Ivancevic_Applied-Diff-Geom

1014 Applied Differential Geometry: A Modern Introductionover gauge fields ∫ D[A µ ]. In other words we will write instead∫∫Z[J] ∝ D[A µ ] δ (some gauge fixing condition) exp{i d 4 xL (A µ )}.A common approach would be to start with the gauge conditionL = − 1 4 F µνF µν − 1 2 (∂µ A µ ) 2where the electrodynamic field tensor is given by F µν = ∂ µ A ν − ∂ ν A µ , andcalculate∫{ ∫}Z[J] ∝ D[A µ ] exp i d 4 x [L(A µ (x)) + J µ (x)A µ (x)]as the generating function for the vacuum expectation values of time orderedproducts of the A µ fields. Note that J µ should be conserved (∂ µ J µ = 0)in order for the full expression L(A µ ) + J µ A µ to be gauge–invariant underthe integral sign when A µ → A µ + ∂ µ Λ. For a proper approach, see [Ryder(1996); Cheng and Li (1984); Gunion (2003)].6.3.3 Riemannian–Symplectic GeometriesIn this section, following [Shabanov and Klauder (1998)], we describe pathintegral quantization on Riemannian–symplectic manifolds. Let ˆq j be aset of Cartesian coordinate canonical operators satisfying the Heisenbergcommutation relations [ˆq j , ˆq k ] = iω jk . Here ω jk = −ω kj is the canonicalsymplectic structure. We introduce the canonical coherent states as |q〉 ≡e iqj ω jk ˆq k |0〉, where ω jn ω nk = δ k j , and |0〉 is the ground state of a harmonicoscillator with unit angular frequency. Any state |ψ〉 is given as a functionon phase–space in this representation by 〈q|ψ〉 = ψ(q). A general operator can be represented in the form  = ∫ dq a(q)|q〉〈q|, where a(q) is thelower symbol of the operator and dq is a properly normalized form of theLiouville measure. The function A(q, q ′ ) = 〈q|Â|q′ 〉 is the kernel of theoperator.The main object of the path integral formalism is the integral kernel ofthe evolution operatorK t (q, q ′ ) = 〈q|e −itĤ|q ′ 〉 =q(t)=q∫D[q] e i R t0 dτ( 1 2 qj ω jk ˙q k −h) . (6.16)q(0)=q ′