Disclaimer:

I don’t have any theoretical physic degree. The posts below are purely my personal opinion and does not reflect any contemporary academic view. Read it for your own amusement.

As a followup from my article in 2020: Do information entropy closely related with physical thermodynamics?, I want to understand more about time.

The idea that time was related with entropy has long since discussed. In thermodynamics, entropy is a quantity that can only increase, so it is naturally irreversible. This is understood as The Second Law of Thermodynamics.

This idea is quite different from the usual equation or formula in mechanics. In mechanics, mostly all of the Physics equation has time symmetry. Meaning, the same equation can be used be it forward in time or backwards. However, only Second Law of Thermodynamics implies a certain direction.

In the pursuit of trying to understand, what generates time, I’m going to try to start from information principle.

Information entropy and mutual understanding

Consider two entities in a closed systems, Alice and Bob. It is humanly familiar to start using human names, but if we want to build bottom up, we can’t do it using human analogy. Human are complex. We need to start from simpler structure. So let’s say Alica and Bob as a hypothetical blackbox structure.

Both Alice and Bob were devices that can communicate with each other. There is no notion of space and time in the beginning. But we accept a notion of communication. These devices can send messages or signals to each other. They can also “records” previous messages. However, there is no concept of time defined here. Only “communication” and “recording”.

We start with the following postulate:

1. Each device can send signal/information, noise free, one at a time.
2. Each device can send signal/information instantaneously
3. Each device can record messages sent and received as “information”
4. Each device will keep communicating with each other, until mutual information has been achieved

With these 4 postulates, we are going to arrive at a concept of continuous and discrete time as independent parameters to track order of events and evolution of information.

We now described the interaction. First let’s take a look at the concept of clock, albeit the classical one.

Throughout history, human invented clock based on different definition. From clock, we invented the notion of time to track events.

The simplest original clock is using astronomical bodies, such as movements of sun, moon, and stars. However, none of this produces linear time. There always a need to some adjustment. To illustrate, we can divide a day into 24 sections based on Sun’s movement. However, at some of the time, the sections has different length. For example, daytime in summer has different length compared with winter. Also, seasons changes over the year. Not to mention that the total number of days also changes in a year. Essentially we defined sections of times based on periodicity of something (in this case, astronomical bodies).

Later on, people invented pendulum clock. The length of time is decided by periodicity of pendulum movements.

Then, people began to standardize clock using atomic oscillation.

As we noticed, we keep making smaller things to define “a second”, in order to make time the most independent definition from all. This is because time is used as the independent parameter to describe other physical quantities.

Things however become a little bit wierd when Relativistic time were introduced. It turns out that time (and space), is a byproduct of gravitational interaction. So geometrically, gravity caused spacetime to count/measure differently depending on the observer. Which means, the definition of “proper time” were reintroduced as the time shown by a clock that is locally within the same geometrical/gravitational coordinate with the observer. Then clock became relative.

From there, I have trouble understanding the concept of time in Quantum Mechanics. Time is usually explained with Schrödinger Equation or SE. But in this SE, time is described as an independent parameter that affects the Wave function. However it doesn’t tells us what and how wave function needs to be described, and why it was affected by time.

Not to mention that SE is usually referred to for the Non-relativistic condition. For the relativistic equation, Physicist uses a slightly different approximation.

Back to topic at hand. We don’t want to start by using oscillation directly. We are going to use probability to generate time.

We are going to treat Alice and Bob as a closed local systems. That means, relativistic interaction will be ignored.

Postulate 1 implies that energy is conserved in this systems.

Postulate 2 implies that we ignore relativistic speed limit because the concept of speed doesn’t exists yet, since concept of time doesn’t exists.

Postulate 3 implies that to record information, we need two states. “Send” and “Receive”.

The interaction is as follows. Now we switch our PoV to Alice.

Alice keeps a “counter” (implementing counter itself needs resources and energy, but we are going to skip this explanation until later). Assume Alice has ideal counter that can always count to arbitrary integer number, even if it is big. She will need one counter for each states. Each time Alice sends message to Bob, she counts 1 for “Send” state. Each time Alice receive message from Bob, she counts 1 for “Receive” state.

It is important to note, that there is no concept of time. So even if hypothetically there is another clock that we have, and it measures delay between Send and Receive. For Alice, the delay between Send and Receive is non-existent. Because for her, time doesn’t tick until she either send or receive. Basically, she can only recognize changes in information, if the counter moves. If it doesn’t move, she essentially doesn’t experience “her time” ticking.

