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Pos(CORFU2018)100 -flux R -flux Back- R ∗ EMPG-19-10 R.J.Szabo@Hw.Ac.Uk Speaker An Introduction to Nonassociative Physics PoS(CORFU2018)100 Richard J. Szabo∗ Department of Mathematics, Heriot-Watt University, Edinburgh, United Kingdom. Maxwell Institute for Mathematical Sciences, Edinburgh, United Kingdom. The Higgs Centre for Theoretical Physics, Edinburgh, United Kingdom. E-mail: [email protected] We give a pedagogical introduction to the nonassociative structures arising from recent develop- ments in quantum mechanics with magnetic monopoles, in string theory and M-theory with non- geometric fluxes, and in M-theory with non-geometric Kaluza-Klein monopoles. After a brief overview of the main historical appearences of nonassociativity in quantum mechanics, string theory and M-theory, we provide a detailed account of the classical and quantum dynamics of electric charges in the backgrounds of various distributions of magnetic charge. We apply Born reciprocity to map this system to the phase space of closed strings propagating in R-flux back- grounds of string theory, and then describe the lift to the phase space of M2-branes in R-flux backgrounds of M-theory. Applying Born reciprocity maps this M-theory configuration to the phase space of M-waves probing a non-geometric Kaluza-Klein monopole background. These four perspective systems are unified by a covariant 3-algebra structure on the M-theory phase space. Corfu Summer Institute 2018 "School and Workshops on Elementary Particle Physics and Gravity" (CORFU2018) 31 August–28 September 2018 Corfu, Greece Preprint: EMPG-19-10 ∗Speaker. c Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). https://pos.sissa.it/ An Introduction to Nonassociative Physics Richard J. Szabo Contents 1. A brief history of nonassociativity in physics 2 1.1 Nonassociative algebra 2 1.2 The octonions 2 1.3 Jordanian quantum mechanics 4 1.4 Nambu mechanics 5 PoS(CORFU2018)100 1.5 Nonassociativity in string theory 6 1.6 Nonassociativity in M-theory 8 1.7 Outline 10 2. Electric charges in magnetic monopole backgrounds 11 2.1 Magnetic Poisson brackets 11 2.2 Magnetic monopoles 12 2.3 Classical motion in fields of magnetic charge 14 2.4 Quantization of magnetic Poisson brackets 16 2.5 Nonassociative quantum mechanics 20 3. Closed strings in locally non-geometric flux backgrounds 23 3.1 The R-flux model 23 3.2 How closed strings see nonassociativity 25 3.3 Nonassociative gravity 26 4. M2-branes in locally non-geometric flux backgrounds 27 4.1 M-theory lift of the R-flux model 27 4.2 Quantization of the M2-brane phase space 29 5. M-waves in non-geometric Kaluza-Klein monopole backgrounds 31 5.1 Magnetic monopoles in quantum gravity 31 5.2 The M-wave phase space 32 5.3 The covariant M-theory phase space 3-algebra 33 1 An Introduction to Nonassociative Physics Richard J. Szabo 1. A brief history of nonassociativity in physics 1.1 Nonassociative algebra Nonassociative algebras first appeared around the middle of the nineteenth century and have since formed an independent branch of mathematics. By far the simplest and best known example of a nonassociative algebra is a Lie algebra. A Lie bracket [·; ·] on a vector space is generally a noncommutative and nonassociative binary operation in the sense that [x;y] 6= [y;x] and [x;[y;z]] 6= [[x;y];z], but rather it is antisymmetric and satisfies the Jacobi identity. If we package the quantity which vanishes by the Jacobi identity into a ternary operation called the ‘Jacobiator’, PoS(CORFU2018)100 1 [x;y;z] := 3 [x;[y;z]] − [[x;y];z] + [[x;z];y] ; (1.1) then Lie algebras are characterized by the feature that this 3-bracket vanishes: [x;y;z] = 0. In our present context this will be considered to be a trivial example of a nonassociative algebra, as the binary operation on the Lie group which integrates the Lie algebra via the exponential map is associative. In fact, in this paper we will be looking at cases of algebras with a multiplication that has non-vanishing Jacobiators, just like the multiplication in a noncommutative algebra can be characterized by a non-vanishing commutator [x;y] := xy − yx 6= 0. A well-known example of a noncommutative and nonassociative algebra in this sense is the algebra of octonions, which con- stitutes an important example of an ‘alternative algebra’, where associativity of the multiplication (and hence the Jacobi identity) is generally violated, but it possesses the alternativity condition which generalizes the Jordan identity (x2 y)x = x2 (yx) of (commutative) Jordan algebras, which we review below. Algebras whose associativity is controlled by identities like the Jacobi or alternative identities were of central mathematical interest in the last century, as arbitrarily relaxing