Batalin-Vilkovisky formalism

13A1 D. Anselmi
Renormalization of gauge theories without cohomology

January 31, 2013

We investigate the renormalization of gauge theories without assuming cohomological properties. We define a renormalization algorithm that preserves the Batalin-Vilkovisky master equation at each step and automatically extends the classical action till it contains sufficiently many independent parameters to reabsorb all divergences into parameter-redefinitions and canonical transformations. The construction is then generalized to the master functional and the field-covariant proper formalism for gauge theories. Our results hold in all manifestly anomaly-free gauge theories, power-counting renormalizable or not. The extension algorithm allows us to solve a quadratic problem, such as finding a sufficiently general solution of the master equation, even when it is not possible to reduce it to a linear (cohomological) problem.

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Eur. Phys. J. C 73 (2013) 2508 | DOI: 10.1140/epjc/s10052-013-2508-5

arXiv:1301.7577 [hep-th]

Posted in Papers, Renormalization of general gauge theories | Tags: Gauge theories, Batalin-Vilkovisky formalism, Generating functionals, Quantum field theory, Renormalization, Field-covariance, Renormalization-group invariance

12A3 D. Anselmi
Master functional and proper formalism for quantum gauge field theory

May 18, 2012

We develop a general field-covariant approach to quantum gauge theories. Extending the usual set of integrated fields and external sources to “proper” fields and sources, which include partners of the composite fields, we define the master functional $\Omega$, which collects one-particle irreducible diagrams and upgrades the usual $\Gamma$-functional in several respects. The functional $\Omega$ is determined from its classical limit applying the usual diagrammatic rules to the proper fields. Moreover, it behaves as a scalar under the most general perturbative field redefinitions, which can be expressed as linear transformations of the proper fields. We extend the Batalin-Vilkovisky formalism and the master equation. The master functional satisfies the extended master equation and behaves as a scalar under canonical transformations. The most general perturbative field redefinitions and changes of gauge-fixing can be encoded in proper canonical transformations, which are linear and do not mix integrated fields and external sources. Therefore, they can be applied as true changes of variables in the functional integral, instead of mere replacements of integrands. This property overcomes a major difficulty of the functional $\Gamma$. Finally, the new approach allows us to prove the renormalizability of gauge theories in a general field-covariant setting. We generalize known cohomological theorems to the master functional and show that when there are no gauge anomalies all divergences can be subtracted by means of parameter redefinitions and proper canonical transformations.

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Eur. Phys. J. C 73 (2013) 2363 | DOI: 10.1140/epjc/s10052-013-2363-4

arXiv:1205.3862 [hep-th]

Posted in Papers, Field-covariant quantum field theory | Tags: Gauge theories, Batalin-Vilkovisky formalism, Dimensional regularization, Functional integral, Generating functionals, Quantum field theory, Renormalization, Field-covariance

94A1 D. Anselmi
More on the subtraction algorithm

July 5, 1994

We go on in the program of investigating the removal of divergences of a generical quantum gauge field theory, in the context of the Batalin-Vilkovisky formalism. We extend to open gauge-algebrae a recently formulated algorithm, based on redefinitions $\delta\lambda$ of the parameters $\lambda$ of the classical Lagrangian and canonical transformations, by generalizing a well-known conjecture on the form of the divergent terms. We also show that it is possible to reach a complete control on the effects of the subtraction algorithm on the space $M_{gf}$ of the gauge-fixing parameters. A principal fiber bundle $E \rightarrow M_{gf}$ with a connection $\omega_1$ is defined, such that the canonical transformations are gauge transformations for $\omega_1$. This provides an intuitive geometrical description of the fact the on shell physical amplitudes cannot depend on $M_{gf}$. A geometrical description of the effect of the subtraction algorithm on the space $M_{ph}$ of the physical parameters lambda is also proposed. At the end, the full subtraction algorithm can be described as a series of diffeomorphisms on $M_{ph}$, orthogonal to $M_{gf}$ (under which the action transforms as a scalar), and gauge transformations on $E$. In this geometrical context, a suitable concept of predictivity is formulated. We give some examples of (unphysical) toy models that satisfy this requirement, though being neither power counting renormalizable, nor finite.

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Class.Quant.Grav. 12 (1995) 319-350 | DOI: 10.1088/0264-9381/12/2/005

arXiv:hep-th/9407023

Posted in Papers, Renormalization of general gauge theories | Tags: Gauge theories, Batalin-Vilkovisky formalism, Renormalization, Nonrenormalizable theories

93A2 D. Anselmi
Removal of divergences with the Batalin-Vilkovisky formalism

September 21, 1993

We consider the problem of removing the divergences in an arbitrary gauge-field theory (possibly nonrenormalizable). We show that this can be achieved by performing, order by order in the loop expansion, a redefinition of some parameters (possibly infinitely many) and a canonical transformation (in the sense of Batalin and Vilkovisky) of fields and BRS sources. Gauge-invariance is turned into a suitable quantum generalization of BRS-invariance. We define quantum observables and study their properties. We apply the result to renormalizable gauge-field theories that are gauge-fixed with a nonrenormalizable gauge-fixing and prove that their predictivity is retained. A corollary is that topological field theories are predictive. Analogies and differences with the formalisms of classical and quantum mechanics are pointed out.

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Class.Quant.Grav. 11 (1994) 2181-2204 | DOI: 10.1088/0264-9381/11/9/005

arXiv:hep-th/9309085

Posted in Papers, Renormalization of general gauge theories, Topological field theory | Tags: Gauge theories, Batalin-Vilkovisky formalism, Renormalization, Nonrenormalizable theories, Topological field theory

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