In other words, for Alice, each consecutive Send/Receive events were spaced in equal sections.

We now assume that the signal she sent to Bob is essentially an energy package.

To reduce the model even further to the most simplest model. We assume that we are using the smallest non-zero energy package possible, that can’t be divided further into several package. By this, we ensure that Alice can only send single package one at a time and can only receive one from Bob, because postulate 1 implies that the system’s energy is conserved, so that the total doesn’t change.

Essentially, Alice and Bob plays ball catching.

This solves our counter issue, because both Alice and Bob now only needs one slot of energy to store this.

Finally for postulate 4, assume that Alice has no information at all about Bob (and how many Bob there is in existence). She only knows that someone is giving her a package of energy after she sends out one. So eventually, Bob and Alice will understand that there only exists one other recipient in the system, based on increasing mutual information each of them have.

Now that we established the interaction, let’s do some simulation.

Simulating order of events

Alice has 0 information at the beginning. Assume that she holds no package. The probability of her holding a package is 0. Inversely, the probability of her not holding a package is one (a certainty). The average of information (the entropy) is 0.

Then she had one. The probability of her holding a package now becomes $\frac{1}{2}$. The probability of her not holding a package is also $\frac{1}{2}$. Because there are only two possible state she was in. The average of information is now $\ln(2)$. It increases.

By now Alice will realize that her entropy is at max, because so far she can only “Send” and “Receive” package. It only covers two possibilities. So from her PoV, the chain of events is only two events at max. She doesn’t experience time when she keeps holding the package. But she experienced time if she do something, even though there are only two possible action she can take. So where does the time came in?

My understanding is that Alice perceive the time abstractly. For this kind of interaction, there are two options for her to “perceive time”. The first time, is to accept that she has discrete cyclic time. Which means her clock is a type of clock that tick-tocking from one state to the other. After each cycle, the information is destroyed.

The second time, is to develop a continuous independent definition of “time”. Which means her clock can be a real number, rather than integer, and it can describe how many cycle she already has.

A discrete time evolution

For the first approach, notice that Alice developed a discrete probability function $p(x)$ where $X$ is the random number variable that corresponds to the state she was in. We only have two. For a mathematical convenience later on, let’s use $X=1$ as Send event and $X=-1$ as Receive event. Since Alice is unable to differentiate the gap between event, it means for her that the probability or frequency between these two states are equally likely (she has no other choice to compare to). Thus $p(x)=\frac{1}{2}$ is a discrete uniform distribution.

The entropy $H(p)$, is essentially an expectance value, the average information: $E[I(p)]$. Enumerating it into formula becomes:

$$\begin{align*} H(p) &= E[I(p)] \\ &=-\sum_X \, p(X) \, \ln(p(X)) \\ &= -\frac{1}{2} \, \ln(\frac{1}{2}) -\frac{1}{2} \, \ln(\frac{1}{2}) \\ &= \ln(2) \end{align*}$$

Since it is the max possible for Alice in this given condition, the entropy only evolves by jumping from $0$ to $\ln(2)$. A discrete jump. Thus Alice perceive a discrete time. Let’s call this notion of time as $N$. In other words, $N$ is a value that generates different probability distribution. We can write it as $p(X, N)$.

A continuous time evolution

We can develop a notion of continuous time by introducing a continuous time parameter. One way to do that is to realize that Probability is a normalized function. It guarantees to converge. The probability that Alice measures is a discrete probability. However, by mathematical theorem, any square integrable function will have unique mapping to another function via Fourier Transform. It means, this distribution will have a uniquely corresponding function in its dual Fourier Domain. Moreover, since the probability function is a discrete distribution. The corresponding Fourier transform is a continuous distribution.

In probability theory the Fourier Transform of a probability distribution is called Characteristic Function. It is defined by the expectance of a certain operation $\varphi(t) = E[e^{itX}]$ this is essentially the same as Fourier Transform that usually been described as integral transform.

In this situation, since $X$ is countable sets of states. We can enumerate it to build the characteristic function. We apply that Send event is $X=1$ and Receive event is $X=-1$.

$$\begin{align*} \varphi(t) &= E[e^{itX}] \\ &= p(X=1) \, e^{it\cdot 1} + p(X=-1) \, e^{it\cdot -1} \\ &= \frac{1}{2}\, e^{it} + \frac{1}{2}\, e^{-it} \\ &= \frac{e^{it}+e^{-it}}{2} \\ &= \cos(t) \end{align*}$$

The characteristic function eventually can be rewritten as simply $\varphi(t) = \cos(t)$. Since both characteristic function and its probability density function is a one to one mapping, we can see that the evolution in $t$ will imply the same entropy changes in its original distribution.

