Acta Polytechnica Vol. 51 No. 4/2011 Erlangen Programme at Large 3.2 Ladder Operators in Hypercomplex Mechanics V. V. Kisil Abstract We revise the construction of creation/annihilation operators in quantum mechanics based on the representation theory of the Heisenberg and symplectic groups. Besides the standard harmonic oscillator (the elliptic case) we similarly treat the repulsive oscillator (hyperbolic case) and the free particle (the parabolic case). The respective hypercomplex numbers turn out to be handy on this occasion. This provides a further illustration to the Similarity andCorrespondence Principle. Keywords: Heisenberg group, Kirillov’s method of orbits, geometric quantisation, quantum mechanics, classical me- chanics, Planck constant, dual numbers, double numbers, hypercomplex, jet spaces, hyperbolic mechanics, interference, Fock-Segal-Bargmann representation, Schrödinger representation, dynamics equation, harmonic and unharmonic oscil- lator, contextual probability, symplectic group, metaplectic representation, Shale-Weil representation. 1 Introduction Harmonic oscillators are treated in most textbooks on quantum mechanics. This is efficiently done through creation/annihilation (ladder) operators [9, 3]. The underlying structure is the representation theory of the Heisenberg and symplectic groups [28, § VI.2], [34, § 8.2], [12, 8]. It is also known that quantum mechanics and field theory can benefit from the in- troduction of Clifford algebra-valued group represen- tations [20, 5, 4, 10]. The dynamics of a harmonic oscillator generates the symplectic transformation of the phase space of the elliptic type. The respective parabolic and hyper- bolic counterparts are also of interest [37, § 3.8], [35]. As we will see, they are naturally connected with the respective hypercomplex numbers. To make this correspondence explicit we recall that the symplectic group Sp(2) [8, § 1.2] consists of 2 × 2 matrices with real entries and the unit determi- nant. It is isomorphic to the group SL2(R) [28,13,30] and provides linear symplectomorphisms of the two- dimensional phase space. It has three types of non- isomorphic one-dimensional subgroups represented by: K = {( cos t sin t − sin t cos t ) = exp ( 0 t −t 0 ) , t ∈ (−π, π] } , (1) N = {( 1 t 0 1 ) = exp ( 0 t 0 0 ) , t ∈ R } , (2) A = {( et 0 0 e−t ) = exp ( t 0 0 −t ) , t ∈ R } . (3) We will refer to them as elliptic, parabolic and hy- perbolic subgroups, respectively. On the other hand, there are three non- isomorphic types of commutative, associative two- dimensional algebras known as complex, dual and double numbers [38, App. C], [29, § 5]. They are represented by expressions x + ιy, where ι stands for one of the hypercomplex units i, ε or j with the prop- erties: i2 = −1, ε2 = 0, j2 = 1. These units can also be labelled as elliptic, parabolic and hyperbolic. In an earlier paper [25], we considered represen- tations of the Heisenberg group which are induced by hypercomplex characters of its centre. The ellip- tic case (complex numbers) describes the traditional framework of quantum mechanics, of course. Double-valued representations, with the imagi- nary unit j2 = 1, are a natural source of hyperbolic quantum mechanics developed for a while [14, 15, 17, 16, 18]. The representation acts on a Krein space with an indefinite inner product [2]. This aroused significant recent interest in connection with PT - symmetric quantum mechanics [10]. However, our approach is different from the classical treatment of Krein spaces: we use the hyperbolic unit j and build the hyperbolic analytic function theory on its own basis [21, 27]. In the traditional approach, the indef- inite metric is mapped to a definite inner product through auxiliary operators. 