DOCSLIB.ORG

Conformal Differential Geometry

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Conformal Differential Geometry NOTES ON CONFORMAL DIFFERENTIAL GEOMETRY y Michael Eastwoo d These notes are in no way meant to b e comprehensive neither in treatment nor in references to the extensive literature They are merely meant as an introduction to a small selection of topics in the eld They were presented as a series of four lectures at th the Winter School on Geometry and Physics SrnCzech Republic January Lecture One Recall that a Riemannian manifold M is really a pair M g consisting of a smo oth manifold M and a metric g a smo oth and everywhere p ositive denite section of J T M the symmetric tensor pro duct of the cotangent bundle Following standard practise we shall usually write g instead of g and more generally we shall adorn ab tensors with upp er and lower indices in corresp ondence with the tangent or cotangent bundle We shall also use the Einstein summation convention to denote the natural a pairing of vectors and covectors Thus V denotes a tangent vector or a vector eld a b and g V V denotes the square of its length with resp ect to g We shall often raise ab a b and lower indices without commentif V is a vector eld then V g V is the a ab corresp onding form More precisely this is Penroses abstract index notation see for details A conformal manifold M is a pair M g where g is a Riemannian metric dened J only up to scale In other words g is a section of R T M the bundle of rays J in T M such that a representative g is p ositive denite In yet other words a conformal manifold is an equivalence class of Riemannian manifolds where two b b metrics g and g are said to b e equivalent if g is a multiple of g In this case ab ab ab ab b it is convenient to write g g for some smo oth function On a conformal ab ab manifold one can measure angles b etween vectors but not lengths This pap er is in nal form and no version of it will b e submitted for publication elsewhere y ARC Senior Research Fellow University of Adelaide MICHAEL EASTWOOD For example in two dimensions an oriented conformal manifold is precisely a Riemann surface ie a onedimensional complex manifold Certainly if M is a Riemann surface with lo cal coordinate z x iy then dx dy is a Riemannian metric and if w u iv is another then the Cauchy Riemann equations u v v u x y y x imply u v v u dx dy dx dy du dv x y x y u v dx dy x x Conversely there is a theorem of Korn and Lichtenstein which says that any Rieman nian metric in two dimensions may b e written lo cally in the form dx dy for some smo oth function and suitable coordinates x y chosen compatible with the orientation Taking z x iy denes a lo cal complex coordinate and any other choice is holomorphically related Indeed the Cauchy Riemann equations are precisely that the Jacobian matrix u x u y v x v y is prop ortional to an orthogonal one Notice that a conformal manifold in two dimensions has no lo cal invariantsall Riemann surfaces are lo cally indistinguishable The only lo cal invariant of a two dimensional Riemannian manifold is its scalar curvature and this has b een eliminated by the freedom to scale the metric In higher dimensions it is reasonable to exp ect some of the rigidity of Riemannian geometry This is indeed the case and evidence in its favour can b e found as follows n Firstly let is consider rigid motions of R ie the connected comp onent of the group n of isometries of R with its usual metric Of course this is wellknown to b e generated by translations and rotations One way of seeing this is to consider an innitesimal a n motion ie a vector eld V on R with the prop erty that a V a son ie V V V ab ab b b x where square brackets around indices denote taking the skew part Consider now a V a son ie V V and V V V abc abc abc abc bc b c x x NOTES ON CONFORMAL DIFFERENTIAL GEOMETRY where round brackets denote the symmetric part However this rst prolongation son vanishes V V V V V V V abc bac bca cba cab acb abc By integration the result concerning rigid motions follows There are some observa tions to b e made The calculation is the principal ingredient in constructing the LeviCivita connection on a Riemannian manifold n The at mo del of Riemannian geometry is R as a homogeneous space n R o SOn rigid motions n R SOn stabiliser of a p oint As an alternative pro of we could show that geo desics ie straight lines with unit sp eed parameterisation are preserved by isometries and by ring out geo desics conclude that an isometry xing a p oint to rst order is necessarily the identity Let us try the same technique for the conformal case in an attempt to identify the n conformal motions of R An innitesimal motion is a vector eld satisfying a V a con ie V g V ab ab b b x Thus a V con ie V g and V V bc abc c ab abc abc This rst prolongation con is no longer zero In fact V g g g a abc c ab b ac a bc n identies con with R However a con ie V g and V V V bcd cd ab abcd abcd abcd and this second prolongation con vanishes if the dimension n is greater than ab ab ab n g V g V g V V cd abcd acbd cabd acbd ab ab g g V g V bd ac cdab cd cdab So cd cd ab cd cd ng g g g g ng cd cd ab cd cd and is tracefree Since n then cd ab n g V cd cdab MICHAEL EASTWOOD implies that the right hand side is symmetric in cd in which case ab ab n g V g g cd ab cd cdab which implies that is pure trace Now V and we nd that V is zero by cd abcd abcd applying on the rst three indices n We may conclude that a conformal motion of