# The Black Hole Firewall Transformation and Realism in Quantum Mechanics

Gerard ’t Hooft

(Faculty of Science, Department of Physics
Institute for Theoretical Physics
Princetonplein 5, 3584 CC Utrecht
The Netherlands
June 21, 2021)

> Abstract
>  A procedure to derive a unitary evolution law for a quantised black hole,
> has been proposed by the author. The proposal requires that one starts off
> with the entire Penrose diagram for the eternal black hole as the background
> metric, after which a procedure was proposed to identify the two asymptotic
> domains of this metric, such that they both refer to the same outside world.
> In this paper we focus on the need to include time reversal in applying this
> identification. This forces us to postulate the existence of an ‘anti-
> vacuum’ state in our world, which is the state where energy density reaches
> a maximal value. We find that this squares well with the deterministic
> interpretation of quantum mechanics, according to which quantum Hilbert
> space is to be regarded as the ‘vector representation’ of a real world.
>
> One has to understand how to deal with gravity in such considerations. The
> non-perturbative component of the gravitational force seems to involve cut-
> and-paste procedures as dynamical features of space and time, of which the
> re-arrangement of space-time into two connected domains in the Penrose
> diagram is a primary example. Thus we attempt to obtain new insights in the
> nature of particle interactions at the Planck scale, as well as quantum
> mechanics itself.
>
> In this newer version of the paper an important correction is made
> concerning the antipodal transformation: applying more insight in the
> situation, as explained in newer publications by the author, we now regard
> region $II$ of the Penrose diagram as an exact quantum copy of region $I$,
> still with time being reversed, but interchanging a region of spacetime with
> its antipodes is now seen to be incorrect.
>

Key words: Black hole, Schwarzschild metric, Penrose diagram, antipodal
mapping, time reversal, horizon, firewall, Cauchy surface, cellular automaton,
determinism, ontology, Planck scale, standard model, fast variables.

##  1 Introduction

There is general agreement that a theoretical study of black holes in a regime
where quantum mechanical effects play a role, is important for a more complete
understanding of General Relativity and/or some modifications of this theme,
in its relation to quantum mechanics. But here already, parts of the universal
agreement end. How do we introduce quantum mechanics in a black hole?
According to one doctrine, when subject to quantum mechanics, even large black
holes exhibit problems with _information loss_ [2]–[6] and _firewalls_[7],
that can only be understood if one invokes superstring theory and/or AdS/CFT
conjectures. However, the result of that is rather sobering: all that can be
said about the black hole quantum states, seems to be that these will be
‘chaotic’, which almost puts a premature end to all attempts to understand how
to characterise these states.

In our approach, we try to be more systematic in pushing the principles of
General Relativity a bit further, postponing the need for ‘new Planck scale
physics’ to be limited to Planck scale black holes only. This supposes that we
should be able to constrain ourselves to black holes whose detailed effects on
space-time are limited to structures with dimensions much larger than the
Planck length. It seems to be obvious that such black holes should be
described by standard General Relativity alone. Quantum mechanical effects
would be limited to the effects of Hawking particles, which have low energies,
and as such have very limited effects on the global space-time structure, at
least when regarded over limited stretches of time. In this paper we show that
there are other topological aspects of joining together patches of space-time,
and these, in turn, will imply important features of the theory of black holes
as well as quantum mechanics itself.

At isolated subregions of space and time, which will be smooth and locally
flat at the Planck scale, one should be able to employ ordinary quantum field
theoretical methods to address all relevant dynamical structures that may
arise. The only thing then left to do is find a systematic way to glue these
isolated pieces of information together, so that everything will be clear.
Beware, this will be a delicate procedure.

It sounds like an obviously correct starting point: if $A$ is large enough,
for black holes with mass111$M_{\mathrm{BH}}$ will often be written simply as
$M$. Here, $A$ is _not_ the surface area but just a coefficient.
$M_{\mathrm{BH}}=A.M_{\mathrm{Planck}}$, and size
$R_{\mathrm{BH}}=2A.L_{\mathrm{Planck}}$, one needs the properties of whatever
field theory the Standard Model evolves to, up to energy scale
$E_{\mathrm{SM}}=E_{\mathrm{Planck}}/A$.

This is the regime that we now focus upon. It should be possible to avoid
(Super)string theory here. It is the same regime where Hawking radiation
becomes exactly calculable, even though it is weak. To derive the existence of
Hawking particles, only standard quantum field theories were needed. Yet, in
the expression for the Hawking temperature,

$\displaystyle k_{B}T_{\mathrm{H}}=\frac{\hbar\,c^{3}}{8\pi
GM_{\mathrm{BH}}}\,,$ (1.1)

we see that thermodynamics ($k_{B}$), Special and General Relativity ($c$ and
$G$), and quantum mechanics ($\hbar$) all come together.222A very explicit
derivation of the Hawking radiation effect can be found in Ref. [8]. Issues
with a factor 2 are postponed to later publications. This indicates that,
indeed at some stages of our understanding, known theories can be used in
combination, to arrive at statements that continue to be valid _at_ the Planck
scale.

In previous accounts, the author explained how to use the physics of
gravitational effects due to particles moving into or out of horizons, to
derive dynamical laws for the quantum black hole. Now here, we wish to
emphasise that _transformations_ should be used in order to arrive at more
precise, generic results333In previous papers we avoided the use of
transformations relating different observers, because at some points, the time
coordinate would have to switch sign. The resulting minus signs were difficult
to handle, but in the present paper we believe to have this right. The price
paid will be some subtle changes in the definitions of particles and
antiparticles, see Sections 2 and 4.. We wish to consider the _quantum data_ ,
at specific isolated points in space and time, as seen by different
‘observers’. The observers do not need to ‘measure’ anything, but rather
‘describe’ what they see and experience, in a language that can be understood
locally. From a philosophical point of view, this should be the thing to
demand: the local language is the best one to use for a local description of
the events, so we must be clear about what these local data may say.

It is here that, if one follows the older text books and lecture courses
literally, one hits upon the black hole information problem and the firewall
problem. According to several authors, these problems imply that the program
just sketched above is bound to fail completely. The present author always
maintained that this does not need to be so, but it would be a mistake to
think that closing one’s eyes for the problems is the appropriate solution. To
the contrary, we have to look more carefully at what we are doing.

Suppose that there exists a Hilbert space for all quantum states a black hole
can be in. According to our views on the interpretation of quantum mechanics
[9], one has to distinguish _realistic_ states from _quantum superpositions_
of realistic states. The mathematical background is summarised in Appendix A.
The realistic444In this paper, we use the words ‘realistic’ and
‘deterministic’ interchangeably. states are the states the system can be in
when we imagine that these states describe, as accurately as possible, what
the most probable truly existing configuration may be. We call those states
_beables_. Together, the beables form an orthonormal set of vectors in Hilbert
space, which can be, but do not have to be, used as a basis. The ‘quantum
superpositions’ are all other states we get when we superimpose two or more
beables.