44 Acta Polytechnica Vol. 51 No. 4/2011 The representation with values in dual numbers provides a convenient description of the classical me- chanics. To this end we do not take any sort of semi- classical limit, rather the nilpotency of the imaginary unit (ε2 = 0) performs the task. This removes the vi- cious necessity to consider the Planck constant tend- ing to zero. Mixing this with complex numbers we get a convenient tool for modelling the interaction between quantum and classical systems [22, 24]. Our construction [25] provides three different types of dynamics and also generates the respec- tive rules for addition of probabilities. In this pa- per we analyse the three types of dynamics produced by transformations (1–3) from the symplectic group Sp(2) by means of ladder operators. As a result we obtain further illustrations to the following: Principle (Similarity and Correspondence) [23, Principle 29] 1. Subgroups K, N and A play a similar role in the structure of the group Sp(2) and its representa- tions. 2. The subgroups shall be swapped simultaneously with the respective replacement of hypercomplex unit ι. Here the two parts are interrelated: without a swap of imaginary units there can be no similarity between different subgroups. In this paper we work with the simplest case of a particle with only one degree of freedom. Higher di- mensions and the respective group of symplectomor- phisms Sp(2n) may require consideration of Clifford algebras [32]. 2 Heisenberg group and its automorphisms Let (s, x, y), where s, x, y ∈ R, be an element of the one-dimensional Heisenberg group H1 [8, 12]. Con- sideration of the general case of Hn will be similar, but is beyond the scope of present paper. The group law on H1 is given as follows: (s, x, y) · (s′, x′, y′) =( s + s′ + 1 2 ω(x, y; x′, y′), x + x′, y + y′ ) , (4) where the non-commutativity is due to ω — the sym- plectic form on R2n [1, § 37]: ω(x, y; x′, y′) = xy′ − x′y. (5) The Heisenberg group is a non-commutative Lie group. The left shifts Λ(g) : f (g′) �→ f (g−1g′) (6) act as a representation of H1 on a certain linear space of functions. For example, an action on L2(H, dg) with respect to the Haar measure dg = ds dx dy is the left regular representation, which is unitary. The Lie algebra hn of H1 is spanned by left- (right-)invariant vector fields Sl(r) = ±∂s, X l(r) = ±∂x − 1 2 y∂s, (7) Y l(r) = ±∂y + 1 2 x∂s on H1 with the Heisenberg commutator relation [X l(r), Y l(r)] = Sl(r) (8) and all other commutators vanishing. We will some- time omit the superscript l for left-invariant field. The group of outer automorphisms of H1, which trivially acts on the centre of H1, is the symplec- tic group Sp(2) defined in the previous section. It is the group of symmetries of the symplectic form ω [8, Thm. 1.22], [11, p. 830]. The symplectic group is isomorphic to SL2(R) [28], [34, Ch. 8]. The explicit action of Sp(2) on the Heisenberg group is: g : h = (s, x, y) �→ g(h) = (s, x′, y′), (9) where g = ( a b c d ) ∈ Sp(2), and ( x′ y′ ) = ( a b c d )( x y ) . The Shale-Weil theorem [8, § 4.2], [11, p. 830] states that any representation ρh̄ of the Heisenberg groups generates a unitary oscillator (or metaplectic) rep- resentation ρSWh̄ of S̃p(2), the two-fold cover of the symplectic group [8, Thm. 4.58]. We can consider the semidirect product G = H 1 ×| S̃p(2) with the standard group law: (h, g) ∗ (h′, g′) = (h ∗ g(h′), g ∗ g′), where h, h′ ∈ H1, g, g′ ∈ S̃p(2), and the stars denote the respective group opera- tions while the action g(h′) is defined as the com- position of the projection map S̃p(2) → Sp(2) and the action (9). This group is sometimes called the Schrödinger group, and it is known as the maxi- mal kinematical invariance group of both the free Schrödinger equation and the quantum harmonic os- cillator [31]. This group is of interest not only in quantum mechanics but also in optics [36, 35]. 