aa r X i v : . [ phy s i c s . c l a ss - ph ] M a r Bracket relations for relativity groups
Thomas F. Jordan ∗ Physics Department, University of Minnesota, Duluth, Minnesota 55812
Abstract
Poisson bracket relations for generators of canonical transformations are derived directly fromthe Galilei and Poincar´e groups of changes of space-time coordinates. The method is simple butrigorous. The meaning of each step is clear because it corresponds to an operation in the group ofchanges of space-time coordinates. Only products and inverses are used; differences are not used.It is made explicitly clear why constants occur in some bracket relations but not in others, and howsome constants can be removed, so that in the end there is a constant in the bracket relations forthe Galilei group but not for the Poincar´e group. Each change of coordinates needs to be only tofirst order, so matrices are not needed for rotations or Lorentz transformations; simple three-vectordescriptions are enough.Conversion to quantum mechanics is immediate. One result is a simpler derivation of the com-mutation relations for angular momentum directly from rotations. Problems are included. . INTRODUCTION The bracket relations of the Galilei group and the Poincar´e group are expressions of rel-ativity in the working language of Hamiltonian dynamics. The relativity is that of Galileoand Newton for the Galilei group in “nonrelativistic” mechanics and that of Einstein forthe Poincar´e group in “relativistic” mechanics. Our goal here is to understand the foun-dation of these bracket relations in classical Hamiltonian mechanics when the brackets arePoisson brackets and the generators in the bracket relations are functions of the canonicalcoordinates and momenta that generate canonical transformations. We will see how to getfrom the groups of changes of space-time coordinates to the Poisson-bracket relations forthe generators of canonical transformations.The derivation of the bracket relations for the rotation generators may be the most usedpart. Almost every course in quantum mechanics uses the commutation relations for angularmomentum and says something about their connection to rotations. That connection isdescribed very directly and simply here, particularly in Section V.B and the paragraph thatcontains Eqs. (6.4) and (6.5). It is clear that quantum mechanics does not play an essentialrole. The equations can be read equally well as quantum mechanics or classical mechanics.It is clear that properties of rotations give the bracket relations only to within constants.The constants can be removed from the bracket relations only by making choices of theconstants that can be added to the generators.Extension to the Galilei group or Poincar´e group brings more applications. When thebracket relations of the generators with the position are included, to ensure that the waythe generators change the position corresponds to the way the relativity group changes co-ordinates, major elements of classical and quantum mechanics can be derived. The “nonrel-ativistic” and “relativistic” forms of the Hamiltonian can be found for an object in classicalmechanics. These give the relation of the canonical momentum to the velocity and theinterpretation of the canonical momentum and Hamiltonian as physical quantities. We cansee how the translation generator is related to mass and velocity and why − i ∇ representsthe momentum in quantum mechanics. We can show that an interaction potential for twoparticles can depend only on the relative position and momentum, not on the center-of-massposition and momentum. Some of these results are reviewed in Section VII. They are allconsequences of relativistic symmetries. They are obtained from the bracket relations. The2racket relations provide a structure that underlies and shapes both classical mechanics andquantum mechanics and is simple enough that it does not depend on either. Our derivationof the bracket relations here can be used in teaching both classical and quantum mechanics.The method used here is simple but rigorous. The meaning of each step is clear because itcorresponds to an operation in the group of changes of space-time coordinates. Only productsand inverses are used; differences are not used. A framework that supports sums anddifferences comes in when we think of functions of the canonical coordinates and momenta asphysical quantities and obtain their brackets from the brackets of the canonical coordinatesand momenta. That is completely separate from what we are doing here. We have twodifferent ways to get the same bracket equations. For example, we can get the bracketrelations for angular momentum by looking at angular-momentum functions as generatorsof rotations, as we are doing here, or by looking at them as physical quantities made frompositions and momenta and using the bracket relations for position and momentum. Ourunderstanding and appreciation will be aided, and our teaching will be clearer, when eachmethod is presented without mixing in operations from the other.It is made explicitly clear why constants occur in some bracket relations but not inothers, and how some constants can be removed, so that in the end there is a constantin the bracket relations for the Galilei group but not for the Poincar´e group. Problem7.5 is stated to show that this constant makes nonrelativistic mechanics unable to describeconservation of momentum without conservation of mass in radioactive decays that wereobserved before Einstein presented his relativity.Conversion to quantum mechanics is simple for most of what is done here. The equationsremain the same for finding the bracket relations with possible constants in Section V andfor eliminating constants in Section VI. What we see when we read the equations is changed,and we use different language to describe it. For the generators and the quantities beingtransformed, we see Hermitian operators instead of real functions of the canonical coordi-nates and momenta. The brackets are commutators divided by i instead of Poisson brackets.There are unitary transformations instead of canonical transformations. The constants thatcan occur in bracket relations and be added to generators are multiples of the identityoperator. The converted procedure is in the Heisenberg picture of quantum mechanics.I wrote this in response to students in my class in classical mechanics asking for “some-thing to read about this.” Something clear and simple was needed. Derivations of bracket3elations in classical mechanics are found in more advanced treatments. Derivations inquantum mechanics are in the Schr¨odinger picture. They cannot be simply converted to afamiliar form of classical mechanics. The Schr¨odinger picture also brings in the ambiguityabout phase factors of state vectors. To set up the closest Schr¨odinger-picture analog ofwhat is done here, it is necessary to at least observe that the phase ambiguity in the productof two changes of state does not depend on the state vector, and prove that the phase factorscan be eliminated for the one-parameter subgroups. An alternative for the Poincar´e groupis to prove that all the phase factors that could affect bracket relations can be eliminated. For the Galilei group, that can be done only in an extension to a larger group.
