4.1 Planck’s Interest

“Einstein once joked that compound interest is the eighth wonder of the world. He may not have realized that this statement applies not only to economics but also to his own physics. The universe can accumulate grand coherent structures from tiny quantum fluctuations precisely because it is performing compound interest calculations at a frequency as high as $10^{43}$ Hz.”

The Origin of $e$: The Banker’s Definition

To understand physics, we must first return to the original definition of $e$ in mathematics textbooks.

Suppose you deposit 1 dollar (principal) in a bank. The bank is very generous, with an annual interest rate of 100%.

• If settled once a year: You get $1+1=2$ dollars.

• If settled twice a year: You get $(1+1/2{)}^2=2.25$ dollars.

• If settled daily: You get $(1+1/365{)}^{365}\approx 2.71$ dollars.

If you are a greedy depositor, you would demand the bank settle interest every moment, and immediately turn the interest into new principal for the next round of interest generation.

When the number of settlements $n$ tends to infinity, your returns do not become infinite but converge to a constant:

$$\underset{n\to \infty }{\lim }{\left(1+\frac{1}{n}\right)}^n=e\approx 2.71828\dots$$

This is the origin of $e$: it is the limit of continuous growth.

The Universe’s Principal and Interest Rate

In Vector Cosmology, we have completely corresponding physical quantities:

1. Principal: Our global vector $|\Psi ⟩$. Its modulus is 1. This is the “base capital” of cosmic existence.

2. Total Capacity: $c_{FS}$. This is the maximum turnover rate allowed by the cosmic bank.

3. Compounding Period: Planck time $t_P$. This is one “frame” of the QCA microscopic engine.

Within every Planck time $t_P$, the universe performs a tiny “interest settlement” operation.

According to the discrete form of the Schrödinger equation, the state update rule is:

$$|\Psi (t+t_P)⟩\approx (1-iH{t}_P)|\Psi (t)⟩$$

Notice that $(1-iH{t}_P)$.

This is identical to the financial formula $(1+\text{Rate})$!

The only difference is that the universe’s interest rate is imaginary ($-iH$).

• Financial compound interest: Real interest rate. Leads to exponential growth (money increases).

• Geometric compound interest: Imaginary interest rate. Leads to exponential rotation (phase changes).

Limit Approximation at Planck Scale

This is why macroscopic physical laws are full of $e$.

Although the underlying QCA is discrete, step-by-step ($n$ is finite), because Planck time is extremely short ($10^{-44}$ seconds), for any macroscopic process, $n$ is an astronomical number.

The universe is actually calculating:

$$U(t)=\underset{N\to \infty }{\lim }{\left(1-iH\frac{t}{N}\right)}^N=e^{-iHt}$$

What we perceive as “continuous time” is actually the compound interest accumulation of countless tiny discrete jumps.

A photon flying one meter does not glide; it has undergone $10^{35}$ “deposit-withdraw-transfer” compound interest calculations, ultimately accumulating that huge phase change $e^{i(kx-\omega t)}$.

$e$ is the bridge between discrete and continuous.

It tells us that smooth physical laws (continuous compound interest) are merely statistical emergence of microscopic discrete transactions (settlement by count) at extremely high frequencies.

Least Action: Maximizing Returns

This perspective also gives us a new economic understanding of the “principle of least action.”

If in finance you want to maximize returns, you need to find the investment portfolio with the strongest compound interest effect.

In physics, if a photon wants to maximize (or stabilize) phase accumulation (geometric returns), it must find that “compound interest path”.

• Straight-line propagation: On this path, the tiny phase interest $\Delta \theta$ generated at each step perfectly adds to the principal of the previous step (in-phase superposition). This is the most efficient mode of “compound interest”.

• Curved path: The interest generated at each step not only fails to increase the principal but cancels it out due to phase disorder (out-of-phase cancellation). This is “investment loss”.

The universe follows least action because the universe is a rational compound interest investor. It always automatically selects the path that allows geometric phase accumulation to be most significant.

Conclusion: Time is Interest

Therefore, we have the ultimate definition of “time”:

Time is not passing; time is generated interest.

When you feel one second has passed, that is your body, your brain, the air around you, after $10^{44}$ settlements in the Planck-scale bank, the accumulated geometric surplus.

We live in a high-frequency trading universe. Precisely because the settlement frequency is so high (the limit of $e$ is so closely approximated), our world appears so continuous, so real, so indestructible.

Since we have built the bridge from discrete QCA to continuous exponential $e$, how does this compound interest mechanism manifest macroscopically as the “thermodynamics” and “temperature” we know?

This leads to the theme of the next section: The Bridge Between Discrete and Continuous. We will further explore why the microscopic pixelated structure (QCA) not only generates time but also generates temperature.