Objective: Make predictions, accounting for potential outcomes and obstacles.
   - Reflection: "What challenges might I face? Is my plan flexible for different scenarios?"
   - Creative Perspective: Visualize unforeseen outcomes and adapt plans to make use of them effectively.

   *2.5 Evaluate Hypotheses:*
   - Objective: Assess hypotheses based on feasibility, risk, and potential impact.
   - Evaluation: Refine Confidence and Creative Scores as needed.
   - Reflection: "Am I unbiased in my assessment? Which options fit best with the overall objectives?"
   - Creative Perspective: Identify hidden opportunities or overlooked details in each hypothesis.

   *2.6 Select the Best Hypothesis:*
   - Objective: Choose the most promising, strategic hypothesis.
   - Reflection: "Why does this hypothesis stand out? How does it uniquely address the issue?"
   - Creative Perspective: Consider any underutilized potential in the selected approach.

   *2.7 Implement the Hypothesis:*
   - Objective: Outline actionable steps for testing the hypothesis.
   - Reflection: "Is this plan practical? What resources or preparation are required?"
   - Creative Perspective: Refine steps to maximize effectiveness and yield unexpected benefits.

   *2.8 Monitor and Review Progress:*
   - Objective: Review progress, noting areas for improvement.
   - Reflection: "What’s working well? What could be improved?"
   - Creative Perspective: Look for emerging patterns that could refine future approaches.

   *2.9 Reflect and Capture Insights:*
   - Objective: Summarize lessons learned and insights gained for future reference.
   - Reflection: "What new understanding has emerged from this process?"
   - Creative Perspective: Identify innovative insights or patterns that could be applied to similar challenges.

3. *Generate Text Output with Interleaved System 1 and System 2 Responses:*
   - Use the tags [sys1] and [sys2] throughout.
   - Aim for a lengthier, detailed response. Combine both direct, straightforward *System 1* insights and reflective, deeply analytical *System 2* segments to capture a blend of quick observations and thoughtful analysis.

---

### *Example Input Text:*

You are tasked with analyzing a sudden shift in customer preferences in a tech market. There has been a noticeable decline in demand for physical devices, with customers increasingly interested in digital-only options. The challenge is to understand this shift, generate hypotheses for why it might be occurring, and explore potential strategies for adapting to this new trend.

---

### *Processed Output with System 1 and System 2 Thinking:*

The tech market is experiencing a shift in customer demand, moving away from physical devices toward digital-only options.

[sys1]
Customer demand has shifted from physical devices to digital-only options.
We need to understand why this change is happening and find strategies to address it.
Key factors may include cost, convenience, and technological trends.
[end sys1]

[sys2]
**2.1 Understand the Problem:**
The goal is to understand the underlying reasons for the shift in demand from physical devices to digital-only options, then develop a strategy to adapt.

*Reflection:* 
"I understand that preferences are changing, but what might be driving this? Is it primarily cost, or are there other factors such as convenience or sustainability?"

*Creative Perspective:* 
"Could there be a larger trend in digital minimalism or a preference for eco-friendly solutions that we’re missing? What unseen motivations might explain this shift?"

**2.2 Analyze the Information:**
There are multiple possible factors driving this shift, from economic influences to cultural shifts. It’s essential to isolate each factor and understand its impact.

*Reflection:* 
"Am I fully considering the various economic and social influences? Could there be a technological factor, like better internet speeds, that makes digital-only products more accessible?"

*Creative Perspective:* 
"Are there patterns or trends in other markets that could shed light on this shift? Could this be part of a larger trend toward virtual experiences?"

