The jones polynomial is a celebrated invariant of a knot or link in ordinary. To a physicist, a quantum field theory defined on a manifold m without any a priori. On the jones polynomial and its applications alan chang abstract. Quantum field theory i chapter 0 eth zurich, hs14 prof. A polynomial quantum algorithm for approximating the. Both these approaches lead to classical algorithms for the computation of the jones polynomial, which are not e cient, that is their complexity grows exponentially with the number of the crossings of the braid or diagram. As the name suggests, relativistic quantum field theory rests really on three pillars, special relativity, quantum mechanics and field theory. Mar 19, 2008 the jones polynomial, discovered in 1984, is an important knot invariant in topology. A polynomial quantum algorithm for approximating the jones. The wave function of a quantum system is a vector f in a hilbert space h, comprising possible wave functions. Here, a the term \ eld refers to one of the following.
The quantum groups and tangle functor formalisms for understanding these. Quantum field theory and the jones polynomial signal lake. The jones polynomial and the kauffman bracket are constructed, and their relation with knot and link theory is described. Frank wilczek institute for advanced study, school of natural science, olden lane, princeton, nj 08540 i discuss the general principles underlying quantum. On the work of edward witten university of california, berkeley. Topological quantum field theories a meeting ground for. The slides are intended for classroom use, with fewer words. Quantum field theory and the jones polynomial inspire. That is, we start with a lagranian density in terms of products of quantum.
It will also cover everything in the \advanced quantum field theory course, much of the \standard model course, and will serve you well if you go on to do research. Field theoretic notions appeared, even though still implicitly, in the modern theory of solvability of polynomial equations, as introduced by abel and galois in the early nineteenth century. States describe properties of prepared ensembles probability distributions of measured values, correlations between observables. The second part is dedicated to topological field theories. On the classical limit of selfinteracting quantum field hamiltonians with cutoffs ammari, zied and zerzeri, maher, hokkaido mathematical journal, 2014. Quantum field theory and the jones polynomial springerlink. Chernsimons theory, conformal field theory, and fractional quantum hall states 37 1. Quantum computing and the jones polynomial louis h. In this version, the jones polynomial can be generalized froms 3 to arbitrary three manifolds, giving invariants of three manifolds that are. Gauge theory and the jones polynomial math berkeley. I discuss the general principles underlying quantum field theory, and attempt to identify its most profound consequences. Unlike the donaldson theory, where a connection with quantum field theory was. In this part, in the rst three chapters i write about scalar elds, elds with spin, and nonabelian elds. Among its many connections to various mathematical and physical areas, it is known due to witten 32 to be intimately connected to topological quantum field theory tqft.
Similarly that of special relativity is demonstrated by experiments showing constancy of speed of light, time dilation etc. This section contains lecture notes and lecture slides. In all this large and exciting field, which involves many of the leading physicists. This multiformat coverage does not occur for all topics, but it happens occasionally.
Any errors or obvious omissions should be communicated to me at j. In a rather informal sense qft is the extension of quantum mechanics qm, dealing with particles, over to fields, i. We discuss the structure of the jones polynomial in relation to representations of the temperley lieb algebra, and give an example of a unitary representation of the braid group. Perturbative algebraic quantum field theory and the renormalization groups brunetti, r. Quantum field theory and the jones polynomial inspirehep. The most elementary ways to calculate vlt use the linear skein theory ideas of 7. Quantum field theory qft is the mathematical and conceptual framework for contemporary elementary particle physics. The positive mass conjecture in general relativity the positive mass conjecture asserts that under appropriate hypotheses the total energy of a gravitating system is positive and can only be. The notes have more words, and are intended to be read as an independent document. A good source of information on the connection between qft and the jones polynominal is. One algorithm that seems to be completely different is the algorithm for evaluating the jones polynomial. The basic starting point is that the axioms of special relativity on the one hand and those of quantum mechanics on the other, should be combined into one theory. This paper is an exploration of relationships between the jones polynomial and quantum computing.
Classical field theory, free fields, interacting fields, the dirac equation, quantizing the dirac field and quantum electrodynamics. In this article khovanov describes a clever way to improve the strength of the jones polynomial by using a trick called categori. Threedimensional quantum gravity, chernsimons theory, and. To a physicist, a quantum field theory defined on a manifold m without any a priori choice of a metric on mis said to be generally covariant.
Quantum field theory i lecture notes by jan louis ii. Relativistic quantum field theory is a mathematical scheme to describe the subatomic particles and forces. Normal surface theory according to the jones polynomial we explore analogies between the classical topology of normal surfaces and quantum knot invariants. It states that the degree of the colored jones polynomial detects boundary slopes of essential surfaces in the knot complement. In this version, the jones polynomial can be generalized from s 3 to arbitrary three manifolds, giving invariants of three manifolds that are computable from a surgery presentation. Similarly to that case this has led to the construction of an axiomatic approach 14.
