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theoretical physics Theoretical physics is a branch of physics that employs mathematical models and abstractions of physical objects and systems to rationalize, explain, and predict List of natural phenomena, natural phenomena. This is in contrast to experimental p ...
, a source is an abstract concept, developed by
Julian Schwinger Julian Seymour Schwinger (; February 12, 1918 – July 16, 1994) was a Nobel Prize-winning American theoretical physicist. He is best known for his work on quantum electrodynamics (QED), in particular for developing a relativistically invariant ...
, motivated by the physical effects of surrounding particles involved in creating or destroying another
particle In the physical sciences, a particle (or corpuscle in older texts) is a small localized object which can be described by several physical or chemical properties, such as volume, density, or mass. They vary greatly in size or quantity, from s ...
. 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 In quantum mechanics, a probability amplitude is a complex number used for describing the behaviour of systems. The square of the modulus of this quantity at a point in space represents a probability density at that point. Probability amplitu ...
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 In mathematics and physics, a tensor field is a function assigning a tensor to each point of a region of a mathematical space (typically a Euclidean space or manifold) or of the physical space. Tensor fields are used in differential geometry ...
and spinorial nature of the source. An example that Julian Schwinger referred to is the creation of \eta^* meson due to the mass correlations among five \pi mesons. Same idea can be used to define source fields. Mathematically, a source field is a ''background'' field J coupled to the original field \phi as S_\text = J\phi. This term appears in the action in
Richard Feynman Richard Phillips Feynman (; May 11, 1918 – February 15, 1988) was an American theoretical physicist. He is best known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics, the physics of t ...
's
path integral formulation The path integral formulation is a description in quantum mechanics that generalizes the stationary action principle of classical mechanics. It replaces the classical notion of a single, unique classical trajectory for a system with a sum, or ...
and 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 \delta J corresponds to the field \phi, i.e. \delta J = \int \mathcal\phi \, \exp\left(-i\!\int\! d^4x \, 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 equation In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives. The function is often thought of as an "unknown" that solves the equation, similar to ho ...
s) for \phi. When the field \phi is the electromagnetic potential or the
metric tensor In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows ...
, the source field is the
electric current An electric current is a flow of charged particles, such as electrons or ions, moving through an electrical conductor or space. It is defined as the net rate of flow of electric charge through a surface. The moving particles are called charge c ...
or the
stress–energy tensor The stress–energy tensor, sometimes called the stress–energy–momentum tensor or the energy–momentum tensor, is a tensor physical quantity that describes the density and flux of energy and momentum in spacetime, generalizing the stress ...
, 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 \mathcal\equiv Z =0/math>, the partition function is given by Z = \mathcal \int \mathcal\phi \, \exp\left i\left(\int dt ~ \mathcal(t;\phi,\dot)+ \int d^4x \, J(x,t) \phi(x,t)\right)\right One can expand the current term in the exponent \mathcal \int \mathcal\phi ~ \exp\left(-i \int d^4x \, J(x,t)\phi(x,t)\right) = \mathcal \sum^_ \frac \int d^4x_1 \cdots \int d^4x_n J(x_1) \cdots J(x_1) \left\langle \phi(x_1) \cdots \phi(x_n) \right\rangle to generate
Green's functions In mathematics, a Green's function (or Green function) is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions. This means that if L is a linear diff ...
( correlators) G(t_1,\cdots,t_n) = ^n \left.\frac\_, where the fields inside the expectation function \langle\phi(x_1)\cdots\phi(x_n)\rangle are in their Heisenberg pictures. On the other hand, one can define the correlation functions for higher order terms, e.g., for \frac m^2 \phi^2 term, the coupling constant like m is promoted to a spacetime-dependent source \mu(x) such that i \frac \left.\frac Z ,\mu\_ = \left\langle \tfrac \phi^2 \right\rangle. One implements the quantum variational methodology to realize that J is an ''external driving source'' of \phi. From the perspectives of probability theory, Z can be seen as the expectation value of the function e^ . This motivates considering the Hamiltonian of forced harmonic oscillator as a toy model \mathcal = E \hat^ \hat - \frac \left(J\hat^ + J^\hat\right) where E^2 = m^2 + \mathbf^2 . In fact, the current is real, that is J=J^. And the Lagrangian is \mathcal=i\hat^\partial_0(\hat)-\mathcal . From now on we drop the hat and the asterisk. Remember that
canonical quantization In physics, canonical quantization is a procedure for quantizing a classical theory, while attempting to preserve the formal structure, such as symmetries, of the classical theory to the greatest extent possible. Historically, this was not quit ...
states \phi\sim (a^+a). In light of the relation between partition function and its correlators, the variation of the vacuum amplitude gives \delta_J\langle0,x'_0, 0,x''_0\rangle_J = i \left\langle0,x'_0\ \int^_dx_0 ~ \delta J _J, where x_0'>x_0> x_0'' . 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 _J = \exp. It is easy to notice that there is a singularity at f=E . Then, we can exploit the i\varepsilon-prescription and shift the pole f-E+i\varepsilon such that for x_0> x_0' the Green's function is revealed \begin &_J = \exp \\ ex&\Delta(x_0-x'_0) = \int \frac\frac \end The last result is the Schwinger's source theory for interacting scalar fields and can be generalized to any spacetime regions. The discussed examples below follow the metric \eta_=\text(1,-1,-1,-1) .


