Physics & Math Primer: A Practical Introduction
Short notes with small visuals for core tools: complex phase, rotations, spacetime metric, two‑state waves, potentials and curvature, and a first look at gauge fields.
Hover here to see how these tools feed the main physics article.
Complex Numbers and Phase
The number $e^{i\alpha}$ is a unit arrow in the plane: it rotates by angle $\alpha$ without changing length. Real and imaginary parts are $\cos\alpha$ and $\sin\alpha$. Euler linked exponentials with sines and cosines in the 1700s; later, Argand’s diagram made the picture standard. Engineers and physicists use this to track oscillations and waves cleanly.
This picture shows $e^{i\alpha}=(\cos\alpha,\,\sin\alpha)$. Re/Im are the x/y components.
Terms in the formula: $\alpha$ is the rotation angle; $i$ rotates by 90° in the plane; $\cos\alpha$ is the horizontal component; $\sin\alpha$ is the vertical component; the length stays 1.
cos(α)=…, sin(α)=…
Rotations as Matrices
A 2D rotation is a matrix acting on a vector. The entries are sines and cosines of the angle. This form became common in 19th–20th century mechanics because matrices compose cleanly: two rotations multiply to give a third. It’s also how computers represent and combine rotations.
This picture shows $R(\theta)=\begin{bmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{bmatrix}$ rotating an arrow by $\theta$.
Terms in the formula: $\theta$ is the rotation angle; $\cos\theta$ keeps the component along the original axis; $\sin\theta$ swaps components with a sign; columns show where unit basis vectors move; determinant $=1$ means no stretching.
R(θ) = …
Euclidean Circles vs. Minkowski Cones
In ordinary space, $x^2+y^2=r^2$ draws a circle. In spacetime (with $c=1$ in 1+1D), $t^2-x^2=s^2$ draws hyperbolas that keep the “spacetime distance” fixed. Einstein’s 1905 postulate about light’s speed led Minkowski in 1908 to fold space and time together; these hyperbolas are the footprints of that geometry.
This picture shows a unit circle (blue) and the hyperbola $t^2-x^2=s^2$ (green). Both express an invariant “distance”, but with different signs.
Terms in the relation: $t$ is time, $x$ is space (with $c=1$); $s$ is the invariant interval (proper time when timelike); the sign flip (minus) encodes that time and space contribute differently; lines $t=\pm x$ are light rays.
In a Lorentz boost, the light cone stays fixed while the axes tilt; these hyperbolas stay the same.
Two‑State Systems (Bloch Picture)
A two‑state wave can be written as $|\psi\rangle=\cos\tfrac{\theta}{2}\,|0\rangle+e^{i\phi}\sin\tfrac{\theta}{2}\,|1\rangle$. Probabilities are $P(0)=\cos^2(\theta/2)$ and $P(1)=\sin^2(\theta/2)$. The Bloch picture dates to the 1940s (NMR): a single point on a sphere encodes the state, with angles setting measurement outcomes. It is now the standard way to think about qubits.
This picture shows the state’s direction on the Bloch circle (equator projection) and reports $P(0)$ and $P(1)$.
Terms in the formula: $\theta$ sets the split between $|0\rangle$ and $|1\rangle$ (probabilities $\cos^2(\theta/2)$ and $\sin^2(\theta/2)$); $\phi$ is the relative phase that matters for interference; $|0\rangle,|1\rangle$ are the measurement outcomes.
P(0)=…, P(1)=…
Potentials and Small Oscillations
A particle tends to roll downhill in a potential $V(x)$. Near a minimum, the curvature $V''(x_\text{min})$ sets the small‑oscillation frequency. From Lagrange onward, this “small oscillations” idea powered mechanics: approximate the bottom of a well by a quadratic and read off the frequency. In high‑energy units ($\hbar=c=1$), a mass behaves like $m^2\sim V''$.
This picture shows $V(x)=\tfrac{\lambda}{4}(x^2-v^2)^2$. The readout shows $m^2=V''(x_\text{min})=2\lambda v^2$.
Terms in the formula: $x$ is the field/position; $v$ sets where the minima sit; $\lambda$ controls how steep the walls are; $V'(x)$ is the slope (force $=-V'$); the curvature $V''$ at a minimum sets the small‑oscillation scale.
Readout: m² = …
Why We Need a Covariant Derivative (1D)
If a wave’s phase $\alpha(x)$ varies with position, an ordinary derivative “sees” that variation even when the amplitude is constant. A gauge field $A_\mu$ fixes this so that the derivative respects the relabeling. This grew out of Maxwell’s equations, where a hidden freedom to change potentials leaves the fields unchanged; Yang–Mills extended the idea to richer “internal labels”.
This picture shows a 1D phase $\alpha(x)$ and compares the average size of $\partial_x\psi$ vs. $D_x\psi$ with $A_x=-(1/g)\,\partial_x\alpha$.
Terms in the formula: $\psi$ is a complex wave with amplitude and phase; $\alpha(x)$ is the position‑dependent phase; $\partial_x$ is the ordinary derivative; $A_x$ is the gauge field that cancels pure phase gradients; $g$ sets how strongly $A_x$ enters; $D_x=\partial_x+i g A_x$ respects the relabeling.
Readout: ⟨|∂xψ|⟩ = …; ⟨|Dxψ|⟩ = …
What You Learned
These are the small tools behind many modern physics ideas:
• Complex phase is a simple rotation in the plane. • Rotations are matrices with sines and cosines. • Spacetime uses a sign‑flipped “distance” that keeps light cones fixed. • A two‑state wave has probabilities set by angles on a circle. • The curvature of a potential near a minimum sets a mass‑like scale. • A covariant derivative keeps calculations honest when phase depends on position.
Use the sliders above to build intuition. Each picture is driven directly by the equation next to it.
Where these tools are used:
- Phase & rotations → Roadmap: Spin‑1/2, Lorentz.
- Two‑state (Bloch) → Roadmap: Neutrino oscillations.
- Potentials/curvature → Roadmap: Higgs.
- Covariant derivative → Roadmap: Gauge (U(1)).