“Philosophical” Intermezzo

Abstract

Inasmuch as quantum mechanics is often claimed to have put the human subject back into science and to have discredited determinism, we review traditional philosophical notions such as idealism and realism, then discuss what determinism means. We also explain how probabilities are used in deterministic physical theories.

Appendices

Appendix

3.A The Law of Large Numbers

Consider first a finite set of real numbers $E=\{ a_k \}^n_{k=1}$ and a random variable $\omega$ taking values in E, with probability distribution $P(\omega =a_k)=p_k$, $k=1,\dots , n$, with $p_k \ge 0$, $\sum ^n_{k=1} p_k=1$. Then, take N independent variables $\omega _i$, $i=1,\dots , N$, with the same distribution as $\omega$, and define the frequency $\rho _N ({\Omega }, k)$ with which the variables $\Omega = (\omega _1, \dots , \omega _N)$ take the value $a_k$ by

$$\begin{aligned} \rho _N (\Omega , k)= \frac{\big |\big \{i= 1,\dots , N | \omega _i= a_k \big \}\big |}{N} = \frac{\sum _{i= 1}^{N} \delta ( \omega _i= a_k)}{N}, \end{aligned}$$

(3.A.1)

where | E | is the number of elements of the set E and $\delta$ the Kronecker delta. The set of frequencies $\{\rho _N (\Omega , k) \}^n_{k=1}$ is called the empirical distribution of the variables $\omega _i$, $i=1,\dots , N$. In the language of Sect. 3.4.4, and of statistical mechanics , the variables $\{ \rho _N (\Omega , k)\}_{k=1}^n$ define the macrostate of the system, while the variables $\Omega = (\omega _1, \dots , \omega _N)$ define its microstate . In the example of coin tossing in that subsection, we have $n=2$, $a_1=0$, $a_2=1$ and $\rho _N (\Omega , 1)= N_0/N$, $\rho _N (\Omega , 2)= N_1/N$.

Let $\mathcal{P}_N$ be the joint probability distribution of the variables $\omega _i$, $i=1,\dots , N$:

$$\begin{aligned} \mathcal{P}_N (\omega _1= a_1, \dots , \omega _N=a_N)= \prod _{i= 1}^{N} p_i. \end{aligned}$$

(3.A.2)

Then, the law of large numbers says that, $\forall \epsilon >0$,

$$\begin{aligned} \lim _{N\rightarrow \infty } \mathcal{P}_N (\max _{k=1, \dots , n} |\rho _N (\Omega , k)- p_k| \ge \epsilon )=0. \end{aligned}$$

(3.A.3)

To prove (3.A.3), let $A_k=\big \{ \Omega | |\rho _N (\Omega , k)- p_k| \ge \epsilon \big \}$, and write

$$\begin{aligned} \mathcal{P}_N (\max _{k=1, \dots , n} |\rho _N (\Omega , k)- p_k| \ge \epsilon ) \le \mathcal{P}_N (\cup _{k=1}^n A_k) \le \sum _{k= 1}^{n}\mathcal{P}_N (|\rho _N (\Omega , k)- p_k| \ge \epsilon ), \end{aligned}$$

(3.A.4)

then use Chebyshev’s inequality to get

$$\begin{aligned} \sum _{k= 1}^{n}\mathcal{P}_N (|\rho _N (\Omega , k)- p_k| \ge \epsilon )&\le n \max _{k=1, \dots , n} \mathcal{P}_N (|\rho _N (\Omega , k)- p_k| \ge \epsilon ) \end{aligned}$$

$$\begin{aligned}&\le n \frac{1}{\epsilon ^2} \max _{k=1, \dots , n} \mathcal{E}_N\big ([\rho _N (\Omega , k)- p_k]^2\big )\,, \end{aligned}$$

(3.A.5)

where $\mathcal{E}_N$ is the expectation with respect to $\mathcal{P}_N$. Using (3.A.1), we get

$$\begin{aligned} \mathcal{E}_N\big ([\rho _N (\Omega , k)- p_k]^2\big )\le \frac{1}{N^2}\sum _{i, j= 1}^{N}\Big [ \mathcal{E}_N \big ( \delta ( \omega _i= a_k) \delta ( \omega _j= a_k)\big )-p_k^2\Big ], \end{aligned}$$

(3.A.6)

where we used $\mathcal{E}_N \big ( \delta ( \omega _i= a_k)\big )= p_k$.

Now insert $\mathcal{E}_N \big ( \delta ( \omega _i= a_k)^2\big )= \mathcal{E}_N \big ( \delta ( \omega _i= a_k)\big )= p_k$ and $\mathcal{E}_N \big ( \delta ( \omega _i= a_k) \delta ( \omega _j= a_k)\big )= p_k^2$ for $i\ne j$ in (3.A.6). Since $p_k-p_k^2 \le 1$, we get

$$ \mathcal{E}_N\big ([\rho _N (\Omega , k)- p_k]^2\big )\le \frac{1}{N}. $$

Inserting this in (3.A.5), and combining (3.A.5) and (3.A.4), we get (3.A.3).

