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.
[1] 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[2] and spinorial[3] nature of the source.
as This term appears in the action in Richard Feynman's path integral formulation and responsible for the theory interactions.
[5] Therefore, the source appears in the vacuum amplitude acting from both sides on the Green's function correlator of the theory.
[8][9] In terms of the statistical and non-relativistic applications, Schwinger's source formulation plays crucial rules in understanding many non-equilibrium systems.
[10][11] Source theory is theoretically significant as it needs neither divergence regularizations nor renormalization.
On the other hand, one can define the correlation functions for higher order terms, e.g., for
This motivates considering the Hamiltonian of forced harmonic oscillator as a toy model
In light of the relation between partition function and its correlators, the variation of the vacuum amplitude gives
As the integral is in the time domain, one can Fourier transform it, together with the creation/annihilation operators, such that the amplitude eventually becomes[6]
The last result is the Schwinger's source theory for interacting scalar fields and can be generalized to any spacetime regions.
The second term of the last amplitude defines the partition function of free scalar field theory.
, yields the equation of motion, one can redefine the Green's function as the inverse of the operator
Despite the "shoes incident", Weinberg gave the credit to Schwinger for catalyzing this theoretical framework.
This method is commonly used in the path integral formulation of quantum field theory.
The general method by which such source fields are utilized to obtain propagators in both quantum, statistical-mechanics and other systems is outlined as follows.
, which behaves as Helmholtz free energy in thermal field theories,[16] to absorb the complex number, and hence
[20] This construction is indispensable in studying scattering (LSZ reduction formula), spontaneous symmetry breaking,[21][22] Ward identities, nonlinear sigma models, and low-energy effective theories.
[16] Additionally, this theoretical framework initiates line of thoughts, publicized mainly be Bryce DeWitt who was a PhD student of Schwinger, on developing a canonical quantized effective theory for quantum gravity.
For a weak source producing a missive spin-1 particle with a general current
For a weak source in a flat Minkowski background, producing then absorbing a missive spin-2 particle with a general redefined energy-momentum tensor, acting as a current,
Together with help of Ward-Takahashi identity, the projector operator is crucial to check the symmetric properties of the field, the conservation law of the current, and the allowed physical degrees of freedom.
[27][28] Interestingly, massive gravity theories have not been widely appreciated until recently due to apparent inconsistencies obtained in the early 1970's studies of the exchange of a single spin-2 field between two sources.
But in 2010 the dRGT approach[29] of exploiting Stueckelberg field redefinition led to consistent covariantized massive theory free of all ghosts and discontinuities obtained earlier.
The symmetric properties of the projection operator make it easier to deal with the vacuum amplitude in the momentum space.
For example in N-dimensions and for a mixed symmetric massless version of Curtright field
The Lagrangian formulations of massive fields and their conditions were studied by Lambodar Singh and Carl Hagen.
[36][37] The non-relativistic version of the projection operators, developed by Charles Zemach who is another student of Schwinger,[38] is used heavily in hadron spectroscopy.
Zemach's method could be relativistically improved to render the covariant projection operators.