[Paper Review] Delocalized membrane model

М. Г. Иванов|ArXiv.org|May 8, 2001

Black Holes and Theoretical Physics13 references3 citations

TL;DR

This paper introduces a covariant, constraint-free formulation of membrane theory using a novel $(\mathbf{M} \to \mathbf{F})$-approach, where membranes are described by scalar fields $\varphi^\alpha: \mathbf{M} \to \mathbf{F}$ mapping spacetime to a target manifold $\mathbf{F}$ of dimension $D-n$. The key contribution is a regularized, delocalized membrane action that generalizes the $p$-brane solution in supergravity, with a Hamiltonian formulation derived directly from the unconstrained action via a matrix $\mathcal{G}^{\alpha\beta}$ of field gradients.

ABSTRACT

A model considered in the paper generalizes membrane theory to the case of delocalized membranes. The model admits covariant formulation, which involves no constraints. It generalizes the notion of membrane to the case of smooth distribution of non-intersecting membranes. A generalization of p-brane solution with delocalized membranes is presented.

Motivation & Objective

• To resolve singularities in standard membrane theories by replacing point-like membranes with a continuous, delocalized distribution of infinitely light membranes.
• To eliminate constraints inherent in the conventional $(\mathbf{V} \to \mathbf{M})$ membrane formulation by redefining the theory in terms of bulk fields on spacetime $\mathbf{M}$.
• To provide a covariant, geometrically natural formulation of membrane dynamics using a target manifold $\mathbf{F}$ of dimension $D-n$, enabling a direct Hamiltonian formulation.
• To generalize the $p$-brane solution in supergravity to include delocalized sources via a new action involving antisymmetric tensor fields and a projective energy-momentum tensor.

Proposed method

• Proposes a new action $S_{\mathbf{M} \to \mathbf{F}} = -\int_{\mathbf{M}} d^D X \sqrt{|g|} \sqrt{\det(g^{MN} \varphi^\alpha_{,M} \varphi^\beta_{,N})}$, which describes delocalized membranes via scalar fields $\varphi^\alpha: \mathbf{M} \to \mathbf{F}$.
• Defines the matrix $\mathcal{G}^{\alpha\beta} = g^{MN} \varphi^\alpha_{,M} \varphi^\beta_{,N}$ and its inverse $\mathcal{G}_{\alpha\beta}$, which encode the induced geometry of the membrane distribution.
• Derives equations of motion from the action via variation with respect to $\varphi^\alpha$, yielding $\partial_M \left( \sqrt{\frac{|g|}{\mathcal{G}}} \mathcal{G}_{\alpha\beta} g^{MN} \varphi^\beta_{,N} \right) = 0$.
• Constructs the energy-momentum tensor as $T_{MN} = -\frac{1}{\sqrt{\mathcal{G}}} \left( g_{MN} - \mathcal{G}_{\alpha\beta} \varphi^\alpha_{,M} \varphi^\beta_{,N} \right)$, satisfying the projective property $T_{MN} = -\rho \mathcal{P}_{MN}$.
• Derives the Hamiltonian density $\mathcal{H} = \sqrt{ \frac{|g|}{\mathcal{G}^0} - \frac{h^{\alpha\beta}}{g^{00}} p_\alpha p_\beta } - \frac{g^{0m}}{g^{00}} \varphi^\alpha_{,m} p_\alpha$ using canonical momenta $p_\alpha = -\sqrt{\frac{|g|}{\mathcal{G}}} \mathcal{G}_{\alpha\beta} \partial^0 \varphi^\beta$.
• Constructs a generalized $p$-brane solution by coupling the delocalized membrane to a $(n+1)$-form field, with metric $ds^2 = H^{-2/q} \eta_{ij} dX^i dX^j + H^{2/(D-q-2)} \delta_{\alpha\beta} dX^\alpha dX^\beta$, where $H$ is a harmonic function.

Experimental results

Research questions

• RQ1 How can singular, point-like membranes in standard membrane theory be regularized to avoid constraints and singular sources?
• RQ2 Can a covariant, constraint-free formulation of membrane dynamics be constructed using a dual field mapping $\varphi: \mathbf{M} \to \mathbf{F}$ instead of $x: \mathbf{V} \to \mathbf{M}$?
• RQ3 What is the geometric and dynamical role of the energy-momentum tensor in the delocalized membrane model, and how does it relate to the projective property?
• RQ4 How does the Hamiltonian formulation emerge directly from the action without solving constraints, and what is its structure in terms of the $\mathcal{G}^{\alpha\beta}$ matrix?
• RQ5 Can the standard $p$-brane solution in supergravity be generalized to include delocalized, continuous distributions of membranes via this new formalism?

Key findings

• The proposed $(\mathbf{M} \to \mathbf{F})$-approach yields a fully covariant, constraint-free action for delocalized membranes, reducing the number of fields from $D$ to $D-n$ and eliminating $n$ constraints present in the $(\mathbf{V} \to \mathbf{M})$ formulation.
• The energy-momentum tensor satisfies the projective property $T_{MN} = -\rho \mathcal{P}_{MN}$, where $\mathcal{P}_{MN}$ is an orthogonal projector of rank $n$, ensuring the correct membrane-like dynamics.
• The Hamiltonian density is derived directly from the unconstrained action, with $\mathcal{H} = \sqrt{ \det(h^{\alpha\beta}) + h^{\alpha\beta} p_\alpha p_\beta }$ in Minkowski space, showing a non-trivial dependence on momenta.
• A generalized $p$-brane solution is constructed for a $D$-dimensional spacetime coupled to a $(n+1)$-form field, with metric components depending on a harmonic function $H$, and source current $J = -h^2 \triangle H \, dX^q \wedge \dots \wedge dX^{D-1}$.
• The solution reduces to the Madjumdar-Papapetru solution in $D=4$, $q=1$ when $H$ is harmonic and $\triangle H = 0$ away from sources, showing consistency with known extremal black hole solutions.
• The action is invariant under diffeomorphisms on both $\mathbf{M}$ and $\mathbf{F}$, ensuring full covariance and supporting the interpretation of $\mathbf{F}$ as a moduli space of membranes.