Special Relativity and its Newtonian Limit from a Group Theoretical Perspective
aa r X i v : . [ phy s i c s . c l a ss - ph ] A ug NCU-HEP-k086May 2020
Special Relativity and its Newtonian Limit from a GroupTheoretical Perspective
Otto C. W. Kong and Jason Payne ∗ Department of Physics and Center for High Energy and High Field Physics,National Central University,Chung-li, Taiwan 32054
Abstract
In this pedagogical article, we explore a powerful language for describing the notion of spacetimeand particle dynamics in it intrinsic to a given fundamental physical theory, focusing on specialrelativity and its Newtonian limit. The starting point of the formulation is the representationsof the relativity symmetries. Moreover, that seriously furnishes – via the notion of symmetrycontractions – a natural way in which one can understand how the Newtonian theory arise asan approximation to Einstein’s theory. We begin with the Poincar´e symmetry underlying specialrelativity and the nature of Minkowski spacetime as a coset representation space of the algebraand the group, as well as how the representation. Then, we proceed to the parallel for the phasespace of a particle, in relation to which we present the full scheme for its dynamics under theHamiltonian formulation illustrating that as essentially the symmetry feature of the phase spacegeometry. Lastly, the reduction of all that to the Newtonian theory as an approximation with itsspace-time, phase space, and dynamics under the appropriate relativity symmetry contraction ispresented.
PACS numbers: ∗ Electronic address: [email protected] . INTRODUCTION Over the past century, the notion of symmetry has become an indispensable feature oftheoretical physics. It no longer merely facilitates the simplification of a difficult calculation,nor lurks behind the towering conservation laws of Newton’s time, but rather unveils funda-mental features of the universe around us, describes how its basic constituents interact, andplaces deep constraints on the kinds of theories that are even possible. In light of this, onenatural approach to contemplating nature would be to take symmetries seriously . In otherwords, one may aim to reformulate as much of our understanding of nature in the languagemost natural for describing symmetries.More specifically, what we will be concerned with here is the notion of a relativity symme-try . Using the archetypal example handed down to us by Einstein, we hope to illustrate thatnot only does a relativity symmetry relate frames of reference in which the laws of physicslook the same, but also captures the structure of physical spacetime itself as well as muchof the theory of particle dynamics on it. Moreover, as we will detail below, formulatingour theory in these terms provides one with a natural language in which approximationsto the theory in various limits can be described. One should note that the term relativitysymmetry, though much like introduced into physics by Einstein, is in fact a valid notionfor Newtonian mechanics too. It just has a different relativity symmetry, the Galilean sym-metry. Newtonian mechanics can be called the theory of Galilean relativity. The symmetryfor the Einstein theory is taken as the Poincar´e symmetry. Note that we neglect the con-sideration of all discrete symmetries like the parity transform in this article. We talk aboutthe relativity symmetries without such symmetries included.In order to parse the details of this fascinating tale, we begin with an examination ofexactly how one can naturally pass from the (classical) relativity symmetry group/algebrato its corresponding geometric counterparts such as the model of the spacetime and the phasespace for a particle in Section II, for the case of the Poincar´e symmetry ISO (1 , Throughout this note we will reserve “spacetime” specifically for the notion of physical spacetime un-derlying Einsteinian special relativity, while using “space” to cover the corresponding notion in general.Spacetime in the Newtonian setting, or space-time used instead whenever admissible, refers to the sum ofthe mathematically independent Newtonian space and time.
II. FROM RELATIVITY TO PHYSICAL SPACETIME AND THE PARTICLEPHASE SPACE
A conventional path to the formulation of a physical theory is to start with a certaincollection of assumptions about the geometry of the physical spacetime objects in this theoryoccupy. That is to say, the theory start with taking a mathematical model for the intuitivenotions of the physical space and time. After all, dynamics means the study of motion, whichis basically the change of position with respect to time. In Newtonian mechanics, Newtonhimself followed the basic definitions in his
Principia with his Scholium arguing for Euclideanspace coupled with absolute time as the foundation of the description of the physical world;the study of special relativity may be introduced via Minkowski space; general relativitytypically assumes the universe is a (torsion-free) Lorentzian manifold, and the list goes on.It is then from this ‘foundation’ that one infers the symmetries present in the model. Notethat in Newton’s time, Euclidean geometry is really the only geometry known. What wehope to convince the reader of in this section is that the opposite path can be just as fruitful,if not more so. In particular, we will start with the relevant (relativity) symmetry, givenby a Lie group (and its associated Lie algebra), and couple it to the representation thatnaturally captures the underlying geometry. Once the basic definitions are in place, we use3pecial relativity as an illustrative example of this procedure. The approach will be extendedto present the full theory of particle dynamics in the next section. Note that the model forthe physical space or spacetime is closely connected to the theory of particle dynamics onit. First of all, Newton introduce the notion of particle as point-mass to serve as the idealphysical object which has a completely unambiguous position in his model of the physicalspace. Putting that perspective up-side-down, in a theory of particle dynamics, there isno other physical notion of the physical space itself rather than the collection of possiblepositions for a free particle (or the center of mass for a closed system of particles which,however, have to be defined based on the full particle theory, for example the three Newton’sLaws). It is and has to be the configuration space for the free particle.
A. The Coset Space Representation
In his seminal paper
Raum und Zeit [2], Hermann Minkowski famously said,The views of space and time which I wish to lay before you have sprung from thesoil of experimental physics, and therein lies their strength. They are radical.Henceforth space by itself, and time by itself, are doomed to fade away intomere shadows, and only a kind of union of the two will preserve an independentreality.In this statement, Minkowski reveals something of tremendous importance: the idea ofLorentz symmetry as the right transformations sending inertial frames to inertial framesdirectly alter the model geometry of the physical space and time, or spacetime, from theNewtonian theory. The model for the physical spacetime itself depends on the explicit formof the
Principle of Relativity being postulated, i.e. the relativity symmetry of the theory.In this subsection, we will take this realization to heart and explore precisely how one goesabout recovering the model for the physical spacetime naturally associated with a givenrelativity structure for the classical theories.Consider a Lie group G , with associated Lie algebra g , which we take as capturing thefinite and infinitesimal transformations, respectively, that we can perform on a given physicalsystem without changing the form of the physical laws. In other words, those transformationswhich take a given (inertial) frame of reference into another equally valid frame. G is then4he relativity symmetry, or the symmetry group of the spacetime model of the theory ofparticle dynamics.The use of the word “transformation” above already hints at the need for a representation-theoretic perspective of what, exactly, the relativity symmetry encodes. Indeed, as it standsthe mathematical group G is merely an abstract collection of symbols obeying certain rules– a representation capturing the group structure is required to illuminate what these rulesreally mean in terms of physical transformations , which are mathematically transformationson a vector space. The best examples of the latter is our Minkowski spacetime and theNewtonian space-time. The first, perhaps prosaic, step in this direction is simply to use thegroup multiplication, thought of as a (left) action of G on itself: g ′ · g g ′ g. In other words, we can try to imagine that what we mean by a location/position in the“physical spacetime” is nothing more than an element g ∈ G , and that a transformation isthen simply furnished directly by the group operation. We have at hand the Poincar´e sym-metry denoted by ISO (1 ,
3) consisting of the rotations and translations. We can take eachelement of the pure translations as a point in the Minkowski spacetime, which is equivalentto saying that each point is to be identified as where you get to after a particular spacetimetranslation from the origin. Note that while the rotations take any point other than theorigin to a different point, they do not move the origin. From the abstract mathematicalpoint of view, what we described here is called a coset space. The Minkowski spacetime isa coset space of the Poincar´e symmetry.From here we consider the coset space M := G/H , defined mathematically as like aquotient of the group G by a closed subgroup H < G . A coset containing the element g isthe collection of all group elements of the form gh where h is any element in H . Note g ′ H = g ( g − g ′ H ) = gH for g − g ′ ∈ H .