In a black hole, the quantum evolution operator will seem to transform beables
into quantum superpositions (superimposables). This may sound mysterious, as
it is not true in some simpler quantum systems.555It does happen in most
conventional quantum models such as a particle in a potential well, see
Appendix A. What this actually means for realism is that, every beable evolves
into one of many different possible future beables, depending on some very
fast fluctuating variables [10]. As soon as we lose sight of the fast
fluctuating variables, the notion of _probability_ enters. A diffusion
process, similar to what was described much earlier by Nelson [11], takes
place. In our theories for quantum realism, summarised in Appendix A, this
tendency is explained. If we could follow each and every one of the fast
oscillating variables, we would see that beables evolve into beables, but as
soon as we lose sight of the fast variables, we have to deal with
probabilities, and probability amplitudes, in accordance with a genuine
Schrödinger equation. Explicit models can be constructed for almost any
Schrödinger equation one would wish to reproduce. In our recent papers [10,
12] it is explained how this can be squared with the apparent contradictions
with Bell’s theorem [13] and the CHSH inequalities [14], by constructing
explicit models.

In short, if we start with any particular realistic state of a black hole at
time $t=t_{1}$, then determinism tells us that the exact moment, and further
details, of the implosion event that has generated the black hole, are
completely fixed, but _only_ if one knows exactly also all fast variables.
Averaging over the fast variables will smear the implosions over large ranges
of the implosion time and all other details, in terms of superpositions of all
possible events, leading to expressions to which the usual quantum Schrödinger
equation applies, in the usual way. In fact, what we get is a picture of the
distant past and the far future of the black hole that is time reversible.

The emergence of superimposed states, where the early implosions and the late
explosions occur at superimposed moments in time, also causes superpositions
in what we intended to use as our background metric, and this explains why we
now avoid the use of a background metric that includes the effects of such
events, replacing it by the metric of an eternal black hole. This formalism
will get its justification a posteriori.

Note that time reversibility appears to be totally absent in most scenarios
that investigators are finding up to this day. In our work, time reversibility
is a central issue. Only if one applies statistical methods in order to handle
thermodynamics, time reversibility gets out of sight.

The doctrine called quantum mechanics is merely the vector space
representation of deterministic events, as described also in Ref. [9]. This
implies that we can employ quantum mechanics as it usually is, while keeping
in mind what it says about reality: the real events taking place follow from
studying the beables that maximally overlap with the vector considered. In
practice, this is the application of Born’s rule. [15] As soon as we are
unable to specify sufficiently precisely the fastest variables, we recover the
usual uncertainty relations, and the standard interpretation of quantum
mechanics then applies.666Note that, if applied with sufficient care, the Born
rule can be extended to past events as easily as to future events, a fact that
can be used to ensure that our effective Schrödinger equation will be time
reversible.

In the next sections, we discuss a couple of technical issues that are
absolutely crucial for understanding the theory that we wish to unfold here.
We claim that the firewall problem is an important feature but it can be
completely handled by using an important transformation, transforming in-going
particles into out-going ones and vice versa [16]. The effect of this
transformation is, that: _A firewall due to an in-going particle may be_
_replaced by out-going particles_ , and, mutatis mutandis, out-going particles
may be replaced by in-going ones. Expanding energy/momenta of in- and out-
going particles in _spherical harmonics_ is crucial.

Then, section 4 explains the theory in detail, step by step. We see that it is
roughly eight precisely shaped jig-saw pieces that have to be put together.
The resulting construction is a package deal that has to be accepted entirely,
or rejected. There is no possibility in-between. Indeed, one finds numerous
publications, in which it was claimed that firewalls [7] would implicate non-
local features in black hole evolution; In our work, this conclusion is
sharpened: the only non-locality we arrive at is the new boundary conditions
at the horizons, which could be called non-local, in a sense. In section 5 we
arrive at our conclusion: black holes can be understood best if one also
accepts a radical attitude towards the interpretation of quantum mechanics.

Our views on quantum mechanics have been explained elsewhere; they are
summarised in Appendix A. I do realise that many readers will not go along
here; they can just ignore this appendix entirely. Although I think they are
useful, these insights in what quantum mechanics really is, are not essential
for the jig-saw puzzle.

Technical aspects of our calculations are to be found in the references, but
these overlap a lot; much of what we need will also be summarised in this
paper.

The newer version of this paper contains some important corrections concerning
antipodal transformations; these do not affect most of our principal
conclusions. See our note at the end of Section 2.

##  2 The Penrose Diagram

To describe any physical state the black hole might be in, we proceed just the
way we are used to in quantum field theory: take a classical solution for the
field equations, and then subsequently describe the quantum states in terms of
small excitations, using perturbation expansions. The first state we consider
is the pure Schwarzschild black hole, where all Standard Model fields take
their local vacuum values. This will be a solution of the pure Einstein
equations without matter. It is usually written as

$\displaystyle{\rm d}s^{2}$ $\displaystyle=$
$\displaystyle-\Big{(}c^{2}-\frac{2GM}{r}\Big{)}{\rm d}t^{2}+\frac{{\rm
d}r^{2}}{1-2GM/rc^{2}}+r^{2}{\rm d}\Omega^{2}\ ,$ (2.1) $\displaystyle{\rm
d}\Omega^{2}$ $\displaystyle\equiv$ $\displaystyle{\rm
d}\vartheta^{2}+\sin^{2}\vartheta\,{\rm d}\phi^{2}\ .$ (2.2)

As argued and motivated in the previous section, we shall not add the effects
of imploding matter or evaporating matter – these will be considered later.

> Extended metric of a stationary black hole. The longitudinal coordinates are
> chosen such that, locally, the light cones are everywhere oriented as
> indicated in the lower right. Curved lines: Cauchy surfaces; local time runs
> from below to top, but time for the distant observer reverses in region $II$
> (long arrows).

Figure 1:

By using different coordinates, one finds that the Schwarzschild solution has
an analytic extension beyond the future and the past horizon. Setting $c=1$,
the Kruskal-Szekeres [17, 18] coordinates $x$ and $y$ are defined by

$\displaystyle\Big{(}\frac{r}{2GM}-1\Big{)}e^{\,r/2GM}$ $\displaystyle=$
$\displaystyle xy\ ;$ (2.3) $\displaystyle e^{\,t/2GM}$ $\displaystyle=$
$\displaystyle x/y\ .$ (2.4)

In these coordinates,

$\displaystyle{\rm d}s^{2}=\frac{4(2GM)^{3}}{r}e^{-r/2GM}{\rm d}x{\rm d}y\ +\
r^{2}{\rm d}\Omega^{2}\ ,$ (2.5)

where we see that the horizon singularity at $r=2GM$ cancels out.

