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\title{\huge \bf {Initial Condition Model from Imaginary Part of Action
and the Information Loss Problem }}
\author{
H.B.~Nielsen ${}^{1}$ \footnote{\large\, hbech@nbi.dk} \\[5mm]
\itshape{${}^{1}$ The Niels Bohr Institute, Copenhagen, Denmark}}
\date{}
\maketitle
\begin{abstract}
We review slightly a work by Horowitz and Maldecena solving the
information loss
problem for black holes by having inside the blackhole - near to the
singularity - a boundary condition, as e.g the no boundary proposal by
Hartle and Hawking. Here we propose to make this boundary condition
come out of our imaginary action model (together with Masao Ninomiya).
This model naturally begins effectively to set up boundaries - whether
it be in future or past! - especially strongly whenever we reach to
high energy physics
regimes, such as near the black hole singularity, or in Higgs producing
machines as LHC or SSC. In such cases one can say our model predicts miracles.
The point is that you may say that the information loss problem, unless you
solve it in other ways, call for such a violation of time causality as in our
imaginary action model!
\end{abstract}
\newpage
\thispagestyle{empty}
\section{Introduction}
The information loss problem \cite{Samir} is essentially this:
1) From one point of view it seems that information falls into
the black hole and
2) the information comming out with the Hawking
radiation seems not so easy to get correlated with the infallen
information.
But then it is not easy as t'Hooft \cite{tHooft} would like: the blackhole
is just
a resonance: you scatter some particles and some other particles come out
connected
by an ordinary S-matrix.
The point of the present talk is to look at the direction of solving
this problem given by an article by Horowitz and Maldacena
\cite{Horowitz:2003he}, in which they propose to
make use of a fixing of the boundary conditions near the or at the
singularity inside the black hole. Let us immediately remark that
such a direction of solving the problem is highly unconventional in the
sense of having influence from the future or should we say backward
causation inside the black hole horizon.
\section{Review of Mathurs putting of the trouble}
In fact we heard Samir Mathur\cite{Samir} put it roughly like this:
%When we take realistic blackhole Penrose diagram as being modified
%by the infalling stuff relative to an ideal Penrose diagram for
%a Schwarzhild solution extended as long one can, we find it natural
%to choose a time coordinate
In the usual Penrose diagram \cite{Penrose} for Black hole \cite{BH} the
Horizon is a lightlike
surface meaning in the Penrose diagram a lightlike directed line.
We now imagine an extension of a curve of given external time t
being the Schwarzhild coordinate but not in the perfect Schwarzhild
solution extended but in one a little bit more realistic model including the
very formation of the blackhole by some material falling in. Now we can
find a surface that is purely spacelike so that we could use it
as description of a moment of time, a given special value of a time coordinate
being in the far outside simply the time $t$ but which would at the end
be seperated by the horizon.
The crux of the matter is that inside the horizon there are moment surfaces
(space like surfaces, that could be taken as corresponding to a single value
of a possible time coordinate) on which the information from the infallen
material
and the information correlated with the emitted Hawking radiation
fall in widely different places. That is to say that the information about
what falls in comes to one place at a certain ``moment of time'' while the
Hawking radiation orginates at that moment of time from far a away region.
Thus it seems against the principle of locallity to get the Hawking radiation
correlated with the incomming material. But this correlation is
what is needed if it should be so that the black hole were simply functioning
as a normal resonance representing the scattering of the incomming material
which then comes out again as Hawking radiation.
\section{Horowitz's and Maldacena's ``solution''}
Since the problem of the information loss as here described is a
matter of a problem with locality, one might think of having some
form of violation of locality. Indeed Horowitz and Madacena
\cite{Horowitz:2003he} has a
proposal
for solving the information loss problem by such a violation. In fact
the idea of Horowitz and Maldacena is that there could exist a boundary
condition
imposed as a law of nature at or close to the singularity inside the
black hole. Such a boundary condition at a time later than the time
for which it is relevant means a restriction of the future, an arrangement
that the future shall be something special and thus a priori opens up the
possibilty for backward causation, meaning that the future influences the
past. Now it were in the Horowitz and Maldacena considerations concerning the
black hole only in the inside the horizon space time region that the backward
causation should take place. That should also be sufficient in order to
solve the problem of the black hole information passing from the incomming
matter to the outgoing Hawking radiation. In the outside the horizon region
the reflection of the inside restriction at the singularity is that
there comes a correlation between the infalling material and the outgoing
Hawking radiation\cite{HR}. If we think of what happens on the above
mentioned surface of events of a special moment of a certain time,
we can see that the future restriction can impose a correlation
between the far away regions with respectively the infalling stuff and the
Hawking radiation related degrees of freedom. Such a restriction at the
singularity could - Horowitz and Maldacena also allude to as the possibility -
be due to the Hartle-Hawking no-boundary boundary condition\cite{noboundary}.
To understand this idea of using a singularity based restriction
``in the future'' to provide the needed correlation it may be needed
to have in mind that the Hawking radiation emmited related degrees of freedom
fall into the black hole and finally end up in the singularity. Thus these
with Hawking radiation related degrees of freedom get via the future
restriction related to the infalling stuff degrees of freedom and so
finally the Hawking radiation comming out ends up related to the
degrees of freedom of the infalling stuff.
\section{Miracles are called for, unless only say string stars}
In the foregoing section we saw that Horowitz and
Maldecena\cite{Horowitz:2003he} could help
on the information loss problem and thus make the black hole easier
consistent with the developments as one usually expects quantum mechanics
or mechanics at all to provide them, by introducing restrictions
at the singularity. But if we first let in the possibility of restrictions on
the future as some law of nature then we have opened up for the
possibility of getting miracles into the theory. In fact if we
have restrictions on what the future shall be then we should expect to see
that some features of the state of the universe should be predestined to
something special. Such arrangements would seem like miraclulously
special arrangements. The restrictions in the future would only come about
typically by happenings which a priori would look so strange that
we would consider them miracles. Thus the type of theory proposed
is a theory with miracles. Now it were in the case of solving the
black hole problem of information loss only inside the black hole
- inside the horizon - that were under the influence from future
physics. Thus it were only inside the black hole that there were
truly the need for the miracles, but if you allow them at all, it may
at the end be difficult to keep them away from the outside region.