R for n which xes the origin to second order is necessarily the identity More precisely we may identify the group P of conformal motions xing the identity as matrices of the form a a n a a for m SOn r R r m b b a a r r r m a a b a n acting on x R by a b c a m x x x r b c a x a b c b r m x x x r r a b c b It is a go o d exercise to check that this really is conformal ie that its derivative is everywhere prop ortional to an orthogonal matrix Notice that the denominator b b c a vanishes when x m r r r On the other hand this formula is forced by the c a prolongation argument In order to allow nonzero r we are therefore obliged to n n n compactify R with a single p oint to obtain the sphere S R fg Further reasoning along these lines identies the full group of conformal motions of the round n n sphere S conformally containing Euclidean R via stereographic pro jection as the identity connected comp onent G of SOn The sphere is realised as the space of future p ointing null rays e n space S of generators n in R The form of P as ab ove is obtained by taking G to preserve the form x x x x x n n n and the basep oint of S to b e represented by the null vector e NOTES ON CONFORMAL DIFFERENTIAL GEOMETRY In summary the at mo del of conformal geometry is n S GP for G SO n and P a suitable parabolic subgroup The other two observations in the Riemannian case will turn out to have analogues in the conformal case We shall nd an invariant connection but not on the tangent bundle In fact we shall nd a somewhat b etter dierential op erator originally found by TY Thomas We shall also nd analogues of geo desics in the conformal case They are known as n conformal circles b ecause in the at case they coincide with the round circles on S equipp ed with their standard pro jective parameterisations Certainly these circles are preserved by G Lecture Two So much for the at case To pro ceed in general the nave approach is to work with a metric in the conformal class and then see how things change when the metric b is scaled If g is replaced by g g then the LeviCivita connection r is ab ab ab a c replaced by the connection r acting on forms according to a c c r r g a b a b a b b a c ab where r Indeed this formula surely denes a torsionfree connection a a and induces d d c r r g g a bc a bc a bc b ac c ba dc ab bd ac b on contravariant tensors clearly annihilating g Corresp ondingly the new ab ab connection on vector elds is b b b b c b c r V r V V V V a a a a c a b where is the Kronecker delta It is convenient to introduce a line bundle E on a M as follows If a metric g in the conformal class is chosen then E is identied ab with the trivial bundle E Equivalently a lo cal section of E may b e regarded as a b b function say f If however g is replaced by g g then the function f repres ab ab ab th b b enting with resp ect to the metric g is given by f f The w p ower of E will ab b e denoted by E w and its sections called conformally weighted functions of weight w w b Such a section may b e represented by a function scaling according to f f We a shall write E and E for the tangent and cotangent bundle resp ectively Other tensor a bundles will b e denoted by adorning E with the corresp onding indices The conformal metric may b e regarded as an invariantly dened section of E Raising and lower ab a ing indices denes a canonical isomorphism of E w with E w for any weight w a Cho osing a metric induces a connection on E w which transforms according to c r r w a a a MICHAEL EASTWOOD under scaling of the metric The Riemann curvature is dened by c c d r r r r V R V a b b a ab d and transforms by b R R g g g g abcd abcd ac bd bc ad bd ac ad bc under scaling of the metric where c r g ab a b a b c ab In particular the variation is entirely through traces The totally
Load more

Recommended publications
  • Abstract Tensor Systems and Diagrammatic Representations
    Abstract tensor systems and diagrammatic representations J¯anisLazovskis September 28, 2012 Abstract The diagrammatic tensor calculus used by Roger Penrose (most notably in [7]) is introduced without a solid mathematical grounding. We will attempt to derive the tools of such a system, but in a broader setting. We show that Penrose's work comes from the diagrammisation of the symmetric algebra. Lie algebra representations and their extensions to knot theory are also discussed. Contents 1 Abstract tensors and derived structures 2 1.1 Abstract tensor notation . 2 1.2 Some basic operations . 3 1.3 Tensor diagrams . 3 2 A diagrammised abstract tensor system 6 2.1 Generation . 6 2.2 Tensor concepts . 9 3 Representations of algebras 11 3.1 The symmetric algebra . 12 3.2 Lie algebras . 13 3.3 The tensor algebra T(g)....................................... 16 3.4 The symmetric Lie algebra S(g)................................... 17 3.5 The universal enveloping algebra U(g) ............................... 18 3.6 The metrized Lie algebra . 20 3.6.1 Diagrammisation with a bilinear form . 20 3.6.2 Diagrammisation with a symmetric bilinear form . 24 3.6.3 Diagrammisation with a symmetric bilinear form and an orthonormal basis . 24 3.6.4 Diagrammisation under ad-invariance . 29 3.7 The universal enveloping algebra U(g) for a metrized Lie algebra g . 30 4 Ensuing connections 32 A Appendix 35 Note: This work relies heavily upon the text of Chapter 12 of a draft of \An Introduction to Quantum and Vassiliev Invariants of Knots," by David M.R. Jackson and Iain Moffatt, a yet-unpublished book at the time of writing.
