|
|
| Line 1: |
Line 1: |
| {{Group theory sidebar |Topological}}
| | If you've gained a few pounds in recent years, you may be wondering, "How do I lose fat?" And, you possibly desire to do so fast plus easily. It's additionally a common (plus usually correct) perception which if you need to get rid of fat, we do so slowly and carefully in purchase to be healthy regarding it. There is a method to lose fat, though, where you can do it swiftly plus be healthy about it -- plus have it be permanent also.<br><br>Ditch the Soda. Colas are loaded with glucose which leads to excess fat gain. Simply one cup of soda has about 180 calories while contributing no dietary value to a diet, thus imagine how numerous empty calories are inside a Big Gulp. Try switching your soda for water if you need to lose weight fast.<br><br>The right foods for weight loss are fresh all-natural food, nutritionally dense, high in fiber, low inside fat plus calories, low inside sodium and refined sugars. They also consist of high amounts of complex carbohydrates, fibers, good quality proteins and significant water content. Some of the greatest foods to eat inside purchase to lose weight are vegetables, fruits, legumes, baked potato, whole grain foods (such as wholemeal pasta, oatmeal, muesli, etc), boiled brown rice, baked potato and fresh fish.<br><br>With all programs you need to recognize which you need to be self disciplined plus follow it correctly. Here are some simple strategies for fat loss in a hurry.<br><br>Eat more usually. This really is possibly the greatest tip of the "7 techniques [http://safedietplansforwomen.com/how-to-lose-weight-fast lose weight] and easy". By eating small plus usually throughout the day, you will naturally speed up your metabolism, meaning you will burn calories more effectively. Of course, it doesn't indicate eat a 3-course meal, 6 occasions a day?snacks such as raisins, an apple, celery or carrot sticks are superior options and considering you are eating continuously, you are less probably to stray towards foods we should be limiting.<br><br>The fact of the matter is that despite that you need to lose pounds plus look better, you nonetheless have to eat. Where the problem comes in is not constantly how much we eat, but what we eat. That is the difference between burning fat or gaining more.<br><br>Please try other treatments plus food guides regarding eating right to help you see the proper system for you to lose fat swiftly. Here's the link to The Fat Loss Factor site should you want to test it. |
| {{Lie groups |Other}}
| |
| | |
| In [[physics]] (and [[mathematics]]), the '''Lorentz group''' is the [[Group (mathematics)|group]] of all [[Lorentz transformation]]s of [[Minkowski spacetime]], the [[classical field theory|classical]] setting for all (nongravitational) [[physics|physical phenomena]]. The Lorentz group is named for the [[Dutch people|Dutch]] physicist [[Hendrik Lorentz]].
| |
| | |
| The mathematical form of
| |
| *the [[kinematics|kinematical laws]] of [[special relativity]],
| |
| *[[Maxwell's equations|Maxwell's field equations]] in the theory of [[electromagnetism]],
| |
| *the [[Dirac equation]] in the theory of the [[electron]],
| |
| are each [[invariant (physics)|invariant]] under the [[Lorentz transformation]]s. Therefore the Lorentz group is said to express the fundamental [[symmetry]] of many of the known fundamental [[Laws of Nature]].
| |
| | |
| == Basic properties ==
| |
| | |
| The Lorentz group is a [[subgroup]] of the [[Poincaré group]], the group of all [[isometry|isometries]] of [[Minkowski spacetime]]. The Lorentz transformations are precisely the isometries which leave the origin fixed. Thus, the Lorentz group is an [[group action#Orbits and stabilizers|isotropy subgroup]] of the [[isometry group]] of Minkowski spacetime. For this reason, the Lorentz group is sometimes called the '''homogeneous Lorentz group''' while the Poincaré group is sometimes called the ''inhomogeneous Lorentz group''. Lorentz transformations are examples of [[linear transformations]]; general isometries of Minkowski spacetime are [[affine transformations]].
| |
| | |
| Mathematically, the Lorentz group may be described as the [[generalized orthogonal group]] O(1,3), the [[matrix Lie group]] which preserves the [[quadratic form]]
| |
| :<math> (t,x,y,z) \mapsto t^2-x^2-y^2-z^2</math>
| |
| on '''R'''<sup>4</sup>. This quadratic form is interpreted in physics as the [[metric tensor]] of Minkowski spacetime, so this definition is simply a restatement of the fact that Lorentz transformations are precisely the linear transformations which are also isometries of Minkowski spacetime.
| |
| | |
| The Lorentz group is a six-[[dimension]]al [[compact space|noncompact]] [[non-abelian group|non-abelian]] [[real Lie group]] which is not [[connected space|connected]]. All four of its [[connected component (topology)|connected component]]s are not [[simply connected]]. The [[identity component]] (i.e. the component containing the identity element) of the Lorentz group is itself a group and is often called the '''restricted Lorentz group''' and is denoted SO<sup>+</sup>(1,3). The restricted Lorentz group consists of those Lorentz transformations that preserve the [[orientation (mathematics)|orientation]] of space and direction of time. The restricted Lorentz group has often been presented through a facility of [[biquaternion]] algebra.
| |
| | |
| The restricted Lorentz group arises in other ways in pure mathematics. For example, it arises as the [[point symmetry group]] of a certain [[ordinary differential equation]]. This fact also has physical significance.
| |
| | |
| === Connected components ===
| |
| Because it is a [[Lie group]], the Lorentz group O(1,3) is both a [[group (mathematics)|group]] and a [[smooth manifold]]. As a manifold, it has four [[connected space|connected component]]s. Intuitively, this means that it consists of four topologically separated pieces.
| |
| | |
| Each of the four connected components can be categorized by which of these two properties its elements have:
| |
| * The element reverses the direction of time, or more precisely, transforms a future-pointing [[timelike vector]] into a past-pointing one.
| |
| * The element reverses the [[orientation (mathematics)|orientation]] of a [[vierbein]] (tetrad).
| |
| Lorentz transformations which preserve the direction of time are called '''{{visible anchor|orthochronous}}'''. The subgroup of orthochronous transformations is often denoted O<sup>+</sup>(1,3). Those which preserve orientation are called '''proper''', and as linear transformations they have determinant +1. (The improper Lorentz transformations have determinant −1.) The subgroup of proper Lorentz transformations is denoted SO(1,3).
