# Quantum dynamics for massless particles in Brinkmann spacetimes

Álvaro Duenas-Vidal<EMAIL_ADDRESS>Departamento de Física, Escuela
Politécnica Nacional, Quito 170143, Ecuador  Jorge Segovia<EMAIL_ADDRESS>Departamento de Sistemas Físicos, Químicos y Naturales,
Universidad Pablo de Olavide, E-41013 Sevilla, Spain

###### Abstract

In Classical Dynamics, Eisenhart lift connects the dynamics of null geodesics
in a Brinkmann spacetime with a continuous family of Hamiltonian systems by
means of a suitable projection. In this work we explore the possibility of
building a model for quantum dynamics of massless particles propagating inside
a Brinkmann spacetime from the Einsenhart lift. As a result, we describe
spatial tunneling between regions classically disconnected for certain class
of null geodesics because of curvature. Also we describe entangled states
arising from observers who have a limited access to the whole Brinkmann space.
Finally we explore the possibility to find a quantum field theory behind these
quantum phenomena.

##  I Introduction

In a microscopic scale, mass particles in a scalar potential evolve showing a
collection of phenomena characteristic of Quantum Mechanics as, for example,
quantum tunneling, barrier penetration, etc. As a particular example we can
choose the scalar potential of Newtonian gravity and, therefore, we would
expect that the microscopic dynamics of a mass particle subjected to gravity
follows the rules of Quantum Mechanics as a response to the Newtonian
gravitational potential. However, gravity is driven by curvature, no by means
of a scalar potential. On the other hand, in the framework of General
Relativity, massless particles follow null geodesics that see curvature. So a
natural question arises: do massless particles response to spacetime
curvature, in a microscopic scale, following the rules of Quantum Mechanics?
Note that we are taking about a scale in which particles behave following
Quantum Mechanics but gravity is still described by classical (_i.e._ no
quantum) General Relativity.

The role of curvature in Quantum Mechanics has been considered many times in
literature. Early works explored the generalization of Quantum Mechanics in
the Euclidean space to Riemannian manifolds. In this way, in PhysRevA.23.1982
the role of the curvature was computing through the constrain of the dynamics
to curved submanifolds of the Euclidean space. On the other hand, in
RevModPhys.29.377  the curvature was incorporated in a natural way by means of
a connection in the fiber bundle of complex number over the space. Each model
brings to different results and problems, and nowadays in not clear what is
the correct answer. With respect to the modeling of the quantum dynamics for
geodesics in curved spacetimes, it has been addressed both in the framework of
Quantum Geometry Beggs:2022bhw ; Liu:2021mgs , where non-commutative spaces
serve as play model for Quantum Gravity, and in the context of quantum
correcting to gravity Miller:2016gwd ; Deng:2020yfm ; Gwynne:2019der ;
Gwynne:2020nex ; Emelyanov:2020wxr ; Dalvit:1999wd . Finally, issues
concerning if gravity sees quantum physics was considered in Colella:1975dq
and, more recently, in Krisnanda:2019glc ; Carlesso:2019cuh .

In this work we propose an approach different of the ones previously
mentioned. Here we do not pretend to develop any quantum model for gravity
either to introduce quantum corrections to gravity. We neither build a
covariant formulation of Quantum Mechanics. Our point of view is to move in a
scale where gravity is well described by General Relativity but particles
exhibit quantum dynamics, and describe this quantum dynamics for a selected
set of observers. For achieving this goal, our point of departure is the
Eisenhart lift.

The Eisenhart lift arises in Classical Dynamics as an attempt to _geometrize_
the dynamics of Hamiltonian systems Cariglia:2015bla ; Cariglia:2016oft .
Being more specific, the Eisenhart mechanism works _lifting_ the dynamics of
some Hamiltonian system to a higher dimensional space by introducing two
addition coordinates. The key point is that the lifted trajectories correspond
to null geodesics in a pseudo-Riemannian space with Lorentzian signature, such
that the coupling constant to scalara and vector potentials appearing in the
original Hamiltonian system corresponds to a conserved quantity over null
geodesics. This conserved quantity is related to a covanriantly conserved
Killing vector and, thus, the null geodesics are in a Brinkmann spacetime
Brinkmann:1925fr ; stephani_2003 ; Blau:2003dz ; Blau:2004yi . As a
consequence, the inverse of the Eisenhart lift gives a projection of the
dynamics of null geodesics in a Brinkamnn space over a continuous
uniparametric family of Hamiltonian systems depending on the coupling
constant.

Based on the connection previously mentioned between null geodesics and a
continuous family of Hamiltonian systems, in this paper we propose to define
quantum dynamics for massless particles in Brinkmann spacetimes by
_quantizing_ the dynamics in each one of the Hamiltonian systems of the
continuous family served by the Eisenhart lift. In some way this is not a new
idea, since second quantization in Quantum Field Theory is carried out by
decomposing the field in a continuum of modes and, then, describing each mode
by means of the quantum harmonic oscillator.

The paper is organized as follow: in Sec. II we discuss briefly the Eisenhart
lift and its relation with null geodesics in Brinkmann spacetimes, setting the
connection between null geodesics and a continuous family of _reduced_
Hamiltonian systems. In Sec. III, as a warm up exercise, we pay some attention
to the Eisenhart lift of autonomous and natural Hamiltonian systems. In Sec.
IV we define quantum dynamics for massless particles in Brinkmann spacetimes
from this continuous family of reduced Hamiltonian systems, doing an special
emphasis over stationary Brinkamnn spacetimes. In Sec. V we extract some
consequences from the model previously developed, and we describe quantum
tunneling and entanglement for null particles in Brinkmann spacetimes.
Finally, in Sec. VI, the possibility to find a Quantum Field Theory behind the
quantum-mechanical model developed is explored, while in Sec. VII some
conclusions and future work are addressed.

##  II Eisenhart lift and Brinkmann spaces

The Eisenhart lift connects the dynamics of Hamiltonian systems with the
dynamics of null particles in a pseudo-Riemannian manifold. In this section we
show briefly how this is done and the connection with Brinkmann spaces. Deeper
details in the Eisenhart lift can be found in Refs. Cariglia:2015bla ;
Cariglia:2016oft  while, for Brinkamnn spaces, canonical texts as Refs.
Ehlers:1962zz ; stephani_2003  can serve as a complete view of the subject.

###  II.1 The Eisenhart lift briefly

Let $(F,\omega,H(\lambda))$ be a Hamiltonian system, where
$\omega=\sum_{i}dq^{i}\wedge dp_{i}$, $i=1,\ldots n$, $F=T^{*}Q_{n}$ and
$\lambda$ the coupling constant to external potentials. The Eisenhart lift
consists in building a new Hamiltonian system
$(\mathcal{F},\Omega,\mathcal{H})$, with $\Omega=\omega+du\wedge
dp_{u}+dv\wedge p_{v}$, $\mathcal{F}=T^{*}\mathcal{Q}_{n+2}$ and,

$\mathcal{H}=\frac{1}{2}\bar{g}^{ij}\left(p_{i}-\frac{p_{v}}{2}\mathcal{A}_{i}\right)\left(p_{j}-\frac{p_{v}}{2}\mathcal{A}_{j}\right)-\frac{p_{v}^{2}}{2}\,\Phi+p_{v}p_{u},$
(1)

such that,

$\mathcal{H}_{\lambda}=\left.H(\lambda)\right|_{t\rightarrow u}+\lambda
p_{u},$ (2)

where the the subscript $\lambda$ means ’valued over $p_{v}=\lambda$’. In the
most general case, $(F,\omega,H(\lambda))$ is a no-autonomous and no-natural
Hamiltonian system with scalar potential $\varphi(t,q)$, vector potential
$A_{i}(t,q)$ and kinetic metric $g_{ij}(t,q)$ given by,

$g_{ij}=\left.\bar{g}_{ij}\right|_{u\rightarrow
t},\quad\varphi=-\frac{1}{2}\left.\Phi\right|_{u\rightarrow t},\quad
A_{i}=\frac{1}{2}\left.\mathcal{A}_{i}\right|_{u\rightarrow t}.$ (3)

We impose the condition that $det(g)\neq 0$ for any value of $t$ and $q^{i}$.
Then, positiveness of the kinetic term fixes $g_{ij}$ as a uniparametric
family of Riemannian metrics over the configuration space $Q_{n}$ for each
value of $t$. From now on we shall call $(\mathcal{F},\Omega,\mathcal{H})$ the
_lifted_ Hamiltonian system, while we shall call $(F,\omega,H(\lambda))$ the
_reduced_ Hamiltonian system with coupling $\lambda$.

