Quantized self-dual Maxwell field on a null surface

The canonical formalism for a self-dual Maxwell field on a null plane is reviewed. After solution of the second class constraints, the transition to the quantum theory is carried out using a representation in which the self-dual Maxwell field is diagonal. The Gauss law constraint allows us to consider the physical state vectors to be holomorphic functionals of one complex function. Application of reality conditions allows us to define an inner product such that the Hermitean adjoint operators are identified with the classical complex conjugate operators. In going over to the Fourier expansion of the operators, we find that the inner product is formally convergent for positive frequency functionals and formally divergent for the negative frequency functionals. Following similar results of Ashtekar, Rovelli, and Smolin, negative frequency states are functional distributions identified with the helicity opposite to that of the positive frequency states.     

Debye Potentials for Self-Dual Fields

It is shown that in a space-time that admits ageodetic and shear-free null vector field which is aprincipal direction of the conformal curvature(therefore, in any algebraically special solution of the Einstein vacuum field equations), any self-dualelectromagnetic field is locally given by a scalar(Debye) potential which obeys a second-orderdifferential equation and, similarly, that any self-dualYang-Mills field is locally given by a matrix-valuedpotential governed by a nonlinear second-orderdifferential equation. Using the fact that any self-dualelectromagnetic field is the self-dual part of a realsolution of the source-free Maxwell equations, it isshown that in any space-time of this class, the solutionof the source-free Maxwell equations is locally given bya Debye potential.     

Dirac brackets for general relativity on a null cone

The Hamiltonian for the Einstein equations is constructed on a outgoing null cone with the help of the usual null tetrad. The resulting null surface constraints are shown to be second class in the terminology of Dirac. These second class constraints are eliminated by use of the starring procedure of Bergmann and Komar.     

Gauge invariance versus masslessness in de Sitter spaces

The connection between gauge invariance, masslessness and null cone propagation is a flat space property which does not persist even in constant curvature geometries. In particular, we show that both the gauge invariant spin $\text{3}\text{2}$ and 2 fields in anti-de Sitter space have support inside the cone, whereas where are conformally invariant, but gauge variant, models which do propagate on the light cone. The Maxwell field in constant curvature spaces of dimension other than four also does not have null cone propagation; again there is a conformally invariant model which does.     

Maxwell fields onH-space

It is shown that theH space associated with a solutionM of the Einstein-Maxwell equations can be endowed with a self-dual Maxwell field which arises from the radiation component ofM's Maxwell field.     

Self-Dual Conformal Supergravity and the Hamiltonian Formulation

In terms of Dirac matrices the self-dual and anti-self-dual decomposition of conformal supergravity is given and a self-dual conformal supergravity theory is developed as a connection dynamic theory in which the basic dynamic variables include the self-dual spin connection i.e. the Ashtekar connection rather than the triad. The Hamiltonian formulation and the constraints are obtained by using the Dirac-Bergmann algorithm.     

The Uncertainty of Fluxes

In the ordinary quantum Maxwell theory of a free electromagnetic field, formulated on a curved 3-manifold, we observe that magnetic and electric fluxes cannot be simultaneously measured. This uncertainty principle reflects torsion: fluxes modulo torsion can be simultaneously measured. We also develop the Hamilton theory of self-dual fields, noting that they are quantized by Pontrjagin self-dual cohomology theories and that the quantum Hilbert space is -graded, so typically contains both bosonic and fermionic states. Significantly, these ideas apply to the Ramond-Ramond field in string theory, showing that its K-theory class cannot be measured.     