45 Acta Polytechnica Vol. 51 No. 4/2011 Consider the Lie algebra sp2 of the group Sp(2). Pick up the following basis in sp2 [34, § 8.1]: A = 1 2 ( −1 0 0 1 ) , B = 1 2 ( 0 1 1 0 ) , Z = ( 0 1 −1 0 ) . The commutation relations between the elements are: [Z, A] = 2B, [Z, B] = −2A, [A, B] = − 1 2 Z. (10) Vectors Z, B + Z/2 and −A are generators of the one-parameter subgroups K, N and A (1–3) respec- tively. Furthermore, we can consider the basis {S, X, Y, A, B, Z} of the Lie algebra g of the Lie group G = H1 ×| S̃p(2). All non-zero commutators besides those already listed in (8) and (10) are: [A, X] = 1 2 X, [B, X] = − 1 2 Y, [Z, X] = Y ; (11) [A, Y ] = − 1 2 Y, [B, Y ] = − 1 2 X, [Z, Y ] = −X. (12) The Shale-Weil theorem allows us to expand any rep- resentation ρh̄ of the Heisenberg group to the repre- sentation ρ̃h̄ = ρh̄ ⊕ ρSWh̄ of group G. Example 1 Let ρh̄ be the Schrödinger representa- tion [8, § 1.3] of H1 in �L2(R), that is [25, (3.5)]: [ρχ(s, x, y)f ](q) = e 2πih̄(s−xy/2)+2πixq · f (q − h̄y). (13) Thus the action of the derived representation on the Lie algebra h1 is: ρh̄(X) = 2πiq, ρh̄(Y ) = −h̄ d dq , ρh̄(S) = 2πih̄I. (14) Then the associated Shale-Weil representation of Sp(2) in L2(R) has the derived action, cf. [35, (2.2)], [8, § 4.3]: ρSWh̄ (A) = − q 2 d dq − 1 4 , ρSWh̄ (B) = − h̄i 8π d2 dq2 − πiq2 2h̄ , (15) ρSWh̄ (Z) = h̄i 4π d2 dq2 − πiq2 h̄ . We can verify commutators (8) and (10–12) for op- erators (14–15). It is also obvious that in this repre- sentation the following algebraic relations hold: ρSWh̄ (A) = i 4πh̄ (ρh̄(X)ρh̄(Y ) − 1 2 ρh̄(S)) = (16) i 8πh̄ (ρh̄(X)ρh̄(Y ) + ρh̄(Y )ρh̄(X)), ρSWh̄ (B) = i 8πh̄ (ρh̄(X) 2 − ρh̄(Y )2), (17) ρSWh̄ (Z) = i 4πh̄ (ρh̄(X) 2 + ρh̄(Y ) 2). (18) Thus it is common in quantum optics to name g as a Lie algebra with quadratic generators, see [9, § 2.2.4]. Note that ρSWh̄ (Z) is the Hamiltonian of the har- monic oscillator (up to a factor). Then we can con- sider ρSWh̄ (B) as the Hamiltonian of a repulsive (hy- perbolic) oscillator. The operator ρSWh̄ (B − Z/2) = h̄i 4π d2 dq2 is the parabolic analog. A graphical repre- sentation of all three transformations is given in Fi- gure 1, and a further discussion of these Hamiltoni- ans can be found in [37, § 3.8]. An important obser- vation, which is often missed, is that the three lin- ear symplectic transformations are unitary rotations in the corresponding hypercomplex algebra. This means that the symplectomorphisms generated by operators Z, B − Z/2, B within time t coincide with the multiplication of hypercomplex number q + ιp by eιt [23, § 3], which is just another illustration of the Similarity and Correspondence Principle. q p q p q p Fig. 1: Three types (elliptic, parabolic and hyperbolic) of linear symplectic transformations on the plane 46 Acta Polytechnica Vol. 51 No. 4/2011 Example 2 There are many advantages of consid- ering representations of the Heisenberg group on the phase space [12, § 1.7], [8, § 1.6], [6]. A convenient expression for Fock-Segal-Bargmann (FSB) represen- tation on the phase space is [25, (3.2)]: [ρF (s, x, y)f ](q, p) = e −2πi(h̄s+qx+py) · (19) f ( q − h̄ 2 y, p + h̄ 2 x ) . Then the derived representation of h1 is: ρF (X) = −2πiq + h̄ 2 ∂p, ρF (Y ) = −2πip − h̄ 2 ∂q, (20) ρF (S) = −2πih̄I. This produces the derived form of the Shale-Weil rep- resentation: ρSWF (A) = 1 2 (q∂q − p∂p) , ρSWF (B) = − 1 2 (p∂q + q∂p) , (21) ρSWF (Z) = p∂q − q∂p. Note that this representation does not contain the pa- rameter h̄, unlike the equivalent representation (15). Thus the FSB model explicitly shows the equivalence of ρSWh̄1 and ρ SW h̄2 if h̄1h̄2 > 0 [8, Thm. 4.57]. As we will also see below, the FSB-type represen- tations in hypercomplex numbers produce almost the same Shale-Weil representations. 