The ”magic” of the Eisenhart lift is that the lifted Hamiltonian covers the
dynamics of the reduced Hamiltonian. For showing this fact, note that $v$ is a
cyclic coordinate in the lifted Hamiltonian Eq. (1) and, therefore, there are
curves from the lifted Hamiltonian flow that satisfies $p_{v}=\lambda$. Then,

$\frac{\partial\mathcal{H}_{\lambda}}{\partial q^{i}}=\left.\frac{\partial
H(\lambda)}{\partial q^{i}}\right|_{t\rightarrow
u},\quad\frac{\partial\mathcal{H}_{\lambda}}{\partial
p_{i}}=\left.\frac{\partial H(\lambda)}{\partial p_{i}}\right|_{t\rightarrow
u},$ (4)

and the Hamiltonian system $(\mathcal{F},\hat{\Omega},\mathcal{H})$ includes
the dynamics of $(F,\omega,H(\lambda))$ for each value of the coupling
$\lambda$. For $p_{v}=\lambda$, the set of equations of motion is completed
with,

$\begin{split}\dot{u}&=\lambda,\\\
\dot{v}&=-\frac{1}{4}\bar{g}_{ij}\mathcal{A}_{i}(2p_{j}-\lambda\mathcal{A}_{j})-\lambda\Phi+p_{u},\\\
\dot{p}_{u}&=-\frac{1}{8}\partial_{u}\bar{g}_{ij}\,(2p_{i}-\lambda\mathcal{A}_{i})(2p_{j}-\lambda\mathcal{A}_{j})+\frac{\lambda}{4}\bar{g}_{ij}\,\partial_{u}\mathcal{A}_{i}\,(2p_{j}-\lambda\mathcal{A}_{j})+\frac{\lambda^{2}}{2}\partial_{u}\Phi,\\\
\dot{p_{v}}&=0.\end{split}$ (5)

Note that the value of the momentum $p_{v}=\lambda$ gives the coupling of the
reduced system to the scalar and vector potentials Eq. (3) and, so, the
coupling becomes dynamical in the lifted system.

Obviously, not all curves of the Hamiltonian flow in
$(\mathcal{F},\Omega,\mathcal{H})$ projects over the Hamiltonian flow of the
reduced system $(F,\omega,H(\lambda_{0}))$ for some fixed coupling
$\lambda_{0}$, since we have to choose the curves with $p_{y}=\lambda_{0}$. In
a mathematical language, defining $\hat{\Gamma}_{\lambda_{0}}$ as the set of
curves in $\mathcal{Q}_{n+2}$ solving the lifted equations of motion for
$p_{v}=\lambda_{0}$, the projection,

$\pi:\mathcal{Q}_{n+2}\rightarrow Q_{n},\quad\pi\left(q^{1},\ldots
q^{n},u,v\right)=(q^{1},\ldots q^{n}),$ (6)

induces a map,

$\hat{\pi}:\hat{\Gamma}_{\lambda_{0}}\rightarrow\Gamma,$ (7)

where $\Gamma$ is the set of curves in $Q_{n}$ solving the equations of motion
in $(F,\omega,H(\lambda_{0}))$. Even under the restriction to the sector
$p_{v}=\lambda_{0}$ of the lifted dynamics, the map Eq. (6) is not a one-to-
one map. Since the $u$ coordinate replaces the time in the original
Hamiltonian system, the lifted Hamiltonian is autonomous and, therefore,

$\varepsilon_{\gamma}=\mathcal{H}|_{\gamma(t)},$ (8)

for any curve $\gamma(t)\in\hat{\Gamma}_{\lambda_{0}}$, is a conserved
quantity. This conserved quantity does not arise from the dynamics in the
configuration space $Q_{n}$ and, therefore, it is a free parameter. Thus, the
inverse of the map Eq. (7) gives and infinity set of curves in
$\hat{\Gamma}_{\lambda_{0}}$ for each curve in $\Gamma$, one curve for each
possible value of $\varepsilon_{\gamma}$. Fixing a value
$\varepsilon_{\gamma}$ gives a one-to-one map. For simplicity we decide to fix
$\varepsilon_{\gamma}=0$ and, then, we get the bijection:

$\hat{\pi}:\hat{\Gamma}^{0}_{\lambda_{0}}\simeq\Gamma,$ (9)

where $\hat{\Gamma}^{0}_{\lambda_{0}}$ is the set of curves in
$\mathcal{Q}_{n+2}$ solving the lifted equations of motion with
$p_{v}=\lambda_{0}$ and $\varepsilon_{\gamma}=0$.

The power of the Eisenhart lift yields over the fact that the lifted
Hamiltonian is homogeneous of second degree in momenta. As a consequence the
lifted Lagrangian formally coincides with the lifted Hamiltonian, and a short
computation gives,

$\mathcal{L}=\frac{1}{2}\bar{g}_{ij}\,\dot{q}^{i}\dot{q}^{j}+\frac{\Phi}{2}\dot{u}^{2}+\dot{u}\left(\dot{v}+\frac{1}{2}\mathcal{A}_{i}\dot{q}^{i}\right).$
(10)

This Lagrangian can be re-interpreted as the one for affine-parametrized
geodesics in $\mathcal{Q}_{n+2}$ with metric,

$ds^{2}=2dvdu+\left(\Phi(u,q)du+\mathcal{A}_{i}(u,q)dq^{i}\right)du+\bar{g}_{ij}(u,q)dq^{i}dq^{j}.$
(11)

In addition, a simple computation gives,

$\mbox{det}(G)=-4\ \mbox{det}(\bar{g}),$ (12)

and, thus, $G_{ab}$ equips $\mathcal{Q}_{n+2}$ with a pseudo-Riemannian
geometry with Lorentzian signature. In this way we gets an effective
_geometrization_ of the classical dynamics of the set of reduced Hamiltonian
systems $(F,\omega,H(\lambda))$, $\lambda\in\mathbb{R}$, by means of the
Eisenhart lift. Since the lifted Lagrangian coincides formally with the lifted
Hamiltonian, we have,

$\varepsilon_{\gamma}=\mathcal{L}(\gamma,\dot{\gamma})=\dot{\gamma}^{a}\dot{\gamma}^{b}G_{ab}.$
(13)

On the other hand, the vector field $\vec{l}=\partial_{v}$ is a Killing field
of the line element Eq. (11) and,

$l^{a}\dot{\gamma}^{b}G_{ab}=p_{v}.$ (14)

Thus,

$\hat{\Gamma}^{0}_{\lambda_{0}}=\\{\gamma(t)\ \mbox{geodesics}\ :\
\dot{\gamma}^{a}\dot{\gamma}^{b}G_{ab}=0,\ \mbox{and}\
l^{a}\dot{\gamma}^{b}G_{ab}=\lambda_{0}\\}.$ (15)

In other words, the curves solving the equations of motions in the reduced
system $(F,\omega,H(\lambda_{0}))$ have a one-to-one correspondence with null
geodesics in $(\mathcal{Q}_{n+2},G_{ab})$ satisfying
$l_{a}\dot{\gamma}^{a}=\lambda_{0}$. Reversely, the set of all null geodesics
in $(\mathcal{Q}_{n+2},G_{ab})$ projects, by means of the map Eq. (6), over
the curves in $Q_{n}$ solving the equations of motion of the continuous family
of reduced Hamiltonian systems $(F,\omega,H(\lambda))$, $\forall\
\lambda\in\mathbb{R}$. So, there is a bridge between the dynamics of null
geodesic in $(\mathcal{Q}_{n+2},G_{ab})$ and the set of reduced Hamiltonian
systems $(F,\omega,H(\lambda))$.

###  II.2 Relation with Brinkmann spaces and pp-waves

Until now we have done somethings that are well know for the physicist that
works in the area of Classical Dynamics. However, thinking as a General
Relativity physicist, $(\mathcal{Q}_{n+2},G_{ab})$ seems much more like some
type of spacetime. Indeed, the line element Eq. (11) appears in the framework
of General Relativity and, also, String Theory Tseytlin:1995fh , as the most
general metric we can build from a covariantly conserved null Killing vector
field. In our case, this covariantly conserved vector is the one in Eq. (14):

$\nabla_{a}l_{b}=0.$ (16)

Spaces with a metric of the form Eq. (11) are called _Binkmann spaces_
Brinkmann:1925fr . Every $u=u_{0}$ hypersurface collapses to a $n$-dimensional
submanifold which is called the transverse space for $u=u_{0}$, and every
transverse space corresponds to the configuration space $Q_{n}$ equipped with
the transverse metric $\bar{g}_{ij}(q,u=u_{0})$.

Following from the point of view of a physicist working in General Relativity,
the next issue calling our curiosity is what about curvature. The first task
is to compute the inverse metric $G^{ab}$. After some algebra,

$\begin{split}G^{uu}&=0,\quad G^{uv}=1,\quad G^{ui}=0\\\
G^{vv}&=-\Phi+\frac{1}{\bar{g}(n-1)!}\varepsilon^{i_{1}\ldots
i_{n}}\mathcal{A}_{i_{1}}\mbox{det}[\vec{\mathcal{A}},\vec{g}_{i_{2}},\ldots,\vec{g}_{i_{n}}],\\\
G^{vi}&=-\frac{1}{\bar{g}(n-1)!}\varepsilon^{ij_{2}\ldots
j_{n}}\mbox{det}[\vec{\mathcal{A}},\vec{g}_{j_{2}},\ldots,\vec{g}_{j_{n}}],\\\
G^{ij}&=\bar{g}^{ij},\end{split}$ (17)

where $\vec{g}_{k}$ is the $k$-th row of $\bar{g}_{ij}$. Now, with some
effort, the Christoffel symbols and, latter, the Ricci tensor components can
be computed. Fortunately, we can write them in terms of geometrical objects
living at $Q_{n}$. Defining,

$\begin{split}C^{i}&=\bar{g}^{ik}\left(\partial_{u}\mathcal{A}_{k}-\frac{1}{2}\partial_{k}\Phi\right),\quad
F_{ij}=\partial_{i}\mathcal{A}_{j}-\partial_{j}\mathcal{A}_{i},\\\
B_{ij}&=\frac{1}{2}(\partial_{u}\bar{g}_{ij}+F_{ij}),\end{split}$ (18)

the non-vanishing components of the Ricci tensor are,

$\begin{split}R_{uu}&=\nabla^{(\bar{g})}_{i}C^{i}+\frac{1}{4}F^{2}-\frac{1}{2}\left(\bar{g}^{ik}\partial_{u}^{2}\bar{g}_{ik}+\frac{1}{2}\partial_{u}\bar{g}_{ik}\partial_{u}\bar{g}^{ik}\right),\\\
R_{ui}&=\bar{g}^{jk}\nabla^{(\bar{g})}_{j}B_{ik}-\frac{1}{2}\partial_{i}\left(\bar{g}^{jk}\partial_{u}\bar{g}_{jk}\right),\\\
R_{ij}&=R_{ij}^{(\bar{g})},\end{split}$ (19)

where the superscript ${}^{(\bar{g})}$ means ”with respect to the metric
$\bar{g}_{ij}$” for each value of $u$. Finally, from Eq. (17) and Eq. (19),

$R=R^{(\hat{g})}.$ (20)

So the scalar curvature is not affected by the scalar and vector potentials in
the set of reduced Hamiltonian systems $(F,\omega,H(\lambda))$,
$\lambda\in\mathbb{R}$.

