Euler + Hilbert = The Language of Strings
Leonhard Euler's exponentials give us rotation and periodicity. David Hilbert's complete inner-product spaces give us quantum states. Together they form the grammar of string theory, M-theory, and the quantum description of black holes we explored with Hawking radiation.
1. Euler Mathematics Toolkit
Euler's Formula
$$ e^{i\theta} = \cos\theta + i\sin\theta $$
This is not a trick. It defines rotation in the complex plane. For \(\theta=2\pi\) we get \(e^{2\pi i}=1\), the periodicity that will later force Hawking temperature and string modular invariance.
Euler-Lagrange
$$ \frac{d}{dt}\frac{\partial L}{\partial \dot q} - \frac{\partial L}{\partial q}=0 $$
For the Polyakov string, \(L = -\frac{1}{4\pi\alpha'}\sqrt{-h}h^{ab}\partial_a X^\mu\partial_b X_\mu\). The E-L equations give the wave equation \(\partial_a\partial^a X^\mu=0\), whose solutions are left/right movers built from Euler exponentials \(e^{in(\tau\pm\sigma)}\).
Euler Characteristic
$$ \chi = V-E+F = 2-2g $$
Worldsheets are classified by genus \(g\). Sphere \(g=0,\chi=2\); torus \(g=1,\chi=0\); genus-2 \(\chi=-2\). String amplitudes sum over topologies weighted by \(g_s^{-\chi}\).
For Calabi–Yau 3-folds: \(\chi = 2(h^{1,1}-h^{2,1})\). In heterotic compactifications, \(|\chi|/2\) = number of particle generations.
Exponential Series
$$ e^z = \sum_{n=0}^\infty \frac{z^n}{n!} $$
Convergence of this series underlies Wick rotation \(e^{iS}\to e^{-S_E}\) and the definition \(U=e^{-iHt/\hbar}\) via power series in the Hilbert space operator algebra.
2. Hilbert Spaces Primer
A Hilbert space \(\mathcal H\) is a complete vector space with inner product \(\langle\psi|\phi\rangle\). Norm: \(\|\psi\|^2=\langle\psi|\psi\rangle\).
- \(L^2(\mathbb{R})\): square-integrable wavefunctions \(\int |\psi(x)|^2dx <\infty\)
- \(\ell^2\): sequences \(\sum |c_n|^2 <\infty\)
- \(\mathbb{C}^2\): qubit, \(|\psi\rangle = \alpha|0\rangle + \beta|1\rangle\)
QM Postulates (Hilbert language)
1. States = rays in \(\mathcal H\). 2. Observables = Hermitian \(A=A^\dagger\). 3. Evolution = unitary \(U = e^{-iHt/\hbar}\). Euler again. 4. Measurement = projection.
In our previous Zeno paradox discussion, repeated projection \(P\) gives survival \(|\langle\psi|PUP|\psi\rangle|^{2N}\approx 1- (\Delta H\,t)^2/N \to 1\). The math lives entirely in \(\langle\cdot|\cdot\rangle\).
Interactive Qubit \(\mathcal H=\mathbb{C}^2\)
\(|\psi\rangle = \cos\frac{\theta}{2}|0\rangle + e^{i\phi}\sin\frac{\theta}{2}|1\rangle\). The Bloch vector is the expectation \(\langle\psi|\vec\sigma|\psi\rangle\).
3. Hilbert Spaces in String Theory
The closed string worldsheet CFT factorizes exactly like our black hole Hilbert space from the previous lecture:
$$ \mathcal H_{\text{string}} = \mathcal H_L \otimes \mathcal H_R $$
Compare to Hawking: \(\mathcal H_{\text{total}} = \mathcal H_{\text{in}} \otimes \mathcal H_{\text{out}}\). Entanglement between left/right movers mirrors entanglement across horizons.
Oscillator Algebra
$$ [\alpha_m^\mu, \alpha_n^\nu] = m\,\eta^{\mu\nu}\,\delta_{m+n,0},\quad [\tilde\alpha_m^\mu,\tilde\alpha_n^\nu]=m\,\eta^{\mu\nu}\,\delta_{m+n,0} $$
Fock space built on momentum eigenket \(|0;k\rangle\):
$$ |\psi\rangle = \left(\prod_{n>0}(\alpha_{-n})^{N_n}(\tilde\alpha_{-n})^{\tilde N_n}\right)|0;k\rangle $$
Virasoro Constraints
Physical states satisfy:
$$ (L_0-1)|\psi\rangle=0,\quad L_{n>0}|\psi\rangle=0 $$
where \(L_0 = \frac{\alpha' p^2}{4} + N\), \(N=\sum n N_n\). This yields the mass shell:
$$ \alpha' m^2 = 4(N-1) $$
Interactive Mode Builder
Each mode \(\alpha_{-n}\) adds a Fourier component \(e^{in\sigma}\). The sum is a vector in the infinite-dimensional Fock Hilbert space.
4. Euler in String / M-Theory
Torus Partition Function
$$ Z(\tau,\bar\tau) = \mathrm{Tr}_{\mathcal H}\, q^{L_0-c/24}\,\bar q^{\tilde L_0-c/24},\quad q=e^{2\pi i\tau} $$
Modular invariance: \(\tau\to\tau+1\) uses \(e^{2\pi i}=1\) (Euler), and \(\tau\to-1/\tau\). This ensures the one-loop string amplitude is well-defined on moduli space \(\mathcal M_1\).
Genus Expansion
$$ \mathcal A = \sum_{g=0}^\infty g_s^{2g-2}\int_{\mathcal M_g} \!\!\! \langle\cdots\rangle_g $$
Weight \(g_s^{-\chi}\) with \(\chi=2-2g\) — the Euler characteristic controls the loop expansion.
