Erlangen Programme at Large 3.2: Ladder Operators in Hypercomplex Mechanics

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 to be handy on this occasion, this provides a further illustration to Similarity and Correspondence Principle. Keywords: Heisenberg group, Kirillov's method of orbits, geometric quantisation, quantum mechanics, classical mechanics, Planck constant, dual numbers, double numbers, hypercomplex, jet spaces, hyperbolic mechanics, interference, Fock--Segal--Bargmann representation, Schr\"odinger representation, dynamics equation, harmonic and unharmonic oscillator, contextual probability, symplectic group, metaplectic representation, Shale--Weil representation


INTRODUCTION
Harmonic oscillators are treated in most textbooks on quantum mechanics. This is efficiently done through creation/annihilation (ladder) operators [3,9]. The underlying structure is the representation theory of the the Heisenberg and symplectic groups [8; 12; 28, § VI.2; 34, § 8.2]. It is also known that quantum mechanics and field theory can benefit from the introduction of Clifford algebra-valued group representations [4,5,10,20].
The dynamics of a harmonic oscillator generates the symplectic transformation of the phase space of the elliptic type. The respective parabolic and hyperbolic counterparts are also of interest [35; 37, § 3.8]. 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 determinant. It is isomorphic to the group SL 2 (R) [13,28,30] and provides linear symplectomorphisms of the two-dimensional phase space. It has three types of non-isomorphic onedimensional subgroups represented by: We will refer to them as elliptic, parabolic and hyperbolic 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 [29, § 5; 38, App. C]. They are represented by expressions x + ιy, where ι stands for one of the hypercomplex units i, ε or j with the properties: These units can also be labelled as elliptic, parabolic and hyperbolic.
In an earlier paper [25],we considered representations of the Heisenberg group which are induced by hypercomplex characters of its centre. The elliptic case (complex numbers) describesthe traditional framework of quantum mechanics, of course.
Double-valued representations, with the imaginary unit j 2 = 1, are atural source of hyperbolic quantum mechanics developed for a while [14][15][16][17][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 indefinite metric is mapped to a definite inner product through an auxiliary operators.
The representation with values in dual numbers provides a convenient description of the classical mechanics. To this end we do not take any sort of semiclassical limit, rather the nilpotency of the imaginary unit (ε 2 = 0) performs the task. This removes the vicious necessity to consider the Planck constant tending 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 respective rules for addition of probabilities. In this paper 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 1 (Similarity and Correspondence). [23,Principle 29] (1) Subgroups K, N and A play a similar rôle in the structure of the group Sp(2) and its representations. (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 dimensions and the respective group of symplectomorphisms Sp(2n) may require consideration of Clifford algebras [32].

HEISENBERG GROUP AND ITS AUTOMORPHISMS
Let (s, x, y), where s, x, y ∈ R, be an element of the one-dimensional Heisenberg group H 1 [8,12]. Consideration of the general case of H n will be similar, but is beyond the scope of present paper. The group law on H 1 is given as follows: , where the non-commutativity is due to ω-the symplectic form on R 2n [1, § 37]: The Heisenberg group is a non-commutative Lie group. The left shifts act as a representation of H 1 on a certain linear space of functions. For example, an action on L 2 (H, dg) with respect to the Haar measure dg = ds dx dy is the left regular representation, which is unitary. The Lie algebra h n of H 1 is spanned by left-(right-)invariant vector fields and all other commutators vanishing. We will sometime omit the superscript l for left-invariant field. The group of outer automorphisms of H 1 , which trivially acts on the centre of H 1 , is the symplectic group Sp(2) defined in the precious 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 SL 2 (R) [28;34,Ch. 8]. The explicit action of Sp(2) on the Heisenberg group is: (2), and The Shale-Weil theorem [ We can consider the semidirect product G = H 1 ⋊ Sp(2) with the standard group law: where h, h ′ ∈ H 1 , g, g ′ ∈ Sp(2), and the stars denote the respective group operations while the action g(h ′ ) is defined as the composition of the projection map Sp(2) → Sp(2) and the action (9). This group is sometimes called the Schrödinger group, and it is known as the maximal kinematical invariance group of both the free Schrödinger equation and the quantum harmonic oscillator [31]. This group is of interest not only in quantum mechanics but also in optics [35,36].
Consider the Lie algebra sp 2 of the group Sp (2). Pick up the following basis in sp 2 [34, § 8.1]: The commutation relations between the elements are: Furthermore, we can consider the basis {S, X, Y, A, B, Z} of the Lie algebra g of the Lie group G = H 1 ⋊ Sp(2). All non-zero commutators besides those already listed in (8) and (10) are: The Shale-Weil theorem allows us to expand any representation ρ h of the Heisenberg group to the representationρ Thus the action of the derived representation on the Lie algebra h 1 is: Then the associated Shale-Weil representation of Sp(2) in L 2 (R) has the derived action, cf. [ We can verify commutators (8) and (10)(11)(12) for operators (14)(15). It is also obvious that in this representation the following algebraic relations hold: Thus it is common in quantum optics to name g as a Lie algebra with quadratic generators, see [ dq 2 is the parabolic analog. A graphical representation of all three transformations is given in Fig. 1, and a further discussion of these Hamiltonians can be found in [37, § 3.8]. An important observation, which is often missed, is that the three linear 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 e ιt [23, § 3], which is just another illustration of the Similarity and Correspondence Principle 1.
Then the derived representation of h 1 is: This produces the derived form of the Shale-Weil representation: Note that this representation does not contain the parameter h, unlike the equivalent representation (15). Thus the FSB model explicitly shows the equivalence of . As we will also see below, the FSB-type representations in hypercomplex numbers produce almost the same Shale-Weil representations.