| |
| | |
| The subgroup of all Lorentz transformations preserving both orientation and the direction of time is called the '''proper, orthochronous Lorentz group''' or '''restricted Lorentz group''', and is denoted by SO<sup>+</sup>(1, 3). (Note that some authors refer to SO(1,3) or even O(1,3) when they actually mean SO<sup>+</sup>(1, 3).)
| |
| | |
| The set of the four connected components can be given a group structure as the [[quotient group]] O(1,3)/SO<sup>+</sup>(1,3), which is isomorphic to the [[Klein four-group]]. Every element in O(1,3) can be written as the [[semidirect product]] of a proper, orthochronous transformation and an element of the [[discrete group]]
| |
| :{1, ''P'', ''T'', ''PT''}
| |
| where ''P'' and ''T'' are the [[P-symmetry|space inversion]] and [[T-symmetry|time reversal]] operators: | |
| :''P'' = diag(1, −1, −1, −1)
| |
| :''T'' = diag(−1, 1, 1, 1).
| |
| Thus an arbitrary Lorentz transformation can be specified as a proper, orthochronous Lorentz transformation along with a further two
| |
| bits of information, which pick out one of the four connected components. This pattern is typical of finite dimensional Lie groups.
| |
| | |
| == The restricted Lorentz group ==
| |
| | |
| The restricted Lorentz group is the [[identity component]] of the Lorentz group, which means that it consists of all Lorentz transformations which can be connected to the identity by a [[continuous function (topology)|continuous]] curve lying in the group. The restricted Lorentz group is a connected [[normal subgroup]] of the full Lorentz group with the same dimension, in this case with dimension six.
| |
| | |
| The restricted Lorentz group is generated by ordinary [[Coordinate rotation|spatial rotations]] and [[Lorentz boost]]s (which can be thought of as [[hyperbolic rotation]]s in a plane that includes a time-like direction). Since every proper, orthochronous Lorentz transformation can be written as a product of a rotation (specified by [[Charts on SO(3)|3 real parameters]]) and a boost (also specified by 3 real parameters), it takes 6 real parameters to specify an arbitrary proper orthochronous Lorentz transformation. This is one way to understand why the restricted Lorentz group is six dimensional. (See also the [[#Lie_algebra|Lie algebra of the Lorentz group]].)
| |
| | |
| The set of all rotations forms a [[Lie subgroup]] isomorphic to the ordinary [[rotation group SO(3)]]. The set of all boosts, however, does ''not'' form a subgroup, since composing two boosts does not, in general, result in another boost. (Rather, a pair of non-colinear boosts is equivalent to a boost and a rotation, and this relates to [[Thomas rotation]].) A boost in some direction, or a rotation about some axis, generates a [[one-parameter subgroup]].
| |
| | |
| === Relation to the Möbius group ===
| |
| {{Seealso|Algebra of physical space}}
| |
| | |
| The restricted Lorentz group SO<sup>+</sup>(1, 3) is [[isomorphic]] to the [[projective special linear group]] PSL(2,'''C'''), which is in turn isomorphic to the [[Möbius group]], the [[symmetry group]] of [[conformal geometry]] on the [[Riemann sphere]]. (This observation was utilized by [[Roger Penrose]] as the starting point of [[twistor theory]].)
| |
| | |
| This may be shown by constructing a [[surjective]] [[homomorphism]] of Lie groups from SL(2,'''C''') to SO<sup>+</sup>(1,3), which we will call the '''spinor map'''. This proceeds as follows:
| |
| | |
| We can define an action of SL(2,'''C''') on Minkowski spacetime by writing a point of spacetime as a two-by-two [[Hermitian matrix]] in the form
| |
| :<math> X = \left[ \begin{matrix} t+z & x-iy \\ x+iy & t-z \end{matrix} \right]. </math>
| |
| This presentation has the pleasant feature that
| |
| :<math> \det \, X = t^2 - x^2 - y^2 - z^2. </math>
| |
| Therefore, we have identified the space of Hermitian matrices (which is four dimensional, as a ''real'' vector space)
| |
| with Minkowski spacetime in such a way that the [[determinant]] of a Hermitian matrix is the squared length of the corresponding vector in Minkowski spacetime.
| |
| SL(2,'''C''') acts on the space of Hermitian matrices via
| |
| :<math> X \mapsto P X P^* </math>
| |
| where <math>P^*</math> is the [[Hermitian adjoint|Hermitian transpose]] of <math>P</math>, and this action preserves the determinant. Therefore, SL(2,'''C''') acts on Minkowski spacetime by (linear) isometries, and so is isomorphic to a subset of the Lorentz group by the definition of the Lorentz group.
| |
| | |
| This completes the proof that there is a [[homomorphism]] from SL(2,'''C''') to SO<sup>+</sup>(1,3). The [[kernel (group theory)|kernel]] of the spinor map is the two element subgroup ±''I''. By the [[Isomorphism_theorem#First_isomorphism_theorem|first isomorphism theorem]], the quotient group PSL(2,'''C''') is isomorphic to SO<sup>+</sup>(1,3).
| |
| | |
| In optics, this construction is known as the [[Polarization (waves)#Propagation, reflection and scattering|Poincaré sphere ]].
| |
| | |
| ==== Appearance of the night sky ====
| |
| | |
| This isomorphism has the consequence that Möbius transformations of the Riemann sphere represent
| |
| the way that Lorentz transformations change the appearance of the night sky, as seen by an observer who is maneuvering at [[special relativity|relativistic]] velocities relative to the "fixed stars".
| |
| | |
| Suppose the "fixed stars" live in Minkowski spacetime and are modeled by points on the celestial sphere. Then a given point on the celestial sphere can be associated with {{math|''ξ {{=}} u + iv''}}, a complex number that corresponds to the point on the [[Riemann sphere]], and can be identified with a [[null vector]] (a [[Minkowski_space#Causal_structure|light-like vector]]) in Minkowski space
| |
| :<math> \left[ \begin{matrix} u^2+v^2+1 \\ 2u \\ -2v \\ u^2+v^2-1 \end{matrix} \right] </math>
| |
| or the Hermitian matrix
| |
| :<math> N = 2\left[ \begin{matrix} u^2+v^2 & u+iv \\ u-iv & 1 \end{matrix} \right]. </math>
| |
| | |
| The set of real scalar multiples of this null vector, called a ''null line'' through the origin, represents a ''line of sight'' from an observer at a particular place and time (an arbitrary event which we can identify with the origin of Minkowski spacetime) to various distant objects, such as stars. Then the points of the [[celestial sphere]] (equivalently, lines of sight) are identified with certain Hermitian matrices.
