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The Square-Root Isometry of Coupled Quadratic Spaces The Square-Root Isometry of Coupled Quadratic Spaces On the relation between vielbein and metric formulations of spin-2 interactions Mikica B Kocic S (or S2) X X X S S (V, f2) (V, h2) (V, g2) X X S2 S2 X X S2 S2 X X X X Ξ(µ) S S Ξ(ν) (V, fˆ) (V, f) (V, h) (V, g) (V, gˆ) X Lˆ Eˆ m m L S (or S2) E et es t ⊕ s ⊕ Λ2 Λ Λ1 (Rm,n, η) (Rm,n, η) (Rm,n, η) (Rm,n, η) 2 ] 2 S = g ◦ f[ = S2 Supervisor: S. F. Hassan Department of Physics Stockholm University Master’s Thesis, 45 HE credits Autumn term 2014 Abstract Bimetric theory is an extension to general relativity that introduces a secondary sym- metric rank-two tensor field. This secondary spin-2 field is also dynamical, and to avoid the Boulware-Deser ghost issue, the interaction between the two fields is obtained through a potential that involes the matrix square-root of the tensors. This square-root “quantity” is a linear transformation, herein referred to as the square-root isometry. In this work we explore the conditions for the existence of the square-root isometry and its group properties. Morever we study the conditions for the simultaneous 3+1 decomposition of two fields, and then, in terms of null-cones, give the (local) causal relations between fields coupled by the square-root isometry. Finally, we show the algebraic equivalency of bimetric theory and its vielbein formulation up to a one-to-one map relating the respective parameter spaces over the real numbers. Sammanfattning Den bimetriska teorin är en utökning av den allmänna relativitetsteorin som introduc- erar ett sekundärt symmetriskt tensorfält av rang-två. Det här sekundära spin-2 fältet är också dynamiskt, och för att undvika Boulware-Deser spöke, erhålls vaxelverkan mellan de två fältena genom en potential som er baserad på kvadratrotsmatris av två tensorfält. Den “kvadratroten” är en linjär avbildning som kallas kvadratrotsisometri. I detta arbete utforskas förutsättningar för existensen av kvadratrotsisometrin och ges dess egenskaper i termer av gruppteori. Därutöver utforskas förutsättningarna för den samtidiga 3+1 dekom- positionen av två tensorfält och sedan, i termer av ljuskoner, ges de (lokala) kausala relation- erna för tensorfält kopplade genom kvadratrotsisometrin. Slutligen bevisas den algebraiska ekvivalensen mellan den bimetriska teorin och dess vielbein formulering upp till en bijektiv relation mellan respektive parameterutrymmen över de reella talen. Contents Introduction iii Notation.......................................... vii 1 Quadratic Spaces, Fundamentals1 1.1 Definitions......................................1 1.1.1 Bilinear Forms...............................1 1.1.2 Quadratic Spaces..............................3 1.2 Matrix and Tensor Notation............................4 1.3 Regular Spaces...................................6 1.3.1 Canonical Dualities.............................6 1.3.2 Inner Product and Norm.........................7 1.4 Transpose and Adjoint Transformations.....................8 1.4.1 Transpose..................................8 1.4.2 Adjoint...................................8 1.5 Isometries and Conformal Maps.......................... 10 1.5.1 Isometries and Pullbacks......................... 10 1.5.2 Conformal Maps.............................. 11 1.6 Orthogonality and Orthogonal Splitting..................... 12 1.6.1 Orthogonal Group............................. 13 1.7 Isotropy and Hyperbolic Splitting......................... 14 1.8 Orthogonal Decomposition............................ 15 1.9 Classification of Real Quadratic Spaces...................... 17 2 The Square-Root Isometry 19 2.1 Coupled Quadratic Spaces............................. 19 2.2 The Square-Root Isometry............................. 20 2.3 The Square-Root Group Actions and Orbits................... 23 2.4 Coupled Symmetrization.............................. 27 3 Geometry of Decompositions 29 3.1 Structure of O(m, n) ................................ 29 3.1.1 Polar Decomposition of O(n) ....................... 30 3.1.2 Polar Decomposition of O(m, n) ..................... 31 3.2 Partitioned Matrices................................ 34 3.3 m+n Decomposition of Quadratic Forms.................... 36 3.4 Vielbeins in Differential Geometry........................ 39 i Mikica Kocic, The Square-Root Isometry, MSc thesis, SU 2014 Printed: 2015-03-11 ii CONTENTS 3.5 Decomposition of Vielbeins............................ 42 3.6 Quadrics....................................... 44 4 Symmetrization Condition 47 4.1 Decomposition of the Symmetrization Condition................ 47 4.2 The Symmetrizing Quadratic Space....................... 50 4.3 Topology of the Symmetrization Condition................... 50 5 Simultaneous Proper Decomposition 53 5.1 Alternative Square-Roots............................. 