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Nonrelativistic Symmetries faculteit Wiskunde en Natuurwetenschappen Nonrelativistic symmetries. Bacheloronderzoek Wis- en Natuurkunde, deel 1: Wiskunde Augustus 2012 Student: P.L.Los Begeleider natuurkunde: D. Boer Begeleider wiskunde: J. Top 1 Introduction Symmetry is, in physics, the invariance of properties of a physical system under coordinate trans- formations of this system. This symmetry often can be used to deduce properties of the system. The main purpose of this bachelor's thesis is, besides giving a general mathematical description of continuous symmetries and Lie groups, to provide the correct mathematical language to study some questions about different physical systems with nonrelativistic symmetries. One of these questions, which will be studied not in this thesis but in a continuation on it, is: What is the exact physical meaning of the central extension with a certain generator M of the Lie algebra of the Schr¨odingergroup? It has something to do with mass, but what does it mean exactly? Similar questions are about the physical meaning of subgoups of the Lorenz group and very spe- cial relativity and about examples of nonrelativistic physical systems and systems with Lifshitz symmetry. In this thesis we first will descibe what symmetry exactly is. Then we will consider continuous symmetries: what are they and how to describe them mathematically? This will bring us to the important concepts of topological group, Lie group and Lie algebra. Further we want to describe Noether's theorem about the relation between symmetry and conser- vation laws. As an important physical example of a system with continuous symmetry, we will derive the maximal symmetry group of a free nonrelativistic point particle, which will be found to be the Schr¨odingergroup. This example is important in studying the question above about the physical meaning of a central extension of the Schr¨odingergroup. Finally we will discuss a list of some common groups and relations between them. 2 What is symmetry? In physics, symmetry is invariance of certain properies of a system under certain transformations of this system. For example: the area of a rectangle (not a square) is invariant under every rota- tion of this rectangle, the rectangle itself is only symmetric under rotations of 180 degrees. y y x x (x; y) −! (x cos θ − y sin θ; y cos θ + x sin θ) Rotation of a rectangle: The rectangle is not invariant under this rotation, but the area of the rectangle stays invariant. Many, but not all symmetries are geometrical symmetries, which means that the symmetry trans- formations are space-time coordinate transformations (for example space-time translations, rota- tions, scalings). The different invertible symmetry transformations of a system form a group: The identity transformation is the identity element of this group, the composition of two invertible symmetry transformations is again an invertible symmetry transformation, and the inverse of a symmetry transformation is a symmetry transformation. 2 2.1 Continuous symmetry and Lie groups: What is continuous symmetry, and how can continuous symmetries be described by Lie groups? Continuous symmetry can be imagined as invariance under \very small" coordinate transforma- tions, in contrast to discrete symmetry such as for example reflection symmetry. More precisely: continuous symmetry is invariance under transformations which depend continu- 2 ously on a set of parameters. For example the translations (x; y) 7! (x + α1; y + α2) in R depends continuous on the parameters α1 and α2. Like ordinary symmetry transformations form an ordinary group, the symmetry transformations of a continuous symmetry form a topological group: A topological group is a topological space in which the elemens form a group, where the compo- sition and the inversion map are continuous functions. Example: (simplified) gravity: Under which spatial coordinate transformations is the gravitational energy of an object invariant? Suppose the gravitational energy of an object is given by the following volume integral: R 3 E = − V (ρ(~x) ∗ g ∗ z)d~x , with ρ(x; y; z) > 0 the mass density, and g the gravitation constant. Then for a coordinate transformation of the form (x; y; z) ! (x − ∆x; y − ∆y; z − ∆z) (a trans- ~ R lation), we have for the gravitational energy in the transformed system: E = − V (~ρ(x − ∆x; y − ∆y; z −∆z)∗g∗(z −∆z))d~x3 withρ ~ the mass density function in the transformed system. Because