A little thought will also convince us that β and γ can only be functions of g : by dimensional analysis they cannot depend upon M , and are clearly independent of either n or { x$_{i}$ } . Using the method of characteristics, we can solve (1.5): with the characteristic equation ρ dg ( ρ ) dρ = β ( g ( ρ )), and the initial condition g ( ρ = 1) = g . Equivalent equations to (1.6) and (1.7) can be derived for all quantum field theories, and the definitions of β and γ are identical to those definitions in (1.6) and (1.7). Next we turn to the significance of β and γ . We defined the beta function, β ( g ) in (1.6), and from its definition we can see that the beta function tells us how the renormalised coupling constant changes as we vary the renormalisation scale. The strength of the coupling constant at any particular energy is an important quantity: it determines when perturbation theory is valid. Perturbation theory is the most suc- cessful tool we have for solving quantum field theories: it was used very successfully in QED, so it would be excellent if we could use the same tech- nique for QCD. At low energies QCD is not weakly coupled: the coupling constant is large and perturbation theory is not applicable. But what about at high energies - perhaps we could use perturbation theory then? To see what happens, let us assume that at high energies the theory is weakly coupled, and check that this doesn’t lead us to a reductio ad 8 1.2.4 The beta function Γ ( n $^{)}$( k$_{i}$ ; g, M ) = Γ ( n $^{)}$( k$_{i}$ ; g ( ρ ); M ) exp ( − n 2 ∫ ρ 1 γ ( g ( ρ )) dx x ) (1.8)