Generalized campbell-baker-hausdorff formula, path-ordering and bernoulli numbers

Grensing, D.; Grensing, Gerhard

doi:10.1007/bf01411151

Cited by 8 publications

(17 citation statements)

References 6 publications

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“…The moments of inertia of this top are taken finally to zero. As we show below this path integral amounts to zero for Wilson loops defined for SU (2). Therefore, it is not a surprise that the application of this erroneous path integral representation to the evaluation of the average value of Wilson loops has led to the conclusion that for large loops the area-law falloff is present for colour charges taken in any irreducible representation r of SU (N ) [7].…”

Section: Path Integral Representation For Wilson Loopsmentioning

confidence: 99%

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Path integral representation for Wilson loops and the non-Abelian Stokes theorem

et al. 2000

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We discuss the derivation of the path integral representation over gauge degrees of freedom for Wilson loops in SU (N ) gauge theory and 4-dimensional Euclidean space-time by using well-known properties of group characters. A discretized form of the path integral is naturally provided by the properties of group characters and does not need any artificial regularization. We show that the path integral over gauge degrees of freedom for Wilson loops derived by Diakonov and Petrov (Phys. Lett. B224 (1989) 131) by using a special regularization is erroneous and predicts zero for the Wilson loop. This property is obtained by direct evaluation of path integrals for Wilson loops defined for pure SU (2) gauge fields and Z(2) center vortices with spatial azimuthal symmetry. Further we show that both derivations given by Diakonov and Petrov for their regularized path integral, if done correctly, predict also zero for Wilson loops. Therefore, the application of their path integral representation of Wilson loops cannot give a new way to check confinement in lattice as has been declared by Diakonov and Petrov (Phys. Lett. B242 (1990) 425). From the path integral representation which we consider we conclude that no new non-Abelian Stokes theorem can exist for Wilson loops except the old-fashioned one derived by means of the path-ordering procedure.

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Section: Path Integral Representation For Wilson Loopsmentioning

confidence: 99%

“…For Wilson loops the contour C defines a closed path C xx . For determinations of the parallel transport operator U (C yx ) the action of the path-ordering operator P Cyx is defined by the following limiting procedure [2] U (C yx ) = P Cxy e i g Cyx dz µ A µ (z) = lim n→∞ n k=1…”

Section: Introductionmentioning

confidence: 99%

Path integral representation for Wilson loops and the non-Abelian Stokes theorem

et al. 2000

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“…(20) proceeds by discretising the Wilson line operator into a product of exponentials, and then combining these using the Baker-Campbell-Hausdorff (BCH) formula, before taking the continuum limit. It is the use of the BCH formula that results in fully nested commutator structures at each order -see ref [113]…”

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confidence: 99%

The non-Abelian exponentiation theorem for multiple Wilson lines

Gardi

Smillie

White

2013

J. High Energ. Phys.

182

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We study the structure of soft gluon corrections to multi-leg scattering amplitudes in a non-Abelian gauge theory by analysing the corresponding product of semi-infinite Wilson lines. We prove that diagrams exponentiate such that the colour factors in the exponent are fully connected. This completes the generalisation of the non-Abelian exponentiation theorem, previously proven in the case of a Wilson loop, to the case of multiple Wilson lines in arbitrary representations of the colour group. Our proof is based on the replica trick in conjunction with a new formalism where multiple emissions from a Wilson line are described by effective vertices, each having a connected colour factor. The exponent consists of connected graphs made out of these vertices. We show that this readily provides a general colour basis for webs. We further discuss the kinematic combinations that accompany each connected colour factor, and explicitly catalogue all three-loop examples, as necessary for a direct computation of the soft anomalous dimension at this order.

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“…This f appears because it is the generating function of the Bernoulli numbers which play an important role in the BCH formula [45]. While it is generally redundant to describe H A , S A , and R A := H A − S A , we choose to do so -and in fact give two expressions for R A -both in order to state definite results and to describe the relative sizes of various contributions.…”

Section: Entropymentioning

confidence: 99%

“…(A.6) Iterating Eq. (12) of[45] in such a way as to obtain all terms with two Y 's gives log(e X e Y ) = X + Y + Rn = (−1) n Bn/n! and AdZ W = [Z, W ].…”

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confidence: 99%

’t Hooft suppression and holographic entropy

Kelly

Kuns

Marolf

2015

J. High Energ. Phys.

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Recent works have related the bulk first law of black hole mechanics to the first law of entanglement in a dual CFT. These are first order relations, and receive corrections for finite changes. In particular, the latter is naively expected to be accurate only for small changes in the quantum state. But when Newton's constant is small relative to the AdS scale, the former holds to good approximation even for classical perturbations that contain many quanta. This suggests that -for appropriate states -corrections to the first law of entanglement are suppressed by powers of N in CFTs whose correlators satisfy 't Hooft large-N power counting. We take first steps toward verifying that this is so by studying the large-N structure of the entropy of spatial regions for a class of CFT states motivated by those created from the vacuum by acting with real-time single-trace sources. We show that 1/N counting matches bulk predictions, though we require the effect of the source on the modular hamiltonian to be non-singular. The magnitude of our sources is N with fixed-but-small as N → ∞. Our results also provide a perturbative derivation -without relying on the replica trick -of the subleading Faulkner-Lewkowycz-Maldacena correction to the Ryu-Takayagi and Hubeny-Rangamani-Takayanagi conjectures at all orders in 1/N .

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Generalized campbell-baker-hausdorff formula, path-ordering and bernoulli numbers

Cited by 8 publications

References 6 publications

Path integral representation for Wilson loops and the non-Abelian Stokes theorem

Path integral representation for Wilson loops and the non-Abelian Stokes theorem

The non-Abelian exponentiation theorem for multiple Wilson lines

’t Hooft suppression and holographic entropy

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