Lecture 4 Harmonic Oscillator and Ladder Operators

Complex valued representations of the Heisenberg group (also known as Weyl or Heisenberg-Weyl group) provide a natural framework for quantum mechanics [139, 104]. This is the most fundamental example of the Kirillov orbit method, induced representations and geometrical quantisation technique [161, 160]. Following the presentation in Section 1.4 we will consider representations of the Heisenberg group which are induced by hypercomplex characters of its centre: complex (which correspond to the elliptic case), dual (parabolic) and double (hyperbolic).

To describe dynamics of a physical system we use a universal equation based on inner derivations (commutator) of the convolution algebra [177] [181]. The complex valued representations produce the standard framework for quantum mechanics with the Heisenberg dynamical equation [336].

The double number valued representations, with the hyperbolic unit є²=1, is a natural source of hyperbolic quantum mechanics developed for a while [143, 142, 154, 151, 155]. The universal dynamical equation employs hyperbolic commutator in this case. This can be seen as a Moyal bracket based on the hyperbolic sine function. The hyperbolic observables act as operators on a Krein space with an indefinite inner product. Such spaces are employed in study of PT-symmetric Hamiltonians and hyperbolic unit є²=1 naturally appear in this setup [121].

The representations with values in dual numbers provide a convenient description of the classical mechanics. For this we do not take any sort of semiclassical limit, rather the nilpotency of the parabolic unit (ε²=0) do the task. This removes the vicious necessity to consider the Planck constant tending to zero. The dynamical equation takes the Hamiltonian form. We also describe classical non-commutative representations of the Heisenberg group which acts in the first jet space.

The approach provides not only three different types of dynamics, it also generates the respective rules for addition of probabilities as well. For example, the quantum interference is the consequence of the same complex-valued structure, which directs the Heisenberg equation. The absence of an interference (a particle behaviour) in the classical mechanics is again the consequence the nilpotency of the parabolic unit. Double numbers creates the hyperbolic law of additions of probabilities, which was extensively investigates [154, 151]. There are still unresolved issues with positivity of the probabilistic interpretation in the hyperbolic case [143, 142].

This section clarifies foundations of quantum and classical mechanics. We recovered the existence of three non-isomorphic models of mechanics from the representation theory. They were already derived in [143, 142] from translation invariant formulation, that is from the group theory as well. It also hinted that hyperbolic counterpart is (at least theoretically) as natural as classical and quantum mechanics are. The approach provides a framework for a description of aggregate system which have say both quantum and classical components. This can be used to model quantum computers with classical terminals [189].

Remarkably, simultaneously with the work [142] group-invariant axiomatics of geometry leaded R.I. Pimenov [284] to description of 3ⁿ Cayley–Klein constructions. The connection between group-invariant geometry and respective mechanics were explored in many works of N.A. Gromov, see for example [115, 116, 119]. They already highlighted the rôle of three types of hypercomplex units for the realisation of elliptic, parabolic and hyperbolic geometry and kinematic.

There is a further connection between representations of the Heisenberg group and hypercomplex numbers. The symplectomorphism of phase space are also automorphism of the Heisenberg group [104]*§ 1.2. We recall that the symplectic group [2] [104]*§ 1.2 is isomorphic to the group SL₂(ℝ)  [240] [140] [251] and provides linear symplectomorphisms of the two-dimensional phase space. It has three types of non-iso­mor­phic one-dimensional continuous subgroups (3-5) with symplectic action on the phase space illustrated by Fig. 11.1. Hamiltonians, which produce those symplectomorphism, are of interest [338]*§ 3.8 [322] [323]. An analysis of those Hamiltonians from Section 3.3 by means of ladder operators recreates hypercomplex coefficients as well [196].

Harmonic oscillators, which we shall use as the main illustration here, are treated in most textbooks on quantum mechanics. This is efficiently done through creation/annihilation (ladder) operators, cf. § 3.3 and [109] [48]. The underlying structure is the representation theory of the Heisenberg and symplectic groups [240]*§ VI.2 [321]*§ 8.2 [139] [104]. As we will see, they are naturally connected with respective hypercomplex numbers. As a result we obtain further illustrations to the Similarity and Correspondence Principle 5.

We work with the simplest case of a particle with only one degree of freedom. Higher dimensions and the respective group of symplectomorphisms [2n] may require consideration of Clifford algebras [166] [69] [66] [121] [289].