Note that a point $(r,t,\vartheta,\phi)$ appears 777In the earliest version of
this paper, the angles $\vartheta$ and $\varphi$ in region $II$ are replaced
by their antipodes, $\pi-\vartheta$ and $\varphi+\pi$. This is now seen as a
mistake, as will be explained. to be mapped onto two points
$(x,y,\vartheta,\phi)$ and $(-x,-y,\vartheta,\phi)$. It is convenient to
define coordinates $\rho\pm\tau=u^{\pm}$, with

$\displaystyle x=\tan({\textstyle\frac{1}{2}}\pi u^{+})\ ,\quad
y=\tan({\textstyle\frac{1}{2}}\pi u^{-})\ ,\qquad-1<u^{\pm}<1\ .$ (2.6)

To understand the resulting picture of the geometry, notice that $x$ and $y$,
or equivalently, $u^{+}$ and $u^{-}$, are light cone coordinates. As time $t$
increases, $\tau$ increases if $x$ and $y$ are positive. But if we switch the
signs of both $x$ and $y$, equations (2.3) and (2.4) stay the same, whereas
$\tau$ decreases as $t$ increases. This sign flip in the flow of time will be
a crucial feature of the theory. Indeed (remark added in version 2), region
$II$ is a CPT image of region $I$.

The _Penrose diagram,_ Fig. 1, is obtained if the longitudinal coordinates $r$
and $t$ are replaced by these $\rho$ and $\tau$, which include the asymptotic
domains of $r$ and $t$.

In Fig. 1, the classically accessible domain is indicated as the diamond
shaped region $I$. The upward arrow labelled as ”time outside” is the arrow of
the Schwarzschild time coordinate $t$ (or $\tau$). The radial Schwarzschild
coordinate $r$ (or $\rho$) is oriented in the horizontal direction.

Analytic extension generates three other domains, labelled $II$, $III$ and
$IV$. Of these, $III$ and $IV$ are of lesser significance. One might now be
tempted to consider the states obtained by just adding a finite number of
particles to this ‘empty’ configuration, but this does not work exactly as one
might expect. This is because _time for an outside observer_ in region $II$
runs in the opposite direction, see Fig. 1: this is where $x$ and $y$ are both
negative. Consequently, any particles we might be inclined to add in region
$II$, would contribute _negatively_ to the total energy obtained as seen by
the outside world. That cannot work correctly. In region $II$, a local
observer would be expected to see particles with negative energies. There are
no particles with negative energies for a local observer.

To see exactly what has to be done, we first consider the effect of time
boosts in the outside regions $I$ and $II$. As explained in the previous
section, the Schwarzschild solution is invariant under time boosts, like
Einstein’s equations themselves. However, the Penrose diagram is not. A time
boost for far-away regions in $I$ and $II$ corresponds to a Lorentz
transformation at the origin of the Penrose diagram. The energy eigenstates
for the distant observer are Lorentz transformation eigenstates for a local
observer near the origin. For small values of the coordinates $\rho$ and
$\tau$, a local Lorentz transformation there is generated by the operator
$L_{\mathrm{lor}}=\rho\,E-\tau\,p_{3}$ , or in a second quantised system,

$\displaystyle L_{\mathrm{lor}}=\int{\rm
d}^{3}\vec{x}\,(\rho\,{\mathcal{H}}(\vec{x})-\tau\,\mathcal{P}_{3}(\vec{x})\,)\,.$
(2.7)

We see that, in region $II$, where $\rho<0$, the local energy density of
particles, generated by the Lorentz transformation (2.7), is negative888Using
time translation invariance, take the case $\tau=0$ to see this.. One
concludes that, if local energies of particles are always positive, then we
seem to be talking of negative energy particles as seen by observers in region
$II$.

> Penrose diagram with Cauchy surface, at three different times, for distant
> observers. Note, that all Cauchy surfaces pivot at the origin. In region $I$
> (white): choose states near the vacuum state, in region $II$ (black), states
> near the antivacuum.

Figure 2:

We now claim that, nevertheless, region $II$ must be assumed to describe the
same universe as region $I$, since it has an asymptotic region at
infinity999Which part of the universe is described by region $II$? We discuss
this at the end of this Section.. It would be tempting to side-step this
difficulty, but we prefer to face it head-on. In order to compare physical
states in the Penrose diagram with physical states in our universe, we must
assume that the particles describing the excited states of the system, not
only have a lowest energy state, the vacuum, but also a _maximum_ energy
state, to be referred to as the ‘antivacuum’. Then we can simply take region
$II$ to be filled almost to the rim with particles, leaving a few holes, which
serve as physical particles in region $II$. We _shift the definition of
energy_ for particles in region $II$, such that their energies take the values
$E^{\mathrm{\,max}}-\Delta E_{\mathrm{part}}$, where only $\Delta
E_{\mathrm{part}}$ represents the locally observed energy of a particle in
$II$. This is exactly the way Dirac introduced the notion of antiparticles. We
emphasise this situation in our choice of colors in Fig. 2.

The reader might wonder how this works out when we add the gravitational
forces due to the particles in region $II$. The answer to this is that we only
consider particles (in region $I$), or holes (in region $II$), with energies
that are sufficiently low to allow us to ignore their gravitational back
reaction. How this condition works out in detail will become clear soon. As
yet, we just declare that gravitational forces will be added as ‘renormalised’
perturbations in terms of gravitons, in order to cancel the effects of
$E^{\mathrm{\,max}}$.

The time evolution operator is the operator that describes how particles
evolve while occupying states on the Cauchy surface. The Cauchy surface is a
space-like surface that propagates in the time direction. For the local
observer, $\rho$ is the space-like coordinate and $\tau$ is time. In Figure 1,
the Cauchy surface for a _local_ observer is shown, at two different values
for the time coordinate. The arrow at the center indicates the evolution with
time. However, this is not the way a Cauchy surface propagates that describes
the data as experienced by a _distant_ observer. She uses the Schwarzschild
coordinate $t$ as her time coordinate. Her Cauchy surfaces, at three different
times, are indicated in Figure 2. Note that these Cauchy surfaces all cross
each other at the origin.

