Revisiting the Fradkin-Vilkovisky Theorem

The status of the usual statement of the Fradkin-Vilkovisky theorem, claiming complete independence of the Batalin-Fradkin-Vilkovisky path integral on the gauge fixing "fermion" even within a nonperturbative context, is critically reassessed. Basic, but subtle reasons why this statement cannot apply as such in a nonperturbative quantisation of gauge invariant theories are clearly identified. A criterion for admissibility within a general class of gauge fixing conditions is provided for a large ensemble of simple gauge invariant systems. This criterion confirms the conclusions of previous counter-examples to the usual statement of the Fradkin-Vilkovisky theorem.Comment: 21 pages, no figures, to appear in Jnl. Phys.

Evaluating Scoped Meaning Representations

Semantic parsing offers many opportunities to improve natural language understanding. We present a semantically annotated parallel corpus for English, German, Italian, and Dutch where sentences are aligned with scoped meaning representations in order to capture the semantics of negation, modals, quantification, and presupposition triggers. The semantic formalism is based on Discourse Representation Theory, but concepts are represented by WordNet synsets and thematic roles by VerbNet relations. Translating scoped meaning representations to sets of clauses enables us to compare them for the purpose of semantic parser evaluation and checking translations. This is done by computing precision and recall on matching clauses, in a similar way as is done for Abstract Meaning Representations. We show that our matching tool for evaluating scoped meaning representations is both accurate and efficient. Applying this matching tool to three baseline semantic parsers yields F-scores between 43% and 54%. A pilot study is performed to automatically find changes in meaning by comparing meaning representations of translations. This comparison turns out to be an additional way of (i) finding annotation mistakes and (ii) finding instances where our semantic analysis needs to be improved.Comment: Camera-ready for LREC 201

Isolating vacuum amplitudes in quantum field calculations at finite temperature

In calculating Feynman diagrams at finite temperature, it is sometimes convenient to isolate subdiagrams which do not depend explicitly on the temperature. We show that, in the imaginary time formalism, such a separation can be achieved easily by exploiting a simple method, due to M. Gaudin, to perform the sum over the Matsubara frequencies. In order to manipulate freely contributions which may be individually singular, a regularization has to be introduced. We show that, in some cases, it is possible to choose this regularization in such a way that the isolated subdiagrams can be identified with analytical continuations of vacuum n-point functions. As an aside illustration of Gaudin's method, we use it to prove the main part of a recent conjecture concerning the relation which exists in the imaginary time formalism between the expressions of a Feynman diagram at zero and finite temperature.Comment: 37 pages, 12 figure

Towards gauge theories in four dimensions

The abundance of infrared singularities in gauge theories due to unresolved emission of massless particles (soft and collinear) represents the main difficulty in perturbative calculations. They are typically regularized in dimensional regularization, and their subtraction is usually achieved independently for virtual and real corrections. In this paper, we introduce a new method based on the loop-tree duality (LTD) theorem to accomplish the summation over degenerate infrared states directly at the integrand level such that the cancellation of the infrared divergences is achieved simultaneously, and apply it to reference examples as a proof of concept. Ultraviolet divergences, which are the consequence of the point-like nature of the theory, are also reinterpreted physically in this framework. The proposed method opens the intriguing possibility of carrying out purely four-dimensional implementations of higher-order perturbative calculations at next-to-leading order (NLO) and beyond free of soft and final-state collinear subtractions.Comment: Final version to appear in JHE

Grover's algorithm on a Feynman computer

We present an implementation of Grover's algorithm in the framework of Feynman's cursor model of a quantum computer. The cursor degrees of freedom act as a quantum clocking mechanism, and allow Grover's algorithm to be performed using a single, time-independent Hamiltonian. We examine issues of locality and resource usage in implementing such a Hamiltonian. In the familiar language of Heisenberg spin-spin coupling, the clocking mechanism appears as an excitation of a basically linear chain of spins, with occasional controlled jumps that allow for motion on a planar graph: in this sense our model implements the idea of "timing" a quantum algorithm using a continuous-time random walk. In this context we examine some consequences of the entanglement between the states of the input/output register and the states of the quantum clock

The Hasse principle for lines on diagonal surfaces

Given a number field $k$ and a positive integer $d$, in this paper we consider the following question: does there exist a smooth diagonal surface of degree $d$ in $\mathbb{P}^3$ over $k$ which contains a line over every completion of $k$, yet no line over $k$? We answer the problem using Galois cohomology, and count the number of counter-examples using a result of Erd\H{o}s.Comment: 14 page