The results in Eqs. (19) and Eq. (20) do not predict easy Einstein field
equations, even for empty spacetime. However, for many many Hamiltonian
systems, the kinetic term does not depends on the time $t$ and, even more, the
kinetic metric $g_{ij}$ is flat. In this case, the Einstein field equations
reduce to some like Riemannian Maxwell equations,

$\triangle^{(\bar{g})}\Phi=-\rho+\frac{1}{2}F^{2}+2\partial_{u}\partial^{k}\mathcal{A}_{k},\quad\nabla^{(\bar{g})k}F_{ik}=J_{i},$
(21)

where the energy density and current are defined as,

$\rho=2\kappa T_{uu},\qquad J_{i}=2\kappa T_{ui},$ (22)

where $\kappa=8\pi G_{D}$, and $G_{D}$ is the $D$-dimensional Newtonian
gravitation constant. Additionally, $R=0$ implies traceless stress tensor,

$T=G^{ab}T_{ab}=0.$ (23)

The latter supposes a significant simplification from the general case.
Indeed, in this case, the Brinkmann metric Eq. (11) reduces to the one of a
_pp-wave_ , revealing a connection between pp-waves and classical Hamiltonian
systems. Under this connection, the scalar potential $\varphi=-\Phi/2$ gives
the profile of the wave, while $A_{i}\neq 0$ is related to the presence of
wave helicity111 Also known as gyratons, spacetimes describing the traveling
of ultrarelativistic energy bunches with helicity. Frolov:2005zq . The energy
density $\rho$ and current $J_{i}$ we need to generate the pp-wave can be read
from the set of reduced Hamiltonian systems $(\mathcal{F},\omega,H(\lambda))$
by means of the Eqs. (21) and (22), so the spacetime and its matter content is
entirely determined from the characteristics of the set of reduced classical
mechanical system.

Finally with respect to Brinkmann spaces , they have a gauge symmetry with
respect to shifts in the $v$ coordinate. The Killing vector $\vec{l}$ means
that we have symmetry under translations $v\rightarrow v+v_{0}$. However, this
symmetry can be gauged to,

$v\rightarrow\bar{v}=v-\frac{1}{2}f(u,q).$ (24)

Then, the line element Eq. (11) remains unchanged provided that we transform
$\Phi$ and $\mathcal{A}_{i}$ according to,

$\Phi\rightarrow\bar{\Phi}=\Phi+\partial_{u}f,\qquad\mathcal{A}_{i}\rightarrow\bar{\mathcal{A}}_{i}=\mathcal{A}_{i}+\partial_{i}f.$
(25)

In terms of the scalar potential $\varphi=-\Phi/2$ and the vector potential
$A_{i}=\mathcal{A}_{i}/2$, these are the gauge transformations of
electromagnetism.

##  III Eisenhart lift of autonomous and natural systems

As a special example of the previous section, let us now consider the case in
which $(F,\omega,H(\lambda))$ is a set of autonomous and natural systems, and
$g_{ij}=\delta_{ij}$ in standard Cartesian coordinates $x^{i}$. Then,
$\mathcal{Q}_{n+2}=\mathcal{Q}_{2}\times\mathbb{R}^{n}$, where
$\mathcal{Q}_{2}$ is a $2$-dimensional manifold spanned by $u,v$ and
$\mathbb{R}^{n}$ is the transverse space, while the line element in Eq. (11)
reduces to,

$ds^{2}=2dvdu+\Phi(x)du^{2}+d\vec{x}^{2}.$ (26)

This line element can be also found for no-natural systems by means of
choosing a suitable null coordinate $\bar{v}$ in Eq. (24), such that the gauge
transformations Eq. (25) can be used to vanishing $\mathcal{A}_{i}$ in Eq.
(11), as long as $\mathcal{A}_{i}$ satisfies,

$\partial_{[i}\partial_{j]}\mathcal{A}_{k}=0,$ (27)

and the line element Eq. (26) is reachable provided that the transformed
$\bar{\Phi}$ does not depend on $u$ and $R^{(g)}=0$. In this situation, by
Eqs. (21) - (23) , the line element Eq. (26) is sourced by a traceless stress
tensor given by a energy density $\rho$ and vanishing current $J_{i}$, with
field equation,

$\nabla^{2}\Phi=-\rho.$ (28)

The independence of $\Phi$ with $u$ then implies $\rho=\rho(x^{i})$.

Under the conditions previously discussed carrying to the line element Eq.
(11), the Brinkmann space $(\mathcal{Q}_{2}\times\mathbb{R}^{n},G_{ab})$ has,
at least, two Killing vectors: the vector field $\vec{l}$ in Eq. (14) and
$\vec{t}=\partial_{u}$. Since,

$t^{a}t^{b}G_{ab}=\Phi=-2\varphi,$ (29)

the vector field $\vec{t}$ will be timelike everywhere provided that $\varphi$
has a global positive minimum. Indeed, the existence of a global minimum of
$\varphi$ is a desirable property we have to protect if we want to keep the
quantum stability of the system, while the value of the minimum can be shifted
to have a positive value without any consequence. In this way, always we can
achieve to promote $(\mathcal{Q}_{2}\times\mathbb{R}^{n},G_{ab})$ to a
stationary spacetime, where $\vec{t}$ can be used to define an observer
$\vec{u}_{p}$ at each event $p\in\mathcal{Q}_{2}\times\mathbb{R}^{n}$ as,

$\vec{u}_{p}=\frac{1}{|\Phi|^{1/2}}\vec{t}.$ (30)

In addition, requiring

$\lim_{x^{i}\rightarrow\infty}\Phi=-1\quad\Rightarrow\quad\lim_{x^{i}\rightarrow\infty}\varphi=\frac{1}{2},$
(31)

$(\mathcal{Q}_{2}\times\mathbb{R}^{n},G_{ab})$ becomes asymptotically flat
and, from Eq. (30), the vector field $\vec{t}$ defines the set of asymptotic
observers. These are the observers measuring the charge and current defined in
Eqs. (22).

The Killing vector field $\vec{t}$ introduces a new constant of motion
$t_{a}\dot{\gamma}^{a}$ that must be related to the conservation of energy in
$(F,\omega,H(\lambda_{0}))$, since $u$ substitutes the time $t$ in Eq. (2).
Indeed it is easy to compute that,

$p_{u}=t^{a}\dot{\gamma}^{b}G_{ab}\,.$ (32)

Then, from Eq. (2),

$t^{a}\dot{\gamma}^{b}G_{ab}=-\frac{E(\lambda_{0})}{\lambda_{0}},\quad\forall\
\gamma\in\hat{\Gamma}^{0}_{\lambda_{0}}(E),$ (33)

where $\hat{\Gamma}^{0}_{\lambda_{0}}(E)\subset\hat{\Gamma}^{0}_{\lambda_{0}}$
is defined as the subset for trajectories with energy $E$ in the Hamiltonian
system $(F,\omega,H(\lambda_{0})$. The value of $p_{u}$ in Eq. (32) is always
measurable for the set of asymptotic observers, so the energy of trajectories
in each reduced Hamiltonian system $(F,\omega,H(\lambda))$ is reachable for
the observers in the Brinkamnn space Eq. (26).

