### Citations

342 | Scaling limits of loop-erased random walks and uniform spanning trees,”
- Schramm
- 2000
(Show Context)
Citation Context ...er looperasure prescription) is in terms of this previously marked territory. We believe a similar LERW model on R2 instead of R could help bridge the gap between the Schramm-Löwner Evolution theory =-=[55]-=- and conformal field theory in its Coulomb gas formulation (see, e.g., [23]). Acknowledgements: This work was supported in part by the National Science 26 Foundation under grant DMS # 0907198. We are ... |

246 |
The P (ϕ)⊂2 Euclidean (quantum) Field Theory
- Simon
- 1974
(Show Context)
Citation Context ...or the construction of the ground state itself. We will not dwell on the the proof of the reduction (1) which is standard. One needs to use the Schrödinger representation of F in Q-space (see, e.g., =-=[32, 56]-=-) as well as the Feynman-Kac-Nelson formula (see, e.g., [56, 42, 37] and also [38] for the simultaneous treatment of spin). Note that the spin part of the interaction can be made to act as a multiplic... |

233 |
Combinatorial Species and Tree-like Structures, Cambrige Univ.
- Bergeron, Labelle, et al.
- 1998
(Show Context)
Citation Context ...orics. Over and over in the course of our proof, we will use relabeling changes of variables, continuous or discrete. This is because behind the scene there is Joyal’s theory of combinatorial species =-=[40, 19]-=- at work. Although we do not resort to this theory explicitly, in this article, it is the proper mathematical framework for much of the combinatorics involved in Feynman/cluster type expansions. The t... |

211 |
Perturbation theory of linear operators (Second edition
- Kato
- 1984
(Show Context)
Citation Context ...onstruct the ground state ΩGS(λ) and its energy E(λ) = inf σ(Hλ). As this energy is at the bottom of the continuous spectrum when λ = 0, the standard perturbation theory tools (as one can find say in =-=[41]-=-) do not apply. More sophisticated methods are needed for the control of the ground state energy for nonzero coupling λ. Such a method was introduced by Bach, Fröhlich and Sigal in their seminal work... |

199 |
Une théorie combinatoire des séries formelles
- Joyal
- 1981
(Show Context)
Citation Context ...orics. Over and over in the course of our proof, we will use relabeling changes of variables, continuous or discrete. This is because behind the scene there is Joyal’s theory of combinatorial species =-=[40, 19]-=- at work. Although we do not resort to this theory explicitly, in this article, it is the proper mathematical framework for much of the combinatorics involved in Feynman/cluster type expansions. The t... |

144 | The multivariate Tutte polynomial (alias Potts model) for graphs and matroids.
- Sokal
- 2005
(Show Context)
Citation Context ...ntioned only consider applications to the field of combinatorial enumeration. The weights used are placeholders such as xn where n counts something. In accordance with Sokal’s multivariate philosophy =-=[57]-=-, allowing more general weights such as contractions of tensors is where the full potential of the theory is. Some steps in this direction were taken in [2]. From the point of view of the latter artic... |

120 | Spectral analysis for systems of atoms and molecules coupled to the quantized radiation field
- Bach, Fröhlich, et al.
- 1999
(Show Context)
Citation Context ... do not apply. More sophisticated methods are needed for the control of the ground state energy for nonzero coupling λ. Such a method was introduced by Bach, Fröhlich and Sigal in their seminal work =-=[17, 18]-=- and it has been further developed over the last decade (see, e.g., [14, 15, 16]). One of its main motivations is to provide rigorous constructive algorithms for the calculation of quantities such as ... |

83 |
Conformal Field Theory. Graduate Texts in Contemporary Physics
- Francesco, Mathieu, et al.
- 1999
(Show Context)
Citation Context ...y. We believe a similar LERW model on R2 instead of R could help bridge the gap between the Schramm-Löwner Evolution theory [55] and conformal field theory in its Coulomb gas formulation (see, e.g., =-=[23]-=-). Acknowledgements: This work was supported in part by the National Science 26 Foundation under grant DMS # 0907198. We are indebted to Ira Herbst for introducing us to the spin-Boson model and for m... |

