221 research outputs found
Combinatorics of 1-particle irreducible n-point functions via coalgebra in quantum field theory
We give a coalgebra structure on 1-vertex irreducible graphs which is that of
a cocommutative coassociative graded connected coalgebra. We generalize the
coproduct to the algebraic representation of graphs so as to express a bare
1-particle irreducible n-point function in terms of its loop order
contributions. The algebraic representation is so that graphs can be evaluated
as Feynman graphs
A differential identity for Green functions
If P is a differential operator with constant coefficients, an identity is
derived to calculate the action of exp(P) on the product of two functions. In
many-body theory, P describes the interaction Hamiltonian and the identity
yields a hierarchy of Green functions. The identity is first derived for scalar
fields and the standard hierarchy is recovered. Then the case of fermions is
considered and the identity is used to calculate the generating function for
the Green functions of an electron system in a time-dependent external
potential.Comment: 14 page
Gell-Mann and Low formula for degenerate unperturbed states
The Gell-Mann and Low switching allows to transform eigenstates of an
unperturbed Hamiltonian into eigenstates of the modified Hamiltonian . This switching can be performed when the initial eigenstate is not
degenerate, under some gap conditions with the remainder of the spectrum. We
show here how to extend this approach to the case when the ground state of the
unperturbed Hamiltonian is degenerate. More precisely, we prove that the
switching procedure can still be performed when the initial states are
eigenstates of the finite rank self-adjoint operator \cP_0 V \cP_0, where
\cP_0 is the projection onto a degenerate eigenspace of
The Hopf algebra of Feynman graphs in QED
We report on the Hopf algebraic description of renormalization theory of
quantum electrodynamics. The Ward-Takahashi identities are implemented as
linear relations on the (commutative) Hopf algebra of Feynman graphs of QED.
Compatibility of these relations with the Hopf algebra structure is the
mathematical formulation of the physical fact that WT-identities are compatible
with renormalization. As a result, the counterterms and the renormalized
Feynman amplitudes automatically satisfy the WT-identities, which leads in
particular to the well-known identity .Comment: 13 pages. Latex, uses feynmp. Minor corrections; to appear in LM
Relating on-shell and off-shell formalism in perturbative quantum field theory
In the on-shell formalism (mostly used in perturbative quantum field theory)
the entries of the time ordered product T are on-shell fields (i.e. the basic
fields satisfy the free field equations). With that, (multi)linearity of T is
incompatible with the Action Ward identity. This can be circumvented by using
the off-shell formalism in which the entries of T are off-shell fields. To
relate on- and off-shell formalism correctly, a map sigma from on-shell fields
to off-shell fields was introduced axiomatically by Duetsch and Fredenhagen. In
that paper it was shown that, in the case of one real scalar field in N=4
dimensional Minkowski space, these axioms have a unique solution. However, this
solution was given there only recursively. We solve this recurrence relation
and give a fully explicit expression for sigma in the cases of the scalar,
Dirac and gauge fields for arbitrary values of the dimension N.Comment: The case of gauge fields was added. 16 page
The Hopf Algebra of Renormalization, Normal Coordinates and Kontsevich Deformation Quantization
Using normal coordinates in a Poincar\'e-Birkhoff-Witt basis for the Hopf
algebra of renormalization in perturbative quantum field theory, we investigate
the relation between the twisted antipode axiom in that formalism, the Birkhoff
algebraic decomposition and the universal formula of Kontsevich for quantum
deformation.Comment: 21 pages, 15 figure
Relativistic corrections in magnetic systems
We present a weak-relativistic limit comparison between the Kohn-Sham-Dirac
equation and its approximate form containing the exchange coupling, which is
used in almost all relativistic codes of density-functional theory. For these
two descriptions, an exact expression of the Dirac Green's function in terms of
the non-relativistic Green's function is first derived and then used to
calculate the effective Hamiltonian, i.e., Pauli Hamiltonian, and effective
velocity operator in the weak-relativistic limit. We point out that, besides
neglecting orbital magnetism effects, the approximate Kohn-Sham-Dirac equation
also gives relativistic corrections which differ from those of the exact
Kohn-Sham-Dirac equation. These differences have quite serious consequences: in
particular, the magnetocrystalline anisotropy of an uniaxial ferromagnet and
the anisotropic magnetoresistance of a cubic ferromagnet are found from the
approximate Kohn-Sham-Dirac equation to be of order , whereas the
correct results obtained from the exact Kohn-Sham-Dirac equation are of order
. We give a qualitative estimate of the order of magnitude of these
spurious terms
On some aspects of the definition of scattering states in quantum field theory
The problem of extending quantum-mechanical formal scattering theory to a
more general class of models that also includes quantum field theories is
discussed, with the aim of clarifying certain aspects of the definition of
scattering states. As the strong limit is not suitable for the definition of
scattering states in quantum field theory, some other limiting procedure is
needed. Two possibilities are considered, the abelian limit and adiabatic
switching. Formulas for the scattering states based on both methods are
discussed, and it is found that generally there are significant differences
between the two approaches. As an illustration of the application and the
features of these formulas, S-matrix elements and energy corrections in two
quantum field theoretical models are calculated using (generalized)
old-fashioned perturbation theory. The two methods are found to give equivalent
results.Comment: 20 page
The structure of Green functions in quantum field theory with a general state
In quantum field theory, the Green function is usually calculated as the
expectation value of the time-ordered product of fields over the vacuum. In
some cases, especially in degenerate systems, expectation values over general
states are required. The corresponding Green functions are essentially more
complex than in the vacuum, because they cannot be written in terms of standard
Feynman diagrams. Here, a method is proposed to determine the structure of
these Green functions and to derive nonperturbative equations for them. The
main idea is to transform the cumulants describing correlations into
interaction terms.Comment: 13 pages, 6 figure
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