Observe that the above action descends to an action of the full group G on M in an obviousway as g ′ · ( gH ) = ( g ′ g ) H .
It is more convenient to use the Lie algebra notation. We write a group element in terms of g = exp( a i X i ) , X i are the generators and a i real parameters (note that, as is typical, we are usingthe Einstein summation convention). X = a i X i as a linear combination of the generators,as basis elements, is an element of the Lie algebra g . Each coset then can be convenientlyidentified with an element exp( s j Y j )where Y j are the generators among the X i set which serves as a basis for the vector subspace p of g complementary to the subalgebra h for H , i.e. g = h + p as a vector space. The realnumbers s j can be seen as coordinates for each coset as a point in the coset space (spaceof the cosets) and the group action as symmetry transformations on the coset space, orequivalently the reference frame transformations. Let us look at such a transformation atthe infinitesimal limit.We are going to need a specific form of the Baker-Campbell-Hausdorff (BCH) series forthe case of products between a coset representative exp( Y ) and an infinitesimal elementexp( ¯ X ). In particular, the resultexp( ¯ X ) exp( Y ) = exp (cid:0) Y − [ Y, ¯ X ] (cid:1) exp( ¯ X ) , (1)can be easily checked to hold in general, though no similarly simple expression can be findfor two operators/matrices neither infinitesimal, with generic commutation relation. B. From the Poincar´e Algebra to Minkowski Space
The protagonists of our story are the Poincar´e group and algebra
ISO (1 ,
3) and iso (1 , iso (1 ,
3) possessesten generators, which are split up into the six generators of rotations, among the spacetimedirections, J µν (where 0 ≤ µ < ν ≤
3) and the four generators of translations along the four6irections E µ , and which satisfy the following commutation relations :[ J µν , J λρ ] = − i ~ ( η νλ J µρ − η µλ J νρ + η µρ J νλ − η νρ J µλ ) , [ J µν , E ρ ] = − i ~ ( η νρ E µ − η µρ E ν ) , [ E µ , E ν ] = 0 , (2)with J µν with µ > ν to be interpreted as − J νµ , and we use η µν = {− , , , } as like theMinkowski metric. For easy reference, we take a notation convention which is essentially thesame as that of the popular text book by Tung [3], besides using E µ and an explicit ~ .It is intuitively clear (and easy to check) that the subset so (1 ,
3) generated by the J µν generators forms a subalgebra of iso (1 ,
3) – the subalgebra of spacetime rotations calledLorentz transformations. Thus, if we are interested in the coset representation introducedin the previous section, the candidate for our Minkowski spacetime should be the coset space M := ISO (1 , /SO (1 , X ∈ iso (1 ,
3) and Y ∈ iso (1 , − so (1 ,
3) (as the complementary space p ) as X = − i ~ (cid:18) ω µν J µν + b µ E µ (cid:19) and Y = − i ~ t ρ E ρ , respectively. Note that we have put in a factor of in the sum ω µν J µν , with ω µν = − ω νµ ,to lift the µ < ν condition for convenience. Next, as we saw in the preceding discussion, wewill pass from this to an action on the corresponding coset space M (which, as we will seebelow, is isomorphic to Minkowski space, R , ). Consider an infinitesimal transformationgiven in the group notation as g ′ = exp( ¯ X H + ¯ Y ) = 1 + ¯ X H + ¯ Y = exp( ¯ Y ) exp( ¯ X H ), with The conventional description of iso (1 ,
3) uses instead the “momentum” P µ as generators, which are relatedto the generators as“energy” used here by E µ = cP µ . As we will see in the following sections, E µ are themore natural choice from the perspective of symmetry contractions. In the mathematicians’ notation, the commutator is really the Lie product defining the real Lie algebrato which the set of generators is a basis more naturally without all the i ~ . Physicists version among torescaling all the generators by the i ~ factor, the mathematically unreasonable i to have the generatorscorrespond (in a unitary representation) to physical observables and ~ to give the proper (SI) units tothem. Strictly speaking, we should be thinking about like − i ~ E µ and − i ~ J µν as our basis vectors are, i.e. the true generators, of the real Lie algebra, which is the real linear combination of them, with parametersin the proper physical dimensions. X H = − i ~ ¯ ω µν J µν and ¯ Y = − i ~ ¯ t µ E µ . We first check that[ ¯ X H , Y ] = − ~ ¯ ω µν t ρ [ J µν , E ρ ]= i ~ t ρ (¯ ω µν η νρ E µ + ¯ ω νµ η µρ E ν )= i ~ ¯ ω µρ t ρ E µ , and [ Y, [ ¯ X H , Y ]] = 0. Applying our BCH formula (1) for the case, we haveexp( ¯ X H ) exp( Y ) = exp( Y − [ Y, ¯ X H ]) exp( ¯ X H )= exp([ ¯ X H , Y ]) exp( Y ) exp( ¯ X H )as exact in the infinitesimal parameters in ¯ X H . Thus, the multiplication g ′ · ( gSO (1 , (cid:18) − i ~ ¯ t µ E µ (cid:19) exp (cid:18) − i ~ ¯ ω µν J µν (cid:19) exp (cid:18) − i ~ t ρ E ρ (cid:19) SO (1 , (cid:18) − i ~ ¯ t µ E µ (cid:19) exp (cid:18) i ~ ¯ ω µρ t ρ E µ (cid:19) exp (cid:18) − i ~ t ρ E ρ (cid:19) exp (cid:18) − i ~ ¯ ω µν J µν (cid:19) SO (1 , − i ~ (cid:18) t µ E µ = original t µ part + (¯ t µ − ¯ ω µρ t ρ ) E µ = infinitesimal change (cid:19) SO (1 , , which is the resulted coset of exp (cid:18) − i ~ ( t µ + dt µ ) E µ (cid:19) SO (1 , t µ is given by dt µ = − ¯ ω µν t ν + ¯ t µ . The lastequation can be seen as giving a representation of iso (1 ,
3) on M by identifying Y with thecolumn vector ( t µ , T and ¯ X = ¯ X H + ¯ Y with the matrix:¯ X = i ~ (cid:0) − ¯ ω µν J µν + ¯ t µ E µ (cid:1) represented by −−−−−−−−→ − ¯ ω µν ¯ t µ so that dt µ = − ¯ ω µν ¯ t µ t ν = − ¯ ω µν t ν + ¯ t µ . (3)We have derived above the representation of the Lie algebra iso (1 ,
3) of infinitesimaltransformations of the coset space M which obviously can be seen as a vector space with8 µ being the four-vector. The infinitesimal transformations with ¯ t µ = 0, i.e. in the Lorentzsubalgebra so (1 , SO (1 ,
3) Lorentz transformation on t µ as − ω µν
00 0 exp −−−−−→ Λ µν
00 1 leads to −−−−−→ the action Λ µν
00 1 t ν = Λ µν t ν . Similarly, the infinitesimal translations exponentiate into the finite translationsexp b µ = δ µν B µ . In fact, the Poincar´e symmetry is given in physics textbooks typically as the transformations x µ → Λ µν x ν + A µ . from which one can obtained the same infinitesimal transformations with d Λ µν = − ω µν and c dA µ = b µ switching from x µ to our t µ = c x µ . That is actually defining a symmetry groupthrough a representation of its generic element. Putting that in the matrix form, we have Λ µν t ν + B µ = δ µρ B µ Λ ρν
00 1 t ν = exp b µ exp − ω ρν
00 0 t ν , from which we can see the infinitesimal limit of the transformation matrix being I + b µ I + − ω µν
