针对带电粒子与电磁场耦合的平移不变 Pauli-Fierz 模型，本文证明在自然假设下零动量纤维 Hamilton 量存在基态。
摘要：
我们考虑描述带电粒子与电磁场相互作用的平移不变 Pauli-Fierz 模型。我们在自然假设下证明，零动量处的纤维 Hamilton 量具有一个基态。
英文摘要：
We consider the translationally invariant Pauli-Fierz model describing a charged particle interacting with the electromagnetic field. We show under natural assumptions that the fiber Hamiltonian at zero momentum has a ground state.
For the Pauli-Fierz operator with a short range potential we study the binding threshold as a function of the fine structure constant $α$ and show that it converges to the binding threshold for the Schrödinger operator in the small $α$ limit.
The spinless semi-relativistic Pauli-Fierz Hamiltonian $H$ in quantum electrodynamics is considered. The self-adjointness and essential self-adjointness of $H$ are shown. It is emphasized that it includes the massless case. Furthermore, the self-adjointness and the essential self-adjointness of the semi-relativistic Pauli-Fierz model with a fixed total momentum is also proven.
By means of functional integrations spectral properties of semi-relativistic Pauli-Fierz Hamiltonians in quantum electrodynamics is considered. Two self-adjoint extensions of a semi-relativistic Pauli-Fierz Hamiltonian are defined. An essential self-adjointness, a spatial decay of bound states, a Gaussian domination of the ground state and the existence of a measure associated with the ground state are shown.
The Pauli-Fierz model with a variable mass $v$ is considered. An ultraviolet cutoff and an infrared regularity condition are imposed on a quantized radiation field. It is shown that the ground state exists for arbitrary values of coupling constants, when $v(x)\leq c\langle x\rangle^{-β}$, $β>3$.
We prove a general theorem about the self-adjointness and domain of Pauli-Fierz type Hamiltonians. Our proof is based on commutator arguments which allow us to treat fields with non-commuting components. As a corollary it follows that the domain of the Hamiltonian of non-relativistic QED with Coulomb interactions is independent of the coupling constant.
We consider the Pauli--Fierz model, which describes a particle (an electron) coupled to the quantized electromagnetic field and limit the number of photons to less than 2. By computing \red{the resolvent explicitly}, we located the spectrum of the Hamiltonian mass. Our results do not depend on the coupling constant $e$ nor on the infrared cutoff parameter $R$.
Generalized Pauli-Fierz Hamiltonian with Kato-class potential $\KPF$ in nonrelativistic quantum electrodynamics is defined and studied by a path measure. $\KPF$ is defined as the self-adjoint generator of a strongly continuous one-parameter symmetric semigroup and it is shown that its bound states spatially exponentially decay pointwise and the ground state is unique.
For a quantum particle interacting with a short-range potential, we estimate from below the shift of its binding threshold, which is due to the particle interaction with a quantized radiation field.
The purpose of this article is to derive a Markovian approximation of the reduced time dynamics of observables for the Pauli-Fierz Hamiltonian with a precise control of the error terms. In that aim, we define a Lindblad operator associated to the corresponding quantum master equation. In a particular case, this allows to study the transition probability matrix.
It is shown that at least one particle is bound in the $N$-particle semi-relativistic Pauli-Fierz model with negative potential $V(\bx)$. It is assumed that the particles have no spin and obey the Bose or Boltzmann statistics, and the one particle Hamiltonian $\sqrt{-Δ+M^2}-M+ V(\bx)$ has a ground state with negative energy $-e_0<0$, where $M>0$ denotes the mass of the particle. We show that the ground state energy of the total system $E^V(N)$ is less than $E^0(N)-e_0$.
The Pauli-Fierz model $H(α)$ in nonrelativistic quantum electrodynamics is considered. The external potential $V$ is sufficiently shallow and the dipole approximation is assumed. It is proven that there exist constants $0<α_-< α_+$ such that $H(α)$ has no ground state for $|α|<α_-$, which complements an earlier result stating that there is a ground state for $|α| > α_+$. We develop a suitable extension of the Birman-Schwinger argument. Moreover for any given $δ>0$ examples of potentials $V$ are provided such that $α_+-α_-<δ$.
We perform the spectral analysis of a zero temperature Pauli-Fierz system for small coupling constants. Under the hypothesis of Fermi golden rule, we show that the embedded eigenvalues of the uncoupled system disappear and establish a limiting absorption principle above this level of energy. We rely on a positive commutator approach introduced by Skibsted and pursued by Georgescu-Gerard-Moller. We complete some results obtained so far by Derezinski-Jaksic on one side and by Bach-Froehlich-Segal-Soffer on the other side.
