On the Representation Theory of Negative Spin

We construct a class of negative spin irreducible representations of the su(2) Lie algebra. These representations are infinite-dimensional and have an indefinite inner product. We analyze the decomposition of arbitrary products of positive and negative representations with the help of generalized characters and write down explicit reduction formulae for the products. From the characters, we define effective dimensions for the negative spin representations, find that they are fractional, and point out that the dimensions behave consistently under multiplication and decomposition of representations.Comment: 21 pages, no figures, Latex2

Difficulties in the comprehension and interpretation of a selection of graph types and subject-specific graphs displayed by senior undergraduate biochemistry students in a South African university

A carefully constructed set of 16 graphical tasks related to key biochemistry concepts was designed and administered to a group of 82 students in their final year of B.Sc. study. The test mean score of 48,3% ( 12,1) was low and characterised by gender and ethnic differences. There was a moderate linear relationship between biochemistry grades obtained by the students over two years of study and their graphical literacy (r = 0,433). The majority of the students exhibited slope/height confusion and only seven students (8,5%) were able to answer the two items corresponding to Kimura‘s Level F, the most complex and difficult level of graphical literacy. Eye tracking data gave valuable insights into different strategies used by students while interpreting graphs and is a valuable tool for assessing graphical literacy. These findings confirmed other studies where researchers have found a widespread lack of graph comprehension among biological science students.Institute of Science and Technology EducationM. Sc. (Science Education

Cohomology and Decomposition of Tensor Product Representations of SL(2,R)

We analyze the decomposition of tensor products between infinite dimensional (unitary) and finite-dimensional (non-unitary) representations of SL(2,R). Using classical results on indefinite inner product spaces, we derive explicit decomposition formulae, true modulo a natural cohomological reduction, for the tensor products.Comment: 22 pages, no figures, Latex2e Added section on product of finite and continuous serie

A Lambda Calculus for Quantum Computation

The classical lambda calculus may be regarded both as a programming language and as a formal algebraic system for reasoning about computation. It provides a computational model equivalent to the Turing machine, and continues to be of enormous benefit in the classical theory of computation. We propose that quantum computation, like its classical counterpart, may benefit from a version of the lambda calculus suitable for expressing and reasoning about quantum algorithms. In this paper we develop a quantum lambda calculus as an alternative model of quantum computation, which combines some of the benefits of both the quantum Turing machine and the quantum circuit models. The calculus turns out to be closely related to the linear lambda calculi used in the study of Linear Logic. We set up a computational model and an equational proof system for this calculus, and we argue that it is equivalent to the quantum Turing machine.Comment: To appear in SIAM Journal on Computing. Minor corrections and improvements. Simulator available at http://www.het.brown.edu/people/andre/qlambda/index.htm

Coordinate-invariant Path Integral Methods in Conformal Field Theory

We present a coordinate-invariant approach, based on a Pauli-Villars measure, to the definition of the path integral in two-dimensional conformal field theory. We discuss some advantages of this approach compared to the operator formalism and alternative path integral approaches. We show that our path integral measure is invariant under conformal transformations and field reparametrizations, in contrast to the measure used in the Fujikawa calculation, and we show the agreement, despite different origins, of the conformal anomaly in the two approaches. The natural energy-momentum in the Pauli-Villars approach is a true coordinate-invariant tensor quantity, and we discuss its nontrivial relationship to the corresponding non-tensor object arising in the operator formalism, thus providing a novel explanation within a path integral context for the anomalous Ward identities of the latter. We provide a direct calculation of the nontrivial contact terms arising in expectation values of certain energy-momentum products, and we use these to perform a simple consistency check confirming the validity of the change of variables formula for the path integral. Finally, we review the relationship between the conformal anomaly and the energy-momentum two-point functions in our formalism.Comment: Corrected minor typos. To appear in International Journal of Modern Physics

Worldsheet Covariant Path Integral Quantization of Strings

We discuss a covariant functional integral approach to the quantization of the bosonic string. In contrast to approaches relying on non-covariant operator regularizations, interesting operators here are true tensor objects with classical transformation laws, even on target spaces where the theory has a Weyl anomaly. Since no implicit non-covariant gauge choices are involved in the definition of the operators, the anomaly is clearly separated from the issue of operator renormalization and can be understood in isolation, instead of infecting the latter as in other approaches. Our method is of wider applicability to covariant theories that are not Weyl invariant, but where covariant tensor operators are desired. After constructing covariantly regularized vertex operators, we define a class of background-independent path integral measures suitable for string quantization. We show how gauge invariance of the path integral implies the usual physical state conditions in a very conceptually clean way. We then discuss the construction of the BRST action from first principles, obtaining some interesting caveats relating to its general covariance. In our approach, the expected BRST related anomalies are encoded somewhat differently from other approaches. We conclude with an unusual but amusing derivation of the value $D= 26$ of the critical dimension.Comment: 64 pages, minor edits in expositio

Semantics of a Typed Algebraic Lambda-Calculus

Algebraic lambda-calculi have been studied in various ways, but their semantics remain mostly untouched. In this paper we propose a semantic analysis of a general simply-typed lambda-calculus endowed with a structure of vector space. We sketch the relation with two established vectorial lambda-calculi. Then we study the problems arising from the addition of a fixed point combinator and how to modify the equational theory to solve them. We sketch an algebraic vectorial PCF and its possible denotational interpretations