Quantum field theory meets Hopf algebra

Brouder, Christian

doi:10.1002/mana.200610828

Cited by 14 publications

(18 citation statements)

References 54 publications

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“…Combinatorial physics is an emerging field that uses modern tools of algebraic combinatorics to solve physical problems. It was born with the investigation of the algebraic structure of renormalization in quantum field theory63 and showed its ability to deal with many‐body problems 64…”

Section: Discussionmentioning

confidence: 99%

“…This superstructure exists for a broad family of Hamiltonians, including finite matrices and differential operators. When the Hamiltonian is described by creation and annihilation operators, then a second combinatorial structure intervenes, which is a Hopf algebra 64. We expect that the interplay of the general superstructure of the Rayleigh series with the Hopf algebraic structure of field operators will provide new insights into MBPT.…”

Section: Discussionmentioning

confidence: 99%

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The Rayleigh‐Schrödinger perturbation series of quasi‐degenerate systems

Brouder¹,

Duchamp²,

Patras³

et al. 2011

Int J of Quantum Chemistry

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16 pages, 3 figuresInternational audienceWe present the first representation of the general term of the Rayleigh-Schrödinger series for quasidegenerate systems. Each term of the series is represented by a tree and there is a straightforward relation between the tree and the analytical expression of the corresponding term. The combinatorial and graphical techniques used in the proof of the series expansion allow us to derive various resummation formulas of the series. The relation with several combinatorial objects used for special cases (degenerate or non-degenerate systems) is established

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Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

The Rayleigh‐Schrödinger perturbation series of quasi‐degenerate systems

Brouder¹,

Duchamp²,

Patras³

et al. 2011

Int J of Quantum Chemistry

Self Cite

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“…• to each monomial O i (for example j 3 x 2 j 5 j 6 x 1 ), we associate the corresponding 1PI partially amputated Green's function (for example PI (5) j 3 x 2 j 5 j 6 x 1 ). The j i are treated as external parameters as they give rise to Feynman propagators in the expansion of PI in terms of Feynman propagators and fully amputated Greens functions.…”

Section: Planar Connected and 1pi Green's Functionsmentioning

confidence: 99%

“…Suppose that A is an algebra with product m A and unit map e A (1) = 1 A , e.g., A = K or A = B, where B is a bialgebra. The vector space L(B, A) of linear maps from B to A together with the convolution product (5) Φ…”

Section: +Wmentioning

confidence: 99%

The combinatorics of Green’s functions in planar field theories

Ebrahimi‐Fard

Patras

2016

Front. Phys.

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Abstract. The aim of this work is to outline in some detail the use of combinatorial algebra in planar quantum field theory. Particular emphasis is given to the relations between the different types of planar Green's functions. The key object is a Hopf algebra which is naturally defined on non-commuting sources, and the fact that its genuine unshuffle coproduct splits into left-and right unshuffle half-coproduts. The latter give rise to the notion of unshuffle bialgebra. This setting allows to describe the relation between planar full and connected Green's functions by solving a simple linear fixed point equation. A modification of this linear fixed point equation gives rise to the relation between planar connected and one-particle irreducible Green's functions. The graphical calculus that arises from this approach also leads to a new understanding of functional calculus in planar QFT, whose rules for differentiation with respect to non-commuting sources can be translated into the language of growth operations on planar rooted trees. We also include a brief outline of our approach in the framework of non-planar theories.

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“…Selected theories (such as the φ 4 model) were studied even in more detail providing many-loop expansions in terms of the Feynman diagrams including closed expressions for their multiplicity factors [9][10][11]. These works fall into a broader effort of the Feynman diagram enumeration [12][13][14][15][16][17], including generalizations to various field theories [18][19][20][21][22][23].…”

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confidence: 99%

A novel approach to perturbative calculations for a large class of interacting boson theories

Brádler

2018

Nuclear Physics B

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A. We present a method of calculating the interacting S-matrix to an arbitrary perturbative order for a large class of boson interaction Lagrangians. The method takes advantage of a previously unexplored link between the n-point Green's function and a certain system of linear Diophantine equations. By finding all nonnegative solutions of the system, the task of perturbatively expanding an interacting S-matrix becomes elementary for any number of interacting fields, to an arbitrary perturbative order (irrespective of whether it makes physical sense) and for a large class of scalar boson theories. The method does not rely on the positionbased Feynman diagrams and promises to be extended to many perturbative models typically studied in quantum field theory. Aside from interaction field calculations we showcase our approach by expanding a pair of Unruh-DeWitt detectors coupled to Minkowski vacuum to an arbitrary perturbative order in the coupling constant. We also link our result to Hafnian as introduced by Caianiello and present a method to list all (2n − 1)!! perfect matchings of a complete graph on 2n vertices.

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Quantum field theory meets Hopf algebra

Cited by 14 publications

References 54 publications

The Rayleigh‐Schrödinger perturbation series of quasi‐degenerate systems

The Rayleigh‐Schrödinger perturbation series of quasi‐degenerate systems

The combinatorics of Green’s functions in planar field theories

A novel approach to perturbative calculations for a large class of interacting boson theories

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