Ivancevic_Applied-Diff-Geom

Geometrical Path Integrals and Their Applications 10136.3.2.3 FieldsLet us now treat the abstract scalar field φ(x) as a coordinate in the sensethat we imagine dividing space up into many little cubes and the averagevalue of the field φ(x) in that cube is treated as a coordinate for that littlecube. Then, we go through the multi–coordinate analogue of the procedurewe just considered above and take the continuum limit. The final result is∫ { ∫Z[J] ∝ D[φ] exp id 4 x(L (φ(x)) + J(x)φ(x) + 1 2 iεφ2 )},where for L we would employ the Klein–Gordon Lagrangian form. In theabove, the dx 0 integral is the same as dτ, while the d 3 ⃗x integral is summingover the sub–Lagrangians of all the different little cubes of space and thentaking the continuum limit. L is the Lagrangian density describing theLagrangian for each little cube after taking the many–cube limit (see [Ryder(1996); Cheng and Li (1984); Gunion (2003)]) for the full derivation).We can now introduce interactions, L I . Assuming the simple form ofthe Hamiltonian, we have∫ { ∫Z[J] ∝ D[φ] exp i}d 4 x (L (φ(x)) + L I (φ(x)) + J(x)φ(x)) ,again using the normalization factor required for Z[J = 0] = 1.For example of Klein Gordon theory, we would useL = L 0 + L I , L 012 [∂ µφ∂ µ φ − µ 2 φ 2 ], L I = L I (φ),where ∂ µ ≡ ∂ x µ and we can freely manipulate indices, as we are working inEuclidean space R 3 . In order to define the above Z[J], we have to includea convergence factor iεφ 2 ,L 0 → 1 2 [∂ µφ∂ µ φ − µ 2 φ 2 + iεφ 2 ], so that∫∫Z[J] ∝ D[φ] exp{i d 4 x( 1 2 [∂ µφ∂ µ φ − µ 2 φ 2 + iεφ 2 ] + L I (φ(x)) + J(x)φ(x))}is the appropriate generating function in the free field theory case.6.3.2.4 GaugesIn the path integral approach to quantization of the gauge theory, we implementgauge fixing by restricting in some manner or other the path integral