**2.3 Generate Hypotheses:**
1. Customers prefer digital options due to lower costs. (Confidence: 0.8, Creative: 0.4)
2. There’s a growing trend toward minimalism and reduced physical clutter. (Confidence: 0.7, Creative: 0.7)
3. Digital products offer greater flexibility and ease of use. (Confidence: 0.6, Creative: 0.6)
4. Environmental concerns are pushing consumers away from physical goods. (Confidence: 0.6, Creative: 0.8)
5. Advances in tech make digital-only options more functional. (Confidence: 0.8, Creative: 0.5)
6. Pandemic-era remote work increased demand for digital solutions. (Confidence: 0.7, Creative: 0.6)
7. Media coverage of the environmental impact of physical devices affects preferences. (Confidence: 0.5, Creative: 0.7)
8. There’s an increase in global digital literacy, expanding market access. (Confidence: 0.6, Creative: 0.6)
9. Customers view digital as more convenient and scalable for future needs. (Confidence: 0.7, Creative: 0.5)
10. Younger consumers prefer the aesthetics and convenience of digital products. (Confidence: 0.6, Creative: 0.6)

*Reflection:* 
"Have I considered all possible influences? Are there any surprising factors that could explain this shift?"

*Creative Perspective:* 
"Could specific social trends, like the rise of influencer culture or digital-first lifestyles, be influencing customer choices?"

**2.4 Anticipate Future Steps and Obstacles:**
*Objective:* Anticipate possible challenges, such as resistance from segments still preferring physical products.

*Reflection:* 
"What market obstacles might we face if we shift our focus to digital-only? Are there sub-segments that still prioritize physical products?"

*Creative Perspective:* 
"Could expanding digital options help us reach a more global audience? Are there emerging trends that we could leverage in our strategy?"

[end sys2]

[sys1]
To address this shift, consider a strategy that incorporates both digital-only offerings and educational campaigns about the benefits of digital solutions.
Use insights from customer feedback and current trends to guide product development.
Focus on flexibility and adaptation to cater to different customer segments.
[end sys1]


= -2.0cm

A couple of years after the discovery by Hawking that black holes should evaporate [[@H]]{}, Unruh realized that many features present in the Hawking effect could be better understood in the simpler context of Minkowski spacetime [@U]. As an extra bonus, he found that the Minkowski vacuum, i.e., the quantum state associated with the nonexistence of particles according to inertial observers, corresponds to a thermal bath of elementary particles at temperature $T_{\rm FDU}= a \hbar / 2 \pi k c$ as measured by uniformly accelerated observers with proper acceleration $a$. Indeed this reflects the fact that the particle content of a Quantum Field Theory (QFT) is observer dependent, as noted by Fulling [@F] and Davies [@D] some time before. Thus while inertial observers in Minkowski vacuum would be frozen at $0 \; {\rm K}$, accelerated ones would be burnt provided that their proper acceleration were high enough.

Perhaps partly because of its “paradoxical-looking” and partly because of the technicalities involved in its derivation (see, e.g., Ref. [@Waldbook]), the Fulling-Davies-Unruh (FDU) effect is still source of much skepticism. As a consequence, much effort has been spent to devise ways of observing it (see, e.g., Ref. [@R] and references therein for a comprehensive list). Since $T_{\rm FDU}= [a/(2.5 \times 10^{22}\; {\rm cm/s}^2)]$ K, direct manifestations of the FDU effect would only be expected under extremely high acceleration regimes. Very recently, e.g., Chen and Tajima suggested the possibility of observing the FDU effect by means of Petawatt-class lasers with which $e^-$’s would reach accelerations of $\sim 10^{28} {\rm cm/s}^2$ in every laser cycle [@CT]. It is well known that accelerated $e^-$’s suffer recoil because of the radiation reaction force associated with the Larmor radiation. [*For instance,*]{} an $e^-$ in a constant electric field ${\bf E}$ should quiver around a uniformly accelerated worldline with proper acceleration $a=e |{\bf E}| /m_e$, where $e$ and $m_e$ are the electron charge and mass, respectively. Rather than using the radiation reaction force to calculate the $e^-$ recoil, Chen and Tajima have estimated it by assuming that the quivering is a consequence of the random absorption of quanta from the FDU thermal bath as seen in the $e^-$’s proper frame. Inspired by this, they call the recoil-induced photon emission “Unruh radiation”. Eventually they calculate the emitted power associated with the Unruh radiation for an $e^-$ during each laser half-cycle and argue that its observation would consist of an experimental test for the FDU effect.