| |
| | |
| === Conjugacy classes ===
| |
| Because the restricted Lorentz group SO<sup>+</sup>(1, 3) is isomorphic to the Möbius group PSL(2,'''C'''), its [[conjugacy classes]] also fall into five classes:
| |
| * '''elliptic''' transformations
| |
| * '''hyperbolic''' transformations
| |
| * '''loxodromic''' transformations
| |
| * '''parabolic''' transformations
| |
| * the trivial '''identity''' transformation
| |
| | |
| In the article on [[Möbius transformation]]s, it is explained how this classification arises by considering the [[fixed point (mathematics)|fixed point]]s of Möbius transformations in their action on the Riemann sphere, which corresponds here to [[null vector|null]] [[eigenspace]]s of restricted Lorentz transformations in their action on Minkowski spacetime.
| |
| | |
| An example of each type is given in the subsections below, along with the effect of the [[one-parameter subgroup]] which it generates (e.g., on the appearance of the night sky).
| |
| | |
| The Möbius transformations are the [[conformal map|conformal transformations]] of the Riemann sphere (or celestial sphere). Then conjugating with an arbitrary element of SL(2,'''C''') obtains the following examples of arbitrary elliptic, hyperbolic, loxodromic, and parabolic (restricted) Lorentz transformations, respectively. The effect on the '''flow lines''' of the corresponding one-parameter subgroups is to transform the pattern seen in the examples by some conformal transformation. For example, an elliptic Lorentz transformation can have any two distinct fixed points on the celestial sphere, but points will still flow along circular arcs from one fixed point toward the other. The other cases are similar.
| |
| | |
| ==== Elliptic ====
| |
| An elliptic element of SL(2,'''C''') is
| |
| :<math> P_1 = \left[ \begin{matrix} \exp(i \theta/2) & 0 \\ 0 & \exp(-i \theta/2) \end{matrix} \right] </math>
| |
| and has fixed points {{mvar|ξ}} = 0, ∞. Writing the action as {{math|''X'' ↦ ''P<sub>1</sub> X P<sub>1</sub>*}} and collecting terms, the spinor map converts this to the (restricted) Lorentz transformation
| |
| :<math> Q_1 = \left[ \begin{matrix} 1 & 0 & 0 & 0 \\
| |
| 0 & \cos(\theta) & -\sin(\theta) & 0 \\
| |
| 0 & \sin(\theta) & \cos(\theta) & 0 \\
| |
| 0 & 0 & 0 & 1 \end{matrix} \right]=\exp \left ( \theta
| |
| \left[ \begin{matrix} 0 & 0 & 0 & 0 \\
| |
| 0 & 0 & -1 & 0 \\
| |
| 0 & 1 & 0 & 0 \\
| |
| 0 & 0 & 0 & 0 \end{matrix} \right]
| |
| \right ) ~. </math>
| |
| This transformation then represents a rotation about the {{mvar|z}} axis, exp({{math|''iθJ<sub>z</sub>''}}). The one-parameter subgroup it generates is obtained by taking {{mvar|θ}} to be a real variable, the rotation angle, instead of a constant.
| |
| | |
| The corresponding continuous transformations of the celestial sphere (except for the identity) all share the same two fixed points, the North and South poles. The transformations move all other points around latitude circles so that this group yields a continuous counterclockwise rotation about the {{mvar|z}} axis as {{mvar|θ}} increases. The ''angle doubling'' evident in the spinor map is a characteristic feature of ''spinorial double coverings''.
| |
| | |
| ==== Hyperbolic ====
| |
| A hyperbolic element of SL(2,'''C''') is
| |
| :<math> P_2 = \left[ \begin{matrix} \exp(\beta/2) & 0 \\ 0 & \exp(-\beta/2) \end{matrix} \right] </math>
| |
| and has fixed points {{mvar|ξ}} = 0, ∞. Under stereographic projection from the Riemann sphere to the Euclidean plane, the effect of this Möbius transformation is a dilation from the origin.
| |
| | |
| The spinor map converts this to the Lorentz transformation | |
| :<math> Q_2 = \left[ \begin{matrix} \cosh(\beta) & 0 & 0 & \sinh(\beta) \\
| |
| 0 & 1 & 0 & 0 \\
| |
| 0 & 0 & 1 & 0 \\
| |
| \sinh(\beta) & 0 & 0 & \cosh(\beta) \end{matrix} \right] = \exp \left ( \beta\left[ \begin{matrix}
| |
| 0 & 0 & 0 & 1 \\
| |
| 0 & 0 & 0 & 0 \\
| |
| 0 & 0 & 0 & 0 \\
| |
| 1 & 0 & 0 & 0
| |
| \end{matrix} \right] \right )~. </math>
| |
| This transformation represents a boost along the {{mvar|z}} axis with [[rapidity]] {{mvar|β}}. The one-parameter subgroup it generates is obtained by taking {{mvar|β}} to be a real variable, instead of a constant. The corresponding continuous transformations of the celestial sphere (except for the identity) all share the same fixed points (the North and South poles), and they move all other points along [[longitude]]s away from the South pole and toward the North pole.
| |
| | |
| ==== Loxodromic ====
| |
| A loxodromic element of SL(2,'''C''') is
| |
| :<math> P_3 = P_2 P_1 = P_1 P_2
| |
| = \left[ \begin{matrix} \exp \left((\beta+i\theta)/2 \right) & 0 \\
| |
| 0 & \exp \left(-(\beta+i\theta)/2 \right)
| |
| \end{matrix} \right] </math>
| |
| and has fixed points {{mvar|ξ}} = 0, ∞. The spinor map converts this to the Lorentz transformation
| |
| :<math>Q_3 = Q_2 Q_1 = Q_1 Q_2. </math>
| |
| The one-parameter subgroup this generates is obtained by replacing ''β+iθ'' with any real multiple of this complex constant. (If ''β, θ'' vary independently, then a ''two-dimensional'' [[abelian group|abelian subgroup]] is obtained, consisting of simultaneous rotations about the {{mvar|z}} axis and boosts along the {{mvar|z}}-axis; in contrast, the ''one-dimensional'' subgroup discussed here consists of those elements of this two-dimensional subgroup such that the '''rapidity''' of the boost and '''angle''' of the rotation have a ''fixed ratio''.)