It should be mentioned that there may be other ways - although it looks
difficult - to solve the problem of information loss: In fact one can
- and this is what stringtheory seems to deliver according to the talk
of Samir Mathur\cite{Mathur} - imagine that a genuine black hole never
truly forms, but that the collapse stops - if not before - in the last
moment before a true black hole is formed. If truly a black hole were
never formed, then of course the problems of information loss would not be
relevant. In a way we can say that when as in Samir Mathurs picture
the size of the string or string theory material forming the potential
black hole remains of the size of the Schwarchild radius \cite{BH} even when
the
mass go so high that this Schwarchild radius has become very big, then
it means that the black hole is not truly realized.
Such a keeping up the size of the stringtheory stuff to remain as
big as the Schwarchild radius even when more and more stuff is being
put on, can solve the information loss problem without need for any miracles.
\section{Ninomiya's and mine miracle model, imaginary part of action.}
The conclusion of the above discussion of the problem of black holes
means that there is a call for a theory of the type with backward
causation as suggested in the Horowitz and Maldacena
article\cite{Horowitz:2003he} reviewed above.
This gives us the motivation and excuse for putting forward the
model of Masao Ninomiya and myself \cite{own}. I made an attempt to a popular
presentation in book of collections of talks on miracles \cite{ownpop} at
AArhus University. Actually our ideas of influence from future are a bit
related to old ideas of such influence mainly for the coupling constants
\cite{old} and for predestining humanity to make a new vacuum called
the`` vacuumbomb''\cite{vacuumbomb}.
This model \cite{own}\cite{ownpop} is characterized by having in it a
prediction
of initial conditions in principle. It is even so that this in principle
predicted initial conditions are arranged so as to minimze a certain
functional $S_I(history)$ depending on the history of the universe
through all times from the beginning to the end, a functional being
an integral over all space time, so that indeed it depends on both past
and future. That is to say that the arrangement of the initial conditions
to appear in our model depends also on the future and thus will appear
as having prearrangement or backward causation in it. If for instance as we
suggest it in our model that Higgs particles being produced will make
$S_I(history)$ bigger than if they are not produced then we should
expect that there would be prearrangements occuring seemingly with
{\em the purpose} of preventing the Higgs production. In fact one can
approximately formulate the result of our predictions about the initial
conditions by saying that they are adjusted so as minimize the functional
$S_I(history)$. That is to say the history of the universe will
in our model be approximately selected among all the histories possible
in accordance with the equations of motion as being that history giving
the smallest (i.e. most negative; it probably will be negative)
$S_I(history)$-value.
This real quantity $S_I(history)$ which by being minimized
determines the initial conditions is in our model actually the
imaginary part of an -unusually - assumed {\em complex action}.
That is to say our model consists actually in postulating that
contrary to what one usually takes it the action for the development
of the universe is fundamentally {\em complex}, i.e.
of the form
\begin{equation}
S(history) = S_R(history) + i S_I(history).
\end{equation}
This is to be understood that the parameters in the action
- such as coupling constants and mass squares (in the case we consider
of a quantum field theory, the standard model say) - are taken to
be complex, while the fields (or the dynamical variables) are
taken as usual, i.e. real if they are real in the usual theory.
Since we now have an a bit unusual model with this complex action,
meaning complex couplings and masses, we shall in principle make the
model precise by setting up - or rather choose - that formalism in which we
want simply to insert the complex action instead of the usual real
action. It is honestly speaking a further assumtion in our model to
choose just what expression into which to insert the new complex action.
We choose to do it in a formalism using the Feynman-Dirac-Wentzel path
way integral but in slightly special way:
Usually one would use the Feynman-Wentzel-Dirac \cite{FH}\cite{Wentzel}
to calculate a development ``matrix'' giving the timedevelopment from
one initial time $t_i$ to a final time $t_f$. Then the transition amplitude
from one initial state $|i>$ to a final state $|f>$ is given as a functional
integral
\begin{equation}
= \int \exp{\frac{i}{\hbar} S_{t_i --> t_f}(path)}
{\cal D}path,
\end{equation}
where it is then understood that the action $S_{t_i--> t_f}(path)$
is the integral of the Lagrangian - taken of course for the $path$
being integrated over - over time from time $t_i$ to time $t_f$.
Also it is understood that the field values of the path at the end
points in time $t_i$ and $t_f$, let us call them $\phi(t_i)$ and
$\phi(t_f)$ respectively, are to be integrated over with a weight
given by the wavefuntion(al)s $<\phi(t_i)|i>$ and $<\phi(t_f)|f>$.
You would of course expect from the physical interpretation
of the Feynaman path integral, that the weight of the contribution
from the part of the integral where the fields at some time take
the values in a certain interval should represent - in some way at least -
the probability for the fields having taken their values in that interval.
However, really the question as to what happens between the preparation
of the state $|i>$ and the measurement of the final state $|f>$ cannot be
answered because it would mean a new experiment to begin to measure on
something in the intermediate time. It would be like in the Einstein Bohr
discussion if Einstein starts measuring through which of the slits in the
double slit experiment the particle goes. Nevertheless
Aharonov et al. \cite{Aharonov} have discussed some weak measurements
being performed in the intermediate time.
One could also for example make an expression for the average of
an operator $O(t_f)$ at the time $t_f$ by means of the Feynman path way
integral like this:
\begin{eqnarray}
* t_f)^{\dagger} O U(t_i --> t_f)|i>& =&\\
= \int \exp{\frac{i}{\hbar}
S_{t_i --> t_f}(path)}O(\phi(t_f)) {\cal D}path \left ( \int \exp{
\frac{i}{\hbar} S_{t_i --> t_f}(path')} {\cal D}path' \right )^{\dagger}&,&
\end{eqnarray}
where it is then to be explained that the boundaries for these two
functional integrals at the end of time interval at $t_i$ should be integrated
over and weighted with the wave function $<\phi(t_i)|i>$ and its complex
conjugate $**$. It is also understood that the $path$ in the
first factor and the $path'$ in the second integral are to be identified
at the time $t_f$,
\begin{equation}
\phi(t_f)_{path} = \phi(t_f)_{path'}.