Thus, what we found out is how to relate states seen by outside observers, to
states observed locally. Cauchy surfaces in region $II$ must be filled or
almost filled with local particles, states in region $I$ are almost empty. It
will be important to realise that, near the origin, ‘quantum gravity effects’
might generate curvatures such that, at very tiny distance scales, the
conditions ‘almost empty’ or ‘almost filled’ become ill-defined, so that
interesting ‘quantum gravity’ effects are to be expected there.101010One could
argue what happens in the following way: due to time reversal symmetry, the
gravitational effects due to $E^{\mathrm{\,max}}$ are to be ignored, but at
the boundary between the vacuum and the antivacuum, in a sense, they may be
seen to explode. One may say that the boundary between the vacuum and the
antivacuum is what remains of the firewall; in Section 3, we succeed to take
the effects of the firewalls into account completely, leaving almost no trace.

We believe that our way of handling the events near the origin of the Penrose
diagram has big advantages as opposed to proposals where some limits to the
_acceleration_ of particles are proposed, see for instance Rovelli [19]. Such
limits require new physics of a kind that is poorly understood. In the
formalism here, particles are not accelerated at all, since the background
metric is smooth, and the distribution of the physical data on the Cauchy
surfaces can be understood directly from flat Minkowski space physics. We do
get rapidly varying data near the origin, and these are carefully considered
and handled by the firewall transformation, in this paper.

There are still three important issues that need to be considered. First, how
can the existence of states with ‘maximal energy’ be reconciled with quantum
mechanics? In contrast to standard quantum field theories, the black hole is
expected to allow only for a finite number of internal quantum states.
Therefore, indeed, one should have expected that there will be a limit to the
energies particles in a black hole are allowed to have, so, we should not be
surprised to encounter the need for a limit in the allowed energy spectra. Our
theory for quantum mechanics has been explained in Refs.[9, 10, 12]. In this
approach, the demand for a maximum in the energy appears to be quite welcome.
It is associated to a lower limit for the time step variables $\Delta t$ in
the classical evolution laws. How this can be squared with Lorentz invariance
is not clear, but then, in the Schwarzschild background, Poincaré invariance
is indeed violated, so, as yet, there seems to be no direct contradiction.

A typical classical dynamical system is as described in our summary of quantum
theory in Appendix A. We find an almost trivial symmetry $t\leftrightarrow-t$
and associated with that, $E\leftrightarrow E^{\mathrm{\,max}}-E$. In the
language of the deterministic system, $E^{\mathrm{\,max}}$ does not play any
other role than being the inverse of the time steps $\Delta t$. See in
particular Eq. (A.3); the vacuum is at $k=0$ and the antivacuum at $k=N-1$,
but both these boundaries are totally artificial, they could be displaced in
many ways. Fearing that the replacement of the zero energy bound by an
$E^{\mathrm{\,max}}$ in region $II$ would distort the gravitational
interactions is unfounded. We must however remember to renormalise the
gravitational force. Then, it is natural to expect that $E^{\mathrm{\,max}}$
cannot generate any anomalous gravitational effects.

The second important issue is the occurrence of two asymptotic regions, one in
region $I$ and one in region $II$. The outside universe, however, has just a
single asymptotic region. This problem does not involve only the time
coordinate. Our problem is that there are two universes that are connected in
the spatial direction.111111And we have to keep them connected, in order to be
able to execute the firewall transformation, see Section 3.

Consider the one-particle states. A local observer can put the particle at a
point in region $I$ or at the corresponding point, with the same Schwarzschild
coordinates, in region $II$. It looks as if the particle in region $II$ is a
_quantum clone_ of the particle in region $I$. This jeopardises our quantum
mapping procedure. We considered to address this problem by dividing 3-space
by an isometry subgroup of the form $\mathbb{Z}_{2}$. This reduction of space
is as a boundary condition, since the physical data are now repeated at the
other side of the horizon(s), much like the data of a flat space region when
mapped on a torus. An important constraint on this boundary condition is that
it should not generate any cusp singularities. In the earlier version of this
paper the _antipodal mapping_ was proposed to be used as $\mathbb{Z}_{2}$ in
order to avoid this, but we now know that this would generate other
difficulties. We now briefly show why the antipodal mapping between regions
$I$ and $II$ is actually not allowed, in the following note added in version 2
of this paper:

In Ref. [22], the antipodal identification is mentioned explicitly as an
elegant solution for the singularity problem at the origin of the Penrose
diagram. We now found not only that the antipodes are not needed, but also
that the transition to antipodes [20, 21] for the physical interpretation of
region $II$ does not obey all our equations. In Ref. [16], it is explained in
more detail how the positions $u^{\pm}(\vartheta,\varphi)$ and the momenta
$p^{\mp}(\vartheta,\varphi)$ of the in- and out going particles, can be
expanded in spherical harmonics, to yield the coefficients $u^{\pm}_{\ell,m}$
and $p^{\mp}_{\ell,m}$, all of which now obey evolution and commutation rules
that are just one-dimensional, since the different components with spherical
wave coefficients $\ell$ and $m$ do not mix. Among other results, the
commutation rules for these coefficients were found:

$\displaystyle[u^{+}_{\ell,m},\,u^{-}_{\ell^{\prime},m^{\prime}}]=i\lambda\,\delta_{\ell\ell^{\prime}}\delta_{mm^{\prime}}\
,\qquad\lambda=\frac{8\pi G/R^{2}}{\ell^{2}+\ell+1}\ .$ (2.8)

The coefficients $u^{\pm}(\vartheta,\varphi)$ must be equal to
$u^{\pm}(\pi-\vartheta,\,\varphi+\pi)$ at the antipodes, so that
$u^{\pm}_{\ell,m}$ are equal to $(-1)^{\ell}u^{\pm}_{\ell,m}$ at the
antipodes. The definition of region $II$ is that this region covers the
negative values of $u^{\pm}(\vartheta,\varphi)$. Taken together, the
coefficients $u^{\pm}_{\ell,m}$ obey

$\displaystyle u^{\pm}_{\ell,m}(-1)^{\ell}=-u^{\pm}_{\ell,m}\ ,$ (2.9)

so that $u^{\pm}_{\ell,m}=0$ if $\ell$ is even. This violates Eq. (2.8) at
even $\ell$. Therefore, we have to abandon the antipodal identification for
region $II$. In our later publications, we do have to, and will, deal with the
cusp singularity at the origin. End of note.

Then, there is a third important issue: the firewalls [7]. We have chosen to
use the Penrose diagram at a given moment $t_{1}$ in time (Sect. 1). At that
moment, the Cauchy surface is the grey straight line in Fig 2. At times later
or earlier than that, the Cauchy surface is distorted, not only as a
coordinate effect, but also because of curvature caused by gravitational back
reactions. Exactly how to handle this situation appropriately, such as when
stationary black holes are considered, where we wish to describe energy
eigenstates for the external observer, is discussed in the next section.