4.1 p-Mechanic Formalism

Here we briefly outline a formalism [169, 294, 177, 53, 181], which allows to unify quantum and classical mechanics.

4.1.1 Convolutions (Observables) on ℍⁿ and Commutator

Using a invariant measure dg=ds dx dy on ℍⁿ we can define the convolution of two functions:

This is a non-commutative operation, which is meaningful for functions from various spaces including L₁(ℍⁿ,dg), the Schwartz space S and many classes of distributions, which form algebras under convolutions. Convolutions on ℍⁿ are used as observables in p-mechanic [169, 181].

A unitary representation of ℍⁿ extends to L₁(ℍⁿ ,dg) by the formula:

This is also an algebra homomorphism of convolutions to linear operators.

For a dynamics of observables we need inner derivations D_k of the convolution algebra L₁(ℍⁿ), which are given by the commutator:

To describe dynamics of a time-dependent observable f(t,g) we use the universal equation, cf. [167, 169]:

where S is the left-invariant vector field (10) generated by the centre of ℍⁿ. The presence of operator S fixes the dimensionality of both sides of the equation (4) if the observable H (Hamiltonian) has the dimensionality of energy [181]*Rem 4.1. If we apply a right inverse of S to both sides of the equation (4) we obtain the equivalent equation

based on the universal bracket k₁k₂=k₁* k₂−k₂* k₁ [181].

4.1.2 States and Probability

Let an observable (k) (2) is defined by a kernel k(g) on the Heisenberg group and its representation at a Hilbert space H. A state on the convolution algebra is given by a vector v∈H. A simple calculation:

can be restated as:

Here the left-hand side contains the inner product on H, while the right-hand side uses a skew-linear pairing between functions on ℍⁿ based on the Haar measure integration. In other words we obtain, cf. [53]*Thm. 3.11:

The addition of vectors in H implies the following operation on states:

The last expression can be conveniently rewritten for kernels of the functional as

for some real number A. This formula is behind the contextual law of addition of conditional probabilities [153] and will be illustrated below. Its physical interpretation is an interference, say, from two slits. Despite of a common belief, the mechanism of such interference can be both causal and local, see [179] [152].

4.2 Elliptic characters and Quantum Dynamics

In this section we consider the representation h of ℍⁿ induced by the elliptic character χ_h(s)=e^{ih s} in complex numbers parametrised by h∈ℝ. We also use the convenient agreement h=2πℏ borrowed from physical literature.

4.2.1 Fock–Segal–Bargmann and Schrödinger Representations

The realisation of h by the left shifts (1) on L₂^h(ℍⁿ) is rarely used in quantum mechanics. Instead two unitary equivalent forms are more common: the Schrödinger and Fock–Segal–Bargmann (FSB) representations.

The FSB representation can be obtained from the orbit method of Kirillov [160]. It allows spatially separate irreducible components of the left regular representation, each of them become located on the orbit of the co-adjoint representation, see [181]*§ 2.1 [160] for details, we only present a brief summary here.

We identify ℍⁿ and its Lie algebra h_n through the exponential map [159]*§ 6.4. The dual h_n^* of h_n is presented by the Euclidean space ℝ²ⁿ⁺¹ with coordinates (ℏ,q,p). The pairing h_n^* and h_n given by

This pairing defines the Fourier transform ^: L₂(ℍⁿ)→ L₂(h_n^*) given by [161]*§ 2.3:

For a fixed ℏ the left regular representation (1) is mapped by the Fourier transform to the FSB type representation (13). The collection of points (ℏ,q,p)∈h_n^* for a fixed ℏ is naturally identified with the phase space of the system.

To recover the Schrödinger representation we use notations and technique of induced representations from § 3.2, see also [173]*Ex. 4.1. The subgroup H={(s,0,y) ∣  s∈ℝ, y∈ℝⁿ}⊂ℍⁿ defines the homogeneous space X=G/H, which coincides with ℝⁿ as a manifold. The natural projection p:G→ X is p(s,x,y)=x and its left inverse s:X→ G can be as simple as s(x)=(0,x,0). For the map r:G→ H, r(s,x,y)=(s−xy/2,0,y) we have the decomposition

For a character χ_h(s,0,y)=e^{ih s} of H the lifting L_χ: L₂(G/H) → L₂^χ(G) is as follows:

Thus the representation χ(g)=P∘Λ (g)∘L becomes:

After the Fourier transform x↦ q we get the Schrödinger representation on the configuration space:

Note that this again turns into a commutative representation (multiplication by an unimodular function) if ℏ=0. To get the full set of commutative representations in this way we need to use the character χ_(h,p)(s,0,y)=e^{2πi(ℏ+ py)} in the above consideration.