\end{equation}
Finally the operator $O(\phi(t_f))$ should be understood as
possibly depending even on the derivative of $\phi(t)$
derived w.r.t. to $t$ which is then identified with $t_f$.
Since in the usual case of the action being real the transition matrices
as $U(t_f --> t_3)$ say is unitary we can even multiply in a product
$U(t_f --> t_3)^{\dagger} U(t_f --> t_3) = 1$, and thus we also write
\begin{eqnarray}
** t_f)^{\dagger} O U(t_i --> t_f)|i>& =& \\
= \int \exp{\frac{i}{\hbar}
S_{t_i --> t_3}(path)}O(\phi(t_f)) {\cal D}path \left ( \int \exp{
\frac{i}{\hbar} S_{t_i --> t_3}(path')} {\cal D}path' \right )^{\dagger}&&,
\end{eqnarray}
where we now instead of at $t_f$ have the identification
\begin{equation}
\phi(t_3)_{path} = \phi(t_3)_{path'}.
\end{equation}
That is to say that it does not matter for calculating the average
of the operator $O(t_f)$ at $t_f$ whether we use the Feynman path integral
with a time interval going up to one $t_3$ or another, so that we could if we
would like take the choice of formulating it with taking $t_3$ to go say to
infinity.
If we wanted we could even replace the initial time $t_i$ state $**$ of the form
\begin{eqnarray}
** t_f)^{\dagger} O U(t_i --> t_f)|i>& =&\\
\int \exp{(\frac{i}{\hbar}
S_{t_0 --> t_3}(path))}``(|i>** t_3}(path')} {\cal D}path' \right )^{\dagger}&&,
\end{eqnarray}
where $t_0$ is a time that can be anything provided it is earlier than
the time $t_i$ at which we have inserted the operator $``(|i>**$ used at first. Since now
this expression does not in the real action case depend on the
times $t_0$ and $t_3$ provided they are outside the time interval
$[t_i,t_f]$, we could choose them to anything we would like as long as
these times $t_0$ and $t_3$ are still outside. For instance we could
take $t_0 = -\infty$ and
$t_3 = + \infty$. We might even imagine as a slight generalization
to insert several operators and think of replacing the special
projection operator $``(|i>** t_3}(path))}O(\phi(t_i)) O(\phi(t_f))
{\cal D}path \left ( \int \exp{
(\frac{i}{\hbar} S_{t_0 --> t_3}(path'))} {\cal D}path' \right )^{\dagger}
,\label{inter}
\end{equation}
as a suggestion for what we can use to extract information from a
Feynaman path integral formulation. Note that
this expression is quadratic in the Feynman path integral in the sence that
it is a product of two Feynman path integrals, one with the dummy path
being denoted $path$ and one complex conjugated with the dummy $path'$.
Now the idea is in the case of a complex action $S=S_R + iS_I$ also
to use this
expression by postulating that this expression obtained by putting
in combinations of operators into a Feynman path integral and then
multiplying that by a complex conjugate path integral without the
operator insertions to deliver expectation values for the to the operators
associated quantities. That is to say we take it that our model
is assumed to deliver the predictions obtained by being extracted
from expressions of this type with $t_0$ and $t_3$ going to respectively
minus and plus infinity.
While in the case of usually assumed real action model the extension
with the time intervals outside the interval $[t_i, t_f]$ used is
irrelevant, this is no longer true in the case of there being an imaginary
part of the action $S_I(path)$. So in our model it becomes important
that we decide to use the whole time axis from the beginning to the end
of all times.
It is strictly speaking an assumption being added into our model
that we postulate just this type of interpretation of what our
complex action model. We think, however, that just such an interpretation
being based on using a Feynman-Wentzel-Dirac path integral extending
a priori over all times from the beginning (big bang or whatever were the
first moment, minus infinity likely) to the end of times is very reasonable.
After all, if we should somehow think of the path integral as the fundamental
theory beyond quantum mechanics then it would not be so nice to choose the
time interval for evaluating the action to be put in the exponent
of the path way integrand to depend on the choice of what we are to calculate,
or even worse on some arbitrary choice. In the case of the real action
when the arbitrary choice of the times $t_0$ and $t_3$ does not matter
it would of course be o.k., but in the complex action where it would depend,
the natural assumption will be to take the maximal time interval
over which to integrate to be the one to use.
We think that it is also very natural to associate the expectation value
of an operator $O$ at a time $t$ to be associated with the
path at the time t and those components of the fields the development
of which are described by the path and associated to the operator $O$.
Thus we claim an interpretation of the form (\ref{inter}) to quite reasonable.
\section{Significance of the imaginary part of the action}
Once we just have decided on an assumtion about the intepretation
of our model using over all time Feynman path integrals - even
without looking too much on the details alluded to in foregoing
section of looking at expectaion values of operators and squaring
the Feynman path integral - it should be rather unavoidable that
only contributions to the Feynman path integral(s) from paths with
the smallest (or rather most negative) imaginary action $S_I(path)$ will
have much significance.
So it is almost obvious without much details that the history of the
universe that will effectively be the one realized must be characterized
by a very negative $S_I(history)$, the ``smallest'' $S_I(history)$.
That it will be so follows from the simple fact that the Feynman path integral
integrand has the factor $\exp{-S_(path)}$, which will only be dominant when
the imaginary part of the action $S_I(path)$ is very negative.
