include zero. This is thought to be very closely linked to the property of confinement , which is simply a statement that gluons cannot exist on their own, but only in colourless bound states. Lattice gauge theories have shown to the satisfaction of most that quarkless QCD exhibits such a phenomena, but solving this problem with due mathematical rigour still remains one of the Millennium Prize Problems. Experimentally it is certainly true that there is a mass gap in QCD. We have already seen in section 1.2.4 that massless QCD anomalously breaks its classical scale invariance. There is a second classical symme- try to the massless QCD Lagrangian: that of chiral symmetry, meaning that the left and right handed quarks transform independently. The Lagrangian of massless QCD is given by taking equation (1.1) and simply setting m = 0: (1.31) possesses a chiral symmetry which can be seen if we write ψ as    ψ$_{L}$ ψ$_{R}$    and choose the chiral representation of the gamma matrices: then (suppressing the gauge field which doesn’t affect the analysis), equation (1.31) becomes 18 1.2.8 Chiral Symmetry Breaking L = ¯ ψ ( i∂$_{ν}$ γ ν + gA a $_{µ}$γ$^{µ}$t a ) ψ − 1 4 F a $_{µν}$F µνa (1.31) γ 0 =    0 GLYPH<BD> GLYPH<BD> 0    γ i =    0 σ i − σ i 0    L = $_{iψ}$† $_{L}$( ∂$_{0}$ − σ. ▽ ) ψ$_{L}$ + $_{iψ}$† $_{R}$( ∂$_{0}$ + σ. ▽ ) ψ$_{R}$ (1.32)