| |
| | |
| The corresponding continuous transformations of the celestial sphere (excepting the identity) all share the same two fixed points (the North and South poles). They move all other points away from the South pole and toward the North pole (or vice versa), along a family of curves called [[Rhumb_line|'''loxodromes''']]. Each loxodrome spirals infinitely often around each pole.
| |
| | |
| ==== Parabolic ====
| |
| A parabolic element of SL(2,'''C''') is
| |
| :<math> P_4 = \left[ \begin{matrix} 1 & \alpha \\ 0 & 1 \end{matrix} \right] </math>
| |
| and has the single fixed point {{mvar|ξ}} = ∞ on the Riemann sphere. Under stereographic projection, it appears as an ordinary [[translation]] along the [[real axis]].
| |
| | |
| The spinor map converts this to the matrix (representing a Lorentz transformation)
| |
| :<math> Q_4 = \left[ \begin{matrix} 1+\vert\alpha\vert^2/2 & Re(\alpha) & Im(\alpha) & -\vert\alpha\vert^2/2 \\
| |
| Re(\alpha) & 1 & 0 & -Re(\alpha) \\
| |
| -Im(\alpha) & 0 & 1 & Im(\alpha) \\
| |
| \vert\alpha\vert^2/2 & Re(\alpha) & Im(\alpha) & 1-\vert\alpha\vert^2/2 \end{matrix} \right] </math>
| |
| :<math> ~ = \exp \left[ \begin{matrix} 0& Re(\alpha) & Im(\alpha) & 0 \\
| |
| Re(\alpha) & 0 & 0 & -Re(\alpha) \\
| |
| -Im(\alpha) & 0 & 0 & Im(\alpha) \\
| |
| 0 & Re(\alpha) & Im(\alpha) & 0 \end{matrix} \right] ~ . </math>
| |
| This generates a two-parameter abelian subgroup which is obtained by considering {{mvar|α}} to be a complex variable rather than a constant. The corresponding continuous transformations of the celestial sphere (except for the identity transformation) move points along a family of circles which are all tangent at the North pole to a certain [[great circle]]. All points other than the North pole itself move along these circles.
| |
| | |
| Parabolic Lorentz transformations are often called '''null rotations''', since they preserve null vectors, just as rotations preserve timelike vectors and boosts preserve spacelike vectors. Since these are likely to be the least familiar of the four types of nonidentity Lorentz transformations (elliptic, hyperbolic, loxodromic, parabolic), it is illustrated here how to determine the effect of an example of a parabolic Lorentz transformation on Minkowski spacetime.
| |
| | |
| The matrix given above yields the transformation
| |
| :<math>
| |
| \left[ \begin{matrix} t \\ x \\ y \\ z \end{matrix} \right]
| |
| \rightarrow
| |
| \left[ \begin{matrix} t \\ x \\ y \\ z \end{matrix} \right]
| |
| + Re(\alpha) \;
| |
| \left[ \begin{matrix} x \\ t-z \\ 0 \\ x \end{matrix} \right]
| |
| + Im(\alpha) \;
| |
| \left[ \begin{matrix} y \\ 0 \\ z-t \\ y \end{matrix} \right]
| |
| + \frac{\vert\alpha\vert^2}{2} \;
| |
| \left[ \begin{matrix} t-z \\ 0 \\ 0 \\ t-z \end{matrix} \right].
| |
| </math>
| |
| | |
| Now, without loss of generality, pick {{math|''Im(α)''}}=0. Differentiating this transformation with respect to the now real group parameter {{mvar|α}} and evaluating at ''α''=0 produces the corresponding vector field (first order linear partial differential operator),
| |
| :<math> x \, \left( \partial_t + \partial_z \right) + (t-z) \, \partial_x. </math>
| |
| | |
| Apply this to a function {{math|''f(t,x,y,z)''}}, and demand that it stays invariant, i.e., it is annihilated by this transformation. The solution of the resulting first order linear partial differential equation can be expressed in the form
| |
| :<math>f(t,x,y,z) = F(y, \, t-z , \, t^2-x^2-z^2),</math>
| |
| where {{mvar|F}} is an ''arbitrary'' smooth function. The arguments of {{mvar|F}} give three ''rational invariants'' describing how points (events) move under this parabolic transformation, as they themselves do not move,
| |
| :<math> y = c_1, ~~~~ t-z = c_2, ~~~~ t^2-x^2-z^2 = c_3. </math>
| |
| Choosing real values for the constants on the right hand sides yields three conditions, and thus specifies a curve in Minkowski spacetime. This curve is an orbit of the transformation.
| |
| | |
| The form of the rational invariants shows that these flowlines (orbits) have a simple description: suppressing the inessential coordinate {{mvar|y}}, each orbit is the intersection of a ''null plane'', {{math| ''t'' {{=}} ''z+c''<sub>2</sub>}}, with a ''hyperboloid'', {{math|''t<sup>2</sup>−x<sup>2</sup>−z<sup>2</sup>'' {{=}} ''c<sub>3</sub>''}}. The case {{mvar|c}}<sub>3</sub> = 0 has the hyperboloid degenerate to a light cone with the orbits becoming parabolas lying in corresponding null planes.
| |
| | |
| A particular null line lying on the light cone is left ''invariant''; this corresponds to the unique (double) fixed point on the Riemann sphere mentioned above. The other null lines through the origin are "swung around the cone" by the transformation. Following the motion of one such null line as {{mvar|α}} increases corresponds to following the motion of a point along one of the circular flow lines on the celestial sphere, as described above.
| |
| | |
| A choice {{math|''Re(α)''}}=0 instead, produces similar orbits, now with the roles of {{mvar|x}} and {{mvar|y}} interchanged.