We prove enhanced binding and increase of binding energies in the non- and semi-relativistic Pauli-Fierz models, for arbitrary values of the fine-structure constant and the ultra-violet cut-off, and discuss the resulting improvement of exponential localization of ground state eigenvectors. For the semi-relativistic model we also discuss the increase of the renormalized electron mass and determine the linear leading order term in the asymptotics of the self-energy, as the ultra-violet cut-off goes to infinity.
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.
In this note, we determine the ground state energy of the translation invariant Pauli-Fierz model to subleading order $O(α^3)$ with respect to powers of the finestructure constant $α$, and prove rigorous error bounds of order $O(α^{4})$. A main objective of our argument is its brevity.
We analyze basic relativistic wave equations for the classical fields, such as Dirac's equation, Weyl's two-component equation for massless neutrinos, and the Proca, Maxwell, and Fierz-Pauli equations, from the viewpoint of the Pauli-Lubanski vector and the Casimir operators of the Poincare group. In general, in this group-theoretical approach, the above wave equations arise in certain overdetermined forms, which can be reduced to the conventional ones by a Gaussian elimination. A connection between the spin of a particle/field and consistency of the corresponding overdetermined system is emph…
Towards the correct prescription of the Problem of divergence of Propagator of Siegal-Zwiebach action from String theory in Fierz-Pauli Gauge in the small mass limit
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 as $\int^\oplus H_p dp$ according to the total momentum $p$. We prove that the ground state subspace of $H_p$ 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.
We consider a hydrogen-like atom in a quantized electromagnetic field which is modeled by means of the semi-relativistic Pauli-Fierz operator and prove that the infimum of the spectrum of the latter operator is an eigenvalue. In particular, we verify that the bottom of its spectrum is strictly less than its ionization threshold. These results hold true for arbitrary values of the fine-structure constant and the ultra-violet cut-off as long as the Coulomb coupling constant (i.e. the product of the fine-structure constant and the nuclear charge) is less than 2/π.
We consider a charged particle, spin 1/2, with relativistic kinetic energy and minimally coupled to the quantized Maxwell field. Since the total momentum is conserved, the Hamiltonian admits a fiber decomposition as $H(P)$, $P\in \BbbR^3$. We study the spectrum of $H(P)$. In particular we prove that, for non-zero photon mass, the ground state is exactly two-fold degenerate and separated by a gap, uniformly in $P$, from the rest of the spectrum.
We consider the polaron of the spinless semi-relativistic Pauli-Fierz model. The Hamiltonian of the model is defined by $H(\mathbf{P}) = \sqrt{(\mathbf{P}-dΓ(\mathbf{k}) + e\bA)^2 + M^2} + dΓ(ω_m)$, where $\mathbf{P}\in\mathbb{R}^3$ is the momentum of the polaron, $dΓ(\cdot)$ denotes the second quantization operator and $ω_m=|\mathbf{k}|+m$ denotes the dispersion relation of the photon with virtual mass $m\geq 0$. Let $E(\mathbf{P})$ be the lowest energy of $H(\mathbf{P})$. In this paper, we prove the inequality $E(\mathbf{P} - \mathbf{k}) - E(\mathbf{P}) + ω_m(\mathbf{k}) \geq m$, for all $\m…
In this article is proved the existence and uniqueness of a minimizer of the energy for the non-relativistic one electron Pauli-Fierz model, within the class of pure quasifree states. The minimum of the energy on pure quasifree states coincides with the minimum of the energy on quasifree states. Infrared and ultraviolet cutoffs are assumed, along with sufficiently small coupling constant and momentum of the dressed electron. A perturbative expression of the minimum of the energy on quasifree states for a small momentum of the dressed electron and small coupling constant is then given. We also …
We consider Dirichlet realizations of Pauli-Fierz type operators generating the dynamics of non-relativistic matter particles which are confined to an arbitrary open subset of the Euclidean position space and coupled to quantized radiation fields. We find sufficient conditions under which their domains and a natural class of operator cores are determined by the domains and operator cores of the corresponding Dirichlet-Schrödinger operators and the radiation field energy. Our results also extend previous ones dealing with the entire Euclidean space, since the involved electrostatic potentials m…