00 0 = I + − ω µν b µ . In fact, we can think of each point ( t µ , T in M as being defined by the action of the abovematrices on the coordinate origin (0 , T by taking B µ = t µ . Indeed t µ ≡ Λ µν t µ = t µ ; (4)hence the t µ -space is essentially isomorphic to the collection of matrices of the form Λ µν t µ . t µ SO (1 , . The latter, therefore, describes a full coset and the vector space of all such cosets is iso-morphic to that of the collection of all e ( − i ~ t µ E µ ) SO (1 ,
3) from the abstract mathematicaldescription we start with.When the Minkowski spacetime is taken as the starting point, it is a homogeneous spacein the physical sense that every point in it is really much the same as another. Each can betaken as the origin on which we can put in a coordinate system fixing a frame of reference.The symmetry of it as a geometric space is caught in the mathematical definition of ahomogeneous space as a space with a transitive group of symmetry, meaning every twopoints in it can be connected through the action of a group element. For a particular pointlike the origin, there is a subgroup of the symmetry that does not move it, which is called thelittle group. It is a mathematical theorem that the homogeneous space is isomorphic to thecoset space of the symmetry group “divided by” the little group. Our result of the Minkowskispacetime as
ISO (1 , /SO (1 , t µ or the x µ coordinates, is just acase example.Indeed, using t µ as the coset space coordinates is really no different from using P µ asgenerators and x µ . This is because we can write Lorentz transformations as x ′ = γ ( x + β i x i ) x ′ i = γ ( x i + β i x ) , (5)or equivalently as t ′ = γ ( t + β i t i ) t ′ i = γ ( t i + β i t ) , (6)with β i = v i c , β i = v i c , and γ = √ − β i β i . Both of the above are equivalent to t ′ = γ ( t + β i c x i ) = γ ( t + v i c x i ) x ′ i = γ ( x i + β i ct ) = γ ( x i + v i t ) , (7)10here t ≡ t . In other words, t µ and x µ describe the same spacetime “position” four-vector,they are simply expressed in time and space units, respectively. Einsteinian relativity saysspace and time are coordinates of a single spacetime, hence they are naturally to be expressedin the same units. It does not say that the spatial units are preferable, or in some sense morenatural, than the time units! Straight to the spirit of special relativity, we should ratheruse the same unit to measure t µ and x µ in which c = 1. With the different units, althoughtextbooks typically use x µ , what we show below is that we should indeed start with t µ ascoordinates for Minkowski spacetime, as we have done above, if we want to directly andnaturally recover t and x i as coordinates of the representation space of Newtonian physicsin the Newtonian limit, i.e. under the symmetry contraction described in the followingsection.In physical terms, J µν has the units of ~ , while the algebra element − i ~ ( ω µν J µν + b µ E µ )has no units (for we do not want to exponentiate something that has units). Hence, ω µν must also have no units, and b µ E µ has the units of ~ , giving b µ the unit of time. Similarly, a µ , and x µ , as well as A µ , have the units of ~ divided by that of P µ . All quantities now havethe right units, and c of course has the units of x µ t µ , i.e. distance over time. C. The Phase Space for Particle Dynamics as a Coset Space
After the Minkowski spacetime M described above, we come to another important cosetspace of the Poincar´e symmetry, one that serves as the phase space for a single particle.Besides the spacetime coordinates, we need also the momentum or equivalently the velocitycoordinates. However, the only parameters in the description of the group elements thatcorrespond to velocity are those for the components of the three-vector β i = ω i . The can-didate coset space is ISO (1 , /SO (3) which is seven-dimensional. An otherwise candidateis ISO (1 , /T H × SO (3) where T H denotes the one-parameter group of (‘time’) transla-tions generated by H = E , which corresponds to the physical energy. That space losesthe time coordinate t which cannot be desirable. There is a further option of extending ISO (1 , /SO (3). Let us first look carefully at the latter coset space. Instead of derivingthat coset space ‘representation’ from the first principle as for the Minkowski spacetimeabove, however, we construct it differently. The coset space here is not a vector space,hence the group action on it is not a representation. Without the linear structure, the group11ransformations cannot be written in terms of matrices acting on vectors representing thestates each as a point in the space. Moreover, obtaining the resultant coset of a genericgroup transformation on a coset following the approach above is a lot more nontrivial. Avector space description of a phase space as a simple extension of the coset space can beconstructed from physics consideration.Newtonian mechanics as the nonrelativistic limit to special relativity has of course asix dimensional vector space as the phase space, each point in which is described by twothree-vectors, the position vector x i and the momentum vector p i . The two parts are infact independent coset space representations of the corresponding relativity symmetry – theGalilean relativity. Or the full phase space can be taken as a single coset space. Goingto special relativity, the three-vectors are to be promoted to Minkowski four-vectors. Afour-vector is an element in the four dimensional irreducible representation of the SO (1 , SO (3) group as a subgroup of SO (1 , x i to x µ we get the Minkowskispacetime M depicted with t µ = x µ c as the ISO (1 , /SO (1 ,
3) coset space. Things for themomentum four-vector p µ are somewhat different. It is a constrained vector with magni-tude square p µ p µ fixed by the particle mass m as − ( mc ) , so long as the theory of specialrelativity is concerned. The actually admissible momenta only corresponds to points on thehyperboloid p µ p µ = − ( mc ) , which is a three-dimensional curved space. This suggests usingthe eight-dimensional vector space of ( x µ , p µ ), our equivalently ( t µ , u µ ) with u µ = p µ mc , thevelocity four-vector in c = 1 unit, for a Lorentz covariant formulation. The dimensionless‘momentum’ u µ is used for the conjugate variables mostly to match better to the groupcoset language. The value of − ( mc ) though is a Casimir invariant of the Poincar´e sym-metry which is a parameter for characterizing a generic irreducible representation of thesymmetry [3]. So, it makes good sense to use the momentum variables, though it reallymakes no difference when only a single particle is considered.The momentum or rather velocity hyperboloid u µ u µ = −
1, recall u µ = ( γ, γβ i ) T , is indeeda homogeneous space of SO (1 ,