| |
| | |
| Parabolic transformations lead to the gauge symmetry of massless particles (like photons) with [[helicity (particle physics)|helicity]] <math>|h|\geq 1</math>. In the above explicit example, a massless particle moving in the {{mvar|z}} direction, so with 4-momentum '''''P'''''=(''p'',0,0,''p''), will not be affected at all by the {{mvar|x}}-boost and {{mvar|y}}-rotation combination {{math|''K<sub>x</sub>−J<sub>y</sub>''}} displayed above, in the "little group" of its motion. This is evident from the explicit transformation law discussed: like any light-like vector, '''''P''''' itself is now invariant, i.e., all traces or effects of {{mvar|α}} have disappeared. {{mvar|c}}<sub>1</sub> = {{mvar|c}}<sub>2</sub> = {{mvar|c}}<sub>3</sub> = 0, in the special case discussed. (The other similar generator, {{math|''K<sub>y</sub>+J<sub>x</sub>''}} as well as it and {{mvar|J}}<sub>''z''</sub> comprise altogether the little group of the lightlike vector, isomorphic to ''E(2)''.)
| |
| | |
| == Lie algebra ==
| |
| As with any Lie group, the best way to study many aspects of the Lorentz group is via its [[Lie algebra]]. The Lorentz group is a subgroup of the [[diffeomorphism group]] of '''R'''<sup>4</sup> and therefore its Lie algebra can be identified with vector fields on '''R'''<sup>4</sup>. In particular, the vectors which generate isometries on a space are its [[Killing vector]]s, which provides a convenient alternative to the [[Maurer–Cartan_form#Intrinsic_construction|left-invariant vector field]] for calculating the Lie algebra. We can write down a set of six generators:
| |
| * vector fields on '''R'''<sup>4</sup> generating three rotations ''i'' '''''J''''',
| |
| :<math> -y \partial_x + x \partial_y \equiv i J_z ~, \qquad -z \partial_y + y \partial_z\equiv iJ_x~, \qquad -x \partial_z + z \partial_x \equiv J_y ~;</math>
| |
| * vector fields on '''R'''<sup>4</sup> generating three boosts ''i'' '''''K''''',
| |
| :<math> x \partial_t + t \partial_x\equiv iK_x ~, \qquad y \partial_t + t \partial_y\equiv iK_y ~, \qquad z \partial_t + t \partial_z\equiv iK_z. </math>
| |
| | |
| It may be helpful to briefly recall here how to obtain a one-parameter group from a [[vector field]], written in the form of a first order [[linear]] [[partial differential operator]] such as
| |
| :<math> -y \partial_x + x \partial_y. </math>
| |
| The corresponding initial value problem is
| |
| :<math> \frac{\partial x}{\partial \lambda} = -y, \; \frac{\partial y}{\partial \lambda} = x, \; x(0) = x_0, \; y(0) = y_0. </math>
| |
| The solution can be written
| |
| :<math> x(\lambda) = x_0 \cos(\lambda) - y_0 \sin(\lambda), \; y(\lambda) = x_0 \sin(\lambda) + y_0 \cos(\lambda) </math>
| |
| or
| |
| :<math> \left[ \begin{matrix} t \\ x \\ y \\ z \end{matrix} \right]
| |
| = \left[ \begin{matrix} 1 & 0 & 0 & 0 \\
| |
| 0 & \cos(\lambda) & -\sin(\lambda) & 0 \\
| |
| 0 & \sin(\lambda) & \cos(\lambda) & 0 \\
| |
| 0 & 0 & 0 & 1 \end{matrix} \right]
| |
| \left[ \begin{matrix} t_0 \\ x_0 \\ y_0 \\ z_0 \end{matrix} \right] </math>
| |
| where we easily recognize the one-parameter matrix group of rotations exp(''i λ J<sub>z</sub>'') about the z axis.
| |
| Differentiating with respect to the group parameter {{mvar|λ}} and setting it ''λ''=0 in that result, we recover the standard matrix,
| |
| :<math>iJ_z= \left[ \begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix} \right] ~,</math>
| |
| which corresponds to the vector field we started with. This illustrates how to pass between matrix and vector field representations of elements of the Lie algebra.
| |
| | |
| Reversing the procedure in the previous section, we see that the Möbius transformations which correspond to our six generators arise from exponentiating respectively ''β''/2 (for the three boosts) or ''iθ''/2 (for the three rotations) times the three [[Pauli matrices]]
| |
| :<math> \sigma_1 = \left[ \begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix} \right], \; \;
| |
| \sigma_2 = \left[ \begin{matrix} 0 & -i \\ i & 0 \end{matrix} \right], \; \;
| |
| \sigma_3 = \left[ \begin{matrix} 1 & 0 \\ 0 & -1 \end{matrix} \right]. </math>
| |
| | |
| For our purposes, another generating set is more convenient. The following table lists the six generators, in which
| |
| *the first column gives a generator of the flow under the Möbius action (after stereographic projection from the Riemann sphere) as a ''real'' vector field on the Euclidean plane,
| |
| *the second column gives the corresponding one-parameter subgroup of Möbius transformations,
| |
| *the third column gives the corresponding one-parameter subgroup of Lorentz transformations (the image under our homomorphism of preceding one-parameter subgroup),
| |
| *the fourth column gives the corresponding generator of the flow under the Lorentz action as a real vector field on Minkowski spacetime.
| |
| Notice that the generators consist of
| |
| *two parabolics (null rotations),
| |
| *one hyperbolic (boost in the ∂<sub>''z''</sub> direction),
| |
| *three elliptics (rotations about the ''x,y,z'' axes, respectively).
