11institutetext:  Center for the Studies of the Glass Bead Game, Notting Hill,
Victoria 3168, Australia
Center for Theoretical Physics, College of Physics, Sichuan University,
Chengdu, 610064, China

# Spin-half bosons with mass dimension three half: Evading the spin-statistics
theorem

Dharam Vir Ahluwalia 11 Cheng-Yang Lee 221122

###### Abstract

By exploiting the freedom in defining the dual of spinors, we report an
unexpected theoretical discovery of a quantum field theory of spin-half
bosons. It fulfils Dirac’s 1969-70 observation that “there must be boson
variables connected with electrons.” The theory is local, Lorentz-invariant,
and has a positive-definite Hamiltonian. We formulate the unitarity-preserving
scattering theory to accommodate the new dual and the associated adjoint. A
model of Yukawa interaction with spin-half bosons and fermions of equal masses
is studied to explicitly show unitarity.

  * There must be such boson variables connected with electrons.

  * P. A. M. Dirac [1]

##  1 Introduction

With ninety five percent of the existence dark, in one way or another, its
particle landscape cannot be a mere copy of one sort or the other of the
Standard Model of the high energy physics (SM). This may appear as an
audacious thought to many physicists. But the unexpected theoretical
developments since 2004 [2, 3, 4], coupled with a lack of empirical signature
for supersymmetry, suggests to consider the unexpected as a first-principle
scenario for their darkness [4].

Recently, a new class of fundamental particles have been reported in [5, 6].
While it is not widely known, Dirac himself argued for electrons to have a
bosonic partner [1]. Here, by exploiting a freedom in defining the dual of
spinors (and the associated adjoint), we construct a theory of spin-half
bosonic particles. These bosons cannot enter the SM doublets because of the
statistics mismatch. The natural place for them is the dark-matter sector. As
a by-product, they cancel the fermionic contributions to the cosmological
constant if for every Dirac fermion of the SM there exists a here-reported
bosonic partner of the same mass.

##  2 Quantum field theory of spin-half bosons

In the SM, the Lagrangian density, at the level of kinematics, is assumed
rather than derived. The dynamics enters through the principle of local gauge
invariance – the choice of symmetry group is then chosen on the basis of the
hints hiding in the phenomenology.

One exception to this general wisdom is to derive, rather than assume, both
the quantum field and the Lagrangian density. This approach was initiated in
1964 by Steven Weinberg, and reached the textbook level in his monograph – The
Quantum Theory of Fields [7].

His approach put the SM on a firm theoretical footing by unifying quantum
mechanics with Poincaré space-time symmetries. An important departure from
this approach occurred when mass dimension one fermions were first constructed
[8, 2]. Here we use the results from both approaches to construct a bosonic
quantum field of spin-half, and the associated Lagrangian density.

Without assuming a Lagrangian density, we begin with the rest spinors for
spin-half derived by Weinberg in the $\mathcal{R}\oplus\mathcal{L}|_{j=1/2}$
representation space [7]

$\xi_{1}(\mbox{\boldmath$\displaystyle\boldsymbol{0}$})=\sqrt{m}\left[\begin{array}[]{c}1\\\
0\\\ 1\\\
0\end{array}\right],~{}\xi_{2}(\mbox{\boldmath$\displaystyle\boldsymbol{0}$})=\sqrt{m}\left[\begin{array}[]{c}0\\\
1\\\ 0\\\ 1\end{array}\right],$ (1)

and

$\xi_{3}(\mbox{\boldmath$\displaystyle\boldsymbol{0}$})=\sqrt{m}\left[\begin{array}[]{c}0\\\
1\\\ 0\\\
-1\end{array}\right],~{}\xi_{4}(\mbox{\boldmath$\displaystyle\boldsymbol{0}$})=\sqrt{m}\left[\begin{array}[]{c}-1\\\
0\\\ 1\\\ 0\end{array}\right].$ (2)

These coincide with the eigenspinors of $\Omega_{5}=\gamma_{0}$, the fifth
square root of the identity matrix in the
$\mathcal{R}\oplus\mathcal{L}|_{j=1/2}$ representation space [6]. The
$\xi_{i}(\mbox{\boldmath$\displaystyle\boldsymbol{p}$})$ for an arbitrary
momentum are obtained by acting the $\mathcal{R}\oplus\mathcal{L}|_{j=1/2}$
boost operator

$\mathcal{D}(\mbox{\boldmath$\displaystyle\boldsymbol{p}$})=\sqrt{\frac{E+m}{2m}}\left[\begin{array}[]{cc}\mathbb{I}+\frac{\boldsymbol{\sigma\cdot
p}}{E+m}&\mbox{\boldmath$\displaystyle\mathbb{O}$}\\\
\mbox{\boldmath$\displaystyle\mathbb{O}$}&\mathbb{I}-\frac{\boldsymbol{\sigma\cdot
p}}{E+m}\end{array}\right]$ (3)

on $\xi_{i}(\mbox{\boldmath$\displaystyle\boldsymbol{0}$})$:

$\xi_{i}(\mbox{\boldmath$\displaystyle\boldsymbol{p}$})=\mathcal{D}(\mbox{\boldmath$\displaystyle\boldsymbol{p}$})\xi_{i}(\mbox{\boldmath$\displaystyle\boldsymbol{0}$}),\quad
i=1,\cdots,4.$ (4)

The crucial point of departure enters by defining a new dual [6, 9]

$\displaystyle\overset{\>{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\xi}}_{i}(\mbox{\boldmath$\displaystyle\boldsymbol{p}$})$
$\displaystyle=\big{[}\mathcal{P}\,\xi_{i}(\mbox{\boldmath$\displaystyle\boldsymbol{p}$})\big{]}^{\dagger}\gamma_{0}$
(5)
$\displaystyle=\begin{cases}+\big{[}\xi_{i}(\mbox{\boldmath$\displaystyle\boldsymbol{p}$})\big{]}^{\dagger}\gamma_{0},\quad
i=1,2\\\
-\big{[}\xi_{i}(\mbox{\boldmath$\displaystyle\boldsymbol{p}$})\big{]}^{\dagger}\gamma_{0},\quad
i=3,4\end{cases}$ (6)

where the covariant parity operator is defined as [10, 3]

$\mathcal{P}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}m^{-1}\gamma_{\mu}p^{\mu}.$
(7)

With the dual thus defined, the orthonormality relations and spin sums are
found to be

$\overset{\>{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\xi}}_{i}(\mbox{\boldmath$\displaystyle\boldsymbol{p}$})\,\xi_{j}(\mbox{\boldmath$\displaystyle\boldsymbol{p}$})=2m\delta_{ij},$
(8)

and

$\displaystyle\sum_{i=1,2}\xi_{i}(\mbox{\boldmath$\displaystyle\boldsymbol{p}$})\overset{\>{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\xi}}_{i}(\mbox{\boldmath$\displaystyle\boldsymbol{p}$})=\gamma_{\mu}p^{\mu}+m\mathbb{I},$
(9)
$\displaystyle\sum_{i=3,4}\xi_{i}(\mbox{\boldmath$\displaystyle\boldsymbol{p}$})\overset{\>{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\xi}}_{i}(\mbox{\boldmath$\displaystyle\boldsymbol{p}$})=-(\gamma_{\mu}p^{\mu}-m\mathbb{I}).$
(10)

We now use the $\xi_{i}(\mbox{\boldmath$\displaystyle\boldsymbol{p}$})$ as
expansion coefficients for a spin-half quantum field

$\displaystyle\mathfrak{a}(x)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\int$
$\displaystyle\frac{d^{3}p}{(2\pi)^{3/2}}\frac{1}{\sqrt{2E}}{\Bigg{[}}\sum_{i=1,2}a_{i}(\mbox{\boldmath$\displaystyle\boldsymbol{p}$})\xi_{i}(\mbox{\boldmath$\displaystyle\boldsymbol{p}$})e^{-ip\cdot
x}$
$\displaystyle+\sum_{i=3,4}b^{\dagger}_{i}(\mbox{\boldmath$\displaystyle\boldsymbol{p}$})\xi_{i}(\mbox{\boldmath$\displaystyle\boldsymbol{p}$})e^{ip\cdot
x}{\Bigg{]}}.$ (11)

This field is same as the usual spin-half field, of Dirac. The associated
statistics follows only after its adjoint is defined. Instead of the Dirac
adjoint, we introduce a new adjoint that exploits the new duals (6)

$\displaystyle\overset{\>{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\mathfrak{a}}}(x)\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\int$
$\displaystyle\frac{d^{3}p}{(2\pi)^{3/2}}\frac{1}{\sqrt{2E}}{\Bigg{[}}\sum_{i=1,2}a^{\dagger}_{i}(\mbox{\boldmath$\displaystyle\boldsymbol{p}$})\overset{\>{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\xi}}_{i}(\mbox{\boldmath$\displaystyle\boldsymbol{p}$})e^{ip\cdot
x}$
$\displaystyle+\sum_{i=3,4}b_{i}(\mbox{\boldmath$\displaystyle\boldsymbol{p}$})\overset{\>{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\xi}}_{i}(\mbox{\boldmath$\displaystyle\boldsymbol{p}$})e^{-ip\cdot
x}{\Bigg{]}}$ (12)

where

$\displaystyle[a_{i}(\mbox{\boldmath$\displaystyle\boldsymbol{p}$}),a_{j}^{\dagger}(\mbox{\boldmath$\displaystyle\boldsymbol{p}$}^{\prime})]_{\pm}=\delta^{3}\left(\mbox{\boldmath$\displaystyle\boldsymbol{p}$}-\mbox{\boldmath$\displaystyle\boldsymbol{p}$}^{\prime}\right)\delta_{ij},$
$\displaystyle[a_{i}(\mbox{\boldmath$\displaystyle\boldsymbol{p}$}),a_{j}(\mbox{\boldmath$\displaystyle\boldsymbol{p}$}^{\prime})]_{\pm}=[a^{\dagger}_{i}(\mbox{\boldmath$\displaystyle\boldsymbol{p}$}),a^{\dagger}_{j}(\mbox{\boldmath$\displaystyle\boldsymbol{p}$}^{\prime})]_{\pm}=0.$
(13)

The upper and lower signs represent anti-commutator and commutator
respectively. We assume the same for
$b_{i}(\mbox{\boldmath$\displaystyle\boldsymbol{p}$})$ and
$b_{i}^{\dagger}(\mbox{\boldmath$\displaystyle\boldsymbol{p}$})$.

To establish the statistics we follow the procedure outlined in [11]. Consider
two space-like separated events $x$ and $y$, and evaluate the amplitude for
the particle, described by the
$\mathfrak{a}(x)$-$\overset{\>{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\mathfrak{a}}}(x)$
system, to propagate from $x$ to $y$ (with $y_{0}>x_{0}$). Since we are
considering space-like separations, there exist a class of observers for whom
$x_{0}>y_{0}$. We call the former observers to be $\mathcal{O}$ and the latter
to be $\mathcal{O}^{\prime}$. For the observers in $\mathcal{O}^{\prime}$, we
calculate the amplitude for an antiparticle to propagate from $y$ to $x$.
Causality requires that these two amplitudes differ at most by a phase [11]

$\displaystyle\mbox{Amp}(x\to y,~{}\mbox{particle})$
$\displaystyle|_{\mathcal{O}}$ $\displaystyle=e^{i\theta}\mbox{Amp}(y\to x,$
$\displaystyle~{}\mbox{antiparticle})|_{\mathcal{O^{\prime}}}$ (14)

where $\theta\in\Re$.111The ‘amplitudes’ in (14) are dimensionful. It is an
artefact of the fact that creation and destruction operators in the quantum
field theory – for historical reasons – carry dimensions. An explicit
calculation of the amplitudes in (14) using spin sums (9) and (10) while
keeping track of the space-like nature of the $(x-y)$ separation, yields
$e^{i\theta}=1$. Thus, we must select the bosonic statistics. This is an
unavoidable consequence of introducing the new dual and adjoint in the
$\mathfrak{a}(x)$-$\overset{\>{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\mathfrak{a}}}(x)$
system. It evades the spin-statistics theorem [12, 13]. The crevice in the
theorem is the following: the new adjoint renders the interacting Hamiltonian
non-hermitian in a manner that still keeps the spectrum real.