81 |
Dynamics of the dissipative two-state system
- Leggett, Chakravarty, et al.
- 1987
(Show Context)
Citation Context ...tic presentation of the model. For ease of reference, we kept the exact same formulation and notations as in the cited article. For more background from a physical perspective, the reader may consult =-=[43]-=- for a good review of the spin-Boson model as well as its Fermionic cousin: the Kondo model. For a review from a mathematical perspective, we recommend [39], where many open problems are listed. It is... |

71 | Renormalization group analysis of spectral problems in quantum field theory
- Bach, Fröhlich, et al.
- 1998
(Show Context)
Citation Context ... do not apply. More sophisticated methods are needed for the control of the ground state energy for nonzero coupling λ. Such a method was introduced by Bach, Fröhlich and Sigal in their seminal work =-=[17, 18]-=- and it has been further developed over the last decade (see, e.g., [14, 15, 16]). One of its main motivations is to provide rigorous constructive algorithms for the calculation of quantities such as ... |

38 | Infrared-finite algorithms in QED: the groundstate of an atom interacting with the quantized radiation field
- Bach, Fröhlich, et al.
- 2006
(Show Context)
Citation Context ...ground state energy for nonzero coupling λ. Such a method was introduced by Bach, Fröhlich and Sigal in their seminal work [17, 18] and it has been further developed over the last decade (see, e.g., =-=[14, 15, 16]-=-). One of its main motivations is to provide rigorous constructive algorithms for the calculation of quantities such as E(λ), in the case of massless Bosons. Such an algorithm based on the renormaliza... |

38 |
Effective action for the Yukawa 2 quantum field Theory
- Lesniewski
- 1987
(Show Context)
Citation Context ...he treatment of Fermionic theories. For instance, early cluster expansions for Fermions used decompositions into cells (see, e.g., [31, 29]). Later, expansions which decouple Feynman diagram vertices =-=[44, 10]-=- allowed to achieve the same convergence bounds in a simpler way. Remark 4: The combinatorial structures which feature in our expansion for the spin-Boson model are more than strikingly similar to the... |

35 | Smooth Feshbach map and operator-theoretic renormalization group methods
- Bach, Chen, et al.
(Show Context)
Citation Context ...ground state energy for nonzero coupling λ. Such a method was introduced by Bach, Fröhlich and Sigal in their seminal work [17, 18] and it has been further developed over the last decade (see, e.g., =-=[14, 15, 16]-=-). One of its main motivations is to provide rigorous constructive algorithms for the calculation of quantities such as E(λ), in the case of massless Bosons. Such an algorithm based on the renormaliza... |

35 |
Gross-Neveu model through convergent perturbation expansions
- Gawedzki, Kupiainen
- 1985
(Show Context)
Citation Context .... A similar distinction can be made between existing techniques for the treatment of Fermionic theories. For instance, early cluster expansions for Fermions used decompositions into cells (see, e.g., =-=[31, 29]-=-). Later, expansions which decouple Feynman diagram vertices [44, 10] allowed to achieve the same convergence bounds in a simpler way. Remark 4: The combinatorial structures which feature in our expan... |

33 |
Convergence of fugacity expansions for classical systems. In: Statistical mechanics: foundations and
- Penrose
- 1967
(Show Context)
Citation Context ...he presentation in [27]. The rooted forest formula has more of a Fermionic flavor. It is 25 related to the matrix-tree theorem and is closer to Penrose’s lemma for the convergence of the Mayer series =-=[47]-=-. Remark 2: Surprisingly, in two dimensions, the obtained series still makes sense, order-by-order, even without infrared cut-off. Its Borel transform has a nonzero radius of analyticity. This can be ... |

32 |
Decay of correlations for infinite range interactions in unbounded spin systems.
- Cammarota
- 1982
(Show Context)
Citation Context ...heir ancestors, and then progressing towards the root. The procedure is somewhat subtle since it is a mixture of straightforward pin and sum with L1 decay and a generalization of Cammarota’s argument =-=[22]-=- to polymers in the continuum instead of on the lattice. A possible choice of cycle opening for the example in the last picture is 1,2 9,10 7,8 5,6 3,4 11,12 13,14 19,20 17,18 15,16 where we indicated... |

