From Fermat to Hilbert to Strings — Euler, Light, and Quantum Paths

1. Euler Mathematics — The Language of Phase

Leonhard Euler gave physics its phase rotor. Every oscillation, every quantum amplitude, every string vibration is a complex exponential.

Core formulas

\[ e^{i\theta} = \cos\theta + i\sin\theta \]

\[ e^{i\theta} = \sum_{n=0}^\infty \frac{(i\theta)^n}{n!} = 1 + i\theta - \frac{\theta^2}{2!} - i\frac{\theta^3}{3!} + \cdots \]

Euler–Lagrange: $\displaystyle \frac{d}{dt}\frac{\partial L}{\partial \dot q} - \frac{\partial L}{\partial q}=0$ — extremizes $S=\int L\,dt$

Euler characteristic: $\displaystyle \chi = V-E+F = 2-2g$ — counts holes in string worldsheets

In QM and strings, time evolution is rotation by Euler: $U(t)=e^{-iHt/\hbar}$. In path integrals, each history carries phase $e^{iS/\hbar}$. Stationary phase picks the classical path — exactly Fermat's principle.

Interactive: Unit Circle

drag the gold dot

2. Hilbert Spaces — The Stage

Quantum states live in a complete inner-product space $\mathcal{H}$. Observables are operators; evolution is unitary.

Definition

Inner product $\langle \psi|\phi\rangle\in\mathbb{C}$ with $\langle\psi|\psi\rangle\ge 0$. Norm $\|\psi\|=\sqrt{\langle\psi|\psi\rangle}$. Complete = limits stay inside.

Schrödinger evolution: $\displaystyle |\psi(t)\rangle = e^{-iHt/\hbar} |\psi(0)\rangle$ — unitary, $\langle\psi(t)|\psi(t)\rangle=1$

For black holes, Hawking's puzzle lives in $\mathcal{H}_{\text{total}} = \mathcal{H}_{\text{in}} \otimes \mathcal{H}_{\text{out}}$. Information loss = apparent non-unitarity on $\mathcal{H}_{\text{out}}$ alone.

Qubit: $|\psi\rangle = \alpha|0\rangle + \beta|1\rangle,\; |\alpha|^2+|\beta|^2=1$ with $\alpha=\cos\frac{\theta}{2},\; \beta=e^{i\phi}\sin\frac{\theta}{2}$

Interactive: Qubit on Bloch Sphere

θ (polar) φ (azimuth)

3. Fermat's Principle — Light Finds the Fastest Path

1662: light extremizes travel time $\delta\int_A^B n\,ds=0$. From wave optics, constructive interference occurs where phase $e^{ik\int n ds}$ is stationary.

Snell's law from Fermat

For two media: minimize $T = \frac{n_1}{c}\sqrt{(x-x_A)^2+y_A^2} + \frac{n_2}{c}\sqrt{(x_B-x)^2+y_B^2}$. Setting $dT/dx=0$ gives $n_1\sin\theta_1 = n_2\sin\theta_2$.

GR Fermat: null geodesics satisfy $ds^2=g_{\mu\nu}dx^\mu dx^\nu=0$, extremizing arrival time $\delta\int dt=0$

Gravitational lensing bend angle (weak field): $\displaystyle \alpha = \frac{4GM}{c^2 b}$ — Einstein 1915, confirmed 1919.

Interactive: Refraction

n₁ n₂

Drag endpoints (white) and the interface point (cyan). Gold = Fermat optimal.

4. Einstein's Light vs Newton

Special Relativity

$c$ is invariant. Lightlike interval: $ds^2 = -c^2dt^2 + dx^2+dy^2+dz^2 = 0$.

General Relativity

Light follows null geodesics: $g_{\mu\nu}k^\mu k^\nu=0,\; \nabla_k k=0$. Spacetime curvature replaces force.

Gravitational redshift: $\omega_\infty = \omega_{\text{emit}}\sqrt{1-\frac{2GM}{rc^2}}$

Shapiro delay: extra coordinate time near mass because $g_{00}$ slows clocks.

Interactive: Light Bending Around Sun

Mass (M☉)

5. Quantum Path Integral — Feynman

Propagator: $\displaystyle K(x_f,t_f;x_i,t_i)=\int \mathcal{D}x(t)\,e^{iS[x]/\hbar}$ with $S=\int L\,dt$

Each path contributes Euler phase $e^{iS/\hbar}$. Writing $K=\langle x_f|e^{-iH\Delta t/\hbar}|x_i\rangle$ reveals the Hilbert operator.

Classical limit $\hbar\to0$: stationary phase → $\delta S=0$ → Euler–Lagrange → for light, Fermat. Quantum = sum; classical = extremum.

Interactive: Sum Over Histories

ħ (quantum-ness)

As ħ→0, paths cluster on the classical (gold) straight line — stationary phase.

6. String Theory & M-Theory

Polyakov action: $\displaystyle S_P=-\frac{1}{4\pi\alpha'}\int d^2\sigma\sqrt{-\gamma}\,\gamma^{ab}\partial_a X^\mu\partial_b X_\mu$

Mode expansion uses Euler: $X^\mu = x_0^\mu + p^\mu\tau + i\sqrt{\frac{\alpha'}{2}}\sum_{n\neq0}\frac{1}{n}\big(\alpha_n^\mu e^{-in(\tau-\sigma)}+\tilde\alpha_n^\mu e^{-in(\tau+\sigma)}\big)$.