Golden Oldie

A form of initial value problem is considered in which the initial hypersurface is not spacelike but null. This approach has the striking advantage over the more usual Cauchy problem that all constraints (initial data equations) are eliminated from the theory, for a wide class of interacting fields in special relativity and also for general relativity. The theory is most naturally described in terms of the two-component spinor calculus, for which an elementary introduction is given here. A general scheme for interacting fields, which holds both in special and general relativity, is presented which describes all fields in terms of sets of irreducible spinors. The concept of an exact set of such spinors is introduced and it is shown that this concept is the appropriate one for an initial value problem on a null cone without constraints. The initial data can be expressed in the form of a complex number, called a null datum, defined at each point of the null cone, one corresponding to each spinor. There is the curious feature of these null data that apparently it is sufficient here, to have onehalf as much information per point as in the corresponding Cauchy problem. The classical Maxwell-Dirac theory and the Einstein-Maxwell theory are two examples that can be put into the form of exact sets. The Einstein empty-space equations are also of particular note, and in this case the null datum describes essentially the intrinsic geometry of the null cone. The argument given here as applied to a general exact set is incomplete in two important respects. Firstly it depends on the null data being analytic, and secondly the initial hypersurface must be a cone. However, both these restrictions are removed in the case of certain elementary fields called basic free fields, examples of which are the Weyl neutrino field, the free Maxwell field, and the linearized gravitational field. For these cases a simple explicit formula is introduced which expresses the field at any point in terms of the null datum, as an integral taken over the intersection of the initial null hypersurface with the null cone of the point.This article originally appeared in 1963 in Aerospace Research Laboratories 63-56 (P.G. Bergmann). It is an important and oft-cited work, but as it has never been published in a widely distributed journal, it is generally inaccessable to the relativity community. This regrettable situation is hereby rectified-Ed.This work was done while the author was at Princeton, Syracuse, and Cornell Universities, visiting under a NATO Fellowship administered by the Department of Scientific and Industrial Research in London. The work at Syracuse was supported by the Aeronautical Research Laboratory and at Cornell by the National Science Foundation.     

Formal Similarity Between Mathematical Structures of Electrodynamics and Quantum Mechanics

Electromagnetic phenomena can be described by Maxwell equations written for the vectors of electric and magnetic field. Equivalently, electrodynamics can be reformulated in terms of an electromagnetic vector potential. We demonstrate that the Schrödinger equation admits an analogous treatment. We present a Lagrangian theory of a real scalar field φ whose equation of motion turns out to be equivalent to the Schrödinger equation with time independent potential. After introduction the field into the formalism, its mathematical structure becomes analogous to those of electrodynamics. The field φ is in the same relation to the real and imaginary part of a wave function as the vector potential is in respect to electric and magnetic fields. Preservation of quantum-mechanics probability is just an energy conservation law of the field φ.     

Hamiltonian Anomalies from Extended Field Theories

We develop a proposal by Freed to see anomalous field theories as relative field theories, namely field theories taking value in a field theory in one dimension higher, the anomaly field theory. We show that when the anomaly field theory is extended down to codimension 2, familiar facts about Hamiltonian anomalies can be naturally recovered, such as the fact that the anomalous symmetry group admits only a projective representation on the Hilbert space, or that the latter is really an abelian bundle gerbe over the moduli space. We include in the discussion the case of non-invertible anomaly field theories, which is relevant to six-dimensional (2, 0) superconformal theories. In this case, we show that the Hamiltonian anomaly is characterized by a degree 2 non-abelian group cohomology class, associated to the non-abelian gerbe playing the role of the state space of the anomalous theory. We construct Dai-Freed theories, governing the anomalies of chiral fermionic theories, and Wess-Zumino theories, governing the anomalies of Wess-Zumino terms and self-dual field theories, as extended field theories down to codimension 2.     

Dirac Quantization of Noncommutative Abelian Proca field

Dirac formalism of Hamiltonian constraint systems is studied for the noncommutative Abelian Proca field. It is shown that the system of constraints are of second class in agreement with the fact that the Proca field is not gauge invariant. Then, the system of second class constraints is quantized by introducing Dirac brackets in the reduced phase space.     

Dual equivalence between self-dual and Maxwell–Chern–Simons models coupled to dynamical U(1) charged matter

We study the equivalence between the self-dual and the Maxwell–Chern–Simons (MCS) models coupled to dynamical, U(1) charged matter, both fermionic and bosonic. This is done through an iterative procedure of gauge embedding that produces the dual mapping of the self-dual vector field theory into a Maxwell–Chern–Simons version. In both cases, to establish this equivalence a current–current interaction term is needed to render the matter sector unchanged. Moreover, the minimal coupling of the original self-dual model is replaced by a non-minimal magnetic like coupling in the MCS side. Unlike the fermionic instance however, in the bosonic example the dual mapping proposed here leads to a Maxwell–Chern–Simons theory immersed in a field dependent medium.     