The following chapters are dedicated to quantum electrodynamics and quantum chromodynamics, followed by the renormalization theory. The validity of quantum mechanics is experimentally demonstrated by the photoelectric e. A particle follows the usual rules of quantum theory. The quantum algorithm of ajl 3 following the work of freedman et al. I decided to use an introduction to quantum field theory by peskin and schroeder, which was, and perhaps still is, one of the standard modern textbooks on the subject. While the analytical foundations of quantum field theory have been intensively studied by mathematicians for many years the new stimulus has involved the more formal algebraic, geometric, topological aspects. Then, at a latter stage, i added some text in oneloop techniques in english. A eld of a classical eld theory, such as electromagnetism. Quantum field theory pdf 155p this book provides a very clear and well written introduction to quantum field theory. The deepest of these consequences result from the infinite number of degrees of freedom invoked to implement locality.
The jones polynomial of a knot and generally a link with an odd number of components is a laurent polynomial in t. In his 1989 paper quantum field theory and the jones polynomial, he brought a powerful organizing principle into both the physics and. This paper is a selfcontained introduction to the jones polynomial that assumes no background in knot theory. In this version, the jones polynomial can be generalized froms3 to arbitrary three manifolds, giving invariants of three manifolds that are. We deal concretely with some concepts that appear in more abstract form in later chapters. Then, i realized that this text could be more useful if it was all in english. Specifically, it is an invariant of an oriented knot or link which assigns to each oriented knot or link a laurent polynomial in the variable with integer coefficients. Find materials for this course in the pages linked along the left. This section provides lecture notes intended to be read as independent documents along with lecture slides intended for classroom use. In theoretical physics, quantum field theory qft is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. This is a basic construction in quantum field theory, though. Chernsimons effective field theories, the jones polynomial, and nonabelian topological phases 33 1.
My favorite the quantum theory of fields by steven weinberg did not suit that well as an introductory course. Topological quantum information and the jones polynomial louis h. The haagkastler axiomatic framework for quantum field theory, introduced by haag and kastler, is an application to local quantum physics of calgebra theory. Because of this it is also known as algebraic quantum field theory aqft. Possible limitations of quantum field theory are viewed in the light of its. There are two distinct logical paths one can follow to arrive at the same goal. The jones polynomial, discovered in 1984 18, is an important knot invariant in topology. On the contrary, in the context of quantum computation the problem admits a solution having polynomial complexity.
Combinatorial evaluation of link invariants and quasiparticle properties 35 d. This is a text for an advanced quantum field theory course that i have been teaching for many years at instituto superior t. Quantum field theory stanford encyclopedia of philosophy. Nonabelian anyons and topological quantum computation. Free quantum field theory books download ebooks online. His work on knot polynomials, with the discovery of what is now called the jones polynomial.
I mention a few of its most striking successes, both achieved and prospective. Quantum field theory and the jones polynomial, comm, math, phys, 121. Introduction to quantum field theory john cardy michaelmas term 2010 version 910 abstract these notes are intendedtosupplementthe lecturecourse introduction toquantum field theory and are not intended for wider distribution. Lecture notes relativistic quantum field theory i physics. Schroeder, an introduction to quantum field theory this is a very clear and comprehensive book, covering everything in this course at the right level. The quantum algorithm of ajl3 following the work of freedman et al. The axioms are stated in terms of an algebra given for every open set in minkowski space, and mappings between those. Qft is used in particle physics to construct physical models of subatomic particles and in condensed matter physics to construct models of quasiparticles. On the work of edward witten university of california. Review of quantum field theory and the jones polynomial. Quantum field theory ii lecture notes by jan louis ii. Modify, remix, and reuse just remember to cite ocw as the source. The jones polynomial, discovered in 1984, is an important knot invariant in topology. To a physicist, a quantum field theory defined on a manifold mwithout any a priori choice of a metric on m is said to be generally covariant.
In the mathematical field of knot theory, the jones polynomial is a knot polynomial discovered by vaughan jones in 1984. These methods are rooted in the bracket state sum model for the jones polynomial. There are many fairly standard quantum algorithms that can all be understood within a very similar framework, from deutschs algorithm simons problem, grovers search, shors algorithm and so on. An important instance of this analogy is the slope conjecture. We know that quantum mechanics and the special theory of relativity are both properties of nature.1269 1265 1457 1519 453 1638 67 1451 1416 121 864 1067 283 1301 1076 1284 672 335 160 79 15 967 888 1034 1443 1074 582 908 1026 762 1045 670 62 1087 16 1481 901 1430 803 1204 1078 965 925