31 | Radiative decay: nonperturbative approaches
- Hübner, Spohn
- 1995
(Show Context)
Citation Context ...hysical perspective, the reader may consult [43] for a good review of the spin-Boson model as well as its Fermionic cousin: the Kondo model. For a review from a mathematical perspective, we recommend =-=[39]-=-, where many open problems are listed. It is well-known that if f√ ω and f ω belong to h, and for real λ, then Hλ is self-adjoint and bounded below. Of particular interest is the massless case where ω... |

26 | Cluster expansion for abstract polymer models. New bounds from an old approach
- Fernández, Procacci
- 2007
(Show Context)
Citation Context ... , this translates (provided Bloch’s formula holds) into a lower bound for the λ radius of analyticity ' 0.34. 6 Closing remarks Remark 1: One could presumably improve the estimates, in the spirit of =-=[30]-=-, using the remaining hardcore constraints which were thrown away in (8). In this case one should use the Rooted Taylor Forest Formula of [9] which forces offspring exclusion. Then one should follow t... |

21 |
A renormalizable field theory: the massive Gross-Neveu model in two dimensions
- Feldman, Magnen, et al.
- 1986
(Show Context)
Citation Context .... A similar distinction can be made between existing techniques for the treatment of Fermionic theories. For instance, early cluster expansions for Fermions used decompositions into cells (see, e.g., =-=[31, 29]-=-). Later, expansions which decouple Feynman diagram vertices [44, 10] allowed to achieve the same convergence bounds in a simpler way. Remark 4: The combinatorial structures which feature in our expan... |

21 | Analytic perturbation theory and renormalization analysis of matter coupled to quantized radiation
- Griesemer, Hasler
(Show Context)
Citation Context ... of the ground state energy with respect to λ (and other parameters) was established for less singular nonrelativistic QED models, using the RG techniques of Bach, Fröhlich and Sigal, in the article =-=[33]-=-. On Page 580 of the latter, one can read: “It seems unlikely that another approach, not based on a renormalization analysis would yield a result similar to ours”. As demonstrated in the present artic... |

20 | Constructive φ4 field theory without tears
- Magnen, Rivasseau
(Show Context)
Citation Context ...in this article, we keep them in reserve for more dire infrared (or ultraviolet) future circumstances. There is also a very promising new set of ideas on how to improve CQFT expansion techniques (see =-=[46, 34]-=-), but 3 this is still at the ‘beta version’ stage. These ideas which originated in [50] unexpectedly came from the study of renormalization in noncommutative quantum field theory (see, e.g., [52]). A... |

20 |
Ground State(s) of the Spin-Boson Hamiltonian
- Spohn
- 1999
(Show Context)
Citation Context ... with unit parameter (i.e., with measure e−tdt). This is the famous Ising model over R with long range ∼ 1 t2 interactions introduced in [11] for the Kondo model and in [25] for spin-Boson model. See =-=[59, 26, 58]-=- for a mathematical study of the spin-Boson model via this route. Our main result is the following. Claim: One has an explicit power series expansion for lim T→∞ −1 T logZ(α, T ) . This quantity is an... |

17 | Interacting Fermi liquid in two dimensions at finite temperature
- Disertori, Rivasseau
- 2000
(Show Context)
Citation Context ...uch optional features as multiscale analysis and renormalization, large versus small field analysis, p-particle irreducibility analysis, and could also handle microlocal sectorial decomposition as in =-=[48, 24]-=-. Since we do not need these in this article, we keep them in reserve for more dire infrared (or ultraviolet) future circumstances. There is also a very promising new set of ideas on how to improve CQ... |

17 | Constructive Matrix Theory
- Rivasseau
(Show Context)
Citation Context ...umstances. There is also a very promising new set of ideas on how to improve CQFT expansion techniques (see [46, 34]), but 3 this is still at the ‘beta version’ stage. These ideas which originated in =-=[50]-=- unexpectedly came from the study of renormalization in noncommutative quantum field theory (see, e.g., [52]). A good pedagogical entry point in the subject of CQFT expansions is [51] (see also the fo... |

17 | Enumerative combinatorics. Vol. 2. With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin - Stanley - 1999 |

16 | Feynman diagrams in algebraic combinatorics - Abdesselam |

16 |
Functional integral representation of a model in quantum electrodynamics
- Hiroshima
- 1997
(Show Context)
Citation Context ...l on the the proof of the reduction (1) which is standard. One needs to use the Schrödinger representation of F in Q-space (see, e.g., [32, 56]) as well as the Feynman-Kac-Nelson formula (see, e.g., =-=[56, 42, 37]-=- and also [38] for the simultaneous treatment of spin). Note that the spin part of the interaction can be made to act as a multiplication operator. This is done by a simple conjugation with u = 1√ 2 (... |

