Pauli Fierz Research Articles

Inconsistency of interacting, multi-graviton theories

We investigate, in any space–time dimension ⩾3, the problem of consistent couplings for a (finite or infinite) collection of massless, spin-2 fields described, in the free limit, by a sum of Pauli–Fierz actions. We show that there is no consistent (ghost-free) coupling, with at most two derivatives of the fields, that can mix the various “gravitons”. In other words, there are no Yang–Mills-like spin-2 theories. The only possible deformations are given by a sum (or integral) of individual Einstein–Hilbert actions. The impossibility of cross-couplings subsists in the presence of scalar matter. Our approach is based on the BRST-based deformation point of view and uses results on the so-called “characteristic cohomology” for massless spin-2 fields which are explained in detail.

Essential Spectrum of a Self-Adjoint Operator on an Abstract Hilbert Space of Fock Type and Applications to Quantum Field Hamiltonians

We establish general theorems on locating the essential spectrum of a self-adjoint operator of the form A⊗I+I⊗dΓ(S)+HI on the tensor product H⊗Fb(K) of a Hilbert space H and the abstract Boson Fock space Fb(K) over a Hilbert space K, where A is a self-adjoint operator on H bounded from below, dΓ(S) is the second quantization of a nonnegative self-adjoint operator S on K, and HI is a symmetric operator on H⊗Fb(K). We then apply the theorems to the generalized spin-boson model by A. Arai and M. Hirokawa (1997, J. Funct. Anal.151, 455–503) and a general class of models of quantum particles coupled to a Bose field including the Pauli–Fierz model in nonrelativistic quantum electrodynamics.

Essential Self-Adjointness of¶Translation-Invariant Quantum Field Models¶for Arbitrary Coupling Constants

The Hamiltonian of a system of quantum particles minimally coupled to a quantum field is considered for arbitrary coupling constants. The Hamiltonian has a translation invariant part. By means of functional integral representations the existence of an invariant domain under the action of the heat semigroup generated by a self-adjoint extension of the translation invariant part is shown. With a non-perturbative approach it is proved that the Hamiltonian is essentially self-adjoint on a domain. A typical example is the Pauli–Fierz model with spin 1/2 in nonrelativistic quantum electrodynamics for arbitrary coupling constants.

Positive Commutators and the Spectrum¶of Pauli--Fierz Hamiltonian of Atoms and Molecules

In this paper we study the energy spectrum of the Pauli–Fierz Hamiltonian generating the dynamics of nonrelativistic electrons bound to static nuclei and interacting with the quantized radiation field. We show that, for sufficiently small values of the elementary electric charge, and under weaker conditions than those required in [3], the spectrum of this Hamiltonian is absolutely continuous, except possibly in small neighbourhoods of the ground state energy and the ionization thresholds. In particular, it is shown that (for a large range of energies) there are no stable excited eigenstates. The method used to prove these results relies on the positivity of the commutator between the Hamiltonian and a suitably modified dilatation generator on photon Fock space.

ASYMPTOTIC COMPLETENESS IN QUANTUM IN FIELD THEORY: MASSIVE PAULI–FIERZ HAMILTONIANS

Spectral and scattering theory of massive Pauli–Fierz Hamiltonians is studied. Asymptotic completeness of these Hamiltonians is shown. The proof consists of three parts. The first is a construction of asymptotic fields and a proof of their Fock property. The second part is a geometric analysis of observables. Its main result is what we call geometric asymptotic completeness. Finally, the last part is a proof of asymptotic completeness itself.

Functional Integral Representation of a Model in Quantum Electrodynamics

This paper presents functional integral representations for heat semigroups with infinitesimal generators given by self-adjoint Hamiltonians (Pauli–Fierz Hamiltonians) describing an interaction of a non-relativistic charged particle and a quantized radiation field in the Coulomb gauge without the dipole approximation. By the functional integral representations, some inequalities are derived, which are infinite degree versions of those known for finite dimensional Schrödinger operators with classical vector potentials.