| |
| | |
| {| border="1" cellspacing="2" cellpadding="5" style="text-align: center; margin: auto; border-collapse: collapse;"
| |
| ! Vector field on '''R'''<sup>2</sup>
| |
| ! One-parameter subgroup of SL(2,'''C'''),<br />representing Möbius transformations
| |
| ! One-parameter subgroup of SO<sup>+</sup>(1,3),<br />representing Lorentz transformations
| |
| ! Vector field on '''R'''<sup>4</sup>
| |
| |-
| |
| ! bgcolor="#DDDDFF" colspan="4" | Parabolic
| |
| |-
| |
| | <math>\partial_u\,\!</math>
| |
| || <math>\left[ \begin{matrix} 1 & \alpha \\ 0 & 1 \end{matrix} \right] </math>
| |
| || <math> \left[ \begin{matrix} 1+\alpha^2/2 & \alpha & 0 & -\alpha^2/2 \\
| |
| \alpha & 1 & 0 & -\alpha \\
| |
| 0 & 0 & 1 & 0 \\
| |
| \alpha^2/2 & \alpha & 0 & 1-\alpha^2/2 \end{matrix} \right] </math>
| |
| || <math>X_1 = \,\!</math><br /><math> x (\partial_t + \partial_z) + (t-z) \partial_x \,\!</math>
| |
| |-
| |
| | <math>\partial_v\,\!</math>
| |
| || <math>\left[ \begin{matrix} 1 & i \alpha \\ 0 & 1 \end{matrix} \right] </math>
| |
| || <math> \left[ \begin{matrix} 1+\alpha^2/2 & 0 & \alpha & -\alpha^2/2 \\
| |
| 0 & 1 & 0 & 0 \\
| |
| \alpha & 0 & 1 & -\alpha \\
| |
| \alpha^2/2 & 0 & \alpha & 1-\alpha^2/2 \end{matrix} \right] </math>
| |
| || <math>X_2 = \,\!</math><br /><math> y (\partial_t + \partial_z) + (t-z) \partial_y \,\!</math>
| |
| |-
| |
| ! bgcolor="#DDDDFF" colspan="4" | Hyperbolic
| |
| |-
| |
| | <math> \frac{1}{2} \left( u \partial_u + v \partial_v \right) </math>
| |
| || <math>\left[ \begin{matrix} \exp \left(\frac{\beta}{2}\right) & 0 \\
| |
| 0 & \exp \left(-\frac{\beta}{2}\right) \end{matrix} \right] </math>
| |
| || <math> \left[ \begin{matrix} \cosh(\beta) & 0 & 0 & \sinh(\beta) \\
| |
| 0 & 1 & 0 & 0 \\
| |
| 0 & 0 & 1 & 0 \\
| |
| \sinh(\beta) & 0 & 0 & \cosh(\beta) \end{matrix} \right] </math>
| |
| || <math>X_3 = \,\!</math><br /><math> z \partial_t + t \partial_z \,\!</math>
| |
| |-
| |
| ! bgcolor="#DDDDFF" colspan="4" | Elliptic
| |
| |-
| |
| | <math> \frac{1}{2} \left( -v \partial_u + u \partial_v \right) </math>
| |
| || <math>\left[ \begin{matrix} \exp \left( \frac{i \theta}{2} \right) & 0 \\
| |
| 0 & \exp \left( \frac{-i \theta}{2} \right) \end{matrix} \right]</math>
| |
| || <math> \left[ \begin{matrix} 1 & 0 & 0 & 0 \\
| |
| 0 & \cos(\theta) & -\sin(\theta) & 0 \\
| |
| 0 & \sin(\theta) & \cos(\theta) & 0 \\
| |
| 0 & 0 & 0 & 1 \end{matrix} \right] </math>
| |
| || <math>X_4 = \,\!</math><br /><math> -y \partial_x + x \partial_y \,\!</math>
| |
| |-
| |
| | <math> \frac{v^2-u^2-1}{2} \partial_u - u v \, \partial_v </math>
| |
| || <math>\left[ \begin{matrix} \cos \left( \frac{\theta}{2} \right) & -\sin \left( \frac{\theta}{2} \right) \\
| |
| \sin \left( \frac{\theta}{2} \right) & \cos \left( \frac{\theta}{2} \right) \end{matrix} \right]</math>
| |
| || <math> \left[ \begin{matrix} 1 & 0 & 0 & 0 \\
| |
| 0 & \cos(\theta) & 0 & \sin(\theta) \\
| |
| 0 & 0 & 1 & 0 \\
| |
| 0 & -\sin(\theta) & 0 & \cos(\theta) \end{matrix} \right] </math>
| |
| || <math>X_5 = \,\!</math><br /><math> -x \partial_z + z \partial_x \,\!</math>
| |
| |-
| |
| | <math> u v \, \partial_u + \frac{1-u^2+v^2}{2} \partial_v </math>
| |
| || <math>\left[ \begin{matrix} \cos \left( \frac{\theta}{2} \right) & i \sin \left( \frac{\theta}{2} \right) \\
| |
| i \sin \left( \frac{\theta}{2} \right) & \cos \left( \frac{\theta}{2} \right) \end{matrix} \right]</math>
| |
| || <math> \left[ \begin{matrix} 1 & 0 & 0 & 0 \\
| |
| 0 & 1 & 0 & 0 \\
| |
| 0 & 0 & \cos(\theta) & -\sin(\theta) \\
| |
| 0 & 0 & \sin(\theta) & \cos(\theta) \end{matrix} \right] </math>
| |
| || <math>X_6 = \,\!</math><br /><math> -z \partial_y + y \partial_z \,\!</math>
| |
| |}
| |
| | |
| Let's verify one line in this table. Start with
| |
| :<math> \sigma_2 = \left[ \begin{matrix} 0 & i \\ -i & 0 \end{matrix} \right]. </math> | |
| Exponentiate:
| |
| :<math> \exp \left( \frac{ i \theta}{2} \, \sigma_2 \right) =
| |
| \left[ \begin{matrix} \cos(\theta/2) & -\sin(\theta/2) \\ \sin(\theta/2) & \cos(\theta/2) \end{matrix} \right]. </math>
| |
| This element of SL(2,'''C''') represents the one-parameter subgroup of (elliptic) Möbius transformations:
| |
| :<math> \xi \mapsto \frac{ \cos(\theta/2) \, \xi - \sin(\theta/2) }{ \sin(\theta/2) \, \xi + \cos(\theta/2) }. </math>
| |
| Next,
| |
| :<math> \frac{d\xi}{d\theta} |_{\theta=0} = -\frac{1+\xi^2}{2}. </math>
| |
| The corresponding vector field on '''C''' (thought of as the image of S<sup>2</sup> under stereographic projection) is
| |
| :<math> -\frac{1+\xi^2}{2} \, \partial_\xi. </math>
| |
| Writing <math>\xi = u + i v</math>, this becomes the vector field on '''R'''<sup>2</sup>
| |
| :<math> -\frac{1+u^2-v^2}{2} \, \partial_u - u v \, \partial_v. </math>
| |
| Returning to our element of SL(2,''C''), writing out the action <math> X \mapsto P X P^* </math> and collecting terms, we find that the image under the spinor map is the element of SO<sup>+</sup>(1,3)
| |
| :<math> \left[ \begin{matrix} 1 & 0 & 0 & 0 \\
| |
| 0 & \cos(\theta) & 0 & \sin(\theta) \\
| |
| 0 & 0 & 1 & 0 \\
| |
| 0 & -\sin(\theta) & 0 & \cos(\theta) \end{matrix} \right]. </math>
| |
| Differentiating with respect to {{mvar|θ}} at {{mvar|θ}}=0, yields the corresponding vector field on '''R'''<sup>4</sup>,
| |
| :<math> z \partial_x - x \partial_z. \,\!</math>
| |
| This is evidently the generator of counterclockwise rotation about the {{mvar|y}} axis.