15 |
Boson quantum field models
- Glimm, Jaffe
- 1972
(Show Context)
Citation Context ...or the construction of the ground state itself. We will not dwell on the the proof of the reduction (1) which is standard. One needs to use the Schrödinger representation of F in Q-space (see, e.g., =-=[32, 56]-=-) as well as the Feynman-Kac-Nelson formula (see, e.g., [56, 42, 37] and also [38] for the simultaneous treatment of spin). Note that the spin part of the interaction can be made to act as a multiplic... |

15 | Functional integral representation of the Pauli-Fierz Hamiltonian with spin 1/2
- Hiroshima, Lőrinczi
(Show Context)
Citation Context ...the reduction (1) which is standard. One needs to use the Schrödinger representation of F in Q-space (see, e.g., [32, 56]) as well as the Feynman-Kac-Nelson formula (see, e.g., [56, 42, 37] and also =-=[38]-=- for the simultaneous treatment of spin). Note that the spin part of the interaction can be made to act as a multiplication operator. This is done by a simple conjugation with u = 1√ 2 ( 1 −1 1 1 ) , ... |

15 | Gibbs measures for Brownian paths under the effect of an external and a small pair potential
- Lőrinczi, Minlos
(Show Context)
Citation Context ...pansion technology which worked so well in the context of constructive quantum field theory (CQFT). We are not the first to use this general approach in the context of nonrelativistic QED, see, e.g., =-=[45]-=-. However, we have the advantage of using the latest ‘stable version’ of the CQFT cluster expansion software [1] which comes with such optional features as multiscale analysis and renormalization, lar... |

14 | Explicit fermionic tree expansions.
- Abdesselam, Rivasseau
- 1998
(Show Context)
Citation Context ...he treatment of Fermionic theories. For instance, early cluster expansions for Fermions used decompositions into cells (see, e.g., [31, 29]). Later, expansions which decouple Feynman diagram vertices =-=[44, 10]-=- allowed to achieve the same convergence bounds in a simpler way. Remark 4: The combinatorial structures which feature in our expansion for the spin-Boson model are more than strikingly similar to the... |

14 |
Infrared-finite algorithms in QED II. The expansion of the groundstate of an atom interacting with the quantized radiation field
- Bach, Fröhlich, et al.
- 2009
(Show Context)
Citation Context ...ground state energy for nonzero coupling λ. Such a method was introduced by Bach, Fröhlich and Sigal in their seminal work [17, 18] and it has been further developed over the last decade (see, e.g., =-=[14, 15, 16]-=-). One of its main motivations is to provide rigorous constructive algorithms for the calculation of quantities such as E(λ), in the case of massless Bosons. Such an algorithm based on the renormaliza... |

14 |
Return to Equilibrium’ for weakly coupled quantum systems: a simple polymer expansion
- Roeck, Kupianien
- 2011
(Show Context)
Citation Context ...ons, inf σ(Hλ) = −∞ and we do not know what the term-by-term infrared-finite perturbation series means in that case. Remark 3: There are similarities between our expansion and the one found, e.g., in =-=[54]-=-, and for a good reason: both essentially are resummations of the same underlying Feynman diagram expansion. Indeed, the massless spinBoson model is a typical example for which the results of [54] can... |

12 |
Quantum tunneling with dissipation and the Ising model over
- Spohn, Dümcke
- 1985
(Show Context)
Citation Context ... with unit parameter (i.e., with measure e−tdt). This is the famous Ising model over R with long range ∼ 1 t2 interactions introduced in [11] for the Kondo model and in [25] for spin-Boson model. See =-=[59, 26, 58]-=- for a mathematical study of the spin-Boson model via this route. Our main result is the following. Claim: One has an explicit power series expansion for lim T→∞ −1 T logZ(α, T ) . This quantity is an... |