##  3 The firewall transformation

In the previous section, it is explained that time translations for a distant
observer correspond to Lorentz transformations for the local observer. We also
explained that we wish to consider only those particle-like excitations whose
energies and momenta are small enough to permit us to neglect temporarily
their gravitational back reaction. This is almost a contradiction; we have to
sharpen our formulations to include the case that particles are boosted to
extraordinary high energies. Here we _cannot_ entirely ignore gravity; in
contrast, including the gravitational force just here, is possible, and it
changes everything; in fact, it becomes the leading effect that dictates how
particles will evolve over slightly longer time periods.

This mechanism has been explained in previous publications by the author[16,
22], but it seems that it still hasn’t been appreciated as we think it should.
The gravitational force that is responsible here is relatively easy to
compute. Since all particles, after strong Lorentz boosts, behave as almost
massless objects moving almost at the speed of light, we may assume as a good
start, that the infinite boost limit is appropriate. The precise calculation
is in the literature [23, 24], but let us first show the hand-waving argument.

Consider a particle going in. At early time $t\lesssim t_{1}$, its energy and
momenta are negligible, but when $t\gg t_{1}$ their gravitational effect
aggravates exponentially. It consists of a shift in the same direction as the
particle is going.121212We refer to this shift as the ‘Shapiro effect’ or
‘Shapiro shift’, after I. Shapiro, who first mentioned the gravitational time
delay of signals grazing past a heavy object.[25] The shift is relatively easy
to calculate. Take $u^{-}$ as the minus coordinate of some out-going
calibration particle. One finds that this particle is shifted by an amount
$\delta u^{-}$ given by

$\displaystyle\delta u^{-}=8\pi Gf(\Omega,\Omega^{\prime})\,\delta p^{-}\
;\qquad(1-\Delta_{\Omega})f(\Omega,\Omega^{\prime})=\delta^{2}(\Omega,\Omega^{\prime})\
,$ (3.1)

where $\Omega$ is the solid angle $(\vartheta,\varphi)$ of the out-going
calibration particle, and $\Omega^{\prime}$ the solid angle of the particle
going in. $\delta p^{-}$ is the momentum of the in-going particle in local
coordinates (normalised to the case $R_{\mathrm{BH}}=1$). The function
$f(\Omega,\Omega^{\prime})$ is a Green function, obeying Eq. (3.1), where
$\Delta_{\Omega}$ is the angular Laplacian. This equation is a linear relation
connecting $\delta u^{-}$ and $\delta p^{-}$, and the fact that it is linear
allows us to derive new equations linking in- and out-going particles.

It is very convenient to ‘renormalise’ the equation (3.1). Instead of
regarding it to relate a _shift_ $\delta u^{-}$ to a _modification_ $\delta
p^{-}$ of the spectrum of in-going particles, we can re-interpret this as a
relation connecting the _average_ position $u^{-}$ of _all_ out-particles, to
the _total_ momentum $p^{-}_{\mathrm{tot}}$ of _all_ particles going in.

These data are dual to one another quantum mechanically, which implies that
the entire angular distribution of all in-going particles is now related to
the entire distribution of the out-going ones. This enables us to derive a
very good approximation for the _black hole time evolution operator._[22].

This discovery also resolves the problem phrased at the beginning of this
section, and this goes as follows. The time-dependence of the momentum of in-
going particles is determined by repeated Lorentz boosts. Thus, the momentum
increases exponentially with time (we talk of time as defined by an external
observer). Consequently, the position of a combination of out-going particles
also begins with increasing exponentially with time. Actually, the positions
are determined by the local velocity of light – as long as the out-particles
stay relativistic. As the momentum of an out-particle decreases exponentially
in time, it sooner or later becomes non-relativistic and continues to obey
standard equations, _while moving out_ and away from the black hole.

As soon as out-going particles have separated from the black hole, we can
remove them from the total picture, at which point also the original
responsible in-going particle may be discarded. It was the exponential
increase of momenta of in-going particles that caused the _firewalls_ there.
Thus, we observe that we actually obtained a proper mechanism for removing the
firewalls. _A firewall due to an in-going particle may be replaced by out-
going particles._ That’s two flies in one blow: the firewall is removed and a
possibly quite accurate evolution operator for the black hole is obtained.

This is one of the reasons for regarding our procedure as a package deal. One
cannot understand the role of the firewall without understanding the black
hole evolution. But it also comes with the other necessary condition: the
relation between in- and out-going particles is as the relation between
momenta and positions in quantum mechanics: one is the Fourier transform of
the other. The Fourier transformation is unitary only if all integrations run
from $-\infty$ to $+\infty$, not from $0$ to $\infty$, which would correspond
to keeping all particles in region $I$. The antipodal identification was
proposed in earlier publications, but, as explained at the end of Section 2 in
the present version of this paper, we decide to disassociate from it; region
$II$ must be a direct image of region $I$. The implications of _that_ will be
further elaborated in a future publication.

The argument for replacing the firewall with out-going particles can also be
time-reversed: the out-going particles are the Hawking particles, which must
originate near the horizon where they start off with huge amounts of momentum
for each particle. What would the effect of these huge momentum values have
been on the metric? It is often claimed that the Hawking particles do not
cause a firewall when extrapolated to the past. For some reason, earlier
researchers had no problem wishing this firewall away. It just could not be
there. Now, we claim that Hawking particles can be regarded as decaying
firewalls, in the sense that the mechanism described above works both forward
in time as backwards, simply because positions and momenta are quantum
mechanically dual to one another.

##  4 The black hole theory as a package deal

We are now in a position to put the various pieces for a sound theory
together. What we find from various strains of reasoning, is to be regarded as
a jig-saw puzzle. Consisting of eight different jig-saw pieces, it is not too
difficult to arrange them in a more transparent structure. First of all, it
seems that, when phrasing the set of rules to understand how black holes
behave, neither quantum mechanics nor general relativity need any major
overhauls, other than considering boundary conditions (such as the firewall
transformation) more carefully. Below, we show one result that appears to make
sense. The overall formulation is more coherent than what we had before, even
if some questions have not yet been answered.

We emphasise that the ensuing black hole theory is a package deal; one has to
unwrap all entities in the package, read their manuals, and understand how to
deal with them, to see the entire picture.131313This situation is not unlike
the early days of special relativity, where scientists had to get used to the
idea that _time_ can not only not be measured independently from the status of
an observer, but it cannot even be _defined_ independently of observers. Only
if one understands that space and time have to be handled as coordinates,
which allow for transformations into different coordinates, one begins to
understand how special relativity works.