We derive Feynman-Kac formulas for Dirichlet realizations of Pauli-Fierz operators generating the dynamics of nonrelativistic quantum mechanical matter particles, which are minimally coupled to both classical and quantized radiation fields and confined to an arbitrary open subset of the Euclidean space. Thanks to a suitable interpretation of the involved Stratonovich integrals, we are able to retain familiar formulas for the Feynman-Kac integrands merely assuming local square-integrability of the classical vector potential and the coupling function in the quantized vector potential. Allowing f…
A semi-relativistic Pauli-Fierz model is defined by the sum of the free Hamiltonian $H_{\rm f}$ of a Boson Fock space, an nuclear potential $V$ and a relativistic kinetic energy: $$ H=\sqrt{[σ\cdot(\mathbf{p}+e\mathbf{A})]^2+M^2} - M + V + H_{\rm f}. $$ Let $-e_0<0$ be the ground state energy of a semi-relativistic Schrödinger operator $$ H_{\rm p}=\sqrt{\mathbf{p}^2+M^2} - M + V. $$ It is shown that the ionization energy $E^{\rm ion}$ of $H$ satisfies $$E^{\rm ion}\geq e_0>0$$ for all values of both of the coupling constant $e\in\mathbb{R}$ and the rest mass $M\geq 0$. In particular our resul…
We study the quasi-classical limit of the Pauli-Fierz model: the system is composed of finitely many non-relativistic charged particles interacting with a bosonic radiation field. We trace out the degrees of freedom of the field, and consider the classical limit of the latter. We prove that the partial trace of the full Hamiltonian converges, in resolvent sense, to an effective Schrödinger operator with magnetic field and a corrective electric potential that depends on the field configuration. Furthermore, we prove the convergence of the ground state energy of the microscopic system to the inf…
A Feynman-Kac-type formula for a Lévy and an infinite dimensional Gaussian random process associated with a quantized radiation field is derived. In particular, a functional integral representation of $e^{-t\PF}$ generated by the Pauli-Fierz Hamiltonian with spin $\han$ in non-relativistic quantum electrodynamics is constructed. When no external potential is applied $\PF$ turns translation invariant and it is decomposed as a direct integral $\PF = \int_\BR^\oplus \PF(P) dP$. The functional integral representation of $e^{-t\PF(P)}$ is also given. Although all these Hamiltonians include spin, ne…
HVZ type theorem for semi-relativistic Pauli-Fierz Hamiltonian, $$\HHH=\sqrt{(p\otimes \one -A)^2+M^2}+V\otimes \one +\one\otimes \hf,\quad M\geq 0,$$ in quantum electrodynamics is studied. Here $H$ is a self-adjoint operator in Hilbert space $\LR\otimes \fff\cong \int^\oplus_{\RR^d}\fff {\rm d}x$, and $A=\int^\oplus_{\RR^d} A(x) {\rm d}x$ a quantized radiation field and $\hf$ the free field Hamiltonian defined by the second quantization of a dispersion relation $ω:\RR^d\to \RR$. It is emphasized that massless case, $M=0$, is included. Let $E=\inf σ(\HHH)$ be the bottom of the spectrum of $\HH…
We study a system composed of a hydrogen atom interacting with an infinite conductor wall. The interaction energy decays like $L^{-3}$, where $L$ is the distance between the atom and the wall, due to the emergence of the van der Waals forces. In this paper we show how, considering the contributions from the quantum fluctuations of the electromagnetic field, the interaction is weakened to a decay of order $L^{-4}$ giving rise to the retardation effects which fall under the name of Casimir-Polder effect. The analysis is done by studying a suitable Pauli-Fierz model associated to the system, in d…
The purpose of these notes is to give a fairly narrow but thorough introduction to the spectral analysis of Hamiltonians and standard Liouvilleans describing finite dimensional small systems linearly coupled to a scalar massless field or reservoir. The Hamiltonians describe the system at zero temperature, and the standard Liouvillean implements unitarily the dynamics of the system at positive temperature. We focus our attention on results valid at arbitrary coupling strength and whose proofs are purely operator theoretic, i.e. for the standard Liouvillean does not make use of the underlying mo…
We consider the semi-relativistic Pauli-Fierz Hamiltonian and a no-pair model of a hydrogen-like atom interacting with a quantized photon field at the respective critical values of the Coulomb coupling constant. For arbitrary values of the fine-structure constant and the ultra-violet cutoff, we prove the existence of normalizable ground states of the atomic system in both models. This complements earlier results on the existence of ground states in (semi-)relativistic models of quantum electrodynamics at sub-critical coupling by E. Stockmeyer and the present authors. Technically, the main new …