10 |
Exact results in the Kondo problem: equivalence to a classical one-dimensional Coulomb gas,” Phys
- Anderson, Yuval
- 1969
(Show Context)
Citation Context ...etween two consecutive spin flips are exponentially distributed with unit parameter (i.e., with measure e−tdt). This is the famous Ising model over R with long range ∼ 1 t2 interactions introduced in =-=[11]-=- for the Kondo model and in [25] for spin-Boson model. See [59, 26, 58] for a mathematical study of the spin-Boson model via this route. Our main result is the following. Claim: One has an explicit po... |

10 | Tree quantum field theory
- Gurau, Magnen, et al.
- 2009
(Show Context)
Citation Context ...in this article, we keep them in reserve for more dire infrared (or ultraviolet) future circumstances. There is also a very promising new set of ideas on how to improve CQFT expansion techniques (see =-=[46, 34]-=-), but 3 this is still at the ‘beta version’ stage. These ideas which originated in [50] unexpectedly came from the study of renormalization in noncommutative quantum field theory (see, e.g., [52]). A... |

10 | G.,Mean Green’s function of the Anderson model at weak disorder with an infra-red cut-off
- Poirot
- 1999
(Show Context)
Citation Context ...uch optional features as multiscale analysis and renormalization, large versus small field analysis, p-particle irreducibility analysis, and could also handle microlocal sectorial decomposition as in =-=[48, 24]-=-. Since we do not need these in this article, we keep them in reserve for more dire infrared (or ultraviolet) future circumstances. There is also a very promising new set of ideas on how to improve CQ... |

9 | Brill-Gordan loci, transvectants and an analogue of the Foulkes conjecture
- Abdesselam, Chipalkatti
(Show Context)
Citation Context ...a shared by such remote areas as quantum field theory and (neo)classical invariant theory. The only difference is that of infinite versus finite dimension respectively. More on this can be found 5 in =-=[6, 7, 3]-=-. This article demonstrates that there is a bright future for ‘the Euclidean strategy + CQFT cluster expansion’ method in the area of nonrelativistic QED. The first part, i.e., the Euclidean reformula... |

9 | Branched polymers and dimensional reduction
- Brydges, Imbrie
(Show Context)
Citation Context ...our expansion for the spin-Boson model are more than strikingly similar to the “canvases” of [6] which are related to the loop-erased random walk (LERW). We suspect a model mapping such as the one in =-=[20]-=- is lurking behind this similarity. Here the LERW seems to be one on the real line where a particle alternately ‘marks its territory’ with an interval of exponentially distributed length, and diffuses... |

9 | Ground states in the spin boson model
- Hasler, Herbst
(Show Context)
Citation Context ...L2(R3,C, d3k). The creation and annihilation operators, a∗(k) and a(k) respectively, satisfy the usual canonical commutation relations. The matrices σx and σz are standard Pauli matrices. We refer to =-=[35]-=- for a more detailed spectral theoretic presentation of the model. For ease of reference, we kept the exact same formulation and notations as in the cited article. For more background from a physical ... |

8 | On the volume conjecture for classical spin networks
- Abdesselam
- 1009
(Show Context)
Citation Context ...a shared by such remote areas as quantum field theory and (neo)classical invariant theory. The only difference is that of infinite versus finite dimension respectively. More on this can be found 5 in =-=[6, 7, 3]-=-. This article demonstrates that there is a bright future for ‘the Euclidean strategy + CQFT cluster expansion’ method in the area of nonrelativistic QED. The first part, i.e., the Euclidean reformula... |

8 | Loop Vertex Expansion for Φ2k Theory in Zero Dimension
- Rivasseau, Wang
(Show Context)
Citation Context ...he study of renormalization in noncommutative quantum field theory (see, e.g., [52]). A good pedagogical entry point in the subject of CQFT expansions is [51] (see also the follow up research article =-=[53]-=-). Our point of departure is Bloch’s formula E(λ) = lim T→∞ − 1 T log ( Ω↓, e−THλΩ↓ ) as well as the Feynman-Kac-Nelson formula followed by integration over the Bosons, with the result ( Ω↓, e−THλΩ↓ )... |