LADDER OPERATORS IN QUANTUM MECHANICS
Let ρ be a representation of the group G = H 1 ⋊ Sp(2) in a space V. Consider the derived representation of the Lie algebra g [28, § VI.1] and denoteX = ρ(X) for X ∈ g. To see the structure of the representation ρ we can decompose the space V into eigenspaces of the operatorX for some X ∈ g. The canonical example is the Taylor series in complex analysis.
We are going to consider three cases corresponding 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. Corresponding symplectomorphisms (9) of the phase space are given by orthogonal rotations with matrices cos t sin t −sint cos t . The Shale-Weil representation (15) coincides with the Hamiltonian of the harmonic oscillator. Since this is a double cover of a compact group, the corresponding eigenspaces Zv k = ikv k are parametrised by a half-integer k ∈ Z/2. Explicitly for a half-integer k: where H k is the Hermite polynomial [7, 8.2(9); 8, § 1.7]. From the point of view of quantum mechanics and the representation theory (which may be the same), it is beneficial to introduce the ladder operators L ± , known as creation/annihilation in quantum mechanics [8, p. 49] or raising/lowering in representation theory [3; 28, § VI.2; 34, § 8.2]. They are defined by the following commutation relations: (23) [Z, L ± ] = λ ± L ± .
In other words, L ± are eigenvectors for operators ad Z of the adjoint representation of g [28, § VI.2].
Remark 4. 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 sl 2 . However, the third case considered in this paper, 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. Moreover, for the hyperbolic vector B, besides the standard ladder operators an additional pair with double number coefficients will also be described.
From the commutators (23) we deduce that if v k is an eigenvector ofZ then L + v k is an eigenvector as well: Thus the action of ladder operators on the respective eigenspaces V k can be visualised by the diagram: (25) . . .
There are two ways to search for ladder operators: in (complexified) Lie algebras h 1 and sp 2 . We will consider them in a sequence.

Ladder Operators from the Heisenberg Group.
Assuming L + = aX + bỸ we obtain from the relations (11)(12) and (23)  Remark 5. Here we have an interesting asymmetric response: due to the structure of the semidirect product H 1 ⋊ Sp(2) it is the symplectic group which acts on H 1 , 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 (26) ρ 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 −πq 2 / h is an eigenvector ofZ with the eigenvalue − i 2 . In addition v −1/2 is annihilated 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 − ) k v −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 [8, § 1.6; 12]. Since v −1/2 is the null solution of L + =X − iỸ, then by the general result [26,Cor. 24] the image of the wavelet transform will be null-solutions of the corresponding linear combination of the Lie derivatives (7): which turns out to be the Cauchy-Riemann equation on a weighted FSB-type space.

Symplectic Ladder Operators.
We can also look for ladder operators within the Lie algebra sp 2 , see [23, § 8]. Assuming L + 2 = aÃ + bB + cZ from the relations (10) and defining condition (23) we obtain the linear equations with unknown a, b and c: The equations have a solution if and only if λ 2 + + 4 = 0, and the raising/lowering operators are L ± 2 = ±iÃ +B. In the Shale-Weil representation (15) they turn out to be: Since this time λ + = 2i the ladder operators L ± 2 produce 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 representations (26) and (28) we get:

LADDER OPERATORS FOR THE HYPERBOLIC SUBGROUP
Consider the case of the Hamiltonian H = 2B, which is a repulsive (hyperbolic) harmonic oscillator [37, § 3.8]. The corresponding one-dimensional subgroup of symplectomorphisms produces hyperbolic rotations of the phase space. The eigenvectors v µ of the operator are Weber-Hermite (or parabolic cylinder) functions v ν = D ν− 1 2 ±2e i π 4 π h q , see [7, § 8.2; 33] for fundamentals 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 2B 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 h 1 and sp 2 separately. However, the additional options will be provided by a choice of the number system: either complex or double.