    [Show full text]
  • Geometric Algebra and Covariant Methods in Physics and Cosmology
    GEOMETRIC ALGEBRA AND COVARIANT METHODS IN PHYSICS AND COSMOLOGY Antony M Lewis Queens' College and Astrophysics Group, Cavendish Laboratory A dissertation submitted for the degree of Doctor of Philosophy in the University of Cambridge. September 2000 Updated 2005 with typo corrections Preface This dissertation is the result of work carried out in the Astrophysics Group of the Cavendish Laboratory, Cambridge, between October 1997 and September 2000. Except where explicit reference is made to the work of others, the work contained in this dissertation is my own, and is not the outcome of work done in collaboration. No part of this dissertation has been submitted for a degree, diploma or other quali¯cation at this or any other university. The total length of this dissertation does not exceed sixty thousand words. Antony Lewis September, 2000 iii Acknowledgements It is a pleasure to thank all those people who have helped me out during the last three years. I owe a great debt to Anthony Challinor and Chris Doran who provided a great deal of help and advice on both general and technical aspects of my work. I thank my supervisor Anthony Lasenby who provided much inspiration, guidance and encouragement without which most of this work would never have happened. I thank Sarah Bridle for organizing the useful lunchtime CMB discussion group from which I learnt a great deal, and the other members of the Cavendish Astrophysics Group for interesting discussions, in particular Pia Mukherjee, Carl Dolby, Will Grainger and Mike Hobson. I gratefully acknowledge ¯nancial support from PPARC. v Contents Preface iii Acknowledgements v 1 Introduction 1 2 Geometric Algebra 5 2.1 De¯nitions and basic properties .
    [Show full text]
  • Two-Spinors, Oscillator Algebras, and Qubits: Aspects of Manifestly Covariant Approach to Relativistic Quantum Information
    Quantum Information Processing manuscript No. (will be inserted by the editor) Two-spinors, oscillator algebras, and qubits: Aspects of manifestly covariant approach to relativistic quantum information Marek Czachor Received: date / Accepted: date Abstract The first part of the paper reviews applications of 2-spinor methods to rela- tivistic qubits (analogies between tetrads in Minkowski space and 2-qubit states, qubits defined by means of null directions and their role for elimination of the Peres-Scudo- Terno phenomenon, advantages and disadvantages of relativistic polarization operators defined by the Pauli-Lubanski vector, manifestly covariant approach to unitary rep- resentations of inhomogeneous SL(2,C)). The second part deals with electromagnetic fields quantized by means of harmonic oscillator Lie algebras (not necessarily taken in irreducible representations). As opposed to non-relativistic singlets one has to dis- tinguish between maximally symmetric and EPR states. The distinction is one of the sources of ‘strange’ relativistic properties of EPR correlations. As an example, EPR averages are explicitly computed for linear polarizations in states that are antisymmet- ric in both helicities and momenta. The result takes the familiar form p cos 2(α β) ± − independently of the choice of representation of harmonic oscillator algebra. Parameter p is determined by spectral properties of detectors and the choice of EPR state, but is unrelated to detector efficiencies. Brief analysis of entanglement with vacuum and vacuum violation of Bell’s inequality is given. The effects are related to inequivalent notions of vacuum states. Technical appendices discuss details of the representation I employ in field quantization. In particular, M-shaped delta-sequences are used to define Dirac deltas regular at zero.
