10.7: Leibniz=Clarke Correspondence: What is space-time?


The problems ("What is space?" and "What is time?") are the most important in modern science as well as the traditional philosophies. In this section, we give my answer to this problem.

10.7.1: "What is space?" and "What is time?"


10.7.1.1: Space in quantum language ( How to describe "space" in quantum language)



In what follows, let us explain "space" in measurement theory (= quantum language ). For example, consider the simplest case, that is,
$(A):$ $ \qquad \qquad \mbox{"space"=${\mathbb R}_q$( one dimensional space)} $
Since classical system and quantum system must be considered, we see
$(B):$ $ \left\{\begin{array}{ll} \mbox{(B$_1$): a classical particle in the one dimensional space ${\mathbb R}_q$} \\ \\ \mbox{(B$_2$): a quantum particle in the one dimensional space ${\mathbb R}_q$} \end{array}\right. $

In the classical case, we start from the following state:

\begin{align} \mbox{ $(q,p)$ $=$ ("position", "momentum") $\in {{\mathbb R}_q \times {\mathbb R}_p }$ } \end{align}

Thus, we have the classical basic structure:

$(C_1):$ $ \qquad [ C_0({\mathbb R}_q \times {\mathbb R}_p) \subseteq L^\infty ( {\mathbb R}_q \times {\mathbb R}_p) \subseteq B( L^2 ( {\mathbb R}_q \times {\mathbb R}_p) ] $

Also, concerning quantum system, we have the quantum basic structure:

$(C_2):$ $ \qquad [ {\mathcal C}( L^2 ( {\mathbb R}_q ) \subseteq B( L^2 ( {\mathbb R}_q ) \subseteq B( L^2 ( {\mathbb R}_q ) ] $

Summing up, we have the basic structure

$(C):$ $ [{\mathcal A} \subseteq \overline{\mathcal A} \subseteq B(H)] \left\{\begin{array}{ll} \mbox{(C$_1$): classical $ [ C_0({\mathbb R}_q \times {\mathbb R}_p) \subseteq L^\infty ( {\mathbb R}_q \times {\mathbb R}_p) \subseteq B( L^2 ( {\mathbb R}_q \times {\mathbb R}_p) ] $ } \\ \\ \mbox{(C$_2$): quantum $ [ {\mathcal C}( L^2 ( {\mathbb R}_q ) \subseteq B( L^2 ( {\mathbb R}_q ) \subseteq B( L^2 ( {\mathbb R}_q ) ] $} \end{array}\right. $

Since we always start from a basic structure in quantum language, we consider that

\begin{align} & \mbox{ How to describe "space" in quantum language} \nonumber \\ \Leftrightarrow & \mbox{ How to describe [(A):space] by [(C):basic structure] } \tag{10.29} \end{align}

This is done in the following steps.

Assertion 10.17 [How to describe "space" in quantum language]
$(D_1):$ Begin with the basic structure: \begin{align} [{\mathcal A} \subseteq \overline{\mathcal A} \subseteq B(H)] \end{align}

$(D_2):$ Next, consider a certain commutative $C^*$-algebra ${\mathcal A}_0 (=C_0(\Omega ))$ such that

\begin{align} {\mathcal A}_0 \subseteq \overline{\mathcal A} \end{align}

$(D_3):$ Lastly, the spectrum $\Omega$ $(\approx {\frak S}^p ({\mathcal A}_* ) )$ is used to represent "space".
\end{Assertion}
For example,
$(E_1):$ in the classical case (C$_1$):

\begin{align} [ C_0({\mathbb R}_q \times {\mathbb R}_p) \subseteq L^\infty ( {\mathbb R}_q \times {\mathbb R}_p) \subseteq B( L^2 ( {\mathbb R}_q \times {\mathbb R}_p)) ] \end{align}

we have the commutative $C_0({\mathbb R}_q )$ such that

\begin{align} C_0({\mathbb R}_q ) \subseteq L^\infty ({\mathbb R}_q \times {\mathbb R}_p ) \end{align}

And thus, {we get the space ${\mathbb R}_q$} as mentioned in (A)

$(E_2):$ in the quantum case (C$_2$):

\begin{align} [ {\mathcal C}( L^2 ( {\mathbb R}_q ) \subseteq B( L^2 ( {\mathbb R}_q )) \subseteq B( L^2 ( {\mathbb R}_q )) ] \end{align}

we have the commutative $C_0({\mathbb R}_q )$ such that

\begin{align} C_0({\mathbb R}_q ) \subseteq B( L^2 ( {\mathbb R}_q )) \end{align}

And thus, {we get the space ${\mathbb R}_q$} as mentioned in (A)

10.7.1.2: Time in quantum language ( How to describe "time" in quantum language)


In what follows, let us explain "time" in measurement theory (= quantum language ). This is easily done in the following steps.
Assertion 10.18 [How to describe "time" in quantum language]
$(F_1):$ Let $T$ be a tree. (Don't mind the finiteness or infinity of $T$. cf. Chapter 14 .) For each $t \in T$, consider the basic structure:

\begin{align} [{\mathcal A}_t \subseteq \overline{\mathcal A}_t \subseteq B(H_t)] \end{align}

$(F_2):$ Next, consider a certain linear subtree $T' (\subseteq T)$, which can be used to represent "time".
\end{Assertion}


10.7.2: Leibniz-Clarke Correspondence


The above argument urges us to recall Leibniz-Clarke Correspondence (1715--1716:), which is important to know both Leibniz's and Clarke's (=Newton's) ideas concerning space and time.