The basic one-parameter subgroups of space and time translations, rotations, and Galileior Lorentz transformations are the most familiar parts of the Galilei and Poincar´e groups.The bracket relations of their generators show how these ten one-parameter subgroups fittogether in the larger groups. For the Galilei or Poincar´e group, a bracket of two of theten generators of the basic one-parameter subgroups never contains a linear combination ofmore than one of these generators; if it is not zero or a constant, it is just plus or minusone of the ten generators. Derivations of bracket relations in quantum mechanics most oftenuse products of three transformations, with the last the inverse of the first and the middleone infinitesimal. Then the product rule shows how the generators are transformed, andthat shows what the bracket relations are. This can be done beautifully, but seeing whatcomes from the product rule requires identification of generators for more than the ten basicone-parameter subgroups. Here we use products of four transformations, with the third theinverse of the first and the fourth the inverse of the second. This pulls out the bracketdirectly and gives just the one of the ten generators, or the minus one or none, that is theanswer. There is no intermediate step that requires identification of another generator.The procedure here is made simpler than the closest quantum parallel by understandingthat, although results are obtained in second order, each step needs to be carried only to firstorder. This allows simple three-vector calculations. In particular, in Section V.B, matricesare not needed for rotations or rotation generators. These simplifications can be made inquantum mechanics as well as in classical mechanics.4 I. CHANGES OF COORDINATES
Our description of a physical system uses a time coordinate t , and at each time t theposition of each object in space is described by a coordinate vector ~r with components x , y , z along axes in three orthogonal directions. We consider several different changes ofcoordinates: Rotations.
The frame of orthogonal axes for the space coordinates is rotated around an axisthrough the origin. The position of each object is described by a new coordinate vector ~r ′ with components x ′ , y ′ , z ′ along the new axes. Space translations.
The origin for the space coordinates is moved a fixed distance − ~d . Eachspace coordinate vector is changed from ~r to ~r ′ = ~r + ~d. (2.1) Galilei transformations.
The position of each object is described by a new coordinate vector ~r ′ = ~r − ~βt (2.2)relative to an origin moving with velocity ~β . Time translations.
The time coordinate is changed from t to t ′ = t − s (2.3)as it would be for clocks set ahead by a fixed amount s .We consider all the changes of coordinates that can be made from these rotations, spacetranslations, Galilei transformations, and time translations. They form a group. The groupproduct of two changes of coordinates is the change of coordinates obtained by doing firstone and then the other. This is called the Galilei group .5or example, a time translation ~r ′ = ~r, t ′ = t − s (2.4)followed by a Galilei transformation ~r ′′ = ~r ′ − ~βt ′ , t ′′ = t ′ (2.5)gives ~r ′′ = ~r − ~βt + ~βs, t ′′ = t − s. (2.6)The product of a time translation and a Galilei transformation includes a space translation;the distance ~d of the space translation is ~βs .We also consider the Poincar´e group . It is obtained by replacing the Galilei transforma-tions with Lorentz transformations. For example, a Galilei transformation with ~β in the z direction is replaced by the Lorentz transformation x ′ = x, z ′ = z cosh α − t sinh αy ′ = y, t ′ = t cosh α − z sinh α (2.7)where | ~β | = tanh α is the velocity dz/dt of the origin of the x ′ , y ′ , z ′ coordinates. We useunits where the velocity of light c is 1.We will look at these groups as composites of one-parameter subgroups. A one-parametersubgroup is a set of changes of coordinates that depend on a parameter u so that when u iszero there is no change of coordinates, the identity element of the group, and the product oftwo changes of coordinates for values u and u of the parameter is the change of coordinatesfor u + u . The time translations form a one-parameter subgroup for which the parameter is s . Rotations around a fixed axis form a one-parameter subgroup for which the parameter isthe angle of rotation. Space translations in a fixed direction form a one-parameter subgroupfor which the parameter is the distance (positive or negative). Galilei transformations forvelocities in a fixed direction form a one-parameter subgroup for which the parameter is thevelocity. Lorentz transformations for velocities in a fixed direction form a one-parametersubgroup. For velocities in the z direction, for example, they are described by Eqs. (2.7). Theparameter is α ; this will be shown in Problem 2.3. The Galilei group, or the Poincar´e group,is a composite of ten one-parameter subgroups: time translations, rotations around the x ,6 , and z axes, space translations in the x , y , and z directions, and Galilei transformationsor Lorentz transformations for velocities in the x , y , and z directions. The way these tenone-parameter subgroups fit together gives the bracket relations for the ten generators H , ~P , ~J , and ~G or ~K to be introduced in Section IV. Problem 2.1.
Every change of coordinates in the Galilei group can be written as ~r ′ = R~r + ~d − ~βt, t ′ = t − s (2.8)where R denotes a rotation of the vector ~r . Show this by showing that the product of twochanges of coordinates of this form is a change of coordinates of the same form: if the firstis for R , ~d , ~β , s and the second is for R , ~d , ~β , s , then the product is for R R , R ~d + ~β s + ~d , R ~β + ~β , s + s with R R the product of the rotations R and R .Equations (2.6) provide an example. Problem 2.2.
Show that the inverse of the change of coordinates described by Eqs. (2.8)in Problem 2.1 is the change of coordinates described by the same equations for R − , R − ~βs − R − ~d , − R − ~β , − s with R − the inverse of the rotation R . Problem 2.3.
