Editing QuasiPhysics (section)

=== Mathematical Frameworks in QuasiPhysics ===
QuasiPhysics relies on several mathematical frameworks to describe and predict the behavior of quasiparticles and other emergent phenomena:

* '''Dispersion Relations''': The relationship between the energy of a quasiparticle and its momentum is described by the dispersion relation. For example, the dispersion relation for phonons in a crystal lattice is given by:

<math>E(\mathbf{k}) = \hbar \omega(\mathbf{k})</math>

where:
* <math>E(\mathbf{k})</math> is the energy of the quasiparticle,
* <math>\mathbf{k}</math> is the wave vector,
* <math>\hbar</math> is the reduced Planck constant,
* <math>\omega(\mathbf{k})</math> is the angular frequency of the quasiparticle.

* '''Bogoliubov Transformation''': In the study of superfluidity and superconductivity, the Bogoliubov transformation is used to diagonalize the Hamiltonian of a many-body system, leading to the creation of quasiparticles:

<math>\alpha_k = u_k \gamma_k + v_k \gamma_{-k}^\dagger</math>

where:
* <math>\alpha_k</math> is the quasiparticle operator,
* <math>u_k</math> and <math>v_k</math> are coefficients determined by the system's parameters,
* <math>\gamma_k</math> and <math>\gamma_{-k}^\dagger</math> are particle annihilation and creation operators.

* '''Heisenberg Model''': The Heisenberg model is used in QuasiPhysics to describe the exchange interactions between spins in magnetic materials, leading to the formation of magnons (spin waves):

<math>H = -J \sum_{\langle i,j \rangle} \mathbf{S}_i \cdot \mathbf{S}_j</math>

where:
* <math>H</math> is the Hamiltonian representing the system's total energy,
* <math>J</math> is the exchange interaction constant,
* <math>\mathbf{S}_i</math> and <math>\mathbf{S}_j</math> are spin vectors at sites <math>i</math> and <math>j</math>.

* '''Electron-Phonon Interaction''': The interaction between electrons and phonons is crucial for understanding electrical resistance and superconductivity in materials:

<math>H_{e-ph} = \sum_{k,q} g_q c_k^\dagger c_{k+q} (a_q + a_{-q}^\dagger)</math>

where:
* <math>H_{e-ph}</math> is the electron-phonon interaction Hamiltonian,
* <math>g_q</math> is the coupling constant,
* <math>c_k^\dagger</math> and <math>c_{k+q}</math> are electron creation and annihilation operators,
* <math>a_q</math> and <math>a_{-q}^\dagger</math> are phonon annihilation and creation operators.