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QCD Vacuum

The QCD vacuum is not empty but a complex state filled with gluons and quark-antiquark pairs, mediating the strong force.
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The statement of the theorem

The QCD vacuum state ∣vac⟩|\text{vac}\rangle is defined by the condition that the vacuum expectation value of the energy-momentum tensor vanishes: ⟨vac∣Tμν∣vac⟩=0\langle \text{vac} | T_{\mu\nu} | \text{vac} \rangle = 0. However, due to non-perturbative effects, the vacuum is characterized by non-zero expectation values of gluon condensates and quark condensates: ⟨vac∣GμνaGaμν∣vac⟩=αsπ⟨vac∣G2∣vac⟩≠0\langle \text{vac} | G^a_{\mu\nu} G^{a\mu\nu} | \text{vac} \rangle = \frac{\alpha_s}{\pi} \langle \text{vac} | G^2 | \text{vac} \rangle \neq 0 and ⟨vac∣ψˉψ∣vac⟩=−12mq2g2≠0\langle \text{vac} | \bar{\psi} \psi | \text{vac} \rangle = -\frac{1}{2} \frac{m_q^2}{g^2} \neq 0. These condensates signal spontaneous symmetry breaking.