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Geometric Methods in Mathematical Physics I: Multi-Linear Algebra, Tensors, Spinors and Special Relativity

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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 . 48 4 Some application to general group theory 51 4.1 Groups . 51 4.1.1 Basic notions on groups and matrix groups . 51 4.1.2 Direct product and semi-direct product of groups . 53 4.2 Tensor products of group representations . 54 4.2.1 Linear representation of groups and tensor representations . 54 4.2.2 An example from Continuum Mechanics . 58 4.2.3 An example from Quantum Mechanics . 60 2 4.3 Permutation group and symmetry of tensors . 60 4.4 Grassmann algebra, also known as Exterior algebra (and something about the space of symmetric tensors) . 65 4.4.1 The exterior product and Grassmann algebra . 68 4.4.2 Interior product . 74 5 Scalar Products and Metric Tools 77 5.1 Scalar products . 77 5.2 Natural isomorphism between V and V ∗ and metric tensor . 80 5.2.1 Signature of pseudo-metric tensor, pseudo-orthonormal bases and pseudo- orthonormal groups . 83 5.2.2 Raising and lowering indices of tensors . 86 6 Pseudo tensors and tensor densities 89 6.1 Orientation and pseudo tensors . 89 6.2 Ricci's pseudo tensor . 91 6.3 Tensor densities . 95 7 Polar Decomposition Theorem in the finite-dimensional case 98 7.1 Operators in spaces with scalar product . 98 7.1.1 Positive operators . 102 7.1.2 Complexification procedure. 104 7.1.3 Square roots . 105 7.2 Polar Decomposition . 108 8 Special Relativity: a Geometric Presentation 109 8.1 Minkowski spacetime . 109 8.1.1 General Minkowski spacetime structure . 110 8.1.2 Time orientation . 111 8.2 Kinematics I: basic notions . 114 8.2.1 Curves as histories or world lines . 114 8.2.2 Minkowskian reference systems and some kinematical effect . 115 8.3 Kinematics II: Lorentz and Poincar´egroups . 121 8.3.1 The Lorentz group . 122 8.3.2 The Poincar´egroup . 125 8.3.3 Coordinate transformations between inertial reference systems . 126 8.3.4 Special Lorentz transformations . 127 8.3.5 Contraction of volumes . 128 8.4 Dynamics I: material points . 130 8.4.1 Relativistic causality and locality . 131 8.4.2 The geometric nature of the relativity principle in the formulation of rel- ativistic dynamics . 132 3 8.4.3 Mass and four momentum . 132 8.4.4 Massless particles . 133 8.4.5 The principle of inertia . 134 8.4.6 Four momentum conservation law . 135 8.4.7 The interpretation of P 0 and the Mass-Energy equivalence . 137 8.4.8 The notion of four force . 140 8.4.9 The relativistic vis viva equation . 141 8.5 Dynamics II: the stress-energy tensor from a macroscopic approach . 142 8.5.1 The non-interacting gas of particles . 142 8.5.2 Mass conservation law in local form for the non-interacting gas . 144 8.5.3 Four-momentum conservation in local form for the non-interacting gas . 145 8.5.4 The stress energy tensor in the general case . 152 8.5.5 The stress-energy tensor of the ideal fluid and of the Electromagnetic field 156 9 Lorentz group decomposition theorem 159 9.1 Lie group structure, distinguished subgroups, and connected components of the Lorentz group . 159 9.2 Spatial rotations and boosts . 161 9.2.1 Spatial rotations . 161 9.2.2 Lie algebra analysis . 162 9.2.3 Boosts . 165 9.2.4 Decomposition theorem for SO(1; 3)" . 168 9.2.5 Proof of decomposition theorem by polar decomposition of SO(1; 3)" . 168 10 The interplay of SL(2; C) and SO(1; 3)" 171 10.1 Elementary properties of SL(2; C) and SU(2) . 171 10.1.1 Almost all on Pauli matrices . 171 10.1.2 Properties of exponentials of Pauli matrices and consequences for SL(2; C) and SU(2) . 173 10.1.3 Polar decomposition theorem and topology of SL(2; C) . 176 10.2 The interplay of SL(2; C) and SO(1; 3)" . 177 10.2.1 Construction of the covering homomorphism Π : SL(2; C) ! SO(1; 3)" . 177 10.2.2 Properties of Π . 179 11 Spinors 183 11.1 The space of Weyl spinors . 183 11.1.1 The metric spinor to lower and raise indices . 184 11.1.2 The metric spinor in the conjugate spaces of spinors . 185 11.1.3 Orthonormal bases and the role of SL(2; C) . 185 11.2 Four-Vectors constructed with spinors in Special Relativity . 187 4 Chapter 1 Introduction and some useful notions and results The content of these lecture notes covers the first part of the lectures of a graduate course in Modern Mathematical Physics at the University of Trento. The course has two versions, one is geometric and the other is analytic. These lecture notes only concern the geometric version of the course. The analytic version regarding applications to linear functional analysis to quantum and quantum relativistic theories is covered by my books [Morettia], [Morettib] and the chapter [KhMo15]. The idea of this first part is to present a quick, but rigorous, picture of the basics of tensor calculus for the applications to mathematical and theoretical physics. In particular I try to cover the gap between the traditional presentation of these topics given by physicists, based on the practical indicial notation and the more rigorous (but not always useful in the practice) approach presented by mathematicians. Several applications are presented as examples and exercises. Some chapters concern the geometric structure of Special Relativity theory and some theoretical issues about Lorentz group. The reader is supposed to be familiar with standard notions of linear algebra [Lang, Sernesi], especially concerning finite dimensional vector spaces. Since the end of Chapter 8 some basic tools of Lie group theory and Lie group representation theory [KNS] are requested. The defini- tion of Hilbert tensor product given at the end of Chapter 2 has to be seen as complementary material and requires that the reader is familiar with elementary notions on Hilbert spaces. Some notions and results exploited several times throughout the text are listed here. First of all, we stress that the notion of linear independence is always referred to finite linear combinations. A possibly infinite subset A ⊂ V of vectors of the vector space V over the field K = C or R is said to be made of linearly independent elements, if for avery finite subset A0 ⊂ A, X cvv = 0 for some cv 2 K, v2A0 5 entails 0 cv = 0 for all v 2 A . Clearly such A cannot contain the zero vector. We remind the reader that, as a consequence of Zorn's lemma and as proved below, every vector space V admits a vector basis, that is a possibly infinite set B ⊂ V made of linearly independent elements such that every v 2 V can be written as n X v = civi i=1 for some finite set of (distinct) elements v1; : : : ; vn 2 B depending on v and a corresponing set of (not necessarily pairwise distinct) scalars c1; : : : ; cn 2 K n f0g depending on v. The sets of elements v1; : : : ; vn 2 B and c1; : : : ; cn 2 K n f0g are uniquely fixed by v itself in view of the 0 0 linear independence of the elements of B. Indeed, if for some distinct elements v1; : : : ; vm 2 B 0 0 and (not necessarily pairwise distinct) c1; : : : ; cm 2 K n f0g it again holds m X 0 0 v = civi ; i=1 we also have n m X X 0 0 = v − v = civi − cjvj : i=1 j=1 0 0 In view of the linear independence of the set fv1; : : : ; vn; v1; : : : ; vmg whose primed and non- 0 primed elements are not necessarily distinct, observing that ci 6= 0 and cj 6= 0 the found identity 0 implies that n = m, the set of vis must coincide with the set of vjs and the corresponding 0 coeffients ci and cj must be equal.
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