| |
| | |
| == Subgroups of the Lorentz group ==
| |
| | |
| The subalgebras of the Lie algebra of the Lorentz group can be enumerated, up to conjugacy, from which we can list the [[closed subgroup]]s of the restricted Lorentz group, up to conjugacy. (See the book by Hall cited below for the details.) We can readily express the result in terms of the generating set given in the table above.
| |
| | |
| The one-dimensional subalgebras of course correspond to the four conjugacy classes of elements of the Lorentz group:
| |
| *<math>X_1</math> generates a one-parameter subalgebra of parabolics SO(0,1),
| |
| *<math>X_3</math> generates a one-parameter subalgebra of boosts SO(1,1),
| |
| *<math>X_4</math> generates a one-parameter of rotations SO(2),
| |
| *<math>X_3 + a X_4</math> (for any <math>a \neq 0</math>) generates a one-parameter subalgebra of loxodromic transformations.
| |
| (Strictly speaking the last corresponds to infinitely many classes, since distinct <math>a</math> give different classes.)
| |
| The two-dimensional subalgebras are:
| |
| *<math>X_1, X_2</math> generate an abelian subalgebra consisting entirely of parabolics,
| |
| *<math>X_1, X_3</math> generate a nonabelian subalgebra isomorphic to the Lie algebra of the affine group A(1),
| |
| *<math>X_3, X_4</math> generate an abelian subalgebra consisting of boosts, rotations, and loxodromics all sharing the same pair of fixed points.
| |
| The three dimensional subalgebras are:
| |
| *<math>X_1,X_2,X_3</math> generate a '''Bianchi V''' subalgebra, isomorphic to the Lie algebra of Hom(2), the group of ''euclidean homotheties'',
| |
| *<math>X_1,X_2,X_4</math> generate a '''Bianchi VII_0''' subalgebra, isomorphic to the Lie algebra of E(2), the [[euclidean group]],
| |
| *<math>X_2,X_2,X_3 + a X_4</math>, where <math>a \neq 0</math>, generate a '''Bianchi VII_a''' subalgebra,
| |
| *<math>X_1,X_3,X_5</math> generate a '''Bianchi VIII''' subalgebra, isomorphic to the Lie algebra of SL(2,'''R'''), the group of isometries of the [[Poincaré half-plane model|hyperbolic plane]],
| |
| *<math>X_4,X_5,X_6</math> generate a '''Bianchi IX''' subalgebra, isomorphic to the Lie algebra of SO(3), the rotation group.
| |
| (Here, the [[Bianchi classification|Bianchi types]] refer to the classification of three dimensional Lie algebras by the Italian mathematician [[Luigi Bianchi]].)
| |
| The four dimensional subalgebras are all conjugate to
| |
| *<math>X_1,X_2,X_3,X_4</math> generate a subalgebra isomorphic to the Lie algebra of Sim(2), the group of Euclidean [[Similarity (geometry)|similitudes]].
| |
| | |
| The subalgebras form a lattice (see the figure), and each subalgebra generates by exponentiation a [[closed subgroup]] of the restricted Lie group. From these, all subgroups of the Lorentz group can be constructed, up to conjugation, by multiplying by one of the elements of the Klein four-group.
| |
| | |
| [[Image:Lorentz group subalgebra lattice.png|300px|thumb|right|The lattice of subalgebras of the Lie algebra SO(1,3), up to conjugacy.]]
| |
| | |
| As with any connected Lie group, the coset spaces of the closed subgroups of the restricted Lorentz group, or [[homogeneous spaces]], have considerable mathematical interest. We briefly describe some of them here:
| |
| *the group Sim(2) is the stabilizer of a ''null line'', i.e. of a point on the Riemann sphere, so the homogeneous space SO<sup>+</sup>(1,3)/Sim(2) is the [[Kleinian geometry]] which represents [[conformal geometry]] on the sphere S<sup>2</sup>,
| |
| *the (identity component of the) Euclidean group SE(2) is the stabilizer of a [[null vector]], so the homogeneous space SO<sup>+</sup>(1,3)/SE(2) is the [[momentum space]] of a massless particle; geometrically, this Kleinian geometry represents the ''degenerate'' geometry of the light cone in Minkowski spacetime,
| |
| *the rotation group SO(3) is the stabilizer of a [[timelike vector]], so the homogeneous space SO<sup>+</sup>(1,3)/SO(3) is the [[momentum space]] of a massive particle; geometrically, this space is none other than three-dimensional [[hyperbolic space]] H<sup>3</sup>.
| |
| | |
| == Covering groups ==
| |
| | |
| In [[#Relation to the Möbius group|a previous section]] we constructed a homomorphism SL(2,'''C''') → SO<sup>+</sup>(1,3), which we called the spinor map. Since SL(2,'''C''') is simply connected, it is the [[universal cover|covering group]] of the restricted Lorentz group SO<sup>+</sup>(1,3). By restriction we obtain a homomorphism SU(2) → SO(3). Here, the [[special unitary group]] SU(2), which is isomorphic to the group of unit [[norm (mathematics)|norm]] [[quaternion]]s, is also simply connected, so it is the covering group of the rotation group SO(3).
| |
| Each of these [[covering map]]s are twofold covers in the sense that precisely two elements of the covering group map to each element of the quotient. One often says that the restricted Lorentz group and the rotation group are '''doubly connected'''. This means that the [[fundamental group]] of the each group is [[isomorphic]] to the two element [[cyclic group]] '''Z'''<sub>2</sub>.
| |
| | |
| (In applications to [[quantum mechanics]], the [[special linear group]] SL(2, '''C''') is sometimes called the Lorentz group.)
