Feynman Path Integral Formulation

4.13 Lattice Regularized Hamiltonian for Gauge Theories 129which then implies that A is transverse.The next step is to separate out the transverse and longitudinal parts in F 0i by settingF L 0i = −∇ if and F T 0i = E i,solveforf using the constraint equation of Eq. (4.134),and eliminate A 0 to obtain the physical Hamiltonian∫H = 1 2d 3 x [ E 2 i + B 2 i +(∇ i f) 2 ] . (4.137)The last terms represents the instantaneous Coulomb interaction that is characteristicof this gauge.4.13 Lattice Regularized Hamiltonian for Gauge TheoriesThe previous section discussed the Hamiltonian formulation for Yang-Mills theoriesin the continuum. In view of applying the Hamiltonian method to quantum gravity, itis of interest to see how such an approach is implemented in SU(N) gauge theories atstrong coupling first. In gauge theories the only known way to deal in a systematicway with the strong coupling problem is to formulate gauge theories on a lattice,obtain an appropriate lattice Hamiltonian by taking the continuous time limit, andfrom it develop a strong coupling expansion for physical states such as glueballsand hadrons. One would hope that a similar procedure could be applied to gravityas well. It is in order to understand the general ideas and issues better that the gaugetheory methods will be described in some detail first.A lattice regularized form of the gauge action in Eq. (4.124) was given in(Wilson, 1974). The theory is defined on a d-dimensional hyper-cubic lattice withlattice spacing a, vertices labeled by an index n and directions by μ (see Fig 4.4).The group elements U nμ = expiagA a μT a are defined in the fundamental representation,and reside on the links of the lattice. The pure gauge Euclidean action involvesa sum of traces of path-ordered products [with U −μ (n + ν) =U † μ(n)] ofunitaryU μ (n) matrices around an elementary square loop (“plaquettes”, here denoted by ✷),I[U] =− a4−d4g 2∑tr [ UUU † U † + h.c. ] . (4.138)✷From now on we will discuss exclusively the case d = 4. The action is locally gaugeinvariant with respect to the changeU μ (n) → V † (n)U μ (n)V (n + ν) , (4.139)where V is an arbitrary SU(N) matrix defined on the lattice sites. The product of fourU matrices around a plaquettes can be shown, using the Baker-Hausdorff formulaexp(A)exp(B) =exp(A + B + 1 2[A,B]+...), to give the exponential of the latticefield strength tensor in the limit of small lattice spacing,