arXiv subjects

Andre van Tonder

Publications and source records attributed to Andre van Tonder.

5 recordsLinked to original sources

arXiv2004

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.

quant-ph↗

arXiv2003

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.

hep-th↗

arXiv2002

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.

hep-th↗

arXiv2002

Ghosts as Negative Spinors

We study the the properties of a BRST ghost degree of freedom complementary to a two-state spinor. We show that the ghost may be regarded as a unit carrier of negative entropy. We construct an irreducible representation of the su(2) Lie algebra with negative spin, equal to -1/2, on the ghost state space and discuss the representation of finite SU(2) group elements. The Casimir operator J^2 of the combined spinor-ghost system is nilpotent and coincides with the BRST operator Q. Using this, we discuss the sense in which the positive and negative spin representations cancel in the product to give an effectively trivial representation. We compute an effective dimension, equal to 1/2, and character for the ghost representation and argue that these are consistent with this cancellation.

hep-th↗

arXiv1996

Finite [q-Oscillator] Description of 2-D String Theory

We present a simple physical representation for states of the two-dimensional string theory. In order to incorporate a fundamental cutoff of the order 1/g we use a picture consisting of q-oscillators at the first-quantized level. In this framework we also find a representation for the (singular) negatively dressed states representing nontrivial string backgrounds.

hep-th↗