    [Show full text]
  • Arxiv:1412.2393V4 [Gr-Qc] 27 Feb 2019 2.6 Geodesics and Normal Coordinates
    Riemannian Geometry: Definitions, Pictures, and Results Adam Marsh February 27, 2019 Abstract A pedagogical but concise overview of Riemannian geometry is provided, in the context of usage in physics. The emphasis is on defining and visualizing concepts and relationships between them, as well as listing common confusions, alternative notations and jargon, and relevant facts and theorems. Special attention is given to detailed figures and geometric viewpoints, some of which would seem to be novel to the literature. Topics are avoided which are well covered in textbooks, such as historical motivations, proofs and derivations, and tools for practical calculations. As much material as possible is developed for manifolds with connection (omitting a metric) to make clear which aspects can be readily generalized to gauge theories. The presentation in most cases does not assume a coordinate frame or zero torsion, and the coordinate-free, tensor, and Cartan formalisms are developed in parallel. Contents 1 Introduction 2 2 Parallel transport 3 2.1 The parallel transporter . 3 2.2 The covariant derivative . 4 2.3 The connection . 5 2.4 The covariant derivative in terms of the connection . 6 2.5 The parallel transporter in terms of the connection . 9 arXiv:1412.2393v4 [gr-qc] 27 Feb 2019 2.6 Geodesics and normal coordinates . 9 2.7 Summary . 10 3 Manifolds with connection 11 3.1 The covariant derivative on the tensor algebra . 12 3.2 The exterior covariant derivative of vector-valued forms . 13 3.3 The exterior covariant derivative of algebra-valued forms . 15 3.4 Torsion . 16 3.5 Curvature .
    [Show full text]
  • 2 Differential Geometry
    2 Differential Geometry The language of General relativity is Differential Geometry. The preset chap- ter provides a brief review of the ideas and notions of Differential Geometry that will be used in the book. In this role, it also serves the purpose of setting the notation and conventions to be used througout the book. The chapter is not intended as an introduction to Differential Geometry and as- sumes a prior knowledge of the subject at the level, say, of the first chapter of Choquet-Bruhat (2008) or Stewart (1991), or chapters 2 and 3 of Wald (1984). As such, rigorous definitions of concepts are not treated in depth —the reader is, in any case, refered to the literature provided in the text, and that given in the final section of the chapter. Instead, the decision has been taken of discussing at some length topics which may not be regarded as belonging to the standard bagage of a relativist. In particular, some detail is provided in the discussion of general (i.e. non Levi-Civita) connections, the sometimes 1+3 split of tensors (i.e. a split based on a congruence of timelike curves, rather than on a foliation as in the usual 3 + 1), and a discussion of submanifolds using a frame formalism. A discussion of conformal geometry has been left out of this chapter and will be undertaken in Chapter 5. 2.1 Manifolds The basic object of study in Differential Geometry is a differentiable man- ifold . Intuitively, a manifold is a space that locally looks like Rn for some n N.
    [Show full text]
  • A Treatise on Quantum Clifford Algebras Contents
    A Treatise on Quantum Clifford Algebras Habilitationsschrift Dr. Bertfried Fauser arXiv:math/0202059v1 [math.QA] 7 Feb 2002 Universitat¨ Konstanz Fachbereich Physik Fach M 678 78457 Konstanz January 25, 2002 To Dorothea Ida and Rudolf Eugen Fauser BERTFRIED FAUSER —UNIVERSITY OF KONSTANZ I ABSTRACT: Quantum Clifford Algebras (QCA), i.e. Clifford Hopf gebras based on bilinear forms of arbitrary symmetry, are treated in a broad sense. Five al- ternative constructions of QCAs are exhibited. Grade free Hopf gebraic product formulas are derived for meet and join of Graßmann-Cayley algebras including co-meet and co-join for Graßmann-Cayley co-gebras which are very efficient and may be used in Robotics, left and right contractions, left and right co-contractions, Clifford and co-Clifford products, etc. The Chevalley deformation, using a Clif- ford map, arises as a special case. We discuss Hopf algebra versus Hopf gebra, the latter emerging naturally from a bi-convolution. Antipode and crossing are consequences of the product and co-product structure tensors and not subjectable to a choice. A frequently used Kuperberg lemma is revisited necessitating the def- inition of non-local products and interacting Hopf gebras which are generically non-perturbative. A ‘spinorial’ generalization of the antipode is given. The non- existence of non-trivial integrals in low-dimensional Clifford co-gebras is shown. Generalized cliffordization is discussed which is based on non-exponentially gen- erated bilinear forms in general resulting in non unital, non-associative products. Reasonable assumptions lead to bilinear forms based on 2-cocycles. Cliffordiza- tion is used to derive time- and normal-ordered generating functionals for the Schwinger-Dyson hierarchies of non-linear spinor field theory and spinor electro- dynamics.
