Enough Numbers to Build a Universe
February 1, 2024
Reading Mode
Wider
mathematics
physics
Enough Numbers to Build a Universe
As a fun excercise, consider the question: What are the numbers that are used to build a universe? Well, we don't know for sure, but we do know that we can build our current understanding of physics out of a handful of fundamental constants. Some we know precisely, and some we know only approximately. Some are universal, and some are determined empirically. Let's start with the mathematical constants.
D is the number of spatial dimensions . Space has three dimensions, which are commonly referred to as length, width, and height. These dimensions are perpendicular to each other, and they provide a framework for understanding the position and motion of objects in space. There might be more, but for now we'll stick with three.
$$
D = 3
$$
The universe also has a single temporal dimension that corresponds to the continuum of time. In spacetime, the temporal dimension is represented by a single number, known as a temporal coordinate, which describes the position of an object in time.
$$
T = 1
$$
Pi is a dimensionless mathematical constant that is defined as the ratio of the circumference of a circle to its diameter.
$$\pi \approx 3.141592653589793238$$
e is a dimensionless mathematical constant that arises in the study of exponential growth and decay. It arises naturally out of calculus.
$$e \approx 2.71828182845904523536$$
Now, physical constants. The complete standard model of physics requires 25 fundamental dimensionless constants. At present, their numerical values are not understood in terms of any overarching theory and are determined only from experimental measurement. These constants are:
The fine structure constant
The strong coupling constant
The four parameters of the Cabibbo-Kobayashi-Maskawa matrix which describe how quarks oscillate between different forms
The four parameters of the Pontecorvo-Maki-Nakagawa-Sakata matrix which describe how neutrinos oscillate between different forms
Fifteen masses of the fundamental particles, which can expressed in terms of the Planck mass
six quarks
six leptons
the Higgs boson
the W boson
the Z boson
NIST has a table of all the physical constants that are used in the standard model of physics.
The Fine Structure Constant (denoted $\alpha$) is a fundamental physical constant that is used in the study of the interaction between electromagnetic radiation and matter. The value of the Fine Structure Constant is a dimensionless quantity.
$$\alpha = \frac{1}{4 \pi \epsilon_0} \frac{e^2}{\hbar c}$$
$$\alpha \approx \frac{1}{137}$$
$$\alpha = 0.0072973525693$$
The Planck constant is denoted by the letter h, and its value is $6.63 \cdot 10^{-34} J \cdot s$. This constant allows us to calculate the amount of energy associated with a particle, given its frequency. The equation for this relationship is given by the formula E = hf, where E is the energy of the particle, f is its frequency, and h is the Planck constant.
$$h = 6.62607015 \times 10^{-34} \quad J s$$
$$\hbar = \frac{h}{2 \pi}$$
The Planck mass is defined as the mass of a particle whose Compton wavelength is equal to the Planck length. The Compton wavelength of a particle is a measure of its size, and the Planck length is the smallest possible length that can be measured in our universe. Therefore, the Planck mass is a measure of the smallest possible mass that can exist in our universe under our current understanding.
$$1.22089 \times 10^{19} \quad GeV/c^2$$
The Cabibbo-Kobayashi-Maskawa matrix , also known as the CKM matrix, is a unitary matrix used in particle physics to describe the mixing of the six quarks that make up protons and neutrons. The CKM matrix describes the mixing of the six quarks, which are the up quark, the down quark, the charm quark, the strange quark, the top quark, and the bottom quark. These six quarks are thought to be different forms or "flavors" of the same fundamental particles, and they can oscillate or "switch" between these different flavors as they interact with other particles.
$$
\begin{bmatrix}
V_{ud}&V_{us}&V_{ub} \\
V_{cd}&V_{cs}&V_{cb} \\
V_{td}&V_{ts}&V_{tb}
\end{bmatrix}
$$
The Pontecorvo-Maki-Nakagawa-Sakata matrix , also known as the PMNS matrix, is a unitary matrix used in particle physics to describe the mixing of three generations of neutrinos.
The PMNS matrix describes the mixing of the three generations of neutrinos, which are the electron neutrino, the muon neutrino, and the tau neutrino. These three generations of neutrinos are thought to be different forms or "flavors" of the same fundamental particle, and they can oscillate or "switch" between these different flavors as they propagate through space.
The PMNS matrix is a 3x3 matrix with three rows and three columns, and it contains six real parameters that describe the mixing of the three generations of neutrinos. These six parameters are...