Reductions, deformations, and resolutions in the service of physics

Outline

These aim of these notes is to summarize the various mathematical structures and constructions employed in physics to build models of classical and quantum field theories. To avoid the (yet non-existent) rigorous construction of non-linear (interacting) field theories, non-linearities are treated as formal perturbations. Theories with gauge invariance (first class constraints) are treated within the BV-BRST formalism.

Reductions, deformations, resolutions

There are multiple operations on classical and quantum field theories that produce new ones. I roughly classify them into three kinds. These terms are not meant to be rigorously defined or taken literally. They mostly reflect how these operations are viewed in the physics literature.

• reductions

: These are easy/natural operations that are essentially uniquely : defined. The uniqueness may not always be mathematically precise, but : precision can be achieved by considering extra physical input. : Suggestive examples are differentiation and computing cohomology. * deformations : These are generally difficult operations inverse to a reduction, that : are themselves essentially uniquely defined. Uniqueness is considered : in the same sense as above. A suggestive examples is integration. * resolutions : These are operations inverse to a reduction that are not uniquely : defined and involve making significant choices in their application. : Suggestive examples are solving underdetermined equations and : choosing a resolution in homological algebra.

The relevant examples that will appear in these notes are the following. A solid arrow represents a reduction, while a dashed arrow represents the inverse deformation or resolution, in the senses described above.

The transition to the classical limit corresponds to taking the limit $\hbar\to0$, which is a parameter explicitly appearing in all physically relevant quantum systems. The inverse operation is quantization, which corresponds to the mathematical field technically called deformation quantization.

Explicit solutions or other kinds of information is readily available for linear field theories. Thus, often, the first step to understanding a non-linear field theory is to consider its linearization. The inverse operation of deforming the linear theory with a non-linear perturbation is often very difficult and thus considered in the physics literature mostly at the level of formal power series in the perturbation parameter.

Many physical theories are difficult to describe without introducing extra/non-physical/redundant so-called gauge degrees of freedom and at the same time so-called gauge symmetry acting on them. However, given such a description, the physical theory must be recovered from the quotient by gauge transformations. The description involving gauge degrees of freedom is highly non-unique, but a particularly convenient (though still not unique) one is provided by the BV-BRST resolution.

Summary of construction

The operations described above can be used as the axes of a three-dimensional space, inhabited by what can be called the resolution-deformation-reduction cube of quantization. Note that gauge fixing can be considered as one of the (non-uniquely defined) steps of the BV-BRST construction.

One corner of the cube corresponds to some classical, non-linear, physical theory. The goal of quantization is to jump to the corner with the quantum version of this theory. This is difficult to do directly because of the non-linearity and when gauge degrees of freedom are present. The usual procedure is then to construct the remaining auxiliary vertices of the cube and to take the indirect path, indicated in pale blue. The meaning of the solid and dashed lines is the same as above.

Scope and contents

• Classical and quantum mechanics

: * algebraic structure of classical mechanics : * algebraic structure of quantum mechanics : * classical limit and (deformation) quantization * Classical and quantum field theory : * geometry of space-time : * algebraic and geometric structure of classical field theory : * algebraic and geometric structure of quantum field theory * Construction of field theories (simplified) : * space-time, field content, dynamics : * (covariant) phase space : * algebra of observables : * (deformation) quantization * Examples of field theories : * scalar field (Klein-Gordon) : * vector field (Maxwell, Yang-Mills) : * gravity (general relatvity) : * spinor field (Dirac) * Fermi fields and supergeometry * Gauge invariance/degeneracy : * Nöther's second theorem : * algebra and geometry of gauge transformations : * physical theory from quotient by gauge * BV-BRST construction of gauge theories : * BV-BRST (Koszul-Tate) resolution of physical theory : * gauge fixing, well-posedness, Poisson structure : * cohomology * BV-BRST, (deformation) quantization and gauge anomalies