4.2.2 Commutator and the Heisenberg Equation

The property (2) of F₂^χ(ℍⁿ) implies that the restrictions of two operators χ (k₁) and χ (k₂) to this space are equal if

In other words, for a character χ(s)=e^{2πi ℏ s} the operator χ (k) depends only on

which is the partial Fourier transform s↦ ℏ of k(s,x,y). The restriction to F₂^χ(ℍⁿ) of the composition formula for convolutions is [181]*(3.5):

Under the Schrödinger representation (16) the convolution (17) defines a rule for composition of two pseudo-differential operators (PDO) in the Weyl calculus [139] [104]*§ 2.3.

Consequently the representation (2) of commutator (3) depends only on its partial Fourier transform [181]*(3.6):

Under the Fourier transform (13) this commutator is exactly the Moyal bracket [341] for of k′ and k on the phase space.

For observables in the space F₂^χ(ℍⁿ) the action of S is reduced to multiplication, e.g. for χ(s)=e^{i h s} the action of S is multiplication by i h. Thus the equation (4) reduced to the space F₂^χ(ℍⁿ) becomes the Heisenberg type equation [181]*(4.4):

based on the above bracket (18). The Schrödinger representation (16) transforms this equation to the original Heisenberg equation.

4.2.3 Quantum Probabilities

For the elliptic character χ_h(s)=e^{ih s} we can use the Cauchy–Schwarz inequality to demonstrate that the real number A in the identity (12) is between −1 and 1. Thus we can put A=cosα for some angle (phase) α to get the formula for counting quantum probabilities, cf. [154]*(2):

4.3 Ladder Operators and Harmonic Oscillator

Let be a representation of the Schrödinger group G=ℍ¹⋊Sp′(2) (17) in a space V. Consider the derived representation of the Lie algebra g [240]*§ VI.1 and denote X′=(X) for X∈g. To see the structure of the representation we can decompose the space V into eigenspaces of the operator X′ 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 (3–5) of [2] starting from the compact case. Let H=Z be a generator of the compact subgroup K. Corresponding symplectomorphisms (4) of the phase space are given by orthogonal rotations with matrices (

). The Shale–Weil representation (16) coincides with the Hamiltonian of the harmonic oscillator in Schrödinger representation.

Since Sp′(2) is a two-fold cover the corresponding eigenspaces of a compact group Z′ v_k=i k v_k are parametrised by a half-integer k∈ℤ/2. Explicitly for a half-integer k eigenvectors are:

where H_k is the Hermite polynomial [104]*§ 1.7 [93]*8.2(9).

From the point of view of quantum mechanics as well as the representation theory it is beneficial to introduce the ladder operators L^± (15), known also as creation/annihilation in quantum mechanics [104]*p. 49 [48]. There are two ways to search for ladder operators: in (complexified) Lie algebras h₁ and sp₂. The later coincides with our consideration in Section 3.3 in the essence.

4.3.1 Ladder Operators from the Heisenberg Group

Assuming L⁺=aX′+bY′ we obtain from the relations (18–19) and (15) the linear equations with unknown a and b:

The equations have a solution if and only if λ₊²+1=0, and the raising/lowering operators are L^±= X′∓iY′.

In the Schrödinger representation (15) the ladder operators are

The standard treatment of the harmonic oscillator in quantum mechanics, which can be found in many textbooks, e.g.  [104]*§ 1.7 [109]*§ 2.2.3, is as follows. The vector v_−1/2(q)=e^{−π q2/ℏ} is an eigenvector of Z′ with the eigenvalue −i/2. In addition v_−1/2 is annihilated by L⁺. Thus the chain (16) 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 [139] [104]*§ 1.6. Since v_−1/2 is the null solution of L⁺=X′−i Y′, then by Cor. 6 the image of the wavelet transform will be null-solutions of the corresponding linear combination of the Lie derivatives (10):

which turns out to be the Cauchy–Riemann equation on a weighted FSB-type space, see Section 5.4 below.