We have to define a way to characterise all states that together form its
Hilbert space, after which we can contemplate the question how these states
evolve. At first sight, this seems to be deceptively simple. The situation
that we wish to get under control is a black hole with mass
$M_{\mathrm{BH}}=AM_{\mathrm{Planck}}$, where $A$ is sufficiently large, so
that it is well distinguishable from the objects that we allow to move in and
out, whose typical energies are of the order of $M_{\mathrm{Planck}}/A$. These
particles move in and out at a sufficiently slow rate to ensure that the black
hole stays stationary on time scales considerably longer than
${\mathcal{O}}(A^{2}T_{\mathrm{Planck}})$.

In that case, we claim that, at first approximation, the description of the
quantum states of the black hole, in a given stretch of time141414This stretch
of time will be kept considerably smaller than the Page time [26], although
the considerations that led Page to define this time period, are of no concern
here. The entanglement issues that brought investigators to consider this
period are handled differently in the present paper: the basis of Hilbert
space must be chosen differently whenever we remove the particles that went
too far away to be relevant., can be formulated as follows:

  * 1)

first ignore the gravitational fields of the in- and out-going objects
altogether. This means that we will be dealing with the space-time of an
eternal black hole, as indicated in Figs. 1 and 2.

  * 2)

This Penrose diagram is not yet suitable for our use because it exhibits two
external regions, region $I$ and region $II$. We need only one such region,
and therefore, as explained at the end of section 2, we divide space-time by a
$\mathbb{Z}_{2}$ isometry. In version 1, this was thought to be the antipodal
mapping. In future publications, more direct identifications between $I$ and
$II$ will be considered.

  * 3)

Regions $I$ and $II$ however must be glued together smoothly, and the gluing
prescription must fit snugly over the entire time period. This means that, as
seen from region $I$, region $II$ seems to be moving backwards in time,
implying that, in region $II$, also the energies are reversed. Ordinary QFT
would not allow this, unless we renormalise the energies in region $II$: given
any small subspace entirely embedded inside region $II$, this subspace must
have a _maximum $E^{\mathrm{\,max}}$ for the total energy it can contain,_ and
if an outside observer sees a particle with energy $E$ in region $I$, an
observer travelling in region $II$, observes an energy $E^{\mathrm{\,max}}-E$
in the same subspace. As seen from region $I$, which may be close to the
conventional vacuum, region $II$ seems to describe it as being almost full; it
is close to its antivacuum. As long as we ignore direct gravitational
interactions, this is completely consistent.151515How large is
$E^{\mathrm{\,max}}$ ? We can estimate a reasonable value for every spherical
harmonic $Y_{\ell m}$. We postulated that the particles considered have
energies of order $E_{\mathrm{Planck}}/A$. Energies much beyond that value
will rarely be relevant. Therefore, we may choose $E^{\mathrm{\,max}}\approx
E_{\mathrm{Planck}}/A$. This means that we are doing quantum mechanics using
time steps of the order of $\delta t\approx A.T_{\mathrm{Planck}}.$ Remember
that each spherical harmonic represents just one physical, particle-like
degree of freedom. They cannot be second-quantized. Of course, this argument
yields energy and mass values typical for Hawking radiation. Adding all
spherical harmonics gives $E^{\mathrm{\,max}}_{\mathrm{total}}\approx
M_{\mathrm{BH}}.$

  * 4)

However, at the point where the two regions border to one another (at
$r\rightarrow 2GM_{\mathrm{BH}}$, or $\rho\rightarrow 0$), the notions of
vacuum and antivacuum are ill-defined. We plan not to ignore the gravitational
interaction close to the origin, where it matters most. We ignore that only at
space-time points sufficiently far separated from the horizons, where all
particles have slowed down sufficiently, so that perturbative, renormalised
gravity, in combination with Standard Model QFT, is all that is needed to see
how these interact. A valid element of the black hole Hilbert space is now
defined by concentrating on one Cauchy surface at fixed time only, say $t=0$.
Left of the origin, where $x$ and $y$ are $<0$, our state rapidly approaches
the anti-vacuum. At the right, where $x>0$ and $y>0$, the ordinary vacuum is
approached. All black hole states are described by specifying how these modes
connect at the origin161616Remember that, at the origin, transverse space is a
projected $S_{2}$ sphere whose size does not vanish as in Minkowski space..
There will always be particles there.

  * 5)

The states of the black hole Hilbert space can now be labelled constructively.
We must start with one template state; say that this is the state where all
fields (except the gravitational metric fields) take their lowest possible
energy values in region $I$ and the highest possible energies in region $II$.
It is not obvious whether this definition would be unique, but in practice,
any one state might do as our template state; we go from there to all other
states by using creation and annihilation operators. The well-known Bogolyubov
transformation [27, 28] will tell us how to do such constructions.

  * 6)

Consider the addition of one particle with inwards momentum $p^{-}$ in region
$I$. Spectator particles moving out will be dragged inwards by the Shapiro
effect. As seen by an observer in region $II$, the spectator particles will be
seen to be moved outwards. The rule here is: the entire set of particle data
on the Cauchy surface spanned by the past event horizon will be preserved. To
be precise: the particle added in region $I$ is seen to move the state there
away from the vacuum state, while in the terminology to be applied in region
$II$, the same particle moves the state there towards the antivacuum. This
sign switch matches the sign switch in our definition of energy in the two
regions (after the shift with $E^{\mathrm{\,max}}$), and its associated
gravitational force.
The _firewall transformation_ tells us that we can remove the in-going
particle if we displace all outgoing particles accordingly. This means that
the momentum operators $p^{-}(\vartheta,\varphi)$ for the in-going particles
can be identified with the position operators $y(\vartheta,\varphi)$ for the
out-going particles, and _mutatis mutandis_ we can identify
$-x(\vartheta,\varphi)$ for the in-going particles with the momenta
$p^{+}(\vartheta,\varphi)$ of the out-going ones. This interchange is to be
performed whenever the momentum operators become so large that we can no
longer ignore their gravitational back reaction effects.

  * 7)

A technical feature that was not yet strongly emphasised in this paper, is
that the equations for the gravitational shift (Shapiro shift) that in-going
particles bring about in the out-going ones – and vice versa – diagonalise if
one subjects the in- and out-momenta, as well as the average in- and out
positions, to expansions in spherical harmonics.