It is interesting to observe what happens with the geodesics of
$\hat{\Gamma}^{0}_{\lambda_{0}}(E)$. In this case, the trajectories in the
reduced system are limited to a finite region of $\mathbb{R}^{n}$. From the
point of view of the Brinkmann space, constant energy $E$ corresponds to the
submanifold,

$\mathcal{S}_{E}(\lambda_{0})=\left\\{(u,v,x^{i})\in\mathcal{Q}_{2}\times\mathbb{R}^{n}:\
2E+\lambda_{0}^{2}\Phi(x)=0\right\\},$ (34)

such that the geodesics $\hat{\Gamma}^{0}_{\lambda_{0}}(E)$ are limited to the
interior of $\mathcal{S}_{E}(\lambda_{0})$. Indeed the normal vector to
$\mathcal{S}_{E}(\lambda_{0})$, $n_{a}=\partial_{a}(2E+\lambda_{0}^{2}\Phi)$,
satisfies,

$n_{a}\dot{\gamma}^{a}=\left.\lambda_{0}^{2}\dot{x}^{i}\partial_{i}\Phi\right|_{\mathcal{S}_{E}(\lambda_{0})}=0,\quad\forall\
\gamma\in\hat{\Gamma}^{0}_{\lambda_{0}}(E).$ (35)

So the geodesics of $\hat{\Gamma}_{\lambda_{0}}^{0}(E)$ reach
$\mathcal{S}_{E}(\lambda_{0})$ tangentially. Also note that,

$n_{a}n^{a}=\bar{g}^{ij}\partial_{i}\Phi\partial_{j}\Phi>0,$ (36)

since $\bar{g}_{ij}$ is a Riemannian metric. Thus,
$\mathcal{S}_{E}(\lambda_{0})$ is a time-like submanifold everywhere.

|  |
---|---|---
Figure 1:  _Left Panel.-_ Pöschl-Teller potential Eq. (38) for $k=1$. _Right
Panel.-_ Projection of surfaces $\mathcal{S}_{E}(\lambda)$ for
$E/\lambda^{2}V_{0}=0.1,0,25,0.4$ in the plane $XY$.

As an example, we can consider the isotropic harmonic oscillator potential,

$\varphi_{{}_{HO}}(x^{i})=\frac{\vec{x}^{2}}{2}.$ (37)

The potential has a global minimum at $x^{i}=0$, with
$\varphi_{{}_{HO}}^{(min)}=0$, and $\mathcal{S}_{E}(\lambda)$ are time-like
hypersurfaces wrapping around $\mathcal{Q}_{2}\times\\{\vec{0}\\}$ which
project as concentric $(n-1)$-spheres in the transverse space
$\mathbb{R}^{n}$. The value of $\lambda^{-1}$ scales the radius of the
hypersurfaces $\mathcal{S}_{E}(\lambda)$ and, for the same energy $E$,
geodesics with lower value of $l^{a}\dot{\gamma}^{b}G_{ab}$ reaches further
points in $\mathbb{R}^{n}$. However, the potential violates Eq. (31) and
$(\mathcal{Q}_{2}\times\mathbb{R}^{n},G_{ab})$ fails to be asymptotically
flat, without a set of asymptotic observers. The situation can be corrected by
considering the Pöschl-Teller potential,

$\varphi_{{}_{PT}}(x^{i})=-V_{0}\left[\frac{1}{2\cosh^{2}(k|\vec{x}|)}-\frac{1}{2}\right],\quad
k>0.$ (38)

This potential has a global minimum at $x^{i}=0$, with
$\varphi_{{}_{PT}}^{(min)}=0$, and also has a finite asymptotic behaviour,

$\lim_{x^{i}\rightarrow\infty}\varphi_{{}_{PT}}(\vec{x})=\frac{V_{0}}{2}.$
(39)

Then, redefining coordinates $u$ and $v$ as,

$\sqrt{V_{0}}\ u\rightarrow u,\quad\frac{v}{\sqrt{V_{0}}}\rightarrow v,$ (40)

also satisfies the asymptotic flatness condition Eq. (31). Therefore, for
fixed $\lambda_{0}$ and energies $0<E<\lambda_{0}^{2}V_{0}/2$, the projection
of the submanifolds $\mathcal{S}_{E}(\lambda_{0})$ over $\mathbb{R}^{n}$ are
concentric $(n-1)$-spheres, while for energies $E>\lambda_{0}^{2}V_{0}/2$ the
null geodesics of $\hat{\Gamma}^{0}_{\lambda_{0}}(E)$ escape to infinite (see
Fig 1).

##  IV Quantum dynamics from Eisenhart lift

In the previous sections we have shown that the projection of the sets of
geodesics $\hat{\Gamma}^{0}_{\lambda}$, $\lambda\in\mathbb{R}$, over the
transverse space of the Brinkmann space $(\mathcal{Q}_{n+2},G_{ab})$
corresponds to the curves solving the equations of motion of the set of
reduced Hamiltonian systems $(F,\omega,H(\lambda))$, with the time $t$ being
the affine parameter of the curves in $\hat{\Gamma}^{0}_{\lambda}$. In this
section we propose to define quantum states for null particles in
$(\mathcal{Q}_{n+2},G_{ab})$ from Quantum Mechanics in the transverse space,
_i.e._ proposing that the space of quatum states for null geodesics in
$(\mathcal{Q}_{n+2},G_{ab})$ projects over the Hilbert space of states in the
transverse space $(Q_{n},\bar{g}_{ij})$ for each value of $\lambda$, such that
the continuous set of reduced Hamiltonian systems $(F,\omega,H(\lambda))$
governs the time evolution of quatum null geodesics in
$(\mathcal{Q}_{n+2},G_{ab})$.

Time evolution must be related to a set of well defined observers. Thus, for
simplicity, from now on we shall constrain our analysis to Brinkmann spaces
with a transverse metric that does not depend on $u$ and, also, such that
$\vec{t}=\partial_{u}$ describes a set of well defined asymptotic observers.
Under these conditions, the transverse space $(Q_{n},\bar{g}_{ij})$
corresponds to ’time slices’ -which do not change with the timelike coordinate
$u$\- where Quantum Mechanics is well posed and states can be built by means
of measures performed by the set of asymptotic observers. Then, in a
coordinate representation, we define operators $\hat{p}_{i}$ as,

$\hat{q}^{i}=q^{i},\quad\hat{p}_{i}=-i\left[\partial_{i}+\sigma_{i}(q)\right]$
(41)

where, following Ref. RevModPhys.29.377 , $\sigma_{i}(q)$ are the coefficients
of the connection on the bundle over $Q_{n}$ with fiber $\mathbb{C}$, such
that wave functions are section of this fiber bundle. They are given by:

$\mathfrak{Re}(\sigma_{i})=\frac{1}{4}\hat{g}^{jk}\partial_{i}\hat{g}_{jk}.$
(42)

The coefficients $\sigma_{i}(q)$ are introduced to guarantee that
$\hat{p}_{i}$ are self-adjoint and the commutation rules,

$[\hat{q}^{i},\hat{q}^{j}]=0,\quad[\hat{p}_{i},\hat{q}^{j}]=i\
\delta_{i}^{j},\qquad[\hat{p}_{i},\hat{p}_{j}]=0.$ (43)

are satisfied222See Ref. kleinert2009path  for a modern introduction to the
subject of Quantum Mechanics in Riemannian manifolds. In addition, in order to
lift the quantum dynamics from the transverse space to the Brinkmann space we
have to introduce also an operator $\hat{p}_{v}$ such that,

$\hat{p}_{v}\ket{\Psi_{\lambda_{0}}}=\lambda_{0}\ket{\Psi_{\lambda_{0}}},$
(44)

where $\ket{\Psi_{\lambda_{0}}}$ is the state of a null particle following a
geodesic with $p_{v}=\lambda_{0}$. In a coordinate representation,

$\hat{p}_{v}=-i\partial_{v}\quad\Rightarrow\quad\Psi_{\lambda_{0}}(v,q)=\frac{1}{\sqrt{2\pi}}e^{i\lambda_{0}v}\varphi(q),$
(45)

such that $[\hat{p}_{v},\hat{v}]=i$ is satisfied. Here $\varphi(q)$ is a wave
function over the transverse space $(Q_{n},\bar{g}_{ij})$ that, under time
evolution in the Schrödinger picture, obeys the Hamiltonian $H(\lambda_{0})$.

The states $\ket{\Psi_{\lambda_{0}}}$ span a Hilbert space,

${\mathfrak{H}}_{\lambda_{0}}=\ket{\lambda_{0}}\otimes\mathfrak{H},$ (46)

where $\mathfrak{H}$ is the Hilbert space of states over the transverse space
$(Q_{n},\bar{g}_{ij})$ and $\ket{\lambda_{0}}$ is the state living over the
line spanned by $v$ with
$\hat{p}_{v}\ket{\lambda_{0}}=\lambda_{0}\ket{\lambda_{0}}$ ,

$\braket{v}{\lambda_{0}}=e^{i\lambda_{0}v},$ (47)

where $\ket{v}$ means ”localized at $v$”. Now we build the bundle
$(\mathcal{B},\bar{\pi},\mathbb{R})$, where,

$\mathcal{B}=\bigcup_{\lambda\in\mathbb{R}}{\mathfrak{H}}_{\lambda}\simeq\mathbb{R}\times\mathfrak{H},$
(48)

such that $\bar{\pi}^{-1}(\lambda_{0})=\mathfrak{H}_{\lambda_{0}}$. Therefore,
each fiber of the bundle Eq. (48) is isomorphic to the Hilbert space
$\mathfrak{H}$ but, with respect to time evolution, states in the fiber at
$\lambda_{0}$ are related to the Hamiltonian $H(\lambda_{0})$. Then, finally
we define null particle states in a Brinkmann space as averaged sections in
$(\mathcal{B},\bar{\pi},\mathbb{R})$ over $\mathbb{R}$, such that the
projection of an state $\ket{\Psi}$ over the transverse space is given by,

$\braket{v}{\Psi}=\frac{1}{\sqrt{2\pi}}\sum_{j}\int_{\mathbb{R}}d\lambda\
C_{j}(\lambda)e^{i\lambda v}\ket{\varphi_{j}}_{\lambda},$ (49)

where $\ket{\varphi_{j}}_{\lambda}$ is a discrete basis of $\mathfrak{H}$
subjected to time evolution by means of the Hamiltonian $H(\lambda)$. In this
way, $\ket{\Psi}$ lives at the space,