Do we have justification to treat $t$ as time?

Here’s one way to look at it.

We have two different ways on observing the interaction between Alice and Bob. The original one we propose, is to measure the energy difference, which is essentially a quantum phenomenon, since the energy package is quantized (has non-zero discrete value). This is like directly detecting the energy.

The other way is to use $t$ from the characteristic function as the generator for the frequency of the distribution. This is like sampling the distribution.

If we are using $t$, we have the characteristic function $\varphi(t)$ behaves as the generator of the sampling. As an example, we can generate the observed frequency of Send event, by reverse transforming with the given parameter. Notice that Send events, correspond to the random variable $X=1$. We are going to use this parameter.

$$\begin{align*} p(X) &= \frac{1}{2\pi}\int_0^{2\pi} \varphi(t) \, e^{-itX} \, dt \\ p(X=1) &= \frac{1}{2\pi}\int_0^{2\pi} \cos(t) \, e^{-it} \, dt \\ &= \frac{1}{2\pi} \left[ \frac{t}{2} + \frac{i e^{-i2t}}{4} \right]_0^{2 \pi} \\ &= \frac{1}{2} \end{align*}$$

We got the same frequency of the probability.

A characteristic function, however, is an abstract quantity not directly related with the measurement itself. It is however refer to the same collection of events. So, if we truly perceive the variable $t$ in the characteristic function as literally time, we have this weird unintuitive notion that the span of the integrals directly affecting the frequency measurement!

The longer the span of the integral, the more precise the frequency measurement will be. More over, this notion of time is independent, continuous, and linear as opposed to $p(X, N)$. It is still possible for Alice to experience these as her time definition. Suppose that she has an abstract clock to measure whether she is in a Send or Receive state, she must simultaneously evaluate the characteristic function as the continuous time in her clock ticking. Between $0$ to $2\pi$, the frequency measurement is not entirely real, and it contains imaginary parts. Check these out:

$$\begin{align*} p(X) &= \frac{2}{\pi}\int_0^{\frac{\pi}{2}} \varphi(t) \, e^{-itX} \, dt \\ p(X=1) &= \frac{2}{\pi} \left[ \frac{t}{2} + \frac{i e^{-i2t}}{4} \right]_0^{\frac{\pi}{2}} \\ &= \frac{2}{\pi} \left[ \frac{t}{2} + \frac{i e^{-i2t}}{4} \right]_0^{\frac{\pi}{2}} \\ &= \frac{2}{\pi} \left[ \frac{\pi}{4} - \frac{i}{2} \right] \\ &= \frac{1}{2} - \frac{i}{\pi} \end{align*}$$

The real part is essentially the same frequency but the imaginary part exists. Which means, we can’t interpret the probability distribution as it is. This way, even though we generate a new parameter $t$, the result is not entirely physical, in the meaning that imaginary numbers can’t be directly observed.

Adding physical constraint to our models

Although we are able to generate a notion of “continuous time” for the duration of our sampling, we still have several problem when transforming back from characteristic function to the distribution. Our goal is to want a local independent definition of time, that can generate the event distribution.

My interpretation in this is to consider that the act of observation in this scenario is basically a two-way interaction. if Alice and Bob pass around energy as the signals/information medium, each of them maintain their own probability distribution. These distribution doesn’t necessarily the same. In fact, it must be the other way around. If the distribution is the same, then we can conclude that Alice and Bob is in a closed systems, because they were only interacting with each other.

Suppose Alice’s distribution of events is symbolized as $p(X)$, and Bob’s distribution as $p(Y)$. Then we have a conditional probability: $p(X,Y)$ meaning the probability of $X$ and $Y$ happens at the same time.

Assuming that Alice and Bob is only interacting locally, and notion of spacetime doesn’t exists yet. Both $p(X)$ and $p(Y)$ needs to be symmetric. Meaning, it doesn’t matter if we swap the observer, both must have observed the same local events. Thus $p(X,Y)=p(X)\,p(Y)$, and it implies that from Bob and Alice’s PoV, the probability distribution were independent of each other.