    [Show full text]
  • Revisiting the Radio Interferometer Measurement Equation IV
    A&A 531, A159 (2011) Astronomy DOI: 10.1051/0004-6361/201116764 & c ESO 2011 Astrophysics Revisiting the radio interferometer measurement equation IV. A generalized tensor formalism O. M. Smirnov Netherlands Institute for Radio Astronomy (ASTRON) PO Box 2, 7990AA Dwingeloo, The Netherlands e-mail: [email protected] Received 22 February 2011 / Accepted 1 June 2011 ABSTRACT Context. The radio interferometer measurement equation (RIME), especially in its 2 × 2 form, has provided a comprehensive matrix- based formalism for describing classical radio interferometry and polarimetry, as shown in the previous three papers of this series. However, recent practical and theoretical developments, such as phased array feeds (PAFs), aperture arrays (AAs) and wide-field polarimetry, are exposing limitations of the formalism. Aims. This paper aims to develop a more general formalism that can be used to both clearly define the limitations of the matrix RIME, and to describe observational scenarios that lie outside these limitations. Methods. Some assumptions underlying the matrix RIME are explicated and analysed in detail. To this purpose, an array correlation matrix (ACM) formalism is explored. This proves of limited use; it is shown that matrix algebra is simply not a sufficiently flexible tool for the job. To overcome these limitations, a more general formalism based on tensors and the Einstein notation is proposed and explored both theoretically, and with a view to practical implementations. Results. The tensor formalism elegantly yields generalized RIMEs describing beamforming, mutual coupling, and wide-field po- larimetry in one equation. It is shown that under the explicated assumptions, tensor equations reduce to the 2 × 2RIME.Froma practical point of view, some methods for implementing tensor equations in an optimal way are proposed and analysed.
    [Show full text]
  • Operators on Di Erential Forms Abstract Index Notation Leibniz Commuting Inner Products
    Differential forms Abstract index notation Coee & Chalk Press (p + q)! Ivica Smolić cbnd (α ^ β)a :::a b :::b = α β 1 p 1 q p!q! [a1:::ap b1:::bq ] 1 (?!) = ! a1:::ap ap+1:::am p! a1:::ap ap+1:::am M is a smooth m-manifold with a metric gab which has s negative b eigenvalues. Metric determinant is denoted by g = det(gµν ). (iX !)a1:::ap−1 = X !ba1:::ap−1 · Base of p-forms via wedge product, (d!)a1:::ap+1 = (p + 1)r[a1 !a2:::ap+1] b µ µ X µ µ (δ!)a :::a = r !ba :::a dx 1 ^ ::: ^ dx p = sgn(π) dx π(1) ⊗ · · · ⊗ dx π(p) 1 p−1 1 p−1 π2Sp p(m−p)+s 2 2 2 Thm. ?? = (−1) ; iX = d = δ = 0 · General p-form ! 2 Ωp, Thm. £X (f) = δ(fX) 1 µ1 µp ! = !µ ...µ dx ^ ::: ^ dx a a p! 1 p Thm. δ(fα) = iX α + fδα with X = r f · Unless otherwise stated, we assume: Thm. iX d?α = ?(X ^ δα) f 2 Ω0 ; α; ! 