Chapter 6 1-D Continuous Groups
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1 Chapter 6 1-D Continuous Groups Continuous groups consist of group eleents labelled by one or ore continuous variables, say a 1, a 2,, a r, where each variable has a well- defined range. This chapter explores: 1. the group of rotations in a plane SO 2, 2. the group of translations in 1-D T 1. These groups are 1-D (i.e. the group eleents only depend on one continuous paraeter) and they are necessarily abelian. A Lie group is an infinite group whose eleents can be paraetrized soothly and analytically. The definition of these properties requires introducing algebraic and geoetric structures beyond group ultiplication in pure group theory (which has been our only concern so far for discrete groups). The precise forulation of the general theory of Lie groups requires considerable care; it involves notions of topology and differential geoetry. All known continuous syetry groups of interest in physics, however, are groups of atrices for which the additional algebraic and geoetric structures are already failiar. These groups are usually referred to as linear Lie groups or classical Lie groups. We shall introduce the ost iportant features and techniques of the theory of linear Lie groups by studying the iportant syetry groups of space and tie, starting fro this chapter. Experience with these physically useful exaples should provide a good foundation for studying the general theory of Lie groups. [ Chevalley, Gilore, Miller, Pontrjagin, Weyl ] Suary by Sections 1. Definition of SO 2 & its general properties. 2. Generator of the group as deterined by the group structure near e.. 3. IRs of SO(2) derived using eigenvectors of the generator. Role of global properties. 4. Invariant integration easure on the group anifold. Orthonorality and copleteness properties of the rep functions. Their relation to Fourier analysis.. 5. Multi- valued reps & their relation to the topology of the group anifold. 6. T Generators are identified with easurable physical quantities. All ideas introduced in this chapter will find significant generalizations in later chapters. 6.1 The Rotation Group SO(2) Consider a syste syetric under rotations in a plane, around a fixed point O.
2 2 6._1DContinuousGroups.nb Let e 1 and e 2 be the Cartesian orthonoral basis vectors [ see Fig. 6.1 ]. A rotation through angle Φ by R Φ gives (6.1-1) R Φ e 1 e 1 cosφ e 2 sinφ or equivalently, R Φ e 2 e 1 sinφ e 2 cosφ (6.1-2) R Φ e i e j R Φ j i with the atrix R Φ j i given by cosφ sinφ (6.1-3) Φ sinφ cosφ Fig. 6.1 Rotation in a plane. Let x e i x i Then x transfors under rotation R Φ according to: Writing x x ' R Φ x R Φ e i x i e j R Φ j i xi x ' e i x ' i we obtain (6.1-4) x ' j R Φ j i xi or in atrix for ' Φ Geoetrically, it is obvious that the length of vectors reains invariant under rotations, i.e. or x 2 x i x i x ' 2 x i ' x ' i T ' T ' T T Φ Φ Thus (6.1-5) T Φ Φ Φ where T denotes the transpose of, and is the unit atrix. Real atrices satisfying the condition (6.1-5) are called orthogonal atrices and denoted by O n, where n is their diension. In the present case, n 2.
3 6._1DContinuousGroups.nb Eq. (6.1-5) iplies that det T Φ Φ det so that det T Φ det Φ 1 [ det det det ] det Φ 2 1 [ det T det ] det Φ ±1 Now, the identity operator is given by E R 0 or in atrix for 0 with det 0 det 1 Operators that evolve continuously fro E ust therefore satisfy (6.1-6) det Φ 1 Φ They are called proper rotations. Matrices satisfying (6.1-6) are said to be special. In the present case, they are special orthogonal atrices of rank 2; and are designated as SO(2) atrices. Operators with det Φ 1 are called iproper rotations. They can always be expressed as a proper rotation followed by a reflection. Theore 6.1: There is a one-to-one correspondence between rotations in a plane and SO(2) atrices. The proof of this theore is left as an exercise. [Proble 6.1] This correspondence is a general one, applicable to SO n atrices and rotations in the Euclidean space of diension n for any n. O(2) Group The O 2 atrices with Φ 0, for a group called O 2. The law of coposition is (6.1-7) R Φ 2 R Φ 1 R Φ 2 Φ 1 with the understanding that any Φ outside 0, is brought back to Φ 0, using (6.1-8) R Φ R Φ n n integer Obviously R Φ 0 E and R Φ 1 R Φ R Φ Note that O 2 is not a Lie group since the iproper rotations are not continuously related to E. Theore 6.2 (2-D Rotation Group): The 2-D proper rotations R Φ for a Lie group called the rotation group R 2 or SO 2 group.