associativity in an algebra does not lead to a good mathematical theory unless some other structures are present, see e.g. [1]. They have recently made their way into the forefront of some recent developments in quantum mechanics, string theory and M-theory. The main aim of these lectures is to survey some of these recent insights and to highlight some of the interesting physical consequences of the theory, together with the many important open avenues awaiting further investigation. Nonassociative algebras actually have a long and diverse history of appearences in physics, which is perhaps not so widely appreciated. The purpose of this opening section is to briefly review some of their occurences with a personally biased point of view: We only discuss develop- ments in physics with an eye to the main topics that we will pursue later on, and to overview the developments that we shall treat in more detail in subsequent sections. 1.2 The octonions For later use, let us begin by recalling the standard mathematical example of the octonions, as it will play an important role in some of the discussions throughout this paper (see e.g. [2] for a nice introduction). A well-known theorem in algebra states that there are only four normed division algebras over the field R of real numbers: The real numbers R themselves, the complex numbers C, the quaternions H, and the octonions O. Both R and C are commutative and associative algebras, H is noncommutative but associative, while O is both noncommutative and nonassociative. The 2 An Introduction to Nonassociative Physics Richard J. Szabo algebra is generated over itself by a central identity element 1, is generated over by 1 and an R p C R imaginary complex unit i = −1, while H is generated by 1 and three imaginary quaternion units i, j and k, familar from quantum mechanics in their representation in terms of Pauli spin matrices p 2 ei = −1si, i = 1;2;3, which obey ei = −1 and ei e j = −di j + ei jk ek ; (1.2) where ei jk is the Levi-Civita symbol in three dimensions. Analogously, the algebra O is generated by 1 and seven imaginary octonion units eA, A = PoS(CORFU2018)100 2 1;:::;7, satisfying eA = −1, so that a generic element O of O is of the form O = a0 + a1 e1 + a2 e2 + ··· + a7 e7 ; (1.3) with a0;a1;:::;a7 2 R. The multiplication in the algebra O can be written in the form eA eB = −dAB + hABC eC ; (1.4) where the structure constants hABC form a completely antisymmetric tensor with only seven non- vanishing components, generalizing the structure constants ei jk (with only one non-zero compo- nent) of the algebra of quaternions H. To specify hABC explicitly, it is convenient to rewrite e4;e5;e6 = f1; f2; f3 and represent the algebra of octonions in terms of the commutators [ei;e j] = 2ei jk ek and [e7;ei] = 2 fi ; [ fi; f j] = −2ei jk ek and [e7; fi] = −2ei ; (1.5) [ei; f j] = 2(di j e7 − ei jk fk) : The first set of commutation relations describes a quaternion subalgebra H ⊂ O (the Lie algebra of SU(2) ' SO(3)), analogous to the hierarchy of embeddings R ⊂ C ⊂ H generated by the inclusions of 1 and i; this follows from the Cayley-Dickson construction of the real normed division algebras. This demonstrates the important feature that, unlike H whose structure constants are invariant under the full three-dimensional rotation group SO(3), the symmetry group of the tensor hABC is only the 14-dimensional simple exceptional Lie subgroup G2 ⊂ SO(7) of the rotation group in seven dimensions. Using these commutators one can derive the non-vanishing Jacobiators [eA;eB;eC] = −4hABCD eD ; (1.6) with antisymmetric structure constants hABCD also having only seven non-vanishing components. The alternative property of the algebra O is then equivalent to the statement that its Jacobiators are proportional to its ‘associators’: [eA;eB;eC] = 2 (eA eB)eC − eA (eB eC) : (1.7) 3 An Introduction to Nonassociative Physics Richard J. Szabo 1.3 Jordanian quantum mechanics The algebra of octonions O is an example of what may be called a noncommutative Jordan algebra, which we now proceed to define. Jordan algebras were the first appearence of nonasso- ciativity in physics, in the early days of quantum theory [3]. Jordan’s motivation for introducing them was to define an algebraic structure on the set of observables in quantum mechanics, and hence an overall new algebraic setting for the foundations of quantum theory. The observables of any quantum system are given by Hermitian elements in a C∗-algebra, which can be represented by operators on a finite- or infinite-dimensional separable Hilbert space. If A and B are Hermi- tian operators, then their operator product is not a Hermitian operator (unless they commute), and PoS(CORFU2018)100 neither is their commutator.
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