Spectral RG and resonances J´ er´ emy Faupin The model Spectral renormaliza- On spectral renormalization group and the theory of tion group resonances in non-relativistic QED Resonances and lifetime of metastable states J´ er´ emy Faupin Institut de Math´ ematiques de Bordeaux September 2012 Conference “Renormalization at the confluence of analysis, algebra and geometry. ”
Spectral RG and resonances Spectral renormalization group: general J´ er´ emy strategy Faupin The model Spectral renormaliza- Problem and general strategy tion group • Want to study the spectral properties of some given Hamiltonian H acting Resonances and lifetime on a Hilbert space H of metastable • Construct an effective Hamiltonian H eff acting in a Hilbert space with fewer states degrees of freedom, such that H eff has the same spectral properties as H • Use a scaling transformation to map H eff to a scaled Hamiltonian H (0) acting on some Hilbert space H 0 • Iterate the procedure to obtain a family of effective Hamiltonians H ( n ) acting on H 0 • Estimate the “renormalized” perturbation terms W ( n ) appearing in H ( n ) and show that W ( n ) vanishes in the limit n → ∞ • Study the limit Hamiltonian H ( ∞ ) • Go back to the original Hamiltonian H using isospectrality of the renormalization map
Spectral RG and resonances Contents of the talk J´ er´ emy Faupin 1 The model The model The atomic system Spectral renormaliza- The photon field tion group Standard model of non-relativistic QED Resonances and lifetime of metastable 2 Spectral renormalization group states Decimation of the degrees of freedom Generalized Wick normal form Scaling transformation Scaling transformation of the spectral parameter Banach space of Hamiltonians The renormalization map 3 Resonances and lifetime of metastable states Existence of resonances Lifetime of metastable states
Spectral RG and resonances J´ er´ emy Faupin The model The atomic system The photon field Standard model of non- Part I relativistic QED Spectral The model renormaliza- tion group Resonances and lifetime of metastable states
Spectral RG and resonances Some references J´ er´ emy Faupin • O. Bratteli and D. W. Robinson. Operator algebras and quantum statistical The model mechanics. 1. Texts and Monographs in Physics. Springer-Verlag, New York, The atomic system (1987). The photon • O. Bratteli and D. W. Robinson. Operator algebras and quantum statistical field mechanics. 2. Texts and Monographs in Physics. Springer-Verlag, Berlin, Standard model of (1997). non- relativistic QED • C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg. Photons et atomes. Spectral Edition du CNRS, Paris, (1988). renormaliza- tion • C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg. Processus d’interaction group entre photons et atomes. Edition du CNRS, Paris, (1988). Resonances and lifetime • E. Fermi, Quantum theory of radiation, Rev. Mod. Phys., 4, 87-132, (1932). of metastable • W. Pauli and M. Fierz, Zur Theorie der Emission langwel liger Lichtquanten, Il, states Nuovo Cimento 15, 167-188, (1938). • M. Reed and B. Simon. Methods of modern mathematical physics. I. Functional analysis. Academic Press, New York, (1972) • M. Reed and B. Simon. Methods of modern mathematical physics. II. Fourier analysis, self-adjointness. Academic Press, New York, (1975). • H. Spohn. Dynamics of charged particles and their radiation field. Cambridge University Press, Cambridge, (2004).
