Abstract
We present a frame- and reparametrisation-invariant formalism for quantum field theories that include fermionic degrees of freedom. We achieve this using methods of field-space covariance and the Vilkovisky–DeWitt (VDW) effective action. We explicitly construct a field-space supermanifold on which the quantum fields act as coordinates. We show how to define field-space tensors on this supermanifold from the classical action that are covariant under field reparametrisations. We then employ these tensors to equip the field-space supermanifold with a metric, thus solving a long-standing problem concerning the proper definition of a metric for fermionic theories. With the metric thus defined, we use well-established field-space techniques to extend the VDW effective action and express any fermionic theory in a frame- and field-reparametrisation-invariant manner.
Highlights
The same theory of physics can often be written in many different ways
All we need is to work exclusively with field-space tensors and ensure that all field-space indices are fully contracted. This formalism led to the reparametrisation invariant Vilkovisky– DeWitt (VDW) effective action [1,2,13]
Connections and invariant volume element determined for the field space in the previous section, we are in a position to make use of the VDW formalism [1,2,13,14,15] and define the quantum effective action for theories with fermionic degrees of freedom in a reparametrisation-invariant manner
Summary
The same theory of physics can often be written in many different ways. By choosing a different parametrisation of the underlying degrees of freedom, one can make the theory appear very different. By constructing the field space for theories with fermionic degrees of freedom, we will complete the formalism and will be able to describe all quantum field theories in a way that is manifestly reparametrisation invariant. This requires the introduction of new mathematics to describe them even at the classical level – namely the introduction of Grassmannian fields [41] To include such anticommuting degrees of freedom in this formalism, we must generalise the field space to a supermanifold [42,43,44,45,46,47,48,49]. This is in contrast to the equations of motion for bosonic fields, which are of second order and, in the absence of potential terms, constitute the geodesic equation of the field space This difference arises from the fact that only single derivatives of fermions appear in the Lagrangian.
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