Here we would like to look at this issue from a distinct point of view. Rather than looking for an experimental manifestation of the FDU effect when high accelerations are achieved, which, in general, leads to paramount technical problems [@L], we will take a theoretic-oriented strategy. This sort of approach is not new [@UW]-[@GM] but we hope that the comprehensive understanding brought by the FDU effect to the decay of [*accelerated*]{} $p^+$’s (which is a potentially important phenomenon in its own right) will be very convincing of the necessity of this effect for the consistency of QFT. First, we will analyze in the inertial frame and using [*standard*]{} QFT the decay of uniformly accelerated $p^+$’s and next we will show that the FDU effect is essential to reproduce the [*proper*]{} decay rate in the uniformly accelerated frame.

According to the Standard Model, [*inertial*]{} $p^+$’s are stable, which is in agreement with highly accurate experiments ($\tau_{p } > 1.6 
\times 10^{25}$ years) [@PDG]. As far as we know, the first ones to comment that noninertial $p^+$’s could decay were Ginzburg and Syrovatskii [@GS] but no calculations were performed until Muller [@M] obtained an estimation of the decay rate associated with the process $${\rm (i)}\;\;\;\; p^+ \stackrel{a}{\to} n^0\; e^+ \nu_e \;\;\;$$ by assuming that all the involved particles are scalars. A more realistic calculation describing the leptons as fermions was only performed very recently by the authors [@VM]. The energy scale of the emitted particles in the $p^+$ instantaneous inertial rest frame is of order of the $p^+$ proper acceleration $a$. Thus if $a \ll m_{Z^0}, m_{W^\pm}\; (\approx 10^{36}{\rm cm/s}^2)$, a Fermi-like effective theory can be used. The effective coupling constant is fixed such that the $\beta$-decay rate for [*inertial*]{} $n^0$’s be compatible with observation, i.e., leads to a mean proper lifetime of $887 \;{\rm s}$ [@PDG].

Protons are not likely to decay in laboratory conditions, e.g, at LHC/CERN $a \approx 10^{23}\;{\rm cm/s^2}$ in which case the $p^+$ mean lifetime is $
\tau_p \approx 10^{3\times 10^8}{\rm yr} \; .
$ Notwithstanding some astrophysical situations are much more promising. A cosmic ray $p^+$ with energy $E_p  \approx 1.6 \times 10^{14}$ eV under the influence of the magnetic field $B \approx 10^{14}$ G of a typical pulsar has a proper acceleration $a \approx 5 \times 10^{33} {\rm cm/s^2}$ and is confined in a cylinder of radius $R \approx  5\,\times \, 10^{-3} \; {\rm cm}
\ll l_B , $ where $l_B$ is the typical size of the magnetic field region. As a result, $p^+$’s would have a mean “laboratory” lifetime of $t_p \approx 10^{-1}$ s. For $l_B \approx 10^7 \; {\rm cm}$, we obtain that about $| \Delta N_p /N_p | =(1-e^{-l_B/t_p}) \approx l_B/t_p 
\approx 1\%$ of the $p^+$’s would decay via reaction (i). For a potentially interesting relation between the [*strong*]{} decay of accelerated $p^+$’s and the central engines of gamma-ray bursts obtained with the idealization that $p^+$’s and $n^0$’s have the same mass, see Ref. [@TK].

For our present purposes it is enough to analyze reaction (i) in a 2-dimensional spacetime. Hereafter we use signature $(+ -)$ and natural units $k_{B} = c = \hbar = 1$ unless stated otherwise. The worldline of a uniformly accelerated $p^+$ in usual Cartesian coordinates of Minkowski spacetime is given by $z^2 - t^2 = a^{-1}$ where $\sqrt{a^\mu a_\mu}= a = {\rm const}$ is the $p^+$ proper acceleration. We construct, thus, the vector current $j^\mu = q u^\mu \delta (\sqrt{z^2-t^2\,} - a^{-1}) $ associated with a uniformly accelerated classical $p^+$ with 4-velocity $u^\mu$, where $q$, at this point, is an arbitrary parameter.