If we enforce the condition that an interaction must happen between Sender and Receiver, and it can’t be only one way. We have a really interesting consequences. $p(X,Y)$ is the joint probability, so the state is a combination of random variable of $X$ and $Y$, so in this case, there are 4 possible combinations. But really, if Alice is sending a message, then Bob must receive the message. It can’t be that both are receiving or both are sending. So it must be either $p(X=1,Y=-1)=\frac{1}{2}$ or $p(X=-1,Y=1)=\frac{1}{2}$, while $p(X=1,Y=1)=0$ and $p(X=-1,Y=-1)=0$. Because the total probability of $p(X,Y)$ must summed up to 1.

The problem is, we stated that $p(X)$ and $p(Y)$ must be symmetric.

In the classical sense, if it were an analog of putting a ball in two boxes. As an outsider, it is easy for us to conclude $p(X=1,Y=-1)=\frac{1}{2}$. But for Alice, the value of $p(X)$ can’t be evaluated, because it is tied with the information from $p(Y)$. From Alice, point of view, it is a certainty that $p(X)$ is a uniform distribution with $\frac{1}{2}$ frequency each. But that is only because Bob’s distribution is a certainty from Alice PoV. If Alice observed her state to be $X=1$ (Sending) with probability $\frac{1}{2}$, then it is a certainty that Bob’s state is $Y=-1$ (Receiving). It would mean the classical joint probability is correct $p(X,Y)=p(X)\,p(Y)=\frac{1}{2}\,1$.

But to satisfy the symmetric nature of physical equation, we are not allowed to evaluate $p(X)$ yet from Alice PoV.

If we borrow the concept from geometry, this is when the concept of linear algebra and basis come into play.

Remember that in geometry, the notion of product (multiplication) is provided by the distance function. That is why in geometry we have inner product as a measure of distance. The distance is measured according to the basis being used in the local geometry.

However, we have a problem here in quantum level because the notion of basis itself doesn’t exists. There is no spatial or temporal coordinate to measure into. We need an abstract basis. Something that is independent with the spatial/temporal dimension that we want to observe.

Neatly, probability function is a converged density, so it has a Fourier dual that is also converging. We then extend the idea that, what if the quantity $p(X)$ doesn’t necessarily mean “a distribution” in classical sense? What if it is just a function that is constructed from linear combination in the Fourier domain? As long as the interaction didn’t happen yet (the probability hasn’t been evaluated), $p(X)$ should take a form that is symmetric with $p(Y)$.

The simplest route we can take now, is by treating both $p(X)$ and $\varphi(t)$ as a function in the same Fourier space. So, we allow both to be constructed from a complex number with Fourier basis as its linear combination.

One important consequence of using Fourier Transform with both function and its dual being square integrable, is that we can use Plancherel Theorem. If we choose this definition of Plancherel Theorem:

$$\begin{align*} \int_{-\infty}^\infty |p(X)|^2 \, dx = \int_{-\infty}^\infty |\varphi(t)|^2 \, dt \end{align*}$$

They were both equal in length! So if we treat $|p(X)|^2$ as a distribution, then we can also treat $|\varphi(t)|^2$ as a distribution, simply because it sums up to 1.

From here, the entropy of the observation is $E[I(X,Y)]$ can be broken down to:

$$\begin{align*} H(X,Y)&=-\int_{-\infty}^\infty |p(X)|^2 \, \ln(|p(X)|^2)\, dx \end{align*}$$

As we can see, if the entropy were maximized, it implies that $p(X)$ and $p(Y)$ were independent and symmetric distribution. The entropy must evolve to a maximum value as Alice gain more information from her surrounding.

In its dual space, since we can treat $|\varphi(t)|^2$ as distribution, we can also quantify some notion of entropy in this domain:

$$\begin{align*} H(t)&=-\int_{-\infty}^\infty |\varphi(t)|^2 \, \ln(|\varphi(t)|^2)\, dt \end{align*}$$

If we treat $t$ as time parameter, notice that $H(t)$ is actually the action. If we want the entropy of $H(t)$ to evolve physically, we can apply the stationary action principle, in which entropy increase by moving in the path min/maxed by $t$. We can then conclude the term inside the integral is the Lagrangian.

