# [time 944] Goedel's incompleteness implies the existence of time

Thu, 21 Oct 1999 20:25:07 +0900

Here is some excerpt from time_VI.tex. LaTeX file is attached, which is
available also at

A key is Goedel's incompleteness theorem, which assures the existence of
(local) time.

A possible solution for the non-existence of time

(October 21, 1999 version)

Abstract. A possible solution for the problem of
non-existence of universal time is given.

In a recent book \cite{Barbour}, Barbour presented
a thought that time is an illusion, by noting that
Wheeler-DeWitt equation yields the non-existence of
time, whereas time around us looks flowing.
However, he does not seem to have a definite idea
to actualize his thought. In the present note I present
a concrete way to resolve the problem of non-existence
of time, which is partly a repetition of my works
\cite{K1}, \cite{K2}, \cite{Ki-Fl1}, \cite{Ki-Fl2}.

1. Time seems not exist

According to equation (5.13) in Hartle \cite{CP},
the non-existence of time would be expressed by
an equation:
\beq
H \Psi=0. \label{1}
\ene
Here $\Psi$ is the state" of the universe
belonging to a suitable Hilbert space $\HH$, and
$H$ denotes the total Hamiltonian of the universe
defined in $\HH$. This equation implies that there
is no global time of the universe, as the state
$\Psi$ of the universe is an eigenstate for the total
Hamiltonian $H$, and hence does not change.
One might think that this implies the non-existence
of local time because any part of the universe is
described by a part of $\Psi$. Then we have no time,
contradicting our observation. This is a restatement
of the problem of time, which is a general problem to
identify a time coordinate with preserving the
diffeomorphism invariance. In fact, equation \eq{1}
follows if one assumes the existence of a preferred
foliating family of spacelike surfaces in spacetime
(see section 5 of \cite{CP}).

We give a solution in the paper to this problem that
on the level of the total universe, time does not exist,
but on the local level of our neighborhood, time looks
existing.

2. G\"odel's theorem

Our starting point is the incompleteness theorem proved by
G\"odel \cite{G}. It states that any formal theory that can
describe number theory includes an infinite number of undecidable
propositions. The physical world we describe includes at least
natural numbers, and it is described by a system of words, which
can be translated into a formal theory. The theory of physics
therefore includes an undecidable proposition, i.e. a proposition
whose correctness cannot be known by human beings until one finds
a phenomenon or observation that supports the proposition or denies
the proposition. Such propositions exist infinitely according to
G\"odel's theorem. Thus human beings can never reach the final
theory that can express the totality of the phenomena in the
universe.

Thus we have to assume that any human observer sees a part
or subsystem $L$ of the universe and never gets the total
Hamiltonian $H$ in \eq{1} by his observation. Here the
total Hamiltonian $H$ is an {\it ideal} Hamiltonian
that might be gotten by God." In other words, a consequence
from G\"odel's theorem is that the Hamiltonian that an
observer assumes with his observable universe
is a part $H_L$ of $H$. Stating explicitly,
the consequence from G\"odel's theorem is the
following proposition
\beq
H=H_L+I+H_E, H_E not = 0, \label{2}
\ene
where $H_E$ is the unknown Hamiltonian describing
the system $E$ exterior to the realm of the observer,
whose existence, i.e. $H_E not = 0$, is assured by G\"odel's
theorem. This unknown system $E$ includes
all what is unknown to the observer.
E.g., it might contain particles which
exist near us but have not been discovered yet.
The term $I$ is an unknown interaction between
the observed system $L$ and the unknown system $E$.
Since the exterior system $E$ is assured to exist
by G\"odel's theorem, the interaction $I$ does not vanish:
In fact if $I$ vanished, then one could not know that
the observed system $L$ and the exterior system $E$ interacts
and hence could not know that the exterior system $E$ exists,
which contradicts G\"odel's theorem. By the same reason,
$I$ is not a constant operator:
\beq
I not = constant operator. \label{3}
\ene
For suppose it is a constant operator. Then
the systems $L$ and $E$ do not change how far or
near they are located because the interaction
between $L$ and $E$ is a constant operator.
Hence the observer cannot know that $E$ exists,

We now arrive at the following observation:
For an observer, the observable universe is a part $L$
of the total universe and it looks following the
Hamiltonian $H_L$, not following the total Hamiltonian $H$.
And the state of the system $L$ is described by a part
$\Psi(.,y)$ of the state $\Psi$ of the total universe,
where $y$ is an unknown coordinate of the system $L$ inside
the total universe, and $.$ is the variable controllable
by the observer, which we will denote by $x$.

