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Bosons have symmetric many-particle states, so any number may occupy the same one-particle state. At low temperature, an ideal Bose gas therefore places a macroscopic population in its lowest one-particle state; below the critical temperature this is Bose-Einstein condensation.

Fermions have antisymmetric many-particle states and obey the Pauli exclusion principle, so each one-particle state can hold at most one fermion of each available internal label. At zero temperature, a noninteracting Fermi gas fills distinct states in increasing order of energy up to the Fermi energy. Its many-particle ground state is thus the filled Fermi sea.

For distinguishable particles with one-particle Hilbert spaces ${\mathcal{H}}_1,\dots ,{\mathcal{H}}_N$, the multiparticle quantum state belongs toA product of one-particle states gives a product vector, and arbitrary linear combinations give entangled states. A noninteracting Hamiltonian is the sum of one-particle Hamiltonians acting on their respective tensor factors, so product-state energies add.

For two identical particles, the particle exchange operator $P_{12}$ swaps every degree of freedom. Since $P_{12}^2=I$, its eigenvalues are $\pm 1$. Particles whose allowed total states satisfyare bosons, while those satisfyingare fermions. Thus bosonic states are symmetric and fermionic states antisymmetric under interchange. The same condition applies to every transposition in a system of more than two identical particles.

Spin-one particles are bosons, so their total two-particle state must be symmetric. Write $|m,n⟩=|m{⟩}_1|n{⟩}_2$ for $m,n\in \{1,0,-1\}$. A basis of symmetric spin states isDenote these six states by ${\chi }_1^{+},\dots ,{\chi }_6^{+}$. A basis of antisymmetric spin states isThis is the six-plus-three decomposition of two identical spin-one bosons.

In the ground orbital both particles occupy $n=0$, so the spatial wavefunctionis symmetric. The complete list of ground states is therefore

At the first excited energy, one particle occupies $n=0$ and the other $n=1$. The normalized symmetric and antisymmetric spatial states areOverall bosonic symmetry permits symmetric spatial times symmetric spin and antisymmetric spatial times antisymmetric spin. Hence all first excited states areThe ground and first excited degeneracies are therefore six and nine.

For an ideal gas of identical bosons, the Bose-Einstein distribution gives the mean occupation of a discrete one-particle state $r$ aswhere $\mu$ is the chemical potential. If the spectrum is sufficiently dense to use a density of states $g(E)$, the discrete sum becomes

Photons are bosons with two polarizations and temperature $T$, so ${\rho }_{\gamma }=({\pi }^2/30)2T^4$. Each relativistic neutrino species includes a neutrino and antineutrino helicity state, is fermionic, and has temperature $T_{\nu }=(4/11{)}^{1/3}T$. ConsequentlyAdding the two contributions gives the defining expression for the effective number of neutrino species:

In natural units, the equilibrium phase-space distribution function iswhere the minus sign gives the Bose-Einstein distribution for bosons and the plus sign gives the Fermi-Dirac distribution for fermions.

For $\mathcal{N}$ supersymmetries and $k$ saturated two-by-two central-charge blocks, $2k$ complex oscillators act trivially. A spin-$j$ Clifford vacuum therefore gives $(2j+1)2^{2\mathcal{N}-2k}$ states and preserves $4k$ of the $4\mathcal{N}$ real supercharges. With $\mathcal{N}=2$ and a scalar vacuum, the shortened multiplet has two boson states and two fermion states, before any CPT completion of a supermultiplet.

A supergravity multiplet is a supermultiplet containing a graviton and its supersymmetric partners. In four-dimensional N=8 supergravity, the irreducible massless supermultiplet contains $128$ boson and $128$ fermion states.

In an unbroken supersymmetric vacuum, physical boson and fermion partners have equal masses because supercharges commute with four-momentum. For one canonical chiral superfield, fluctuations about $W^{\prime }(v)=0$ have mass $|W^{\prime \prime }(v)|$ for both the complex scalar field and Weyl spinor.

Supersymmetry extends spacetime symmetry by odd supercharges that interchange boson and fermion states. Their Super-Poincaré algebra pairs positive-energy states into supermultiplets and gives energy positivity in global supersymmetry. Superspace and superfields package the component fields into representations of this symmetry.