Update theory.md

docs/src/theory.md

@@ -63,22 +63,22 @@ Hence, we can keep the pure state description and avoid moving to density matric
 Assuming a unitary evolution for both the system and environment, the system's dynamics can be isolated by tracing out the environmental degrees of freedom. The density operator for the system at time $t$ is described as:
 
 ```math
-\hat{\rho}_S(t) = \Tr_E\left\{\hat{U}(t) \hat{\rho}_S(0) \otimes \hat{\rho}_E(0) \hat{U}^\dagger(t)\right\}.
+\hat{\rho}_S(t) = \text{T}r_E\left\{\hat{U}(t) \hat{\rho}_S(0) \otimes \hat{\rho}_E(0) \hat{U}^\dagger(t)\right\}.
 ```
 
-Initially, the system state, ``\hat{\rho}_S(0)``, can be pure or mixed, and the environment is in a thermal state defined by the inverse temperature ``\beta = (k_B T)^{-1}``. This state is represented by a product of Gaussian states:
+Initially, the system state, $\hat{\rho}_S(0)$, can be pure or mixed, and the environment is in a thermal state defined by the inverse temperature $\beta = (k_B T)^{-1}$. This state is represented by a product of Gaussian states:
 
 ```math
 \hat{\rho}_E(0) = \bigotimes_\omega \frac{e^{-\beta \omega \hat{b}_\omega^\dagger \hat{b}_\omega}}{Z_\omega(\beta)},
 ```
 
-The system's evolution is dictated by the environment's two-time correlation function, which in turn is determined by the spectral density function ``J`` and the temperature ``\beta``:
+The system's evolution is dictated by the environment's two-time correlation function, which in turn is determined by the spectral density function $J$ and the temperature $\beta$:
 
 ```math
 \hat{S}(t) = \int_0^\infty d\omega J(\omega)\left[e^{-i\omega t}(1 + \hat{n}_\omega(\beta)) + e^{i\omega t} \hat{n}_\omega(\beta)\right],```
 ```
 
-To simulate finite temperature effects using a zero-temperature model, we extend the spectral density function to cover both positive and negative frequencies, allowing us to use a pure state description for the environment. This is achieved by defining a new spectral density function ``J(\omega, \beta)`` that incorporates the Boltzmann factors, supporting the entire real axis:
+To simulate finite temperature effects using a zero-temperature model, we extend the spectral density function to cover both positive and negative frequencies, allowing us to use a pure state description for the environment. This is achieved by defining a new spectral density function $J(\omega, \beta)$ that incorporates the Boltzmann factors, supporting the entire real axis:
 
 ```math
 J(\omega, \beta) = \frac{\text{sign}(\omega)J(\left|\omega\right|)}{2} \Big(1 + \coth\Big(\frac{\beta \omega}{2}\Big)\Big).