2 Ωp ; β 2 Ωq ; γ 2 Ωp−1 ; X; Y 2 TM Leibniz p · Generalized Kronecker delta iX (α ^ β) = (iX α) ^ β + (−1) α ^ (iX β) p δa1···an := n! δ [a1 δ a2 ··· δ an] d(α ^ β) = (dα) ^ β + (−1) α ^ (dβ) b1··· bn b1 b2 bn £X (α ^ β) = (£X α) ^ β + α ^ (£X β) 1 2 n 1 n µ1···µn det(A) := n! A [1A 2 ··· A n] = A µ1 ··· A µn δ1 ··· n 1 2 n 1 ··· n n! A [µ1 A µ2 ··· A µn] = det(A) δµ1··· µn ! a ···a a ···a m − n + k a ···a Commuting δ 1 k k+1 n = k! δ k+1 n a1···akbk+1··· bn k bk+1··· bn α ^ β = (−1)pqβ ^ α m p 1 m m(p+1)+s · Volume form 2 Ω , := jgj dx ^ ::: ^ dx iX ?α = ?(α ^ X) ; iX α = (−1) ?(X ^ ?α) s p 1··· m µ1···µm (−1) µ1···µm µ ···µ = jgj δµ ···µ ; = δ 1 m 1 m pjgj 1 ··· m [£X ; £Y ] = £[X;Y ] ; [£X ; iY ] = i[X;Y ] ; [£X ; d] = 0 a1···akak+1···am s ak+1···am a ···a b ···b = (−1) k! δ 1 k k+1 m bk+1···bm £X ?α − ? £X α = (δX) ?α + ?αb with α := p( gbc)g α ba1:::ap £X b[a1 jcja2:::ap] Operators on dierential forms d ? = (−1)p+1 ? δ ; δ ? = (−1)p ? d p p−1 ?∆ = ∆?; d∆ = ∆d ; δ∆ = ∆δ · Contraction with vector iX :Ω ! Ω iX f = 0 ; (iX !)(X1;:::;Xp−1) = !(X; X1;:::;Xp−1) Inner products · Hodge dual ? :Ωp ! Ωm−p 1 a1:::ap !µ ...µ (α j !) := αa1:::ap ! ?! = 1 p µ1...µp dxµp+1 ^ ::: ^ dxµm p! p!(m − p)! µp+1...µm Normalization of the volume form, ( j ) = (−1)s.
    [Show full text]
  • Birdtracks for SU(N)
    SciPost Phys. Lect. Notes 3 (2018) Birdtracks for SU(N) Stefan Keppeler Fachbereich Mathematik, Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany ? [email protected] Abstract I gently introduce the diagrammatic birdtrack notation, first for vector algebra and then for permutations. After moving on to general tensors I review some recent results on Hermitian Young operators, gluon projectors, and multiplet bases for SU(N) color space. Copyright S. Keppeler. Received 25-07-2017 This work is licensed under the Creative Commons Accepted 19-06-2018 Check for Attribution 4.0 International License. Published 27-09-2018 updates Published by the SciPost Foundation. doi:10.21468/SciPostPhysLectNotes.3 Contents Introduction 2 1 Vector algebra2 2 Birdtracks for SU(N) tensors6 Exercise: Decomposition of V V 8 ⊗ 3 Permutations and the symmetric group 11 3.1 Recap: group algebra and regular representation 14 3.2 Recap: Young operators 14 3.3 Young operators and SU(N): multiplets 16 4 Colour space 17 4.1 Trace bases vs. multiplet bases 18 4.2 Multiplet bases for quarks 20 4.3 General multiplet bases 22 4.4 Gluon projectors 23 4.5 Some multiplet bases 26 5 Further reading 28 References 29 1 SciPost Phys. Lect. Notes 3 (2018) Introduction The term birdtracks was coined by Predrag Cvitanovi´c, figuratively denoting the diagrammatic notation he uses in his book on Lie groups [1] – hereinafter referred to as THE BOOK. The birdtrack notation is closely related to (abstract) index notation. Translating back and forth between birdtracks and index notation is achieved easily by following some simple rules.