With the
$\mathfrak{a}(x)$-$\overset{\>{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\mathfrak{a}}}(x)$
system now fully specified, the Feynman-Dyson propagator for the new bosons,
up to a factor of $i$, is simply the vacuum expectation value of the time
order product

$\displaystyle\langle~{}|\mathfrak{T}[\mathfrak{a}(x^{\prime})\overset{\>{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\mathfrak{a}}}(x)]|~{}\rangle=$
$\displaystyle\langle~{}|\mathfrak{a}(x^{\prime})\overset{\>{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\mathfrak{a}}}(x)|~{}\rangle\theta(t^{\prime}-t)$
$\displaystyle+\langle~{}|\overset{\>{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\mathfrak{a}}}(x)\mathfrak{a}(x^{\prime})|~{}\rangle\theta(t-t^{\prime})$

with $|~{}\rangle$ as the vacuum state and $\mathfrak{T}$ as the time ordering
operator. The two vacuum expectation values that appear in the right-hand side
of the above expression evaluate, respectively to

$\displaystyle\langle~{}|\mbox{$\displaystyle\mathfrak{a}$}(x^{\prime})\mbox{$\overset{\>{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\displaystyle\mathfrak{a}}}$}(x)|~{}\rangle=\int\frac{d^{3}p}{(2\pi)^{3}}$
$\displaystyle\left(\frac{1}{2E}\right)e^{-ip\cdot(x^{\prime}-x)}$
$\displaystyle\times(\gamma_{\mu}p^{\mu}+m\mathbb{I})$ (15)

and

$\displaystyle\langle~{}|\mbox{$\overset{\>{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\displaystyle\mathfrak{a}}}$}(x)\mbox{$\displaystyle\mathfrak{a}$}(x^{\prime})|~{}\rangle=\int\frac{d^{3}p}{(2\pi)^{3}}$
$\displaystyle\left(\frac{1}{2E}\right)e^{-ip\cdot(x-x^{\prime})}$
$\displaystyle\times\left[-(\gamma_{\mu}p^{\mu}-m\mathbb{I})\right]$ (16)

where we have used the spin sums (9) and (10). Substituting the two vacuum
expectation values into the expression for
$\langle~{}|\mathfrak{T}[\mathfrak{a}(x^{\prime})\overset{\>{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\mathfrak{a}}}(x)]|~{}\rangle$
and using the integral representation of the step function

$\theta(t)=\frac{1}{2\pi
i}\int^{\infty}_{-\infty}ds\frac{e^{ist}}{s-i\epsilon},$ (17)

we obtain

$\displaystyle\langle~{}|\mathfrak{T}[\mathfrak{a}(x^{\prime})\overset{\>{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\mathfrak{a}}}(x)]|~{}\rangle=$
$\displaystyle\frac{i}{(2\pi)^{4}}\int d^{4}q\,e^{-iq\cdot(x^{\prime}-x)}$
$\displaystyle\times\frac{\gamma_{\mu}q^{\mu}+m\mathbb{I}}{q\cdot
q-m^{2}+i\epsilon}.$ (18)

Consequently, the mass dimension of the new field – as referred to the
$\mathfrak{a}(x)$-$\overset{\>{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\mathfrak{a}}}(x)$
system – is three half, with the Lagrangian density

$\mathcal{L}(x)=\mbox{$\overset{\>{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\displaystyle\mathfrak{a}}}$}(x)\left(i\gamma^{\mu}\partial_{\mu}-m\mathbb{I}\right)\mbox{$\displaystyle\mathfrak{a}$}(x)$
(19)

from which we obtain the canonical equal-time commutators (thus establishing
locality)

$\displaystyle\left[\mbox{$\displaystyle\mathfrak{a}$}(t,\mbox{\boldmath$\displaystyle\boldsymbol{x}$}),\mbox{$\displaystyle\mathfrak{a}$}(t,\mbox{\boldmath$\displaystyle\boldsymbol{y}$})\right]=\mbox{\boldmath$\displaystyle\mathbb{O}$}=\left[\mathfrak{p}(t,\mbox{\boldmath$\displaystyle\boldsymbol{x}$}),\mathfrak{p}(t,\mbox{\boldmath$\displaystyle\boldsymbol{y}$})\right],$
(20)
$\displaystyle\left[\mbox{$\displaystyle\mathfrak{a}$}(t,\mbox{\boldmath$\displaystyle\boldsymbol{x}$}),\mathfrak{p}(t,\mbox{\boldmath$\displaystyle\boldsymbol{y}$})\right]=i\delta^{3}(\mbox{\boldmath$\displaystyle\boldsymbol{x}$}-\mbox{\boldmath$\displaystyle\boldsymbol{y}$})\mathbb{I},$
(21)

where
$\mathfrak{p}(x)=i\mbox{$\overset{\>{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\displaystyle\mathfrak{a}}}$}(x)\gamma_{0}$
is the conjugate momentum. The free Hamiltonian evaluates to

$\displaystyle H_{0}=$ $\displaystyle\int
d^{3}p\sqrt{|\mbox{\boldmath$\displaystyle\boldsymbol{p}$}|^{2}+m^{2}}$
$\displaystyle\times\left[\sum_{i=1,2}a^{{\dagger}}_{i}(\mbox{\boldmath$\displaystyle\boldsymbol{p}$})a_{i}(\mbox{\boldmath$\displaystyle\boldsymbol{p}$})+\sum_{i=3,4}b_{i}(\mbox{\boldmath$\displaystyle\boldsymbol{p}$})b^{{\dagger}}_{i}(\mbox{\boldmath$\displaystyle\boldsymbol{p}$})\right].$

The free Hamiltonian is positive-definite if and only if the annihilation and
creation operators satisfy bosonic statistics.

The above expression can be rewritten as

$\displaystyle H_{0}=$ $\displaystyle 2\delta^{3}(0)\int
d^{3}p\,\sqrt{|\mbox{\boldmath$\displaystyle\boldsymbol{p}$}|^{2}+m^{2}}+\int
d^{3}p\sqrt{|\mbox{\boldmath$\displaystyle\boldsymbol{p}$}|^{2}+m^{2}}$
$\displaystyle\times\left[\sum_{i=1,2}a^{{\dagger}}_{i}(\mbox{\boldmath$\displaystyle\boldsymbol{p}$})a_{i}(\mbox{\boldmath$\displaystyle\boldsymbol{p}$})+\sum_{i=3,4}b^{\dagger}_{i}(\mbox{\boldmath$\displaystyle\boldsymbol{p}$})b_{i}(\mbox{\boldmath$\displaystyle\boldsymbol{p}$})\right].$

Since

$\delta^{3}(\mbox{\boldmath$\displaystyle\boldsymbol{p}$})=\frac{1}{(2\pi)^{3}}\int
d^{3}x\exp(i\mbox{\boldmath$\displaystyle\boldsymbol{p}$}\cdot\mbox{\boldmath$\displaystyle\boldsymbol{x}$}),$

$\delta^{3}(0)$ may be replaced by $\left[1/(2\pi)^{3}\right]\int d^{3}x$.
With this observation, the zero-point energy takes the form

$H_{\text{vac}}=4\times\frac{1}{(2\pi)^{3}}\int d^{3}x\int
d^{3}p\,\frac{1}{2}\sqrt{|\mbox{\boldmath$\displaystyle\boldsymbol{p}$}|^{2}+m^{2}}.$
(22)

In natural units $\hbar=1$ implies $2\pi=h$, so $H_{\text{vac}}$ represents a
zero-point energy assignment of
$+\frac{1}{2}\sqrt{|\mbox{\boldmath$\displaystyle\boldsymbol{p}$}|^{2}+m^{2}}$
to each unit-size cell $(1/h^{3})d^{3}x\,d^{3}p$ in the sense of statistical
mechanics. The factor of $4$ is consistent with the four degrees of freedom
(when spinorial degrees of freedom associated with particles and antiparticles
are accounted for). The second term in $H$ shows that each of the four degrees
of freedom contributes exactly the same energy to the field for a given
momentum $\displaystyle\boldsymbol{p}$. If for each of the SM Dirac fermions
there exists a new boson of equal mass then the zero-point energies exactly
cancel. This is a natural consequence of our formalism.

##  3 Unitary evolution and scattering theory

The bilinear invariants constructed from $\mathfrak{a}(x)$ and
$\overset{\>{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\mathfrak{a}}}(x)$ that
enter the interacting $H$ are not hermitian; see (26) and (27) below. Despite
this, the interacting $H$ still carries a real spectrum provided the norm of
states is appropriately modified.

To develop the formalism, we introduce

${H}^{\\#}\stackrel{{\scriptstyle\textrm{def}}}{{=}}\eta^{-1}{H}^{{\dagger}}\eta={H}$
(23)

where $\eta$ is a hermitian operator, to be determined below. The work of
Mostafazadeh then assures that ${H}$ has a real spectrum provided that its
eigenstates $|\psi\rangle$ have non-vanishing $\eta$-norms
$\langle\psi|\eta|\psi\rangle\neq 0$. To see this, we note that (23) yields

$\displaystyle 0$ $\displaystyle=\langle\psi|(H^{{\dagger}}\eta-\eta
H)|\psi\rangle$ $\displaystyle=(E^{*}-E)\langle\psi|\eta|\psi\rangle.$ (24)

Therefore, when $\langle\psi|\eta|\psi\rangle\neq 0$, the eigenvalues are real
[14, 16, 15, 17].

To preserve unitarity, we construct $\eta$ to have the following properties
[16]

$\eta
a_{i}(\mbox{\boldmath$\displaystyle\boldsymbol{p}$})\eta^{-1}=a_{i}(\mbox{\boldmath$\displaystyle\boldsymbol{p}$}),\quad\eta
b_{i}(\mbox{\boldmath$\displaystyle\boldsymbol{p}$})\eta^{-1}=-b_{i}(\mbox{\boldmath$\displaystyle\boldsymbol{p}$}).$
(25)

That is, $\eta$ commutes with
$a_{i}(\mbox{\boldmath$\displaystyle\boldsymbol{p}$})$, and anti-commutes with
$b_{i}(\mbox{\boldmath$\displaystyle\boldsymbol{p}$})$. Combined with the
definition of the dual (5), this gives a relation between
$\overset{\>{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\mathfrak{a}}}(x)$ and
$\overline{\mathfrak{a}}(x)$

$\eta\overset{\>{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\mathfrak{a}}}(x)\eta^{-1}=\overline{\mathfrak{a}}(x).$
(26)

Similarly, the bilinear invariant
$\overset{\>{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\mathfrak{a}}}(x)\mathfrak{a}(x)$
is related to its hermitian conjugate by

$\eta[\overset{\>{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\mathfrak{a}}}(x)\mathfrak{a}(x)]\eta^{-1}=[\overset{\>{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\mathfrak{a}}}(x)\mathfrak{a}(x)]^{{\dagger}}.$
(27)

Since $[\eta,H_{0}]=\mbox{\boldmath$\displaystyle\mathbb{O}$}$, the free
Hamiltonian is hermitian. But in general, the full interacting Hamiltonian is
non-hermitian and does not commute with $\eta$, so we require (23) to ensure
the reality of the spectra. The simplest contributions from spin-half bosons
to $H$ are of the form
$\mbox{$\overset{\>{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\displaystyle\mathfrak{a}}}$}(x)\mbox{$\displaystyle\mathfrak{a}$}(x)$
and
$\mbox{$\overset{\>{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\displaystyle\mathfrak{a}}}$}(x)\gamma^{5}\mbox{$\displaystyle\mathfrak{a}$}(x)$.

