12 research outputs found
On the connection between Hamilton and Lagrange formalism in Quantum Field Theory
The connection between the Hamilton and the standard Lagrange formalism is
established for a generic Quantum Field Theory with vanishing vacuum
expectation values of the fundamental fields. The Effective Actions in both
formalisms are the same if and only if the fundamental fields and the momentum
fields are related by the stationarity condition. These momentum fields in
general differ from the canonical fields as defined via the Effective Action.
By means of functional methods a systematic procedure is presented to identify
the full correlation functions, which depend on the momentum fields, as
functionals of those usually appearing in the standard Lagrange formalism.
Whereas Lagrange correlation functions can be decomposed into tree diagrams the
decomposition of Hamilton correlation functions involves loop corrections
similar to those arising in n-particle effective actions. To demonstrate the
method we derive for theories with linearized interactions the propagators of
composite auxiliary fields and the ones of the fundamental degrees of freedom.
The formalism is then utilized in the case of Coulomb gauge Yang-Mills theory
for which the relations between the two-point correlation functions of the
transversal and longitudinal components of the conjugate momentum to the ones
of the gauge field are given.Comment: 25 pages, 24 figures, revised and extended version with an explicit
application of the formalism to Coulomb gauge QC
Gauge-Independent Off-Shell Fermion Self-Energies at Two Loops: The Cases of QED and QCD
We use the pinch technique formalism to construct the gauge-independent
off-shell two-loop fermion self-energy, both for Abelian (QED) and non-Abelian
(QCD) gauge theories. The new key observation is that all contributions
originating from the longitudinal parts of gauge boson propagators, by virtue
of the elementary tree-level Ward identities they trigger, give rise to
effective vertices, which do not exist in the original Lagrangian; all such
vertices cancel diagrammatically inside physical quantities, such as current
correlation functions or S-matrix elements. We present two different, but
complementary derivations: First, we explicitly track down the aforementioned
cancellations inside two-loop diagrams, resorting to nothing more than basic
algebraic manipulations. Second, we present an absorptive derivation,
exploiting the unitarity of the S-matrix, and the Ward identities imposed on
tree-level and one-loop physical amplitudes by gauge invariance, in the case of
QED, or by the underlying Becchi-Rouet-Stora symmetry, in the case of QCD. The
propagator-like sub-amplitude defined by means of this latter construction
corresponds precisely to the imaginary parts of the effective self-energy
obtained in the former case; the real part may be obtained from a (twice
subtracted) dispersion relation. As in the one-loop case, the final two-loop
fermion self-energy constructed using either method coincides with the
conventional fermion self-energy computed in the Feynman gauge.Comment: 30 pages; uses axodraw (axodraw.sty included in the src); final
version to appear in Phys. Rev.
Pinch Technique and the Batalin-Vilkovisky formalism
In this paper we take the first step towards a non-diagrammatic formulation
of the Pinch Technique. In particular we proceed into a systematic
identification of the parts of the one-loop and two-loop Feynman diagrams that
are exchanged during the pinching process in terms of unphysical ghost Green's
functions; the latter appear in the standard Slavnov-Taylor identity satisfied
by the tree-level and one-loop three-gluon vertex. This identification allows
for the consistent generalization of the intrinsic pinch technique to two
loops, through the collective treatment of entire sets of diagrams, instead of
the laborious algebraic manipulation of individual graphs, and sets up the
stage for the generalization of the method to all orders. We show that the task
of comparing the effective Green's functions obtained by the Pinch Technique
with those computed in the background field method Feynman gauge is
significantly facilitated when employing the powerful quantization framework of
Batalin and Vilkovisky. This formalism allows for the derivation of a set of
useful non-linear identities, which express the Background Field Method Green's
functions in terms of the conventional (quantum) ones and auxiliary Green's
functions involving the background source and the gluonic anti-field; these
latter Green's functions are subsequently related by means of a Schwinger-Dyson
type of equation to the ghost Green's functions appearing in the aforementioned
Slavnov-Taylor identity.Comment: 45 pages, uses axodraw; typos corrected, one figure changed, final
version to appear in Phys.Rev.