$(G):$ [The realistic space-time]
Newton's absolutism says that the space-time should be regarded as a receptacle of a "thing." Therefore, even if "thing" does not exits, the space-time exists.
On the other hand,
$(H):$ [The metaphysical space-time]
Leibniz's relationalism says that
$(H_1):$ Space is a kind of state of "thing".
$(H_2):$ Time is an order of occurring in succession which changes one after another.


Therefore, I regard this correspondence as

\begin{align} \underset{\mbox{ (realistic view)}}{\fbox{Newton ($\approx$ Clarke)}} \quad \underset{\mbox{v.s.}}{\longleftrightarrow} \quad \underset{\mbox{ (linguistic view)}}{\fbox{Leibniz}} \end{align}

which should be compared to

\begin{align} \underset{\mbox{ (realistic view)}}{\fbox{Einstein}} \quad \underset{\mbox{v.s.}}{\longleftrightarrow} \quad \underset{\mbox{ (linguistic view)}}{\fbox{Bohr}} \end{align}

(also, recall Note 4.4).

$\fbox{Note 10.6}$ Many scientists may think that
$\bullet$ Newton's assertion is understandable, in fact, his idea was inherited by Einstein. On the other, Leibniz's assertion is incomprehensible and literary. Thus, his idea is not related to science.
However, recall the classification of the world-description (Figure 1.1 in $\S$1.1):




$ \left\{\begin{array}{ll} \mbox{①}: \underset{\scriptsize{\mbox{ (realistic world view)}}}{\mbox{Newton, Clarke}} & \cdots \overset{\scriptsize{\mbox{ (space-time in physics)}}}{ \underset{\scriptsize{\mbox{ "What is space-time?"}}}{ \fbox{realistic space-time}} } \quad \qquad \mbox{(successors: Einstein, etc.)} \\ \\ \mbox{②}: \underset{\scriptsize{\mbox{ (linguistic world view)}}}{\mbox{Leibniz}} & \cdots \overset{\scriptsize{\mbox{ (space-time in measurement theory)}}}{ \underset{\scriptsize{\mbox{ "How should space-time be represented?"}}}{ \fbox{linguistic space-time}} } \;\; \mbox{(i.e., spectrum, tree)} \end{array}\right. $





in which Newton and Leibniz respectively devotes himself to ① and ②. Although Leibniz's assertion is not clear, we believe that
$\bullet$ Leibniz found the importance of "linguistic space and time" in science,

$\qquad \quad $Fig. 1.1: The history of world-descriptions
Also, it should be noted that
$(\sharp):$ Newton proposed the scientific language called Newtonian mechanics,
on the other hand,
Leibniz could not propose a scientific language


$\fbox{Note 10.7}$ I want to believe that "realistic" vs. "linguistic" is always hidden behind the greatest disputes in the history of the world view. That is,

\begin{align} \underset{\mbox{ }}{\fbox{$\mbox{realistic world view}$}} \quad \underset{\mbox{v.s.}}{\longleftrightarrow} \quad \underset{\mbox{ (idealistic)}}{\fbox{$\mbox{linguistic world view}$}} \end{align}

For example,

The realistic world view vs the linguistic world view

It is usally said that the Problem of universals is not easy to understand. The reason is due to the fact that the two problems ( i,e., " Dualism (i.e., Trialism by Thomas Aquinas)" and "realistic view (Nominalisme) or linguistic view (Realismus)" in the above table") were simultaneously discussed and confused in the history.


$\fbox{Note 10.8}$ The space-time in measuring object is well discussed in the above. However, we have to say something about "observer's time". We conclude that observer's time is meaningless in measurement theory as mentioned the linguistic interpretation in Chap. 1. That is,
$(\sharp_1):$ "When and where does an observer take a measurement?" is nonsense. If it is meaningful, the pardox of Wigner's friend appears.
$(\sharp_2):$ Therefore, there is no tense (present, past, future) in sciences.
Thus, some may recall
McTaggart's paradox: "Time does not exist"
Although McTaggart's logic is not clear, we believe that his assertion is the same as "Subjective time (e.g., Augustinus' times, Bergson's times, etc. ) does not exist in science". If it be so,
$(\sharp_3):$ McTaggart's assertion as well as Leibniz' assertion are one of the linguistic interpretation.
After all, we conclude that
$(\sharp_4):$ the cause of philosophers' failure is not to propose a language.
Talking cynically, we say that
$(\sharp_5):$ Philosophers continued investigating "linguistic interpretation" (="how to use Axioms 1 and 2") without language (i.e., Axiom 1(measurement:$\S$2.7) and Axiom 2(causality:$\S$10.3)).