Source theory for scalar fields

Causal perturbation theory explains how sources weakly act. For a weak source emitting spin-0 particles J_e by acting on the
vacuum state In quantum field theory, the quantum vacuum state (also called the quantum vacuum or vacuum state) is the quantum state with the lowest possible energy. Generally, it contains no physical particles. However, the quantum vacuum is not a simple ...
with a probability amplitude \langle 0, 0\rangle_\sim1, a single particle with momentum p and amplitude \langle p, 0\rangle_ is created within certain spacetime region x'. Then, another weak source J_a absorbs that single particle within another spacetime region x such that the amplitude becomes \langle 0, p\rangle_. Thus, the full vacuum amplitude is given by _ \sim 1 + \frac \int dx \, dx' \, J_a(x) \Delta(x-x') J_e(x') where \Delta(x-x') is the propagator (correlator) of the sources. The second term of the last amplitude defines the partition function of free scalar field theory. And for some interaction theory, the Lagrangian of a scalar field \phi coupled to a current J is given by \mathcal = \tfrac \partial_ \phi \partial^ \phi - \tfrac m^2 \phi^2 + J\phi. If one adds -i\varepsilon to the mass term then Fourier transforms both J and \phi to the momentum space, the vacuum amplitude becomes \langle 0, 0\rangle = \exp, where \tilde(p) = \phi(p) + \frac. It is easy to notice that the \tilde(p) \left(p_ p^ - m^2 + i \varepsilon\right) \tilde(-p) term in the amplitude above can be Fourier transformed into \tilde(x) \left(\Box + m^2\right) \tilde(x) = \tilde(x) \, J(x) , i.e., the equation of motion \left(\Box + m^2\right) \tilde = J . As the variation of the free action, that of the term \frac \partial_ \phi \partial^ \phi - \frac m^2 \phi^2 , yields the equation of motion, one can redefine the Green's function as the inverse of the operator G(x_1,x_2) \equiv ^ such that \left(\Box_ + m^2\right) G(x_1,x_2) = \delta(x_1-x_2)
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
\left(p_ p^ - m^2\right) G(p) = 1, which is a direct application of the general role of functional derivative \frac=\delta(x_1-x_2). Thus, the generating functional is obtained from the partition function as follows. The last result allows us to read the partition function as Z = Z \exp\left(\tfrac \left\langle J(y) \Delta(y-y') J(y')\right\rangle\right) , where Z = \int \mathcal\tilde \, \exp\left(-i \int dt \left tfrac \partial_ \tilde \partial^ \tilde - \tfrac \left(m^2 - i \varepsilon\right) \tilde^2\rightright), and \langle J(y)\Delta(y-y')J(y')\rangle is the vacuum amplitude derived by the source \langle0, 0\rangle_ . Consequently, the propagator is defined by varying the partition function as follows. \begin _ &= \frac \frac _ \\ .5ex&= _ \\ .5ex&= \Delta(x-x'). \end This motivates discussing the mean field approximation below.