Really we should have in mind that most likely the very long time intervals
after $t_f$ ``the future'' and before $t_i$ ``the past'' will give big
contributions to the imaginary part of the action, that will suppress most
possible developments so much that essentially almost only one development,
one history of the universe, comes to dominate. The best may actually be to
think of performing a classical approximation. It is wellknow how in the
Feynman path integral formulation obtains the classical approximation as
a saddle point approximation to the functional integral. Pedagogically
- and to avoid a complicated discussion to extract at all a
classical approximation, which is not so obvious at first - we may
assume the imaginary part is taken (at first) to be small. Then we would
in first approximation be allowed to think of the Feynman path integral
being in the classical approximation be given by the saddle point
contributions calculated at first as if we only had the real part
$S_R(path)$, which would mean quite usual classical solutions to the
equations of motion would be all that would contributes in this approximation.
But even a small imaginary part compared to the real part could give
enormously big factors of the form $\exp{-S_I(history)/\hbar}$. We shall in
fact
not forget that in practice one expects $\hbar$ to be very small, so that for
this reason already the exponent gets huge. When we think about that we have
to do with integrals over time regions of the size of the whole lifetime
of the universe, these imaginary action values for the whole life span
of the unviverse will easily suppress almost all but one single classical
solution. That is to say that even a in some sense small imaginary part
would be far sufficient to drastically select almost only one sadle point
contribution to survive being of significance.
We thus arrive at the first approximation description of the prediction
of our model namely in a classical approximation: First imagine calculate
all the classical solutions using just the real part of the action. This
delivers a set of all the classical solutions. Then calculate for each of
these soltions (in practice of course we cannot do that, but think of
it in principle), these possible histories $history$, the imaginary part
of the action $S_I(history)$. Then our model predicts that that history
$reahist$ which gets the one recognized as the realized one, the one
that truly happens, is the one for which $S_I(reahist)$ is {\em minimal}.
This is what we could refer to as the formula ``for the will of God''
being
\begin{equation}
S_I(reahist) \hbox{ shall be {\bf minimal}}.
\label{Godwill}\end{equation}
It should be had in mind that this type of determination of the
initial state to be realized depending on an integral $_I(history)$
which invloves all times, means that the way the universe development has been
started in a way depending on what could happen or not happen at much later
times. But that then means that it would be like miracles, namely as if
things have been prearranged in a statistically unexpected way so as to just
arrange that especially negative contributions to $S_I(hisory)$, i.e.
negative imaginary part of the Lagrangian $L_I(history(t))$ (where
$history(t)$ means the state and development derivatives in the history
$history$ at time equal to $t$). One of our speculations to be discussed
in section \ref{Higgs} is that production of Higgs particles should
cause a relatively huge positive contribution to $L_I$ so that histories
leading to
Higgs production become disfavoured.
\subsection{Really strong assumption, if we take action real}
First let us, however, now give an argument that it would be very nice
estetically to have the action not being restricted to be real,
but rather has also an imaginary part. Indeed we may simply think of the
Feynmann-Wentzel path integral - even written only very abstractly without
going in detail - written in the form
\begin{equation}
\int exp(\frac{i}{\hbar} S(path)) {\cal D}path.
\end{equation}
Now we come with the remark that it is completely obvious that the integrand
$exp(\frac{i}{\hbar} S(path])$ is {\em complex}. There is namely
even simply an $i$ present as an over all factor in the exponent.
If we therefore take the point of view that the most fundamental and
important quantity is the integrand rather than say the action itself,
then we could say: if something should be assumed to be real rather
than complex then it should be this most fundamental quantity
that should be assumed real.
To take the integrand to be real would of course be completely unacceptable,
if one would have any connection to the usual theory. So the natural
possibility is that there is {\em no restricton to reality at all},
so that both $S(path)$ and the integrand are allowed to be complex.
This argumentation may also be made more concrete by imagining that
one would find some theory behind the Feyman Wentzel path way
descripton of quantum mechanics, i.e. some model from which one derives
quantum mechanics and arrive to a Feynman Wentzel path way formulation.
Then if one would hope for the usual theory with the {\em real}
action it would be a very delicate mechanism that would be needed to
ever get the
integrand
become just a quantity of norm unity - as is what the real action means-.
For example we attempted\cite{bled} such derivation of quantum mechanics
in the
path way formulation and indeed did not at first find any reason
why the integrand should be of norm unity.
\subsection{How to hide roughly the imaginary part of action}
At first it would look that our model with
the complex action would lead to too many prearranged happenings to
agree with what we observe, there would be too many ``miracles''
or ``antimiracles''(repectively good or bad a priori unlikely
events). Now, however, we have found some mechanism that might indeed
help to reduce the predicted number of such at first unlikely arrangements
in practice. Let us here in this discussion already accept the above mentioned
classical approximation that we just have the effect of the imaginary part of
the action, $S_I(path)$ in our model so that it just delivers the formula
(\ref{Godwill}) to select the realized solution to the classical equations
of motion.
The important point that brings down dramaticly the number of strange
events, miracles or anti miracles, is that with the restriction from the
equation of motions it is made troublesome to make too many miracles.
If the initial state so to speak has to be adjusted to make certain special
event at one moment of time then the degrees of freedom of this initial
state are partly fixed by this arrangement and there is less freedom to
adjust them to make - arrange for - more miracles. Thus it looks that the
longer time the universe exists the more competition there will be
about getting arrangements to each indivdual era of times. Only
the ``miracles'' or ``anti miracles'' in the human history has a good chance
to be spotted by poeple, and even then probably mainly the ones close to
our own times, if we shall get aware of them. But since the universe has
an age of the order of 13 milliard years already alone, the fact
that there were $10^{8}$ human age periodes in even just the certainly
existing time periode of for the universe, each arrangement would have to
be shared by at least these $10^8$ periodes. Actually we believe
from consideration of an action ansatz analogous the real part $S_R$
already known from phenomenology of the equations of motions, that it is
likely that high energy scale physics contributes the most. When we think
of the action as being written as a four dimensional space time integral
$\int {\cal L} d^4x$ with the Lagrangian density having dimension mass to the
fourth power, it should be obvious that in order to get a big contribution
to the action from space time volume of a given size we should involve physics
with so high energies (per particles) involved as possible. Now the universe
were of
smaller size in the time shortly after big bang and the time scales of the
eras were smaller so that this presumed higher contribution from the high
energy scale being high is partly compensated for by smaller space time
volume.
