Classical phase space and Hadamard states in the BRST formalism for gauge field theories on curved spacetime

We investigate linearized gauge theories on globally hyperbolic spacetimes in the BRST formalism. A consistent definition of the classical phase space and of its Cauchy surface analogue is proposed. We prove that it is isomorphic to the phase space in the ‘subsidiary condition’ approach of Hack and Schenkel in the case of Maxwell, Yang–Mills, and Rarita–Schwinger fields. Defining Hadamard states in the BRST formalism in a standard way, their existence in the Maxwell and Yang–Mills case is concluded from known results in the subsidiary condition (or Gupta–Bleuler) formalism. Within our framework, we also formulate criteria for non-degeneracy of the phase space in terms of BRST cohomology and discuss special cases. These include an example in the Yang–Mills case, where degeneracy is not related to a non-trivial topology of the Cauchy surface.

Necessity

This chapter examines the doctrine of necessity as an element of the prohibition of the use of force and as a subsidiary condition of the legality of self-defence. It begins by discussing the thesis of necessity as a general justification of the use of force within the context of the international law of responsibility. It then analyses necessity as a condition enshrined in self-defence and in the United Nations collective security system. The chapter also considers the methodological problems that arise from any interpretation of the concept of necessity, especially with respect to the use of force. It highlights the fact that the International Law Commission, the International Court of Justice, and state practice have never recognized necessity as a general justification to use force.

LOCAL CHARGED STATES OF THE GAUGE FIELD IN THREE-DIMENSIONAL MAXWELL-TYPE THEORIES

Gauge-invariant local creation operators of charged states are introduced and studied in pure gauge theories of the Maxwell-type in (2+1) dimensions. These states are usually unphysical because of the subsidiary condition imposed on the physical subspace by Gauss’ law. A dual Maxwell theory which possesses a topological electric charge is introduced. Pure electrodynamics lies in the sector where the topological charge identically vanishes. Charge bearing operators completely expressed in terms of the gauge field, however, can create physical states in the nontrivial topological sectors which thereby generalize QED. An order–disorder structure exists relating the charged operators and the magnetic flux creating (vortex) operators, both through commutation rules and correlation functions. The relevance of this structure for bosonization in 2+1 dimensions is discussed.

CLASSICAL SOLUTIONS OF THE p-BRANES

An appropriate subsidiary condition is introduced in the classical actions of the p-branes (p arbitrary). A general class of exact solutions of the resulting nonlinear equations of motion are obtained which yield a broad class of characteristics for the original covariant equations of the p-branes.