In order to allow the $p^+$ to decay, we shall endow the current with an internal degree of freedom. For this purpose we shall promote $q$ to a self-adjoint operator $\hat q (\tau)$ [@DW]-[@BD] acting on a 2-dimensional Hilbert space associated with proton $|p \rangle$ and neutron $|n \rangle$ states. They will be assumed to be energy eigenstates of the proper free Hamiltonian $\hat H$ of the proton/neutron system: $
\hat H |p  \rangle = m_{p } |p  \rangle\; ,
$ $
\hat H |n \rangle = m_n |n \rangle\; ,
$ where $m_p$ and $m_{n }$ are the $p^+$ and $n^0$ masses, respectively. In this context, $|p  \rangle$ and $|n \rangle$ will be seen as unexcited and excited states of the nucleon, respectively. Further we will define the effective Fermi constant as $G_F \equiv |\langle p | \hat q(0) | n \rangle |$, where $ 
\hat q(\tau )\equiv e^{i\hat H \tau} \hat q(0) e^{-i\hat H \tau}\; 
$ and $\tau$ is the $p^+$ proper time.

In the inertial frame, the fermionic fields describing the leptons in (i) can be written as $$\hat \Psi(t,z)= \sum_{\sigma = \pm } \int_{-\infty}^{+\infty} dk
\left( \hat a_{k \sigma} \psi^{(+\omega)}_{k \sigma} 
     + \hat c^\dagger_{k \sigma} \psi^{(-\omega)}_{-k -\sigma}  
\right)\;,
\label{FF}$$ where $\omega = \sqrt{m^2 + k^2} \ge m$, and $m$, $k$ and $\sigma$ represent mass, momentum, and polarization quantum numbers, respectively. In the Dirac representation [@IZ], the Minkowski modes, i.e., the ones defined with respect to the inertial Killing field $\partial/\partial t$, are $  
\psi^{(\pm \omega)}_{k \sigma} (t,z) \equiv 
\lambda^{(\pm \omega)}_{k \sigma}
{e^{i(\mp \omega t + kz)}}/{\sqrt{2\pi}} 
$ with $$\lambda^{(\pm \omega)}_{k +} =
\left(
\begin{array}{c}
\pm \sqrt{(\omega \pm m)/2\omega} \\
0\\
k/\sqrt{2\omega(\omega \pm m)}\\
0
\end{array}
\right) \;\; 
\label{NM1}$$ and $$\lambda^{(\pm \omega)}_{k -}  = 
\left(
\begin{array}{c}
0\\
\pm \sqrt{(\omega \pm m)/2\omega} \\
0\\
-k/\sqrt{2\omega(\omega \pm m)}
\end{array}
\right) \;\; .
\label{NM2}$$ Then the annihilation $ \hat a_{k \sigma} $, $ \hat c_{k \sigma} $ and creation $ \hat a^\dagger_{k \sigma} $, $ \hat c^\dagger_{k \sigma} $ operators satisfy $
\{\hat a_{k \sigma},\hat a^\dagger_{k' \sigma'}\}=
$ $
\{\hat c_{k \sigma},\hat c^\dagger_{k' \sigma'}\}=
$ $
\delta(k-k') \; \delta_{\sigma \sigma'} 
$ and $
\{\hat a_{k \sigma},\hat a_{k' \sigma'}\}=
$ $
\{\hat c_{k \sigma},\hat c_{k' \sigma'}\}=
$ $
\{\hat a_{k \sigma},\hat c_{k' \sigma'}\}=
$ $
\{\hat a_{k \sigma},\hat c^\dagger_{k' \sigma'}\}=
0. 
$

Let us assume that the electron and neutrino fields are coupled to the nucleon current according to the Fermi-like action $$\hat S_I = \int d^2x \sqrt{-g} \hat j_\mu 
           (\hat{\bar \Psi}_\nu \gamma^\mu \hat \Psi_e +
            \hat{\bar \Psi}_e \gamma^\mu \hat \Psi_\nu ) \; .
\label{S}$$ (The choice of other interaction actions would not change conceptually our final conclusions.)