Applying the Euler Lagrangian method, assuming some sort of constraints exists:

$$\begin{align*} \frac{d}{d\tau}\left(\frac{\partial L}{\partial \varphi}\right) &= \lambda \\ \frac{d}{d\tau}\frac{\partial}{\partial \left|\varphi\right|} \left( -|\varphi(t)|^2 \, \ln(|\varphi(t)|^2) \right)&= \\ \frac{d}{d\tau}\left( -2\left|\varphi\right|\,\ln(|\varphi|^2) - |\varphi|^2 \, \frac{2 \left|\varphi\right|}{|\varphi|^2} \right) &= \\ \frac{d}{d\tau}\left( 2\left|\varphi\right|\,\ln(|\varphi|^2) + 2 \left|\varphi\right| \right) &= -\lambda \\ \left|\varphi\right| \, \left( \ln(|\varphi|^2) + 1 \right)&= - \frac{\lambda \tau}{2} + C \end{align*}$$

We arrive at an interesting results. How we define the function constraint $\lambda$ defines how $\varphi$ works for some new independent parameter $\tau$. We can treat this new parameter as the proper time of this system.

The simplest constraint is if $\lambda = 0$, which means the Lagrangian is free to move, as long as the entropy action is stationary. From there, we derive:

$$\begin{align*} \left|\varphi\right| \, \left( \ln(|\varphi|^2) + 1 \right)&= - \frac{\lambda \tau}{2} + C \\ \left|\varphi\right| \, \left( \ln(|\varphi|^2) + 1 \right)&= C \\ \end{align*}$$

Suppose that $p(X)$ (which is now, not the probability density), is a function with Fourier basis with $|p(X)|^2$ as the probability density of the event. We are going to use the delta function $\delta(x)$ as a basis. Since we have two events for $X$, we can define:

$$\begin{align*} |p(x)|= \frac{\delta(x-1)}{\sqrt{2}} + \frac{\delta(x+1)}{\sqrt{2}} \end{align*}$$

The intention is clear, when we try to calculate the squared norm.

$$\begin{align*} |p(x)| &= \frac{\delta(x-1)}{\sqrt{2}} + \frac{\delta(x+1)}{\sqrt{2}} \\ |p(x)|^2 &= \frac{\delta(x-1)}{\sqrt{2}} \frac{\delta(x-1)}{\sqrt{2}} + \frac{\delta(x+1)}{\sqrt{2}}\frac{\delta(x+1)}{\sqrt{2}} + 2 \frac{\delta(x-1)}{\sqrt{2}} \frac{\delta(x+1)}{\sqrt{2}} \end{align*}$$

The delta basis has important property, that it is zero everywhere except at $\delta(0)$. This implies that $\delta(x-1) \delta(x+1)$ is always zero, because $(x-1)$ and $(x+1)$ cannot both be zero at the same time. The above expression can then be simplified:

$$\begin{align*} |p(x)|^2 &= \frac{\delta(x-1)^2}{2} + \frac{\delta(x+1)^2}{2} \end{align*}$$

So, the above expression works for our distribution which is $|p(X=1)|^2=\frac{1}{2}$ and $|p(X=-1)|^2=\frac{1}{2}$

Then the Fourier transform of $|p(x)|$:

$$\begin{align*} |p(x)| &= \frac{\delta(x-1)}{\sqrt{2}} + \frac{\delta(x+1)}{\sqrt{2}} \\ \mathcal{F}\left\{|p(x)|\right\} &= \mathcal{F}\left\{\frac{\delta(x-1)}{\sqrt{2}}\right\} + \mathcal{F}\left\{\frac{\delta(x+1)}{\sqrt{2}}\right\} \varphi(t) &= \frac{e^{-i\tau t}}{\sqrt{2}} + \frac{e^{i\tau t}}{\sqrt{2}} \\ &= \sqrt{2} \cos(\tau t) \end{align*}$$

With this interpretation, the parameter $t$ can be thought of as the local clock “experienced” by Alice (or Bob) in this systems. Whenever the clock reached a full cycle, new information gained by Alice, thus changing the information entropy. This solves our previous issue when transforming back this time function back to event probability, but now we beg another different question.

Remarks

If we treat $H(t)$ as information entropy over time, there is two possible way to observe physical events. Track the probability of the event, then measure the entropy of the Fourier Transformed function (the $\varphi(t)$). Or treat the time entropy as “Action” then from the Lagrangian predict the $\varphi(t)$ function to get the $p(X)$ function.

The latter seems to be far more general, but we need to put more constraint to make it more “physically” possible.

Note that since this is only an “interpretation” and models. You actually need to have actual physical observation to make it a theory. So take this interpretation as just a rambling, or a fun way to make a toy model of alternate universes.