Effective action, mean field approximation, and vertex functions

Based on Schwinger's source theory,
Steven Weinberg Steven Weinberg (; May 3, 1933 – July 23, 2021) was an American theoretical physicist and Nobel laureate in physics for his contributions with Abdus Salam and Sheldon Glashow to the unification of the weak force and electromagnetic inter ...
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 In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
of the partition sum considered as a function of the source fields. This method is commonly used in the
path integral formulation The path integral formulation is a description in quantum mechanics that generalizes the stationary action principle of classical mechanics. It replaces the classical notion of a single, unique classical trajectory for a system with a sum, or ...
of
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines Field theory (physics), field theory and the principle of relativity with ideas behind quantum mechanics. QFT is used in particle physics to construct phy ...
. 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 W -i\ln(\langle 0, 0 \rangle_) , the partition function becomes Z e^ . One can introduce F iW , which behaves as
Helmholtz free energy In thermodynamics, the Helmholtz free energy (or Helmholtz energy) is a thermodynamic potential that measures the useful work obtainable from a closed thermodynamic system at a constant temperature ( isothermal). The change in the Helmholtz ene ...
in thermal field theories, to absorb the complex number, and hence \ln Z F . The function F is also called ''reduced quantum action''. And with help of Legendre transform, we can invent a "new" ''effective energy'' functional, or effective action, as \Gamma bar= W - \int d^4x \, J(x) \bar(x), with the transforms \begin &\frac = \bar~, & &\frac\Bigg, _ = \langle\phi\rangle~ , \\ .2ex&\frac\Bigg, _ = -J ~,& &\frac\Bigg, _ = 0. \end The integration in the definition of the effective action is allowed to be replaced with sum over \phi, i.e., \Gamma bar= W - J_a(x) \bar^a(x) . The last equation resembles the thermodynamical relation F=E-TS 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 \langle\phi\rangle is called '' mean field'' obviously because \langle\phi\rangle=\frac, while \bar is a background classical field. A field \phi is decomposed into a classical part \bar and fluctuation part \eta, i.e., \phi=\bar+\eta, so the vacuum amplitude can be reintroduced as e^ = \mathcal \int \exp\left \left( S[\phi- \frac \eta \right) \right">phi.html" ;"title=" \left( S[\phi"> \left( S[\phi- \frac \eta \right) \rightd\phi, and any function \mathcal[\phi] is defined as \langle\mathcal[\phi]\rangle = e^ ~ \mathcal \int \mathcal[\phi] \exp \left[i \left(S[\phi] - \frac \eta\right)\right] d\phi, where S[\phi] 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 Spontaneous symmetry breaking is a spontaneous process of symmetry breaking, by which a physical system in a symmetric state spontaneously ends up in an asymmetric state. In particular, it can describe systems where the equations of motion o ...
, 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. Back to Green functions of the actions. Since \Gamma bar/math> is the Legendre transform of F /math>, and F /math> defines N-points ''
connected Connected may refer to: Film and television * ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular'' * '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film * ''Connected'' (2015 TV ...
'' correlator G^_=\frac\Big, _, then the corresponding correlator obtained from F /math>, known as
vertex function In quantum electrodynamics, the vertex function describes the coupling between a photon and an electron beyond the leading order of perturbation theory (quantum mechanics), perturbation theory. In particular, it is the one particle irreducible cor ...
, is given by G^_ = \left.\frac\_. Consequently in the one particle irreducible graphs (usually acronymized as 1PI), the connected 2-point F -correlator is defined as the inverse of the 2-point \Gamma -correlator, i.e., the usual reduced correlation is G^_=\frac\Big, _=\frac , and the effective correlation is G^_=\frac\Big, _=p_p^-m^2 . For J_i =J(x_i), the most general relations between the N-points connected F /math> and Z /math> are \begin \frac =& \frac \frac - \Big\ + \big\ + \cdots \\ & - \Big\ + \Big\ - \cdots \end and \begin \frac\frac = & \frac + \Big\ + \Big\ + \cdots \\ & + \Big\ + \Big\ + \cdots \end


Source theory for fields


Vector fields

For a weak source producing a missive spin-1 particle with a general current J=J_e+J_a acting on different causal spacetime points x_0> x_0', the vacuum amplitude is \langle 0, 0\rangle_=\exp In momentum space, the spin-1 particle with rest mass m has a definite momentum p_=(m,0,0,0) in its rest frame, i.e. p_p^=m^2 . Then, the amplitude gives \begin (J_(p))^T ~ J^(p) - \frac (p_J^(p))^T ~ p_J^(p) & = (J_(p))^T ~ J^(p) - (J^(p))^T ~ \frac\bigg, _\text ~ J^(p) \\ &= (J^(p))^T ~ \left eta_-\frac\right~ J^(p) \end where \eta_=\text(1,-1,-1,-1) and (J_(p))^T is the transpose of J_(p) . The last result matches with the used propagator in the vacuum amplitude in the configuration space, that is, \left\langle 0\ T A_(x) A_(x') \left, 0\right\rangle = -i\int\frac \frac \left \eta_ - \left(1 - \xi\right) \frac \righte^. When \xi = 1 , the chosen Feynman–'t Hooft gauge-fixing makes the spin-1 massless. And when \xi = 0 , the chosen Landau gauge-fixing makes the spin-1 massive. The massless case is obvious as studied in
quantum electrodynamics In particle physics, quantum electrodynamics (QED) is the Theory of relativity, relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quant ...
. 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 W -i\ln(\langle 0, 0\rangle_)=\frac\int dx~dx'\left _(x)\Delta(x-x')J^(x')+\frac\partial_J^(x)\Delta(x-x')\partial'_J^(x')\right One can apply integration by part on the second term then single out \int dx J_(x) to get a definition of the massive spin-1 field A_(x)\equiv\int dx'\Delta(x-x')J^(x')-\frac\partial_\left int dx'\Delta(x-x')\partial'_J^(x')\right Additionally, the equation above says that \partial_A^ = \tfrac \partial_J^ . Thus, the equation of motion can be written in any of the following forms \begin &\left(\Box + m^2\right) A_ = J_ + \tfrac \partial_\partial_J^, \\ ex&\left(\Box + m^2\right) A_ + \partial_\partial_A^ = J_. \end