7 |
forests and jungles: a botanical garden for cluster expansions
- Trees
- 1995
(Show Context)
Citation Context ...on, we set F (s) = ∏ X∈Π(P∗+P ) ∏ {A,B}∈X(2) 1l {tA ∩ tB = ∅} × ∏ {X,Y }∈Π(P∗+P )(2) ∏ A∈X,B∈Y ( 1− s{X,Y }1l {tA ∩ tB 6= ∅} ) . Now use the Brydges-Kennedy-Abdesselam-Rivasseau formula =-=[21, 9]-=- (see [4] for a gentle introduction and detailed proof of this identity). The outcome is F (1) = ∑ F forest on Π(P∗+P ) ∫ [0,1]F ∏ l∈F dul ∂|F|F∏ l∈F ∂sl (s(F, u)) . Here the sum is over all forests F... |

7 |
expansions and the HamiltonJacobi equation
- Mayer
- 1987
(Show Context)
Citation Context ...on, we set F (s) = ∏ X∈Π(P∗+P ) ∏ {A,B}∈X(2) 1l {tA ∩ tB = ∅} × ∏ {X,Y }∈Π(P∗+P )(2) ∏ A∈X,B∈Y ( 1− s{X,Y }1l {tA ∩ tB 6= ∅} ) . Now use the Brydges-Kennedy-Abdesselam-Rivasseau formula =-=[21, 9]-=- (see [4] for a gentle introduction and detailed proof of this identity). The outcome is F (1) = ∑ F forest on Π(P∗+P ) ∫ [0,1]F ∏ l∈F dul ∂|F|F∏ l∈F ∂sl (s(F, u)) . Here the sum is over all forests F... |

7 |
An expression of the ground state energy of the Spin-Boson model
- Hirokawa
- 1999
(Show Context)
Citation Context ...nother approach, not based on a renormalization analysis would yield a result similar to ours”. As demonstrated in the present article, there is another way ! Another expression for E(λ) was found in =-=[36]-=- but it is far from explicit, nor is it clear (at least to us) how one can use it to prove analyticity. In this article, we obtain a completely explicit combinatorial expansion for E(λ). It is explici... |

6 |
A connected graph identity and convergence of cluster expansions
- Faris
(Show Context)
Citation Context ...ng hardcore constraints which were thrown away in (8). In this case one should use the Rooted Taylor Forest Formula of [9] which forces offspring exclusion. Then one should follow the presentation in =-=[27]-=-. The rooted forest formula has more of a Fermionic flavor. It is 25 related to the matrix-tree theorem and is closer to Penrose’s lemma for the convergence of the Mayer series [47]. Remark 2: Surpris... |

6 | Constructive Field Theory in Zero Dimension
- Rivasseau
(Show Context)
Citation Context ...ch originated in [50] unexpectedly came from the study of renormalization in noncommutative quantum field theory (see, e.g., [52]). A good pedagogical entry point in the subject of CQFT expansions is =-=[51]-=- (see also the follow up research article [53]). Our point of departure is Bloch’s formula E(λ) = lim T→∞ − 1 T log ( Ω↓, e−THλΩ↓ ) as well as the Feynman-Kac-Nelson formula followed by integration ov... |

5 |
Notes on the Brydges-Kennedy-AbdesselamRivasseau forest interpolation formula. Notes for a graduate course at the University of Virginia. Available at http://people.virginia.edu/∼aa4cr/Math845.html
- Abdesselam
(Show Context)
Citation Context ...(s) = ∏ X∈Π(P∗+P ) ∏ {A,B}∈X(2) 1l {tA ∩ tB = ∅} × ∏ {X,Y }∈Π(P∗+P )(2) ∏ A∈X,B∈Y ( 1− s{X,Y }1l {tA ∩ tB 6= ∅} ) . Now use the Brydges-Kennedy-Abdesselam-Rivasseau formula [21, 9] (see =-=[4]-=- for a gentle introduction and detailed proof of this identity). The outcome is F (1) = ∑ F forest on Π(P∗+P ) ∫ [0,1]F ∏ l∈F dul ∂|F|F∏ l∈F ∂sl (s(F, u)) . Here the sum is over all forests F on the v... |

5 |
Singular perturbations of positivity preserving semigroups via path space techniques
- Klein, Landau
- 1975
(Show Context)
Citation Context ...l on the the proof of the reduction (1) which is standard. One needs to use the Schrödinger representation of F in Q-space (see, e.g., [32, 56]) as well as the Feynman-Kac-Nelson formula (see, e.g., =-=[56, 42, 37]-=- and also [38] for the simultaneous treatment of spin). Note that the spin part of the interaction can be made to act as a multiplication operator. This is done by a simple conjugation with u = 1√ 2 (... |