3 Ladder operators in quantum mechanics Let ρ be a representation of the group G = H 1 ×| S̃p(2) in a space V . Consider the derived rep- resentation of the Lie algebra g [28, § VI.1] and de- note X̃ = ρ(X) for X ∈ g. To see the structure of the representation ρ we can decompose the space V into eigenspaces of the operator X̃ for some X ∈ g. The canonical example is the Taylor series in complex analysis. We are going to consider three cases correspond- ing to three non-isomorphic subgroups (1–3) of Sp(2) starting from the compact case. Let H = Z be a generator of the compact subgroup K. Cor- responding symplectomorphisms (9) of the phase space are given by orthogonal rotations with matri- ces ( cos t sin t − sin t cos t ) . The Shale-Weil representa- tion (15) coincides with the Hamiltonian of the har- monic oscillator. Since this is a double cover of a compact group, the corresponding eigenspaces Z̃vk = ikvk are parametrised by a half-integer k ∈ Z/2. Explicitly for a half-integer k: vk (q) = Hk+12 (√ 2π h̄ q ) e− π h̄ q2, (22) where Hk is the Hermite polynomial [8, § 1.7], [7, 8.2(9)]. From the point of view of quantum mechanics and representation theory (which may be the same), it is beneficial to introduce the ladder operators L±, known as creation/annihilation in quantum mechan- ics [8, p. 49] or raising/lowering in representation the- ory [28, § VI.2], [34, § 8.2], [3]. They are defined by the following commutation relations: [Z̃, L±] = λ±L ±. (23) In other words, L± are eigenvectors for operators ad Z of the adjoint representation of g [28, § VI.2]. Remark 1 The existence of such ladder operators follows from the general properties of Lie algebras if the Hamiltonian belongs to a Cartan subalgebra. This is the case for vectors Z and B, which are the only two non-isomorphic types of Cartan subalgebras in sl2. However, the third case considered in this pa- per, the parabolic vector B + Z/2, does not belong to a Cartan subalgebra, yet a sort of ladder operators is still possible with dual number coefficients. More- over, for the hyperbolic vector B, besides the stan- dard ladder operators an additional pair with double number coefficients will also be described. From the commutators (23) we deduce that if vk is an eigenvector of Z̃ then L+vk is an eigenvector as well: Z̃(L+vk) = (L +Z̃ + λ+L +)vk = L+(Z̃vk) + λ+L +vk = ikL+vk + λ+L +vk = (ik + λ+)L +vk. (24) Thus the action of ladder operators on the respective eigenspaces Vk can be visualised by the diagram: (25) There are two ways to search for ladder operators: in (complexified) Lie algebras h1 and sp2. We will consider them in sequence. 3.1 Ladder operators from the Heisenberg group Assuming L+ = aX̃ + bỸ we obtain from the re- lations (11–12) and (23) the linear equations with unknown a and b: 47 Acta Polytechnica Vol. 51 No. 4/2011 a = λ+b, −b = λ+a. The equations have a solution if and only if λ2+ + 1 = 0, and the raising/lowering operators are L± = X̃ ∓ iỸ . Remark 2 Here we have an interesting asymmet- ric response: due to the structure of the semidirect product H1 ×| S̃p(2) it is the symplectic group which acts on H1, not vise versa. However, the Heisenberg group has a weak action in the opposite direction: it shifts eigenfunctions of Sp(2). In the Schrödinger representation (14) the ladder operators are ρh̄(L ±) = 2πiq ± ih̄ d dq . (26) The standard treatment of the harmonic oscillator in quantum mechanics, which can be found in many textbooks, e.g. [8, § 1.7], [9, § 2.2.3], is as follows. The vector v−1/2(q) = e −πq2/h̄ is an eigenvector of Z̃ with the eigenvalue − i 2 . In addition v−1/2 is an- nihilated by L+. Thus the chain (25) terminates to the right and the complete set of eigenvectors of the harmonic oscillator Hamiltonian is presented by (L−)kv−1/2 with k = 0, 1, 2, . . . We can make a wavelet transform generated by the Heisenberg group with the mother wavelet v−1/2, and the image will be the Fock-Segal-Bargmann (FSB) space [12], [8, § 1.6]. Since v−1/2 is the null solution of L+ = X̃ − iỸ , then by the general re- sult [26, Cor. 24] the image of the wavelet transform will be null-solutions of the corresponding linear com- bination of the Lie derivatives (7): D = X r − iY r = (∂x + i∂y) − πh̄(x − iy), (27) which turns out to be the Cauchy-Riemann equation on a weighted FSB-type space. 