Spectral RG and resonances Physical system and model J´ er´ emy Faupin The model The atomic Physical System system The photon • Non-relativistic matter: atom, ion or molecule composed of non-relativistic field Standard quantum charged particles (electrons and nuclei) model of non- • Interacting with the quantized electromagnetic field, i.e. the photon field relativistic QED Spectral renormaliza- Model: Standard model of non-relativistic QED tion group • Obtained by quantizing the Newton equations (for the charged particles) Resonances and lifetime minimally coupled to the Maxwell equations (for the electromagnetic field) of • Restriction: charges distribution are localized in small, compact sets. metastable states Corresponds to introducing an ultraviolet cutoff suppressing the interaction between the charged particles and the high-energy photons • Goes back to the early days of Quantum Mechanics (Fermi, Pauli-Fierz). Largely studied in theoretical physics (see e.g. books by Cohen-Tannoudji, Dupont-Roc and Grynberg)
Spectral RG and resonances Physical system and model J´ er´ emy Faupin The model The atomic Physical System system The photon • Non-relativistic matter: atom, ion or molecule composed of non-relativistic field Standard quantum charged particles (electrons and nuclei) model of non- • Interacting with the quantized electromagnetic field, i.e. the photon field relativistic QED Spectral renormaliza- Model: Standard model of non-relativistic QED tion group • Obtained by quantizing the Newton equations (for the charged particles) Resonances and lifetime minimally coupled to the Maxwell equations (for the electromagnetic field) of • Restriction: charges distribution are localized in small, compact sets. metastable states Corresponds to introducing an ultraviolet cutoff suppressing the interaction between the charged particles and the high-energy photons • Goes back to the early days of Quantum Mechanics (Fermi, Pauli-Fierz). Largely studied in theoretical physics (see e.g. books by Cohen-Tannoudji, Dupont-Roc and Grynberg)
Spectral RG and resonances Description of the atomic system (I) J´ er´ emy Faupin Simplest physical system The model The atomic • Hydrogen atom with an infinitely heavy nucleus fixed at the orign system The • Spin of the electron neglected photon field • Units such that � = c = 1 Standard model of non- relativistic QED Hilbert space and Hamiltonian for the electron Spectral renormaliza- • Hilbert space tion H el = L 2 ( R 3 ) group Resonances and lifetime • Hamiltonian of metastable p 2 V α ( x el ) = − α states el H el = 2 m el + V α ( x el ) , | x el | , where p el = − i ∇ x el , m el is the electron mass, and α = e 2 is the fine-structure constant ( α ≈ 1 / 137) • H el is a self-adjoint operator in L 2 ( R 3 ) with domain D ( H el ) = D ( p 2 el ) = H 2 ( R 3 )
Spectral RG and resonances Description of the atomic system (I) J´ er´ emy Faupin Simplest physical system The model The atomic • Hydrogen atom with an infinitely heavy nucleus fixed at the orign system The • Spin of the electron neglected photon field • Units such that � = c = 1 Standard model of non- relativistic QED Hilbert space and Hamiltonian for the electron Spectral renormaliza- • Hilbert space tion H el = L 2 ( R 3 ) group Resonances and lifetime • Hamiltonian of metastable p 2 V α ( x el ) = − α states el H el = 2 m el + V α ( x el ) , | x el | , where p el = − i ∇ x el , m el is the electron mass, and α = e 2 is the fine-structure constant ( α ≈ 1 / 137) • H el is a self-adjoint operator in L 2 ( R 3 ) with domain D ( H el ) = D ( p 2 el ) = H 2 ( R 3 )
Spectral RG and resonances Description of the atomic system (II) J´ er´ emy Faupin The model Spectrum of H el The atomic system The • An infinite increasing sequence of negative, isolated eigenvalues of finite photon field multiplicities { E j } j ∈ N Standard model of • The semi-axis [0 , ∞ ) of continuous spectrum non- relativistic QED Spectral Bohr’s condition renormaliza- tion group • According to the physical picture, the electron jumps from an initial state of Resonances energy E i to a final state of lower energy E f by emitting a photon of energy and lifetime of E i − E f metastable states • To capture this image mathematically, we need to take into account the interaction between the electron and the photon field • The ground state energy E 0 is expected to remain an eigenvalue (stability of the system) • The excited eigenvalues E j , j ≥ 1, associated with bound states are expected to turn into resonances associated with metastable states of finite lifetime
Spectral RG and resonances Description of the atomic system (II) J´ er´ emy Faupin The model Spectrum of H el The atomic system The • An infinite increasing sequence of negative, isolated eigenvalues of finite photon field multiplicities { E j } j ∈ N Standard model of • The semi-axis [0 , ∞ ) of continuous spectrum non- relativistic QED Spectral Bohr’s condition renormaliza- tion group • According to the physical picture, the electron jumps from an initial state of Resonances energy E i to a final state of lower energy E f by emitting a photon of energy and lifetime of E i − E f metastable states • To capture this image mathematically, we need to take into account the interaction between the electron and the photon field • The ground state energy E 0 is expected to remain an eigenvalue (stability of the system) • The excited eigenvalues E j , j ≥ 1, associated with bound states are expected to turn into resonances associated with metastable states of finite lifetime
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