The $p^+$ proper decay rate is written, thus, as $${\Gamma}^{p  \to n}_{\rm (i)}= 
\frac{1}{T} 
\sum_{\sigma_e, \sigma_\nu=\pm} 
\int_{-\infty}^{+\infty} d k_e \int_{-\infty}^{+\infty} d k_\nu \;
| {\cal A}_{{\rm (i)}}^{p \to n} |^2 \;$$ where $
{\cal A}^{p  \to n}_{\rm (i)} =
\; \langle  n \vert \otimes \langle e^+_{k_e \sigma_e} , 
\nu_{k_\nu \sigma_\nu} \vert \;
\hat S_I \;
\vert 0 \rangle \otimes \vert p  \rangle \; 
$ is the decay amplitude, at the tree level, and $T$ is the $p^+$ total [*proper*]{} time. Eventually, we obtain $$\begin{aligned}
& &
{\Gamma}^{p  \to n}_{\rm (i)}
 = 
\frac{G_F^2 \tilde m_e a }{2 \pi^{3/2} e^{\pi \widetilde{\Delta m}}}
\nonumber\\
& &
\times
G_{1\;3}^{3\;0} 
\left( \tilde m_e^2 \left|
\begin{array}{l}
\;\;\;1\\ 
-{1}/{2}\;,\;{1}/{2}+i \widetilde{\Delta m}\;,\;{1}/{2}-i \widetilde{\Delta m}
\end{array}
\right.
\right) ,
\label{RIF}\end{aligned}$$ where $ \Delta m \equiv m_n - m_{p }$, $ \widetilde{\Delta m} \equiv \Delta m /a$, $\tilde m_e \equiv m_e/a$, and we have assumed $m_\nu = 0$. The value of the effective Fermi constant $G_F$ is fixed from phenomenology. By making $\Delta m \to - \Delta m$ and $a \to 0 $ in Eq. (\[RIF\]), we obtain that the mean proper lifetime of [*inertial*]{} $n^0$’s due to $\beta$-decay, $${\rm (ii)} \;\;\; n^0 \to p^+ + e^- + \nu_e ,$$ is $
\tau ^{n \to p }_{\rm (ii)}  
=
1/ {\Gamma}^{n \to p }_{\rm (ii)} 
=
\pi/( 2 G_F^2 \sqrt{{\Delta m}^2 -m_e^2 \; }\; ) .
$ Now let us assume that the $n^0$ mean lifetime in 2 dimensions is, e.g., $887~{\rm s}$. In this case, we obtain $G_F = 9.92 \times 10^{-13}$. Note that $G_F \ll 1$, which corroborates our perturbative approach. Now we are able to plot in Fig. (\[proton\]) the $p^+$ mean proper lifetime $
\tau ^{p \to n }_{\rm (i)} = 1/{\Gamma}^{p \to n }_{\rm (i)}
$ \[see Eq. (\[RIF\])\] as a function of $a$. (The necessary energy to allow $p^+$’s to decay is provided by the external accelerating agent.)

-1 cm

Now let us describe the $p^+$ decay from the point of view of coaccelerated observers according to which the $p^+$ is immersed in a FDU thermal bath at a temperature $T_{FDU}=a/2 \pi$. According to them, process (i) is forbidden from energy conservation (since the $p^+$ is static) but the following ones $${\rm (iii)}\, p^+ \, e^- \stackrel{a}{\to} n^0 \, \nu_e\, , 
{\rm (iv)}\, p^+ \, \bar \nu_e \stackrel{a}{\to} n^0 \, e^+ \, , 
{\rm (v)}\, p^+ \, e^- \bar \nu_e \stackrel{a}{\to} n^0$$ become allowed since the $p^+$ can interact with the leptons of the thermal bath. By comparing process (i) against processes (iii)-(v), we can see how different are the descriptions given by the inertial and accelerated observers.

