Justification by Infinite Loops

In an earlier paper we have shown that a proposition can have a well-defined probability value, even if its justification consists of an infinite linear chain. In the present paper we demonstrate that the same holds if the justification takes the form of a closed loop. Moreover, in the limit that the size of the loop tends to infinity, the probability value of the justified proposition is always well-defined, whereas this is not always so for the infinite linear chain. This suggests that infinitism sits more comfortably with a coherentist view of justification than with an approach in which justification is portrayed as a linear process

SLEs as boundaries of clusters of Brownian loops

In this research announcement, we show that SLE curves can in fact be viewed as boundaries of certain simple Poissonian percolation clusters: Recall that the Brownian loop-soup (introduced in the paper arxiv:math.PR/0304419 with Greg Lawler) with intensity c defines a Poissonian collection of (simple if one focuses only on the outer boundary) loops in a domain. This random family of (possibly intersecting) loops is conformally invariant (and there are almost surely infinitely many small loops in any sample). We show that there exists a critical value a in (0,1] such that if one colors all the interiors of the loops, the obtained clusters are bounded when ca, one single cluster fills the domain. We prove that for small c, the outer boundaries of the clusters are SLE-type curves where $\kappa \le 4$ and $c$ related by the usual relation $c=(3\kappa-8)(6-\kappa)/2\kappa$ (i.e. c corresponds to the central charge of the model). Conjecturally, the critical value a is equal to one and corresponds to SLE4 loops, so that this should give for any c in (0,1] a construction of a natural countable family of random disjoint SLE$_\kappa$ loops (i.e. $\kappa$ should span $(8/3,4]$), that behaves ``nicely'' under perturbation of the domain. A precise relation between chordal SLE and the loop-soup goes as follows: Consider the sample of a certain restriction measure (i.e. a certain union of Brownian excursions) in a domain, attach to it all the above-described clusters that it intersects. The outer boundary of the obtained set is exactly an SLE$_\kappa$, if the restriction measure exponent is equal to the highest-weight of the corresponding representation with central charge c.Comment: Research anouncement, to appear in C. R. Acad. Sci. Pari

Ferromagnetism, antiferromagnetism, and the curious nematic phase of S=1 quantum spin systems

We investigate the phase diagram of S=1 quantum spin systems with SU(2)-invariant interactions, at low temperatures and in three spatial dimensions. Symmetry breaking and the nature of pure states can be studied using random loop representations. The latter confirm the occurrence of ferro- and antiferromagnetic transitions and the breaking of SU(3) invariance. And they reveal the peculiar nature of the nematic pure states which MINIMIZE \sum_x (S_x^i)^2.Comment: 12 pages, 6 figure

Multi-Loop Zeta Function Regularization and Spectral Cutoff in Curved Spacetime

We emphasize the close relationship between zeta function methods and arbitrary spectral cutoff regularizations in curved spacetime. This yields, on the one hand, a physically sound and mathematically rigorous justification of the standard zeta function regularization at one loop and, on the other hand, a natural generalization of this method to higher loops. In particular, to any Feynman diagram is associated a generalized meromorphic zeta function. For the one-loop vacuum diagram, it is directly related to the usual spectral zeta function. To any loop order, the renormalized amplitudes can be read off from the pole structure of the generalized zeta functions. We focus on scalar field theories and illustrate the general formalism by explicit calculations at one-loop and two-loop orders, including a two-loop evaluation of the conformal anomaly.Comment: 85 pages, including 17 pages of technical appendices; 4 figures; v2: typos and refs correcte

Phase transition for loop representations of Quantum spin systems on trees

We consider a model of random loops on Galton-Watson trees with an offspring distribution with high expectation. We give the configurations a weighting of $\theta^{\#\text{loops}}$. For many $\theta>1$ these models are equivalent to certain quantum spin systems for various choices of the system parameters. We find conditions on the offspring distribution that guarantee the occurrence of a phase transition from finite to infinite loops for the Galton-Watson tree.Comment: 16 pages, 1 figur

The formal path integral and quantum mechanics

Given an arbitrary Lagrangian function on \RR^d and a choice of classical path, one can try to define Feynman's path integral supported near the classical path as a formal power series parameterized by "Feynman diagrams," although these diagrams may diverge. We compute this expansion and show that it is (formally, if there are ultraviolet divergences) invariant under volume-preserving changes of coordinates. We prove that if the ultraviolet divergences cancel at each order, then our formal path integral satisfies a "Fubini theorem" expressing the standard composition law for the time evolution operator in quantum mechanics. Moreover, we show that when the Lagrangian is inhomogeneous-quadratic in velocity such that its homogeneous-quadratic part is given by a matrix with constant determinant, then the divergences cancel at each order. Thus, by "cutting and pasting" and choosing volume-compatible local coordinates, our construction defines a Feynman-diagrammatic "formal path integral" for the nonrelativistic quantum mechanics of a charged particle moving in a Riemannian manifold with an external electromagnetic field.Comment: 33 pages, many TikZ diagrams, submitted to _Journal of Mathematical Physics

Quantum cohomology of flag manifolds and Toda lattices

We discuss relations of Vafa's quantum cohomology with Floer's homology theory, introduce equivariant quantum cohomology, formulate some conjectures about its general properties and, on the basis of these conjectures, compute quantum cohomology algebras of the flag manifolds. The answer turns out to coincide with the algebra of regular functions on an invariant lagrangian variety of a Toda lattice.Comment: 35 page