**Hitoshi Kitada** (*hitoshi@kitada.com*)

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

**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ]**Next message:**Matti Pitkanen: "[time 945] Re: [time 944] Goedel's incompleteness implies the existence of time"**Previous message:**Hitoshi Kitada: "[time 943] Re: [time 938] RE: [time 937] Does time really exist?"**Next in thread:**Matti Pitkanen: "[time 945] Re: [time 944] Goedel's incompleteness implies the existence of time"

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

available also at

http://www.kitada.com/time_VI.tex (the link is not yet made in index.html)

A key is Goedel's incompleteness theorem, which assures the existence of

(local) time.

Hitoshi Kitada

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,

contradicting G\"odel's theorem.

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,

http://www.freelance-academy.org/).

\bibitem{Ki-Fl2} H. Kitada and L. Fletcher,

Comments on the Problem of Time.

(http://xxx.lanl.gov/abs/gr-qc/9708055,

http://kims.ms.u-tokyo.ac.jp/time_V.tex).

- application/x-tex attachment: time_VI.tex

**Next message:**Matti Pitkanen: "[time 945] Re: [time 944] Goedel's incompleteness implies the existence of time"**Previous message:**Hitoshi Kitada: "[time 943] Re: [time 938] RE: [time 937] Does time really exist?"**Next in thread:**Matti Pitkanen: "[time 945] Re: [time 944] Goedel's incompleteness implies the existence of time"

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