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Concept
Source field
Summary
In theoretical physics, a source field is a background field coupled to the original field as
This term appears in the action in Feynman's path integral formulation and responsible for the theory interactions. In Schwinger's formulation the source is responsible for creating or destroying (detecting) particles. In a collision reaction a source could the other particles in the collision. Therefore, the source appears in the vacuum amplitude acting from both sides on Green function correlator of the theory.
Schwinger's source theory stems from Schwinger's quantum action principle and can be related to the path integral formulation as the variation with respect to the source per se corresponds to the field , i.e.
Also, a source acts effectively in a region of the spacetime. As one sees in the examples below, the source field appears on the right-hand side of the equations of motion (usually second-order partial differential equations) for . When the field is the electromagnetic potential or the metric tensor, the source field is the electric current or the stress–energy tensor, respectively.
In terms of the statistical and non-relativistic applications, Schwinger's source formulation plays crucial rules in understanding many non-equilibrium systems. Source theory is theoretically significant as it needs neither divergence regularizations nor renormalization.
In the Feynman's path integral formulation with normalization , partition function
generates Green's functions (correlators)
One implements the quantum variational methodology to realize that is an external driving source of . From the perspectives of probability theory, can be seen as the expectation value of the function . This motivates considering the Hamiltonian of forced harmonic oscillator as a toy model
where .
In fact, the current is real, that is . And the Lagrangian is . From now on we drop the hat and the asterisk. Remember that canonical quantization states . In light of the relation between partition function and its correlators, the variation of the vacuum amplitude gives
where .
Official source
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