3) corresponding to the coset space SO (1 , /SO (3). SO (3)which keeps the point u µ = (1 , , , T fixed is the little group. A simple way to see that is toidentify each point in the hyperboloid by the Lorentz boost that uniquely takes the referencepoint u µ = (1 , , , T to it, hence equivalently by the coset represented by the boosts.Matching with the group notation as we have above, each coset is an exp( − i ~ ω i J i ) SO (3).12n fact, the coordinate for the coset ω i = − ω i can be identified with − β i , for example from t ′ = γ ( t + β i t i ) giving dt = ¯ β i t i = − ¯ ω i t i . Putting together the ‘phase space’ as a productof the configuration space and the momentum space, we have ISO (1 , /SO (1 , × SO (1 , /SO (3) , which is mathematically exactly ISO (1 , /SO (3). We can use as the actual phase spacein the Hamiltonian formulation of the particle dynamics, which has to have coordinatesin conjugate pairs, either the vector space of ( t µ , u µ ). Note that no parameter in the fullPoincar´e group can correspond to u and β i cannot be part of a four-vector. But there isno harm using the redundant coordinates u µ to describe points in the velocity hyperboloid.That is mathematically a natural embedding of the velocity hyperboloid into the Minkowsifour-vector velocity space M v .Let us write down the explicit infinitesimal action of SO (1 ,
3) on SO (1 , /SO (3). Notethat the translations generated by E µ in the Poincar´e group do not act on the velocity four-vector u µ . The action hence can be seen as the full action of the Poincar´e group. Obviously,we have simply du µ = − ¯ ω µν u ν . Rewriting that by taking out a γ = u factor, we have dβ i + β i dγγ = − ¯ ω ij β j + ¯ β i , (8)and dγγ = ¯ β k β k . The latter as the extra term in the dβ i expression shows the complicationof the description in terms of the coset coordinates β i or ω i versus the simple picture interms of u µ . III. SPECIAL RELATIVITY AS A THEORY OF HAMILTONIAN DYNAMICS
The Hamiltonian formulation of a dynamical theory is a powerful one which is also par-ticularly good for a symmetry theoretical formalism. Here, we consider a coset space ofthe relativity symmetry group as the particle phase space, one bearing the geometric struc-ture of a so-called symplectic space. The structure can be seen as given by the existenceof a Poisson bracket as a antisymmetry bilinear structure on the algebra of differentiablefunctions F on the space to be given under local coordinates z n as { F ( z n ) , F ′ ( z n ) } = Ω mn ∂F∂z m ∂F ′ ∂z n , Ω mn = − Ω nm , det Ω = 1 .
13n terms of canonical coordinates, for example the position and momentum of a singleNewtonian particle, we have { F ( x i , p i ) , F ′ ( x i , p i ) } = δ ij (cid:18) ∂F∂x i ∂F ′ ∂p j − ∂F∂p j ∂F ′ ∂x i (cid:19) . General Hamiltonian equation of motion for any observable F ( z n ) is given by ddt F ( z n ) = { F ( z n ) , H t ( z n ) } , (9)where H t ( z n ) is the physical Hamiltonian as the energy function on the phase space, whichfor case of F being x i or p i reduces to ddt x i = ∂ H t ∂p i , ddt p i = − ∂ H t ∂x i . Note that the configuration/position variables x i and momentum variables p i are to beconsidered the basic independent variables while the Newtonian particle momentum beingmass times velocity is to be retrieved from the equations of motion for the standard casewith the p i dependent part of H t being p i p i / m . A. Dynamics as Symmetry Transformations
The key lesson here is to appreciate that the phase space (symplectic) geometric structureguarantees that for any generic Hamiltonian function H s , points on the phase space havingthe same value for the function lie on a curve of the Hamiltonian flow characterized bythe monotonically increasing real parameter s on which any observables F ( z n ) satisfy theequation dds F ( z n ) = { F ( z n ) , H s ( z n ) } . (10)The equation of motion for the usual case is simply the case for H t , i.e. time evolution. Infact, the Hamiltonian flow is the one-parameter group of symmetry transformations with H s the generator function. We have the Hamiltonian vector field X s = −{H s ( z n ) , ·} (11)as a differential operator being the generator and the collection of such X s being a repre-sentation of the basis vectors of the symmetry Lie algebra. Hence, we have dFds = X s ( F ) . (12)14he group action is realized as the canonical transformations. For the relativity symmetryin particular, the symmetry realized is actually the U (1) central extension of relativitysymmetry for the Newtonian case [4, 5]. The symmetry extension can be seen to have itsorigin in quantum mechanics [6].The structure works at least for any theory of particle dynamics with any backgroundrelativity symmetry including for examples Newtonian and Einsteinian ones of our focushere as well as quantum mechanics. Mathematics for the latter case is quite a bit moreinvolved and in many ways more natural and beautiful from the symmetry point of view.Interested readers are referred to Ref.[6]. B. Particle Dynamics of Special Relativity
For the phase space formulation of particle dynamics of special relativity, we can have apicture of the particle phase space as the coset space P := ISO (1 , /T H × SO (3) with canon-ical coordinates ( t k , u k ). The standard Hamilton’s equations in our canonical coordinatesare dt i dt = ∂ H t ( t k , u k ) ∂u i , du i dt = − ∂ H t ( t k , u k ) ∂t i , (13)where Hamiltonian function H t ( t k , u k ) = √ u k u k , which is basically energy per unit massin the dimensionless velocity unit ( mc H t = c p m c + p k p k = c p ). The equations are onlyspecial cases of Eq.(10). Note that the first equation really gives dt i dt = u i √ u k u k = β i as √ u k u k = γ , and the second du i dt = 0. For the extended phase space P e := M × M v , withcanonical coordinates ( t µ , u µ ), we have dt µ dζ = ∂ ˜ H ζ ( t ν , u ν ) ∂u µ , du µ dζ = − ∂ ˜ H ζ ( t ν , u ν ) ∂t µ , (14)with the extended Hamiltonian ˜ H ζ ( t ν , u ν ) = H t − u giving, besides the same results as from H t above, du dζ = 0 for consistency and dt dζ = −
1, hence ζ as essentially the coordinate time t ≡ t , and the same dynamics [7]. Alternatively, we can have a covariant description withthe proper time evolution dt µ dτ = ∂ ˜ H τ ( t ν , u ν ) ∂u µ = u µ , du µ dτ = − ∂ ˜ H τ ( t ν , u ν ) ∂t µ = 0 , (15)15here ˜ H τ ( t ν , u ν ) = u ν u ν . All formulations have equations of the form (10). In fact, they canbe seen all as special case of the single general equation from the symmetry of the symplecticmanifold coordinated by ( t µ , u µ ).We only write free particle dynamics here. The reason being special relativity actuallydoes not admit motion under a nontrivial x µ or t µ dependent potential without upsetting u µ u µ = −
1. Motion under gauge field, like electromagnetic field, modifies the nature of theconjugate momentum and the story is somewhat different.
