Fractal Morphisms of theWorldSheet

Quantum Mechanics: 1. Hilbert Spaces and State Vectors: - Quantum states are vectors in a complex Hilbert space H. - For a single particle in one dimension, the state $|\psi\rangle$ can be represented in the position basis as: $$\psi(x)=\langle x \mid \psi\rangle$$ - In coordinate space, this can be written as a wave function: $$\psi(x)=\int_R \psi\left(x^{\prime}\right) \delta\left(x-x^{\prime}\right) d x^{\prime}$$ 2. Operators: - Observables are represented by Hermitian operators. For example, the position operator $\hat{X}$ and momentum operator $\hat{P}$ : $$\hat{x}=x, \quad \hat{P}=-i \hbar \frac{d}{d x}$$ - These operators follow the commutation relation: $$[\hat{x}, \hat{P}]=i \hbar \hat{T}$$ 3. Time Evolution: - The Schrödinger equation governs the time evolution of a quantum state: $$i \hbar \frac{\partial}{\partial t}|\psi(t)\rangle=\hat{H}|\psi(t)\rangle$$ Where A is the Hamiltonian operator.Supersymmetry (SUSY): 1. Supersymmetric Algebra: - SUSY extends the Poincaré algebra with supercharges $Q_\alpha$ and $Q_\alpha$ : $$\left\{Q_\alpha, \bar{Q}_\beta\right\}=2 \sigma_{\alpha \beta}^\mu P_\mu$$ Where $\sigma^\mu$ are the Pauli matrices in Lorentz indices. 2. Superfields: - In SUSY, one often deals with superfields, which combine bosonic and fermionic fields. For example, a chiral superfield $\Phi$ might include: $$\Phi(z)=\phi(y)+\sqrt{2} \theta \psi(y)+\theta \theta F(y)$$ where $\theta$ are Grassmann numbers, $\phi(y)$ is a bosonic field, $\psi(y)$ is a fermionic field, and $F(y)$ is an auxiliary field. String Theory: 1. Worldsheet Formalism: - Strings are described by their embedding in spacetime, typically via coordinates $X^\mu(\sigma, \tau)$, where $\sigma$ and $\tau$ parameterize the string's worldsheet. The action might be: