Loop Space Representation Research Articles

Large N optimization for multi-matrix systems

In this work we revisit the problem of solving multi-matrix systems through numerical large N methods. The framework is a collective, loop space representation which provides a constrained optimization problem, addressed through master-field minimization. This scheme applies both to multi-matrix integrals (c = 0 systems) and multi-matrix quantum mechanics (c = 1 systems). The complete fluctuation spectrum is also computable in the above scheme, and is of immediate physical relevance in the later case. The complexity (and the growth of degrees of freedom) at large N have stymied earlier attempts and in the present work we present significant improvements in this regard. The (constrained) minimization and spectrum calculations are easily achieved with close to 104 variables, giving solution to Migdal-Makeenko, and collective field equations. Considering the large number of dynamical (loop) variables and the extreme nonlinearity of the problem, high precision is obtained when confronted with solvable cases. Through numerical results presented, we prove that our scheme solves, by numerical loop space methods, the general two matrix model problem.

Generalized loop space and TMDs

The Standard Model describes the three (of four) basic interactions known in Nature in terms of the quantum fields which are constituted by representations of special unitary gauge groups of symmetry. However, the physical observables do not always coincide with the fundamental degrees of freedom of the Standard Model. Therefore it can be useful to switch to the loop space representation of the gauge theory, where the variables are inherently gauge invariant but the degrees of freedom are absorbed in the path/loop dependence. Over-completeness of this space requires the introduction of an equivalence relation which is provided by Wilson loop functionals operating on piecewise regular paths. It is well known that certain Wilson loops show the same singularity structure as some Transverse Momentum Dependent PDFs (TMDs), which are not renormalizable by the common methods due to exactly this singularity structure. By introducing geometrical operators, like the area-derivative, we were able to derive an evolution equation for these Wilson loops and we hope to apply this method in the future to find some renormalization schemes for TMDs.

String propagator: A loop space representation.

The string quantum kernel is normally written as a functional sum over the string coordinates and the world-sheet metrics. As an alternative to this quantum field-inspired approach, we study the closed bosonic string propagation amplitude in the functional space of loop configurations. This functional theory is based entirely on the Jacobi variational formulation of quantum mechanics, without the use of a lattice approximation. The corresponding Feynman path integral is weighed by a string action which is a reparametrization invariant version of the Schild action. We show that this path integral formulation is equivalent to a functional ``Schr\"odinger'' equation defined in loop space. Finally, for a free string, we show that the path integral and the functional wave equation are exactly solvable. \textcopyright{} 1996 The American Physical Society.

Loop space representation of quantum fermions and gravity

We study the quantum fermions + gravity system, that is, the gravitational counterpart of QED. We start from the standard Einstein-Weyl theory, reformulated in terms of Ashtekar variables; and we construct its non-perturbative quantum theory by extending the loop representation of general relativity. To this aim, we construct the fermion equivalent of the loop variables, and we define the quantum theory as a representation of their Poisson algebra. Not surprisingly, fermions can be accounted for in the loop representation simply by including open curves into loop space, as expected from lattice Yang-Mills theory. We explicitly construct the diffeomorphism and Hamiltonian operators. The first can be fully solved as in pure gravity. The second is constructed by using a background independent regularization technique. The theory retains the clean geometrical features of pure quantum gravity. In particular, the Hamiltonian constraint admits the same simple geometrical interpretation as its pure gravity counterpart: it is the operator that shifts curves along themselves (shift operator). Quite surprisingly, we believe, this simple action codes the full dynamics of the interacting fermion-gravity theory. To unravel the dynamics of the theory we study the evolution of the fermion-gravity system in the physical time defined by an additional coupled (clock-) scalar field. We explicitly construct the Hamiltonian operator that evolves the system in this physical time. We show that this Hamiltonian is finite, diffeomorphism invariant, and has a simple geometrical action confined to the intersections and the end points of the “loops”. The quantum theory of fermions + gravity evolving in the clock time is finally given by the combinatorial and geometrical action of this Hamiltonian on a set of graphs with a finite number of end points. This geometrical action defines the topological Feynman rules of the theory.

Graph invariants of Vassiliev type and application to 4D quantum gravity

We consider a special class of Kauffman's graph invariants of rigid vertex isotopy (graph invariants of Vassiliev type). They are given by a functor from a category of colored and oriented graphs embedded into a 3-space to a category of representations of the quasi-triangular ribbon Hopf algebra $U_q(sl(2,\bf C))$. Coefficients in expansions of them with respect to $x$ ($q=e^x$) are known as the Vassiliev invariants of finite type. In the present paper, we construct two types of tangle operators of vertices. One of them corresponds to a Casimir operator insertion at a transverse double point of Wilson loops. This paper proposes a non-perturbative generalization of Kauffman's recent result based on a perturbative analysis of the Chern-Simons quantum field theory. As a result, a quantum group analog of Penrose's spin network is established taking into account of the orientation. We also deal with the 4-dimensional canonical quantum gravity of Ashtekar. It is verified that the graph invariants of Vassiliev type are compatible with constraints of the quantum gravity in the loop space representation of Rovelli and Smolin.

Loop space hamiltonians and field theory of non-critical strings

We consider the loop space representation of multi-matrix models. Explaining the origin of a time variable through stochastic quantization we make contact with recent proposals of Ishibashi and Kawai. We demonstrate how collective field theory with its loop space interactions generates a field theory of non-critical strings.

General covariance, loops, and matter.

A four-dimensional generally covariant field theory is presented which describes nondynamical three-geometries coupled to scalar fields. The theory has an infinite number of physical observables (or constants of the motion) which are constructed from loops made from scalar field configurations. The Poisson algebra of these observables is closed, and is the same as that for the 3+1 gravity loop variables in the Ashtekar formalism. The theory also has observables that give the areas of open surfaces and the volumes of finite regions. Solutions to all the Hamilton-Jacobi equations for the theory, and the Dirac quantization conditions in the coordinate representation are given. These solutions are holonomies based on matter loops. A brief discussion of the loop space representation for the quantum theory is also given, together with some implications for the quantization of 3+1 gravity.

Ashtekar formulation of general relativity and loop-space nonperturbative quantum gravity: a report

The formulation of general relativity discovered by Ashtekar (1986, 1987) and the recent results obtained in non-perturbative quantum gravity using loop-space techniques are reviewed. The new formulation is based on the choice of a set of Lagrangian (and Hamiltonian) variables, instead of the spacetime metric. In terms of these new variables, the dynamical equations are remarkably simplified and a structural identity between general relativity and the Yang-Mills theories is revealed. The formalism has proven to be useful in numerous problems in gravitational physics. In quantum gravity, the new formalism has overcome long-standing difficulties and led to unexpected results. A nonperturbative approach to quantum theory has been constructed in terms of the Wilson loops of the Ashtekar connection. This approach, denoted as loop-space representation, has led to the complete solution of the quantum diffeomorphism constraint in terms of knot states, to the discovery of an infinite-dimensional class of solutions to the quantum gravitational dynamics, and to certain surprising indications on the existence of a discrete structure of spacetime around the Planck length. These results are presented in a compact self-contained form. The basic Ashtekar formalism is presented and its applications are outlined. The loop-space representation and the non-perturbative knot states of quantum gravity are described in detail, with particular regard to their physical interpretation and to the information they may provide on the microstructure of spacetime.

Intersecting N loop solutions of the hamiltonian constraint of quantum gravity

Following the introduction by Ashtekar of a new set of canonical variables for the gravitational field and a new representation of quantum gravity based on them, Jacobson and Smolin presented a large class of solutions to the quantum hamiltonian constraint of general relativity based on the holonomy of the Ashtekar connection along simple loops and two intersecting loops. For all these solutions the metric is degenerate everywhere, including the point of intersection. This motivated Husain to extend the results to consider the case of three intersecting loops. However, the metric was degenerate as well on the three-loop solutions found since the loops were only allowed to have two independent tangent vectors at the point of intersection. We have developed a computer algebra code capable of generating solutions for an arbitrary number of loops. We explicitly present new four- and five-loop solutions. These are the first solutions known to have three independent tangent vectors at the intersection point. In spite of this they fail, however, to have a nondegenerate metric. We present a general argument that shows that the same situation arises for an arbitrary finite number of loops. Implications of this degeneracy for the interpretation of the connection and loop space representation of quantum gravity are also discussed.

The loop space representation of 2+1 gravity: Physical observables, variational principles, and the issue of time

We consider the loop space representation of Rovelli and Smolin as the starting point for the analysis of quantum gravity in 2+1 dimensions. To do so, we introduce observables directly within this representation and consider the problem of its Lagrangian formulation. The implications of this framework for the issue of time are discussed and the extension of these results to 3+1 gravity is also considered.

Loop space representation of quantum general relativity and the group of loops

The action of the constraints of quantum general relativity on a general state in the loop representation is coded in terms of loop derivatives. These differential operators are related to the infinitessimal generators of the group of loops and generalize the area derivative first considered by Mandelstam. A new sector of solutions of the physical states space of nonperturbative quantum general relativity is found.

Hermitian operators on quantum general relativity loop space

The author discusses a more precise mathematical formulation of the Rovelli and Smolin loop space representation of quantum general relativity. Their representation space based on wavefunctions defined on sets of loops can be made more definitive by using either subsets or Cartesian products of the loop space of the 3-manifold Sigma . This enables him to consider the problem of defining an inner product on the wavefunctions. He constructs a 'model' inner product on a subspace of the representation space using a discrete sum instead of an integral over elements of the domain space of the wavefunctions. With respect to this inner product, Hermitian conjugates to the T operators can be constructed. The Hermitian operators he obtains are significant from the point of view of their action on the sets rather than the individual loops.

Loop space representation of quantum general relativity

We define a new representation for quantum general relativity, in which exact solutions of the quantum constraints may be obtained. The representation is constructed by means of a noncanonical graded Poisson algebra of classical observables, defined in terms of Ashtekar's new variables. The observables in this algebra are nonlocal and involve parallel transport around loops in a three-manifold Σ. The theory is quantized by constructing a linear representation of a deformation of this algebra. This representation is given in terms of an algebra of linear operators defined on a state space which consists of functionals of sets of loops in Σ. The construction is general and can be applied also to Yang-Mills theories. The diffeomorphism constraint is defined in terms of a natural representation of the diffeomorphism group. The hamiltonian constraint, which contains the dynamics of quantum gravity, is constructed as a limit of a sequence of observables which incorporates a regularization prescription. We give the general solution of the diffeomorphism constraint in closed form. It is spanned by a countable basis which is in one-to-one correspondence with the diffeomorphism equivalence classes of multiple loops, which are a generalization of the link classes studied in knot theory. Then we explicitly construct, in closed form, a large space of solution of the entire set of constraints, including the hamiltonian constraint. These turn out to be classified by the ordinary knot and link classes of Σ. The space of solutions that we find is a sector of the physical states space of nonperturbative quantum general relativity. The failure of perturbation theory is thus shown to be not relevant to the problem of the existence of a nontrivial physical state space in quantum gravity. The relationship between this new loop representation and the self-dual representation of Ashtekar is illuminated by means of a functional transform between states in the two representations. Questions of the completeness of the solution space, the meaning of the physical operators and the physical inner product, are discussed, but not, so far, resolved.

Loop-space representation and the large-Nbehavior of the one-plaquette Kogut-Susskind Hamiltonian

Loop-space representation and the large-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>N</mml:mi></mml:math>behavior of the one-plaquette Kogut-Susskind Hamiltonian