4 4 6._1DContinuousGroups.nb The set Φ 0, corresponds to all points on the unit circle which defines the "topology" of the group paraeter space ( anifold ) [ cf Fig. 6.2 ]. Obviously, this paraeterization is not unique. Any onotonic function Ξ Φ of Φ over the above doain can serve as an alternative label for the group eleent. The structure of the group and its reps should not be affected by the labelling schee. We shall coe back to the "naturalness" of the variable Φ and the question of topology in a later section. Both issues are iportant for the analysis of general continuous groups. Note that Eq. (6.1-7) iplies R Φ 1 R Φ 2 R Φ 2 R Φ 1 Φ 1, Φ 2 Thus, the group SO(2) is abelian. Fig. 6.2 SO(2) group anifold. 6.2 Generator of SO(2) Consider an infinitesial SO(2) rotation by the angle dφ. Differentiability of R Φ in Φ requires that R dφ differs fro R 0 E by only a quantity of order dφ which we define by the relation (6.2-1) R E i dφ J where the factor i is included by convention and for later convenience. The quantity J is independent of the rotation angle dφ. Next, consider the rotation R Φ dφ, which can be evaluated in two ways: (6.2-2) R Φ dφ R Φ R dφ R Φ i dφ R Φ J R Φ dφ R Φ dφ d R Φ Coparing the two equations, we obtain the differential equation, d R Φ (6.2-3) i R Φ J with the boundary condition R 0 E. The solution to Eq. (6.2-3) is therefore unique, we present it in the for of a theore.
5 6._1DContinuousGroups.nb Theore 6.3 (Generator of SO(2)): All 2-D proper rotations can be expressed in ters of the operator J as: (6.2-4) R Φ e i Φ J J is called the generator of the group. Thus the general group structure and its reps are deterined ostly by J which, in turn, is specified by local behavior of the group near the identity. This is typical of Lie groups. Note however that certain global properties of the group, such as Eq. (6.1-8), cannot be deduced fro (6.2-3). These global properties, ostly topological in nature, also play a role in deterining the IRs of the group, as we shall see in the next section. Let us turn fro this abstract discussion to the explicit representation of R Φ given by (6.1-3). We have, to first order in dφ, dφ 1 dφ dφ 1 Coparing with Eq. (6.2-1), we deduce (6.2-5) 0 i i 0 Thus is a traceless heritian atrix. It is easy to show that 2, 3, etc. Therefore, e i Φ i Φ 1 2 Φ2 i Φ3 cosφ i sinφ which reproduces Eq. (6.1-3). cosφ sinφ sinφ cosφ IRs of SO(2) Consider any rep of SO(2) defined on a finite diensional vector space V. Let U Φ be the operator on V which corresponds to R Φ. Then, according to Eq. (6.1-7), we ust have U Φ 2 U Φ 1 U Φ 2 Φ 1 U Φ 1 U Φ 2 with the sae understanding that U Φ U Φ n. For an infinitesial transforation, we can again define an operator corresponding to the generator J in Eq. (6.2-1). In order to avoid a proliferation of sybols, we use the sae letter J to denote this operator, (6.3-1) U dφ E i dφ J Repeating the arguents of the last section, we obtain (6.3-2) U Φ e i Φ J which is now an operator equation on V. If U Φ is to be unitary for all Φ, J ust be heritian with real eigenvalues.
