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# Introduction to State (Discrete States)

"Coin States"
Coin States

Coins and dice are usually used in probability discussions as devices supposed to provide random results. Here they merely are examples of objects which have a small number of possible states.

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Consider a (mathematical) space in which in which every possible state of a system is represented by a point. (Configuration is a common synonym for state.)

Susskind's coin is an example of the simplest possible example of a system in such a space (leaving aside a one-sided coin). It is an imaginary (abstract) coin on the table. It is either heads (head side up) or tails (tail side up). This is the only fact about the coin that is relevant to anything.

This depicts the two sides of the actual imaginary coin:

{$$\Huge \style{color:red} \bigodot_\text{heads} \; \style{color:blue} \bigoplus_\text{tails}$$}

There is just one coin, but it has two sides. The totally imaginary coin is not the state space. The state space is represented as {$P$} below. ({$P$} is for ''phase space.")

"Coin States"
Coin States

Now we consider a system with laws of motion over stroboscopic time. Stroboscopic time is just an expression for discrete time, or in other words, For two times {$t_1, t_2,$}{$$\left\vert t_1 - t_2 \right\vert = n, \text{ for } n \in \left\{0,1,2,\dots\right\}.$$}

Rule 1

One possible law of motion is if the coin is heads it goes to heads and if it is tails it goes to tails.

{$$\left\{ \begin{matrix} \style{color:red} H \to \style{color:red} H \\ \style{color:blue} T \to \style{color:blue}T \end{matrix} \right.$$}

This is not especially interesting, but it does say something, namely that if we know the state of the coin at any time, we know its state for any time in the future. So if the initial state is {$\style{color:red} H,$} the history of the world will be {$\style{color:red} H, \style{color:red} H, \style{color:red} H, \dots,$} and if the initial state is {$\style{color:blue} T,$} the history of the world will be {$\style{color:blue} T, \style{color:blue} T, \style{color:blue} T, \dots.$} depending on the initial conditions.

Rule 2

Another possible law of motion is if the coin is heads it goes to tails and if it is tails it goes to heads.

{$$\left\{ \begin{matrix} \style{color:red} H \to \style{color:blue} T \\ \style{color:blue} T \to \style{color:red}H \end{matrix} \right.$$}

Now the history of the world will be {$\style{color:blue} T, \style{color:red} H, \style{color:blue} T, \style{color:red} H, \dots$} or {$\style{color:red} H, \style{color:blue} T, \style{color:red} H, \style{color:blue} T, \dots,$}

Rule 1 and Rule 2 are two possible laws of motion for the coin system. They are alternatives, not simultaneously true.

They can be expressed mathematically. {$\sigma$} is a variable name traditionally used for a two-valued system with typical values 1 and -1, so the rules may be summarized:

{\large \bf \begin{align} \text{Rule 1: } \sigma_{t+1} &= \sigma_t \\ \bf \text{Rule 2: } \sigma_{t+1} &= -\sigma_t \end{align}}

In other words, Rule 1 says whichever state the coin is in at time t, it is in the same state at time t+1, and Rule 2 says whichever state the coin is in at time t, it is in the opposite state at time t+1.

These rules, or laws of motion in this world, have been deterministic. They make it possible to predict the state at any time in the future. If the state is known at one time, the state at the next time can be predicted. They are also retrodictive in that if you know what the state is a one time, you know what the state was at the previous time. If you know where you are, you know with certainty where you are going, and you know where you came from with certainty.

Other possible rules can be imagined. One (illustrated as Bad Rule) is

{$$\left\{ \begin{matrix} \style{color:red} H \to \style{color:blue} T \\ \style{color:blue} T \to \style{color:blue}T \end{matrix} \right.$$}

And another (not illustrated) is:

{$$\left\{ \begin{matrix} \style{color:red} H \to \style{color:red} H \\ \style{color:blue} T \to \style{color:red} H \end{matrix} \right.$$}

Although these rules are deterministic, in that if you know the state at one time, you know the state in a future time, they are not retrospective. You cannot be sure what the state was at a previous time. These systems have lost information, and this violates the principle of conservation of information.

In sum, we had one object with two possible states. There were four possible rules (laws of motion) for the configuration, but two of them lost information which is not allowable.

Dice

#### Die Facts

• Die is the singular of dice.
• Opposite faces sum to 7.
• Western dice are "right handed." If you curl your fingers from 1 toward 2, your thumb points in the direction of 3. Asian dice are left-handed.

Now, we consider the states of an abstract die. We were not flipping the coin, and we are not rolling the die. For present purposes, how the die changes states is a deep and perhaps unknowable mystery. We are just observing the die in discrete, stroboscopic flashes of time. So first we describe the states of the die in a state space. The numbers 1 through 6 are used to label the observable states. So far as the state space is concerned, these numbers are just labels and have no computational implications. They could be labeled Fire, Air, Earth, Water, Holy, and Dark. They do not have any particular order in state space.

