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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 meson mesons.

The same idea can be used to define source fields. Mathematically, a source field is a background 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, 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 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

, the partition function is given by

to generate Green's functions (correlators are in their Heisenberg pictures term, the coupling constant

is an external driving source. From the perspectives of probability theory. This motivates considering the Hamiltonian of forced harmonic oscillator as a toy model

. . From now on we drop the hat and the asterisk. Remember that canonical quantization. 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

the Green's function is revealed

The last result is the Schwinger's source theory for interacting scalar fields and can be generalized to any spacetime regions..

Source theory for scalar fields

Causal perturbation theory by acting on the vacuum state. Thus, the full vacuum amplitude is given by

is the propagator (correlator) of the sources. The second term of the last amplitude defines the partition function of free scalar field theory is given by

to the momentum space, the vacuum amplitude becomes

if and only if. Thus, the generating functional is obtained from the partition function as follows.. Consequently, the propagator is defined by varying the partition function as follows.

This motivates discussing the mean field approximation below.

Effective action, mean field approximation, and vertex functions

Based on Schwinger's source theory, Steven Weinberg established the foundations of the effective field theory, which is widely appreciated among physicists. Despite the "shoes incident", Weinberg gave the credit to Schwinger for catalyzing this theoretical framework.

All Green's functions may be formally found via Taylor expansion of the partition sum considered as a function of the source fields. 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. Upon redefining the partition function in terms of Wick-rotated, which behaves as Helmholtz free energy in thermal field theories, to absorb the complex number is also called reduced quantum action. And with help of Legendre transform, we can invent a "new" effective energy functional, or effective action, as

with the transforms

. between Helmholtz free energy and entropy. It is now clear that thermal and statistical field theories stem fundamentally from functional integrations and functional derivatives. Back to the Legendre transforms,

is called mean field is a background classical field., so the vacuum amplitude can be reintroduced as

is defined as

is the action of the free Lagrangian. The last two integrals are the pillars of any effective field theory. This construction is indispensable in studying scattering (LSZ reduction formula), spontaneous symmetry breaking, Ward identities, nonlinear sigma models, and low-energy effective theories. 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.

defines N-points connected, known as vertex function. Consequently, in the one particle irreducible graphs (usually acronymized as 1PI are obtained from Faà di Bruno's formula

and

Source theory for fields

Vector fields

For a weak source producing a massive spin-1 particle, the vacuum amplitude is

in its rest frame. Then, the amplitude gives

. The last result matches with the used propagator in the vacuum amplitude in the configuration space, that is,

, the chosen Feynman–'t Hooft gauge-fixing, the chosen Landau gauge-fixing makes the spin-1 massive. The massless case is obvious as studied in quantum electrodynamics. The massive case is more interesting as the current is not demanded to conserved. However, the current can be improved in a way similar to how the Belinfante-Rosenfeld tensor is improved so it ends up being conserved. And to get the equation of motion for the massive vector, one can define

to get a definition of the massive spin-1 field

. Thus, the equation of motion can be written in any of the following forms

Massive totally symmetric spin-2 fields

For a weak source in a flat Minkowski background, producing then absorbing a massive spin-2 particle with a general redefined energy-momentum tensor is the vacuum polarization tensor, the vacuum amplitude in a compact form is

This amplitude in momentum space gives (transpose is imbedded)

, where the projection operator, or the Fourier transform of Jacobi field operator obtained by applying Peierls braket on Schwinger's variational principle,.

In N-dimensional flat spacetime, 2/3 is replaced by 2/(N−1). And for massless spin-2 fields, the projection operator is defined as.

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.

appear in the early versions of massive gravity theories. Interestingly, massive gravity theories have not been widely appreciated until recently due to apparent inconsistencies obtained in the early 1970s studies of the exchange of a single spin-2 field between two sources. But in 2010 the dRGT approach of exploiting Stueckelberg field redefinition led to consistent covariantized massive theory free of all ghosts and discontinuities obtained earlier.

and follows the same procedure used to define massive spin-1 fields, then it is easy to define massive spin-2 fields as

according to Belinfante-Rosenfeld construction. Thus, the equation of motion

becomes

, so the equation of motion is simplified as

Massive totally symmetric arbitrary integer spin fields

higher-spin . of the quantized electromagnetic vector potential, the addition theorem of spherical harmonics states that

Also, the representation theory of the space of complex-valued homogeneous polynomials on a unit (N-1)-sphere defines the polarization tensor as

in configuration space, we write

Mixed symmetric arbitrary spin fields

Also, it is theoretically consistent to generalize the source theory to describe hypothetical gauge fields with antisymmetric and mixed symmetric properties in arbitrary dimensions and arbitrary spins. But one should take care of the unphysical degrees of freedom in the theory. For example, in N-dimensions and for a mixed symmetric massless version of Curtright field which for a theory in N=4 makes the source eventually reveal that it is a theory of a non physical field. However, the massive version survives in N≥5.

Arbitrary half-integer spin fields

For spin- fermion propagator as defined above, the vacuum amplitude is

In momentum space the reduced amplitude is given by

For spin- Rarita-Schwinger to get

, the above results can be generalized to

is obtained from the properties of the projection operator, the tracelessness of the current, and the conservation of the current after being projected by the operator. These conditions can be derived from the Fierz-Pauli and the Fang-Fronsdal conditions on the fields themselves. The Lagrangian formulations of massive fields and their conditions were studied by Lambodar Singh and Carl Hagen. The non-relativistic version of the projection operators, developed by Charles Zemach who is another student of Schwinger, is used heavily in hadron spectroscopy. Zemach's method could be relativistically improved to render the covariant projection operators.