6 6 6._1DContinuousGroups.nb Since SO(2) is an abelian group, all its IRs are 1-D. This eans that given any Α in a inial invariant subspace under SO(2) we ust have: (6.3-3) J Α Α Α (6.3-4) U Φ Α Α e i Φ Α where Α is an eigenvalue of J. It is easy to show that the for given by Eq. (6.3-4) autoatically satisfies the group ultiplication rule Eq. (6.1-7) for an arbitrary Α. However, in order to satisfy the global constraint Eq. (6.1-8), a restriction ust be placed on the eigenvalue Α. Indeed, we ust have e i n Α 1 n integer which iplies that Α is an integer. We denote this integer by, and the corresponding representation by U : (6.3-5) J (6.3-6) U Φ e i Φ Let us take a closer look at these reps: (i) When 0, This is the identity rep; (ii) When 1, R Φ U 0 Φ 1. R Φ U 1 Φ e i Φ This is an isoorphis between SO(2) group eleents and ordinary nubers on the unit circle in the coplex nuber plane. As R Φ ranges over the group space, U 1 Φ covers the unit circle once, in the clockwise sense; (iii) When 1, R Φ U 1 Φ e i Φ The situation is the sae as above, except that the unit circle is covered once in the counter- clockwise direction; (iv) When ±2, R Φ U ±2 Φ e 2 i Φ These are appings of the group paraeter space to the unit circle on the coplex nuber plane covering the latter twice in opposite directions. The general case follows in an obvious anner fro these exaples. We suarize these results in the for of a theore. Theore 6.4 ( IRs of SO(2) ): The single valued IRs of SO(2) are given by J where is any integer, and (6.3-7) U Φ e i Φ Of these, only the ±1 ones are faithful reps. The defining equation for R Φ, Eq. (6.1-3), is a 2-D rep of the group. It has to be reducible. Indeed, it is equivalent to a direct su of the ±1 representations. To see this. we note that due to Eq. (6.3-2), it suffices to diagonalize the atrix corresponding to the generator. J 0 i i 0
7 6._1DContinuousGroups.nb It is obvious that J has two eigenvalues, ±1; and the corresponding eigenvectors are: e ± 1 2 e 1 i e 2 [ Proble 6.2 ] Thus, with respect to the new basis. (6.3-8) J e ± ± e ± R Φ e ± e ± e i Φ 6.4 Invariant Integration Measure, Orthonorality & Copleteness Relations We now derive the orthonorality and copleteness relations for the representation functions U Φ e i Φ. Because Φ is a continuous variable, the suation over group eleents ust be replaced by an integration, and the integration easure ust be well defined. Now, any function Ξ Φ, onotonic in 0 Φ, can also serve as label. However, for an arbitrary function f of the group eleents, d Ξ f R Ξ Ξ ' Φ f R Φ f R Φ [ Ξ ' d Ξ ] Hence, "integration" of f over the group anifold is not well defined a priori. Now, the Rearrangeent Lea lies at the heart of the proof of ost iportant results of the representation theory. Therefore we seek to define an integration easure such that, (6.4-1) dτ R f R dτ R f S 1 R S G dτ SR f R If the group eleents are labelled by the paraeter Ξ, then dτ R Ρ R Ξ dξ where Ρ R Ξ is soe appropriately defined "weight function". Definition 6.1 (Invariant Integration Measure): A paraeterization R Ξ in group space with an associated weight function Ρ R Ξ is said to provide an invariant integration easure if Eq. (6.4-1) holds. The validity of Eq. (6.4-1) requires dτ R dτ SR which iposes the condition on the weight function, Ρ R Ξ (6.4-2) Ρ SR Ξ d Ξ SR d Ξ R This condition is autoatically satisfied if we define (6.4-3) Ρ R Ξ d Ξ E d Ξ R where Ξ E is the group paraeter around the identity eleent E and Ξ R Ξ ER is the corresponding paraeter around R. 2 In evaluating the right- hand side of the above equation, R is to be regarded as fixed; the dependence of Ξ ER on Ξ E is deterined by the group ultiplication rule.