An explicit solution for $\eta$ that satisfies the above requirements may be
constructed following [18]. To do that, we introduce

$\eta\stackrel{{\scriptstyle\mathrm{def}}}{{=}}e^{i\alpha\chi},\quad\alpha\in\mbox{\boldmath$\displaystyle\mathbb{R}$},$
(28)

where

$\chi=\int
d^{3}p\sum_{i=3,4}b^{{\dagger}}_{i}(\mbox{\boldmath$\displaystyle\boldsymbol{p}$})b_{i}(\mbox{\boldmath$\displaystyle\boldsymbol{p}$}).$
(29)

Thus, $\eta$ is a state-dependent relative phase between the particle-
antiparticle sector. A short calculation reveals that

$\eta
b^{\dagger}_{i}(\mbox{\boldmath$\displaystyle\boldsymbol{p}$},\sigma)\eta^{-1}=e^{i\alpha}b^{\dagger}_{i}(\mbox{\boldmath$\displaystyle\boldsymbol{p}$},\sigma)$
(30)

and helps us fix $\alpha$. For equation (25) to be satisfied, we must have
$\alpha=\pi$ (modulo $\pm$ integral multiples of $2\pi$). This gives us
$\eta=e^{i\pi\chi}$, which is hermitian and unitary. To see that $\eta$ is
indeed hermitian, notice $\eta^{\dagger}=e^{-i\pi\chi^{\dagger}}$, but for
bosons $\chi^{\dagger}=\chi$; therefore $\eta^{\dagger}=e^{-i\pi\chi}$. Now
$\chi$ acting on physical states $|\psi\rangle$ gives $n$ where $n$ is either
zero, or an integer: $\chi|\psi\rangle=n|\psi\rangle$. Therefore,
$\eta^{\dagger}|\psi\rangle=e^{-i\pi n}|\psi\rangle$, while
$\eta|\psi\rangle=e^{i\pi n}|\psi\rangle$. Thus while acting on physical
states $\eta^{\dagger}=\eta$, since $e^{-i\pi}$ = $e^{i\pi}$. It is readily
seen that $\eta^{2}|\psi\rangle=|\psi\rangle$, so $\eta^{2}=\mathbb{I}$.
Consequently, while acting on $|\psi\rangle$: $\eta^{\dagger}=\eta^{-1}=\eta$.

We now develop the scattering theory for spin-half bosons. Since for the
states with time evolution governed by the generalized hermitian Hamiltonian –
of the class defined above in equation (23) – their hermitian inner-product is
not invariant under time-translation

$\langle\phi|\psi\rangle\big{|}_{t>0}=\langle\phi|e^{iH^{{\dagger}}t}e^{-iHt}|\psi\rangle\neq\langle\phi|\psi\rangle\big{|}_{t=0}.$
(31)

It is necessary to introduce the $\eta$-norm [14, 16]

$\langle\phi|\psi\rangle_{\eta}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}\langle\phi|\eta|\psi\rangle$
(32)

where $\eta=e^{i\pi\chi}$. At $t>0$, we find

$i\frac{d}{dt}\langle\phi|\eta|\psi\rangle\big{|}_{t>0}=\langle\phi|(\eta
H-H^{{\dagger}}\eta)|\psi\rangle\big{|}_{t>0}.$ (33)

Therefore, the $\eta$-norm is invariant if and only if $H$ satisfies (23).

Using the $\eta$-norm, the $S$-matrix is given by

$\displaystyle S_{\beta\alpha}$ $\displaystyle\equiv$
$\displaystyle\langle\beta_{\text{out}}|\alpha_{\text{in}}\rangle_{\eta}$ (34)
$\displaystyle=$
$\displaystyle\lim_{\tau\rightarrow\infty}\lim_{\tau_{0}\rightarrow-\infty}\langle\beta_{0}|\Omega^{{\dagger}}(\tau)\eta\Omega(\tau_{0})|\alpha_{0}\rangle$

where $\Omega(\tau)=e^{iH\tau}e^{-iH_{0}\tau}$ and we have taken the free
Hamiltonian to be hermitian. Therefore,

$\displaystyle
S_{\beta\alpha}=\lim_{\tau\rightarrow\infty}\lim_{\tau_{0}\rightarrow-\infty}\langle\beta_{0}|U(\tau,\tau_{0})|\alpha_{0}\rangle$
(35)

where

$\displaystyle
U(\tau,\tau_{0})=\eta\,e^{iH_{0}\tau}e^{-iH(\tau-\tau_{0})}e^{-iH_{0}\tau_{0}}.$
(36)

In obtaining (36), we have used the identity

$e^{iH^{{\dagger}}\tau}\eta e^{-iH\tau}=\eta$ (37)

which is equivalent to (23).

Taking $H=H_{0}+V$ and differentiating $U(\tau,\tau_{0})$ with respect to
$\tau$, we obtain

$\displaystyle i\frac{d}{d\tau}U(\tau,\tau_{0})=\left[\eta
V(\tau)\eta^{-1}\right]U(\tau,\tau_{0}),$ (38) $\displaystyle
V(\tau)=e^{iH_{0}\tau}Ve^{-iH_{0}\tau}.$ (39)

Using the initial condition $U(\tau_{0},\tau_{0})=\eta$, the solution to $U$
is given by

$\displaystyle U(\tau,\tau_{0})$
$\displaystyle=\eta-i\int^{\tau}_{\tau_{0}}d\tau^{\prime}\left[\eta
V(\tau^{\prime})\eta^{-1}\right]U(\tau^{\prime},\tau_{0})$
$\displaystyle=\eta\left[I-i\int^{\tau}_{\tau_{0}}d\tau^{\prime}V(\tau^{\prime})+\cdots\right].$
(40)

Taking $V(t)=\int d^{3}x\mathcal{H}(x)$, the Dyson series is

$\displaystyle S_{\beta\alpha}=$
$\displaystyle\eta_{\beta\alpha}+\sum^{\infty}_{n=1}\frac{(-i)^{n}}{n!}\int
d^{4}x_{1}\cdots d^{4}x_{n}$
$\displaystyle\times\langle\beta_{0}|\eta\mathfrak{T}\left[\mathcal{H}(x_{1})\cdots\mathcal{H}(x_{n})\right]|\alpha_{0}\rangle$
(41)

where $\eta_{\beta\alpha}\equiv\langle\beta_{0}|\eta|\alpha_{0}\rangle$.

Next, we address the issue of unitarity. For hermitian Hamiltonians
$H^{{\dagger}}=H$, unitarity implies $\int d\beta
S^{{\dagger}}_{\gamma\beta}S_{\beta\alpha}=\delta(\gamma-\alpha)$ where $\int
d\beta$ integrates over momentum, sums over particle species and internal
degrees of freedom. As for generalized hermitian Hamiltonians, we have
$H^{\\#}=H$ so it suggests the following generalization to unitarity [19]

$\int d\beta S^{\\#}_{\gamma\beta}S_{\beta\alpha}=\delta(\gamma-\alpha),$ (42)

where the matrix element of $S^{\\#}_{\gamma\beta}$ is given by

$\displaystyle
S^{\\#}_{\gamma\beta}=\lim_{\tau\rightarrow\infty}\lim_{\tau_{0}\rightarrow-\infty}\langle\gamma_{0}|U^{\\#}(\tau,\tau_{0})|\beta_{0}\rangle,$
(43) $\displaystyle
U^{\\#}(\tau,\tau_{0})\equiv\eta^{-1}U^{{\dagger}}(\tau,\tau_{0})\eta.$ (44)

To prove (42), we start with orthonormality and completeness relations
associated with the free (and hermitian) Hamiltonian $H_{0}$, and then
implement evolution due to the full Hamiltonian $H$ satisfying (23) as
follows:

$\int
d\alpha|\alpha_{0}\rangle\langle\alpha_{0}|=I,\quad\langle\beta_{0}|\alpha_{0}\rangle=\delta(\beta-\alpha).$
(45)

Therefore, equation (42) is satisfied when

$U^{\\#}(\tau,\tau_{0})U(\tau,\tau_{0})=I$ (46)

for all $\tau$ and $\tau_{0}$. Rewriting $U(\tau,\tau_{0})$ as

$U(\tau,\tau_{0})=e^{iH_{0}\tau}e^{-iH^{{\dagger}}(\tau-\tau_{0})}e^{-iH_{0}\tau_{0}}\eta,$
(47)

we find

$\displaystyle U^{\\#}(\tau,\tau_{0})U(\tau,\tau_{0})$
$\displaystyle=e^{iH_{0}\tau_{0}}\left[e^{iH(\tau-\tau_{0})}\eta
e^{-iH^{{\dagger}}(\tau-\tau_{0})}\right]e^{-iH_{0}\tau_{0}}\eta.$ (48)

Taking the inverse of (37), we get

$e^{iH(\tau-\tau_{0})}\eta^{-1}e^{-iH^{{\dagger}}(\tau-\tau_{0})}=\eta^{-1}.$
(49)

Since $\eta=\eta^{-1}$, we obtain (46) as required.

From (38) and (39), we obtain

$-i\frac{d}{d\tau}U^{\\#}(\tau,\tau_{0})=U^{\\#}(\tau,\tau_{0})V^{{\dagger}}(\tau).$
(50)

Following the same procedure in deriving the Dyson series for
$S_{\beta\alpha}$, we obtain

$\displaystyle S^{\\#}_{\gamma\beta}=$
$\displaystyle\eta_{\gamma\beta}+\sum^{\infty}_{n=1}\frac{i^{n}}{n!}\int
d^{4}x_{1}\cdots d^{4}x_{n}$
$\displaystyle\times\langle\gamma_{0}|\eta\mathfrak{T}\left[\mathcal{H}^{{\dagger}}(x_{n})\cdots\mathcal{H}^{{\dagger}}(x_{1})\right]|\beta_{0}\rangle.$

Normalizing the $S$-matrix as

$\displaystyle S_{\beta\alpha}\equiv\eta_{\beta\alpha}-2\pi
iM_{\beta\alpha}\delta^{4}(p_{\beta}-p_{\alpha}),$ (51) $\displaystyle
S^{\\#}_{\gamma\beta}\equiv\eta_{\gamma\beta}+2\pi
iM^{\\#}_{\gamma\beta}\delta^{4}(p_{\gamma}-p_{\beta}),$ (52)

equation (42) becomes

$\displaystyle i\int
d\beta\left[\eta_{\gamma\beta}M_{\beta\alpha}\delta^{4}(p_{\beta}-p_{\alpha})-M^{\\#}_{\gamma\beta}\eta_{\beta\alpha}\delta^{4}(p_{\gamma}-p_{\beta})\right]$
$\displaystyle=2\pi\int
d\beta\left[\delta^{4}(p_{\beta}-p_{\gamma})\delta^{4}(p_{\beta}-p_{\alpha})M^{\\#}_{\gamma\beta}M_{\beta\alpha}\right].$
(53)

Setting $\gamma=\alpha$, the term $\delta^{4}(p_{\beta}-p_{\alpha})$ cancels
from both side of (53) so we obtain

$\displaystyle i\int
d\beta\left(\eta_{\alpha\beta}M_{\beta\alpha}-M^{\\#}_{\alpha\beta}\eta_{\beta\alpha}\right)$
$\displaystyle=2\pi\int
d\beta\left[\delta^{4}(p_{\beta}-p_{\alpha})M^{\\#}_{\alpha\beta}M_{\beta\alpha}\right].$
(54)

Equations (53) and (54) can be seen as a generalization to the optical
theorem. In the subsequent section, we will demonstrate that (54) is satisfied
for a model of Yukawa interaction involving spin-half bosons.

To compute physical observables, we first need to define the transition
probability for the process $\alpha\rightarrow\beta$. Since the unitarity of
the $S$-matrix is defined using the generalized hermitian adjoint $\\#$, the
transition probability should be appropriately modified. In a finite volume
$V$ with duration $T$, we define the transition probability to be [7]

$\displaystyle{\color[rgb]{0,1,1}P(\alpha\rightarrow\beta)\equiv}$
$\displaystyle{\color[rgb]{0,1,1}\left[\frac{(2\pi)^{3}}{V}\right]^{N_{\alpha}+N_{\beta}}\frac{VT}{(2\pi)^{2}}\left(\wp_{\beta\alpha}M^{\\#}_{\alpha\beta}M_{\beta\alpha}\right)}$
$\displaystyle{{\color[rgb]{0,1,1}\times\left[\delta^{3}_{V}(p_{\beta}-p_{\alpha})\delta_{T}(E_{\beta}-E_{\alpha})\right].}}$
(55)

Because $M^{\\#}_{\alpha\beta}M_{\beta\alpha}$ is not guaranteed to be
positive-definite, we multiply it by a constant phase
$\wp_{\beta\alpha}\in\Re$ to ensure that $P(\alpha\rightarrow\beta)\geq 0$.
From the definition of $\eta$ and the Dyson series of $S_{\beta\alpha}$ and
$S^{\\#}_{\alpha\beta}$, we find that the
$M^{\\#}_{\alpha\beta}=e^{i\pi(n_{\beta}-n_{\alpha})}M^{{\dagger}}_{\alpha\beta}$
where $n_{\alpha}$ and $n_{\beta}$ are the number of spin-half anti-bosons
present in states $|\alpha_{0}\rangle$ and $|\beta_{0}\rangle$ respectively.
Therefore, by choosing $\wp_{\alpha\beta}=e^{-i\pi(n_{\beta}-n_{\alpha})}$,
equation (55) reduces to the standard transition probability and the physical
observables defined using hermitian conjugation can be applied without
difficulty.