Complex Ladder Operators.
Assuming L + h = aX + bỸ from the commutators (11)(12), we obtain the linear equations: The equations have a solution if and only if λ 2 + −1 = 0. Taking the real roots λ = ±1 we obtain that the raising/lowering operators are L ± h =X ∓Ỹ. In the Schrödinger representation (14) the ladder operators are The null solutions v ± 1 2 (q) = e ± πi h q 2 to operators ρ h (L ± ) are also eigenvectors of the Hamiltonian ρ SW h (2B) with the eigenvalue ± 1 2 . However the important 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 sp 2 , that is in the form L + 2h = aÃ + bB + cZ for the commutator [2B, L + h ] = λL + h . We will get the system: A solution again exists if and only if λ 2 = 4. Within complex numbers we get only the values λ = ±2 with the ladder operators L ± 2h = ±2Ã +Z/2, see [13, § II.1; 30, § 1.1]. Each indecomposable h 1 -or sp 2 -module is formed by a one-dimensional chain of eigenvalues with a transitive action of ladder operators L ± h or L ± 2h respectively. And we again have a quadratic relation between the ladder operators:

Double Ladder Operators.
There are extra possibilities in in the context of hyperbolic quantum mechanics [16][17][18]. Here we use the representation of H 1 induced by a hyperbolic character e jht = cosh(ht) + j sinh(ht), see [25, (4.5)], and obtain the hyperbolic representation of H 1 , cf. (13): Then the associated Shale-Weil derived representation of sp 2 in the Schwartz space S(R) is, cf. (15): Note that ρ SW h (B) now generates a usual harmonic oscillator, not the repulsive one like ρ SW h (B) in (15). However, the expressions in the quadratic algebra are still the same (up to a factor), cf. (16)(17)(18): This is due to the Principle 1 of similarity and correspondence: we can swap operators Z and B with simultaneous replacement of hypercomplex units i and j.
The compatibility condition for a ladder operator within the Lie algebra h 1 will be (29) as before, since it depends only on the commutators (11)(12). Thus we still have the set of ladder operators corresponding to values λ = ±1: Admitting double numbers, we have an extra way to satisfy λ 2 = 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): Pairs L ± h and L ± j shift eigenvectors in the "orthogonal" directions changing their eigenvalues by ±1 and ±j. Therefore an indecomposable sp 2 -module can be parametrised by a two-dimensional lattice of eigenvalues in double numbers, see Table 1.
are null solutions to the operators L ± h and L ± j , respectively. They are also eigenvectors of 2ρ SW h (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]: and v ±j 1 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 multiplication by a zero, as it 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 sp 2 and admitting double numbers we will again find two sets of them [23, § 8]: Again the operators L ± 2h and L ± 2h produce double shifts in the orthogonal directions on the same two-dimensional lattice in Tab. 1.

LADDER OPERATOR FOR THE NILPOTENT SUBGROUP
Finally, we look for ladder operators for the HamiltonianB +Z/2 or, equivalently, −B +Z/2. It can be identified with a free particle [37, § 3.8].
We can look for ladder operators in the representation (14-15) within the Lie algebra h 1 in the form L ± ε = aX + bỸ. This is possible if and only if (37) − b = λa, 0 = λb.
The compatibility condition λ 2 = 0 implies λ = 0 within complex numbers. However, such a "ladder" operator produces only the zero shift on the eigenvectors, cf. (24). Another possibility appears if we consider the representation of the Heisenberg group induced by dual-valued characters. On the configurational space such a representation is [25, (4.11)]: (38) [ρ ε The corresponding derived representation of h 1 is 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): f ′ p (q, p))). Then the derived representation of h 1 is: εhI. An advantage of the FSB representation is that the derived form of the parabolic Shale-Weil representation coincides with the elliptic one (21).
Eigenfunctions with the eigenvalue µ of the parabolic HamiltonianB +Z/2 = q∂ p have the form (42) v µ (q, p) = e µp/q f(q), with an arbitrary function f(q).
Finally, consider the ladder operator for the same element B + Z/2 within the Lie algebra sp 2 . According to the above procedure we get the equations: which can again be resolved if and only if λ 2 = 0. There is the only complex root λ = 0 with the corresponding operators L ± p =B +Z/2, which does not affect the eigenvalues. However the dual number roots λ = ±ελ 2 with λ 2 ∈ C lead to the operators L ± ε = ±ελ 2Ã +B +Z/2 = ± ελ 2 2 (q∂ q − p∂ p ) + q∂ p .

CONCLUSIONS: SIMILARITY AND CORRESPONDENCE
We wish to summarise our findings. Firstly, the appearance of hypercomplex numbers in ladder operators for h 1 follows exactly the same pattern as was already noted for sp 2 [23, Rem. 32]: • the introduction of complex numbers is a necessity 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 usability of ladder operators, but they do provide an enhancement. In the spirit of the Similarity and Correspondence Principle 1 we have the following extension of Prop. 33 from [23]: Proposition 6. Let a vector H ∈ sp 2 generates the subgroup K, N ′ or A ′ , that is H = Z, B + Z/2, or 2B, respectively. Let ι be the respective hypercomplex unit. Then the ladder operators L ± satisfying to the commutation relation: [H, L ± 2 ] = ±ιL ± are given by: (1) Within the Lie algebra h 1 : L ± =X ∓ ιỸ. It is worth continuing this investigation and describing in detail hyperbolic and parabolic versions of FSB spaces.
Acknowledgements: I am grateful to the anonymous referees for their helpful remarks.