A new approach to BRST operator cohomologies: Exact results for the BRST-fock theories

Horuzhy, S. S.; Voronin, A. V.

doi:10.1007/bf01083529

“…The operator N has the following permutation relations with Q and ∆: We can add the commutation relations (3.8) for the generators Z α with the usual quadratic Casimir operator Z α Z α for the semi-simple Lie group. Note that the Lie superalgebra for the quantities Q, ∆, N , T α T α and Z α Z α determined by the relations (3.8), (4.1)-(4.11) can be used for the calculation of the BRST operator cohomologies [22]. JHEP00(2002)000…”

Section: Lie Superalgebra For the Brst And Anti-brst Chargesmentioning

confidence: 99%

Exterior differentials in superspace and Poisson brackets

Soroka

¹

,

Soroka

²

2003

J. High Energy Phys.

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It is shown that two definitions for an exterior differential in superspace, giving the same exterior calculus, yet lead to different results when applied to the Poisson bracket. A prescription for the transition with the help of these exterior differentials from the given Poisson bracket of definite Grassmann parity to another bracket is introduced. It is also indicated that the resulting bracket leads to generalization of the Schouten-Nijenhuis bracket for the cases of superspace and brackets of diverse Grassmann parities. It is shown that in the case of the Grassmann-odd exterior differential the resulting bracket is the bracket given on exterior forms. The above-mentioned transition with the use of the odd exterior differential applied to the linear even/odd Poisson brackets, that correspond to semi-simple Lie groups, results, respectively, in also linear odd/even brackets which are naturally connected with the Lie superalgebra. The latter contains the BRST and anti-BRST charges and can be used for calculation of the BRST operator cohomology.

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“…It should be noted that the BRST charge for the Schwinger models has the same form as that for BRST-QEM when written in terms of creators and annihilators which is why we get the correct physical result. The BRST physical algebra is not discussed in this example, however the selection of the physical algebra is considered in [60] where it is shown that for theories with structures similar to QEM, the operator cohomology is what we expect. The proof of this fact relies on using an 'operator dsp-decomposition' and is entirely algebraic.…”

Section: Introductionmentioning

confidence: 99%

“…As BRST theory has existed for 30 years, work in this direction has of course been done. General BRST structures has been examined by Horuzhy et al in a series of papers [57,6,60,58,61,62,103], in particular in Horuzhy and Voronin [57] the BRST charge is analysed Q as a possibly unbounded Hermitian operator acting on a Krein space, and various results including the dsp-decomposition are obtained. Ghost number operators are studied in Azizov and Khoruzhii [6] and conditions for the existence of these operators are given.…”

Section: Introductionmentioning

confidence: 99%

“…The proof of this fact relies on using an 'operator dsp-decomposition' and is entirely algebraic. It is conjectured at the end of [60] that the spatial dsp-decomposition can be used to simplify the operator cohomology calculation. We do this here using a structure theorem similar to the one mentioned in H&T [55], but extended to cover the infinite dimensional case.…”

Section: Introductionmentioning

confidence: 99%

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The Mathematical Structure of the Quantum BRST Constraint Method

Costello

2009

Preprint

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This thesis describes mathematical structures of the quantum BRST constraint method. Ultimately, the quantum BRST structures are formulated in a C * -algebraic context, leading to comparison of the quantum BRST and the Dirac constraint method in a mathematically consistent framework.Rigorous models are constructed for the heuristic examples of BRST for quantum electromagnetism (BRST-QEM) and Hamiltonian BRST with a finite number of constraints. This facilitates comparison between the results produced by the BRST method, and the results of the T -procedure of Grundling and Hurst for the quantum Dirac constraint method.The different constraint methods are shown not to be equivalent for the examples of Hamiltonian BRST with a finite number of constraints that close, and a BRST-QEM model constructed using the Resolvent Algebra of Buchholz and Grundling with covariant test function space.Moreover, this leads to the following three consequences:The quantum BRST method, and quantum Dirac method of constraints, are not equivalent in general.Examples of quantum Hamiltonian BRST can be constructed to show that the BRST method does not remove the ghosts in the BRST physical algebra. This occurs since quantum Hamiltonian BRST selects multiple copies of the physical state space selected by the Dirac algorithm, and the ghosts are not removed from the BRST-physical state space. Extra selection criteria are required to select the correct physical space, which do not gaurantee correspondence between the Dirac and BRST physical algebras.Conversely, the BRST physical algebra and Dirac physical algebra coincide when QEM is encoded in the auxiliary Resolvent Algebra. This is a rigorous example of Lagrangian BRST, hence quantum Lagrangian and quantum Hamiltonian BRST are not equivalent constraint methods.iii Acknowledgments I would like to say my thanks and express sincere appreciation to my supervisor Dr. Hendrik Grundling for all his time, effort and patience. I would like to express my gratitude to Professor Klaus Fredenhagen for his insights into the correct statement of the BRST charge. Thanks also to Dr. Ben Warhurst for many stimulating discussions along the way. Finally, I am very grateful to the School of Mathematics and Statistics at the University of New South Wales and the University for their support.

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“…Note that the Lie superalgebra for the quantities Q, ∆, N , T α T α and Z α Z α determined by the relations (3.8), (4.1)-(4.11) can be used for the calculation of the BRST operator cohomologies [22].…”

Section: Lie Superalgebra For the Brst And Anti-brst Chargesmentioning

confidence: 99%

Exterior Differentials in Superspace and Poisson Brackets of Diverse Grassmann Parities

Soroka¹,

Soroka²

2002

Supersymmetry and Unification of Fundamental Interactions

0

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It is shown that two definitions for an exterior differential in superspace, giving the same exterior calculus, yet lead to different results when applied to the Poisson bracket. A prescription for the transition with the help of these exterior differentials from the given Poisson bracket of definite Grassmann parity to another bracket is introduced. It is also indicated that the resulting bracket leads to generalization of the Schouten-Nijenhuis bracket for the cases of superspace and brackets of diverse Grassmann parities. It is shown that in the case of the Grassmann-odd exterior differential the resulting bracket is the bracket given on exterior forms. The above-mentioned transition with the use of the odd exterior differential applied to the linear even/odd Poisson brackets, that correspond to semi-simple Lie groups, results, respectively, in also linear odd/even brackets which are naturally connected with the Lie superalgebra. The latter contains the BRST and anti-BRST charges and can be used for calculation of the BRST operator cohomology.

show abstract

A new approach to BRST operator cohomologies: Exact results for the BRST-fock theories

Cited by 4 publications

References 6 publications

Exterior differentials in superspace and Poisson brackets

Exterior differentials in superspace and Poisson brackets

The Mathematical Structure of the Quantum BRST Constraint Method

Exterior Differentials in Superspace and Poisson Brackets of Diverse Grassmann Parities

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