C. Hamiltonian Flows Generated by Elements of the Poincar´e Symmetry
We first look at the ( t µ , u µ ) phase space picture. With the canonical coordinates, we havefrom Eq.(11) dt µ = ¯ s ∂ ˜ H s ∂u µ , du µ = − ¯ s ∂ ˜ H s ∂t µ , (16)where ¯ s = ds is the infinitesimal parameter in line with the notation of our coset descrip-tions above. We can see that the canonical transformations given by the equations forthe generators of the Poincar´e symmetry exactly agree with our coset picture above. For˜ H ω µν = t µ u ν − t ν u µ , we have dt ρ = − δ ρµ ¯ ω µν t ν + δ ρν ¯ ω νµ t µ and du ρ = − δ ρµ ¯ ω µν u ν + δ ρν ¯ ω νµ u µ , whilefor ˜ H b µ = u µ , we have dt ρ = δ ρµ ¯ b µ , du ρ = 0 — note that here we are talking about specific˜ H s functions with specific infinitesimal parameters ¯ s on specific phase space variables andthere is no summation over any of the indices involved in the expressions.The ten Hamiltonian functions ˜ H ω µν and ˜ H b µ combined together gives a full realizationof the action of the Poincar´e symmetry as transformations on the covariant phase space of( t µ , u µ ). One can check that with the Poisson bracket as the Lie bracket, they span a Liealgebra: { ˜ H ω µν , ˜ H ω λρ } = − ( η νλ ˜ H ω µρ − η µλ ˜ H ω νρ + η µρ ˜ H ω νλ − η νρ ˜ H ω µλ ) , { ˜ H ω µν , ˜ H b ρ } = − ( η νρ ˜ H b µ − η µρ ˜ H b ν ) , { ˜ H b µ , ˜ H b ν } = 0 , (17)where we have explicitly { ˜ H , ˜ H } = η µν ∂ ˜ H ∂t µ ∂ ˜ H ∂u ν − ∂ ˜ H ∂u ν ∂ ˜ H ∂t µ ! . H ω µν to i ~ J µν and ˜ H b µ to i ~ E µ we can see that the Lie algebra is that of thePoincar´e symmetry given by Eq.(2). In fact, it is a representation of the symmetry on thespace of functions of the phase space variables. If the phase space P is taken, however, wecan have only as Hamiltonian functions H ω ij and H b i , with identical expressions to ˜ H ω ij and ˜ H b i , illustrating only the ISO (3) symmetry of translations and rotations, with the Lieproduct { H , H } = η ij (cid:18) ∂ H ∂t i ∂ H ∂u j − ∂ H ∂u j ∂ H ∂t i (cid:19) . The time translation symmetry can be added with H t given above, which has the rightvanishing Lie product as {H ω ij , H t } = 0 and {H b i , H t } = 0. Not being able to have theboosts as Hamiltonian transformations is one of the short-coming of not using the covariantphase space. IV. CONTRACTIONS AS APPROXIMATIONS OF PHYSICAL THEORIES
With an understanding of how the principle of relativity informs our notion of physicalspacetime and the theory of particle dynamics behind us, we can move on to the importantconnection this language provides us between different theories from the relativity symmetryperspective. Broadly speaking this can be put thusly: it is commonplace to find phraseslike “Newtonian physics arises from special relativity when c → ∞ ” and we will place suchcomments on a firm mathematical foundation within the relativity theoretical symmetrysetting. A. A Crash Course on Symmetry Contractions
Imagine we are standing on a perfectly spherical, uninhabited planetary body . Thetransformation that arise as symmetries of said body are nothing more than the SO (3)group elements as rotations about the center. Now consider what we can say if this bodybegan to rapidly expand without limit. It is intuitively clear that as the radius of thesphere becomes larger and larger, making the surface of the sphere more and more flat, the This example, and indeed the entire examination of contractions found here, is strongly influenced bythe wonderful discussion found in [5]. i.e.