###  3.1 Phenomenologies

We now consider two spin-half bosonic and fermionic fields of equal mass
coupled to a real scalar field $\phi(x)$

$\displaystyle V(t)=g\int
d^{3}x\left[\overline{\psi}(x)\psi(x)+\mbox{$\overset{\>{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\displaystyle\mathfrak{a}}}$}(x)\mbox{$\displaystyle\mathfrak{a}$}(x)\right]\phi(x)$
(56)

where $\psi(x)$ is a fermionic field whose expansion coefficients are given by
(1) and (2). By the construction of $\eta$, it commutes with the fermionic and
scalar fields

$\left[\eta,\psi(x)\right]=\left[\eta,\phi(x)\right]=O.$ (57)

Therefore, the Yukawa interaction (56) is a generalized hermitian operator
$\eta^{-1}V^{{\dagger}}(t)\eta=V(t)$.

We now demonstrate that the Yukawa interaction satisfies (42), the generalized
unitarity relation. Taking the initial state to be a single scalar boson, the
matrix element of $\eta$ is
$\eta_{\phi\beta}=\eta_{\beta\phi}=\delta(\beta-\phi)$ so (54) becomes

$i(M_{\phi\phi}-M^{\\#}_{\phi\phi})=2\pi\int
d\beta\left[\delta^{4}(p_{\beta}-p_{\phi})M^{\\#}_{\phi\beta}M_{\beta\phi}\right].$
(58)

We will evaluate both sides of (58) independently. Starting with the left-hand
side, the matrix elements $M_{\phi\phi}$ and $M^{\\#}_{\phi\phi}$ are obtained
by evaluating the amplitude for $\phi\rightarrow\phi^{\prime}$ and
subsequently setting $\phi^{\prime}=\phi$.

Using the solution of $\eta$, we obtain $\eta|\phi\rangle=|\phi\rangle$ where
$|\phi\rangle$ denotes a single scalar boson state. Setting
$\phi=\phi^{\prime}$, we find $M^{{\dagger}}_{\phi\phi}=M^{\\#}_{\phi\phi}$.
Therefore, we may rewrite (58) as

$\text{Im}(M_{\phi\phi})=-\pi\int
d\beta\left[\delta^{4}(p_{\beta}-p_{\phi})M^{\\#}_{\phi\beta}M_{\beta\phi}\right].$
(59)

Here, $\mbox{Im}(M_{\phi\phi})$ is proportional to the imaginary part of the
loop corrections to the scalar propagator [20]. Due to the difference in
statistics and the fact that the spin-half bosons and fermions have equal
mass, the bosonic and fermionic loop corrections identically cancel to all
orders in perturbation so $\text{Im}(M_{\phi\phi})=0$. Next, we evaluate the
right-hand side of (58). It expands to

$\displaystyle\int
d\beta\left[\delta^{4}(p_{\beta}-p_{\alpha})M^{\\#}_{\phi\beta}M_{\beta\phi}\right]$
$\displaystyle=\int
d\beta_{\psi}\sum_{\text{spins}}M^{\\#}_{\phi(\overline{\psi}\psi)}M_{(\overline{\psi}\psi)\phi}\delta^{4}(p_{\,\overline{\psi}}+p_{\psi}-p_{\phi})$
$\displaystyle+\int
d\beta_{\mathfrak{a}}\sum_{\text{spins}}M^{\\#}_{\phi(\overset{\>{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\mathfrak{a}}}\mathfrak{a})}M_{(\overset{\>{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\mathfrak{a}}}\mathfrak{a})\phi}\delta^{4}(p_{\overset{\>{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\mathfrak{a}}}}+p_{\mathfrak{a}}-p_{\phi})$
(60)

where $d\beta_{\psi}=d^{3}p_{\psi}d^{3}p_{\,\overline{\psi}}$ and
$d\beta_{\mathfrak{a}}=d^{3}p_{\mathfrak{a}}d^{3}p_{\overset{\>{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\mathfrak{a}}}}$.
The first and second term of (60) comes from the fermionic and bosonic
interactions respectively.  Acting $\eta$ on the relevant states in (60), we
obtain

$\displaystyle\eta|\phi\rangle=|\phi\rangle,\quad\eta|\overline{\psi}\psi\rangle=|\overline{\psi}\psi\rangle,\quad\eta|\mbox{$\overset{\>{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\displaystyle\mathfrak{a}}}$}\mbox{$\displaystyle\mathfrak{a}$}\rangle=-|\mbox{$\overset{\>{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\displaystyle\mathfrak{a}}}$}\mbox{$\displaystyle\mathfrak{a}$}\rangle.$

where $|\overline{\psi}\psi\rangle$ and
$|\mbox{$\overset{\>{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\displaystyle\mathfrak{a}}}$}\mbox{$\displaystyle\mathfrak{a}$}\rangle$
denote the pair of spin-half fermion-anti-fermion and boson-anti-boson states
respectively. The actions of $\eta$ on $|\overline{\psi}\psi\rangle$ and
$|\mbox{$\overset{\>{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\displaystyle\mathfrak{a}}}$}\mbox{$\displaystyle\mathfrak{a}$}\rangle$
have opposite signs which induces a relative phase in (60). In effect, we find

$M^{\\#}_{\phi(\overline{\psi}\psi)}=M^{{\dagger}}_{\phi(\overline{\psi}\psi)},\quad
M^{\\#}_{\phi(\overset{\>{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\mathfrak{a}}}\mathfrak{a})}=-M^{{\dagger}}_{\phi(\overset{\>{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\mathfrak{a}}}\mathfrak{a})}.$
(61)

Using the decay rate formula [7]

$\displaystyle\Gamma(1\rightarrow 1^{\prime}2^{\prime})=2\pi\int$
$\displaystyle
d\beta_{1^{\prime},2^{\prime}}\sum_{\text{spins}}|M_{(1^{\prime}2^{\prime})1}|^{2}$
$\displaystyle\times\delta^{4}(p_{1^{\prime}}+p_{2^{\prime}}-p_{1}),$ (62)

where $d\beta_{1^{\prime},2^{\prime}}=d^{3}p^{\prime}_{1}d^{3}p^{\prime}_{2}$,
we obtain

$\displaystyle\int
d\beta\left[\delta^{4}(p_{\beta}-p_{\alpha})M^{\\#}_{\phi\beta}M_{\beta\phi}\right]$
$\displaystyle\qquad\qquad=\frac{1}{2\pi}\left[\Gamma(\phi\rightarrow\overline{\psi}\psi)-\Gamma(\phi\rightarrow\overset{\>{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\mathfrak{a}}}\mathfrak{a})\right]$
(63)

so the unitarity condition becomes

$\text{Im}(M_{\phi\phi})=\frac{1}{2}\left[\Gamma(\phi\rightarrow\overset{\>{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\mathfrak{a}}}\mathfrak{a})-\Gamma(\phi\rightarrow\overline{\psi}\psi)\right].$
(64)

Since the bosons and fermions have the same mass and their quantum fields are
expanded the same spinors, we have
$\Gamma(\phi\rightarrow\overline{\psi}\psi)=\Gamma(\phi\rightarrow\overset{\>{}^{{}^{\boldsymbol{\neg}}}}{\smash[t]{\mathfrak{a}}}\mathfrak{a})$.
Therefore, the right-hand side of (64) identically vanishes to all orders in
perturbation. This is in agreement with $\text{Im}(M_{\phi\phi})=0$ so the
theory is unitary.

##  4 Conclusion

Spin-statistics theorem has a very venerated position in the theory of quantum
fields, as does Dirac equation. The works in the past have shown that the
Dirac formalism with mass dimension three half, needs to be updated with a new
formalism of mass dimension one fermions. Here we have argued that the spin-
statistics theorem can also be evaded by exploiting the freedom in defining
new dual for spinors and the ensuing adjoint. This results in a new quantum
theory of spin half bosons. We then formulated a generalisation of the
S-matrix to calculate some elementary processes.

###### Acknowledgements.

We are grateful to James Brister, Julio M. Hoff da Silva, Ali Mostafazadeh,
Takaaki Nomura, Zheng Sun and Cong Zhang for useful discussions. CYL is
supported by The Sichuan University Post-doctoral Research Fund No.
2022SCU12119.

## References

  * [1] P. A. M. Dirac, Spinors in Hilbert Space.  Plenum Press, 1974.  A series of lectures given in 1969 and revised in 1970.
  * [2] D. V. Ahluwalia and D. Grumiller, Spin half fermions with mass dimension one: Theory, phenomenology, and dark matter, JCAP 07 (2005) 012, [hep-th/0412080].
  * [3] D. Ahluwalia, Mass Dimension One Fermions.  Cambridge Monographs on Mathematical Physics. Cambridge University Press, 2019.
  * [4] D. V. Ahluwalia, J. M. H. da Silva, C.-Y. Lee, Y.-X. Liu, S. H. Pereira, and M. M. Sorkhi, Mass dimension one fermions: Constructing darkness, Physics Reports 967 (2022) 1–43.
  * [5] D. V. Ahluwalia, A new class of mass dimension one fermions, Proc. Roy. Soc. Lond. A 476 (2020) 20200249, [arXiv:2008.0152].
  * [6] D. V. Ahluwalia, Spin-half bosons with mass dimension three half: towards a resolution of the cosmological constant problem, EPL 131 (2020) 41001, [arXiv:2008.0263].
  * [7] S. Weinberg, The Quantum theory of fields. Vol. 1: Foundations.  Cambridge University Press, 6, 2005.
  * [8] D. V. Ahluwalia and D. Grumiller, Dark matter: A Spin one half fermion field with mass dimension one?, Phys. Rev. D 72 (2005) 067701, [hep-th/0410192].
  * [9] J. M. Hoff da Silva, R. J. Bueno Rogerio, and N. C. R. Quinquiolo, Spinorial discrete symmetries and adjoint structures, arXiv:2203.0206.
  * [10] L. D. Sperança, An identification of the Dirac operator with the parity operator, Int. J. Mod. Phys. D23 (2014) 1444003, [arXiv:1304.4794].
  * [11] D. Ahluwalia and A. Nayak, Elko and mass dimension one field of spin one-half: Causality and fermi statistics, International Journal of Modern Physics D 23 (2014), no. 14.
  * [12] R. F. Streater and A. S. Wightman.  W. A. Benjamin Inc., 1989.
  * [13] I. Duck and E. C. G. Sudarshan, Pauli and the spin-statistics theorem.  1997\.
  * [14] A. Mostafazadeh, Pseudo Hermiticity versus PT symmetry. The necessary condition for the reality of the spectrum, J. Math. Phys. 43 (2002) 205–214, [math-ph/0107001].
  * [15] C. M. Bender and S. Boettcher, Real spectra in nonHermitian Hamiltonians having PT symmetry, Phys. Rev. Lett. 80 (1998) 5243–5246, [physics/9712001].
  * [16] A. LeClair and M. Neubert, Semi-Lorentz invariance, unitarity, and critical exponents of symplectic fermion models, JHEP 10 (2007) 027, [arXiv:0705.4657].
  * [17] W. Pauli, On dirac’s new method of field quantization, Rev. Mod. Phys. 15 (Jul, 1943) 175–207.
  * [18] D. J. Robinson, E. Kapit, and A. LeClair, Lorentz Symmetric Quantum Field Theory for Symplectic Fermions, J. Math. Phys. 50 (2009) 112301, [arXiv:0903.2399].
  * [19] M. A. Simón, A. Buendía, A. Kiely, A. Mostafazadeh, and J. G. Muga, $S$-matrix pole symmetries for non-Hermitian scattering Hamiltonians, Phys. Rev. A 99 (2019), no. 5 052110, [arXiv:1811.0627].
  * [20] M. D. Schwartz, Quantum Field Theory and the Standard Model.  Cambridge University Press, 3, 2014.