Nevertheless it is highly possible that a major contribution could
have come to the imaginary part $S_I(history)$ from the era of inflation.
One could even imagine that the as slow roll problem presented
phenomenological call for a somewhat suspicially long time during
which the inflaton field remained in a special region could be one of the
``miracles'' in our model. That should mean that because a special value
for the inflaton field would give especially numerically high but negative
imaginary part $L_I$ this value of the inflaton field would be (pre)arranged
to take on over the biggest possible space time volume the specially favoured
value giving the very negative $L_I$. That might indeed favour what would
look like a miraculously long stand in the inflation state.
If indeed some contribution from the time around big bang might dominate
numerically $S_(history)$ then the initial state would be dominantly
arranged to make the dominant imaginary action contribution possibly
most negative and then there would be less freedom of adjusting to
make miracles at other times in the development of the universe. The point
of course is that if first the initial state of the universe has been
adjusted to give the smallest or most negative $S_(history)$
in some era near big bang like the inflation era, then there is less
freedom to arrange miracles in later time. The equations of motion will
namely determine what happens later once it has been determined
with some``purpose'' related to the inflation era, say.
So the hypotesis of a dominant era different from our own concerning the
imaginary action $S_I(history)$ would help to reduce the number and degree
of remarkableness of the miracles or anti miracles to be predicted to
occur to day.
\section{Our prediction of the failure of LHC or other Higgs producing
machines}
\label{Higgs}
It is the suggested to be the major experimental test of our model of
imaginary part for the action that production of many Higgs particles
should be suppressed in the sense that machines destined to make big
amounts of Higgs particles should have bad luck. We shall therefore now
review
this part of our model, i.e. the arguments for this bad luck comming out
of our model:
The most natural assumption about the order of magnitude of the
imaginary part would be that we take the various coefficients in the
expression for the Lagrangian density such as coupling constants and
mass squares ($m_h^2$ say) should have rather random phases of order
unity. That would
imply that we would expect the imaginary part $S_I$ and the real part $S_R$
of the action to be of similar order of magnitude for some random
field development. For the special development, which the universe
should perform in our model, and which is selected by minimizing the imaginary
part it could be different. There is, however, one coefficient in the
Standard Model Lagrangian density which requires a special consideration
concerning the
order of magnitude of real versus imaginary part, namely the mass square
of the Higgs particle. The special point about this mass square of the Higgs
particle is that it is a very wellknow {\em mystery} why the mass square
of the Higgs particle defined in a renormalized way is so enormously
small compared to the magnitude,which we would expect to be the fundamental
mass square scale of physics namely the Plack mass squared. This mystery
we may call the `scale problem'' - why so different scales ?! -. It would
be even more a problem, if one would like to have a unified gauge theory
like $SU(5)$ or $SO(10)$ or the like since the unifying scales
would also be far away from the Higgs mass scale- one would then even have the
doublet triplet seperation problem-. It is really this scale problem
that manifests itself by giving rise to the hierarchy problem: How to
avoid that by each new perturbative correction to the
Higgs VEV or Higgs renormalized mass quadratic divergences -
supposedly cut off by some fundamental physics at the Plack scale -
does not reshuffle the Higgs mass (square) by enourmous amounts
recalling the scale problem mystery order by order again.
The part of the Higgs mass square coefficient for which we know
the order of magnitude phenomenologically is the {\em real} part
$m_H^2|_R$ of this coefficient to the Higgs field squared $m_H^2$
in the Lagrangian density ${\cal L} = ...+ m_H^2/2 * \phi_H^2(x)+ ...$,
where we have split up the coefficient
\begin{equation}
m_H^2 = m_H^2|_R + i* m_H^2|_I.
\end{equation}
But now, if it is so mysterious, why the real part $m_H^2|_R$ is so
small compared to the Plack scale mass square (the square of the
Planck mass), and we do not really understand yet the true
mechanism for it being so small, then how can we know whether this
``mysterious'' mechanism works to also make the imaginary part
of the Higgs mass square $m_H^2|_I$ surprisingly small? Very likely
it will actually not make also the imaginary part small, because what is
truly what is small concerning the real part is not simply the bare
real part but rather the by several corrections modified - i.e. relative
to that
dressed or renormalized - real part. But renormalizing the imaginary
part would likely be a quite different story so that a mysterious finetuning
tuning the renormalized real part to be exceptionally small compared to the
a priori expectation, the Planck mass square, would most likely not
hit to make the imaginary part small. So we actually expect the
imaginary part $m_H^2|_I$ still to be of the order of the Planc mass square.
But now from the point of view of the typical energy scales of say the
LHC accelerator
- a few TeV - the mysterious small Higgs mass and
thus the real part $m_H^2|_R$ is of a rather normal order of magnitude,
while an imaginary part of Plack scale size would seem enourmously big!
This then means that as soon as the imaginary part of the square of the
Higgs mass comes in, it will completely dominate. Now in the experiments
we have so far studied, not even seeing the Higgs yet at all, the couplings
and masses relevant have only been what came out of the dimensionless
couplings/coefficients in the Standard model Lagrangian density
and the {\em Higgs vacuum expectation value}. The latter is determined from
the real part $m_H^2|_R$ and thus the imaginary part would get so far
no influence. There would as long as no Higgses
are truly produced only be a constant vacuum contribution from the term
$ m_H^2|_I |\phi(x)|^2 $ to the imaginary Lagrangian density ${\cal L}(x)$.
Only when the Higgs field is modified relative to its uusual vacuum
value VEV= $<\phi_H>$ will the imaginary part come into play in a variable
way. But that is
typically the Higgs production and the existence of genuine Higgs particles
flowing arround. We therefore expect that it is the flowing arround
of produced Higgses that will contribute the very likely very huge
contribtuion to the ${\cal L}(x)$ and thus to $S_I(hitory]$. Now presumably
the appearance of geuine Higgs particles flowing arround is presumably
a positive contribution to the imaginary action so that it would be
disfavoured in the selection of the truly realized solution to the equations
of motion to have Higgs arround; if it were namely instead very favoured
we should already have Higgses all over.