"Die States"
Die States

"Die Rule 1"
Die Rule 1

Now we can consider the laws of motion for the states of the die. The simplest is that everything stays the same. Nothing happens. If it was state 5 to begin with, it is state 5 the next time and it was state 5 the previous time.
{$$\left\{ \begin{matrix} 1 \to 1 \\ 2 \to 2 \\ 3 \to 3 \\ 4 \to 4 \\ 5 \to 5 \\ 6 \to 6 \end{matrix} \right. \tag{Die Rule 1}$$}

Usually my naming of the rules is completely arbitrary, but I have called this one Die Rule 1 because it has much in common with multiplying by 1. These states do not have values yet. Recall that the numbers 1 through 6 are just labels and do not necessarily have anything to do with whatever value we might assign or discover for the states. But if we had the value of the states at one time, we could discover the value of the states at the next time or the previous time by multiplying by 1. The state does not change, so its value does not change, and multiplying by 1 does not change the value. Moreover, it does not matter if we reverse the order of the arrows in the diagram or in the bracket definition of Dice Rule 1. However we look at it, it is still the same rule.

"Die States Hexagon"
Die Rule a

There are numerous possible rules (or laws of motion). One of them is to rotate around the states in the numeric order of their labels almost. The almost is because in the intergers, 1 does not follow 6, but in the rule state 1 does come after state 6.

{$$\left\{ \begin{matrix} 1 \to 2 \\ 2 \to 3 \\ 3 \to 4 \\ 4 \to 5 \\ 5 \to 6 \\ 6 \to 1 \end{matrix} \right. \tag{Die Rule a}$$}

It is important here to distinguish retrospection from reversal. Retrospection is looking back and reversal is going back. Retrospection is looking at the arrows backwards: if you are 1, you can see you came from 6, and see that to get to 6 you had to have been at 5 before that. Time reversal means turning the arrows around. The reversal of Dice Rule a is not the same rule as Dice Rule a. All that we are requiring of the laws of motion for now is that if you know where you are, you know where you are going and where you came from.

"Die Rule b"
Die Rule b

Die Rule b looks as if it ought to be very different from Die Rule a, but logically they are the same. This brings home the point that the numbers on the state diagram are just labels. The real imaginary observation we are making is the number of pips showing on the die. The pips on the die are the phenomena, and we have to answer to them. But there is nothing sacrosanct about the labels. We can rearrange the relation between the labels and the observed the values so that Die Rule b looks like Die Rule a.

We can do this by switching the labels 3 and 5 and then the labels 3 and 4.

If in observations the die really does show the values 1, then 2, then 5, then 3, then 4, then 6, and then continues to cycle through those values in the same order so that in Die Rule b the names of the states (1 through 6) really do correspond to the number of pips on the die, we can just use the label 5 for the observation that 4 pips are showing, the label 4 for the observation that 3 pips are showing, and the label 3 for the condition that 5 pips are showing, leaving labels 1, 2, and 6 to represent the observations of 1, 2, and 6 pips respectively, then we can display a phase space that looks exactly like Die Rule a. This has not changed the phenomena at all, but has only changed the way we label them.

dieLabel bLabel a
111
222
553
334
445
666

There are 6! (720) ways labels can be rearranged so that the state space diagrams look like Die Rule a. These do not change the phenomena or the laws of motion, they only change the labels. All these rules are perfectly good, although they are redundant. If you know the state at one time you know the next state uniquely and you know the previous state uniquely. Information is preserved. So there is no reason they would not be perfectly good laws of motion.

But there are other possible rules which are not equivalent to Dice Rule a.

"Die Rule c"
Die Rule c
"Die Rule d"
Die Rule d

Die Rule c and Die Rule d are not equivalent to Die Rule a, or to each other. No switching of labels can make them look like Die Rule a, but the point of them is that they are example of conservation laws.

A quantity that is preserved regardless of time is conserved. I have mentioned that a principle of physics is that information is conserved. Die Rule c and Die Rule d illustrate conservation laws. In both rules the quantity of being in the cycle 4 goes to 6, 6 goes to 5, 5 goes to 4 is preserved. So state 5 always goes to goes to 4 and always is preceded by state 6, but there is no way state 5 evolves to state 2 no matter how many steps ahead we look and never evolved from state 1 no matter how many steps we look back. Die rule a has two cycles and Die rule b has three cycles, and in both cases the quantity of belonging to a particular cycle is conserved.

"Conserve Evenness"
Conserving Evenness

There is no need to confine the consideration of state to cases in which there are a finite number of states.

There could be a infinite number of states. That does mean an infinite number of things, but could mean one thing capable of having an infinite number of states.

Consider an infinite line and a particle which can occupy any integer position on the line. There could be a law that whatever position the particle is in, it moves to the next position in the positive direction which could be expressed mathematically as {$n \to n + 1 .$} In other words, if the particle is at n, it next goes to n+1. An obvious conserved quantity is being an integer, but this is not saying much because there are only integer positions in the state space. Position 1.5 just does not exist in the state space.

But instead the law of motion could be that a particle must move to the second next position, {$n \to n + 2 .$} In this case, the conserved quantity is oddness or evenness. A particle that is in an even position must always go to an even position, and has always been in an even position. If a particle starts odd, it will always be odd, and always been odd.

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This is a student's notebook. I am not responsible if you copy it for homework, and it turns out to be wrong.