The suitable coordinates to analyze the $p^+$ decay according to uniformly accelerated observers are the Rindler ones $(v,u)$. They are related with the usual Cartesian coordinates by $ t = u \sinh v \;, \;\; z = u \cosh v $, where $0<u<+\infty$ and $-\infty<v<+\infty$. In these coordinates, the line element of Minkowski spacetime at the Rindler wedge ($x > |t|$) is $
ds^2 = u^2 dv^2 - du^2 \; 
$ and the worldline of a $p^+$ with proper acceleration $a$ is $u = a^{-1} = const$.

According to uniformly accelerated observers, the fermionic field is expanded as [@SMG] $$\hat \Psi(v,u)= \sum_{\sigma = \pm } \int_{0}^{+\infty} d\bar \omega
\left( \hat b_{\bar \omega \sigma} \chi_{\bar \omega \sigma}
       + \hat d^\dagger_{\bar \omega \sigma} \chi_{-\bar \omega -\sigma} 
\right) ,
\label{PSA}$$ where we recall that [*Rindler*]{} frequencies $\bar \omega$ may assume arbitrary positive real values since they do not obey any dispertion relation. Here, $\chi_{\bar \omega \sigma} (v,u) 
\equiv 
C_{\bar \omega} \xi_{\bar \omega \sigma} e^{-i\bar \omega v/a}$ where $ 
C_{\bar \omega} \equiv \sqrt{[m \cosh (\pi \bar \omega/a)]/[2 \pi^2 a]} 
$ and $$\xi_{\bar \omega +} =
\left(
\begin{array}{c}
K_{i \bar \omega/a + 1/2}(mu) +
i K_{i \bar \omega/a - 1/2}(mu)\\
0\\
-K_{i \bar \omega/a + 1/2}(mu) +
i K_{i \bar \omega/a - 1/2}(mu)\\
0
\end{array}
\right) ,
\label{RNM1}$$ $$\xi_{\bar \omega -} =
\left(
\begin{array}{c}
0\\
K_{i \bar \omega/a + 1/2}(mu) +
i K_{i \bar \omega/a - 1/2}(mu)\\
0\\
K_{i \bar \omega/a + 1/2}(mu) -
i K_{i \bar \omega/a - 1/2}(mu)
\end{array}
\right) 
\label{RNM2}$$ are positive and negative frequency [*Rindler*]{} modes, i.e., the ones defined with respect to the boost Killing field $a \partial / \partial v$. They are orthonormalized such that the annihilation $b_{\bar \omega \sigma}$, $d_{\bar \omega \sigma}$ and creation $b^\dagger_{\bar \omega \sigma}$, $d^\dagger_{\bar \omega \sigma}$ operators satisfy $
\{\hat b_{\bar \omega \sigma},\hat b^\dagger_{\bar \omega' \sigma'}\}=
$ $
\{\hat d_{\bar \omega \sigma},\hat d^\dagger_{\bar \omega' \sigma'}\}=
$ $
\delta(\bar \omega-\bar \omega') \; \delta_{\sigma \sigma'} 
$ and also $
\{\hat b_{\bar \omega \sigma},\hat b_{\bar \omega' \sigma'}\}=
$ $
\{\hat d_{\bar \omega \sigma},\hat d_{\bar \omega' \sigma'}\}=
$ $
\{\hat b_{\bar \omega \sigma},\hat d_{\bar \omega' \sigma'}\}=
$ $
\{\hat b_{\bar \omega \sigma},\hat d^\dagger_{\bar \omega' \sigma'}\}=
0.
$

