A boson is a particle (or field) with integer spin \(s=0,1,2,\dots\) that obeys Bose–Einstein statistics. The force carriers of the Standard Model are spin-1 gauge bosons — the gluons of QCD, the photon, and the \(W^\pm, Z\) of the electroweak sector — while the Higgs is a spin-0 boson.

The multiparticle state of identical bosons is symmetric under exchange,

\[ \phi(x_1,x_2) = +\,\phi(x_2,x_1), \]

so there is no exclusion principle: any number of bosons can occupy the same quantum state. This underlies collective phenomena such as Bose–Einstein condensation and the classical limit of gauge fields (a coherent state of many photons is an ordinary electromagnetic wave).

By the spin–statistics theorem, bosonic fields are quantized with commutators,

\[ [\phi(\vec x,t),\pi(\vec y,t)] = i\,\delta^{3}(\vec x-\vec y), \]

in contrast to the anticommutators of a fermion. In the path integral formulation a boson is represented by ordinary c-number fields, and Gaussian integration gives an inverse-determinant factor \((\det M)^{-1/2}\).

At finite temperature, the trace defining the partition function forces bosonic fields to be periodic in euclidean time,

\[ \phi(\vec x,\tau+\beta) = +\,\phi(\vec x,\tau), \]

which is the boundary condition used for the gauge field \(A_\mu\) in thermal QCD. The allowed Matsubara frequencies are therefore \(\omega_n = 2\pi n T\) (integer), whereas fermions carry the odd frequencies \(\omega_n=(2n+1)\pi T\).