In theoretical physics, a source is an abstract concept, developed by Julian Schwinger, motivated by the physical effects of surrounding particles involved in creating or destroying another particle. So, one can perceive sources as the origin of the physical properties carried by the created or destroyed particle, and thus one can use this concept to study all quantum processes including the spacetime localized properties and the energy forms, i.e., mass and momentum, of the phenomena. The probability amplitude of the created or the decaying particle is defined by the effect of the source on a localized spacetime region such that the affected particle captures its physics depending on the tensorial and spinorial nature of the source. An example that Julian Schwinger referred to is the creation of η ∗ {\displaystyle \eta ^{*}} meson due to the mass correlations among five π {\displaystyle \pi } mesons. The same idea can be used to define source fields. Mathematically, a source field is a background field J {\displaystyle J} coupled to the original field ϕ {\displaystyle \phi } as
S source = J ϕ . {\displaystyle S_{\text{source}}=J\phi .}
This term appears in the action in Richard Feynman's path integral formulation and is responsible for the theory interactions. In a collision reaction a source could be other particles in the collision. Therefore, the source appears in the vacuum amplitude acting from both sides on the Green's 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 δ J {\displaystyle \delta J} corresponds to the field ϕ {\displaystyle \phi } , i.e.
δ J = ∫ D ϕ exp ( − i ∫ d 4 x J ( x , t ) ϕ ( x , t ) ) . {\displaystyle \delta J=\int {\mathcal {D}}\phi \,\exp \left(-i\!\int \!d^{4}x\,J(x,t)\phi (x,t)\right).}
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 ϕ {\displaystyle \phi } . When the field ϕ {\displaystyle \phi } 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.
Relation between path integral formulation and source formulation In the Feynman's path integral formulation with normalization N ≡ Z [ J = 0 ] {\displaystyle {\mathcal {N}}\equiv Z[J=0]} , the partition function is given by
Z [ J ] = N ∫ D ϕ exp [ − i ( ∫ d t L ( t ; ϕ , ϕ ˙ ) + ∫ d 4 x J ( x , t ) ϕ ( x , t ) ) ] . {\displaystyle Z[J]={\mathcal {N}}\int {\mathcal {D}}\phi \,\exp \left[-i\left(\int dt~{\mathcal {L}}(t;\phi ,{\dot {\phi }})+\int d^{4}x\,J(x,t)\phi (x,t)\right)\right].}
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