Massive totally symmetric spin-2 fields

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, \bar^ = T^ - \tfrac \eta_ \bar_T^, where \bar_(p) = \eta_ - \tfrac p_p_ is the vacuum polarization tensor, the vacuum amplitude in a compact form is \begin \langle 0, 0\rangle_ = \exp\Biggl( -\frac \int \biggl & \bar_(x)\Delta(x-x')\bar^(x') \\ &+\frac \eta_ \partial_ \bar^(x) \Delta(x-x') \partial'_ \bar^(x') \\ &+\frac \partial_ \partial_ \bar^(x) \Delta(x-x') \partial'_ \partial'_ \bar^(x')\biggrdx \, dx' \Biggr), \end or \begin \langle 0, 0\rangle_ = \exp\Biggl( - \frac \int \biggl[ & T_(x) \Delta(x-x') T^(x') \\ & + \frac \eta_ \partial_ T^(x) \Delta(x-x') \partial'_ T^(x') \\ & + \frac \partial_ \partial_ T^(x) \Delta(x-x') \partial'_\partial'_ T^(x') \\ & - \frac \left( \eta_ T^(x) - \frac \partial_ \partial_ T^(x) \right) \Delta(x-x') \left( \eta_ T^(x') - \frac \partial'_ \partial'_ T^(x') \right) \biggr]dx~dx' \Biggr). \end This amplitude in momentum space gives (transpose is imbedded) \begin \bar_(p)\eta^\eta^\bar_(p) & -\frac\bar_(p)\eta^p^p^\bar_(p)\\ &-\frac\bar_(p)\eta^p^p^\bar_(p)+\frac\bar_(p)p^p^p^p^\bar_(p)= \end \begin \eta^ \biggl(\bar_(p) \eta^ \bar_(p) & - \frac \bar_(p) p^ p^\bar_(p)\biggr) \\ & - \frac p^ p^ \left(\bar_(p) \eta^\bar_(p) - \frac\bar_(p) p^ p^\bar_(p)\right) \\ = \left(\eta^-\fracp^ p^\right) & \left( \bar_(p)\eta^ \bar_(p) - \frac \bar_(p)p^p^\bar_(p)\right) \\ = & \bar_(p) \left(\eta^-\fracp^ p^\right) \left(\eta^ - \fracp^p^\right) \bar_(p). \end And with help of symmetric properties of the source, the last result can be written as T^(p)\Pi_(p)T^(p) , where the projection operator, or the Fourier transform of Jacobi field operator obtained by applying Peierls braket on Schwinger's variational principle, is \Pi_(p) = \tfrac \left(\bar_(p) \bar_(p) + \bar_(p) \bar_(p) - \tfrac \bar_(p) \bar_(p)\right). 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 \Pi^_ = \tfrac \left(\eta_ \eta_ + \eta_ \eta_ - \tfrac \eta_ \eta_\right) . 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. It is worth noting that the vacuum polarization tensor \bar_ and the improved energy momentum tensor \bar^ 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 1970's 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. If one looks at \langle0, 0\rangle_ and follows the same procedure used to define massive spin-1 fields, then it is easy to define massive spin-2 fields as \begin h_(x) = & \int\Delta(x-x')T_(x') dx' \\ & - \frac \partial_ \int\Delta(x-x') \partial'^ T_(x')dx' \\ & - \frac \partial_ \int\Delta(x-x') \partial'^ T_(x')dx' \\ & + \frac \partial_ \partial_ \int \Delta(x-x') \partial'_\partial'_ T^(x')dx' \\ & -\frac\left(\eta_-\frac\partial_\partial_\right)\int\Delta(x-x')\left eta_ T^(x')-\frac\partial'_\partial'_T^(x')\rightdx'. \end The corresponding divergence condition is read \partial^h_-\partial_h=\frac\partial^T_, where the current \partial^T_ is not necessarily conserved (it is not a gauge condition as that of the massless case). But the energy-momentum tensor can be improved as \mathfrak_=T_-\frac\eta_\mathfrak such that \partial^\mathfrak_=0 according to Belinfante-Rosenfeld construction. Thus, the equation of motion \begin \left(\square + m^2\right) h_ = T_ & + \dfrac\left( \partial_ \partial^ T_ + \partial_ \partial^ T_ - \frac \eta_ \partial^ \partial^ T_ \right) \\ &+ \frac \left(\partial_ \partial_ - \frac \eta_ \square\right) \partial^\partial^ T_ \end becomes \left( \square+m^\right) h_=\mathfrak_-\frac ~\eta_\mathfrak-\dfrac\left( \partial_\partial_-\frac~\eta_\square\right) \left( \square+3m^\right) \mathfrak. One can use the divergence condition to decouple the non-physical fields \partial^h_ and h, so the equation of motion is simplified as \left( \square+m^\right) h_=\mathfrak_-\frac ~\eta_\mathfrak-\frac~\partial_\partial_ \mathfrak.