3.2 Symplectic ladder operators We can also look for ladder operators within the Lie algebra sp2, see [23, § 8]. Assuming L + 2 = aà + bB̃ + cZ̃ from the relations (10) and defining condition (23) we obtain the linear equations with unknown a, b and c: c = 0, 2a = λ+b, −2b = λ+a. The equations have a solution if and only if λ2+ + 4 = 0, and the raising/lowering operators are L±2 = ±ià + B̃. In the Shale-Weil representation (15) they turn out to be: L±2 = ±i ( q 2 d dq + 1 4 ) − h̄i 8π d2 dq2 − πiq2 2h̄ = − i 8πh̄ ( ∓2πq + h̄ d dq )2 . (28) Since this time λ+ = 2i the ladder operators L ± 2 pro- duce a shift on the diagram (25) twice bigger than the operators L± from the Heisenberg group. After all, this is not surprising since from the explicit rep- resentations (26) and (28) we get: L±2 = − i 8πh̄ (L±)2. 4 Ladder operators for the hyperbolic subgroup Consider the case of the Hamiltonian H = 2B, which is a repulsive (hyperbolic) harmonic oscilla- tor [37, § 3.8]. The corresponding one-dimensional subgroup of symplectomorphisms produces hyper- bolic rotations of the phase space. The eigenvectors vμ of the operator ρSWh̄ (2B)vν = −i ( h̄ 4π d2 dq2 + πq2 h̄ ) vν = iνvν , are Weber-Hermite (or parabolic cylinder) functions vν = Dν−12 ( ±2ei π 4 √ π h̄ q ) , see [7, § 8.2], [33] for fun- damentals of Weber-Hermite functions and [35] for further illustrations and applications in optics. The corresponding one-parameter group is not compact and the eigenvalues of the operator 2B̃ are not restricted by any integrality condition, but the raising/lowering operators are still important [13, § II.1], [30, § 1.1]. We again seek solutions in two subalgebras h1 and sp2 separately. However, the ad- ditional options will be provided by a choice of the number system: either complex or double. 4.1 Complex ladder operators Assuming L+h = aX̃ + bỸ from the commuta- tors (11–12), we obtain the linear equations: − a = λ+b, −b = λ+a. (29) The equations have a solution if and only if λ2+ − 1 = 0. Taking the real roots λ = ±1 we obtain that the raising/lowering operators are L±h = X̃ ∓ Ỹ . In the Schrödinger representation (14) the ladder operators are L±h = 2πiq ± h̄ d dq . (30) The null solutions v±12 (q) = e ± πi h̄ q2 to operators ρh̄(L ±) are also eigenvectors of the Hamiltonian 48 Acta Polytechnica Vol. 51 No. 4/2011 ρSWh̄ (2B) with the eigenvalue ± 1 2 . However the im- portant distinction from the elliptic case is that they are not square-integrable on the real line anymore. We can also look for ladder operators within the sp2, that is in the form L + 2h = aà + bB̃ + cZ̃ for the commutator [2B̃, L+h ] = λL + h . We will get the system: 4c = λa, b = 0, a = λc. A solution again exists if and only if λ2 = 4. Within complex numbers we get only the values λ = ±2 with the ladder operators L±2h = ±2à + Z̃/2, see [13, § II.1], [30, § 1.1]. Each indecomposable h1- or sp2-module is formed by a one-dimensional chain of eigenvalues with a transitive action of ladder op- erators L±h or L ± 2h respectively. And we again have a quadratic relation between the ladder operators: L±2h = i 4πh̄ (L±h ) 2. 