Physics Reports 589 (2015) 1–71 Contents lists available at ScienceDirect Physics Reports journal homepage: www.elsevier.com/locate/physrep Spacetime algebra as a powerful tool for electromagnetism Justin Dressel a,b,∗, Konstantin Y. Bliokh b,c, Franco Nori b,d a Department of Electrical and Computer Engineering, University of California, Riverside, CA 92521, USA b Center for Emergent Matter Science (CEMS), RIKEN, Wako-shi, Saitama, 351-0198, Japan c Interdisciplinary Theoretical Science Research Group (iTHES), RIKEN, Wako-shi, Saitama, 351-0198, Japan d Physics Department, University of Michigan, Ann Arbor, MI 48109-1040, USA article info a b s t r a c t Article history: We present a comprehensive introduction to spacetime algebra that emphasizes its Accepted 3 June 2015 practicality and power as a tool for the study of electromagnetism. We carefully develop Available online 9 June 2015 this natural (Clifford) algebra of the Minkowski spacetime geometry, with a particular focus editor: D.K. Campbell on its intrinsic (and often overlooked) complex structure. Notably, the scalar imaginary that appears throughout the electromagnetic theory properly corresponds to the unit Keywords: 4-volume of spacetime itself, and thus has physical meaning. The electric and magnetic Spacetime algebra fields are combined into a single complex and frame-independent bivector field, which Electromagnetism Dual symmetry generalizes the Riemann–Silberstein complex vector that has recently resurfaced in studies Riemann–Silberstein vector of the single photon wavefunction. The complex structure of spacetime also underpins Clifford algebra the emergence of electromagnetic waves, circular polarizations, the normal variables for canonical quantization, the distinction between electric and magnetic charge, complex spinor representations of Lorentz transformations, and the dual (electric–magnetic field exchange) symmetry that produces helicity conservation in vacuum fields. This latter symmetry manifests as an arbitrary global phase of the complex field, motivating the use of a complex vector potential, along with an associated transverse and gauge-invariant bivector potential, as well as complex (bivector and scalar) Hertz potentials. Our detailed treatment aims to encourage the use of spacetime algebra as a readily available and mature extension to existing vector calculus and tensor methods that can greatly simplify the analysis of fundamentally relativistic objects like the electromagnetic field. ' 2015 Elsevier B.V. All rights reserved. Contents 1. Introduction.............................................................................................................................................................................................3 1.1. Motivation...................................................................................................................................................................................3 1.2. Insights from the spacetime algebra approach.........................................................................................................................5 2. A brief history of electromagnetic formalisms......................................................................................................................................7 3. Spacetime algebra...................................................................................................................................................................................9 3.1. Spacetime.................................................................................................................................................................................... 10 3.2. Spacetime product...................................................................................................................................................................... 10 3.3. Multivectors ................................................................................................................................................................................ 12 3.3.1. Bivectors: products with vectors................................................................................................................................ 14 3.3.2. Reversion and inversion.............................................................................................................................................. 15 ∗ Corresponding author at: Department of Electrical and Computer Engineering, University of California, Riverside, CA 92521, USA. E-mail address: [email protected] (J. Dressel). http://dx.doi.org/10.1016/j.physrep.2015.06.001 0370-1573/' 2015 Elsevier B.V. All rights reserved. 2 J. Dressel et al. / Physics Reports 589 (2015) 1–71 3.4. Reciprocal bases, components, and tensors.............................................................................................................................. 16 3.5. The pseudoscalar I, Hodge duality, and complex structure..................................................................................................... 17 3.5.1. Bivectors: canonical form............................................................................................................................................ 18 3.5.2. Bivectors: products with bivectors............................................................................................................................. 18 3.5.3. Complex conjugation................................................................................................................................................... 19 3.6. Relative frames and paravectors................................................................................................................................................ 19 3.6.1. Bivectors: spacetime split and cross product ............................................................................................................ 20 3.6.2. Paravectors................................................................................................................................................................... 21 3.6.3. Relative reversion........................................................................................................................................................ 22 3.7. Bivectors: commutator bracket and the Lorentz group ........................................................................................................... 23 3.7.1. Spinor representation.................................................................................................................................................. 25 3.8. Pauli and Dirac matrices............................................................................................................................................................. 25 4. Spacetime calculus.................................................................................................................................................................................. 26 4.1. Directed integration ................................................................................................................................................................... 27 4.2. Vector derivative: the Dirac operator........................................................................................................................................ 27 4.2.1. Coordinate-free definition .......................................................................................................................................... 28 4.2.2. The relative gradient and d'Alembertian ................................................................................................................... 28 4.2.3. Example: field derivatives........................................................................................................................................... 28 4.3. Fundamental theorem................................................................................................................................................................ 29 4.3.1. Example: Cauchy integral theorems .......................................................................................................................... 29 5. Maxwell's equation in vacuum .............................................................................................................................................................. 30 5.1. Relative frame form.................................................................................................................................................................... 30 5.2. Global phase degeneracy and dual symmetry.......................................................................................................................... 32 5.3. Canonical form............................................................................................................................................................................ 33 5.4. Electromagnetic waves............................................................................................................................................................... 34 6. Potential representations ....................................................................................................................................................................... 36 6.1. Complex vector potential..........................................................................................................................................................