Thus we now expect that it would make the imaginary part of the
action $S_I(history)$ appreciably bigger (less negative) if in the
history of th universe many Higgses come to exist, thus accelerators
like SSC, the Tevatron, and the LHC producing Higgses should
preferably for minimizing $S_I$ be avoided by not comming to work
or quckly be stopped again once working. One should of course also then
expect that cosmic rays should miraculously or somehow from the
initial conditions of the universe have been arranged to produce
as few Higgses as can easily be organized without spoiling too
much the possibilies for the appropriate miracles in other eras
so that their $S_I$-contributions are not too much reduced from what
they can maximally be. But we humans have little understanding of
how much cosmic rays there would have been under slightly different
choices of the initial conditions, so we do not know if there should be
a fine tuning of the initial conditions so as to make few or many cosmic
radiation particles. Contrarily we have, however, good understandings of
that when one has built about the quarter of the tunnel of the
planned SSC (= superconducting super collider)\cite{SSC} in Texas then there
is apriori a very high expectation that there should soon be produced
a lot of Higgs particles (if it exist at all of course as we assume here).
Then it were really like an anti-miracle when the Congress stopped the machine
from being built and led the tunnel be only used for champignon growing
or the like.
In our model we actually take this somewhat surprising fate of bad luck
for the great SSC-project as an anti-miracle confirming our model.
Also the accident of a bad connection stopping for soon
a year the LHC just when it were about to start functioning we take as
a symptom of our model! It should be stressed that our predictions
about bad luck for Higgs producing accelerators were made after the bad fate of
SSC, but {\em before} the accident at LHC that have delayed it
by now soon a year!
.
\section{Conclusion}
The main point of the presnt article were to call attention to that
by the ideas of Horowitz and Maldacena \cite{Horowitz:2003he} for solving
the problem with black holes
of correlating the infall information with the outgoing Hawking radiation
a backward causation theory is called for. In competition with for instance
the Hartle Hawking no-boundary postulate\cite{noboundary} replacing
the singularity with
a special condition - say no boundary - thereby imposing ``final conditions''
leading to backward causation we presnted ``the imaginary part of action
model'' of Masa Ninomiya and myself\cite{own}. The crux of this matter
were that
really as explained in the talk by Samir Mathur the problem of getting the
information from the infalling stuff into the black hole transfered to the
outgoing Hawking radiation is a problem of causality- like the problem of
tranfering information from one place to another one faster than with
speed of light-. The problem would therefore possibly be avoided, if we
have a theory with backward causation, so that future can influence past and
therefore no causality principle can be truly valid. For phenomenological
reasons
it is of course needed that under ``normal'' conditions the amount of
backward causation - or as we also refered to cases of backward causation,
miracles or anti miracles - should be seldom. This is indeed the case
both by thinking of Hartle Hawking no-boundary (mainly showing up in
black holes, which are phenomenologically badly known) and in our
``imaginary part of action model'', in which it is though needed a
somewhat speculative argumentation to argue that the cases
of backward causation get so seldom as needed for agreement with dayly
life experience. We think, however, that there {\em is} a good chanse
that the restriction from the history of the universe having to obey the
(classical) equations of motion (at least approximately) could impose
so strong restrictions on the amount of backward causation or miracles
or anti miracles that it would not disagree with present knowledge.
In this way we want to claim that our model is viable so far.
\begin{thebibliography}{99}
% \bibliographystyle{style}
% \bibliography{bibfile}
\bibitem{Samir}
%\cite{Mathur:2009hf}
%\bibitem{Mathur:2009hf}
S.~D.~Mathur,``The information paradox: A pedagogical introduction,''
arXiv:0909.1038 [hep-th].
%%CITATION = ARXIV:0909.1038;%%
\bibitem{tHooft}
Frontiers of Fundamental Physics
Proceedings of the Sixth International Symposium Frontiers of
Fundamental and Computational Physics, Udine, Italy, 26
September 2004
10.1007/1-4020-4339-2\_4
B.G. SIDHARTH, F. HONSELL and A. DE ANGELIS
%\cite{Horowitz:2003he}
\bibitem{Horowitz:2003he}{Horowitz:2003he}
G.~T.~Horowitz and J.~M.~Maldacena,
``The black hole final state,''
JHEP {\bf 0402} (2004) 008
[arXiv:hep-th/0310281].
%%CITATION = JHEPA,0402,008;%%
\bibitem{BH}
On the Means of Discovering the Distance, Magnitude, &c. of the
Fixed Stars, in Consequence of the D...;
John Michell,
Philosophical Transactions of the
Royal Society of London, Vol. 74, (1784), pp. 35-57,
Published by: The Royal Society
^ a b Schwarzschild, Karl (1916), "Über das Gravitationsfeld eines
Massenpunktes nach der Einsteinschen Theorie", Sitzungsber.
Preuss. Akad. D. Wiss.: 189\u2013196 and Schwarzschild, Karl (1916),
"Über das Gravitationsfeld eines Kugel aus inkompressibler
Flüssigkeit nach der Einsteinschen Theorie", Sitzungsber. Preuss. Akad. D.
Wiss.: 424\u2013434 .
# ^ "Dark Stars (1783)". Thinkquest. http://library.thinkquest.
org/25715/discovery/conceiving.htm#darkstars. Retrieved 2008-05-28.
# ^ Laplace; see Israel, Werner (1987), "Dark stars: the evolution of an idea",
in Hawking, Stephen W. & Israel, Werner, 300 Years of Gravitation,
Cambridge University Press, Sec. 7.4
\bibitem{Penrose}# d'Inverno, Ray (1992). Introducing Einstein's Relativity.
Oxford: Oxford University Press. ISBN 0-19-859686-3.