  * 8)

Finally, we have the question how a black hole, with or without an exotic,
antipodal, boundary condition, can ever form – or evaporate completely – and
what the rules are here. This question has not yet been addressed, but we can
sketch a reasonably plausible scenario, starting from the conventional
description of a black hole being formed by matter. The imploding matter will
rapidly form a firewall on the past horizon. A firewall transformation will be
needed to replace the imploding matter by Hawking particles going out. The
question is then, how the boundary condition is re-generated. How to formulate
this elegantly is not yet known; we had been imagining that first a black hole
_seed_ is formed, a Planck-sized black hole, close to where the singularity
would arise, before we apply the first firewall transformation. This latter
point may require Planck length physics.

##  5 Conclusion

As stated in the introduction, one important motivation for the theoretical
study of black holes and their quantum features, was to arrive at new insights
concerning physics at the Planck scale.

In our attempts to do things accurately, one of the first things noted, is the
reversal of time where regions $I$ and $II$ in the Penrose diagram, are glued
together. This is because a time translation for the external observer
generates a Lorentz transformation near the center of the diagram.
Consequently, if one regards the Penrose diagram after a (long or short) lapse
of time, one sees the Cauchy surfaces pivot around the origin, see Figure 2.

It may sound odd that this time reversal takes place during the entire
lifetime of the black hole, which is much longer than the time scale that can
be visualised in the diagram. This is because Lorentz transformations form a
non-compact group. It also implies that one cannot ignore its consequences. In
particular, if one wants to study the stationary modes – the energy
eigenstates – of the black hole.

After a strong Lorentz boost, one finds that particles may accumulate towards,
or separate from, the centre of the diagram. One of our observations is that
the accumulation of particles near the origin may – and should – be
transformed away, a procedure described here as the ‘firewall transformation’.
It replaced these accumulated in-particles by out particles or _vice versa_.
The procedure works, but there may be an important side effect. We ignored
this in earlier papers, but it seems inevitable that, when distant observers
send in particles with positive energies, the energies may switch sign as
well, when a local observer sees them appearing in region $II$. The sign
switch in the energy comes about when one wishes to write the mapping from the
familiar world around a black hole to the local world surrounding its
horizons, by means of a unitary operation. The time-energy relation in the
quantum states, expressed by their phases $e^{-iEt}$ would ruin unitarity if
$t$ changes sign but not $E$. Particles can make transitions through the
wormhole from $I$ to $II$ and back (just because of the Shapiro effect), which
is why we do not believe that the energy sign flip can be avoided. Total
energy must be conserved.

We were forced to speculate on a limiting local energy, $E^{\mathrm{\,max}}$,
so that the particles in region $II$ can have positive energies,
$E^{\mathrm{\,max}}-E$, while energy conservation is maintained.
$E^{\mathrm{\,max}}$ can be related to the smallest time step, $\delta t$, and
this is where deterministic quantum mechanics may enter. Deterministic quantum
mechanics leaves some freedom in defining what the smallest time step is, so
there may be still some room for speculations concerning the value of
$E^{\mathrm{\,max}}$. The theory is entirely invariant under $E\leftrightarrow
E^{\mathrm{\,max}}-E$.

The state with energy $E^{\mathrm{\,max}}$ will be called the ‘antivacuum’.
The antivacuum in region $II$ may actually represent all those particles that
filled the black hole during its implosion in the distant past.

As yet in a modest way, our work seems to pay off. Quantum mechanics is likely
to demand such a thing as an upper limit for the energy on any compact subset
of 3-space – even when disregarding the direct effects of gravity – leading to
the notion of an antivacuum. In elementary exercises involving the nature of
quantum theory in general, the notion of antivacuum was hinted at, if only to
account for the existence of a smallest time scale for defining local
interactions. Limits on the energy density of quantum states near the Planck
scale have been speculated about a lot; here we see the necessity of this,
although the mathematical details are still not quite clear.

How does gravity act in region $II$? The answer should follow in a
straightforward way by analysing the transformations from the asymptotic
Minkowski space to the geometry of region $II$, or even simpler, by
transforming $I$ to $II$ and back. The doctrine demands that we start by
ignoring the gravitational effects of the quantum excitations that are being
considered, just because the masses and energies of these quantum particles
were systematically assumed to be orders of magnitude smaller than that of the
black hole itself. As the first step towards further refinement, one can then
add the gravitational force perturbatively as one can add the other Standard
Model interactions. The non-perturbative aspects of gravity are most acute in
the immediate vicinity of the horizons, and those were taken care of
accurately by taking the Shapiro shifts into account: the firewall
transformation.

We found that in describing interacting beams of particles at Planckian
energies, firewalls may be created and subsequently removed by utilising the
firewall transformation. A super-Planckian particle may come with the
gravitational analogue of a shock wave. General Relativity will allow us to
remove this shock wave by a cut-and-paste procedure combined with a local
coordinate transformation, after which particles going one way (inwards or
outwards) are replaced by particles going in the opposite direction, the rules
here being precisely dictated by standard physics.

Needless to say, there are still many questions left: a _systematic and
complete_ formalism for the entire set of physical degrees of freedom at
Planckian scales, is still missing. Unlike some procedures in string theory
and $M$ theory, we insist on maintaining fundamental aspects of _locality_ ,
which should considerably constrain the amount of freedom we have in
constructing our theory. More than half a century ago, physicists had similar
difficulties in reconciling quantum mechanics with _special_ relativity. The
insight of how to do this right became clear gradually: quantum field theory
(QFT). This theory became a beautiful synthesis of simplicity and complexity;
the theory was simple enough to enable us to categorise all possible
scenarios, and sufficiently complex to allow something as intricate as an
entire universe with galaxies, stars, planets and life all controlled by its
equations.

We can only hope that the eventual synthesis of quantum mechanics with
_general_ relativity will be equally simple and complex to give us an even
more detailed understanding.

In the Standard Model of the subatomic particles, the different energy
eigenstates of its Hamiltonian appear to have quite distinctively different
physical properties. The importance of its interpretation as a vector
representation of a theory with local realism is that, all energy eigenstates
just differ from one another by a man-made phase coefficient, which does
depend on time. The lowest energy state is the vacuum state, which in our new
picture appears not at all to differ from the highest energy state, the
‘antivacuum’. Because of this, it now looks much more natural than before, to
treat vacuum and antivacuum as each other’s mirror image.