8 8 6._1DContinuousGroups.nb The situation is siplest when Ξ SR linear in Ξ R,. This is the case when Ξ Φ is the rotation angle. The group ultiplication rule, Eq. (6.1-7), requires Φ ER Φ E Φ R Ρ R E ER R 1 Theore 6.5 (Invariant Integration Measure of SO(2) ): The rotation angle Φ, Fig. 6.1, and the volue easure dτ R dφ, provide the proper invariant integration easure over the SO(2) group space. If Ξ is a general paraeterization of the group eleent, then dτ R Ρ R Ξ dξ Ρ R Φ dφ dφ We ust have, therefore, Ρ R Ξ d Ξ The above discussion ay appear to be rather long-winded just to arrive at a relatively obvious conclusion. The otivation for including so uch detail is to set up a line of reasoning which can be applied to general continuous groups in later Chapters. Once an invariant easure is found, it is straightforward to write down the expected orthonorality and copleteness relations. Theore 6.6: The SO(2) representation functions U n Φ of Theore 6.4 satisfy the following orthonorality and copleteness relations: (6.4-4) 1 0 dφ U n Φ U Φ n (orthogonality) U n Φ U n Φ' Φ Φ' n (copleteness) Three siple rearks of general iportance are in order here: (i) These relations are natural generalizations of Theore 3.5 and 3.6 (for finite groups) to a continuous group; the only change is the replaceent of the finite su over group eleents by the invariant integration over the continuous group paraeter; (ii) Theore 6.6. with U n Φ given by Eq. (6.3-7), is equivalent to the classical Fourier Theore for periodic functions, the continuous group paraeter Φ and the discrete representation label n are the "conjugate variables" ; (iii) These relations are also identical to the results encountered in Chap.1, Eqs. (1.4-1) (1.4-2), in connection with the discrete translation group T d. Note, however, the roles of the group eleent label (discrete) and the representation label (continuous) are exactly reversed in coparison to the present case. 6.5 Multi-Valued Reps For later reference, we ention here a new feature of continuous groups the possibility of having ulti- valued representations. To introduce the idea, consider the apping (6.5-1) R Φ U 1 2 Φ e i 2 Φ
9 6._1DContinuousGroups.nb This is not a unique representation of the group, as (6.5-2) U 1 2 Φ e i Π i 2 Φ U 1 2 Φ whereas, we expect, on physical grounds, R 2 n Φ R Φ. However, since U Π Φ U 1 2 Φ Eq.(6.5-1) does define a one-to-two apping where each R Φ is assigned to two coplex nubers e i 2 Φ differing by a factor of 1. This is a two- valued representation in the sense that the group ultiplication law for SO(2) is preserved if either one of the two nubers corresponding to R Φ can be accepted. Clearly, we can also consider general appings, (6.6-3) R Φ U n Φ e i n Φ where n and are integers with no coon factors. For any pair n, this apping defines a "-valued representation" of SO(2) in the sae sense as described above. The following questions naturally arise: (i) Do continuous groups always have ulti-valued irreducible representation; (ii) How do we know whether (and for what values of do) ulti-valued iepresentations exist; (iii) When ulti-valued representations exist, are they realized in physical systes? In other words, does it ake sense to restrict our attention to solutions of classical and/or quantu- echanical systes only to those corresponding to singlevalued representations of the appropnate syetry groups? It turns out that the existence of ulti- valued representations is intiately tied to connectedness" a global topological property of the group paraeter space. In the case of SO(2), the group paraeter space (the unit circle) is "ultiplyconnected" 3, which iplies the existence of ulti- valued representations. Thus, it is possible to deterine the existence and the nature of ulti- valued representations fro an intrinsic property of the group. In regard to the last question posed above, so far as we know, both single- and double- valued representations, but no others, are realized in quantu echanical systes, and only single- valued representations appear in classical solutions to physical probles. The occurrence of double- valued representations can be traced to the connectedness of the group anifolds of syetries associated with the physical 3- and 4- diensional spaces. This observation will becoe clearer after we discuss the full rotation group and the Lorentz group in the next few chapters. 