$\ket{\Psi}\in\bigoplus_{\lambda\in\mathbb{R}}\mathfrak{H}_{\lambda}.$ (50)

The Hilbert product extends naturally to the kets $\ket{\Psi}$ as,

$\braket{\Psi_{1}}{\Psi_{2}}=\int_{-\infty}^{\infty}dv\,\braket{\Psi_{1}}{v}\braket{v}{\Psi_{2}}=\sum_{j,k}\int_{\mathbb{R}}d\lambda\
C^{(1)*}_{j}(\lambda)C_{k}^{(2)}(\lambda)\braket{\varphi_{j}}{\varphi_{k}}_{\lambda},$
(51)

In particular, we recover the wave function Eq. (45) for
$C_{j}(\lambda)=\delta(\lambda-\lambda_{0})$ for some $j$. Respect to
normalization of kets,

$\braket{\Psi}{\Psi}=\sum_{j,k}\int_{\mathbb{R}}d\lambda\
C^{*}_{j}(\lambda)C_{k}(\lambda)\braket{\varphi_{j}}{\varphi_{k}}_{\lambda},$
(52)

which fixes,

$\sum_{j}\int_{\mathbb{R}}d\lambda\ |C_{j}(\lambda)|^{2}=1,$ (53)

if the basis $\\{\ket{\varphi_{j}}\\}$ of each $\mathfrak{H}_{\lambda}$ is
orthonormal. Finally, given some observable $\hat{O}(v)$,

$\begin{split}&\braket{\hat{O}}_{\Psi}=\\\
&\frac{1}{2\pi}\sum_{j,k}\int_{-\infty}^{\infty}dv\int_{\mathbb{R}^{2}}d\lambda_{1}d\lambda_{2}\
C^{(1)*}_{j}(\lambda_{1})C_{k}^{(2)}(\lambda_{2})e^{i(\lambda_{2}-\lambda_{1})v}\braket{\varphi_{j}}{\hat{O}(v)}{\varphi_{k}}_{\lambda}.\end{split}$
(54)

Note that, for $\hat{O}(v)$ changing slowly with $v$ with characteristic
length $\Delta v$, this expected value can be approximated by,

$\braket{\hat{O}}_{\Psi}=\frac{1}{\Delta v}\sum_{j,k}\int_{-\frac{\Delta
v}{2}}^{\frac{\Delta v}{2}}dv\int_{\mathbb{R}}d\lambda\
C^{(1)*}_{j}(\lambda)C_{k}^{(2)}(\lambda)\braket{\varphi_{j}}{\hat{O}(v)}{\varphi_{k}}_{\lambda}+F.O.T.$
(55)

where $F.O.T$ means fast oscillating terms.

Together with pure states Eq. (49), mixed states can be also considered. For a
mixed state, the density operator will be,

$\bar{\rho}=\sum_{j}\int_{\mathbb{R}}d\lambda\
w_{j}(\lambda)\ket{\Psi_{j}}_{\lambda}\bra{\Psi_{j}}_{\lambda},$ (56)

where,

$\braket{v}{\Psi_{j}}_{\lambda}=e^{i\lambda
v}\ket{\varphi_{j}}_{\lambda},\qquad\sum_{j}\int_{\mathbb{R}}d\lambda\
w_{j}(\lambda)=1.$ (57)

Thus,

$\bar{\rho}=\sum_{j}\int_{\mathbb{R}}d\lambda\
w_{j}(\lambda)\ket{\varphi}_{\lambda}\bra{\varphi}_{\lambda}$ (58)

For the special case that $w_{j}(\lambda)$ factorizes as,

$w_{j}(\lambda)=p_{j}\ w(\lambda),\quad\sum_{j}p_{j}=1,$ (59)

The density operators take the more simple form,

$\bar{\rho}=\int_{\mathbb{R}}d\lambda\ w(\lambda)\rho_{\lambda},\quad
w(\lambda)\in[0,\infty)\ :\ \int_{\mathbb{R}}d\lambda\ w(\lambda)=1,$ (60)

where,

$\rho_{\lambda}=\sum_{j}p_{j}\ket{\varphi}_{\lambda}\bra{\varphi}_{\lambda},$
(61)

is a density matrix on the Hilbert space $\mathfrak{H}$ evolving with
$H(\lambda)$, such that the state Eq. (60) projects over density matrices
$\rho_{\lambda}$ with the same probabilities $\\{p_{j}\\}$ for each value of
$\lambda$.

###  IV.1 Massless particle states in stationary Brinkmann spacetimes

In general terms, we cannot add much more to the quantumness of null geodesics
in Brinkmann spaces only subjected to the constrains of well defined
asymptotic observers and the no $u$-dependence of the transverse metric
$\bar{g}_{ij}$. However, by adding the condition that
$(\mathcal{Q}_{n+2},G_{ab})$ be stationary with $\vec{t}=\partial_{u}$ a time-
like Killing, we can also discuss ground states, stationary states and time
evolution since, under these conditions, the set $\\{H(\lambda)\\}$ is a
family of autonomous Hamiltonians.

Naively we could think that the ground state for the QM dynamics of null
particles we are constructing can be defined from the ground state in each
fiber of Eq. (48). However, each fiber $\mathfrak{H}_{\lambda}$ has a ground
state $\ket{0}_{\lambda}$ and, thus, there is no way to define uniquely a
ground state for the Hilbert space (50). That makes sense since time evolution
in (50) does not come from one unique Hamiltonian, but from the complete
family of reduced Hamiltonians $\\{H(\lambda),\ \lambda\in\mathbb{R}\\}$. For
example, for any fixed $\lambda_{0}$, the state given by,

$\braket{v}{\Omega}_{\lambda_{0}}=\frac{1}{\sqrt{2\pi}}e^{iv\lambda_{0}}\ket{0}_{\lambda_{0}},$
(62)

is a ground state, with definite momentum $p_{v}=\lambda_{0}$, in the sense
that there is no possibility to connect it with a lower energy state by means
of operators in our model. In general, any ground state of the model will
respond to an expression of the form:

$\braket{v}{\Omega}=\frac{1}{\sqrt{2\pi}}\int_{\mathbb{R}}d\lambda
C_{0}(\lambda)e^{i\lambda v}\ket{0}_{\lambda},$ (63)

such that,

$\int_{\mathbb{R}}d\lambda\,\left|C_{0}(\lambda)\right|^{2}=1.$ (64)

As an example, the expect value of $\hat{p}_{v}$ over Eq. (63),

$\braket{\hat{p}_{v}}_{\Omega}=\int_{\mathbb{R}}d\lambda|C_{0}(\lambda)|^{2}\lambda,$
(65)

And , therefore, it is possible to build ground states with zero expected
value of $\hat{p}_{v}$ providing $|C_{0}(\lambda)|$ be a defined parity
function.

Together with the fact that, in our model, there is an infinite of ground
states for the quantum dynamics of null particles in Brinkmann spaces, the
existence of a continuous family of Hamiltonians $\\{H(\lambda)\\}$ over the
transverse space $(Q_{n},\bar{g}_{ij})$ also leads to different time evolution
in the Schrödinger picture for each $\lambda$-component in Eq. (49). For each
state $\ket{\Psi_{\lambda_{0}}}$ of definite momentum $p_{v}=\lambda_{0}$,

$i\partial_{t}\braket{v}{\Psi_{\lambda_{0}}}=\frac{1}{\sqrt{2\pi}}e^{i\lambda_{0}v}\hat{H}(\lambda_{0})\ket{\varphi},$
(66)

where $t$ is the affine parameter over null geodesics satisfying the Eqs. (4)
and (5). Now, since $t$ runs with the coordinate $u$ as $\dot{u}=\lambda_{0}$,

$i\partial_{u}\braket{v}{\Psi}_{\lambda_{0}}=\frac{1}{\sqrt{2\pi}}\frac{e^{i\lambda_{0}v}}{\lambda_{0}}\hat{H}(\lambda_{0})\ket{\varphi}.$
(67)

For the case that $\ket{\varphi}$ be an eigenvector of $H(\lambda_{0})$ with
energy $E_{0}$,

$i\partial_{u}\braket{v}{\Psi}_{\lambda_{0}}=\frac{1}{\sqrt{2\pi}}\frac{e^{i\lambda_{0}v}}{\lambda_{0}}E_{0}\ket{\varphi}=\frac{E_{0}}{\lambda_{0}}\braket{v}{\Psi}_{\lambda_{0}},$
(68)

and $\braket{v}{\Psi_{\lambda_{0}}}$ describe a stationary state. As a
particular example, we can consider the ground state Eq. (62). Since near a
minimum of $\varphi(q^{i})$, we can approximate the potential by the harmonic
oscillator one,

$i\partial_{u}\braket{v}{\Omega}_{\lambda_{0}}=\frac{\lambda_{0}\omega_{0}}{2}\braket{v}{\Omega}_{\lambda_{0}},\qquad\omega_{0}^{2}=\sum_{i=1}^{n}h_{i},$
(69)

where $h_{i}$ are the eigenvalues of the Hessian of $\varphi(q_{i})/2$ in the
minimum. In general, for a pure state responding to the expression Eq. (49),

$i\partial_{u}\braket{v}{\Psi}=\frac{1}{\sqrt{2\pi}}\sum_{j}\int_{\mathbb{R}}d\lambda\frac{C_{j}(\lambda)}{\lambda}e^{i\lambda
v}\hat{H}(\lambda)\ket{\varphi_{j}}_{\lambda}.$ (70)

Therefore, for $\ket{\varphi_{j}}_{\lambda}$ eigenvectors of
$\hat{H}(\lambda)$ satisfying,

$\hat{H}(\lambda)\ket{\varphi_{j}}_{\lambda}\sim\lambda\ket{\varphi_{j}}_{\lambda},\quad\forall\
\lambda\in\mathbb{R}$ (71)

the pure state Eq.(49) is stationary.