Show that the Lorentz transformations described by Eqs. (2.7) form a one-parameter group for which α is the parameter. III. RELATIVITY
We assume that each of these changes of coordinates leads to an equivalent description ofthe system and its dynamics; physical quantities and the way they change can be describedwith the new coordinates as well as with the old. In quantum mechanics, we assume thatfor each change of coordinates there is a unitary operator that changes the operators thatrepresent physical quantities in the Heisenberg picture, and also changes the Hamiltonianoperator that generates the changes described by dynamics. In classical Hamiltonian me-chanics, we assume that for each change of coordinates there is a canonical transformationthat changes the canonical coordinates and momenta, and also changes the Hamiltonianfunction that generates the changes described by dynamics.7hen the change of coordinates is a time translation, the canonical transformation is thesame as for a change in time described by the dynamics. The change of canonical coordinatesand momenta from q n , p n to q ′ n , p ′ n corresponding to the change of the time coordinate from t to t ′ is simply that q ′ n ( t ′ ) = q n ( t ) , p ′ n ( t ′ ) = p n ( t ) . (3.1)There is no change beyond referral to a different time. When t ′ is t − s , this means that q ′ n ( t ′ ) = q n ( t ′ + s ) , p ′ n ( t ′ ) = p n ( t ′ + s ) . (3.2)We assume that after one canonical transformation for one change of coordinates, a secondcanonical transformation can be made the same way for a second change of coordinates.This means that the product of the two canonical transformations, the result of followingone by the other, is the canonical transformation that corresponds to the product of thetwo changes of coordinates. This assumption is not always valid. For example, if there is atime-dependent external force, the canonical transformations that make the changes in timedescribed by the dynamics will be different from one time to another. The assumption holdsat least for closed systems that are isolated from outside influences. IV. GENERATORS
We consider classical Hamiltonian dynamics for a Hamiltonian H that is a function ofthe canonical coordinates and momenta and does not depend on time. A function F ofthe canonical coordinates and momenta that represents a physical quantity at time zero ischanged by the dynamics between time zero and time t to a function F ( t ) of the canonicalcoordinates and momenta that is determined by the equation of motion dF ( t ) dt = [ F ( t ) , H ] (4.1)and the boundary condition that F ( t ) is F at time zero. We write [ F, G ] for the Poissonbracket of any two functions F and G of the canonical coordinates and momenta. Whenthe series converges, the solution of the equation of motion (4.1) is F ( t ) = F + t [ F, H ] + 12 t [[ F, H ] , H ] ... + 1 k ! t k [ ... [ F, H ] ...H ] + ... (4.2)in which the bracket with H is taken k times in the term with t k . The change of functionsof the canonical coordinates and momenta between time zero and time s is a canonical8ransformation. The same canonical transformation makes the change determined by thedynamics between time t ′ and t ′ + s . This is the canonical transformation that correspondsto the time translation, the change in the time coordinate from t to t − s , as is explainedin the discussion leading to Eqs. (3.2). As a function of s , these canonical transformationsform a one-parameter group; this is to be shown in Problem 4.1.The Hamiltonian function H generates a one-parameter group of canonical transforma-tions. Conversely, every one-parameter group of canonical transformations has a generator,a function of the canonical coordinates and momenta, that acts like the Hamiltonian; thisis shown in the Appendix. For each of the ten basic one-parameter groups of changes ofcoordinates, we assume there is an identified one-parameter group of corresponding canon-ical transformations. We give the generators different names. The Hamiltonian H is thegenerator of the canonical transformations that correspond to time translations. We let P , P , P be the generators of the one-parameter groups of canonical transformations thatcorrespond to space translations in the x , y , z directions, let J , J , J be the generators ofthe one-parameter groups of canonical transformations that correspond to rotations aroundthe x , y , z axes, let G , G , G be the generators of the one-parameter groups of canoni-cal transformations that correspond to Galilei transformations for velocities in the x , y , z directions, and let K , K , K be the generators of the one-parameter groups of canonicaltransformations that correspond to Lorentz transformations for velocities in the x , y , z di-rections. We write ~P , ~J , ~G , and ~K for the sets of three generators. We are making plusand minus sign conventions by saying that the canonical transformations generated by H , ~P , ~G , and ~K with signs as in Eq. (4.2) correspond to the changes of coordinates with thesigns in Eqs. (2.3), (2.1), (2.2), and (2.7). The sign convention for ~J is that the canonicaltransformations generated by J , for example, correspond to y ′ = y cos θ − z sin θz ′ = z cos θ + y sin θ. (4.3)A one-parameter group of canonical transformations does not completely determine agenerator; adding a constant to a generator does not change the transformations it gener-ates. The transformations do determine a generator to within addition of constants; addinga function of the canonical coordinates and momenta to a generator does change the trans-formations it generates when the added function is not a constant.9he one-parameter group of canonical transformations that corresponds to a one-parameter group of changes of coordinates can be identified by seeing that the way thecanonical transformations change particular functions of the canonical coordinates and mo-menta that describe particular physical quantities is the way those quantities are supposedto be changed by the changes of coordinates. Examples are worked out in Section VII. Problem 4.1.