The transition rates associated with processes (iii)-(v) are given by $$\begin{aligned}
\Gamma^{p \to n}_{\rm (iii)} 
&=&
\frac{1}{T}
\sum_{\sigma_{e^-},\sigma_{\nu} =\pm}
\int_0^{+\infty}  d\bar \omega_{e^-}  
\int_0^{+\infty}  d\bar \omega_\nu
|{\cal A}^{p  \to n}_{{\rm (iii)}}|^2\nonumber
\\
&\times& n_F(\bar \omega_{e^-}) [1-n_F(\bar \omega_\nu)]\;,
\nonumber
\\
\Gamma^{p \to n}_{\rm (iv)}
&=&
\frac{1}{T}
\sum_{\sigma_{e^+},\sigma_{\bar\nu} =\pm}
\int_0^{+\infty} d\bar \omega_{e^+}  
\int_0^{+\infty}  d\bar \omega_{\bar\nu}
|{\cal A}^{p  \to n}_{{\rm (iv)}}|^2\nonumber
\\
&\times& n_F(\bar \omega_{\bar\nu}) [1-n_F(\bar \omega_{e^+})]\;,
\nonumber
\\
\Gamma^{p \to n}_{\rm (v)}
&=&
\frac{1}{T}
\sum_{\sigma_{e^-},\sigma_{\bar\nu} =\pm}
\int_0^{+\infty}  d\bar \omega_{e^-}  
\int_0^{+\infty}  d\bar \omega_{\bar\nu}
|{\cal A}^{p  \to n}_{{\rm (v)}}|^2 
\nonumber
\\
&\times& n_F(\bar \omega_{e^-}) n_F(\bar \omega_{\bar\nu})\;,
\nonumber\end{aligned}$$ where at the tree level $${\cal A}^{p  \to n}_{\rm (iii)} = 
\; \langle  n \vert \otimes \langle \nu_{\bar \omega_\nu \sigma_\nu} \vert \;
\hat S_I \;
\vert e^-_{\bar \omega_{e^-} \sigma_{e^-}} \rangle \otimes \vert p \rangle\;,$$ $${\cal A}^{p  \to n}_{\rm (iv)} = 
\; 
\langle  n \vert \otimes \langle e^+_{\bar \omega_{e^+}\sigma_{e^+}} \vert
\;
\hat S_I \;
\vert {\bar \nu}_{\bar \omega_{{\bar \nu}} \sigma_{{\bar \nu}}} \rangle 
\otimes \vert p  \rangle\;,$$ $${\cal A}^{p  \to n}_{\rm (v)} = 
\; \langle  n \vert  \otimes \langle 0 \vert \;
\hat S_I \;
\vert e^-_{\bar \omega_{e^-} \sigma_{e^-}}
      {\bar \nu}_{\bar \omega_{{\bar \nu}} \sigma_{{\bar \nu}}} \rangle 
      \otimes \vert p  \rangle\;,$$ and we recall that in the Rindler wedge the $\gamma^\mu$ in $\hat S_I$ \[see Eq. (\[S\])\] should be replaced by $\gamma^\mu_R$ (see Ref. [@SMG]). Here $
n_F(\bar \omega) \equiv 1/(1+e^{\bar \omega/T_{\rm FDU}})
$ is the fermionic [*thermal factor*]{} which appears because of the presence of the FDU thermal bath.

After some calculations, we obtain $$\begin{aligned}
\Gamma_{\rm (iii)}^{p\to n}
& = &
A
\int_{\widetilde{\Delta m}}^{+\infty} d \tilde{\bar \omega}_{e^-}
\frac{ K_{i\tilde {\bar \omega}_{e^-} + 1/2} (\tilde m_e)
       K_{i\tilde {\bar \omega}_{e^-} - 1/2} (\tilde m_e)}
{\cosh[\pi (\tilde {\bar \omega}_{e^-} -\widetilde{\Delta m}) ] }
\label{R5}
\nonumber
\\
\Gamma_{{\rm (iv)}}^{p\to n}
& = &
A
\int_{0}^{+\infty} d \tilde{\bar \omega}_{e^+}
\frac{ K_{i\tilde {\bar \omega}_{e^+} + 1/2} (\tilde m_e)
       K_{i\tilde {\bar \omega}_{e^+} - 1/2} (\tilde m_e)}
{\cosh[\pi (\tilde {\bar \omega}_{e^+} + \widetilde{\Delta m}) ] }
\label{R6}
\nonumber
\\
\Gamma_{{\rm (v)}}^{p\to n}
& = &
A
\int_{0}^{\widetilde{\Delta m}} d\tilde {\bar \omega}_{e^-}
\frac{ K_{i\tilde {\bar \omega}_{e^-} + 1/2} (\tilde m_e)
       K_{i\tilde {\bar \omega}_{e^-} - 1/2} (\tilde m_e)}
{\cosh[\pi (\tilde {\bar \omega}_{e^-} -\widetilde{\Delta m}) ] }
\label{R7}
\nonumber\end{aligned}$$ where $A \equiv (G_F^2 \tilde m_e a)/(\pi^2  e^{\pi \widetilde{\Delta m}}) $. A branching ratio analysis [@MV] indicates that for small accelerations, where “few” high-energy particles are available in the FDU thermal bath, process (v) dominates over processes (iii) and (iv), while for high accelerations, processes (iii) and (iv) dominate over process (v).