We consider the semi-relativistic Pauli-Fierz Hamiltonian $$ H_m = |{\bf p}-{\bf A}({\bf x})| + H_{f,m} + V({\bf x}),\quad m\geq0, $$ and prove the existence of the ground state of $H_m$ for $m=0$. Here ${\bf A}({\bf x})$ denotes a quantized radiation field and $H_{f,m}$ the free field Hamiltonian with the dispersion relation $\sqrt{|{\bf k}|^2+m^2}$ with $m\geq0$. This paper is the sequel of [HH16], where the existence of the ground state $Φ_m$ of $H_m$ for $m>0$ is proven. In order to show the existence of the ground state for $m=0$ we estimate a singular and non-local pull-through formula a…
We consider the semi-relativistic Pauli-Fierz model for a single free electron interacting with the quantized radiation field. Employing a variant of Pizzo's iterative analytic perturbation theory we construct a sequence of ground state eigenprojections of infra-red cutoff, dressing transformed fiber Hamiltonians and prove its convergence, as the cutoff goes to zero. Its limit is the ground state eigenprojection of a certain Hamiltonian unitarily equivalent to a renormalized fiber Hamiltonian acting in a coherent state representation space. The ground state energy is an exactly two-fold degene…
We consider the spinless Pauli-Fierz Hamiltonian which describes a quantum system of non-relativistic identical particles coupled to the quantized electromagnetic field. We study the time evolution in a mean-field limit where the number $N$ of charged particles gets large while the coupling to the radiation field is rescaled by $1/\sqrt{N}$. At time zero we assume that almost all charged particles are in the same one-body state (a Bose-Einstein condensate) and we assume also the photons to be close to a coherent state. We show that at later times and in the limit $N \rightarrow \infty$ the cha…
We consider the weak coupling limit for a quantum system consisting of a small subsystem and reservoirs. It is known rigorously since [Dav74] that the Heisenberg evolution restricted to the small system converges in an appropriate sense to a Markovian semigroup. In the nineties, Accardi, Frigerio and Lu [AFL90] initiated an investigation of the convergence of the unreduced unitary evolution to a singular unitary evolution generated by a quantum Langevin equation. We present a version of this convergence which is both simpler and stronger than the formulations which we know. Our main result say…
We consider a finite quantum system coupled to quasifree thermal reservoirs at different temperatures. Under the assumptions of small coupling and exponential decay of the reservoir correlation function, the large deviation generating function of energy transport into the reservoirs is shown to be analytic on a bounded set. Our method is different from the spectral deformation technique which was employed recently in the study of spin-boson-like models. As a corollary, we derive the Gallavotti-Cohen fluctuation relation for the entropy production and a central limit theorem for energy transpor…
Degeneracy of the eigenvalues of the Pauli-Fierz Hamiltonian with spin 1/2 is proven by the Kramers degeneracy theorem. The Pauli-Fierz Hamiltonian at fixed total momentum is also investigated.
The present paper is devoted to the detailed study of quantization and evolution of the point limit of the Pauli-Fierz model for a charged oscillator interacting with the electromagnetic field in dipole approximation. In particular, a well defined dynamics is constructed for the classical model, which is subsequently quantized according to the Segal scheme. To this end, the classical model in the point limit is reformulated as a second order abstract wave equation, and a consistent quantum evolution is given. This allows a study of the behaviour of the survival and transition amplitudes for th…
Two decades ago, Lieb and Loss proposed to approximate the ground state energy of a free, nonrelativistic electron coupled to the quantized radiation field by the infimum $E_{α, Λ}$ of all expectation values $\langle ϕ_{el} \otimes ψ_{ph} | H_{α, Λ} (ϕ_{el} \otimes ψ_{ph}) \rangle$, where $H_{α, Λ}$ is the corresponding Hamiltonian with fine structure constant $α>0$ and ultraviolet cutoff $Λ< \infty$, and $ϕ_{el}$ and $ψ_{ph}$ are normalized electron and photon wave functions, respectively. Lieb and Loss showed that $c α^{1/2} Λ^{3/2} \leq E_{α, Λ} \leq c^{-1} α^{2/7} Λ^{12/7}$ for some consta…
We derive the ground state energy up to the fourth order in the fine structure constant $α$ for the translation invariant Pauli-Fierz Hamiltonian for a spinless electron coupled to the quantized radiation field. As a consequence, we obtain the non-analyticity of the ground state energy of the Pauli-Fierz operator for a single particle in the Coulomb field of a nucleus.