ISO (2). It might not, however, be immediately clear how exactly this is encoded in thestructure of the Lie algebras. How might one achieve such a description? It is ultimatelythis question, applied to a general Lie algebra g , that we are concerned with in this section.The notion of a contraction is precisely the answer we are looking for.In particular, we will focus on the simplest form of contractions: the so-called In¨on¨u-Wigner contractions [8]. The setup is as follows: consider a Lie algebra g with a decom-position g = h + p , where h is an n -dimensional subalgebra and p the complementary m -dimensional vector subspace. In terms of our example above, the idea is that we col-lect the portion of the symmetries that do not change in the limit (which are the rotationsaround the vertical axis through where we stand on the planet, for the example at hand)and call their Lie algebra h . The rest, or the span of the independent generators is p . Thenwe can form a one-parameter sequence of base changes, corresponding directly to the changein scale of the physical system, of the form hp ′ = I n R I m hp for any nonzero value of R here taken conveniently as positive. For any finite R , the Liealgebra hence our symmetry is not changed. In the R → ∞ limit, however,we obtain thecontracted algebra g ′ = h ⊕ p ′ . Note that, although the change of basis matrix is singularin the limit, the commutation relations still make sense:[ h , h ] = [ h , h ] ⊆ h R →∞ −−−−−−−→ h , [ h ′ , p ′ ] = 1 R [ h , p ] ⊆ R ( h + p ) = 1 R h + p ′ R →∞ −−−−−−−→ p ′ , [ p ′ , p ′ ] = 1 R [ p , p ] ⊆ R ( h + p ) = 1 R h + 1 R p ′ R →∞ −−−−−−−→ . Though the vector space is the same, the Lie products, or commutators, change. p is ingeneral not even a subalgebra of g . p ′ is however an Abelian subalgebra of g ′ and is aninvariant one.Take the explicit example we have. The Lie algebra so (3) for the group SO (3) is givenby the commutation relations[ J x , J y ] = i ~ J z , [ J y , J z ] = i ~ J x , and [ J z , J x ] = i ~ J y . P x = R J x , P y = R J y , and J z as the generator of h is not changed (takinga coordinate system with where we stand as the on the positive z -axis), the commutatorsbecome [ P x , P y ] = 1 R [ J x , J y ] = i ~ R J z → , [ J z , P x ] = 1 R [ J z , J x ] = i ~ P y , [ J z , P y ] = 1 R [ J z , J y ] = − i ~ P x , in the limit as R → ∞ . Therefore, we recover precisely commutation relations of theLie algebra iso (2). From the physical geometric perspective, we see that what is reallyhappening in the limit is that the ratio of the characteristic distance scales we have chosen,like the length of our foot step or the distance we can travel and that of the radius, isbecoming zero. The radius R is effectively infinity to us if we can only manage to explore adistance tiny in comparison. The planet is as good as flat to us then, though it is only anapproximation. B. The Poincar´e to Galilean Symmetry Contraction
Our starting point for describing the transition from Einsteinian relativity to Galileanrelativity is the following natural choice of a contraction of the Poincar´e algebra to theGalilean algebra. Moreover, we will see that this takes Minkowski spacetime, viewed as acoset space of
ISO (1 , G (3).Actually, it goes all the way to take the full dynamical theory as given by the symplecticgeometry of the phase space as a representation space from that of special relativity to theNewtonian one.The contraction is performed via the new generators K i = c J i and P i = c E i , keeping J ij and E is renamed − H . Then we have[ J ij , J hk ] = − i ~ ( δ jh J ik − δ ih J jk + δ ik J jh − δ jk J ih ) , [ J jk , H ] = 0 , (18) Our notation is such that it have a nice matching to the Poincar´e symmetry ones used above, with theidentification of J x , J y , J z , P x and P y as J , J , J , E and E , respectively. η ij = δ ij ). As for the other commutators, wehave [ J ij , K k ] = 1 c [ J ij , J k ] = − i ~ ( δ jk K i − δ ik K j ) , [ K i , K j ] = 1 c [ J i , J j ] = − i ~ c J ij , [ J ij , P k ] = 1 c [ J ij , E k ] = − i ~ ( δ jk P i − δ ik P j ) , [ J ij , H ] = 0 , [ K i , P j ] = 1 c [ J i , E j ] = − i ~ c η ij H , [ K i , H ] = − c [ J i , E ] = − i ~ P i , [ H, P i ] = − c [ E , E i ] = 0 , [ P i , P j ] = 1 c [ E i , E j ] = 0 . (19)When we take the c → ∞ limit, we have [ K i , K j ] = 0 and [ K i , P j ] = 0. That is, we recoverthe Galilean symmetry algebra. Note that we need the c factor in K i = c J i in order toget [ K i , K j ] = 0, hence Lorentz boosts becoming commutating Galilean boosts. Moreover,this will give [ K i , P j ] = 0 as well if we simply take P i = E i . However, this will also yield[ K i , H ] = 0 in the contraction limit which cannot be the Galilean symmetry. By taking P i = c E i though, one can see that this saves [ K i , H ] = − i ~ P i , as needed. This is actuallyprecisely the reason we wanted to start with E µ , instead of P µ ! Indeed, the momentum P i are not the generators of the Poincar´e algebra we started with before the introduction ofthe nontrivial factor of c .The mathematical formulation of the contraction above can also be understood froma geometric picture. It is about an approximation when the relevant velocities of particlemotion have magnitude small relative to the speed of light c , i.e. β i <<
1. The velocity spacefor particle motion under special relativity is the three-dimensional hyperboloid of ‘radius’ c – the four-velocity cu µ is a timelike vector of magnitude c . When we are only looking at asmall region around zero motion of u µ = (1 , , , T , the velocity space seems to be flat, likethe Euclidean space of Newtonian three-velocity v i , and the boosts as commuting velocitytranslations. 20 . Retrieving Newtonian Space-Time from Minkowski Spacetime Now we can parse the changes in the Minkowski spacetime coordinates t µ , as a represen-tation, under the contraction. First of all, we have to write our algebra elements in termsof these new generators, in order to paint a coherent picture. We have − i ~ (cid:18) ω µν J µν + b µ E µ (cid:19) = − i ~ (cid:18) ω ij J ij + b E + ω i J i + b i E i (cid:19) = − i ~ (cid:18) ω ij J ij + b E + c ω i K i + c b i P i (cid:19) = − i ~ (cid:18) ω ij J ij + b E + v i K i + a i P i (cid:19) , (20)where v i = c ω i and a i = c b i are the new parameters for the boosts and spatial translations(i.e. the x i translations). The representation for the algebra is given by dtdx i = c dt i ≡ − c ¯ ω j ¯ b − c ¯ ω i − ¯ ω ij ¯ a i = c ¯ b i tx i = c t i = − c ¯ ω j x j + ¯ b − c ¯ ω i t − ¯ ω ij x j + ¯ a i = c ¯ v j ¯ b ¯ v i − ¯ ω ij ¯ a i tx i = c ¯ v j x j + ¯ b ¯ v i t − ¯ ω ij x j + ¯ a i . (21)where we have used c ¯ ω i = − c ¯ ω i = − c ¯ β i = − ¯ v i , ¯ ω j = ¯ ω j = − c v j . Lastly, we take the limit c → ∞ and get dtdx i = b ¯ v i − ¯ ω ij ¯ a i tx i = ¯ b ¯ v i t − ¯ ω ij x j + ¯ a i . (22)The group of finite transformations can be written in the form t ′ x ′ i = BV i R ij A i tx i = t + BV i t + R ij x j + A i . (23)21ewtonian space-time with transformations under a generic element in the Galilean grouphas been retrieved. Now we can see that the Newtonian space-time ‘points’ can be describedby the coset tV i R ij x i = t δ ik x i V k R kj