##  V Quantum mechanical phenomena in stationary Brinkmann spacetimes

If the quantum dynamics of null particles in Brinkmann spaces works according
to the model developed in the previous section, quantum phenomena would appear
for low enough energies. For the case of the Brinkmann space given by Eq.
(26), the only nonvanishing component of the Ricci tensor Eq. (19), $R_{uu}$,
scales as second derivatives of $\Phi$. Thus,

$T_{uu}\sim\Phi^{\prime\prime}/2\kappa.$ (72)

On the other hand, near a minimum $q_{0}$ of $V(q)$ (therefore a maximum of
$\Phi$), the energy in the transverse space of the null geodesic with
$p_{v}=\lambda$ scales as,

$E^{2}\sim\lambda^{2}\Phi^{\prime\prime}.$ (73)

Therefore, in Planck units, the scale of the energy density to observe quantum
phenomena at an energy scale $E$ in the transverse space is,

$\frac{T_{uu}}{\rho_{p}}\sim\frac{1}{\lambda^{2}}\left(\frac{E}{E_{p}}\right)^{2},$
(74)

where $\rho_{p}\sim 10^{96}$ (SI) is the Planck density and $E_{p}\sim 10^{9}$
(SI) the Planck energy. In terms of the energy density defined in Eq. (22),

$\rho\sim\rho_{p}\
\frac{\kappa}{\lambda^{2}}\left(\frac{E}{E_{p}}\right)^{2},$ (75)

or, alternatively, from the Ricci tensor in Eq. (19),

$R_{uu}\sim\rho_{p}\frac{\kappa}{\lambda^{2}}\left(\frac{E}{E_{p}}\right)^{2}.$
(76)

So, for large enough $\lambda$, quantum phenomena for null particles appears
in a regime where the gravity can still be classical.

With respect to quantum phenomena, there are two characteristic phenomena of
quantum mechanics we can consider: quantum tunneling and entanglement.

###  V.1 Quantum tunneling

In Sec. III we showed the example of a stationary Brinkmann space, with a well
defined set of asymptotic observers, where null geodesics of
$\hat{\Gamma}^{0}_{\lambda_{0}}(E)$ were trapped inside the region bounded by
the hypersurface $\mathcal{S}_{E}(\lambda_{0})$ (see Fig. 1). For that we
propose the Pöschl-Teller potential Eq. (38) for the set of reduced
Hamiltonian systems $(Q_{n},\omega,H(\lambda))$. A more interesting situation
arises for the case the potential $\varphi$ has several minima. In this case,
$\mathcal{S}_{E}(\lambda_{0})$ could be a disconnected submanifold composed of
two or more hypersurfaces deppending on the energy $E$. As an example,
consider the double well potential,

$\varphi_{{}_{2PT}}(x^{i})=\frac{1}{2}\left[\varphi_{{}_{PT}}(x^{i}-x_{0}^{i})+\varphi_{{}_{PT}}(x^{i}+x_{0}^{i})\right],$
(77)

This potential satisfies the condition Eq. (31) and, for $k|\vec{x_{0}}|>>1$,
has two minima located at,

$x^{i}_{\pm}\simeq\pm x^{i}_{0},$ (78)

such that,

$\varphi_{{}_{2PT}}(x^{i}_{\pm})=\frac{V_{0}}{2}\left[\frac{1}{2}-\frac{1}{2\cosh(2k|\vec{x_{0}}|)}\right]\simeq\frac{V_{0}}{4}.$
(79)

Also, the potential has a maximum located between the to minima, at $x=0$,
with value,

$\varphi_{{}_{2PT}}^{(max)}\simeq V_{0}/2,$ (80)

for $k|\vec{x_{0}}|>>1$. Then, for fixed $\lambda_{0}$ and energies satisfying
$\lambda_{0}^{2}V_{0}/4<E<\lambda_{0}^{2}V_{0}/2$,
$\mathcal{S}_{E}(\lambda_{0})$ could splits in two disconnected hypersurfaces,
as it is shown in Fig. 2, which enclose separate regions, $A$ and $B$ of the
Brinkmann space $(\mathcal{Q}_{2}\times R^{n},G_{ab})$. For two observers Eq.
(30) located at events $p_{1}\in A$ and $p_{2}\in B$ that pretend to
communicate through light signals, they cannot do if they use the
”$\lambda_{0}$-channel” (geodesics satisfying $p_{v}=\lambda_{0}$) for
energies below some threshold $E_{0}=\varphi_{{}_{2PT}}^{(max)}$.

|  |
---|---|---
|  |
Figure 2:  Double well potential Eq. (77) (up) and projection of surfaces
$\mathcal{S}_{E}$ (down) for $E/\lambda^{2}V_{0}=0.45,0.375,0.3$ in the plane
$XY$ for $\vec{x}_{0}=(1,0)$. _Left column ( $k=3$)_.- Observers in regions
$A$ and $B$ cannot communicate by means of geodesics in
$\hat{\Gamma}_{\lambda}^{0}(E)$ for any of the three proposed values of $E$.
Surfaces $\mathcal{S}_{E}$ split in two pieces for any of the three values of
energy. _Right column ( $k=1.5$)_.- Observers in region $A$ and $B$ can
communicate by means of geodesics in $\hat{\Gamma}^{0}_{\lambda}(E)$ for
$E/\lambda^{2}V_{0}=0.3$, but they cannot for $E/\lambda^{2}V_{0}=0.375,0.45$.
As a consequence, regions $A$ and $B$ become together for
$E/\lambda^{2}V_{0}=0.3$.

The classical behaviour we have exposed changes in the framework of the
quantum mechanical model developed in Sec. IV since, then, quantum tunneling
between regions $A$ and $B$ appears. Let us consider the case of a generic
double-well potential $\varphi(x)$ over a one-dimensional transverse space
$\simeq\mathbb{R}$, and such that $\varphi(x)$ possesses two symmetric minima
at $\pm x_{0}$. Then, the tunneling between the two minima for null particles
with $p_{v}=\lambda$, in the semiclassical approximation, can either be
approached from the paradigm of instanton physics or the WKB method
Coleman:1978ae ; PhysRevD.16.423 ; SONG20082991 . Choosing the first option,
the penetration factor of the classically forbidden region in a (Euclidean)
time $T$ is given by,

$\Delta=e^{-\lambda S_{0}T},$ (81)

where $S_{0}$ is the one-instanton action minus the action of the solution
corresponding to the equilibrium ($x=\pm x_{0}$)333Usually instantons in the
double-well potential problem are computed for potentials satisfying
$\varphi(\pm x_{0})=0$ and, thus, $S[x=\pm x_{0}]=0$. That is not our case..
In terms of the $uu$-component of the metric Eq. (26), $\Phi(x)$, it is:

$S_{0}=\int_{-x_{0}}^{x_{0}}dx\ \left[\Phi(x_{0})-\Phi(x)\right]^{1/2},$ (82)

Then, it is a known result that under the dilute instanton gas condition and
for large $T$ ,

$\begin{split}\braket{\pm
x_{0}}{e^{-H(\lambda)T}}{-x_{0}}&\simeq\frac{1}{2}\left(\frac{\lambda\omega}{\pi}\right)^{1/2}e^{-\lambda\omega
T/2}\\\ &\times\left[\mbox{exp}\left(KTe^{-\lambda
S_{0}}\right)\mp\mbox{exp}\left(-KTe^{-\lambda
S_{0}}\right)\right],\end{split}$ (83)

where $\omega$ is given by,

$\omega^{2}=-\frac{1}{2}\Phi^{\prime\prime}(x_{0}),$ (84)

and the constant factor $K$ is computed from the one-instanton solution,

$t=t_{0}+\frac{1}{\lambda}\int_{0}^{\bar{x}(t)}dx\
\left[\Phi(x_{0})-\Phi\left(x\right)\right]^{-1/2}$ (85)

as,

$K=\left(\frac{\lambda
S_{0}}{2\pi}\right)^{1/2}\left|\frac{\mbox{det}\left(-\partial^{2}_{t}+\lambda^{2}\omega^{2}\right)}{\mbox{det'}\left(-\partial^{2}_{t}-\frac{\lambda^{2}}{2}\Phi^{\prime\prime}(\bar{x})\right)}\right|.$
(86)

where the prime determinant means it may be computed excluding the zero
eigenvalue444See Coleman:1978ae  for more details..