2022

1]Stefan Meyer Institute for Subatomic Physics, Austrian Academy of Sciences,
Kegelgasse 27, Vienna, 1030, Austria

2]Institute for Particle Physics and Astrophysics, eth, Zurich, Zurich, 8093,
Switzerland

3]Department of Physics and Astronomy, University of Turku, Turku, 20014,
Finland

4]Institut Max von Laue - Paul Langevin, 71 avenue des Martyrs, Grenoble,
38042, France

5]Laboratoire Kastler Brossel, Sorbonne Université, CNRS, ENS-PSL Université,
Collège de France, Paris, 75252, France

#  Grasian: Towards the first demonstration of gravitational quantum states of
atoms with a cryogenic hydrogen beam

Carina Killian  Zakary Burkley  Philipp Blumer  Paolo Crivelli  Fredrik
Gustafsson  Otto Hanski  Amit Nanda  François Nez  Valery Nesvizhevsky  Serge
Reynaud  Katharina Schreiner  Martin Simon  Sergey Vasiliev  Eberhard Widmann
Pauline Yzombard  [  [  [  [  [

###### Abstract

At very low energies, a light neutral particle above a horizontal surface can
experience quantum reflection. The quantum reflection holds the particle
against gravity and leads to gravitational quantum states (gqs). So far, gqs
were only observed with neutrons as pioneered by Nesvizhevsky and his
collaborators at ill. However, the existence of gqs is predicted also for
atoms.
The Grasian-collaboration pursues the first observation and studies of gqs of
atomic hydrogen. We propose to use atoms in order to exploit the fact that
orders of magnitude larger fluxes compared to those of neutrons are available.
Moreover, recently the q-Bounce collaboration, performing gqs spectroscopy
with neutrons, reported a discrepancy between theoretical calculations and
experiment which deserves further investigations. For this purpose, we set up
a cryogenic hydrogen beam at $6\text{\,}\mathrm{K}$. We report on our
preliminary results, characterizing the hydrogen beam with pulsed laser
ionization diagnostics at $243\text{\,}\mathrm{nm}$.

##  1 Introduction

Quantum bouncers were first predicted in 1928 Bre:1928pr . Nearly 75 years
later, this phenomenon was demonstrated through the observation of neutron
($n$) gravitational quantum states (gqs) Lus:1978jl ; Nes:2000nima ;
Nes:2002Nat ; Nes:2003prd ; Nes:2003prdbis ; Nes:2005epjc ; Wes:2007epjc .
Confined by the gravitational- and the mirror potential, the $n$ are settled
in gravitationally bound quantum states.

Studies of $n$ gqs have a broad impact on fundamental and applied physics.
They serve as a unique method to study the interaction of a particle in a
quantum state with a gravitational field. For example, paired with more recent
measurements of $n$ whispering gallery states (wgs) - quantum states trapped
by the centrifugal- and the mirror potential Nes:2010NatPhys , they result in
the first direct demonstration of the validity of the weak equivalence
principle for a particle in a pure quantum state.

The observation of gqs initiated active analysis of the pecularities of this
phenomenon Rob:2004pr ; Ber:2005jmc ; Mat:2006pra ; ber:2006plb ; Rom:2007prl
; Del:2009prl ; Gar:2012pr ; Bel:2014pr  and their application to the search
for new physics, such as the searches for extra fundamental short-range
interactions Abe:2003lnp ; Fra:2004plb ; Bra:2006prd ; Bui:2007cqg ;
Bae:2007prd ; shortrangef ; Bra:2011prl , verification of the weak equivalence
principle in the quantum regime Ber:2003CQG ; Kie:2005AP ; Kaj:2010APB ,
extensions of quantum mechanics Ber:2005PRD ; Sah:2007EPJC , extensions of
gravity and space theories Noz:2010EPL ; Ped:2011JHEP ; Kob:2011PRD ;
Cha:2011PLB  or tests of Lorentz invariance Mar:2018PRD ; Esc:2019PRD ;
Iva:2019PLB .

New fundamental short-range interactions are predicted in extensions of the
Standard Model such as grand unified, supersymmetric and string theories that
could alter the weak gravitational potential. They also appear in certain
models explaining dark matter and dark energy shortrangef . Additionally,
studies on gqs can provide extremely sensitive measurements of the mirror’s
surface potential and shape, which is of high interest to the surface physics
community.

Spectroscopy and interferometry methods of observation of gqs of $n$ have been
analyzed theoretically and implemented experimentally over the previous two
decades Nes:2006TQGR ; Nes:2010PU ; Jen:2011NP ; Jen:2014PRL ; Ich:2014PRL .
However, the existence of gqs is predicted also for atoms and anti-atoms
Vor:2011PRA ; WOS:000395981900005 ; WOS:000312994700008 ; WOS:000225326300020
; WOS:000186422700011 ; WOS:A1993LW23100029 ; WOS:A1989AU53000009 ;
WOS:A1972N687600001 ; WOS:000222126500001 . Those are expected to have
essentially identical properties for particles of almost equal mass such as
$n$, atomic hydrogen ($H$) or even anti-hydrogen ($\bar{H}$).

A major constraint to improve the precision of the current measurements of gqs
of $n$ is the limited density of ultracold $n$ (ucns). It looks natural to
exploit the much higher fluxes available for atoms, namely the high densities
of existing $H$-beams CooperHBeam2020 . However, all the projects concerning
the use and study of gqs of atoms are currently based only on theoretical
estimations since those, in contrast to $n$, have never been observed
experimentally. Only a direct experiment can prove the existence of gqs of
atoms, evaluate the systematic and statistical uncertainties of such
experiments and develop the experimental techniques needed for more precise
measurements in the future.

In section 2, a theoretical derivation of gqs is given. The method used
earlier for the observation of $n$ gqs and the planned implementation of a
measurement with $H$ is presented in 3. A detailed description of the Grasian
experimental setup and the recent measurements is given in 4.

##  2 Theoretical framework

A sufficiently slow particle trapped by the gravitational field on one side
and a horizontal reflective surface ("mirror") on the other side settles in
gqs. The particle’s wave function $\psi(z)$ in the Earth’s gravitational field
above a mirror is governed by the Schrodinger equation
$\frac{\hbar^{2}}{2m}\frac{d^{2}\psi(z)}{dz^{2}}+(E-mgz)\psi(z)=0$, where
$\hbar$ is the reduced Planck constant, $m$ is the particles mass, $z$ is the
height, $E$ is the energy of the vertical motion of the particle, and $g$ is
the acceleration in the Earth’s gravitational field. The only constant related
to the particle’s identity is it’s mass, which is nearly exactly the same for
$n$, $H$ or $\bar{H}$. For simplicity, $n$ over an ideal mirror will be
considered in the following derivation.

An ideal horizontal mirror at the height $z=0$ can be approximated as an
infinitely high and abrupt potential step. This approximation is justified by
the characteristic values of energies and lengths in our problem. The energy
of neutrons in low quantum states $\sim$${10}^{-12}\text{\,}\mathrm{eV}$, is
much smaller than the optical potential of the mirror material
$\sim$${10}^{-7}\text{\,}\mathrm{eV}$ golub1991ultra , and the characteristic
range of increase in the optical potential for a polished mirror
$\sim$${10}^{-9}\text{\,}\mathrm{m}$, is much smaller than the wavelength of
neutrons in low quantum states $\sim$${10}^{-5}\text{\,}\mathrm{m}$. Such an
infinitely high and abrupt optical potential corresponds to the zero boundary
condition for the wave function, $\psi(z=0)=0$.

A solution of the Schrodinger equation can be written in terms of the Airy
function Ai, $\psi(z)=C\text{Ai}(z/z_{0})$, where
$z_{0}=[\hbar^{2}/(2m^{2}g)]^{1/3}=\,$$5.87\text{\,}\mathrm{\SIUnitSymbolMicro
m}$ is the characteristic length scale of the problem and $C$ is a
normalization constant. The Airy function zeros $\lambda_{k}$ define the
quantum state energies $E_{k}=mgz_{0}\lambda_{k}$, where
$\varepsilon_{0}=mgz_{0}=\,$$0.602\text{\,}\mathrm{peV}$ is the characteristic
energy of the problem and
$f_{0}=\varepsilon_{0}/(2\pi\hbar)=\,$$145\text{\,}\mathrm{Hz}$ is its
characteristic frequency.

The five lowest zeros of the Airy function Ai are
$\lambda_{k}=\\{2.34,4.09,5.52,6.79,7.94...\\}$. The eigenfunctions of the
quantum states are $\psi_{k}(\xi(z))\sim C_{k}\text{Ai}(\xi_{k}(z))$, where
$\xi_{k}(z)=z/z_{0}-\lambda_{k}$, and $C_{k}$ are normalization constants.

The energy eigenvalues $E_{k}$ depend only on $m$, $g$ and $\hbar$, and are
independent of the ideal mirror properties. Within the classical description,
a neutron with the energy $E_{k}$ can rise in the gravitational field up to
the height $z_{k}=E_{k}/{mg}$. In quantum mechanics, the probability of
observing a neutron with the energy $E_{k}$ in the $k^{th}$ quantum state at a
height $z$ is equal to the squared modulus of its wave function (see Fig. 1).

Figure 1: Squared modules of the neutron wavefunctions $\abs{\psi_{k}(z)}^{2}$
as a function of the height $z$ for the five lowest quantum states; they
correspond to the probabilities of observing neutrons at a height $z$.

##  3 Measurement method

###  3.1 gqs measurement with $n$

In this section, the methods developed for the first observation of gqs of $n$
at ill will be described. The experimental installation is a one-component
gravitational ucn spectrometer with a high energy and spatial resolution
Nes:2000nima . The principle of its operation, illustrated in Fig. 2, is the
measurement of the $n$ flux through a slit between the mirror on bottom and
the flat scatterer on top as a function of the slit height $\Delta z$ which
can be changed and precisely measured.

Figure 2: Schematic of the experimental setup in the flow-through mode. 1 are
the bottom and top entrance collimator plates, arrows 2 correspond to neutron
classical trajectories between the collimator and the entrance to the slit
between mirror 3 and scatterer 4. Dotted horizontal arrows 5 illustrate
neutron quantum motion above the mirror. 6 is the neutron detector. The height
of the slit between the mirror and the scatterer can be varied and precisely
measured.

The scatterer surface is smooth on a large scale but rough on the
$\mathrm{\SIUnitSymbolMicro m}$ scale. The roughness amplitude is about a few
$\mathrm{\SIUnitSymbolMicro m}$, and is comparable to the characteristic scale
$z_{0}$ of the problem. The scatterer’s surface reflects $n$ which reach it
non-specularly, mixing the vertical and horizontal velocity components of the
$n$. Because the $n$ horizontal velocity components are much larger than their
vertical velocity components, such mixing causes numerous collisions of the
$n$ with the scatterer, thus resulting in a prompt loss of those $n$.

The length of the bottom mirror is chosen based on the energy-time uncertainty
relation $\Delta\tau\Delta E\geq\hbar/2$ . The observation of the $k^{th}$
quantum state is possible if the difference between the Eigenenergies of state
$k+1$ and state $k$, $\Delta E_{k+1,k}$ is bigger than the width of the
$k^{th}$ quantum state, $\delta E_{k}$: $\Delta E_{k+1,k}>\delta E_{k}$. As
the state number $k$ increases, $\Delta E_{k+1,k}\sim k^{-1/3}$ decreases
until the levels pass into the classical continuum. Evidently, measurements of
low quantum states are easier and more convenient. $\delta E_{k}$ is defined
by the time of flight of $n$ above the mirror. Therefore, the mirror length is
determined by the time interval needed to observe a $n$ in a gqs:
$\Delta\tau\sim$0.5\text{\,}\mathrm{ms}$$. It follows, that the mirror length
should be $L\sim$10\text{\,}\mathrm{cm}$$ for low states and for $n$
velocities $v_{hor}\sim 5-$$10\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$.

The vertical scale in the problem is defined by the relation between momentum
$p$, velocity $v$ and wavelength $\lambda$: $p=mv=h/\lambda$ and the momentum-
position uncertainty relation $\Delta p\Delta z\geq\hbar/2$. The smaller the
$n$ vertical velocity component, the larger the $n$ wavelength associated with
this velocity component. But the classical height up to which a $n$ can rise
in the gravitational field cannot be smaller than the quantum mechanical
uncertainty of its vertical coordinate, i.e., the $n$ wavelength. This
relation determines the lowest bound state of $n$ in the Earth’s gravitational
field. The height uncertainty is then $\Delta z\sim z_{0}$, and the vertical
velocity uncertainty is $\Delta v_{z}\sim
v_{0}=\sqrt{2\varepsilon_{0}/m}=$$1.07\text{\,}\mathrm{cm}\text{\,}{\mathrm{s}}^{-1}$,
the characteristic velocity in the problem.