4.3.2 Symplectic Ladder Operators

We can also look for ladder operators within the Lie algebra sp₂, see Subsection 4.3.1 and [194]*§ 8. Assuming L₂⁺=aA′+bB′+cZ′ from the relations (13) and defining condition (15) 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 L₂^±=±i A′+B′. In the Shale–Weil representation (16) they turn out to be:

Since this time λ₊=2i the ladder operators L₂^± produce a shift on the diagram (16) twice bigger than the operators L^± from the Heisenberg group. After all, this is not surprising since from the explicit representations (28) and (30) we get:

4.4 Hyperbolic Quantum Mechanics

Now we turn to double numbers also known as hyperbolic, split-complex, etc. numbers [339]*App. C [325] [157]. They form a two dimensional algebra ℝ_h spanned by 1 and є with the property є²=1. There are zero divisors:

Thus double numbers algebraically isomorphic to two copies of ℝ spanned by є_±. Being algebraically dull double numbers are nevertheless interesting as a homogeneous space [191, 194] and they are relevant in physics [151, 325, 326]. The combination of p-mechanical approach with hyperbolic quantum mechanics was already discussed in [53]*§ 6.

For the hyperbolic character χ_{є h}(s)=e^{є h s}=coshh s +єsinhh s of ℝ one can define the hyperbolic Fourier-type transform:

It can be understood in the sense of distributions on the space dual to the set of analytic functions [155]*§ 3. Hyperbolic Fourier transform intertwines the derivative d/dx and multiplication by є q [155]*Prop. 1.

An elegant theory of hyperbolic Fourier transform may be achieved by a suitable adaptation of [138], which uses representation theory of the Heisenberg group.

4.4.1 Hyperbolic Representations of the Heisenberg Group

Consider the space F_h^є(ℍⁿ) of ℝ_h-valued functions on ℍⁿ with the property:

and the square integrability condition (3). Then the hyperbolic representation is obtained by the restriction of the left shifts to F_h^є(ℍⁿ). To obtain an equivalent representation on the phase space we take ℝ_h-valued functional of the Lie algebra h_n:

The hyperbolic Fock–Segal–Bargmann type representation is intertwined with the left group action by means of the Fourier transform (13) with the hyperbolic functional (32). Explicitly this representation is:

For a hyperbolic Schrödinger type representation we again use the scheme described in § 3.2. Similarly to the elliptic case one obtains the formula, resembling (15):

Application of the hyperbolic Fourier transform produces a Schrödinger type representation on the configuration space, cf. (16):

The extension of this representation to kernels according to (2) generates hyperbolic pseudodifferential operators introduced in [155]*(3.4).

4.4.2 Hyperbolic Dynamics

Similarly to the elliptic (quantum) case we consider a convolution of two kernels on ℍⁿ restricted to F_h^є(ℍⁿ). The composition law becomes, cf. (17):

This is close to the calculus of hyperbolic PDO obtained in [155]*Thm. 2. Respectively for the commutator of two convolutions we get, cf. (18):

This the hyperbolic version of the Moyal bracket, cf. [155]*p. 849, which generates the corresponding image of the dynamic equation (4).

4.4.3 Hyperbolic Probabilities

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(a)  (b)

Figure 4.2: Hyperbolic probabilities: the blue (dashed) graph shows the addition of probabilities without interaction, the red (solid) graph present the quantum interference. Left picture shows the Gaussian state (23), with the same distribution as in quantum mechanics, cf. Fig. 4.1(a). The right picture shows the rational state (26), note the absence of interference oscillations in comparison with the quantum state on Fig. 4.1(b).

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To calculate probability distribution generated by a hyperbolic state we are using the general procedure from Section 4.1.2. The main differences with the quantum case are as follows:

1. The real number A in the expression (12) for the addition of probabilities is bigger than 1 in absolute value. Thus it can be associated with the hyperbolic cosine coshα , cf. Rem. 8, for certain phase α∈ℝ [155].
2. The nature of hyperbolic interference on two slits is affected by the fact that e^{є h s} is not periodic and the hyperbolic exponent e^{є t} and cosine cosht do not oscillate. It is worth to notice that for Gaussian states the hyperbolic interference is exactly the same as quantum one, cf. Figs. 4.1(a) and 4.2(a). This is similar to coincidence of quantum and hyperbolic dynamics of harmonic oscillator.

   The contrast between two types of interference is prominent for the rational state (26), which is far from the minimal uncertainty, see the different patterns on Figs. 4.1(b) and 4.2(b).