From Eq. (83) it is also possible to read the ground state energy for each
$\lambda$ PhysRevD.16.423 . Naively, we could think that we have a double
degenerate ground state for each $\lambda$, since $V_{2PT}$ has two symmetric
minima. Let us to call $\ket{0}_{R}$ and $\ket{0}_{L}$ to the ground states
corresponding to each minima. However, because tunneling, this expected
degeneracy breaks. By the symmetry $\pm x_{0}\leftrightarrow\mp x_{0}$,
$\ket{0}_{R}$ and $\ket{0}_{L}$ combine into odd and even states,

$\ket{-}=\frac{1}{\sqrt{2}}\left(\ket{0}_{R}-\ket{0}_{L}\right),\qquad\ket{+}=\frac{1}{\sqrt{2}}\left(\ket{0}_{R}+\ket{0}_{L}\right),$
(87)

that correspond to the ground and first excited states, with energies
$E_{\pm}$. On the other hand, in a (discrete) basis of $H(\lambda)$,

$\braket{+x_{0}}{e^{-H(\lambda)T}}{-x_{0}}=\sum_{n}e^{-E_{n}(\lambda)T}\braket{x_{0}}{n}\braket{n}{x_{0}},$
(88)

Thus, the leading order term and the first next-to-leading-order term in Eq.
(88) are,

$\braket{+x_{0}}{e^{-H(\lambda)T}}{-x_{0}}\simeq
e^{-E_{-}T}\Psi^{*}_{-}(-x_{0})\Psi_{-}(x_{0})+e^{-E_{+}T}\Psi^{*}_{+}(-x_{0})\Psi_{+}(x_{0}).$
(89)

Then, by comparing with Eq. (83), in the semi-classical limit the ground state
energy and first excited energy are,

$E_{\pm}(\lambda)\simeq\frac{\lambda\omega}{2}\pm KTe^{-\lambda S_{0}}.$ (90)

Thus, the energy split because of tunneling between the ground state and the
first excited state for each $\lambda$ is proportional to the penetration
factor in the classically forbidden region.

As an example, we can compute the tunneling splitting for the special case of
the double-well potential given at Eq. (77). The minima of the potential are
given by the zeroes of:

$\Phi^{\prime}(x)=-2kV_{0}\left\\{\frac{\sinh\left[2k(x-x_{0})\right]}{\left(\cosh\left[2k(x-x_{0})\right]+1\right)^{2}}+\frac{\sinh\left[2k(x+x_{0})\right]}{\left(\cosh\left[2k(x+x_{0})\right]+1\right)^{2}}\right\\}.$
(91)

For large $kx_{0}>>1$, at $x=\pm x_{0}$ we have,

$\Phi^{\prime}(\pm x_{0})\simeq-2kV_{0}\frac{\tanh(\pm
4kx_{0})}{\cosh(4kx_{0})}=0+\mathcal{O}\left(1/\cosh(4kx_{0})\right).$ (92)

So, we can take $x=\pm x_{0}$ as the minima until order $1/\cosh(4kx_{0})$.
Then,

$\Phi^{\prime\prime}(\pm x_{0})=-\simeq
k^{2}V_{0}\left[\frac{1}{2}+\frac{4}{\cosh(4kx_{0})}\right],$ (93)

and, as a consequence,

$\omega^{2}\simeq\frac{V_{0}k^{2}}{4}+\mathcal{O}\left(1/\cosh(4kx_{0})\right).$
(94)

On the other hand, also for large $kx_{0}>>1$, near $x_{0}$,

$\Phi(x)\simeq
V_{0}\left\\{\frac{1}{\cosh\left[2k(x-x_{0})\right]+1}-1+\mathcal{O}\left(1/\cosh(2kx_{0})\right)\right\\}.$
(95)

Then, in a similar way to Eq. (94),

$S_{0}\simeq 2V_{0}\int_{0}^{x_{0}}dx\
\left\\{\frac{3}{2}-\frac{1}{\cosh\left[2k(x-x_{0})\right]+1}\right\\}^{1/2}+\mathcal{O}(1/\cosh(2kx_{0})).$
(96)

Then, substituting in Eq. (90),

$\begin{split}E_{\pm}(\lambda)\simeq&\frac{\lambda k\sqrt{V_{0}}}{4}\\\ &\pm
KT\mbox{exp}\left\\{-2\lambda V_{0}\int_{0}^{x_{0}}dx\
\left\\{\frac{3}{2}-\frac{1}{\cosh\left[2k(x-x_{0})\right]+1}\right\\}^{1/2}\right\\}\\\
&+\mathcal{O}(1/\cosh(2kx_{0})).\end{split}$ (97)

###  V.2 Entanglement

Another typically quantum-mechanical phenomenon that would appear in the
context of the model for quantum dynamics proposed in Sec. IV is entanglement.
It must appear as long as an asymptotic observer in Brinkmann spacetime has
limited access to the whole space. That could be because of she/he only know a
patch of the whole Brinkmann space. For a small enough patch, the observer
effectively lives in a lower dimensional time-like submanifold of the
Brinkmann space.

As an example, let us to start from the ground state given in Eq. (63). For
this state, the density matrix is,

$\braket{v_{2}}{\bar{\rho}_{{}_{\Omega}}}{v_{1}}=\frac{1}{2\pi}\int_{\mathbb{R}}d\lambda_{1}\int_{\mathbb{R}}d\lambda_{2}C_{0}(\lambda_{1})C_{0}^{*}(\lambda_{2})e^{i\lambda_{1}v_{1}}e^{-i\lambda_{2}v_{2}}\ket{0}_{\lambda_{1}}\bra{0}_{\lambda_{2}}.$
(98)

Now, let us to consider an asymptotic observer who lives in a region
$v_{0}-\varepsilon<v<v_{0}+\varepsilon$. For small enough $\varepsilon$, the
observer does not see the degrees of freedom related to the $v$-coordinate,
and her/his spacetime is spanned by the coordinates $(u,q^{i})$. This observer
sees a reduced density matrix,

$\left.\bar{\rho}_{{}_{\Omega}}\right|_{v_{0}}=\int_{-\infty}^{v_{0}-\varepsilon}dv\,\braket{v}{\bar{\rho}_{{}_{\Omega}}}{v}+\int_{v_{0}+\varepsilon}^{\infty}dv\,\braket{v}{\bar{\rho}_{{}_{\Omega}}}{v}.$
(99)

Strictly, this equation only satisfies for $\epsilon\rightarrow 0$, but we are
going to take this limit at the end. After some computations,

$\begin{split}&\left.\bar{\rho}_{{}_{\Omega}}\right|_{v_{0}}=\int_{\mathbb{R}}d\lambda\,\left|C_{0}(\lambda)\right|^{2}\ket{0}_{\lambda}\bra{0}_{\lambda}\\\
&{\quad}-\frac{1}{\pi}\int_{\mathbb{R}^{2}}d\lambda_{1}d\lambda_{2}\,\frac{C_{0}(\lambda_{1})C_{0}^{*}(\lambda_{2})}{\lambda_{1}-\lambda_{2}}e^{iv_{0}(\lambda_{1}-\lambda_{2})}\sin{\left[\varepsilon(\lambda_{1}-\lambda_{2})\right]}\ket{0}_{\lambda_{2}}\bra{0}_{\lambda_{1}}.\end{split}$
(100)

Because Eq. (64), the trace of this reduced density matrix is,

$\mbox{Tr}\left[\left.\bar{\rho}_{{}_{\Omega}}\right|_{v_{0}}\right]=1-\frac{\epsilon}{\pi},$
(101)

and, in the limit $\varepsilon\rightarrow 0$, we recover
$\mbox{Tr}\left[\left.\bar{\rho}_{{}_{\Omega}}\right|_{v_{0}}\right]=1$, as it
was expected. On the other hand, from Eq. (100),

$\left.{\bar{\rho}}^{2}_{{}_{\Omega}}\right|_{v_{0}}=\int_{\mathbb{R}}d\lambda\,\left|C_{0}(\lambda)\right|^{4}\ket{0}_{\lambda}\bra{0}_{\lambda}+\mathcal{O}(\varepsilon^{2}),$
(102)

and, therefore,

$\mbox{Tr}\left[\left.\bar{\rho}^{2}_{{}_{\Omega}}\right|_{v_{0}}\right]=\int_{\mathbb{R}}d\lambda\,\left|C_{0}(\lambda)\right|^{4}+\mathcal{O}(\varepsilon^{2}).$
(103)

Since the function $|C_{0}(\lambda)|^{2}$ must be normalized to $1$, the
function $C_{0}(\lambda)$ satisfies:

$0<|C_{0}(\lambda)|^{2}<1.$ (104)

Therefore, in the $\varepsilon\rightarrow 0$ limit,

$\mbox{Tr}\left[\left.\bar{\rho}^{2}_{{}_{\Omega}}\right|_{v_{0}}\right]<\mbox{Tr}\left[\left.\bar{\rho}^{2}_{{}_{\Omega}}\right|_{v_{0}}\right]=1.$
(105)

Thus, the observer limited to live inside the region
$v_{0}-\varepsilon<v<v_{0}+\varepsilon$, $\epsilon<<1$, sees an entangled
state coming from the pure ground state Eq. (63). This state lives at the
transverse space and, in addition, the observer should measure an entangled
entropy,

$\begin{split}S\left(\left.\bar{\rho}_{{}_{\Omega}}\right|_{v_{0}}\right)&=-\mbox{Tr}\left[\left.\bar{\rho}_{{}_{\Omega}}\right|_{v_{0}}\log\left.\bar{\rho}_{{}_{\Omega}}\right|_{v_{0}}\right]\\\
&=-\int_{\mathbb{R}}d\lambda\,\left|C_{0}(\lambda)\right|^{2}\log\left|C_{0}(\lambda)\right|^{2}+\mathcal{O}(\varepsilon\log\varepsilon).\end{split}$
(106)

Note that Eq. (103)as well as Eq. (106) do not depend on $v_{0}$. That is
because the translation symmetry in the $v$ coordinate in the Brinkmann space
generated by the Killing field Eq. (16).