For each function F of the canonical coordinates and momenta let C t ( F ) bethe F ( t ) given by Eq. (4.2). Show that C u ( C t ( F )) = C t + u ( F ) . (4.4)Hint: Compare the power series you get for C u ( C t ( F )) with the(Power Series) × (Power Series) = Power Seriesfor e t e u = e t + u . This shows that the canonical transformations generated by a Hamiltonianform a one-parameter group. V. BRACKET RELATIONS
We will show that, when the correct constants are added to them, the generators H , ~P , ~J , ~G for the Galilei group satisfy the Poisson-bracket relations[ J j , J k ] = ǫ jkm J m , [ J j , P k ] = ǫ jkm P m , (5.1)[ J j , G k ] = ǫ jkm G m , [ G j , H ] = P j , (5.2)[ G j , P k ] = δ jk M (5.3)with M a real number, the generators H , ~P , ~J , ~K for the Poincar´e group satisfy the Poisson-bracket relations (5.1), the same as for the Galilei group, and[ J j , K k ] = ǫ jkm K m , [ K j , H ] = P j , (5.4)[ K j , K k ] = − ǫ jkm J m , [ K j , P k ] = δ jk H, (5.5)and all the other Poisson brackets of the generators H , ~P , ~J , and ~G or ~K are zero. Firstwe show that each bracket relation is true if a constant is added to its right side. Then, inSection VI, we will show that all the constants except M can be eliminated when constantsare added to the generators. 10 . The pattern Our first calculation of a Poisson bracket of two generators will be a pattern for the others.To find [ G , H ], we consider a product of four infinitesimal canonical transformations madefrom generators G , H , G , H with parameters ǫ , δ , − ǫ , − δ . We calculate power seriesto second order in ǫ and δ . If either ǫ or δ is zero, the product of the four canonicaltransformations will be just the product of one canonical transformation and its inverse,which is the identity transformation, the same as when ǫ and δ are both zero, so the lowest-order terms in the power series for the product will be proportional to ǫδ , not ǫ , ǫ , δ , or δ .We need to go only to first order in ǫ and first order in δ . The product of the four canonicaltransformations takes each function F of the canonical coordinates and momenta throughthe sequence of transformations F → F ′ = F + ǫ [ F, G ] → F ′′ = F ′ + δ [ F ′ , H ′ ] → F ′′′ = F ′′ − ǫ [ F ′′ , G ′′ ] → F ′′′′ = F ′′′ − δ [ F ′′′ , H ′′′ ] . (5.6)The generators are changed the same as any other functions of the canonical coordinatesand momenta. To the first order that we need, the transformed generators are H ′ = H + ǫ [ H, G ] G ′′ = G + δ [ G , H ] H ′′′ = H. (5.7)Using these, the Jacobi identity[[ A, B ] , C ] = [[ C, B ] , A ] + [[ A, C ] , B ] , (5.8)and the antisymmetry property of the Poisson bracket,[ A, B ] = − [ B, A ] , (5.9)we find that the result of the product of the four canonical transformations described byEq. (5.6) is that F → F ′′′′ = F − ǫδ [ F, [ G .H ]] . (5.10)11his is the same as the lowest-order term of the canonical transformation generated by[ G , H ] with parameter − ǫδ . We assume this must be the lowest-order term of the canonicaltransformation that corresponds to the product of the four changes of coordinates thatcorrespond to the four canonical transformations generated by G and H . By calculatingthis product of changes of coordinates, we can find [ G , H ].The sequence of four changes of coordinates corresponding to the four canonical trans-formations generated by G , H , G , and H with parameters ǫ , δ , − ǫ , − δ the same as forthe canonical transformations in Eq. (5.6) is x, t → x ′ = x − ǫt, t ′ = t → x ′′ = x ′ , t ′′ = t ′ − δ → x ′′′ = x ′′ + ǫt ′′ , t ′′′ = t ′′ → x ′′′′ = x ′′′ , t ′′′′ = t ′′′ + δ (5.11)with no changes in y and z . The product of these four changes of coordinates, which is theresult of the sequence, is x → x ′′′′ = x − ǫδ (5.12)with no change in y , z and t . It is a space translation in the x direction. The canonicaltransformation that corresponds to this change of coordinates gives F → F − ǫδ [ F, P ] (5.13)to lowest order. Comparing with Eq. (5.10), we see that [ G , H ] and P must generate thesame canonical transformations. This implies that [ G , H ] is either P or P plus a constant.We can show similarly that [ G j , H ] is either P j or P j plus a constant for j = 1 , ,
3. Asimilar calculation for [ K j , H ] is to be done as Problem 5.3.To find the Poisson bracket of another pair of generators, we can use Eq. (5.10) with G and H replaced by that pair, and calculate the product of the corresponding four changesof coordinates. When the changes of coordinates commute, the product of the four changesof coordinates is the identity change of coordinates, no change at all, and the generator ofthe corresponding canonical transformation is either zero or a constant. Thus we find thateach of the Poisson brackets [ H, P k ] , [ H, J k ] , [ P j , P k ] , (5.14)12 P k , J k ] , [ G k , J k ] , [ K k , J k ] , (5.15)[ G j , G k ] , [ G j , P k ] , (5.16)is either zero or a constant. For all of these brackets except the last, the different changesof coordinates commute simply because they change different coordinates.When a calculation is needed, it needs to be only to second order in ǫ and δ . We need togo only to first order in ǫ and first order in δ ; just as in the product of the four canonicaltransformations, if either ǫ or δ is zero, the product of the four changes of coordinates willbe just the product of one change and its inverse, which is the identity change, the same aswhen ǫ and δ are both zero, so the lowest-order terms in the power series for the productwill be proportional to ǫδ , not ǫ , ǫ , δ , or δ . B. Rotations