##  Appendix A Quantum mechanics as the vector representation of a real,
deterministic theory

The author’s approach to viewing quantum mechanics as a deterministic theory
in disguise, has been unfolded in several publications [9, 10, 12, 29, 30].
Consider a classical, deterministic system, which only allows for ‘states’
that are as ‘real’ as the planets in their elliptical orbits. There is nothing
yet to remind us of quantum mechanics, except that the physical variables may
refer to phenomena at the tiniest distance - and time scales. We label the
real states by using the Dirac notation – but here it is meant to be nothing
more than just a notation – :

$\displaystyle|0\rangle,\ |1\rangle,\ \cdots\ |n\rangle,\
\dots,\quad\hbox{where\ \ }n\ \hbox{\ is integer.}\ $ (A.1)

It will then often be convenient to assume a world where many of its variables
are almost periodic. ‘Almost’ will mean in practice: the variable returns to
one of its previous values regularly, unless something in its direct
environment will cause it te deviate from the rule in exceptional situations.
What this means is that we can use the assumption of periodicity as an
approximation, and worry about the – rare – exceptions later171717The rare
exceptions are to be expressed in terms of interaction Hamiltonians,
$H^{\mathrm{int}}$, which will have only tiny effects on the states
considered.. Thus, let us start with an ‘unperturbed’ world where everything
is periodic (though the periods may differ). A variable $|\psi\rangle$ then
obeys the evolution law

$\displaystyle|\psi\rangle_{t}=U_{\Delta t}\,|\psi\rangle_{t-\Delta t}\
,\qquad U_{\Delta t}\,|n\rangle=|n+1\rangle\ ,\quad|N\rangle=|0\rangle\ .$
(A.2)

Since these states refer to ‘reality’, we call the states $|n\rangle$
‘beables’, in honour of J.S. Bell [31] The theory known as ‘quantum mechanics’
is now nothing but the _vector representation_ of such evolution laws. This
means that we regard the beables as elementary vectors that form the basis of
a Hilbert space. It allows us to define _energy_ $E$, by re-arranging the
beables in eigenstates $|k\rangle_{E}$ of the evolution operator $U_{\Delta
t}$:

$\displaystyle|n\rangle=\frac{1}{\sqrt{N}}\sum_{k=0}^{N-1}e^{-2\pi
ikn/N}|k\rangle_{E}\
,\qquad|k\rangle_{E}=\frac{1}{\sqrt{N}}\sum_{n=0}^{N-1}e^{2\pi
ikn/N}|n\rangle\ ,$ (A.3)

Consequently,

$\displaystyle U_{\Delta t}|k\rangle_{E}$ $\displaystyle=$ $\displaystyle
e^{-2\pi ik/N}|k\rangle_{E}\ ,\quad\hbox{so that}$ (A.4) $\displaystyle\quad
U_{t}|k\rangle_{E}$ $\displaystyle=$ $\displaystyle e^{-iEt}|k\rangle_{E}\
,\quad\hbox{with}\quad E=2\pi k\,/\,N\Delta t\ ,$ (A.5)

where $U_{t}$ describes the evolution over $t/\Delta t$ time steps. Note that
the maximum energy is

$\displaystyle E^{\mathrm{\,max}}=2\pi(1-1/N)\,/\,\Delta t\ .$ (A.6)

We can define the Schrödinger equation as

$\displaystyle\frac{{\rm d}}{{\rm d}t}|k\rangle_{E}=-\frac{2\pi ik}{N\Delta
t}\,|k\rangle_{E}\ ,$ (A.7)

and evaluating the operator $E$ in the basis of the beables $|n\rangle$, using
transformations (A.3), is straightforward.

By regarding the entire theory of quantum mechanics as the vector
representation of a deterministic system, one may conclude that there is
absolutely nothing mysterious about quantum mechanics. In particular, _any_
discrete, realistic, deterministic, classical system can be regarded as a
large set of periodic subunits, which will always allow us to set up a
Schrödinger equation along lines as sketched above. One may _always_
‘postulate’ Born’s rule of regarding the absolute squares of amplitudes as
probabilities, and one may always ‘postulate’ the rule that a wave function
collapses after a measurement, and explain the outcomes of experiments along
such lines. Thus, what used to be distinct axioms of quantum mechanics, can
actually be seen as man-made rules to interpret superpositions as probability
distributions. All mysteries then evaporate.

To illustrate this, the author set up a generic program to mimic any
Schrödinger equation, to any accuracy[10, 12], by starting out with periodic
variables that interact weakly (an interaction that is as real and
deterministic as the periodic ones above). This program works by noting that
variables that are periodic at the tiniest conceivable time scale, will always
turn out to be in their lowest energy states, and this can be used to express
all dynamics in terms of the energy eigen modes, leading to complete freedom
to derive any Schrödinger equation one wants. An interesting aspect of this
derivation is that some Schrödinger equations will be easier to obtain than
others, and this might become a lead in future model building attempts.

The resulting theory for quantum mechanics as resulting from smearing the
fastest variables shows strong similarities with Nelson’s stochastic theory of
quantum mechanics [11], although we believe to have more universal
mathematical justifications and procedures.

There are, however, a number of questions that are still quite legitimate. One
may ask: “How come that J.S. Bell arrived at very disturbing contradictions by
using his theorem?”[13, 31, 32]. This question can be answered in various
ways. One answer is that Bell assumed his observers Alice and Bob to have
‘free will’, whereas the notion of free will cannot be defined along the lines
above. Bell’s _actual_ problem was his fear of ‘conspiracy’, or
superdeterminism: the laws of physics seem to conspire to overthrow his
carefully derived theorem. Answer: we can choose our classical system as
complicated as we want, but at the classical level there is never any
conspiracy. Conspiracies appear to arise if one tries to write all physical
states, including all their probabilities, as quantum superpositions with well
defined phases. But these are man-made. The quantum superposition coefficients
are not entirely observable, and consequently, they, and particularly the
phases of the superposition coefficients, may seem to behave not quite
logically. The classical system does not know about superpositions and phases.

In view of this, we think the source of the contradiction Bell arrived at, is
that he still described the atoms and photons as if the quantum states they
were in should be, to some extent, observable. This contradiction disappears
completely only if one uses _nothing but_ the language of the really
observable beables.

And yet, questions still remain. One may ask: Bell’s theorem, and numerous
similar mysteries, arise in particular whenever we consider spin states that
can be rotated. Rotation symmetry itself _is_ a mystery. How can discrete,
finite systems such as fields on a lattice, display invariance under
continuous symmetry groups? Now this is a hard question. It is hard to rotate
beable states over fractions of angles. We could argue that continuous
symmetries may be ‘emergent phenomena’, but detailed, rigorous mathematical
investigations that elucidate these phenomena have not yet been performed. So
we haven’t completed our job. There is one observation concerning continuous
rotation symmetries: we are blessed with computer programs that allow us to
rotate any photograph any way we like, as if there were no pixels, which are
certainly not rotation invariant. Thus we note that emergent, continuous
symmetries perhaps should not worry us, they may well occur in nature, but a
more precise mathematical treatment would be welcome.

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