Interactive Modular Domain
Compactification
For CY3, \(\chi = 2(h^{1,1}-h^{2,1})\). Standard embedding gives generations \(=|\chi|/2\). Euler topology becomes particle physics.
M-Theory Circle
\(R_{11}=g_s\ell_s\), KK modes \(e^{iny/R_{11}}\) give masses \(m_n=n/R_{11}\). Periodicity \(y\sim y+2\pi R\) is pure Euler phase \(e^{2\pi i n}=1\).
5. Connection to Quantum Mechanics & Hawking
Black Hole Factorization = String Factorization
From our prior black hole notes: \(\mathcal H = \mathcal H_{\text{in}}\otimes\mathcal H_{\text{out}}\). In strings: \(\mathcal H_L\otimes\mathcal H_R\). Both describe entangled pure states with thermal reduced density matrices \(\rho = \mathrm{Tr}_{R}|\Psi\rangle\langle\Psi| = e^{-\beta H}/Z\).
Hawking Temperature from Euler
Euclidean Schwarzschild requires no conical deficit at horizon. Fields periodic: \(\phi(t_E+\beta)=\phi(t_E)\). With \(\phi\sim e^{-i\omega t_E}\), periodicity demands:
$$ e^{-i\omega(t_E+\beta)} = e^{-i\omega t_E} \;\Rightarrow\; e^{-i\omega\beta}=1 $$
By Euler, \(\omega\beta = 2\pi n\). The lowest \(n=1\) with surface gravity \(\kappa\) gives \(\beta = 2\pi/\kappa\). Hence \(T_H = \hbar\kappa/2\pi\). This is the exact same \(e^{2\pi i}=1\) as modular \(\tau\to\tau+1\).
Wick Rotation & Path Integrals
$$ \int \mathcal D g\, e^{iS/\hbar} \xrightarrow{t\to -i\tau} \int \mathcal D g\, e^{-S_E/\hbar} = \mathrm{Tr}_{\mathcal H} e^{-\beta H} $$
The rotation uses \(e^{i\theta}\) analytically continued. Convergence comes from the Euler series. The trace is over the Hilbert space — string theory makes this trace finite via modular invariance.
Information Paradox Resolution
AdS/CFT identifies black hole Hilbert space with boundary CFT Hilbert space built from string oscillators. Evolution \(U=e^{-iHt}\) is unitary, preserving \(\langle\psi(t)|\phi(t)\rangle = \langle\psi(0)|\phi(0)\rangle\). Hawking radiation is not mixed fundamentally — the apparent mixedness is tracing over \(\mathcal H_{\text{in}}\), exactly as we trace over \(\mathcal H_R\) in strings.
Time Reversal
Anti-unitary \(T\): \(TiT^{-1}=-i\), so \(T e^{-iHt}T^{-1}=e^{+iHt}\). On the worldsheet, orientation reversal \(\sigma\to -\sigma\) swaps \(\alpha_n \leftrightarrow \tilde\alpha_n\), i.e. \(\mathcal H_L \leftrightarrow \mathcal H_R\). This mirrors our earlier discussion of T-symmetry in quantum measurement.
6. GNU Octave Lab
Copy-paste these into Octave to explore Euler and Hilbert spaces numerically.
theta = linspace(0,2*pi,1000);
z = exp(1i*theta); % Euler's formula
plot(real(z), imag(z), 'linewidth',2);
axis equal; grid on;
title('e^{i\theta} traces unit circle');
xlabel('Re'); ylabel('Im');
% check: max error vs cos+i sin
err = max(abs(z - (cos(theta)+1i*sin(theta))))psi = [1/sqrt(2); 1i/sqrt(2)]; % |psi> in C^2
phi = [1; 0]; % |0>
ip = phi' * psi; % = conj(phi)^T psi
norm_psi = sqrt(psi' * psi); % should be 1
proj = psi * (psi' * phi); % projection of |phi> onto |psi>
disp([' = ', num2str(ip)]);
disp(['||psi|| = ', num2str(norm_psi)]); tau = 0.1 + 1.1i; % modular parameter, Im>0
q = exp(2*pi*1i*tau); % Euler exponential
Nmax = 50;
eta = q^(1/24) * prod(1 - q.^(1:Nmax)); % Dedekind eta approx
Z = 1/(abs(eta)^48); % toy bosonic 1-loop |eta|^{-48}
disp(['q = ', num2str(q), ', |q| = ', num2str(abs(q))]);
disp(['|Z| ~ ', num2str(Z)]);
% modular T: tau -> tau+1 leaves q invariant because e^{2πi}=1H = [0 1; 1 0]; % Pauli X
psi0 = [1;0]; % start |0>
T = pi/2; N = 100; dt = T/N;
U = expm(-1i*H*dt); % Euler evolution U = e^{-iHdt}
P = [1 0;0 0]; % projector onto |0>
psi = psi0;
for k=1:N
psi = U*psi;
psi = P*psi; % frequent measurement
psi = psi / norm(psi); % renormalize
end
survival = abs(psi0'*psi)^2
% As N→∞, survival →1 (Zeno freeze)g = 0:4; % genus
chi = 2 - 2*g; % Euler characteristic
gs = 0.3;
weight = gs.^(2*g-2); % g_s^{-chi}
plot(g, chi, 'o-', 'linewidth',2); grid on;
xlabel('genus g'); ylabel('\chi = 2-2g');
title('String loop suppression by Euler characteristic');
disp([g' chi' weight']);