Long-time behavior of an electron interacting with a quantized radiation field

The long-time behavior of an electron coupled to a quantized radiation field is discussed in the ground state and in equilibrium states at finite temperatures. The electron is not confined in an external potential. The model used is a d-dimensional extension of a standard model (the Pauli–Fierz model) in nonrelativistic quantum electrodynamics: The electron moves in Rd (d≥2) and the radiation field is over Rd. Further, the energy function ω of one free photon is taken to be a general one. In defining the interaction part of the Hamiltonian of the model, an ultraviolet cutoff is introduced for photon momenta with a cutoff function ρ̂ and the dipole approximation is used. It is proved that at each finite temperature T>0, the mean-square displacement of the electron behaves like (kBTd/m)t2 as time t tends to infinity, where kB is the Boltzmann constant and m>0 is a renormalized mass of the electron which should be identified with the observed mass of the electron. The long-time asymptotics of the mean square displacement of the electron in the ground state is different from that at the finite temperatures; it depends on the space dimension d and on the infrared behavior of ω and ρ̂.

An asymptotic analysis and its application to the nonrelativistic limit of the Pauli–Fierz and a spin-boson model

An abstract asymptotic theory of a family of self-adjoint operators {Hκ}κ>0 acting in the tensor product of two Hilbert spaces is presented and it is applied to the nonrelativistic limit of the Pauli–Fierz model in quantum electrodynamics and of a spin-boson model. It is proven that the resolvent of Hκ converges strongly as κ→∞ and the limit is a pseudoresolvent, which defines an ‘‘effective operator’’ of Hκ at κ≊∞. As corollaries of this result, some limit theorems for Hκ are obtained, including a theorem on spectral concentration. An asymptotic estimate of the infimum of the spectrum (the ground state energy) of Hκ is also given. The application of the abstract theory to the above models yields some new rigorous results for them.

On the connection between acausal propagation and the non-local structure of higher spin fields

We show that a free spin 3/2 theory of the Pauli–Fierz type displays nonlocal structure. Similarly, a spin 1 theory coupled via an electric quadrupole to an external field also shows nonlocal structure. This is in sharp contrast to the cases of spin 0, 1/2, and 1 which do not have any nonlocality even in the presence of minimal coupling to an electromagnetic field. This leads us to conjecture that the acausal propagation discovered for spin 3/2 and spin 1 is due to additional structure of these fields.

Recursive numerical integration of multi-particle phase space with peripheral matrix element

On the basis of the Wigner unitary representations of the covering group ISL(2,C) of the Poincaré group, we obtain spin-tensor wave functions of free massive particles with arbitrary spin. The wave functions automatically satisfy the Dirac–Pauli–Fierz equations. In the framework of the two-spinor formalism we construct spin-vectors of polarizations and obtain conditions that fix the corresponding relativistic spin projection operators (Behrends–Fronsdal projection operators). With the help of these conditions we find explicit expressions for relativistic spin projection operators for integer spins (Behrends–Fronsdal projection operators) and then find relativistic spin projection operators for half integer spins. These projection operators determine the numerators in the propagators of fields of relativistic particles. We deduce generalizations of the Behrends–Fronsdal projection operators for arbitrary space–time dimensions D>2.

Ground state degeneracy of the Pauli–Fierz Hamiltonian with spin

We consider an electron, spin 1/2, minimally coupled to the quantized radiation field in the nonrelativistic approximation, a situation defined by the Pauli-Fierz Hamiltonian $H$. There is no external potential and $H$ fibers according to the total momentum $p$. We prove that the ground state subspace of $H$ with a total momentum $p$ is two-fold degenerate provided the charge $e$ and the total momentum $p$ are sufficiently small. We also establish that the total angular momentum of the ground state subspace is $\pm1/2$ and study the case of a confining external potential.