See Chapter 17 (and various succeeding sections) for a very readable
introduction to the concept of conformal infinity plus examples.
# Frauendiener, Jörg. "Conformal Infinity". Living Reviews in Relativity.
http://relativity.livingreviews.org/Articles/lrr-2004-1/index.html.
Retrieved February 2, 2004.
# Carter, Brandon (1966). "Complete Analytic Extension of the Symmetry Axis
of Kerr's Solution of Einstein's Equations". Phys. Rev. 141: 1242\u20131247.
doi:10.1103/PhysRev.141.1242. See also on-line version (requires a
subscription to access)
# Hawking, Stephen; and Ellis, G. F. R. (1973).
The Large Scale Structure of Space-Time. Cambridge: Cambridge University
Press. ISBN 0-521-09906-4. See Chapter 5 for a very clear discussion of
Penrose diagrams (the term used by
Hawking & Ellis) with many examples.
# Kaufmann, William J. III (1977). The Cosmic Frontiers of General
Relativity. Little Brown & Co. ISBN 0-316-48341-9.
Really breaks down the transition
from simple Minkowski diagrams, to Kruskal-Szekeres diagrams to
Penrose diagrams,
and goes into much detail the facts and fiction concerning wormholes. Plenty
of easy to understand illustrations. A less involved, but still very
informative
book is his William J. Kaufmann (1979)). Black Holes and Warped Spacetime.
W H Freeman & Co (Sd). ISBN 0-7167-1153-2.
\bibitem{HR}
Hawking, S. W. (1974). "Black hole explosions?".
Nature 248 (5443): 30. doi:10.1038/248030a0. Hawking's first article
on the topic
Page, Don N. (1976). "Particle emission rates from a black hole:
Massless particles from an uncharged, nonrotating hole".
Physical Review D 13 (2): 198-206. doi:10.1103/PhysRevD.13.198.
first detailed studies of the evaporation mechanism
\bibitem{noboundary}
%\cite{Hartle:1983ai}
%\bibitem{Hartle:1983ai}
J.~B.~Hartle and S.~W.~Hawking,
%``Wave Function Of The Universe,''
Phys.\ Rev.\ D {\bf 28} (1983) 2960.
%%CITATION = PHRVA,D28,2960;%%
%\cite{Vilenkin:1998rp}
%\bibitem{Vilenkin:1998rp}
A.~Vilenkin,
%``The quantum cosmology debate,''
arXiv:gr-qc/9812027.
%%CITATION = GR-QC/9812027;%%
%\bibitem{HR}
%# Hawking, S. W. (1974). "Black hole explosions?".
%Nature 248 (5443): 30. doi:10.1038/248030a0. \u2192 Hawking's first article
%on the topic
%# Page, Don N. (1976). "Particle emission rates from a black hole:
%Massless particles from an uncharged, nonrotating hole".
%Physical Review D 13 (2): 198\u2013206. doi:10.1103/PhysRevD.13.198.
% \u2192 first detailed studies of the evaporation mechanism
\bibitem{own}
%\cite{Nielsen:2008zz}
%\bibitem{Nielsen:2008zz}
H.~B.~Nielsen and M.~Ninomiya,``Nonexistence of irreversible processes
in compact space-time,''
Int.\ J.\ Mod.\ Phys.\ A {\bf 22} (2008) 6227.
%%CITATION = IMPAE,A22,6227;%%
%\cite{Nielsen:2008cm}
%\bibitem{Nielsen:2008cm}
H.~B.~Nielsen and M.~Ninomiya,
``Test of Influence from Future in Large Hadron Collider: A Proposal,''
arXiv:0802.2991 [physics.gen-ph].
%%CITATION = ARXIV:0802.2991;%%
%\cite{Nielsen:2007mj}
%\bibitem{Nielsen:2007mj}
H.~B.~Nielsen and M.~Ninomiya,
``Complex Action, Prearrangement for Future and Higgs Broadening,''
arXiv:0711.3080 [hep-ph].
%%CITATION = ARXIV:0711.3080;%%
%\cite{Nielsen:2007ak}
%\bibitem{Nielsen:2007ak}
H.~B.~Nielsen and M.~Ninomiya,
``Search for Future Influence from L.H.C,''
Int.\ J.\ Mod.\ Phys.\ A {\bf 23} (2008) 919
[arXiv:0707.1919 [hep-ph]].
%%CITATION = IMPAE,A23,919;%%
%\cite{Nielsen:2007kz}
%\bibitem{Nielsen:2007kz}
H.~B.~Nielsen and M.~Ninomiya,
``Degenerate vacua from unification of second law of thermodynamics with
%other laws,''
arXiv:hep-th/0701018.
%%CITATION = HEP-TH/0701018;%%
%\cite{Nielsen:2006pz}
%\bibitem{Nielsen:2006pz}
H.~B.~Nielsen and M.~Ninomiya,
``Future dependent initial conditions from imaginary part in lagrangian,''
arXiv:hep-ph/0612032.
%%CITATION = HEP-PH/0612032;%%
%\cite{Nielsen:2006td}
%\bibitem{Nielsen:2006td}
H.~B.~Nielsen and M.~Ninomiya,
``Trouble with irreversible processes in non-boundary postulate. and
perfect
match of equation of motions and number of fields,''
arXiv:hep-th/0602186.
%%CITATION = HEP-TH/0602186;%%
%\cite{Nielsen:2006th}
\bibitem{Nielsen:2006th}
H.~B.~Nielsen and M.~Ninomiya,
%``Compactified time and likely entropy: World inside time machine: Closed
%time-like curve,''
arXiv:hep-th/0601048.
%%CITATION = HEP-TH/0601048;%%
%\cite{Nielsen:2006vc}
%\bibitem{Nielsen:2006vc}
H.~B.~Nielsen and M.~Ninomiya,
%``Intrinsic periodicity of time and non-maximal entropy of universe,''
Int.\ J.\ Mod.\ Phys.\ A {\bf 21} (2006) 5151
[arXiv:hep-th/0601021].