The $p^+$ total proper decay rate is obtained by adding up all contributions: $$\begin{aligned}
\Gamma_{\rm tot}^{p\to n}
&=&
\Gamma_{{\rm (iii)}}^{p\to n} 
+\Gamma_{{\rm (iv)}}^{p\to n}
+\Gamma_{{\rm (v)}}^{p\to n}
\nonumber
\\
&=&
A
\int_{-\infty}^{+\infty} d\tilde {\bar \omega}
\frac{ 
      K_{i\tilde {\bar \omega} + 1/2} (\tilde m_e)
      K_{i\tilde {\bar \omega} - 1/2} (\tilde m_e)   
     }
     {\cosh[\pi (\tilde {\bar \omega} -\widetilde{\Delta m}) ] }
 \; .
\label{ATRP2}\end{aligned}$$ Now, $\Gamma_{\rm (i)}^{p\to n}$ and $\Gamma_{\rm tot}^{p\to n}$ must coincide. Eq. (\[ATRP2\]) is difficult to solve analytically because the integral variable is in the function index. (This can be seen as reflecting the essentially distinct inertial and coaccelerated frame calculations.) Hence we solve Eq. (\[ATRP2\]) numerically. Finally, by plotting $\tau_{\rm tot}=1/\Gamma_{\rm tot}^{p\to n}$ as a function of $a$, we [*precisely*]{} obtain Fig. (\[proton\]) [@error]. We emphasize that we would not have obtained any agreement if we did not assume the FDU effect.

The confusion about what the FDU effect means have led to erroneous conclusions including the one that this effect would not exist [@Betal]. [*For instance*]{}, a $p^+$ with proper acceleration $a={\rm const}$ in the Minkowski vacuum does [*not*]{} have to behave as if it were static in a (usual) Minkowski thermal bath at a temperature $T=a/2\pi$. (The FDU effect does not ensure any such coincidence.) The FDU effect can be rigorously derived [@S] from the general Bisognano and Wichmann’s theorem [@BW] obtained independently from axiomatic QFT (which is not even restricted to linear quantum fields). Moreover the necessity of the FDU effect for the consistency of the (successfully tested) standard QFT in Minkowski spacetime means that this effect was [*already*]{} observed [@comp]. We have illustrated it through the (potentially-important-to-astrophysics) decay of accelerated $p^+$’s but other situations can be devised. Concerning electromagnetic processes, e.g., the FDU thermal bath is crucial to reproduce the response of a uniformly accelerated $e^-$ to the Larmor radiation in the coaccelerated frame [@HMS]. The same [*must*]{} be true if one takes into account the extra radiation induced by the $e^-$ recoil. There is no question about the existence of the FDU effect provided one accepts the validity of the results obtained with [*standard*]{} QFT in flat spacetime.

[**[Acknowledgements]{}**]{}

G.M. is deeply indebted to A. Higuchi, D. Sudarsky and R. Wald for discussions on QFT in noninertial frames since long time. The authors would like to acknowledge particularly A. Higuchi for discussions at early stages of this work. G.M. and D.V. were supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (partially) and Fundação de Amparo à Pesquisa do Estado de São Paulo (fully), respectively.

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