The method used in the first observation of $n$ gqs Nes:2002Nat  consisted in
measuring $n$ transmission through the narrow slit $\Delta z$ between the
horizontal mirror and the scatterer above it. If $\Delta z\gg z_{k}$, neutrons
in the $k^{th}$ quantum state pass through the slit with no significant loss.
But as $\Delta z$ decreases, the neutron wave function $\psi_{k}(z)$ starts
penetrating the scatterer, and the $n$ loss probability increases. If $\Delta
z\leq z_{k}$, the slit is practically nontransparent to neutrons in the
$k^{th}$ quantum state. In an "ideal" experiment with an infinitely high
energy resolution, the $n$ flux $N_{QM}(\Delta z)$ through the slit would
sharply change at the height $\Delta z\sim z_{k}$. In reality, the idealized
step-like dependence is smoothed due to two factors: the spectrometer
experimental resolution and the smooth shape of $n$ wave functions. The latter
is due to the tunneling of $n$ through the gravitational barrier separating
the classically allowed heights and the scatterer height. An example of the
experimental data is shown in Fig. 3.

Figure 3: The data points correspond to the measured $n$ flux through the slit
between mirror and absorber versus the slit width at low width-values. The
dashed curve is a fit using the quantum-mechanical calculation, where all
level populations and the height resolution are extracted from the
experimental data. The solid curve is the full classical treatment. The dotted
line is a truncated fit in which it is assumed that only the lowest quantum
state - which leads to the first step - exists. Figure taken from Nes:2002Nat
.

###  3.2 gqs measurement with $H$

The method, developed for the observation of gqs of $n$, will be used to
demonstrate gqs of $H$. In the following section, the feasibility of such an
experiment will be analysed and the characteristic parameters of the two
experiments will be compared.

The observation time is a key parameter. The time of observation is defined by
the mean particle velocity and the mirror length. For the characteristic
velocity of $n$,
$v_{n}\sim\,$$10\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$ and the mirror
length $L_{n}=\,$$10\text{\,}\mathrm{cm}$, the observation time was
$\tau_{n}\sim\,$$10\text{\,}\mathrm{ms}$. $\tau_{n}/\tau_{0}\sim 20$, i.e. the
observation time is much larger than the formation time
($\tau_{n}\gg\tau_{0}$) and the gqs were well resolved. In order to provide
the same conditions for the resolution of gqs, the time of observation of $H$
has to be the same $\tau_{H}\sim\tau_{n}\sim\,$$10\text{\,}\mathrm{ms}$. For
the planned gqs-mirror length $L_{H}\sim\,$$30\text{\,}\mathrm{cm}$, the mean
velocity of $H$ has to be $v_{H}\sim
L_{H}/$10\text{\,}\mathrm{ms}$\sim$30\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$$.
All $H$ with significantly higher velocities do not settle in gqs, they only
increase the background and have to be eliminated. Velocities of up to
$v_{H}\sim$100\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$$ can still be
tolerated, at the cost of a worse energy resolution of the experiment. This is
still acceptable for the first observation of gqs of $H$. However, low
velocities, good control over the velocity selection and sufficient
background-suppression is the key condition for the observation of gqs of $H$.

A major difference in the behavior of $n$ and $H$, is the mechanism of their
interaction with the rough surface of the scatterer. Scattered $n$ are lost in
the bulk of the mirror or scatterer after several reflections from their
surfaces. Scattered $H$ atoms have higher chances to leak through the slit and
produce background. They escape from the slit into a broad angular
distribution, in contrast to $H$, which pass the slit setteld in a gqs. $H$
atoms escaping to larger angles have to be eliminated to decrease the
background. Therefore, it is very important to implement a proper background-
suppression.

The expected count rate of $H$ is much higher than that of $n$. A simple
comparison of the "brightness" of the $n$ source at ill and the $H$ source at
the Grasian experiment at eth Zurich is the following. The total number of the
particles produced at ill is $\sim${10}^{19}\text{\,}{\mathrm{s}}^{-1}$$,
while it is $\sim${10}^{17}\text{\,}{\mathrm{s}}^{-1}$$ at eth. A
characteristic temperature of $n$ spectrum at ill is
$\sim$40\text{\,}\mathrm{K}$$, while it is $\sim$6\text{\,}\mathrm{K}$$ at
eth. An effective surface area of the source at ill is
$\sim${10}^{5}\text{\,}{\mathrm{cm}}^{2}$$, while it is
$\sim$1\text{\,}{\mathrm{cm}}^{2}$$ at eth. These values result in a $\sim
10^{4}$ higher flux of particles at eth. This estimation does not account for
the velocity and spatial distributions of the particles at the two sources,
the detection efficiencies, the transport losses, the experiment geometries
etc. but gives a fair overall gain factor and the conclusion that $H$ fluxes
are much higher than those available at the best existing ucn sources. A more
detailed calculation provides the same estimate of the count rate for this
setup: $\sim 10^{3}$ $H${\mathrm{s}}^{-1}$$ for the lowest gqs, over four
orders of magnitude larger than measured with $n$. With these increased count
rates, and the minimal background with the $H$ detection method based on the
photo-ionization described in section 4.1, one can measure the height
distribution covering the first four gqs with a
$1\text{\,}\mathrm{\SIUnitSymbolMicro m}$ resolution and 5$\%$ accuracy in a
matter of hours (compared to several days with the original $n$ experiment).
These robust statistics will give fast feedback, enabling us to solve
experimental challenges quickly and efficiently.

##  4 Grasian cryogenic $H$-beam

In this section the Grasian cryogenic $H$-beam, located at eth Zurich, will be
described. It was originally developed in the context of the ASACUSA Diermaier
, Mu-MASS 2018-MuMASS ; PhysRevLett.128.011802  and GBAR Perez:2015zya
experiments to test and commission detectors, microwave- and laser-setups. It
will be used for our attempt to demonstrate GQS of atoms for the first time.

The eth $H$-beam consists of an $H$-source, a cryogenic chamber, a beamline
and a detection chamber. The GQS-chamber, which will contain the mirror-
absorber system described in section 3, will be installed in the near future.

Figure 4: Rendering of the eth cryogenic $H$-beam setup.

In the $H$-source, molecular hydrogen gas ($H_{2}$) is injected into a
microwave cavity with a flow of
$\sim$0.8\text{\,}\mathrm{m}\mathrm{L}\mathrm{/}\min$$. Within the cavity,
microwaves with 20-$30\text{\,}\mathrm{W}$ at $2400\text{\,}\mathrm{MHz}$,
dissociate the $H_{2}$ into atomic $H$. The cracking efficiency was measured
to be $\sim 80\%$, resulting in an $H$-flux of more than
${10}^{17}\text{\,}\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{m}\mathrm{s}\mathrm{/}\mathrm{s}$.

The generated atomic $H$ is directed into the cryogenic chamber through a
glass-Teflon tube. The tube is inserted into a bent nozzle, directly connected
with a coldhead to thermalize the atoms to $\sim$6\text{\,}\mathrm{K}$$ with
minimal recombination. After the cryogenic chamber, the beam of atomic $H$ is
divided into bunches of
$\sim$${10}^{14}\text{\,}\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{m}\mathrm{s}\mathrm{/}\mathrm{s}$
by a rotating chopper wheel. The $H$ bunches flow through a system of velocity
selecting apertures into the detection chamber at the end of the beamline.

A $243\text{\,}\mathrm{nm}$ ultraviolet (UV) laser is directed into the
detection chamber in a perpendicular direction with respect to the atomic beam
and is retro-reflected by a mirror on the other side. This setup creates two
counter propagating beams, as required to excite the hydrogen atoms from the
1S to the 2S state via a two-photon transition. A third photon from the same
laser induces photo-ionization, the resulting protons can be detected by a
microchannel plate (MCP).

###  4.1 The laser system for hydrogen detection

The $H$ 1S-2S transition is dipole forbidden, hence it is only possible to
drive it with two photons. The energy of this transition is split between the
two photons involved, meaning that the wavelength of the laser driving this
transition has to be $\sim$243\text{\,}\mathrm{nm}$$. As a dipole forbidden
transition is very unlikely, a high photon density and hence a high-intensity
laser is needed for efficient excitation and subsequent ionization of $H$
atoms. The most efficient realization is a pulsed laser. Such a laser system
has been set up at eth Zurich. It will be described in the following section.

The laser system consists of two lasers - a continuous wave (CW) and a pulsed
laser, and a pulsed dye amplifier (pda). The CW laser has a wavelength of
$486\text{\,}\mathrm{nm}$ and an average power of
$\sim$$250\text{\,}\mathrm{mW}$. With exactly double of the desired wavelength
for the two-photon 1S-2S transition, the CW laser acts as the seeder laser.
The pulsed laser is the $355\text{\,}\mathrm{nm}$ third harmonic of a
$10\text{\,}\mathrm{Hz}$ pulsed Nd:YAG laser. In the Q-switch mode, the pulses
are $10\text{\,}\mathrm{ns}$ long and have a pulse energy of
$\sim$$100\text{\,}\mathrm{mJ}$.

Within the pda, three quartz cuvettes are circulated with the dye, Coumarin
102, dissolved in ethanol. The absorption and fluorescence of Coumarin 102 fit
the application well: when pumped with $355\text{\,}\mathrm{nm}$-light, the
emission is centered around $473\text{\,}\mathrm{nm}$.

The pulsed laser is split up and focused onto the three cuvettes, and pumps
the Coumarin molecules. The CW beam seeds the pda by passing through the three
cuvettes overlapping with the pulsed laser beam, and hence with the pumped dye
molecules. This stimulates the emission of $486\text{\,}\mathrm{nm}$ photons
with every pulse that pumps the dye. After three cuvettes,
$\sim$10-$12\text{\,}\mathrm{mJ}$ of $486\text{\,}\mathrm{nm}$ pulsed laser
light is generated. Like the pulsed pump laser, it runs at
$10\text{\,}\mathrm{Hz}$ with a pulse length of
$\sim$10\text{\,}\mathrm{ns}$$. The output of the pda is frequency doubled
with a barium borate (bbo) crystal to generate $\sim$1-$2\text{\,}\mathrm{mJ}$
of pulsed UV radiation at $243\text{\,}\mathrm{nm}$.

The $243\text{\,}\mathrm{nm}$ beam is then sent into the detection chamber. A
mirror is mounted behind the photo ionization region to produce counter-
propagating beams for the doppler free two-photon excitation. The
$243\text{\,}\mathrm{n}\mathrm{m}$ photons efficiently ionize the $H$ and the
$H^{+}$ ions are detected by an MCP. The MCP creates a voltage signal, which
is read out by an oscilloscope. In figure 5, such a waveform of $H^{+}$ signal
is shown.

Figure 5: Waveform captured by the oscilloscope showing the $H^{+}$ signal.
The purple region determines the offset in the signal strength evaluation. The
magenta curve corresponds to the signal induced by the UV light in the
detection chamber. The red curve corresponds to the $H^{+}$ induced voltage
change in the expected time-of-flight window between $640\text{\,}\mathrm{ns}$
to $1.15\text{\,}\mathrm{\SIUnitSymbolMicro s}$.

Scanning the laser frequency in the $400\text{\,}\mathrm{MHz}$ range around
the resonance, shows that we are capable to resolve the hyperfine splitting
(HFS) of $H$. The two peaks, shown in figure 6, correspond to the difference
of the HFS of the 1S and the 2S state. The measured value is
$\Delta\nu_{meas.}=$$1.23(2)\text{\,}\mathrm{GHz}$, which agrees with the
literature value ($\Delta\nu_{lit.}=$$1.24\text{\,}\mathrm{GHz}$) within 1
$\sigma$ Parthey:2011lfa . This proves, that we detect $H$ atoms, and not any
other potential pollutants in the detection chamber.

Figure 6: Laser frequency sweep - Observation of the 1S and 2S HFS of $H$.

A scan of the laser pulse-intensity dependency of the signal resulted in the
same conclusion, showing the expected behavior. The intensity $I$ is
determined by the pulse energy $E$ and the beam waist
$\omega_{0}\sim$0.75\text{\,}\mathrm{mm}$$: $I=E/(\omega_{0}^{2}\pi)$.

The data can be taken from figure 7.