%%CITATION = IMPAE,A21,5151;%%
%\cite{Nielsen:2005ub}
%\bibitem{Nielsen:2005ub}
H.~B.~Nielsen and M.~Ninomiya,
``Unification of Cosmology and Second Law of Thermodynamics: Solving
Cosmological Constant Problem, and Inflation,''
Prog.\ Theor.\ Phys.\ {\bf 116} (2007) 851
[arXiv:hep-th/0509205].
%%CITATION = PTPKA,116,851;%%
%\bibitem{ownpop}
%@MISC{nielsen-2008,
% author = {Holger B. Nielsen},
% title = {Model for Laws of Nature with Miracles},
% url = {http://www.citebase.org/abstract?id=oai:arXiv.org:0811.0304},
% year = {2008}
%}
\bibitem{ownpop}
Holger B. Nielsen, Model for Laws of Nature with Miracles,\\
{\it http://www.citebase.org/abstract?id=oai:arXiv.org:0811.0304},
(2008)
\bibitem{old}
%\bibitem{nonlocalfirst}
%\cite{Bennett:1995ag}
%\bibitem{Bennett:1995ag}
D.~L.~Bennett, C.~D.~Froggatt and H.~B.~Nielsen,
``Nonlocality as an explanation for fine tuning in nature,''
(CITATION = C94-08-30);
%\bibitem{nonlocalsec}
%\cite{Bennett:1994yx}
%\bibitem{Bennett:1994yx}
D.~L.~Bennett, C.~D.~Froggatt and H.~B.~Nielsen,
``Nonlocality as an explanation for fine tuning and field replication in
nature,''
arXiv:hep-ph/9504294.
(CITATION = HEP-PH/9504294;)
\bibitem{vacuumbomb}
%VACUUM BOMB
%\bibitem{vacuumbomb1sv}
D.~L.~Bennett, ``Who is Afraid of the Past'' ( A resume of discussions
with H.B. Nielsen during the summer 1995 on Multiple Point Criticallity
and the avoidance of Paradoxes in the Presence of Non-Locality in
Physical Theories), talk given by D. L. Bennett at the meeting of the
Cross-displiary Initiative at Niels Bohr Institute on September 8, 1995.
QLRC-95-2.
%\bibitem{vacuumbomb2th}
%\cite{Bennett:1996hx}
%\bibitem{Bennett:1996hx}
D.~L.~Bennett,
``Multiple point criticality, nonlocality and fine tuning in fundamental
physics: Predictions for gauge coupling constants gives alpha**(-1) = 136.8
+- 9,''
arXiv:hep-ph/9607341.
(CITATION = HEP-PH/9607341;)
%\bibitem{vacuumbomb3co}
%\cite{Nielsen:1995rs}
%\bibitem{Nielsen:1995rs}
H.~B.~Nielsen and C.~Froggatt,
``Influence from the future,''
arXiv:hep-ph/9607375.
(CITATION = HEP-PH/9607375;)
\bibitem{FH} Feynman and Hibbs,
Quantum Mechanics and Path Integrals (Hardcover)
by Richard P. Feynman (Author), A. R. Hibbs (Author)
\bibitem{Wentzel} Wentzel,
Physics Letters A
Volume 324, Issues 2-3, 12 April 2004, Pages 132-138;
Salvatore Antoci and Dierck-E. Liebscher,
Wentzel's Path Integrals,International Journal of Theoretical Physics
Publisher Springer Netherlands
ISSN 0020-7748 (Print) 1572-9575 (Online)
Issue Volume 37, Number 1 / January, 1998
DOI 10.1023/A:1026628515300
Pages 531-535
Subject Collection Physics and Astronomy
SpringerLink Date Wednesday, December 29, 2004;
International Journal of Theoretical Physics,
Volume 37, Number 1, 1 January 1998 , pp. 531-535(5)
\bibitem{Aharonov}
%\cite{Aharonov:1981zz}
%\bibitem{Aharonov:1981zz}
%\cite{Aharonov:1990zza}
%\bibitem{Aharonov:1990zza}
Y.~Aharonov and L.~Vaidman,
%``Properties of a quantum system during the time interval between two
%measurements,''
Phys.\ Rev.\ A {\bf 41} (1990) 11.
%%CITATION = PHRVA,A41,11;%%%\cite{Aharonov:2009bu}
%\bibitem{Aharonov:2009bu}
Y.~Aharonov, T.~Kaufherr and S.~Nussinov,
%``Some Aspects of Classical and Quantum Phases,''
arXiv:0907.1666 [quant-ph].
%%CITATION = ARXIV:0907.1666;%%
Y.~Aharonov, D.~Z.~Albert and C.~K.~Au,
%``New Interpretation of the Scalar Product in Hilbert Space,''
Phys.\ Rev.\ Lett.\ {\bf 47} (1981) 1765.
%%CITATION = PRLTA,47,1765;%%
\bibitem{Bled}%\cite{Bennett:2007zz}
%\bibitem{Bennett:2007zz}
D.~Bennett, A.~Kleppe and H.~B.~Nielsen,
``Random dynamics in starting levels,''
{\it http://www.slac.stanford.edu/spires/find/hep/www?irn=7560230}
SPIRES entry
{\it Prepared for 10th Workshop on What Comes Beyond the Standard Model,
Bled, Slovenia, 17-27 Jul 2007}
\end{thebibliography}
%\thispagestyle{empty}
%\section{Our monopole coupling versus $d$ curve and successful agreement}
%\begin{tabular}[h]{|l|c|c|c|c|}
%\hline
%Group &$ d_0$ &$ d$ &$ 3\tilde{g}^2/\pi|_{pred}$ &
%$3\tilde{g}^2/\pi`_{exp}$ \\
%\hline
%$U(1)$ & 0 & 0 & ...& 27.7 \\
%$SU(2)$ & $11/3 = 3.67$ &$3.67$& ... & $196._{12}$\\
%$SU(3)$ & $11\sqrt{65/108}=8.534$ &...& ...& $212$\\
%\hline
%\end{tabular}
\end{document}*