4.2 Double ladder operators There are extra possibilities in the context of hyper- bolic quantum mechanics [17,16,18]. Here we use the representation of H1 induced by a hyperbolic charac- ter ejht = cosh(ht) + j sinh(ht), see [25, (4.5)], and obtain the hyperbolic representation of H1, cf. (13): [ρjh(s ′, x′, y′)f̂ ](q) = ejh(s ′−x′y′/2)+jx′q · f̂ (q − hy′). (31) The corresponding derived representation is ρ j h(X) = jq, ρ j h(Y ) = −h d dq , (32) ρ j h(S) = jhI. Then the associated Shale–Weil derived representa- tion of sp2 in the Schwartz space S(R) is, cf. (15): ρSWh (A) = − q 2 d dq − 1 4 , ρSWh (B) = jh 4 d2 dq2 − jq2 4h , (33) ρSWh (Z) = − jh 2 d2 dq2 − jq2 2h . Note that ρSWh (B) now generates a usual harmonic oscillator, not the repulsive one like ρSWh̄ (B) in (15). However, the expressions in the quadratic algebra are still the same (up to a factor), cf. (16–18): ρSWh (A) = − j 2h (ρjh(X)ρ j h(Y ) − 1 2ρ j h(S)) = (34) − j 4h (ρjh(X)ρ j h(Y ) + ρ j h(Y )ρ j h(X)), ρSWh (B) = j 4h (ρjh(X) 2 − ρjh(Y ) 2), (35) ρSWh (Z) = − j 2h (ρjh(X) 2 + ρjh(Y ) 2). (36) This is due to the Principle of similarity and corre- spondence: we can swap operators Z and B with simultaneous replacement of hypercomplex units i and j. The eigenspace of the operator 2ρSWh (B) with an eigenvalue jν are spanned by the Weber-Hermite functions D−ν−12 ( ± √ 2 h x ) , see [7, § 8.2]. Functions Dν are generalisations of the Hermit functions (22). The compatibility condition for a ladder operator within the Lie algebra h1 will be (29) as before, since it depends only on the commutators (11–12). Thus we still have the set of ladder operators correspond- ing to values λ = ±1: L±h = X̃ ∓ Ỹ = jq ± h d dq . Admitting double numbers, we have an extra way to satisfy λ2 = 1 in (29) with values λ = ±j. Then there is an additional pair of hyperbolic ladder operators, which are identical (up to factors) to (26): L±j = X̃ ∓ jỸ = jq ± jh d dq . Pairs L±h and L ± j shift eigenvectors in the “orthogo- nal” directions changing their eigenvalues by ±1 and ±j. Therefore an indecomposable sp2-module can be parametrised by a two-dimensional lattice of eigen- values in double numbers, see Table 1. The following functions v±h1 2 (q) = e∓jq 2/(2h) = cosh q2 2h ∓ j sinh q2 2h , v ±j 1 2 (q) = e∓q 2/(2h) are null solutions to the operators L±h and L ± j , re- spectively. They are also eigenvectors of 2ρSWh (B) with eigenvalues ∓ j 2 and ∓ 1 2 , respectively. If these functions are used as mother wavelets for the wavelet transforms generated by the Heisenberg group, then the image space will consist of the null-solutions of the following differential operators, see [26, Cor. 24]: Dh = X r − Y r = (∂x − ∂y) + h 2 (x + y), Dj = X r − jY r = (∂x + j∂y ) − h 2 (x − jy), 49 Acta Polytechnica Vol. 51 No. 4/2011 Table 1: The action of hyperbolic ladder operators on a 2D lattice of eigenspaces. Operators L±h move the eigenvalues by 1, making shifts in the horizontal direction. Operators L±j change the eigenvalues by j, shown as vertical shifts for v±h1 2 and v±j1 2 , respectively. This is again in line with the classical result (27). However annihilation of the eigenvector by a ladder operator does not mean that the part of the 2D-lattice becomes void, since it can be reached via alternative routes. Instead of mul- tiplication by a zero, as happens in the elliptic case, a half-plane of eigenvalues will be multiplied by the divisors of zero 1 ± j. We can also search ladder operators within the algebra sp2 and admitting double numbers we will again find two sets of them [23, § 3]: L±2h = ±Ã + Z̃/2 = ∓ q 2 d dq ∓ 1 4 − jh 4 d2 dq2 − jq2 4h = − j 4h (L±h ) 2, L±2j = ±jà + Z̃/2 = ∓ jq 2 d dq ∓ j 4 − jh 4 d2 dq2 − jq2 4h = − j 4h (L±j ) 2. Again these operators L±2h and L ± 2h produce double shifts in the orthogonal directions on the same two- dimensional lattice in Tabular 1. 