the object itself is not changed, we haveρ ~(x − ∆x; y − ∆y; z − ∆z) = ρ(x; y; z), and hence: Z E~ = − (ρ(x; y; z) ∗ g ∗ (z − ∆z))d~x3 V Therefore, the gravitational energy is invariant under this transformation if and only if there is no translation in the z-direction. In formulas: ~ R 3 E = E () V (ρ(~x)∆z)d~x = 0 () ∆z = 0. Hence this gravitational energy is invariant under translation transformations which leave z in- variant, these symmetry transformations depend continuously on the parameters ∆x and ∆y. The topological group of symmetry transformations of a continuous symmetry often is a Lie-group. A Lie group is a smooth manifold, from which the elements form a group where the composition and inversion map are not only continuous but even smooth functions. 2.2 Group Generators and the Lie Algebra A system with symmetry transformations which form a Lie group is invariant under “infinitesimal small transformations". Such infinitesimal small transformations can be described by linearising the symmetry transformations of the system around the identity transformation. Suppose we ~ ~ have a coordinate transformation ~x ! ξ(~x;α1; α2; :::), where ξ depends differentiable on the set of ~ parameters ~α = α1; α2;::: , and ξ(~x;~0) = ~x is the identity transformation. We can linearise this transformation around the identity transformation, the linearized transformation ~x ! ζ~(~x;~α) is ~ ~ ~ @ξ~(~x;~a) then given by: ζ(~x;~α) = ξ(~x; 0) + ( @~a j~a=~0)~α. The infinitesimal generators of the group are now given by: n o ζ~(~x; 1; 0;:::; 0) − ~x; ζ~(~x; 0; 1;:::; 0) − ~x; : : : ; ζ~(~x; 0; 0;:::; 1) − ~x ( ) @ξ~(~x;~α) @ξ~(~x;~α) @ξ~(~x;~α) = ( j )(1; 0;:::; 0); ( j )(0; 1;:::; 0);:::; ( j )(0; 0;:::; 1) @α ~α=~0 @α ~α=~0 @α ~α=~0 This set of generators can be seen as a basis of a vectorspace V which is the tangent space to the Lie group in ~x. We now will give the definition of Lie algebra to be able to describe what the Lie algebra of a group is. A Lie algebra s is a vectorspace X over some field F together with some operation [:; :]: X×X ! X 3 which is called the Lie bracket and which satisfies the following properties: Bilinearity: [α~x+β~y; ~z] = α[~x;~z]+β[~y; ~z] and [~z; α~x+β~y] = α[~z; ~x]+β[~z; ~y] 8α; β 2 F; 8~x;~y; ~z 2 X; Alternating property: [~x;~x] = 0; Jacobi identity: [[~x;~y]; ~z] + [[~y; ~z]; ~x] + [[~z; ~x]; ~y] = 0. For a matrix group, the Lie algebra is given by the vectorspace spanned by the infinitesimal generators of the group, with the Lie bracket given by the commutator: [A; B] = AB − BA. 2.3 Example: Rotations (in 3 dimensions): As an example of a Lie group and its Lie algebra we will look at the group of rotations in 3 dimensions: An arbitrary rotation in 3 dimensions is given by the coordinate transformation: ~x ! A~x, with A an orthogonal matrix with det(A) = 1. Which is: A 2 SO(3) = fA 2 GL(3) j AT A = I; det(A) = 1g. (GL(3) is the general linear group in 3 dimensions, which is: the group of all 3x3 invertible ma- trices, with matrix multiplication as group operation) The group of all rotations in 3 dimensions is a 3-dimensional Lie group. We want to linearize this group around the identity element to get the Lie algebra of this group. A rotation over an angle 'z around the z-axis is given by: 0 1 cos 'z − sin 'z 0 @sin 'z cos 'z 0A 0 0 1 We can linearise this with respect to 'z: 0 0cos ' − sin ' 01 1 0 1 −' 01 @ z z z I + ' sin ' cos ' 0 j = ' 1 0 z @@' @ z z A 'z =0A @ z A z 0 0 1 0 0 1 The same we can do for rotations 'x and 'y around resp. the x- and the y-axis. Hence, lineari- sation of the rotations around the identity gives: 0 1 1 −'z 'y @ 'z 1 −'xA −'y 'x 1 In - more or less - the same way, the infinitesimal generator of rotations around the z-axis is given by: 0cos ' − sin ' 01 00 −1 01 @ z z J = sin ' cos ' 0 j = 1 0 0 z @' @ z z A 'z =0 @ A z 0 0 1 0 0 0 And analogue the infintitesimal generators around the x- and the y-axis are given by: 00 0 0 1 0 0 0 11 Jx = @0 0 −1A ;Jy = @ 0 0 0A 0 1 0 −1 0 0 The Lie algebra of the group of rotations is generated by Jx, Jy and Jz with the commutation relations: 4 [Jx;Jy] = JxJy − JyJx 00 0 0 1 0 0 0 11 0 0 0 11 00 0 0 1 = @0 0 −1A @ 0 0 0A − @ 0 0 0A @0 0 −1A 0 1 0 −1 0 0 −1 0 0 0 1 0 00 −1 01 = @1 0 0A 0 0 0 = Jz Similarly: [Jy;Jz] = Jx and [Jz;Jx] = Jy. 3 Noether's Theorem[24];[32];[12];[35] 3.1 Noether's Theorem and its proof Noether's Theorem is about the relation between symmetry and conservation laws.
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