5 |
Vignes-Tourneret F., Renormalisation of non-commutative field theories
- Rivasseau
(Show Context)
Citation Context ...[46, 34]), but 3 this is still at the ‘beta version’ stage. These ideas which originated in [50] unexpectedly came from the study of renormalization in noncommutative quantum field theory (see, e.g., =-=[52]-=-). A good pedagogical entry point in the subject of CQFT expansions is [51] (see also the follow up research article [53]). Our point of departure is Bloch’s formula E(λ) = lim T→∞ − 1 T log ( Ω↓, e−T... |

4 |
Renormalisation Constructive Explicite
- Abdesselam
- 1997
(Show Context)
Citation Context ...e first to use this general approach in the context of nonrelativistic QED, see, e.g., [45]. However, we have the advantage of using the latest ‘stable version’ of the CQFT cluster expansion software =-=[1]-=- which comes with such optional features as multiscale analysis and renormalization, large versus small field analysis, p-particle irreducibility analysis, and could also handle microlocal sectorial d... |

2 |
A.: Scattering and Bound States in Euclidean Lattice Quantum Field Theories
- Auil, Barata
- 2001
(Show Context)
Citation Context ...this endevor succeeds, would be to apply the lessons learned in such simpler models to the scattering theory of P (φ)2. The latter is the missing piece of the glorious CQFT work in the seventies (see =-=[12]-=- for a recent update and list of references). We would like to dream that this achievement is possible. 2 The jump process Let N(t), t ≥ 0, be the usual Poisson process with parameter 1. Let B = ±1 be... |

2 |
Translating the spin-boson model into a classical system
- Fannes, Nachtergaele
- 1988
(Show Context)
Citation Context ... with unit parameter (i.e., with measure e−tdt). This is the famous Ising model over R with long range ∼ 1 t2 interactions introduced in [11] for the Kondo model and in [25] for spin-Boson model. See =-=[59, 26, 58]-=- for a mathematical study of the spin-Boson model via this route. Our main result is the following. Claim: One has an explicit power series expansion for lim T→∞ −1 T logZ(α, T ) . This quantity is an... |

2 | Coulomb interactions symmetries and the Mayer series in the two-dimensional dipole gas - Procacci, Pereira, et al. - 1997 |

1 |
The ground state of the massless spin-Boson model
- Abdesselam
(Show Context)
Citation Context ...och’s formula requires a nonzero overlap (Ω↓,ΩGS(λ)) 6= 0 which is known to hold for small coupling, for instance by the results of [35], but with no explicit range of validity. This will be fixed in =-=[5]-=- where the present expansion technique is used for the construction of the ground state itself. We will not dwell on the the proof of the reduction (1) which is standard. One needs to use the Schrödi... |

1 | Cramer’s rule and loop ensembles
- Abdesselam, Brydges
- 2006
(Show Context)
Citation Context ...a shared by such remote areas as quantum field theory and (neo)classical invariant theory. The only difference is that of infinite versus finite dimension respectively. More on this can be found 5 in =-=[6, 7, 3]-=-. This article demonstrates that there is a bright future for ‘the Euclidean strategy + CQFT cluster expansion’ method in the area of nonrelativistic QED. The first part, i.e., the Euclidean reformula... |

1 |
Low-temperature properties of the Kondo Hamiltonian
- Emery, Luther
- 1974
(Show Context)
Citation Context ...s are exponentially distributed with unit parameter (i.e., with measure e−tdt). This is the famous Ising model over R with long range ∼ 1 t2 interactions introduced in [11] for the Kondo model and in =-=[25]-=- for spin-Boson model. See [59, 26, 58] for a mathematical study of the spin-Boson model via this route. Our main result is the following. Claim: One has an explicit power series expansion for lim T→∞... |

1 |
Review of “Quantum Physics. A Functional Integral Point of View” by Glimm and Jaffe
- Federbush
- 1981
(Show Context)
Citation Context ...QED. The first part, i.e., the Euclidean reformulation of the problems posed by the interaction of matter with radiation should be straightforward. To paraphrase the prophetic words of Paul Federbush =-=[28]-=-, the second part is where the action is. One can try to use techniques similar to ours, in order to treat progressively more difficult and more realistic models. However, we think it is more importan... |