Figure 7: Laser intensity scan - Observation of the $I^{3}-$ and $I^{2}$
dependency of $H$ ionization.

The signal shows an $I^{3}$\- dependence as expected for a three-photon
process. The two-photon 1S-2S excitation is $I^{2}$-dependent and the
ionization of the 2S state adds another $I$-dependency to the overall process.
At $\sim${10}^{7}\text{\,}\mathrm{W}\text{\,}{\mathrm{cm}}^{-2}$$, the
2S-ionization process starts to saturate, as the ionization rate
$\Gamma_{i}=(I/{h\nu})\sigma$, where $\sigma$ is the 2S $H$ ionization cross
section and $h\nu$ the photon energy, reaches $1/\tau_{2S}$, where $\tau_{2S}$
is the lifetime of the 2S state. From here, the overall process follows an
$I^{2}$-dependency. At
$\sim$3\text{\times}{10}^{7}\text{\,}\mathrm{W}\text{\,}{\mathrm{cm}}^{-2}$$,
also the 1S-2S excitation process will start to saturate and the overall
process becomes $I$-independent Downey:89 .

It would be ideal to run the $H$ detection on saturation, because the
ionization process would become independent of energy- or frequency-
instabilities of the laser. In order to reach the point of saturation, the
laser beam size has to be decreased. Different combinations of convex and
concave lenses were already used to compress the beam. But, the mirrors did
not withstand the increased intensity for long and were damaged. It seemed
like the point of saturation overlapped with the damage threshold of the UV
mirrors, which were used at that point. We replaced the mirrors and will
hopefully be able to increase the intensity until saturation is reached.

###  4.2 Hydrogen beam characterization and rate estimation

To characterise the velocity distribution of the $H$-beam, a time of flight
(ToF) measurement was performed. The delay between the opening of the chopper
and the firing of the laser was varied while the $H$-count rate was measured.

The expected signal $S(t)$ is a convolution of the chopper kernel $h(t)$ and
the atomic ToF distribution $P_{t}(t)$ (assumed to follow a Maxwellian
distribution) and is given by

$\displaystyle S(t)$ $\displaystyle=h(t)*P_{t}(t)\,,$ (1) $\displaystyle
P_{v}(t)$ $\displaystyle\propto
v^{3}\exp{\left(-\frac{mv^{2}}{2kT}\right)}\,,$ (2) $\displaystyle P_{t}(t)$
$\displaystyle=P_{v}\left(\frac{\Delta x}{t}\right)\frac{\Delta x}{t^{2}}\,,$
(3)

where $\Delta x$ is the distance between chopper and detection region, $m$ is
the $H$-mass, $k$ is the Boltzmann constant and $T$ the temperature.

The data taken in 2021 and a fit are shown in figure 8. The fit resulted in a
temperature of $T=$6.07(74)\text{\,}\mathrm{K}$$, meaning, that the $H$-gas
thermalizes well with the cryogenic nozzle at $6\text{\,}\mathrm{K}$.

Figure 8: Upper figure: ToF data and fit. The fit results in a temperature of
$T=$6.07(74)\text{\,}\mathrm{K}$$. Lower figure: Standardized residuals of the
fit and corresponding histogram.

The measurement shows that the maximum of the atom flux appears after around
$5\text{\,}\mathrm{ms}$ delay, relating to an atomic velocity of
$250\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$ with a significant fraction
of atoms below $100\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$.

As mentioned in section 3.2, $H$ velocities of up to
$100\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$ can be tolerated. For the
upcoming gqs measurement, a certain velocity interval will be selected by
setting the delay between chopper opening and firing of the laser to a certain
value. The width of this interval is determined by the duration of the chopper
opening $t_{open}\sim$6.1\text{\,}\mathrm{ms}$$. With the current velocity
distribution of the $H$ beam, it makes sense, to set the upper bound of the
velocity interval to
$v_{max}=$100\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$$ which leads to a
lower bound of $v_{min}=\Delta
x/(t_{100}+t_{open})=$62\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$$. The
mean velocity of this interval is
$\bar{v}=$81\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$$ with a
corresponding ToF of $t_{\bar{v}}=$12.3\text{\,}\mathrm{ms}$$.

With the fit result, it is possible to estimate the rate of $H$ atoms passing
through the future GQS-region. It is composed of the $H$ input rate
$R_{in}\sim${10}^{17}\text{\,}{\mathrm{s}}^{-1}$$, the chopper duty cycle
$d_{c}\sim 0.061$, the form of the distribution after the chopper, assuming a
cone-like distribution with an opening angle of $\theta\sim\frac{3}{4}\pi$,
the cross-sectional area of the gqs region
$A\sim$0.5\text{\,}{\mathrm{mm}}^{2}$$ ($\Delta
z\sim$20\text{\,}\mathrm{\SIUnitSymbolMicro m}$$, $\Delta
y\sim$25\text{\,}\mathrm{mm}$$), the beam waist of the laser
$\omega_{0}\sim$0.75\text{\,}\mathrm{mm}$$ and the probability of the atoms
with the given velocity distribution to have a velocity within the selected
velocity interval, $P_{v}(v_{min}\geq v_{x}\geq v_{max})=3.9\times 10^{-3}$.
These parameters yield an estimated rate of $H$ passing through the GQS
chamber of

$R=R_{in}dP_{v}(v_{min}\geq v_{x}\geq v_{max})\\\ \frac{A}{2\pi\Delta
x^{2}\left(1-\cos{\theta/2}\right)}\frac{2\omega_{0}}{(v_{max}-v_{min})t_{\bar{v}}}\cong${10}^{4}\text{\,}{\mathrm{s}}^{-1}$\,.$
(4)

Multiplying $R$ by the ionization efficiency of $\epsilon_{ion}\sim 8\%$
(determined by $\omega_{0}$ and an assumed laser energy of
$1\text{\,}\mathrm{mJ}$) and the MCP efficiency $\epsilon_{MCP}\sim 50\%$
yields the signal rate $R_{sig}\cong$400\text{\,}{\mathrm{s}}^{-1}$$. This is
$4\times 10^{3}$ times more $H$ signal, as compared to the ucn signal.

##  5 Outlook

There are currently three major improvements being implemented and tested.

New UV mirrors with a higher damage threshold were installed. It can be
expected, that the laser intensity will be improved by an order of magnitude:
when the beam size is compressed to $\omega_{0}\sim$0.3\text{\,}\mathrm{mm}$$,
with a UV laser energy of $\sim$1\text{\,}\mathrm{mJ}$$, the intensity becomes
$\sim$3.6\text{\times}{10}^{7}\text{\,}\mathrm{W}\text{\,}{\mathrm{cm}}^{-2}$$.
At this level, saturation is reached and the signal becomes independent of
laser energy- or frequency fluctuations. Furthermore the ionization efficiency
will be improved dramatically. A beam size of $0.3\text{\,}\mathrm{mm}$ yields
an ionization efficiency of $\epsilon_{ion}=98.22\%$, which would improve the
estimated rate by a factor of 12.

The estimated rate will further be improved, by the installation of a new
coldhead and an additional heatshield. This is currently being implemented,
and first measurements show, that temperatures around
$\sim$4\text{\,}\mathrm{K}$$ can be expected. This would improve our estimated
rate for the velocity interval
$[$62\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$,\
$100\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$]$ by a factor of $\sim
2.2$. It would alternatively be possible to select slower velocities in the
interval $[$55\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$,\
$83\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$]$ while maintaining the same
countrate as with the old cryo system.

It would be preferable to go to even lower velocities. But, as can be seen in
figure 8 at around $20\text{\,}\mathrm{ms}$, the residual hydrogen gas in the
chamber prevents the measurements to be sensitive to atoms with lower
velocities. This could be improved by an aperture system between the source
and the chamber where the gqs region will be installed. Such a system is
currently being installed and tested. It consists of three height-adjustable,
vertical slits with a width of $200\text{\,}\mathrm{\SIUnitSymbolMicro m}$ for
the first slit and $1\text{\,}\mathrm{mm}$ for the second and third. Two more
vacuum pumps will be installed in between the first and the second and the
second and the third slit. This system has two purposes: It will decrease the
background, due to the separation of the different vacuum regions of the
cryogenic chamber, the beamline and the detection chamber. It will also act as
a velocity selecting aperture. As the slit height is adjustable, different
trajectories of the atoms can be selected. With the three slits, it will be
possible to select the low energy tail of the $H$ atoms with a vertical
velocity component $v_{z}\sim 0$ at the entrance of the gqs-spectrometer as
described in section 3.1.

As soon as those new implementations are completed and characterized, the gqs-
chamber will be installed at the end of the beamline replacing the detection
chamber. It will contain the gqs-spectrometer and the viewports for the UV
laser. In this way, the atoms passing through the spectrometer will be photo
ionized at the end of the mirror and the $H^{+}$ detected in the MCP.

##  6 Conclusions

We conclude that a gqs measurement with $H$ is a very promising but
challenging endeavor. The expected count rate exceeds the count rates
accessible with ucns by orders of magnitudes.

An extension and improvement of the existing gqs measurements is highly
interesting for multiple fields.

In the course of realizing a gqs measurement with $H$, we set up a cryogenic
$H$-beam. A highly efficient $H$ detection system was developed. New UV
mirrors, an improved cryo-system and an aperture system which will reduce the
background and select ideal velocity components are currently being
implemented and tested.

We aim to demonstrate the existence of gqs of $H$ within the next measurement
campaign, starting in 2023.

##  7 Acknowledgments

This project was supported by the Austrian Science Fund (FWF) [W1252-N27]
(Doktoratskolleg Particles and Interactions) and the ETH Zurich Career Seed
Grant [SEED-17 20-1]. François Nez and Pauline Yzombard acknowledge support
from CNRS (IEA 2021-2022 QRECH). Paolo Crivelli acknowledges the support of
the European Research Council (grant 818053-Mu-MASS) and the Swiss National
Science Foundation (grant 197346).