6.6 T 1 Rotations in the 2-diensional plane (by the angle Φ) can be interpreted as translations on the unit circle (by the arc length Φ). This fact accounts for the siilarity in the for of the irreducible representation function, U n Φ e i n Φ, in coparison to the case of discrete translation, t k n e i n k b, discussed in Chap. 1. The "copleentary" nature of these results has been noted in Sec We now extend the investigation to the equally iportant and basic continuous translation group in one diension, which we shall refer to as T 1. Let the coordinate axis of the one- diensional space be labelled x. An arbitrary eleent of the group T 1 corresponding to translation by the distance x will be denoted by T x. Consider "states" x 0 4 of a "particle" localized at the position x 0. 5 The action of T x on x is: (6.6-1) T x x 0 x x 0 It is easy to see that T x ust have the following properties: (6.6-2a) T x 1 T x 2 T x 1 x 2 (6.6-2b) T 0 E (6.6-2c) T x 1 T x
10 10 6._1DContinuousGroups.nb These are just the properties that are required for T x, x to for a group [ cf. Eqs. (1.2-1abc) ]. For an infinitesial displaceent denoted by dx, we have (6.6-3) T dx E i dx P which defines the (displaceent-independent) generator of translation P. Next, we write T x dx in two different ways: d T x (6.6-4a) T x dx T x dx d x and (6.6-4b) T x dx T dx T x Substituting (6.6-3) in (6.6-4b), and coparing with (6.6-4a), we obtain d T x (6.6-5) i P T x d x Considering the boundary condition (6.6-2b), this differential equation yields the unique solution, (6.6-6) T x e i P x It is straightforward to see that with T x written in this for, all the required group properties, (6.6-2a,b,c), are satisfied. This derivation is identical to that given for the rotation group SO(2). [ cf. Theore 6.3 ] The only difference is that the paraeter x in T x is no longer restricted to a finite range as for Φ in R Φ. As before, all irreducible representations of the translation group are onediensional. For unitary representations, the generator P corresponds to a heritian operator with real eigenvalues, which we shall denote by p. For the representation T x U p x, We obtain: (6.6-7) P p p p (6.6-8) U p x p p e i p x It is easy to see that all the group properties, Eqs. (6.6-2a,b,c), are satisfied by this representation function for any given real nuber p. Therefore the value of p is totally unrestricted. Coparing these results with those obtained in Chap. 1 for the discrete translation group T d and in Sections ( ) for SO(2), we reark that: (i) The representation functions in all these cases take the exponential for [ cf. Eqs. (1.3-3), (6.3-6), (6.6-8) ], reflecting the coon group ultiplication rule [ cf Eqs. (1.2-la),(6.1-7),(6.6-2a) ]; (ii) For T d, the group paraeter (n in Eq. (1.3-3)) is discrete and infinite, the representation label (k) is continuous and bounded. For SO(2), the forer (Φ in Eq. (6.3-6)) is continuous and bounded, the latter () is discrete and infinite. Finally, for T 1 the forer (x in Eq. (6.6-8)) is continuous and unbounded, so is the latter (p). The conjugate role of the group paraeter and the representation label in the sense of Fourier analysis was discussed in Sec The case is strengthened ore by applying the orthonorality and copleteness relations of representation functions to the present case of full one- diensional translation. For this purpose we ust again define an appropriate invariant easure for integration over the group eleents. Just as in the case of SO(2), one needs only to pick the natural Cartesian displaceent x as the integration variable. Because the range of integration is now infinite, not all integrals are strictly convergent in the classical sense. But for our purposes, it suffices to say that all previous results hold in the sense of generalized functions. We obtain: (6.6-9) (6.6-10) dx U p x U p ' x N p p' dp U p x U p x ' N x x '