3. Local Time Exists

Assume now, as is expected usually, that there is no
local time of $L$, i.e. that the state $\Psi(x,y)$
is an eigenstate of the local Hamiltonian $H_L$ for
some $y=y_0$:
\beq
H_L\Psi(x,y_0)=0. \label{4}
\ene
Then from \eq{1}, \eq{2} and \eq{4} follows that
\beq
&&0=H\Psi(x,y_0)
=H_L\Psi(x,y_0)+I(x,y_0)\Psi(x,y_0)+H_E\Psi(x,y_0)\nonumber\\
&&\ \hskip5pt=I(x,y_0)\Psi(x,y_0)+H_E\Psi(x,y_0). \label{5}
\ene
Here $x$ varies over the possible positions of the particles
inside $L$. On the other hand, since $H_E$ is the Hamiltonian
describing the system $E$ exterior to $L$, it does not
affect the variable $x$ and acts only on the variable $y$.
Thus $H_E\Psi(x,y_0)$ varies as a bare function $\Psi(x,y_0)$
insofar as the variable $x$ is concerned.
Equation \eq{5} is now written: For all $x$
\beq
H_E\Psi(x,y_0)=-I(x,y_0)\Psi(x,y_0). \label{6}
\ene
As we have seen in \eq{3}, the interaction $I$
is not a constant operator and varies when $x$
varies\footnote[2]{Note that G\"odel's theorem
applies to any fixed $y=y_0$ in \eq{3}. Namely,
for any position $y_0$ of the system $L$ in the
universe, the observer must be able to know
that the exterior system $E$ exists because
G\"odel's theorem is a universal statement
valid throughout the universe.
Hence $I(x,y_0)$ is not a constant operator
with respect to $x$ for any fixed $y_0$.},
whereas the action of $H_E$ on $\Psi$ does not.
Thus there is a nonempty set of points $x_0$
where $H_E\Psi(x_0,y_0)$ and $-I(x_0,y_0)\Psi(x_0,y_0)$
are different, and \eq{6} does not hold at such points
$x_0$. If $I$ is assumed to be continuous in the variables
$x$ and $y$, these points $x_0$ constitutes a set of
positive measure. This then implies that our assumption
\eq{4} is wrong. Thus a subsystem $L$ of the universe cannot
be a bound state with respect to the observer's Hamiltonian
$H_L$. This means that the system $L$ is observed as
a non-stationary system, therefore there must be observed
a motion inside the system $L$. This proves that the
time" of the local system $L$ {\it exists for the
observer} as a measure of motion, whereas the total
universe is stationary and does not have time."

4. A refined argument

(A rather technical part and is omitted.)

5. Conclusion

G\"odel's proof of the incompleteness theorem relies on the
following type of proposition $P$ insofar as the meaning is concerned:
\beq
P = P cannot be proved." \label{8}
\ene
Then if P is provable it contradicts P itself, and if P is not
provable, P is correct and seems to be provable. Both cases lead
to contradiction, which makes this kind of propositions undecidable
in a given formal theory.

This proposition reminds us the following type of self-referential
statements:
\beq
A person P says I am telling a lie." \label{9}
\ene
Both of this and proposition P in \eq{8} are non-diagonal statements
in the sense that both denies themselves. Namely the core of
G\"odel's theorem is in proving the existence of non-diagonal
elements" (i.e. propositions) in any formal theory that includes
number theory. By constructing such propositions in number theory,
G\"odel's theorem shows that any formal theory has a region exterior
to the knowable world.