00 0 1 , as tV i R ij x i = tx i . Indeed, the matrix expressed as that product of two is exactly in the form of the first matrixrepresenting a particular element exp (cid:0) − i ~ ( tH + x i P i ) (cid:1) of pure translations multiply to anyelement with the rotations and Galilean boosts, as translations on the space of Newtonianvelocity, only, hence any element of the coset exp (cid:0) − i ~ ( tH + x i P i ) (cid:1) ISO v (3). The Newtonianspace-time as a coset space is given by G (3) /ISO v (3), and the ISO v (3) subgroup is exactlythe result of the contraction from SO (1 , i.e. we have ISO (1 , /SO (1 , −→ G (3) /ISO v (3) . The infinitesimal action of the G (3) group on the coset here obtained from the contrac-tion may also be obtained directly from first principle. The simpler commutation relationsactually make the calculation easier.In Einstein relativity, spacetime should be described by coordinates with the same units.The natural units are given by the c = 1 units, which identifies each spatial distance unitwith a time unit, and vice versa. If one insists on using different units for the time and spaceparts, c has then the unit of distance over time and can be written as any value in differentunits, like ∼ × ms − , or ∼ × A yr − , or ∼ × − km ps − , or ∼ − M pc ps − .The exact choice of units is arbitrary. The structure of the physical theory is independentof that. Hence, any finite value of c describes the same symmetry represented by spacetimecoordinates in different units. The c → ∞ limit is different. Infinity is infinity in any units,and the algebra becomes the contracted one, which is to say that the relativity symmetrybecomes Galilean. The latter is practical as an approximation for physics at velocity muchless than c . Pictured in the Minkowski spacetime, such lines of motion hardly deviate from22he time axis, giving the idea of the Newtonian absolute time. The relativity symmetrycontraction picture gives a coherent description of all aspects of that approximate theory,including the dynamics to which we will turn below. D. Hamiltonian Transformations and Particle Dynamics at the Newtonian Limit
Turning to the phase space pictures, we have already dt µ = − ¯ ω µν t ν + ¯ t µ giving at thecontraction limit dt = ¯ t and dx i = ¯ v i t − ω ij x j + ¯ x i . Similarly, we can see that du = − ¯ ω i u i = ¯ β i u i = ⇒ dγ = ¯ v i v i γc → ,du i = − ¯ ω iν u ν = ⇒ dv i = − v i dγγ − ¯ ω ij v j + ¯ v i → − ¯ ω ij v j + ¯ v i , (24)where we have used Eq.(8). The phase space P at the contraction limit should be describedwith coordinates ( x i , v i ). The coset space of ISO (1 , /SO (3) or P e with ( t, x i , v i ) as γ → x i , v i ).To look at the Hamiltonian symmetry flows or the dynamics at the contraction limit, thenotation of the Hamiltonian vector field is convenient. On P with {· , ·} we have X (3) s = −{H s ( t i , u i ) , ·} = − η ij (cid:18) ∂ H s ( t i , u i ) ∂t i ∂∂u j − ∂ H s ( t i , u i ) ∂u j ∂∂t i (cid:19) = − δ ij (cid:18) ∂c H s ( x i , v i ) ∂x i ∂∂ ( γv j ) − ∂c H s ( x i , v i ) ∂ ( γv j ) ∂∂x i (cid:19) . (25)For H t = √ u k u k in particular, we have c H t = c + γ v k v k + . . . where the terms notshown contains negative powers of c and vanishes at the c → ∞ contraction limit. Multiplyby the mass m and take the expression to the contraction limit, the first term is diverging,but is really the constant rest mass contribution to energy, while the finite second term isthe kinetic energy mH t = mv k v k . Anyway, the c term being constant does not contributeto X (3) t , which then reduces to X (3) t → X t = − δ ij (cid:18) ∂H t ∂x i ∂∂v j − ∂H t ∂v j ∂∂x i (cid:19) . (26)The Hamilton’s equations of motion are more directly giving dx i dt = v i and dv i dt = 0. Wehave retrieved free particle dynamics of the Newtonian theory, though with the mass m dropped from the description. The case with a nontrivial potential energy V can obviously23e given by taking H t = v k v k + Vm . The fact that the case cannot be retrieved from thecontraction limit of special relativity is a limitation of the latter which cannot describepotential interaction other than those from gauge fields [9].On the Lorentz covariant phase space, we have X (4) s = −{ ˜ H s ( t µ , u µ ) , ·} = −{ ˜ H s , ·} − η ∂ ˜ H s ∂t ∂∂u − ∂ ˜ H s ∂u ∂∂t ! . (27)This, together with the above, shows that for c → ∞ X (4) ζ = X (3) t + ∂∂t → X t + ∂∂t , (28)giving the same dynamics. Similarly, we have X (4) τ giving the same limit, as c ˜ H τ = H t + c γ → H t + c . The exact limits of the X (4) s are generally vector fields on the space of( t, x i , v i ) though. The space can be seen as an extension of the Newtonian phase spacewith the time coordinate, and a Poisson bracket defined independent of the latter. The trueHamiltonian vector field as a vector field on the phase space should have the t part droppedfrom consideration, like X t + ∂∂t projected onto X t .Further extending the analysis to the Hamiltonian functions H ω ij = ˜ H ω ij , H b i = ˜ H b i , H t ,˜ H ω i , and ˜ H b , one can retrieve { H ω ij , H ω hk } = − ( δ jh H ω ik − δ ih H ω jk + δ ik H ω jh − δ jk H ω ih ) , { H ω ij , H v k } = − ( δ jk H v i − δ ik H v j ) , { H v i , H v j } = 0 , { H ω ik , H a k } = − ( δ jk H a i − δ ik H a j ) , { H a i , H a j } = 0 , { H v i , H a j } = − δ ij , { H v i , H t } = − H a i , { H t , H a i } = 0 , { H ω ij , H t } = 0 , (29)with H ω ij = x i v j − x j v i , H v i = − x i , and H a i = v i . We have already looked at H t from c H t .˜ H b = u can of course be rewritten as −H t , which is in line with the Galilean generator H as − E . As mentioned above, the Hamiltonian functions give a representation of the U (1)central extension of Galilean Lie algebra on the functional space with the Poisson bracketas the Lie bracket. { H v i , H a j } = − δ ij as versus [ K i , P j ] = 0 is the central extension. H ω ij and H a i = v i are from the limit of c ˜ H ω ij and c ˜ H b i , respectively. We can see from theabove Hamiltonian vector field analysis that using the ( x i , v i ) instead of ( t i , u i ) as canonicalcoordinates implies a c factor for the matching Hamiltonian function. The c extra factor24n H a i is from the symmetry contraction of P i = c E i . The a bit more complicated case iswith the Hamiltonian generator for the Galilean boosts H v i . We are supposed to take again c ˜ H ω i to the c → ∞ limit, which gives v i t − x i and the Hamiltonian vector field as ∂∂v i − t ∂∂x i .Projecting that vector field on space of ( t, x i , v i ) to a Hamiltonian vector field on the phasespace gives only ∂∂v i , which corresponds to our Hamiltonian function of H v i = − x i . If v i t − x i is naively taken, all expression would be the same except { H v i , H v j } which would then be − tδ ij . V. CONCLUDING REMARKS