| |
| | |
| Twofold coverings are characteristic of [[spin group]]s. Indeed, in addition to the double coverings
| |
| :Spin<sup>+</sup>(1,3)=SL(2,'''C''') → SO<sup>+</sup>(1,3)
| |
| :Spin(3)=SU(2) → SO(3)
| |
| we have the double coverings
| |
| :[[pin group|Pin(1,3)]] → O(1,3)
| |
| :Spin(1,3) → SO(1,3)
| |
| :Spin<sup>+</sup>(1,2) = SU(1,1) → SO(1,2)
| |
| | |
| These spinorial [[Double covering group|double coverings]] are all closely related to [[Clifford algebras]].
| |
| | |
| == Topology ==
| |
| The left and right groups in the double covering
| |
| :SU(2) → SO(3)
| |
| are [[deformation retract]]s of the left and right groups, respectively, in the double covering
| |
| :SL(2,'''C''') → SO<sup>+</sup>(1,3).
| |
| But the homogeneous space SO<sup>+</sup>(1,3)/SO(3) is [[homeomorphic]] to [[hyperbolic 3-space]] H<sup>3</sup>, so we have exhibited the restricted Lorentz group as a [[fiber bundle|principal fiber bundle]] with fibers SO(3) and base H<sup>3</sup>. Since the latter is homeomorphic to '''R'''<sup>3</sup>, while SO(3) is homeomorphic to three-dimensional [[real projective space]] '''R'''P<sup>3</sup>, we see that the restricted Lorentz group is ''locally'' homeomorphic to the product of '''R'''P<sup>3</sup> with '''R'''<sup>3</sup>. Since the base space is contractible, this can be extended to a global homeomorphism.
| |
| | |
| ==Generalization to higher dimensions==
| |
| | |
| The concept of the Lorentz group has a natural generalization to spacetime of any number of dimensions. Mathematically, the Lorentz group of ''n''+1-dimensional Minkowski space is the group O(''n'',1) (or O(1,''n'')) of linear transformations of '''R'''<sup>''n''+1</sup> which preserve the quadratic form
| |
| :<math> (x_1,x_2,\ldots ,x_n,x_{n+1})\mapsto x_1^2+x_2^2+\cdots +x_n^2-x_{n+1}^2.</math>
| |
| Many of the properties of the Lorentz group in four dimensions (where {{nowrap|1=''n'' = 3}}) generalize straightforwardly to arbitrary ''n''. For instance, the Lorentz group O(''n'',1) has four connected components, and it acts by conformal transformations on the celestial (''n''−1)-sphere in ''n''+1-dimensional Minkowski space. The identity component SO<sup>+</sup>(''n'',1) is an SO(''n'')-bundle over hyperbolic ''n''-space H<sup>''n''</sup>.
| |
| | |
| The low dimensional cases {{nowrap|1=''n'' = 1}} and {{nowrap|1=''n'' = 2}} are often useful as "toy models" for the physical case {{nowrap|1=''n'' = 3}}, while higher dimensional Lorentz groups are used in physical theories such as [[string theory]] that posit the existence of hidden dimensions. The Lorentz group O(''n'',1) is also the isometry group of ''n''-dimensional [[de Sitter space]] dS<sub>''n''</sub>, which may be realized as the homogeneous space O(''n'',1)/O(''n''−1,1). In particular O(4,1) is the isometry group of the [[de Sitter universe]] dS<sub>4</sub>, a cosmological model.
| |
| ==See also==
| |
| {{columns-list|2|
| |
| *[[Lorentz transformation]]
| |
| *[[Representation theory of the Lorentz group]]
| |
| *[[Poincaré group]]
| |
| *[[Möbius group]]
| |
| *[[Minkowski space]]
| |
| *[[Biquaternion]]s
| |
| *[[Indefinite orthogonal group]]
| |
| *[[Quaternions and spatial rotation]]
| |
| *[[Center of mass (relativistic)]]
| |
| *[[Special relativity]]
| |
| *[[Symmetry in quantum mechanics]]
| |
| }}
| |
| | |
| == References ==
| |
| *{{cite book | authorlink=Emil Artin |author=Artin, Emil | title=Geometric Algebra | location=New York | publisher=Wiley | year=1957 | isbn=0-471-60839-4}} ''See Chapter III'' for the orthogonal groups O(p,q).
| |
| *{{cite book | author=Carmeli, Moshe
| |
| |title=Group Theory and General Relativity, Representations of the Lorentz Group and Their Applications to the Gravitational Field
| |
| |publisher=McGraw-Hill, New York
| |
| |year=1977
| |
| |isbn=0-07-009986-3}} A canonical reference; ''see chapters 1–6'' for representations of the Lorentz group.
| |
| *{{cite book | author=Frankel, Theodore | title=The Geometry of Physics (2nd Ed.) | location=Cambridge | publisher=Cambridge University Press | year=2004 | isbn=0-521-53927-7}} An excellent resource for Lie theory, fiber bundles, spinorial coverings, and many other topics.
| |
| *{{Fulton-Harris}} ''See Lecture 11'' for the irreducible representations of SL(2,'''C''').
| |
| *{{cite book | author=Hall, G. S. | title=Symmetries and Curvature Structure in General Relativity | location=Singapore | publisher=World Scientific | year=2004 | isbn=981-02-1051-5}} ''See Chapter 6'' for the subalgebras of the Lie algebra of the Lorentz group.
| |
| *{{cite book | author=Hatcher, Allen | title=Algebraic topology | location=Cambridge | publisher=Cambridge University Press | year=2002 | isbn=0-521-79540-0}} ''See also'' the {{cite web | title=online version | url=http://www.math.cornell.edu/~hatcher/AT/ATpage.html | accessdate=July 3, 2005 }} ''See Section 1.3'' for a beautifully illustrated discussion of covering spaces. ''See Section 3D'' for the topology of rotation groups.
| |
| *{{cite book | author=Naber, Gregory | title=The Geometry of Minkowski Spacetime | location=New York | publisher=Springer-Verlag | year=1992 | isbn=0486432351}} (Dover reprint edition.) An excellent reference on Minkowski spacetime and the Lorentz group.
| |
| *{{cite book | author=Needham, Tristan |authorlink=Tristan Needham| title=Visual Complex Analysis | location=Oxford | publisher=Oxford University Press | year=1997 | isbn=0-19-853446-9}} ''See Chapter 3'' for a superbly illustrated discussion of Möbius transformations.
| |
| | |
| {{DEFAULTSORT:Lorentz Group}}
| |
| [[Category:Lie groups]]
| |
| [[Category:Special relativity]]
| |
| [[Category:Group theory]]
| |