To first order, the rotation described by Eq. (4.3) is also described by ~r ′ = ~r + ǫ ˆ x × ~r (5.17)with ǫ the infinitesimal value of the angle θ . We write ˆ x , ˆ y , ˆ z for unit vectors in the x , y , z directions. To find the Poisson bracket [ J , J ], we calculate the product of four rotations,around the x , y , x , y axes, through the angles ǫ , δ , − ǫ , − δ : ~r → ~r ′ = ~r + ǫ ˆ x × ~r → ~r ′′ = ~r ′ + δ ˆ y × ~r ′ → ~r ′′′ = ~r ′′ − ǫ ˆ x × ~r ′′ → ~r ′′′′ = ~r ′′′ − δ ˆ y × ~r ′′′ . (5.18)To lowest order, the result is ~r → ~r ′′′′ = ~r − ǫδ ˆ x × (ˆ y × ~r ) + ǫδ ˆ y × (ˆ x × ~r )= ~r − ǫδ (ˆ x · ~r ) ˆ y + ǫδ (ˆ y · ~r ) ˆ x = ~r − ǫδ (ˆ x × ˆ y ) × ~r = ~r − ǫδ ˆ z × ~r. (5.19)13he change of coordinates that is the product of the four rotations is rotation by − ǫδ aroundthe z axis. The corresponding canonical transformation gives F → F − ǫδ [ F, J ] (5.20)in the lowest order. Comparing this with Eq. (5.10) with G and H replaced by J and J ,we conclude that [ J , J ] is either J or J plus a constant. We can find similarly that theEq. (5.1) for each [ J j , J k ] is true with a constant added. C. Rotations and translations
For [ J , P ], the product of the four changes of coordinates is ~r → ~r ′ = ~r + ǫ ˆ x × ~r → ~r ′′ = ~r ′ + δ ˆ y → ~r ′′′ = ~r ′′ − ǫ ˆ x × ~r ′′ → ~r ′′′′ = ~r ′′′ − δ ˆ y. (5.21)To lowest order, the result is that ~r → ~r ′′′′ = ~r − ǫδ (ˆ x × ˆ y )= ~r − ǫδ ˆ z. (5.22)It is a space translation in the z direction. The corresponding canonical transformation gives F → F − ǫδ [ F, P ] (5.23)to lowest order. Comparing this with Eq. (5.10) for J and P , we conclude that [ J , P ] iseither P or P plus a constant. We can do a similar calculation for each [ J j , P k ] with j and k different. For [ J k , P k ], the changes of coordinates commute, because the rotation and thespace translation change different coordinates, so the product of the four changes of coordi-nates is no change at all, and the generator of the corresponding canonical transformationis either zero or a constant. Thus we find that the Eq. (5.1) for each [ J j , P k ] is true with aconstant added. Similar calculations for [ J j , G k ] and [ J j , K k ] are to be done as Problems 5.1and 5.2. 14 . Lorentz transformations To first order, the Lorentz transformation described by Eq. (2.7) is also described by ~r ′ = ~r − ǫ ˆ z t, t ′ = t − ǫ ˆ z · ~r (5.24)with ǫ the infinitesimal value of the parameter α . For [ K , K ], the product of the fourchanges of coordinates is ~r, t → ~r ′ = ~r − ǫ ˆ x t, t ′ = t − ǫ ˆ x · ~r → ~r ′′ = ~r ′ − δ ˆ y t ′ , t ′′ = t ′ − δ ˆ y · ~r ′ → ~r ′′′ = ~r ′′ + ǫ ˆ x t ′′ , t ′′′ = t ′′ + ǫ ˆ x · ~r ′′ → ~r ′′′′ = ~r ′′′ + δ ˆ y t ′′′ , t ′′′′ = t ′′′ + δ ˆ y · ~r ′′′ . (5.25)To lowest order, the result is that ~r → ~r ′′′′ = ~r − ǫδ (ˆ y · ~r ) ˆ x + ǫδ (ˆ x · ~r ) ˆ y = ~r − ǫδ (ˆ y × ˆ x ) × ~r = ~r + ǫδ ˆ z × ~r (5.26)and t is not changed. The product of the four Lorentz transformations is rotation by ǫδ around the z axis. The corresponding canonical transformation gives F → F + ǫδ [ F, J ] (5.27)to lowest order. Comparing this with Eq. (5.10) for K and K , we conclude that [ K , K ] iseither − J or − J plus a constant. We can find similarly that the Eq. (5.5) for each [ K j , K k ]is true with a constant added. E. Lorentz transformations and translations
For [ K , P ], the product of the four changes of coordinates is z, t → z ′ = z − ǫ t, t ′ = t − ǫ z → z ′′ = z ′ + δ, t ′′ = t ′ → z ′′′ = z ′′ + ǫ t ′′ , t ′′′ = t ′′ + ǫ z ′′ → z ′′′′ = z ′′′ − δ, t ′′′′ = t ′′′ (5.28)15ith no changes in x and y . To lowest order, the result is that x , y , z are not changed and t → t ′′′′ = t + ǫδ. (5.29)It is a time translation. The corresponding canonical transformation gives F → F − ǫδ [ F, H ] (5.30)to lowest order. Comparing this with Eq. (5.10) for K and P , we conclude that [ K , P ] iseither H or H plus a constant. We can find similarly that the Eq. (5.5) for each [ K k , P k ] istrue with a constant added. For [ K j , P k ] with j and k different, the four changes of coordi-nates commute, so their product is no change at all, and the generator of the correspondingcanonical transformation is either zero or a constant. Thus we find that the Eq. (5.5) foreach [ K j , P k ] is true with a constant added.At every step, we see a another new simplification that the method brings. Because eachchange of coordinates needs to be only to first order, matrices are not needed for eitherrotations or Lorentz transformations. It is enough to use simple three-vector descriptions,and they fit readily with translations. When the calculations to be done in problems areincluded, we can see that every one of the bracket relations for the Galilei group and thePoincar´e group is true with a constant added. Problem 5.1.
Use language from Eqs. (2.2) and (5.21) to calculate the product of the fourchanges of coordinates for [ J , G ] and see that [ J , G ] is either G or G plus a constant.Consider the similar calculation for [ J , G ] and see that [ J , G ] is either zero or a constant.We can find similarly that the Eq. (5.2) for each [ J j , G k ] is true with a constant added. Problem 5.2.
Use language from Eqs. (5.21) and (5.25) for the product of the four changesof coordinates to show that [ J , K ] is either K or K plus a constant. Show that [ J , K ]is zero or a constant and that the Eq. (5.4) for each [ J j , K k ] is true with a constant added. Problem 5.3.