11 6._1DContinuousGroups.nb where N is a yet unspecified noralization constant. Since U p x e i p x, these equations represent a stateent of the Fourier theore for arbitrary (generalized) functions. This correspondence also gives the correct value of N, i.e. N. 6.7 Conjugate Basis Vectors In Chap. 1 we described two types of basis vectors: x, defined by Eq. (1.1-6), and u E, k, defined by Eq. (1.3-3). The first represents "localized states" at soe position x ; the second corresponds to noral odes which fill the entire lattice and have siple translational properties. Each one has its unique features and practical uses. State functions expressed in ters of these two bases are related by a Fourier expansion. Analogous procedures can be applied in the representation space of the rotation group SO(2) and the continuous translation group. We describe the in turn. Consider a particle state localized at a position represented by polar coordinates r, Φ On the 2-diensional plane. The value of r will not be changed by any rotation; therefore we shall not be concerned about it in subsequent discussions. Intuitively, (6.7-1) U Φ Φ 0 Φ Φ 0 so that, (6.7-2) Φ U Φ 0 0 Φ where 0 represents a "standard state" aligned with a pre-chosen x-axis. How are these states related to the eigenstates of J defined by Eqs. (6.3-5) and (6.3-6)? If we expan in ters of the vectors ; 0, ±1,, Φ Φ then Φ U Φ 0 U Φ 0 0 e i Φ States with different values of are unrelated by rotation, and we can choose their phases (i.e. a ultiplicative factor e i Α for each ) such that 0 1 for all thus obtaining (6.7-3) Φ e i Φ To invert this equation, ultiply by e i Φ and integrate over Φ. We obtain (6.7-4) Φ Φ ei 0 We see that by this convention, the "transfer atrix eleents" functions. Φ between the two are just the group representation An arbitrary state Ψ in the vector space can be expressed in either of the two bases: (6.7-5) Ψ Ψ Φ Ψ Φ 0 The "wave functions" Ψ and Ψ Φ are related by (6.7-6) Ψ Φ Φ Ψ Φ Φ e i Φ Ψ and (6.7-7) Ψ 0 Ψ Φ e i Φ
12 12 6._1DContinuousGroups.nb Let us exaine the action of the operator J on the states (6.7-8) J Φ J e i Φ e i Φ d i Φ Φ. Fro Eq. (6.7-3), we obtain For an arbitrary state, we have: (6.7-9) Φ J Ψ J Φ Ψ 1 i 1 i d Φ Ψ d Ψ Φ Readers who have had soe quantu echanics [ Messiah, Schiff ] will recognize that J is the angular oentu operator (easured in units of ħ ). The above purely group- theoretical derivation underlines the general, geoetrical origin of these results. The above discussion can be repeated for the continuous translation group. The "localized states" "translationally covariant" states p, Eq. (6.6-7) are related by (6.7-10) x and d p p x p e i (6.7-11) p d x x e i p x x, Es. (6.6-1), and the where the noralization of the states is chosen, by convention, as x ' x x x ' p ' p p p ' The transfer atrix eleents are, again, the group representation functions [ Eq. (.6-8) ], (6.7-12) p x e i p x As before, if (6.7-13) Ψ x Ψ x d x p Ψ p d p then (6.7-14) Ψ x Ψ p e i p x d p (6.7-15) Ψ p Ψ x e i p x d x and (6.7-16) x P Ψ P x Ψ 1 i d d x Ψ x Thus, the generator P can be identified with the linear oentu operator in quantu echanical systes. [ Messiah, Schiff ]
13 6._1DContinuousGroups.nb Probles 6.1 Show that the rotation atrix R Φ, Eq. (6.1-3), is an orthogonal atrix and prove that every SO(2) atrix represents a rotation in the plane. 6.2 Show that e ± 1 2 e 1 i e 2 are eigenvectors of J with eigenvalues ±1 respectively [ cf. Eq. (6.3-8) ]. FootNotes Notes 1: Matrices satisfying Eq. (h.1-5) but with deterinant equal to -1 correspond physically to rotations cobined with spatial inversion or irror reflection. This set of atrices is not connected to the identity transforation by a continuous change of paraeters. We shall include spatial inversion in our group theoretical analysis in Chap. 11. Notes 2: We note that Ρ R d Ξ E d Ξ ER d Ξ E d Ξ SR d Ξ SR d Ξ R Ρ SR d Ξ SR d Ξ R Notes 3: This eans that there exist closed "paths" on the unit circle which wind around it ties (for all Integers ) and which cannot be continuously defored into each other. Notes 4: The "state" can be interpreted in the sense of either classical echanics or quantu echanics. We use the state- vector convention of quantu echanics only for the sake of clarity in notation. By "particle" we siply ean an entity with no spatial extension, which can be represented by a atheati- Notes 5: cal point.
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