Massive totally symmetric arbitrary integer spin fields

One can generalize T^(p) source to become S^(p) higher-spin source such that T^(p)\Pi_(p)T^(p) becomes S^(p) \Pi_(p) S^(p) . The generalized projection operator also helps generalizing the electromagnetic polarization vector e^_(p) of the quantized electromagnetic vector potential as follows. For spacetime points x and x' , the addition theorem of spherical harmonics states that x^\cdots x^ \Pi_(p) x'^\cdots x'^=\frac\frac\sum\limits^_Y_(x)Y_^(x'). Also, the representation theory of the space of complex-valued
homogeneous polynomial In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. For example, x^5 + 2 x^3 y^2 + 9 x y^4 is a homogeneous polynomial of degree 5, in two variables ...
s of degree \ell on a unit (N-1)-sphere defines the polarization tensor ase_(x_1,\dots,x_n) = \sum_ e_x_\cdots x_,~ \forall x_i\in S^.Then, the generalized polarization vector ise^(p)~ x_\cdots x_=\sqrt~~Y_(x). And the projection operator can be defined as \Pi^(p)=\sum\limits^_ ^_(p) ^_(p)*. 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 \Delta(x-x') in configuration space, we write \langle0, 0\rangle_S=\exp.


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 T_ and a source S_=\partial_\partial^T_ , the vacuum amplitude is\langle 0, 0\rangle_=\exp 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 S(x-x')=(p \!\!\!/+m)\Delta(x-x') and current J=J_e+J_a as defined above, the vacuum amplitude is \begin \langle 0, 0\rangle_J & =\exp\\ &=\langle 0, 0\rangle_ \exp\langle 0, 0\rangle_. \end In momentum space the reduced amplitude is given by W_=-\frac\int \frac~J(-p)\left gamma^0\frac\rightJ(p). For spin- Rarita-Schwinger fermions, \Pi_ = \bar_ - \tfrac \gamma^ \bar_ \gamma^ \bar_. Then, one can use \gamma_=\eta_\gamma^ and the on-shell p\!\!\!/=-m to get \begin W_ &= - \frac \int \frac \, J^(-p) \left gamma^0 \frac\rightJ^(p)\\ &= - \frac \int \frac \, J^(-p) \left gamma^0 \frac\rightJ^(p). \end One can replace the reduced metric \bar_ with the usual one \eta_ if the source J_ is replaced with \bar_(p)=\frac\gamma^\Pi_\gamma^J^(p). For spin-(j + \tfrac) , the above results can be generalized to W_ = - \frac \int \frac \, J^(-p) ~ \left gamma^0 \frac\rightJ^(p). The factor \frac 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 form 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.


See also

* Keldysh-Schwinger formalism *
Schwinger function In quantum field theory, the Wightman distributions can be analytically continued to analytic functions in Euclidean space with the domain restricted to ordered ''n''-tuples in \mathbb R^d that are pairwise distinct. These functions are called ...
* Wigner-Bargmann equations * Joos–Weinberg equation


References

{{DEFAULTSORT:Source Field Quantum field theory