Calculate the product of the four changes of coordinates for [ K , H ] andsee that [ K , H ] is either P or P plus a constant. This can be done simply by puttingthe Lorentz transformations of time coordinates from Eqs. (5.25) into Eqs. (5.11); this is allthat is needed to change the Galilei transformations to Lorentz transformations. We canfind similarly that the Eq. (5.4) for each [ K j , H ] is true with a constant added.16 I. ELIMINATING CONSTANTS
We have shown that every one of the bracket relations for the Galilei group and thePoincar´e group is true with a constant added. Now we will show that all the constantsexcept M can be eliminated when constants are added to the generators.The presence of constants is limited because the Poisson brackets are antisymmetric andsatisfy the Jacobi identity. For example, from the antisymmetry (5.9), the Jacobi identity(5.8), and bracket relations possibly with constants, we can see that[ P , P ] = [[ J , P ] , P ]= [[ P , P ] , J ] + [[ J , P ] , P ]= 0 (6.1)because inside the brackets the constants give the same result as zero. We can calculatesimilarly that [ P j , P k ] = 0 , [ G j , G k ] = 0 , (6.2)[ P j , H ] = 0 , [ J j , H ] = 0 . (6.3)From the antisymmetry of the brackets, we have[ J j , J k ] = ǫ jkm J m + ǫ jkm b m (6.4)where b , b , b are real numbers. By adding these constants to the generators J , J , J ,we get [ J j , J k ] = ǫ jkm J m . (6.5)From the antisymmetry, the Jacobi identity, and bracket relations possibly with constants,we get [ J , P ] = [[ J , J ] , P ]= [[ P , J ] , J ] + [[ J , P ] , J ]= [ J , P ] + [ J , P ] (6.6)and similarly [ J , P ] = [ J , P ] + [ J , P ] (6.7)17rom which we see that [ J , P ] is zero and conclude that similarly each [ J k , P k ] is zero. Inthe same way, we get [ J , P ] = [[ J , J ] , P ]= [[ P , J ] , J ] + [[ J , P ] , J ]= − [ J , P ] (6.8)and conclude that each [ J j , P k ] is − [ J k , P j ] so that[ J j , P k ] = ǫ jkm P m + ǫ jkm b m (6.9)with real numbers b , b , b . By adding these constants to the generators P , P , P , we get[ J j , P k ] = ǫ jkm P m . (6.10)We can see similarly that by adding constants to the generators G , G , G and K , K , K we can get [ J j , G k ] = ǫ jkm G m (6.11)and [ J j , K k ] = ǫ jkm K m . (6.12)In the same way, we get[ G , H ] = [[ J , G ] , H ]= [[ H, G ] , J ] + [[ J , H ] , G ]= [ J , P ] = P (6.13)and conclude that, similarly, [ G j , H ] = P j (6.14)and [ K j , H ] = P j . (6.15)In the same way, we get[ G , P ] = [[ J , G ] , P ]= [[ P , G ] , J ] + [[ J , P ] , G ]= 0 (6.16)18nd [ G , P ] = [[ J , G ] , P ]= [[ P , G ] , J ] + [[ J , P ] , G ]= [ G , P ] (6.17)and conclude that [ G j , P k ] = δ jk M (6.18)with M a real number. The same equations (6.16) and (6.17) hold with ~K in place of ~G .From them, we conclude that [ K j , P k ] = δ jk H + δ jk M (6.19)with M a real number. By adding M to H we get[ K j , P k ] = δ jk H. (6.20)The one remaining step, to see that there are no constants in the equations for [ K j , K k ],is to be done as Problem 6.1. When that is included, we can see that all the constants exceptthe M for [ G k , P k ] can be eliminated by adding constants to the generators. When that isdone, the generators ~P , ~J , ~G , and ~K are completely determined; we cannot add constantsto them without putting constants back in the bracket relations. For the Poincar´e group, H is also completely determined; we can not add a constant to H without putting a constantback in the equation for [ K k , P k ]. For the Galilei group, H is not completely determined.We can still add a constant to H . It will not change the bracket relations for the Galileigroup because in them H never occurs outside on the right. In examples, we will see thatthe M of [ G k , P k ] is a mass; it is the nonrelativistic limit of the H of [ K k , P k ].Removing the constants puts the bracket relations into the simple standard forms that aregenerally used. When we look at examples in Section VII, we will see that the adjustmentof constants leaves the generators in familiar forms that can be identified with physicalquantities. Then the bracket relations for the generators correspond to bracket relationsfor physical quantities. Constants in the bracket relations for generators do not changethe group structure, but constants in the bracket relations for physical quantities can beimportant, as we know from the example of Planck’s constant in the commutation relationsfor position and momentum. 19 roblem 6.1. Use Eq. (6.5) for [ J , J ], other bracket relations possibly with constants,the antisymmetry of Poisson brackets, and the Jacobi identity applied to [[ K , K ] , J ], toshow that [ K , K ] is − J without a constant. We can conclude that similarly [ K j , K k ] is − ǫ jkm J m without a constant. VII. EXAMPLES
Consider a single object moving in three-dimensional space. There are three degreesof freedom. We use canonical coordinates and momenta q , q , q and p , p , p that arecomponents of three-dimensional vectors ~q and ~p . Functions of the canonical coordinatesand momenta that satisfy the bracket relations are ~P = ~p, ~J = ~q × ~p,~G = M ~q, H = ~p M (7.1)for the Galilei group and ~P = ~p, ~J = ~q × ~p,~K = H~q, H = q ~p + M (7.2)for the Poincar´e group. Checking that these are solutions of Eqs. (5.1)-(5.5) is to be doneas Problem 7.1.Physical interpretation is established by identifying a three-dimensional vector functionof the canonical coordinates and momenta that can represent the position of the object. Weassume it is changed by the canonical transformations generated by H , ~P , ~J , and ~G or ~K the way the position coordinate vector should be changed by the corresponding coordinatechanges in the Galilei or Poincar´e group. The only possibility is that ~q represents theposition. Then, in the nonrelativistic case, for the Galilei group, the velocity is ~V = [ ~q, H ] = ~pM (7.3)so ~p = M ~V is the momentum of an object with mass M and velocity ~V , the angularmomentum ~q × ~p is ~J , and H = 12 M ~V (7.4)20s the kinetic energy. In the relativistic case, for the Poincar´e group, the velocity is ~V = [ ~q, H ] = ~p √ ~p + M (7.5)so ~p = M ~V / q − ~V is the relativistic momentum of an object with mass M and velocity ~V , the relativistic angular momentum ~q × ~p is ~J , and H = M q − ~V (7.6)is the relativistic energy.Conversely, if it is assumed that ~q represents the position of the object and is changedby the canonical transformations generated by H , ~P , ~J , and ~G or ~K the way the positioncoordinate vector should be changed by the corresponding coordinate changes in the Galileior Poincar´e group, then the generators H , ~P , ~J , and ~G or ~K can be put in the forms (7.1) or(7.2) by a canonical transformation that changes only the canonical momenta ~p and leavesthe canonical coordinates ~q unchanged. This is a gauge transformation. For two objects, we use canonical coordinates and momenta ~q (1) , ~q (2) and ~p (1) , ~p (2) . Forthe Galilei group, we can let ~P = ~p (1) + ~p (2) , ~J = ~q (1) × ~p (1) + ~q (2) × ~p (2) ,~G = m ~q (1) + m ~q (2) , H = ( ~p (1) ) m + ( ~p (2) ) m + V (7.7)in which V is the potential energy that describes the interaction between the two objects.The bracket relations for the Galilei group imply that m + m is M and that [ V, ~P ], [
V, ~G ],and [
V, ~J ] are zero. There are restrictions on V . To describe them, we use center-of-massand relative coordinates and momenta ~Q = m m + m ~q (1) + m m + m ~q (2) , ~P ( tot ) = ~p (1) + ~p (2) ,~q = ~q (1) − ~q (2) , ~p = m m + m ~p (1) − m m + m ~p (2) . (7.8)In terms of these, the generators (7.7) are ~P = ~P ( tot ) , ~J = ~Q × ~P + ~q × ~p,~G = M ~Q, H = ~P M + ~p µ + V (7.9)21ith µ = m m / ( m + m ). We will write ~P for ~P ( tot ) ; it will be the same whether we arethinking about generators or canonical momenta. The ~Q , ~P , ~q , ~p are canonical coordinatesand momenta. Poisson brackets can be written with derivatives with respect to them. Thusthe restrictions on V from the bracket relations give ∂V∂Q j = [ V, P j ] = 0 ∂V∂P j = − M [ V, G j ] = 0 . (7.10)This means that V can not depend on ~Q or ~P ; it can depend only on ~q and ~p . Then [ V, ~J ]is [
V, ~q × ~p ]. The bracket relations imply it is zero. It is to be shown in Problem 7.2 that thisimplies that V is a function only of ~q , ~p and ~q · ~p .For the Poincar´e group, describing interactions is not so simple. Generators can bewritten as sums of generators for single objects, describing objects that do not interact.If H is changed to describe an interaction, either ~K or ~P must be changed too, because H is [ K k , P k ]. Generators that describe interactions can be made to satisfy the bracketrelations, but if it is also assumed that the position coordinates are changed by thecanonical transformations the way they should be changed, corresponding to coordinatechanges in the Poincar´e group, then there are no interactions; the accelerations of all theobjects are zero. This shows that there are limits to the use of canonical transformationsfor a representation of the Poincar´e group that contains a Hamiltonian description of therelativistic dynamics of different objects at the same time without fields.
Problem 7.1.
Show that the functions H , ~P , ~J , ~G given by Eqs.(7.1) satisfy the Poisson-bracket relations (5.1) - (5.3) for the Galilei group and that the other Poisson brackets ofthese functions are zero. Show that the functions H , ~P , ~J , ~K given by Eqs.(7.2) satisfy thePoisson-bracket relations (5.1), (5.4) and (5.5) for the Poincar´e group and that the otherPoisson brackets of these functions are zero. Problem 7.2.
Consider changes of canonical coordinates and momenta ~q and ~p made byrotations around a fixed axis along the direction of a unit vector ˆ e . To first order, for aninfinitesimal value ǫ of the angle of rotation, ~q → ~q + ǫ ˆ e × ~q p → ~p + ǫ ˆ e × ~p (7.11)as in Eq. (5.17). These changes of ~q and ~p form a one-parameter group of canonical trans-formations. The generator is ˆ e · ~q × ~p ; show this by showing thatˆ e × ~q = [ ~q, ˆ e · ~q × ~p ] , ˆ e × ~p = [ ~p, ˆ e · ~q × ~p ] . (7.12)This implies that if V is a function of ~q and ~p and [ V, ~q × ~p ] is zero, then V is not changedwhen ~q and ~p are rotated. This means that V can only be a function of ~q , ~p and ~q · ~p . Problem 7.3.
For one-dimensional space, the Galilei group has generators H , P , G , andbracket relations [ P, H ] = F, [ G, H ] = P, [ G, P ] = M (7.13)with F and M constants. Neither F nor M can generally be eliminated by adding constantsto the generators. Show this with an example. Specifically, consider a single object withcanonical coordinates and momenta q and p . Let P be p , let G be M q , and find a function H of q and p so that the bracket relations (7.13) are satisfied with constants F and M thatcan not be removed. Show that then p is M V with V the velocity [ q, H ]. Since F is [ p, H ],it is the time derivative of the momentum. It is a force. A constant force is allowed becauseGalilei transformations do not change accelerations. It is not allowed for three-dimensionalspace where there are also rotations. Problem 7.4.
For one-dimensional space, the Poincar´e group has generators H , P , K , andbracket relations [ P, H ] = F, [ K, H ] = P, [ K, P ] = H (7.14)with F a constant. This constant F generally can not be eliminated by adding constantsto the generators. Show this with an example. Specifically, consider a single object withcanonical coordinates and momenta q and p . Let P be p . Find functions H and K of q , p so that p is M V / √ − V , with V the velocity [ q, H ], and the bracket relations (7.14)are satisfied with a constant F that can not be removed. Since F is [ p, H ], it is the timederivative of the momentum. It is a force. A constant force is allowed because dp ′ /dt ′ after a Lorentz transformation is the same as dp/dt before. It is not allowed by Lorentztransformations for three-dimensional space.23 roblem 7.5. Consider a nonrelativistic particle with canonical coordinates and momenta ~q and ~p and generators of the Galilei group given by Eqs.(7.1). Suppose physical interpretationis established as described in the second and third paragraphs of this Section VII, so thevelocity and energy are described by Eqs.(7.3) and (7.4), the momentum ~p is M ~V , theangular momentum ~J is ~q × ~p , and the mass is M . Show that [ G j , P k ] = δ jk M impliesthat the momentum is changed by − M ~β when the space coordinates are changed by aGalilei transformation described by Eqs.(2.2) to coordinates relative to an origin movingwith velocity ~β . Suppose all the particles are described this way in the decay of a particle ofmass m into two particles with masses m and m . Show that conservation of momentumfor the two coordinate systems with the origin of the second moving relative to the firstimplies that m = m + m . That mass is not conserved in radioactive decays that werebeing observed before Einstein presented his relativity could have been seen as showing aneed for revision of the relativity of Galileo and Newton. Appendix: From subgroups to generators