As a particular example, let us to choose the ground state given by:

$C_{0}(\lambda)=\frac{1}{\sqrt{\pi}}e^{-\lambda^{2}/2}.$ (107)

As exposed in Eq. (65), this choice guarantees that the pure state
$\ket{\Omega}$ has zero expectation value of $\hat{p}_{v}$. Then,

$\mbox{Tr}\left[\left.\bar{\rho}^{2}_{{}_{\Omega}}\right|_{v_{0}}\right]=\frac{1}{\pi^{2}}\int_{\mathbb{R}}d\lambda\,e^{-2\lambda^{2}}+\mathcal{O}(\varepsilon^{2})=\frac{\sqrt{\pi}}{2\pi^{2}}+\mathcal{O}(\varepsilon^{2})\approx
0.4+\mathcal{O}(\varepsilon^{2}).$ (108)

And, as expected, the observer living at
$v_{0}-\varepsilon<v<v_{0}+\varepsilon$, $\varepsilon\ll 1$, sees an entangled
state. Substituting Eq. (107) in Eq. (106), the corresponding entangled
entropy is,

$S\left(\left.\bar{\rho}_{{}_{\Omega}}\right|_{v_{0}}\right)=-\frac{1}{\pi}\int_{\mathbb{R}}d\lambda\,e^{-\lambda^{2}}\log\left(\frac{1}{\pi}e^{-\lambda^{2}}\right)+\mathcal{O}(\varepsilon\log\varepsilon).$
(109)

In the $\varepsilon\rightarrow 0$ limit,

$S\left(\left.\bar{\rho}_{{}_{\Omega}}\right|_{v_{0}}\right)=\frac{\sqrt{\pi}}{\pi}\left(\frac{1}{2}+\log\pi\right).$
(110)

##  VI The field viewpoint

In Sec. IV we set a model to build quantum states for null particles in
Brinkmann spaces and, latter, we made use of the model to study some quantum
phenomena (tunneling and entanglement) in stationary Brinkmann spaces in Sec.
V. This is the quantum-mechanical point of view. However, if the reader goes
back to IV, she/he will realize that Eq. (50) looks quite similar to the Fock
space of some quantum field theory. In this section we explore the possibility
to find a quantum field theory for null particles in a stationary Brinkmann
space.

Let us consider a stationary Brinkmann space, with $\mathcal{A}_{i}=0$. The
reduced Hamiltonian for each value of $\lambda$ is then,

$H(\lambda)=\frac{1}{2}g^{ij}p_{i}p_{j}-\frac{\lambda^{2}}{2}\Phi(q^{k}).$
(111)

As we discussed in Sec. IV, $\Phi(q^{k})$ must have a global maximum to
guarantee the quantum stability of the system. Therefore, let us consider that
$\Phi(q^{k})$ has a unique maximum at $q_{0}^{k}$. Under this condition, at
the low energy regime the projection of the null particles motion over the
transverse space $Q_{n}$ is trapped in a region near to $q_{0}^{k}$, and we
can expand $H(\lambda)$ as,

$\begin{split}H(\lambda)=\frac{1}{2}g^{ij}p_{i}p_{j}-\frac{\lambda^{2}}{2}&\left(\Phi_{0}+\frac{1}{2}H^{(0)}_{ij}(q^{i}-q^{i}_{0})(q^{j}-q^{j}_{0})\right.\\\
&\left.+\frac{1}{6}W^{(0)}_{ijk}(q^{i}-q^{i}_{0})(q^{j}-q_{0}^{j})(q^{k}-q^{k}_{0})\right)+\ldots\end{split}$
(112)

Where $\Phi_{0}=\Phi(q^{k}_{0})$, $H_{ij}^{(0)}$ is the Hessian at $q^{k}_{0}$
and $W_{ijk}^{(0)}$ is,

$W_{ijk}^{(0)}=\left.\partial_{i}\Phi\partial_{j}\Phi\partial_{k}\Phi\right|_{q^{k}_{0}}.$
(113)

The contribution of $\Phi_{0}$ can be dropped off because it has not dynamical
consequences. In addition, normal coordinates $\\{\eta^{i}\\}$ can be then
introduced, at least in a neighbourhood of $q^{k}_{0}$, such that the reduced
Hamiltonian $H(\lambda)$ in the low energy regime takes the form,

$H(\lambda)=\frac{1}{2}\delta^{ij}\Pi_{i}\Pi_{j}+\frac{\lambda^{2}}{2}\left(\omega_{i}^{2}{\eta^{i}}^{2}-\frac{1}{6}\bar{W}^{0}_{ijk}\eta^{i}\eta^{j}\eta^{k}\right)+\ldots$
(114)

For conjugate canonical momenta $\Pi_{i}$ of $\eta^{i}$, where
$-2(\omega^{i})^{2}$ are the eigenvalues of the Hessian $H_{ij}^{(0)}$ and
$\bar{W}^{(0)}_{ijk}$ are the components of Eq. (113) in normal coordinates
$\\{\eta^{i}\\}$. Therefore, at the low energy regime, the reduced Hamiltonian
for each $\lambda$ can be factorized as,

$H(\lambda)=H_{HO,\lambda}(\Pi_{i},\eta^{i})+H_{int,\lambda}(\eta^{i}),$ (115)

where $H_{int,\lambda}<H_{HO,\lambda}$ in the low energy regime, for each
$\lambda$, and $H_{HO,\lambda}$ is the Hamiltonian of the $n$-dimensional
harmonic oscillator with proper frequencies $\lambda\omega_{i}$.
$H_{int,\lambda}$ will bring to an interaction term at the end of the
calculus. Without this interaction term, for each $\lambda$, normal
coordinates satisfy the equation of motion,

$\ddot{\eta}^{i}_{\lambda}+\lambda^{2}\omega_{i}^{2}\eta^{i}_{\lambda}=0.$
(116)

Now, if we define,

$\bar{\eta}^{i}(v,t)=\int_{\mathbb{R}}d\lambda\
\eta^{i}(\lambda,t)e^{-i\lambda\omega_{i}v},$ (117)

where $\eta^{i}(\lambda,t)\equiv\eta_{\lambda}^{i}(t)$, Eq. (116) takes the
form,

$\partial_{t}^{2}\bar{\eta}^{i}-\partial_{v}^{2}\bar{\eta}^{i}=0,$ (118)

which remembers the bosonic string if we take $\eta^{i}$ as the coordinate
fields over a world sheet spanned by coordinates $v$ and $t$. Eq. (118) gives
the low energy regime for null particles as viewed form the transverse space
$Q_{n}$. Including the interaction term $H_{int}$ in the calculus we get the
first correction to the low energy regime,

$\partial_{t}^{2}\bar{\eta}^{i}-\partial_{v}^{2}\bar{\eta}^{i}+V^{i}\left(\vec{\bar{\eta}}\right)=0,$
(119)

where $V^{i}\left(\vec{\bar{\eta}}\right)$ is an interaction term coupling the
$n$ bosonic fields $\bar{\eta}^{i}$. It is given by,

$V^{i}\left(\vec{\bar{\eta}}\right)=-\frac{\delta^{il}\bar{W}^{(0)}_{ljk}}{4}\int_{\mathbb{R}}d\lambda\
\lambda^{2}\eta^{j}(\lambda,t)\eta^{k}(\lambda,t)e^{-i\lambda\omega_{i}v}.$
(120)

because of Eq. (113), this interaction term between the fields $\eta^{i}(v,t)$
appears provided $(\partial_{i}\Phi)^{3}|_{0}$ has a relevant value, _i.e._
the gravitational field changes enough sharply near $q_{0}^{k}$. The next
correction term would be proportional to $(\partial_{i}\Phi)^{4}|_{0}$ and it
would require a more suddenly change in the gravitational field at $q_{0}^{k}$
in order to have consequences at the low energy regime.

##  VII Conclusions and future work

In this work we have proposed a way to build a quantum-mechanical model for
massless particles in Brinkmann spacetimes. As a consequence of the model,
quantum tunneling and entanglement of states for massless particles have been
described. Also the possibility of finding a field theory at the low energy
regime has been discussed. However, the massless particles has been described
by means of null geodesics, without any internal degree of freedom. In this
sense, the present work must be assumed as a benchmark, and the incorporation
of helicity in subsequent works is a necessity to have a model approaching
photons in curved spacetimes. On the other hand, generalizations to other
spacetimes could be faced through the Penrose limit. In particular, the
application to weak gravity in the Newtonian gauge and FLRW spacetimes could
be of interest.

###### Acknowledgements.

This work has been partially funded by the Escuela Politécnica Nacional under
projects PIS-22-04 and PIM-19-01; the Ministerio Español de Ciencia e
Innovación under grant No. PID2019-107844GB-C22; the Junta de Andalucía under
contract Nos. Operativo FEDER Andalucía 2014-2020 UHU-1264517, P18-FR-5057 and
also PAIDI FQM-370.

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