As we have seen above, quite a lot of information about our description of physicalsystems is actually encoded in the underlying relativity symmetry algebra. What we hopeto emphasize here is that this is really a great, if not the correct, perspective from whichone can classify a physical paradigm, as well as the possible extensions and approximations.It is also important to note that this story is not unique to special relativity and theNewtonian limit. There is, indeed, an additional question motivating this note, namely,how the symmetry perspective can be used to understand better quantum mechanics andits classical limit. The relativity symmetry contraction picture can be seen as a way tounderstand the classical phase space as an approximation to the quantum phase space [6],and even suggests a notion of a quantum model for the physical space [10]. In a broaderscope, relativity symmetry deformations has been much pursued as a probe to possibledynamical at the more fundamental levels [11–14].Concerning the classical theories with some coset spaces serving essentially as the phasespaces for dynamical theories under the corresponding symmetries, it should be mentionedthat the so-called coadjoint orbits of Lie groups are essentially the only nontrivial math-ematical candidates for symplectic geometries. The full structures of all such symplecticgeometries and hence dynamical theories can be derived [4, 15, 16] though the detail math-ematics is not so easily appreciable to many students which are a major part of the targetreaders for this article. Coset spaces are also the natural candidates for homogeneous geo-metric spaces. 25 . DEFORMATIONS AS PROBES OF MORE FUNDAMENTAL PHYSICS
We would expect the symmetry of a physical system to be robust under small pertur-bations, as otherwise our limited precise in measurements would imply that we can never actually correctly determine or identify the symmetry of a given system. Indeed, the factthat a minute perturbation – too small to be detected by our best measuring apparatuses –could yield a different symmetry Lie algebra than the actual one means that we are episte-mologically blind to the underlying physics. As such, it makes sense to focus our attentionon algebras that are significantly robust under small perturbations.For a Lie algebra, a perturbation can be taken as a (small) modification of the struc-tural constants. For example, taking the Lorentz symmetry of SO (1 ,
3) with generators atthe standard physical units, the commutators/Lie brackets among the infinitesimal Lorentzboosts is proportional to 1 /c , as can be seen in the main text. Actually, for any finitevalue of c , the symmetry is the same mathematical group/algebra. Again, at 1 /c = 0, i.e. speed of light being infinity, it is a different symmetry, the ISO (3) of rotations and Galileanboosts. If we have not measured the finite speed of light, we would only be able to havean experimental lower bound for it. Confirming 1 /c = 0 requires infinite precision, whichcan never be available. It makes sense then to prefer the Lorentz symmetry and see theGalilean one as probably only an approximation at physical velocities small compared tothe yet undetermined large speed of light. That is more or less the argument Minkowski had[1, 2] on one could have discover special relativity from mathematical thinking alone. Hereit is only about the idea of the zero structural constant in the Galilean symmetry makingit unstable upon perturbations, or deformations. The physical identification of the constantas 1 /c is not even necessary.One can argue that we have a similar situation with the commutator between a pair ofposition and momentum operators as generators for the Heisenberg-Weyl symmetry behindquantum mechanics. Actually, the zero commutator limit of which can be essentially iden-tified as that between the K i and P i generators of the Galilean symmetry, with K i = mX i and m being the particle mass. Then, one can also further contemplate deforming the zerocommutators among the position and momentum operators, all the way till reaching a stableLie algebra, one that no further deformation to a different Lie algebra is mathematicallypossible [13]. Within the Lie group/algebra symmetry framework, the scheme may suggest26 bottom-up approach to construct some plausible more fundamental theories. Acknowledgments
The authors wish to thank the students from their
Group Theory and Symmetry courseat the National Central University, in which a preliminary version of the above material wasfirst prepared as a supplementary lecture note. [1] F.J. Dyson, Bull. Amer. Math. Soc. 78, 635-652 (1972)[2] H. Minkowski, Lecture to the 80th Assembly of Natural Scientists (Koln, 1908), Phys. Z. 10(1909), 104-111. English transi., The principle of Relativity, Aberdeen Univ. Press, Aberdeen,1923.[3] W-K. Tung,
Group Theory in Physics , World Scientific (1985).[4] J.A. de Azc´arraga and J.M. Izquierdo,
Lie Groups, Lie Algebras, Cohomology and SomeApplications in Physics , Cambridge (1995).[5] R. Gilmore,
Lie Groups, Lie Algebras, and Some of Their Applications , Dover (2005).[6] C.S. Chew, O.C.W. Kong, and J. Payne, Adv. High Energy Phys. Grav. Cosmo. 5, (2019).[7] O.D. Johns,
Analytical Mechanics for Relativity and Quantum Mechanics , Oxford UniversityPress (2005).[8] E. In¨on¨u and E.P. WignerProc. Natl. Acad. Sci. USA, 39(6): 510-524, (1953).[9] E.C.G. Sudarshan and N. Mukunda,
Classical Dynamics: A Modern Perspective , World Sci-entific (2015).[10] C.S. Chew, O.C.W. Kong, and J. Payne, Adv. High Energy Phys. 2017, 4395918, 9pp (2017).[11] H.S. Snyder, Phys. Rev. , 38 (1947); C.N. Yang, Phys. Rev. , 874 (1947).[12] H. Bacry and J.-M. Levy-Leblond, Possible kinematics, J. Mathematical Phys. 9 (1968), 1605-1614.[13] R.V. Mendes, J. Phys. A , 8091 (1994); C. Chryssomalakos and E. Okon, Int. J. Mod. Phys.D , 1817 (2004); ibid. D ~ . Phys. Lett. B 665, 58-61[14] Amelino-Camelia, G. (2001). Testable Scenario for Relativity with Minimum Length. Phys. ett. B 510, 255-263. Amelino-Camelia, G. (2002). Relativity in Spacetimes with Short-Distance Structure Governed by an Observer-Independent (Plankian) Length Scale. Int. J.Mod. Phys. D 11, 35-59. Magueijo, J. and Smolin, L. (2002). Lorentz Invariance with anInvariant Energy Scale. Phys. Rev. Lett. 88, 190403. Magueijo, J. and Smolin, L. (2003).Generalized Lorentz Invariance with an Invariant Energy Scale. Phys. Rev. D 67, 044017.Kowalski-Glikman, J. and Smolin, L. (2004). Triply Special Relativity. Phys. Rev. D 70,065020.[15] V. Guillemin and S. Sternberg, Symplectic Techniques in Physics , Cambridge (1984).[16] J.E. Marsden and T.S. Ratiu,
Introduction to Mechanics and Symmetry ,,