Functional Integral Calculation For Correlation Functions In QFT

The path-integral formalism quantum mechanics, created by Feynman[1,2] is an importantbenchmark in the development of quantum theory. Many varieties of this formalism and newmethods to solve problems, which seemed before unsolvable, have been suggested since then[3-9].The extension of the path-integral formalism to the quantum field theory was started already in [2],where field variables play the role of quantum spacial coordinates, and the resulting path integralis becoming the functional integral. The functional integration formalism is another representationof quantum field theory. The fields can be quantized and any n-point correlation functions andFeynman rules can be derived directly by using the method of functional integral and the Lorentzinvariance is maintained in this process.It is however important that the usual path integral concerns spacial coordinates or fieldvariables, but not the time coordinate which plays only the ordering role. Based on the propertime coordinate introduced by Fock and Schwinger[10,11], the space and time coordinate canbe unified in the path integral. This formalism of path integral, which is based on the Fock-Schwinger proper time and Feynman path integral, is called the Fock-Feynman-Schwinger repre-sentation(FFSR)[12,13]. Nowadays, the FFSR has been widely applied in many fields. In quantumelectrodynamics (QED), an more accurate description for the composite particles existing as boundstates can be obtained by using the FFSR instead of B-S equation (Up to date, actual solutions tothe B-S equation have only been constructed in the ladder approximation). In addition, the FFSRis explicitly gauge invariant and thus readily extend to QCD to describe the hadronic systems withquark-gluon confinement. The extension of FFSR to nonzero temperature field theory forms thebasis of a systematic study of the role of nonperturbative configurations in the temperature phasetransitions.As stated above, the path integtral (or functional integral) is very important in quantumphysics. The knowledge on path integral make us have a better understanding of the quantumfield theory. Furthermore, the FFSR based on proper time coordinate is an widely used methoddealing with the nonperturbative effects in hadronic states. In most books on quantum field theory,the correlation functions and Feynman rules are derived by using canonical quantization and Wicktheorem, while the functional integral method is seldom involved comparatively. In this thesis,we give a detailed derivation for the correlation functions and Feynman rules by using functionalintegral.We calculate the correlation functions and give the Feynman rules for the real scalar field,electromagnetic field, Dirac field, quantum electrodynamics (QED) via the functional integral. Firstly, the functional integral is introduced briefly and corresponding representation for the am-plitude in quantum mechanism is derived with a particle moving in one dimension as an example.Then, the real scalar field, electromagnetic field and Dirac field are considered respectively. Start-ing from the generating functional of correlation functions, we evaluate the2-point,3-point and4-point correlation functions and give related Feynman rules. For the real scalar field, the free andinteracting situation with φ4coupling are considered. while for electromagnetic and Dirac field,due to no self-coupling, only the free situation is considered. Finally, we discuss the interaction be-tween electromagnetic and Dirac field (QED), calculate the2-point,3-point, some typical4-pointcorrelation functions and the Feynman rule for electromagnetic coupling.For the interacting situation (the real scalar field, QED), the interacting term is expressedwith derivatives acting on the”source” term, and then the interacting generating functional ofcorrelation functions can be derived from the free one via perturbative expansion. For intuition,the Feynman diagrams are drawn after the expressions of correlation functions and some analysisis given briefly. It is found that the correlation functions obtained by this method contain manyunconnected parts. For this reason, we define the generating functional of connected correlationfunctions and calculate the correlation functions containing only the connected parts.