We slightly extend prior results about the derivation of the Maxwell-Schrödinger equations from the bosonic Pauli-Fierz Hamiltonian. More concretely, we show that the findings from [SIAM J. Math. Anal. 52(5), 4900-4936 (2020)] about the coherence of the quantized electromagnetic field also hold for soft photons with small energies. This is achieved with the help of an estimate from [arXiv:2202.05015] which proves that the domain of the number of photon operator is invariant during the time evolution generated by the Pauli-Fierz Hamiltonian.
Pauli-Fierz approach to description of a massless spin-2 particle is investigated in the framework of 30-component first order relativistic wave equation theory on a curved space-time background. It is shown that additional gauge symmetry of massless equations established by Pauli-Fierz can be extended only to curved space-time regions where Ricci tensor vanishes. In all such space-time models the generally covariant S=2 massless wave equation exhibits gauge symmetry property, otherwise it is not so.
We consider two different models of a hydrogenic atom in a quantized electromagnetic field that treat the electron relativistically. The first one is a no-pair model in the free picture, the second one is given by the semi-relativistic Pauli-Fierz Hamiltonian. We prove that the no-pair operator is semi-bounded below and that its spectral subspaces corresponding to energies below the ionization threshold are exponentially localized. Both results hold true, for arbitrary values of the fine-structure constant, $e^2$, and the ultra-violet cut-off, $Λ$, and for all nuclear charges less than the cri…
The hydrogen binding energy in the Pauli-Fierz model with the spin Zeeman term is determined up to the order alpha cube, where alpha denotes the fine-structure constant.
A general approach to description of multigravity models in D-dimensional space-time is presented. Different possibilities of generalization of the invariant volume are given. Then a most general form of the interaction potential is constructed, which for bigravity coincides with the Pauli-Fierz model. A thorough analysis of the model along the 3+1 expansion formalism is done. It is shown that the absence of ghosts the considered bigravity model is equivalent in the weak field limit to the massive gravity (the Pauli-Fierz model). Thus, on the concrete example it is shown, that the interaction …
We consider resonances in the Pauli-Fierz model of non-relativistic QED. We use and slightly modify the analysis developed by Bach, Froehlich, and Sigal to obtain an upper and lower bound on the lifetime of quasi-stationary states.
We review the non-trivial issue of the relativistic description of a quantum mechanical system that, contrary to a common belief, kept theoreticians busy from the end of 1920s to (at least) mid 1940s. Starting by the well-known works by Klein-Gordon and Dirac, we then give an account of the main results achieved by a variety of different authors, ranging from de Broglie to Proca, Majorana, Fierz-Pauli, Kemmer, Rarita-Schwinger and many others. A particular interest comes out for the general problem of the description of particles with \textit{arbitrary} spin, introduced (and solved) by Majoran…
We consider a spinless particle coupled to a photon field and prove that even if the Schrödinger operator $p^2 + V$ does not have eigenvalues the system can have a ground state. We describe the coupling by means of the Pauli-Fierz Hamiltonian and our result holds in the case where the coupling constant $α$ is small.
We consider the Pauli-Fierz Hamiltonian with dynamical nuclei and investigate the transitions between the resonant electronic energy levels under the assumption that there are no free photons in the beginning. Coupling the limits of small fine structure constant and of heavy nuclei allows us to prove the validity of the Born-Oppenheimer approximation at leading order and to provide a simple formula for the rate of spontaneous decay.
We consider a Pauli-Fierz Hamiltonian for a particle coupled to a photon field. We discuss the effects of the increase of the binding energy and enhanced binding through coupling to a photon field, and prove that both effects are the results of the existence of the ground state of the self-energy operator with total momentum $P = 0$.
We consider a (semi-)relativistic spin-1/2 particle interacting with quantized radiation. The Hamiltonian has the form $\hat{H}_c^V:=\{c^2[(\mathbf{p}+{\bf A})^2+{\bf σ}\cdot{\bf B}]+(mc^2)^2\}^{1/2}-mc^2+V+H_f$. Assuming that the potential $|V|$ is bounded with respect to the momentum $|\mathbf{p}|$, we show that $\hat{H}_c^V$ converges in norm-resolvent sense to the usual Pauli-Fierz operator when $c$, the speed of light, tends to $\infty$.
We consider a free hydrogen atom composed of a spin-1/2 nucleus and a spin-1/2 electron in the standard model of non-relativistic QED. We study the Pauli-Fierz Hamiltonian associated with this system at a fixed total momentum. For small enough values of the fine-structure constant, we prove that the ground state is unique. This result reflects the hyperfine structure of the hydrogen atom ground state.