Boundary conditions as Dirac constraints

In this article we show that boundary conditions can be treated as Lagrangian and Hamiltonian constraints. Using the Dirac method, we find that boundary conditions are equivalent to an infinite chain of second class constraints, which is a new feature in the context of constrained systems. Constructing the Dirac brackets and the reduced phase space structure for different boundary conditions, we show why mode expanding and then quantizing a field theory with boundary conditions is the proper way. We also show that in a quantized field theory subjected to the mixed boundary conditions, the field components are non-commutative. Received: 16 October 2000 / Revised version: 8 January 2001 / Published online: 23 February 2001     

Review: Dirac's Observables for the Rest-Frame Instant Form of Tetrad Gravity in a Completely Fixed 3-Orthogonal Gauge

We define the rest-frame instant form of tetrad gravity restricted to Christodoulou-Klainermann spacetimes. After a study of the Hamiltonian group of gauge transformations generated by the 14 first class constraints of the theory, we define and solve the multitemporal equations associated with the rotation and space diffeomorphism constraints, finding how the cotriads and their momenta depend on the corresponding gauge variables. This allows to find a quasi-Shanmugadhasan canonical transformation to the class of 3-orthogonal gauges and to find the Dirac observables for superspace in these gauges. The construction of the explicit form of the transformation and of the solution of the rotation and supermomentum constraints is reduced to solve a system of elliptic linear and quasi-linear partial differential equations. We then show that the superhamiltonian constraint becomes the Lichnerowicz equation for the conformal factor of the 3-metric and that the last gauge variable is the momentum conjugated to the conformal factor. The gauge transformations generated by the superhamiltonian constraint perform the transitions among the allowed foliations of spacetime, so that the theory is independent from its 3+1 splittings. In the special 3-orthogonal gauge defined by the vanishing of the conformal factor momentum we determine the final Dirac observables for the gravitational field even if we are not able to solve the Lichnerowicz equation. The final Hamiltonian is the weak ADM energy restricted to this completely fixed gauge.     

On singular Lagrangian underlying the Schrödinger equation

We analyze the properties that manifest Hamiltonian nature of the Schrödinger equation and show that it can be considered as originating from singular Lagrangian action (with two second class constraints presented in the Hamiltonian formulation). It is used to show that any solution of the Schrödinger equation with time independent potential can be presented in the form , where the real field ?(t,xi) is some solution of nonsingular Lagrangian theory being specified below. Preservation of probability turns out to be the energy conservation law for the field ?. After introduction the field into the formalism, its mathematical structure becomes analogous to those of electrodynamics.     

Einstein-Maxwell Spacetime with Two Commuting Spacelike Killing Vector Fields and Newman-Penrose Formalism

Einstein-Maxwell spacetimes endowed with twocommuting spacelike Killing vector fields areconsidered. Subject to the hypotheses that one of thetwo null geodesic congruence orthogonal to thetwo-surface generated by the two commuting spacelikeKilling vector fields is shearfree and theelectromagnetic field is non null, it is shown that,with a specific choice of null tetrad, theNewman-Penrose equations together with the Maxwell equations for theclass of spacetime considered may be reduced to asecond-order ode of Sturm-Liouville type, from whichexact solutions of the class of spacetimes consideredmay be constructed. Examples of exact solutions arethen given. Exact solutions with distribution-valuedWeyl curvature describing the scattering ofelectromagnetic shock wave with gravitational impulsiveor shock wave of variable polarisation are also constructed.     

The Hamiltonian of general relativity on a null surface

The Hamiltonian for the Einstein equations is constructed on an outgoing null cone with the help of the usual null tetrad used in the study of the asymptotical gravitational radiation field.     

Linking covariant and canonical general relativity via local observers

Hamiltonian gravity, relying on arbitrary choices of ‘space,’ can obscure spacetime symmetries. We present an alternative, manifestly spacetime covariant formulation that nonetheless distinguishes between ‘spatial’ and ‘temporal’ variables. The key is viewing dynamical fields from the perspective of a field of observers—a unit timelike vector field that also transforms under local Lorentz transformations. On one hand, all fields are spacetime fields, covariant under spacetime symmeties. On the other, when the observer field is normal to a spatial foliation, the fields automatically fall into Hamiltonian form, recovering the Ashtekar formulation. We argue this provides a bridge between Ashtekar variables and covariant phase space methods. We also outline a framework where the ‘space of observers’ is fundamental, and spacetime geometry itself may be observer-dependent.