Hilbert space is a Fock space built by oscillators: $|0\rangle, \alpha_{-n}|0\rangle$. Left/right movers give $\mathcal{H}_{\text{string}}=\mathcal{H}_L\otimes\mathcal{H}_R$ — exactly the tensor structure Hawking discussed for in/out.

Partition function: $Z(\tau)=\mathrm{Tr}\,q^{L_0-c/24}$, $q=e^{2\pi i\tau}$ — Euler exponential again. Genus expansion weighted by $g_s^{-\chi}$, $\chi=2-2g$.

M-theory (11D): $R_{11}=g_s\ell_s$. BFSS matrix model: Hilbert space of $N\times N$ matrices, Hamiltonian $H\sim\mathrm{Tr}(P^2-[X^i,X^j]^2)$.

Interactive: String Mode Builder

7. Tying to Hawking and Our Previous Conversations

Hawking temperature from Euler periodicity

Euclidean black hole requires $\phi(\tau+\beta)=\phi(\tau)$. Modes $e^{i\omega\tau}$ give $e^{i\omega\beta}=1\Rightarrow \omega\beta=2\pi n$. For Schwarzschild, $\beta=8\pi GM/\hbar c^3$.

$T_H=\frac{\hbar c^3}{8\pi GM k_B}$ — periodicity is Euler's formula in imaginary time.

Entropy = log dim Hilbert

Strominger–Vafa (1996): string microstates count gives $S_{BH}= \log \dim \mathcal{H}_{BH}=2\pi\sqrt{c\,n_L/6}+L\leftrightarrow R$. Same left-right tensor product as strings.

Information paradox

Unitary $S$-matrix acts on $\mathcal{H}_{rad}\otimes\mathcal{H}_{BH}$. Measurement and Zeno effect are projections $P|\psi\rangle$ in Hilbert space — no cloning, only entanglement transfer.

Concept Map

Fermat extremum → Euler phase → path integral stationary phase → string worldsheets → Hawking periodicity. Hilbert space carries the unitary evolution throughout.

8. GNU Octave Laboratory

Copy-paste these five self-contained scripts to explore the mathematics directly.

% a) Euler and Fermat refraction (Snell's law)
n1 = 1.0; n2 = 1.5;
theta1 = pi/6;  % 30 degrees
theta2 = asin(n1/n2 * sin(theta1));
fprintf('Snell: n1 sinθ1 = %.4f, n2 sinθ2 = %.4f\n', n1*sin(theta1), n2*sin(theta2));
% Euler phase
theta = linspace(0,2*pi,400);
z = exp(1i*theta);
plot(real(z), imag(z)); axis equal; title('Euler e^{iθ}');

% b) Hilbert inner product for qubit
psi = [1; 1i]/sqrt(2);      % |+i>
phi = [1; 0];               % |0>
ip = phi' * psi;            % 
prob = abs(ip)^2;
fprintf('|<0|psi>|^2 = %.3f\n', prob);
% Unitary evolution
H = [0 1; 1 0];             % sigma_x
U = expm(-1i*H*0.5);
psi_t = U*psi;

% c) GR light deflection
G = 6.67430e-11; M = 1.9885e30; c = 299792458; b = 6.9634e8;
alpha_rad = 4*G*M/(c^2*b);
alpha_arcsec = alpha_rad * 180/pi * 3600;
fprintf('Einstein deflection at solar limb: %.3f arcsec\n', alpha_arcsec);
% ~1.75"

% d) Path integral Monte Carlo (free particle)
Npaths = 2000; Nt = 80;
paths = cumsum(randn(Npaths,Nt),2); % Wiener
S = sum(paths.^2,2);                 % toy action ~∫ v^2
hbar = 0.2;
K = mean(exp(1i*S/hbar));
fprintf('|K| = %.3f, arg(K)=%.3f\n', abs(K), angle(K));

% e) String partition (Dedekind eta, bosonic 24 transverse)
tau = 0.5 + 0.8i;
q = exp(2*pi*1i*tau);
eta = q^(1/24) * prod(1 - q.^(1:200));
Z = abs(eta)^(-48);  % |η|^{-2(d-2)} with d=26
fprintf('Bosonic partition |η(τ)|^{-48} = %.3e\n', Z);
% Euler form q appears via e^{2π i τ}

Conclusion — The Rosetta Stone

Pillar	Math Form	QM	GR	Strings	Hawking BH
Euler	$e^{i\theta}$, $e^{iS/\hbar}$	Phase rotation $U=e^{-iHt/\hbar}$	Null phase $e^{ikx}$	Modes $e^{-in\sigma}$, $q=e^{2\pi i\tau}$	Euclidean $e^{i\omega\beta}=1$
Hilbert	$\langle\psi\|\phi\rangle$, $\\|\psi\\|=1$	State space $\mathcal{H}$	QFT on curved space	Fock $\mathcal{H}_L\otimes\mathcal{H}_R$	$\mathcal{H}_{in}\otimes\mathcal{H}_{out}$
Fermat	$\delta\int nds=0$	Stationary phase	Null geodesics	Worldsheet extremum	Extremal surface RT

The same three ideas carry you from a lifeguard minimizing time to a photon bending around the Sun, to a quantum amplitude, to a vibrating string, to Hawking radiation. Euler provides the phase, Hilbert provides the home, Fermat provides the classical path that emerges when many phases interfere constructively.