Here we show that every one-parameter group of canonical transformations has a gener-ator function that acts like a Hamiltonian. We use the canonical coordinates and momenta q n and p n together as variables we call ζ j . We do not need to distinguish the half of thevariables ζ j that are q n from the half that are p n . The generator, which we call H , is to bea function of these variables. By letting F be ζ j in Eq. (4.2), we can see from the first-orderterm that the canonical transformations determine what the Poisson bracket [ ζ j , H ] is tobe. If there is an H , each ∂H/∂ζ k will be either [ ζ j , H ] or [ − ζ j , H ] for some j . We concludethat the canonical transformations determine what the ∂H/∂ζ k are to be. We do not as-sume there is a function H , so we can not write [ ζ j , H ] and ∂H/∂ζ k for the things that aredetermined by the canonical transformations. We will write [ ζ j , H ?] and ∂H ? /∂ζ k for themuntil we have proved there is a function H . The canonical transformations do not changethe Poisson brackets [ ζ j , ζ k ]. This means that[ ζ j , ζ k ] = [ ζ j + t [ ζ j , H ?] + ... , ζ k + t [ ζ k , H ?] + ... ]= [ ζ j , ζ k ] + t [[ ζ j , H ?] , ζ k ] + t [ ζ j , [ ζ k , H ?]] + ... (A.1)24hich implies that ∂∂ζ j ∂H ? ∂ζ k = ∂∂ζ k ∂H ? ∂ζ j (A.2)for all j, k . Let H ( ζ ) = Z ζζ (0) X j ∂H ? ∂ζ ′ j dζ ′ j . (A.3)The ζ stands for the set of variables ζ j . The ζ (0) stands for a particular set of values ofthe ζ j , which mark a particular point in the “phase” space for which the ζ j are coordinates.The integral is along a path in that space from the point ζ (0) to the point ζ . We will showthat the equations (A.2) imply that the integral does not depend on the path. It defines H as just a function of ζ . It gives ∂H∂ζ k = ∂H ? ∂ζ k . (A.4)Changing the point ζ (0) where the integral begins adds only a constant to H .We can show that the integral is the same along two different paths because it is zeroaround the closed loop going forward on one path and back on the other. Let η and λ bevariables that change along the two paths so that every point on the closed loop is markedby a different set of values of η and λ . Along the loop, the ζ j are functions of η and λ andthey change only when η or λ changes. The integral (A.3) around the closed loop is theintegral over the area enclosed by the loop in the plane of η and λ , I ( A η dη + A λ dλ ) = Z Area ( ∂A η ∂λ − ∂A λ ∂η ) dη dλ, (A.5)with A η = X j ∂H ? ∂ζ j ∂ζ j ∂η , A λ = X j ∂H ? ∂ζ j ∂ζ j ∂λ , (A.6)and ∂A η ∂λ − ∂A λ ∂η = X j X k ∂∂ζ j ∂H ? ∂ζ k − ∂∂ζ k ∂H ? ∂ζ j ! ∂ζ k ∂η ∂ζ j ∂λ , (A.7)which is zero from Eq. (A.2). ∗ email: [email protected] T. F. Jordan, “How relativity determines the Hamiltonian description of an object in classicalmechanics,” Phys. Lett. A , 123-130 (2003). T. F. Jordan, “Why -iDel is the momentum,” Am. J. Phys. , 1089-1093 (1975). T. F. Jordan,
Linear Operators for Quantum Mechanics (Wiley, New York, 1969; Dover, NewYork, 2007), Chapters 6 and 7. References to the original work of Wigner and Bargmann canbe found here. E. C. G. Sudarshan and N. Mukunda,
Classical Dynamics: A Modern Perspective (Wiley, NewYork, 1974). M. Pauri and G. M. Prosperi, “Canonical realizations of the Poincar group. I. General theory,”J. Math. Phys. , 1503-1521 (1975), and references therein. F. R. Halpern,
Special Relativity and Quantum Mechanics (Prentice-Hall, Engelwood Cliffs,NJ, 1968), Chapter 3. V. Bargmann, “On unitary ray representations of continuous groups,” Ann. Math. , 1-46(1954). J. Voisin, “On Some Unitary Representations of the Galilei Group I. Irreducible Representa-tions,” J. Math. Phys. , 1519-1529 (1965). S. Weinberg,
The Quantum Theory of Fields, Volume 1 (Cambridge University Press, Cam-bridge, U. K., 1995), Chapter 2. References to the original work of Wigner and Bargmann canbe found here. B. L. van der Waerden,
Group Theory and Quantum Mechanics (Springer-Verlag, New York,1974). T. F. Jordan, “Steppingstones in Hamiltonian dynamics,” Am. J. Phys. , 1095-1099 (2004). A. Das and T. Ferbel,
Introduction to Nuclear and Particle Physics (World Scientific, Singapore,2003), particularly Section 10.3. This introduces symmetries with Poisson brackets in classicalmechanics and then converts to quantum mechanics. P. A. M. Dirac, “Forms of Relativistic Dynamics,” Rev. Mod. Phys. , 392-399 (1949). L. H. Thomas, “The relativistic dynamics of a system of particles interacting at a distance,”Phys. Rev. , 868-872 (1952). B. Bakamjian and L. H. Thomas, “Relativistic particle dynamics. II,” Phys. Rev. , 1300-1310(1953). B. Bakamjian, “Relativistic particle dynamics,” Phys. Rev. , 1849-1851 (1961). L. L. Foldy, “Relativistic particle systems with interaction,” Phys. Rev. , 275-288 (1961). D. G. Currie, T. F. Jordan, and E. C. G. Sudarshan, “Relativistic invariance and Hamiltoniantheories of interacting particles,” Revs. Modern Phys. , 350-375 (1963). D. G. Currie, “Interaction contra classical relativistic Hamiltonian particle mechanics,” J. Math.Phys. , 1470-1488 (1963). J. T. Cannon and T. F. Jordan, “A no-interaction theorem in classical relativistic Hamiltonianparticle dynamics,” J. Math. Phys. , 299-307 (1964). H. Leutwyler, “A no-interaction theorem in classical relativistic Hamiltonian particle mechan-ics,” Nuovo Cimento , 556-567 (1965)., 556-567 (1965).