Thus, the law of large numbers says that, for N large, the empirical distribution $\{\rho _N (\Omega , k) \}^n_{k=1}$ of the variables $\omega _i$, $i=1,\dots , N$ is almost equal to the numbers $(p_k)^n_{k=1}$, with a probability close to 1. This gives a physical meaning to the probabilities $(p_k)^n_{k=1}$. For a single event, the probability of an outcome can be any number between 0 and 1, since it reflects partly our knowledge and partly our ignorance. But when we have N copies of the same random event, we can make statements about empirical distributions whose probabilities are close to 1, for N large, and which thus become near certainties.

One can even prove a stronger result (the strong law of large numbers ): the probability distribution (3.A.2) can be extended⁵⁷ to infinite sequences $\Omega = (\omega _1, \omega _2, \omega _3, \dots )$ and, for that probability distribution, denoted $\mathcal P$, there is a set of probability equal to 1 on which $\lim _{N \rightarrow \infty }\rho _N (\Omega _N, k)=p_k$, $\forall k= 1, \dots , n$, where $\Omega _N$ denotes the first N elements of $\Omega$.

Let us define a sequence of events $\Omega = (\omega _1, \omega _2, \omega _3, \dots )$ to be typical if, for every $k=1,\dots , n$, $\lim _{N \rightarrow \infty }\rho _N (\Omega _N, k)=p_k$. Then, the set of typical sequences has $\mathcal P$-probability 1.

Consider now a continuous variable $\omega \in {{\mathbb {R}}}$ with a probability density p(x): for $B\subset {{\mathbb {R}}}$, $P(\omega \in B)= \int _B p(x) dx$. Let $\Omega =(\omega _i)_{i=1}^N$ be N independent variables with identical distribution whose density is p(x). We define the empirical distribution of those N variables, for any (Borel) subset $B\subset {{\mathbb {R}}}\,$:

$$\begin{aligned} \rho _N ( \Omega , B)= \frac{|\{i= 1,\dots , N | \omega _i \in B \}|}{N} = \frac{\sum _{i= 1}^{N} \mathbbm {1}_{B} (\omega _i) }{N}, \end{aligned}$$

(3.A.7)

where $\mathbbm {1}_{B}$ is the indicator function of the set B.

Let $\mathcal{P}_N$ be the joint probability distribution of the variables $\omega _i$, $i=1,\dots , N$, whose density in ${{\mathbb {R}}}^N$ is $\mathcal{P}_N (x_1, \dots , x_N)= \prod _{i= 1}^{N} p (x_i)$. Then the law of large numbers says that, for any set B, and $\forall \epsilon >0$,

$$\begin{aligned} \lim _{N\rightarrow \infty } \mathcal{P}_N \Big ( |\rho _N ( \Omega , B)- P_B | \ge \epsilon \Big )=0, \end{aligned}$$

(3.A.8)

where $P_B= \int _B p(x) dx$. To prove this result, one argues as in (3.A.4)–(3.A.6), using the fact that $\mathcal{E}_N ( \mathbbm {1}_{B}^2 (\omega _i))= \mathcal{E}_N (\mathbbm {1}_{B} (\omega _i))= P_B$ and $\mathcal{E}_N ( \mathbbm {1}_{B} (\omega _i) \mathbbm {1}_{B} (\omega _j))= P_B^2$ for $i\ne j$.

For continuous variables, it is the family of functions $\rho _N (\Omega , B)$ which is the empirical distribution of the variables $\Omega =(\omega _i)_{i=1}^N$ and which defines the macrostate of the system, while its microstate is given by $\Omega =(\omega _i)_{i=1}^N$.

One can again strengthen this result by extending, as in the discrete case, the probability distribution $\mathcal{P}_N$ to a probability $\mathcal{P}$ defined on infinite sequences $\Omega = (\omega _1, \omega _2, \omega _3, \dots )$ and then considering the family of sets B of the form $]- \infty , z]$, for $z \in {{\mathbb {R}}}$. There is a set of probability $\mathcal{P}$ equal to 1 on which $\lim _{N \rightarrow \infty }\rho _N (\Omega _N, ]- \infty , z])=P_{]- \infty , z]}$, $\forall z \in {{\mathbb {R}}}$, where $\Omega _N$ denotes the first N elements of $\Omega$. The convergence is even uniform in z, by the Glivenko–Cantelli theorem (see, e.g., [490, p. 266]).

Since $P_{]- \infty , z]}= \int _{- \infty }^z p(x) dx$, one can consider p(x) as the empirical density distribution of the variables $\Omega = (\omega _1, \omega _2, \omega _3, \dots )$. One can also define a sequence of events $\Omega =(\omega _1, \omega _2, \omega _3, \dots )$ to be typical if $\lim _{N \rightarrow \infty }\rho _N (\Omega _N, ]- \infty , z])=P_{]- \infty , z]}$, $\forall z \in {{\mathbb {R}}}$. Then the set of typical sequences has $\mathcal{P}$-probability 1. So, just as in the discrete case, the empirical distribution $\rho _N (\Omega , B)$, for B an interval, is, for typical configurations of many variables, close to the probability $P_B$ of a single variable belonging to B.