On the other hand, what we have deduced from G\"odel's theorem in
section 2 is that the interaction term $I$ is not a constant operator.
Moreover the argument there implies that $I$ does not commute with
at least one of $H_L$ and $H_E$. For suppose that $I$ commutes with
both of $H_L$ and $H_E$. Then by spectral theory for
selfadjoint operators, $I$ is decomposed as $I=f(H_L)+g(H_E)$ for
some functions $f(H_L)$ and $g(H_E)$ of $H_L$ and $H_E$.
Thus $H$ is decomposed as a sum of mutually commuting operators:
$H=(H_L+f(H_L))+(H_E+g(H_E))$. Here the Hilbert space $\HH$ is
decomposed as a direct sum of two direct integrals:
\beq
\HH=\int^\oplus \HH_L(\lambda)d\lambda\oplus
\int^\oplus \HH_E(\mu)d\mu, \label{10}
\ene
where the first term on the RHS is the decomposition of $\HH$ with
respect to the spectral representation of $H_L$, and the second is
the one with respect to that of $H_E$. In this decomposition,
$H$ is decomposed as a diagonal operator:
$$H=\int^\oplus (\lambda+f(\lambda))d\lambda\oplus \int^\oplus (\mu+g(\mu))d\mu.$$
Namely the total Hamiltonian $H$ is decomposed into a sum of
mutually independent operators in the decomposition of the
total system into the observable and unobservable systems $L$
and $E$. This means that there are no interactions between $L$
and $E$, contradicting G\"odel's theorem as in section 2.
Thus $I$ does not commute with one of $H_L$ and $H_E$.
Therefore $I$ is not diagonalizable with respect to the direct
integral decomposition \eq{10} of the space $\HH$.

Now a consequence of G\"odel's theorem in the context
of the decomposition of the total universe into observable and
unobservable systems $L$ and $E$ is the following:

In the spectral decomposition \eq{10} of
$\HH$ with respect to a decomposition
of the total system into the observable and
unobservable ones, $I$ is non-diagonalizable.
In particular so is the total Hamiltonian
$H=H_L+I+H_E$.

Namely G\"odel's theorem yields the existence of non-diagonal
elements in the spectral representation of $H$ with respect to
the decomposition of the universe into observable and
unobservable systems. The existence of non-diagonal
elements in this decomposition is the cause that the
observable state $\Psi(.,y)$ is not a stationary state
and local time arises, and that decomposition is inevitable by
the existence of the region unknowable to human beings.

From the standpoint of the person P in \eq{9}, his universe needs
to proceed to the future for his statement to be decided true or
not, the decision of which requires his system an infinite time."
This is due to the fact that his self-destructive statement does
not give him satisfaction in his own world and forces him
to go out to the region exterior to his universe.
Likewise, the interaction $I$ in the decomposition above
forces the observer to anticipate the existence of the
region exterior to his knowledge. In both cases the unbalance
caused by the existence of the exterior region yields time.
In other words, time is an indefinite desire to reach the
balance that only the universe has.

References

\bibitem{Barbour} Julian Barbour, The End of Time,"
Weidenfeld & Nicolson, 1999.

\bibitem{G} K. G\"odel, On formally undecidable propositions of
Principia mathematica and related systems I, in Kurt G\"odel
Collected Works, Volume I, Publications 1929-1936," Oxford University
Press, New York, Clarendon Press, Oxford, 1986, pp.144-195,
translated from \"Uber formal unentsceidebare S\"atze der
Principia mathematica und verwandter Systeme I, Monatshefte
fur Mathematik und Physik, 1931.

\bibitem{CP} J. B. Hartle, Time and Prediction in
Quantum Cosmology, in Conceptual
Problems of Quantum Gravity," Einstein Studies Vol. 2,
Edited by A. Ashtekar and J. Stachel, Birkh\"auser,
Boston-Basel-Berlin, 1991.

\bibitem{K1} H. Kitada, Theory of local times,
Il Nuovo Cimento 109 B, N. 3 (1994), 281-302.
(http://xxx.lanl.gov/abs/astro-ph/9309051,
http://kims.ms.u-tokyo.ac.jp/time_I.tex).

\bibitem{K2} H. Kitada, Quantum Mechanics and Relativity
--- Their Unification by Local Time,
in Spectral and Scattering Theory,"
Edited by A.G.Ramm, Plenum Publishers, New York,
pp. 39-66, 1998. (http://xxx.lanl.gov/abs/gr-qc/9612043,
http://kims.ms.u-tokyo.ac.jp/time_IV.tex).

\bibitem{Ki-Fl1} H. Kitada and L. Fletcher, Local time and
the unification of physics, Part I: Local time, Apeiron
3 (1996), 38-45. (http://kims.ms.u-tokyo.ac.jp/time_III.tex,