5 Ladder operator for the nilpotent subgroup Finally, we look for ladder operators for the Hamil- tonian B̃ + Z̃/2 or, equivalently, −B̃ + Z̃/2. It can be identified with a free particle [37, § 3.8]. We can look for ladder operators in the represen- tation (14–15) within the Lie algebra h1 in the form L±ε = aX̃ + bỸ . This is possible if and only if − b = λa, 0 = λb. (37) The compatibility condition λ2 = 0 implies λ = 0 within complex numbers. However, such a “ladder” operator produces only the zero shift on the eigen- vectors, cf. (24). Another possibility appears if we consider the rep- resentation of the Heisenberg group induced by dual- valued characters. On the configurational space such a representation is [25, (4.11)]: [ρεχ(s, x, y)f ](q) = e 2πixq (( 1 − εh ( s − 1 2 xy )) · f (q) + εhy 2πi f ′(q) ) . (38) The corresponding derived representation of h1 is ρ p h(X) = 2πiq, ρ p h(Y ) = εh 2πi d dq , (39) ρ p h(S) = −εhI. However the Shale-Weil extension generated by this representation is inconvenient. It is better to consider the FSB-type parabolic representation [25, (4.9)] on the phase space induced by the same dual-valued character, cf. (19): [ρεh(s, x, y)f ](q, p) = e −2πi(xq+yp) · (40) (f (q, p) + εh(sf (q, p) + y 4πi f ′q(q, p) − x 4πi f ′p(q, p))). Then the derived representation of h1 is: ρ p h(X) = −2πiq − εh 4πi ∂p, ρ p h(Y ) = −2πip + εh 4πi ∂q, (41) ρ p h(S) = εhI. 50 Acta Polytechnica Vol. 51 No. 4/2011 An advantage of the FSB representation is that the derived form of the parabolic Shale–Weil representa- tion coincides with the elliptic one (21). Eigenfunctions with the eigenvalue μ of the parabolic Hamiltonian B̃ + Z̃/2 = q∂p have the form vμ(q, p) = e μp/qf (q), (42) with an arbitrary function f (q). The linear equations defining the corresponding ladder operator L±ε = aX̃ + bỸ in the algebra h1 are (37). The compatibility condition λ2 = 0 implies λ = 0 within complex numbers again. Admitting dual numbers, we have additional values λ = ±ελ1 with λ1 ∈ C with the corresponding ladder operators L±ε = X̃ ∓ ελ1Ỹ = −2πiq − εh 4πi ∂p ± 2πελ1ip = −2πiq + εi ( ±2πλ1p + h 4π ∂p ) . For the eigenvalue μ = μ0 + εμ1 with μ0, μ1 ∈ C the eigenfunction (42) can be rewritten as: vμ(q, p) = e μp/qf (q) = eμ0p/q ( 1 + εμ1 p q ) f (q) (43) due to the nilpotency of ε. Then the ladder action of L±ε is μ0 + εμ1 �→ μ0 + ε(μ1 ± λ1). Therefore, these operators are suitable for building sp2-modules with a one-dimensional chain of eigenvalues. Finally, consider the ladder operator for the same element B + Z/2 within the Lie algebra sp2. Accord- ing to the above procedure we get the equations: −b + 2c = λa, a = λb, a 2 = λc, which can again be resolved if and only if λ2 = 0. There is the only complex root λ = 0 with the corre- sponding operators L±p = B̃ +Z̃/2, which does not af- fect the eigenvalues. However the dual number roots λ = ±ελ2 with λ2 ∈ C lead to the operators L±ε = ±ελ2à + B̃ + Z̃/2 = ± ελ2 2 (q∂q − p∂p) + q∂p. 6 Conclusions: similarity and correspondence We wish to summarise our findings. Firstly, the ap- pearance of hypercomplex numbers in ladder opera- tors for h1 follows exactly the same pattern as was already noted for sp2 [23, Rem. 32]: • the introduction of complex numbers is a neces- sity for the existence of ladder operators in the elliptic case; • in the parabolic case, we need dual numbers to make ladder operators useful; • in the hyperbolic case, double numbers are not required neither for the existence or for the us- ability of ladder operators, but they do provide an enhancement. In the spirit of the Similarity and Correspondence Principle we have the following extension of Prop. 33 from [23]: Proposition Let a vector H ∈ sp2 generate the subgroup K, N ′ or A′, that is H = Z, B + Z/2, or 2B, respectively. Let ι be the respective hypercom- plex unit. Then the ladder operators L± satisfying the commutation relation: [H, L±2 ] = ±ιL ± are given by: 1. Within the Lie algebra h1: L ± = X̃ ∓ ιỸ . 2. Within the Lie algebra sp2: L ± 2 = ±ιÃ+Ẽ. 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