Degrees of freedom

The four basic degrees of freedom of homogeneous INM are the isoscalar scalar density $\rho_0$, the isovector scalar density $\rho_1$, the isoscalar vector density s₀, and the isovector vector density s₁. They can be expressed through the usual neutron and proton, spin-up, and spin-down densities in the following way.

$$\rho_0 \rho_{{\rm n}\uparrow}+ \rho_{{\rm n}\downarrow}+ \rho_{{\rm p}\uparrow}+ \rho_{{\rm p}\downarrow},$$ =

$$\rho_1 \rho_{{\rm n}\uparrow}+ \rho_{{\rm n}\downarrow}- \rho_{{\rm p}\uparrow}- \rho_{{\rm p}\downarrow},$$ =

$$\rho_{{\rm n}\uparrow}- \rho_{{\rm n}\downarrow}+ \rho_{{\rm p}\uparrow}- \rho_{{\rm p}\downarrow},$$ s₀ =

$$\rho_{{\rm n}\uparrow}- \rho_{{\rm n}\downarrow}- \rho_{{\rm p}\uparrow}+ \rho_{{\rm p}\downarrow}.$$ s₁ = (35)

Similarly, densities of protons and neutrons with spin up and down can be expressed as:

$$\rho_{{\rm n}\uparrow} {\textstyle\frac{{1}}{{4}}} ( \rho_0 + \rho_1 + s_0 + s_1 ) = {\textstyle\frac{{1}}{{4}}} ( 1 + I_\tau + I_\sigma + I_{\tau \sigma} ) \; \rho_0 ,$$ =

$$\rho_{{\rm n}\downarrow} {\textstyle\frac{{1}}{{4}}} ( \rho_0 + \rho_1 - s_0 - s_1 ) = {\textstyle\frac{{1}}{{4}}} ( 1 + I_\tau - I_\sigma - I_{\tau \sigma} ) \; \rho_0 ,$$ =

$$\rho_{{\rm p}\uparrow} {\textstyle\frac{{1}}{{4}}} ( \rho_0 - \rho_1 + s_0 - s_1 ) = {\textstyle\frac{{1}}{{4}}} ( 1 - I_\tau + I_\sigma - I_{\tau \sigma} ) \; \rho_0 ,$$ =

$$\rho_{{\rm p}\downarrow} {\textstyle\frac{{1}}{{4}}} ( \rho_0 - \rho_1 - s_0 + s_1 ) = {\textstyle\frac{{1}}{{4}}} ( 1 - I_\tau - I_\sigma + I_{\tau \sigma} ) \; \rho_0 ,$$ =

where $I_\tau = \rho_1 / \rho_0$ is the relative isospin excess, $I_\sigma = s_0 / \rho_0$ is the relative spin excess, and $I_{\sigma \tau} = s_1 / \rho_0$ is the relative spin-isospin excess, with $-1 \leq I_i \leq +1$.

In symmetric unpolarized INM I_i = 0, while in asymmetric INM $\rho_1 \neq 0$. Polarized INM has $s_0 \neq 0$, and spin-isospin polarized nuclear matter has $s_1 \neq 0$.