    [Show full text]
  • Two-Spinors and Symmetry Operators
    Two-spinors and Symmetry Operators An Investigation into the Existence of Symmetry Operators for the Massive Dirac Equation using Spinor Techniques and Computer Algebra Tools Master’s thesis in Physics Simon Stefanus Jacobsson DEPARTMENT OF MATHEMATICAL SCIENCES CHALMERS UNIVERSITY OF TECHNOLOGY Gothenburg, Sweden 2021 www.chalmers.se Master’s thesis 2021 Two-spinors and Symmetry Operators ¦ An Investigation into the Existence of Symmetry Operators for the Massive Dirac Equation using Spinor Techniques and Computer Algebra Tools SIMON STEFANUS JACOBSSON Department of Mathematical Sciences Division of Analysis and Probability Theory Mathematical Physics Chalmers University of Technology Gothenburg, Sweden 2021 Two-spinors and Symmetry Operators An Investigation into the Existence of Symmetry Operators for the Massive Dirac Equation using Spinor Techniques and Computer Algebra Tools SIMON STEFANUS JACOBSSON © SIMON STEFANUS JACOBSSON, 2021. Supervisor: Thomas Bäckdahl, Mathematical Sciences Examiner: Simone Calogero, Mathematical Sciences Master’s Thesis 2021 Department of Mathematical Sciences Division of Analysis and Probability Theory Mathematical Physics Chalmers University of Technology SE-412 96 Gothenburg Telephone +46 31 772 1000 Typeset in LATEX Printed by Chalmers Reproservice Gothenburg, Sweden 2021 iii Two-spinors and Symmetry Operators An Investigation into the Existence of Symmetry Operators for the Massive Dirac Equation using Spinor Techniques and Computer Algebra Tools SIMON STEFANUS JACOBSSON Department of Mathematical Sciences Chalmers University of Technology Abstract This thesis employs spinor techniques to find what conditions a curved spacetime must satisfy for there to exist a second order symmetry operator for the massive Dirac equation. Conditions are of the form of the existence of a set of Killing spinors satisfying some set of covariant differential equations.
    [Show full text]
  • The Poor Man's Introduction to Tensors
    The Poor Man’s Introduction to Tensors Justin C. Feng Preface In the spring semester of 2013, I took a graduate fluid mechanics class taught by Philip J. Morrison. On thefirst homework assignment, I found that it was easier to solve some of the problems using tensors in the coordinate basis, but I was uncomfortable with it because I wasn’t sure if I was allowed to use a formalism that had not been taught in class. Eventually, I decided that if I wrote an appendix on tensors, I would be allowed to use the formalism. Sometime in 2014, I converted1 the appendix into a set of introductory notes on tensors and posted it on my academic website, figuring that they might be useful for someone interested in learning thetopic. Over the past three years, I had become increasingly dissatisfied with the notes. The biggest issue with these notes was that they were typeset in Mathematica—during my early graduate career, I typeset my homework assignments in Mathematica, and it was easiest to copy and paste the appendix into another Mathematica notebook (Mathematica does have a feature to convert notebooks to TeX, but much of the formatting is lost in the process). I had also become extremely dissatisfied with the content—the notes contained many formulas that weren’t sufficiently justified, and there were many examples of sloppy reasoning throughout. At one point, I had even considered removing the notes from my website, but after receiving some positive feedback from students who benefited from the notes, I decided to leave it up for the time being, vowing to someday update and rewrite these notes in TeX.
    [Show full text]
  • Geometric Methods in Mathematical Physics I: Multi-Linear Algebra, Tensors, Spinors and Special Relativity
    Geometric Methods in Mathematical Physics I: Multi-Linear Algebra, Tensors, Spinors and Special Relativity by Valter Moretti www.science.unitn.it/∼moretti/home.html Department of Mathematics, University of Trento Italy Lecture notes authored by Valter Moretti and freely downloadable from the web page http://www.science.unitn.it/∼moretti/dispense.html and are licensed under a Creative Commons Attribuzione-Non commerciale-Non opere derivate 2.5 Italia License. None is authorized to sell them 1 Contents 1 Introduction and some useful notions and results 5 2 Multi-linear Mappings and Tensors 8 2.1 Dual space and conjugate space, pairing, adjoint operator . 8 2.2 Multi linearity: tensor product, tensors, universality theorem . 14 2.2.1 Tensors as multi linear maps . 14 2.2.2 Universality theorem and its applications . 22 2.2.3 Abstract definition of the tensor product of vector spaces . 26 2.2.4 Hilbert tensor product of Hilbert spaces . 28 3 Tensor algebra, abstract index notation and some applications 34 3.1 Tensor algebra generated by a vector space . 34 3.2 The abstract index notation and rules to handle tensors . 37 3.2.1 Canonical bases and abstract index notation . 37 3.2.2 Rules to compose, decompose, produce tensors form tensors . 39 3.2.3 Linear transformations of tensors are tensors too . 41 3.3 Physical invariance of the form of laws and tensors . 43 3.4 Tensors on Affine and Euclidean spaces . 43 3.4.1 Tensor spaces, Cartesian tensors . 45 3.4.2 Applied tensors . 46 3.4.3 The natural isomorphism between V and V ∗ for Cartesian vectors in Eu- clidean spaces .
    [Show full text]