Лагранж функциясындағы көздік өріс

Type of field appearing in the Lagrangian
In theoretical physics, a source field is a background field coupled to the original field as This term appears in the action in Richard Feynman's path integral formulation and responsible for the theory interactions. In Julian Schwinger's formulation the source is responsible for creating or destroying (detecting) particles. 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 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. 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
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

If one adds to the mass term then Fourier transforms both and to the momentum space, the vacuum amplitude becomes

,

where It is easy to notice that the term in the amplitude above can be Fourier transformed into , i. e.,
Thus, the generating functional is obtained from the partition function as follows. 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 amplitude , the partition function becomes One can introduce , which behaves as Helmholtz free energy in thermal field theories, to absorb the complex number, and hence The function 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

The integration in the definition of the effective action is allowed to be replaced with sum over , i. e., The last equation resembles the thermodynamical relation 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,

The is called mean field obviously because , while is a background classical field. Ward identities, nonlinear sigma models, and low energy effective theories. Back to Green functions of the actions. Since is the Legendre transform of , and defines N points connected correlator , then the corresponding correlator obtained from , known as vertex function, is given by Consequently in the one particle irreducible graphs (usually acronymized as 1PI), the connected 2 point correlator is defined as the inverse of the 2 point correlator, i. e., the usual reduced correlation is , and the effective correlation is For , the most general relations between the N points connected and are

and

One can generalize source to become higher spin source such that becomes Then, the generalized polarization vector is
And the projection operator can be defined as
The symmetric properties of the projection operator make it easier to deal with the vacuum amplitude in the momentum space. Therefore rather that we express it in terms of the correlator in configuration space, we write

.

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 and a source , the vacuum amplitude is 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.

For spin fermion propagator and current as defined above, the vacuum amplitude is 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.