## References

  *   * (1) G. Breit, The propagation of schroedinger waves in a uniform field of force.  Phys. Rev. 32, 273–276 (1928).  10.1103/PhysRev.32.273.  URL https://link.aps.org/doi/10.1103/PhysRev.32.273
  * (2) V.I. Luschikov, A.I. Frank, Quantum effects occurring when ultracold neutrons are stored on a plane.  JETP Lett. (USSR) (Engl. Transl.); (United States) 28:9 (1978).  URL https://www.osti.gov/biblio/6125370
  * (3) V. Nesvizhevsky, H. Borner, A. Gagarski, G. Petrov, A. Petukhov, H. Abele, S. Bassler, T. Stoferle, S. Soloviev, Search for quantum states of the neutron in a gravitational field: gravitational levels.  Nucl. Instr. Meth. A 440, 754 (2000).  10.1016/S0168-9002(99)01077-3
  * (4) V. Nesvizhevsky, H. Boerner, A. Petukhov, H. Abele, S. Baessler, F. Ruess, T. Stoferle, A. Westphal, A. Gagarski, G. Petrov, A. Strelkov, Quantum states of neutrons in the Earth’s gravitational field.  Nature 415, 297 (2002).  10.1038/415297a
  * (5) V. Nesvizhevsky, H. Borner, A. Gagarski, A. Petoukhov, G. Petrov, H. Abele, S. Baessler, G. Divkovic, F. Ruess, T. Stoferle, A. Westphal, A. Strelkov, K. Protasov, A. Voronin, Measurement of quantum states of neutrons in the earth’s gravitational field.  Phys. Rev. D 67, 102,002 (2003).  10.1103/PhysRevD.67.102002
  * (6) V. Nesvizhevsky, A. Petoukhov, H. Borner, K. Protasov, A. Voronin, A. Westphal, S. Baessler, H. Abele, A. Gagarski, Reply to “comment on ‘measurement of quantum states of neutrons in the earth’s gravitational field’ “.  Phys. Rev. D 68, 108,702 (2003).  10.1103/PhysRevD.68.108702
  * (7) V. Nesvizhevsky, A. Petukhov, H. Borner, T. Baranova, A. Gagarski, G. Petrov, K. Protasov, A. Voronin, S. Baessler, H. Abele, A. Westphal, L. Lucovac, Study of the neutron quantum states in the gravity field.  Europ. Phys. J. C 40, 479 (2005).  10.1140/epjc/s2005-02135-y
  * (8) A. Westphal, H. Abele, S. BaeBler, V. Nesvizhevsky, K. Protasov, A. Voronin, A quantum mechanical description of the experiment on the observation of gravitationally bound states.  Europ. Phys. J. C 51, 367 (2007).  10.1140/epjc/s10052-007-0283-x
  * (9) V. Nesvizhevsky, A. Voronin, R. Cubitt, K. Protasov, Neutron whispering gallery.  Nature Phys. 6, 114 (2010)
  * (10) R. Robinett, Quantum wave packet revivals.  Phys. Rep. 392, 1 (2004).  10.1016/j.physrep.2003.11.002
  * (11) M. Berberan-Santos, E. Bodunov, L. Pogliani, Classical and quantum study of the motion of a particle in a gravitational field.  J. Math. Chem. 37, 101 (2005).  10.1007/s10910-004-1443-y
  * (12) W. Mather, R. Fox, Coherent-state analysis of the quantum bouncing ball.  Phys. Rev. A 73, 032,109 (2006).  10.1103/PhysRevA.73.032109
  * (13) O. Bertolami, J. Rosa, Quantum and classical divide: the gravitational case.  Phys. Lett. B 633, 111 (2006).  10.1016/j.physletb.2005.11.041
  * (14) E. Romera, F. de los Santos, Identifying wave-packet fractional revivals by means of information entropy.  Phys. Rev. Lett. 99, 263,601 (2007).  10.1103/PhysRevLett.99.263601
  * (15) G.D. Valle, M. Savoini, M. Ornigotti, P. Laporta, V. Foglietti, M. Finazzi, L. Duo, S. Longhi, Experimental observation of a photon bouncing ball.  Phys. Rev. Lett. 102, 180,402 (2009).  10.1103/PhysRevLett.102.180402
  * (16) I. Garanovich, S. Longhi, A. Sukhorukov, Y. Kivshar, Light propagation and localization in modulated photonic lattices and waveguides.  Phys. Rep. 518, 1 (2012).  10.1016/j.physrep.2012.03.005
  * (17) M. Belloni, R. Robinett, The infinite well and dirac delta function potentials as pedagogical, mathematical and physical models in quantum mechanics.  Phys. Rep. 540, 25 (2014).  10.1016/j.physrep.2014.02.005
  * (18) H. Abele, S. Baessler, A. Westphal, (2003), p. 355
  * (19) A. Frank, P.V. Isacker, J. Gomez-Camacho, Probing additional dimensions in the universe with neutron experiments.  Phys. Lett. B 582, 15 (2004).  10.1016/j.physletb.2003.12.026
  * (20) F. Brau, F. Buisseret, Minimal length uncertainty relation and gravitational quantum well.  Physical Review D 74(3) (2006).  10.1103/physrevd.74.036002.  URL https://doi.org/10.1103%2Fphysrevd.74.036002
  * (21) F. Buisseret, B. Silvestre-Brac, V. Mathieu, Modified newton’s law, braneworlds and the gravitational quantum well.  Class. Quant. Grav. 24, 855 (2007).  10.1088/0264-9381/24/4/006
  * (22) S. Baessler, V. Nesvizhevsky, K. Protasov, A. Voronin, Constraint on the coupling of axionlike particles to matter via an ultracold neutron gravitational experiment.  Phys. Rev. D 75, 075,006 (2007).  10.1103/PhysRevD.75.075006
  * (23) I. Antoniadis, S. Baessler, M. Buechner, V. Fedorov, S. Hoedl, A. Lambrecht, V. Nesvizhevsky, G. Pignol, K. Protasov, S. Reynaud, Y. Sobolev, Short-range fundamental forces.  Compt. Rend. Phys. 12, 755 (2011).  10.1016/j.crhy.2011.05.004
  * (24) P. Brax, G. Pignol, Strongly coupled chameleons and the neutronic quantum bouncer.  Phys. Rev. Lett. 107 (2011).  10.1103/PhysRevLett.107.111301
  * (25) O. Bertolami, F. Nunes, Ultracold neutrons, quantum effects of gravity and the weak equivalence principle.  Class. Quant. Grav. 20, L61 (2003).  10.1088/0264-9381/20/5/103
  * (26) C. Kiefer, C. Weber, On the interaction of mesoscopic quantum systems with gravity.  Ann. Phys. 14, 253 (2005).  10.1002/andp.200410119
  * (27) E. Kajari, N. Harshman, E. Rasel, S. Stenholm, G. Suessmann, W. Schleich, Inertial and gravitational mass in quantum mechanics.  Appl. Phys. B 100, 43 (2010).  10.1007/s00340-010-4085-8
  * (28) O. Bertolami, J. Rosa, C. de Aragao, P. Castorina, D. Zappala, Noncommutative gravitational quantum well.  Phys. Rev. D 72 (2005).  10.1103/PhysRevD.72.025010
  * (29) A. Saha, Time-space non-commutativity in gravitational quantum well scenario.  Europ. Phys. J. C 51, 199 (2007).  10.1140/epjc/s10052-007-0274-y
  * (30) K. Nozari, P. Pedram, Minimal length and bouncing-particle spectrum.  EPL 92 (2010).  10.1209/0295-5075/92/50013
  * (31) P. Pedram, K. Nozari, S. Taheri, The effects of minimal length and maximal momentum on the transition rate of ultra cold neutrons in gravitational field.  J. High En. Phys. (2011).  10.1007/JHEP03(2011)093
  * (32) A. Kobakhidze, Gravity is not an entropic force.  Phys. Rev. D 83 (2011).  10.1103/PhysRevD.83.021502
  * (33) M. Chaichian, M. Oksanen, A. Tureanu, On gravity as an entropic force.  Phys. Lett. B 702, 419 (2011).  10.1016/j.physletb.2011.07.019
  * (34) A. Martin-Ruiz, C. Escobar, Testing lorentz and cpt invariance with ultracold neutrons.  Phys. Rev. D 97 (2018).  10.1103/PhysRevD.97.095039
  * (35) C. Escobar, A. Martin-Ruiz, Gravitational searches for lorentz violation with ultracold neutrons.  Phys. Rev. D 99 (2019).  10.1103/PhysRevD.99.075032
  * (36) A. Ivanov, M. Wellenzohn, H. Abele, Probing of violation of lorentz invariance by ultracold neutrons in the standard model extension.  Phys. Lett. B 797 (2019).  10.1016/j.physletb.2019.134819
  * (37) V. Nesvizhevsky, K. Protasov, _in: Trends in Quantum Gravity Research, Editor: David C. Moore, pp. 65-107_ (2006)
  * (38) V. Nesvizhevsky, Near-surface quantum states of neutrons in the gravitational and centrifugal potentials.  Phys. Usp. 53, 645 (2010).  10.3367/UFNe.0180.201007a.0673
  * (39) T. Jenke, P. Geltenbort, H. Lemmel, H. Abele, Realization of a gravity-resonance-spectroscopy technique.  Nat. Phys. 7, 468 (2011).  10.1038/NPHYS1970
  * (40) T. Jenke, G. Cronenberg, J. Burgdoerfer, L. Chizhova, P. Geltenbort, A. Ivanov, T. Lauer, T. Lins, S. Rotter, H. Saul, U. Schmidt, H. Abele, Gravity resonance spectroscopy constrains dark energy and dark matter scenarios.  Phys. Rev. Lett. 112 (2014).  10.1103/PhysRevLett.112.151105
  * (41) G. Ichikawa, S. Komamiya, Y. Kamiya, Y. Minami, M. Tani, P. Geltenbort, K. Yamamura, M. Nagano, T. Sanuki, S. Kawasaki, M. Hino, M. Kitaguchi, Observation of the spatial distribution of gravitationally bound quantum states of ultracold neutrons and its derivation using the wigner function.  Phys. Rev. Lett. 112 (2014).  10.1103/PhysRevLett.112.071101
  * (42) A. Voronin, P. Froelich, V. Nesvizhevsky, Gravitational quantum states of antihydrogen.  Phys. Rev. A 83, 032,903 (2011).  10.1103/PhysRevA.83.032903
  * (43) P.P. Crepin, G. Dufour, R. Guerout, A. Lambrecht, S. Reynaud, Casimir-polder shifts on quantum levitation states.  PHYSICAL REVIEW A 95(3) (2017).  10.1103/PhysRevA.95.032501
  * (44) G. Dufour, A. Gerardin, R. Guerout, A. Lambrecht, V.V. Nesvizhevsky, S. Reynaud, A.Y. Voronin, Quantum reflection of antihydrogen from the casimir potential above matter slabs.  PHYSICAL REVIEW A 87(1) (2013).  10.1103/PhysRevA.87.012901
  * (45) T. Pasquini, Y. Shin, C. Sanner, M. Saba, A. Schirotzek, D. Pritchard, W. Ketterle, Quantum reflection from a solid surface at normal incidence.  PHYSICAL REVIEW LETTERS 93(22) (2004).  10.1103/PhysRevLett.93.223201
  * (46) V. Druzhinina, M. DeKieviet, Experimental observation of quantum reflection far from threshold.  PHYSICAL REVIEW LETTERS 91(19) (2003).  10.1103/PhysRevLett.91.193202
  * (47) I. YU, J. DOYLE, J. SANDBERG, C. CESAR, D. KLEPPNER, T. GREYTAK, Evidence for universal quantum reflection of hydrogen from liquid-he-4.  PHYSICAL REVIEW LETTERS 71(10), 1589–1592 (1993).  10.1103/PhysRevLett.71.1589
  * (48) J. BERKHOUT, O. LUITEN, I. SETIJA, T. HIJMANS, T. MIZUSAKI, J. WALRAVEN, Quantum reflection - focusing of hydrogen-atoms with a concave mirror.  PHYSICAL REVIEW LETTERS 63(16), 1689–1692 (1989).  10.1103/PhysRevLett.63.1689
  * (49) M. BERRY, K. MOUNT, Semiclassical approximations in wave mechanics.  REPORTS ON PROGRESS IN PHYSICS 35(4), 315+ (1972).  10.1088/0034-4885/35/1/306
  * (50) H. Friedrich, J. Trost, Working with wkb waves far from the semiclassical limit.  PHYSICS REPORTS-REVIEW SECTION OF PHYSICS LETTERS 397(6), 359–449 (2004).  10.1016/j.physrep.2004.04.001
  * (51) S.F. Cooper, A.D. Brandt, C. Rasor, Z. Burkley, D.C. Yost, Cryogenic atomic hydrogen beam apparatus with velocity characterization.  Rev Sci Instrum 91(1), 013,201 (2020)
  * (52) R. Golub, D. Richardson, S. Lamoreaux, _Ultra-Cold Neutrons_ (Taylor & Francis, 1991).  URL https://books.google.at/books?id=QPUjlcWNg78C
  * (53) M. Diermaier, C. Jepsen, B. Kolbinger, C. Malbrunot, O. Massiczek, C. Sauerzopf, M. Simon, J. Zmeskal, E. Widmann, In-beam measurement of the hydrogen hyperfine splitting and prospects for antihydrogen spectroscopy.  Nature Communications 8 (2017).  10.1038/ncomms15749
  * (54) P. Crivelli, The mu-mass (muonium laser spectroscopy) experiment.  Hyperfine Interactions 239(1), 49 (2018)
  * (55) B. Ohayon, G. Janka, I. Cortinovis, Z. Burkley, L.d.S. Borges, E. Depero, A. Golovizin, X. Ni, Z. Salman, A. Suter, C. Vigo, T. Prokscha, P. Crivelli, Precision measurement of the lamb shift in muonium.  Phys. Rev. Lett. 128, 011,802 (2022).  10.1103/PhysRevLett.128.011802.  URL https://link.aps.org/doi/10.1103/PhysRevLett.128.011802
  * (56) P. Pérez, et al., The GBAR antimatter gravity experiment.  Hyperfine Interact. 233(1-3), 21–27 (2015).  10.1007/s10751-015-1154-8
  * (57) C.G. Parthey, et al., Improved Measurement of the Hydrogen 1S - 2S Transition Frequency.  Phys. Rev. Lett. 107, 203,001 (2011).  10.1103/PhysRevLett.107.203001.  arXiv:1107.3101 [physics.atom-ph]
  * (58) S.W. Downey, R.S. Hozack, Saturation of three-photon ionization of atomic hydrogen and deuterium at 243 nm.  Opt. Lett. 14(1), 15–17 (1989).  10.1364/OL.14.000015.  URL https://opg.optica.org/ol/abstract.cfm?URI=ol-14-1-15