54 5.2 Reflection Group of the Square-Root Isometry.................. 56 5.3 Symmetrizing Space Covered by Null-Cones................... 59 5.4 Intersecting Null-Cones.............................. 60 5.5 Simultaneous Diagonalization by Congruence.................. 61 6 Application 65 6.1 Review of Bimetric Theory............................ 65 6.2 Algebraic Relation to Vielbein Formulation................... 69 7 Summary and Discussion 73 Appendix A Tedious Derivations 75 References 81 Index 87 Introduction For almost a century it has been known that the theory of general relativity (GR) describes the gravitational force with immaculate agreement with observations. Likewise, as a pillar of the standard model of particle physics (SM), since around the 1950s it has been widely accepted that physical laws are shaped in the form of a quantum theory of fields (QFT). The significance of these theories, upon which all modern physics rests, is well pointed out by Steven Weinberg [Wei95]: “If it turned out that some physical system could not be described by a quantum field theory, it would be a sensation; if it turned out that the system did not obey the rules of quantum mechanics and relativity, it would be a cataclysm.” Nonetheless, it is evident that we need the theoretical developments which are beyond the standard model and general relativity. The origins of dark energy and dark matter are just to name some of the unresolved topics in present-day physics. One of the approaches towards new physics is to build novel theoretical models from a top-down perspective, like string the- ory, which might eventually lead us to the ultimate theory of nature. From this viewpoint, GR and the contemporary quantum field theories are mere low-energy approximations and ‘effective field theory’ limits of a more fundamental theory. However, although our current field theories are certainly not ‘fundamental,’ we know that any relativistic quantum theory will take the form of a field theory when describing particles at sufficiently low energies [Wei95]. An equally viable approach is therefore to reuse the existing tools of field theories, and then from a bottom-up perspective search for a consistent theory of interacting spin-2 fields which will extend general relativity. One such classical field theory is a ghost-free bi- metric theory, which is an extension to GR that introduces a dynamical secondary symmetric rank-two tensor field. The meaning of ‘ghost-free’ and ‘spin-2 field’ will be highlighted by reviewing the methodology used for constructing a quantum field theory. In general, fields are physical quantities that have a value at each point in spacetime. Subject to whether these values are characterized by numbers or quantum operators, we can distinguish between classical and quantum fields. The energy of the system, which is represented by the Hamiltonian, has a crucial role for the consistency of a field theory. Provided that we have a consistent classical field for which the Hamiltonian has a lower bound (that is, for which the energy of the system is bounded from below), the process of quantization is ‘fairly’ simple either using the canonical quantization or the path integrals formalism; this process will consequently convert classical fields into operators acting on quantum states of QFT.1 However, if the Hamiltonian is not bounded from below, a classical 1Of course, one must be careful not to introduce inconsistencies and violate the unitarity during quanti- zation (which is not always possible). iii Mikica Kocic, The Square-Root Isometry, MSc thesis, SU 2014 Printed: 2015-03-11 iv INTRODUCTION field will be inherently unstable allowing the transfer of an infinite amount of energy to coupled fields, which is considered unphysical. In such case, the negative kinetic term will give rise to the negative probability states, i.e., to the loss of unitarity in the resulting quantum theory. Therefore, a consistent classical field with a Hamiltonian having the ‘healthy’ kinetic term is a necessary condition for a consistent QFT. A field with the inconsistent kinetic term bearing the wrong sign is called a ghost field.2 After constructing a QFT, we can formally define particles as “representations of the inhomogeneous Lorentz group” [Wig39, Wei95]. This rather cumbersome statement is equivalent to saying that particles are represented by the degrees of freedom in the flat spacetime characterized by mass and spin (or helicity), and carried by fields as quanta. What is important is that the origin of the classification by mass and spin are the space- time symmetries of a field relative to inhomogeneous Lorentz transformations. In particular, depending on whether the value of a specific field at each point of spacetime is given by (or, transforms as) a scalar, a vector, a
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