Feynman Path Integral Formulation

4.12 Quantum Hamiltonian for Gauge Theories 127
and cannot be considered small in any sense (in particular they diverge as x → x ′ ,
or, equivalently, become sensitive to an ultraviolet cutoff at short distances).
Furthermore, since one is dealing with a finite number of degrees of freedom,
there are no radiative corrections, no renormalization effects, and thus no scale dependent
couplings arising from the quantum corrections. In other words, the model
is still in many ways essentially classical, in spite of the appearance of some quantum
variables such as ˆπ φ . In particular the short distance, and therefore high particle
density, behavior of the theory, which in the full treatment becomes strongly coupled
at short distance due to the UV growth of the coupling λ, isnot described correctly
by the minisuperspace model.
It would seem that similar concerns could be raised regarding the minisuperspace
approximation to quantum gravity. The infinitely many degrees of freedom of the
metric in this case are just reduced to a few, such as a(t). As a result, the transversetraceless
nature of quantum fluctuations in the linearized limit is no longer apparent.
The spatial quantum fluctuations of the metric, which are expected to acquire divergent
contributions due to the ultraviolet renormalization effects of four-dimensional
quantum field theories, and quantum gravity in particular, are set to zero, and quantum
correlations in the spatial directions are entirely neglected. Finally the highly
degenerate (and therefore genuinely quantum mechanical) nature of the strongly
coupled graviton gas is not taken into account, one more aspect which could, possibly,
be quite relevant for early time cosmology.
4.12 Quantum Hamiltonian for Gauge Theories
It is of interest to see how the Hamiltonian approach has fared for ordinary SU(N)
gauge theories, whose non-trivial infrared properties (confinement, chiral symmetry
breaking) cannot be seen to any order in perturbation theory, and require therefore
some sort of non-perturbative approach, based for example on a strong coupling
expansion. In the continuum one starts from the Yang-Mills action
I = − 1 ∫
4g 2 d 4 xFμν a F μνa , (4.124)
with field strength
F a μν = ∂ μ A a ν − ∂ ν A a μ + gf abc A b μA c ν , (4.125)
and gauge fields are A a μ with (a = 1...N 2 − 1), where the quantities f abc are the
structure constants of the Lie group, such that the generators satisfy [T a ,T b ]=
if abc T c . The gauge invariant energy-momentum tensor for this theory is given by
T μν
= F μρ
a
F ν ρa − 1 4 η μν F ρσ
a
F σρ
a , (4.126)
and in terms of the fields strengths Ea i = Fa
i0
1,2,3) one has
and B i a = − 1 2 ε ijkF jk
a (with i, j,k =