Quantum Kinetic Theory

3. Conservation laws

Let us now go back to the particle densities (2.4). In general these densities will not correspond to a conservation law, because of chemical, nuclear or other reactions taking place in the system. First we must ask ourselves which particle properties are conserved, i.e. what charges are carried by the particles. Assume, for instance, that the particles carry an electromagnetic charge $q_{k}$ . The conserved charge density is then

N (x) = \sum_{k} q_{k} N_{k} (x),

and the corresponding current density

J (x) = \sum_{k} q_{k} J_{k} (x),

J_{k} (x) = \int {d ω}_{k} v_{k} f_{k} (x, p),

with the velocity derived from the kinetic energy $ϵ_{k} (p)$ of the particles:

v_{k} (p) = \frac{\partial ϵ_{k} (p)}{\partial p} .

In general, particles will carry more than one conserved charge $q_{A k}$ , $A = 1, 2, . . .,$ like atomic weight, lepton charge, baryon charge, etc. Moreover, for non-relativistic systems, total mass is strictly conserved which means that the mass number $m_{k} / m$ , with $m$ some convenient reference mass, is one of the conserved quantum numbers. We take this into account by defining

J_{A}^{μ} (x) = \sum_{k} q_{A k} J_{k}^{μ} (x) .

For the sake of a compact notation, we combined charge and current density into one four-vector $J_{A}^{μ} = (N_{A}, J_{A})$ . This allows us to write the continuity equation

\partial_{μ} J_{A}^{μ} (x) = 0 .

Any conserved quantum number implies the existence of a conserved macroscopic current density.

Exercise 2

Consider a $ν$ npe system. Electromagnetic charge, lepton number and baryon number are conserved. Write down the corresponding quantum numbers and conserved currents, and tabulate the allowed two-particle reactions.

Quantum Kinetic Theory

3. Conservation laws

Conserved charges

Exercise 2

Momentum and energy