Bose-Einstein Condensation in the Relativistic Ideal Bose Gas

The Bose-Einstein condensation (BEC) critical temperature in a relativistic ideal Bose gas of identical bosons, with and without the antibosons expected to be pair-produced abundantly at sufficiently hot temperatures, is exactly calculated for all boson number densities, all boson point rest masses, and all temperatures. The Helmholtz free energy at the critical BEC temperature is lower with antibosons, thus implying that omitting antibosons always leads to the computation of a metastable state.

Figure 1

BEC $T_c$s (in units $m{c}^2/k_B$) as function of boson number-density $n$ expressed in dimensionless form as ${\hslash }^{3}n/m^3{c}^3$. Thick curve labeled “exact $B\overline{B}$” is exact numerical result of (8) that corresponds to BEC $T_c$ in a RIBG with both bosons $B$ and antibosons $\overline{B}$. Thin full straight line labeled “UR-$B\overline{B}$” is the ultrarelativistic limit (9) for $d=3$ with both kinds of bosons. Dashed straight line labeled “NR-$B\overline{B}$” is its corresponding nonrelativistic limit (5) for $d=3$ and tends asymptotically to the “exact $B\overline{B}$” curve at smaller $n$.

Figure 2

Same as Fig. 1 comparing exact $B\overline{B}$ RIBG $T_c$ extracted from (8) against that of exact $B$ from (4).

Figure 3

Difference between exact Helmholtz free energy density $F/V$ (in units of $nm{c}^2$ which is the total rest-mass energy density) (full curve labeled “exact $B\overline{B}$”) and that without antibosons (dashed curve labeled “exact $B$”) using same horizontal axes as in Figs. 1, 2. The inset figure displays the two free energies.