4.4.4 Ladder Operators for the Hyperbolic Subgroup

Consider the case of the Hamiltonian H=2B, which is a repulsive (hyperbolic) harmonic oscillator [338]*§ 3.8. The corresponding one-dimensional subgroup of symplectomorphisms produces hyperbolic rotations of the phase space, see Fig. 11.1. 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/ℏ q), see [93]*§ 8.2 [312] for fundamentals of Weber–Hermite functions and [322] 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 [140]*§ II.1 [251]*§ 1.1. We again seek solutions in two subalgebras h₁ and sp₂ separately. However the additional options will be provided by a choice of the number system: either complex or double.

4.4.5 Double Ladder Operators

There are extra possibilities in in the context of hyperbolic quantum mechanics [154] [151] [155]. Here we use the representation of ℍ¹ induced by a hyperbolic character e^{є h t}=cosh(h t)+єsinh(h t), see [199]*(4.5), and obtain the hyperbolic representation of ℍ¹, cf. (16):

The corresponding derived representation is

Then the associated Shale–Weil derived representation of sp₂ in the Schwartz space S(ℝ) is, cf. (16):

Note that SWh(B) now generates a usual harmonic oscillator, not the repulsive one like SWℏ(B) in (16). However the expressions in the quadratic algebra are still the same (up to a factor), cf. (17–19):

This is due to the Principle 5 of similarity and correspondence: we can swap operators Z and B with simultaneous replacement of hypercomplex units i and є.

The eigenspace of the operator 2SWh(B) with an eigenvalue є ν are spanned by the Weber–Hermite functions D_−ν−1/2(±√2/hx), see [93]*§ 8.2. Functions D_ν are generalisations of the Hermit functions (27).

The compatibility condition for a ladder operator within the Lie algebra h₁ will be (37) as before, since it depends only on the commutators (18–19). Thus we still have the set of ladder operators corresponding to values λ=±1:

Admitting double numbers we have an extra way to satisfy λ²=1 in (37) with values λ=±є. Then there is an additional pair of hyperbolic ladder operators, which are identical (up to factors) to (28):

Pairs L_h^± and L_є^± shift eigenvectors in the “orthogonal” directions changing their eigenvalues by ±1 and ±є. Therefore an indecomposable sp₂-module can be para­metrised by a two-dimensional lattice of eigenvalues in double numbers, see Fig. 3.2.

The following functions

are null solutions to the operators L_h^± and L_є^± respectively. They are also eigenvectors of 2SWh(B) with eigenvalues ∓є/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 Cor. 6:

for v_1/2^±h and v_1/2^±є respectively. This is again in line with the classical result (29). 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 on this lattice. 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±є.

We can also search ladder operators within the algebra sp₂ and admitting double numbers we will again find two sets of them, cf. § ??:

Again the operators L_2h^± and L_2h^± produce double shifts in the orthogonal directions on the same two-dimensional lattice in Fig. 3.2.

4.5 Parabolic (Classical) Representations on the Phase Space

After the previous two cases it is natural to link classical mechanics with dual numbers generated by the parabolic unit ε²=0. Connection of the parabolic unit ε with the Galilean group of symmetries of classical mechanics is around for a while [339]*App. C.

However the nilpotency of the parabolic unit ε make it difficult if we will work with dual number valued functions only. To overcome this issue we consider a commutative real algebra C spanned by 1, i, ε and iε with identities i²=−1 and ε²=0. A seminorm on C is defined as follows:

4.5.1 Classical Non-Commutative Representations

We wish to build a representation of the Heisenberg group which will be a classical analog of the Fock–Segal–Barg­mann representation (13). To this end we introduce the space F_h^ε(ℍⁿ) of C-valued functions on ℍⁿ with the property:

and the square integrability condition (3). It is invariant under the left shifts and we restrict the left group action to F_h^ε(ℍⁿ).

An equivalent form of the induced representation acts on F_h^ε(ℝ²ⁿ), where ℝ²ⁿ) is the homogeneous space of ℍⁿ over its centre. The Fourier transform (x,y)↦(q,p) intertwines the last representation with the following action on C-valued functions on the phase space:

We may also write a classical Schrödinger type representation. According to § 3.2 we get a representation formally very similar to the elliptic (15) and hyperbolic versions (34):

However due to nilpotency of ε the (complex) Fourier transform x↦ q produces a different formula for parabolic Schrödinger type representation in the configuration space, cf. (16) and (39):

This representation shares all properties mentioned in Rem. 16 as well.

4.5.2 Hamilton Equation

The identity e ^{ε t}−e ^{−ε t}= 2ε t can be interpreted as a parabolic version of the sine function, while the parabolic cosine is identically equal to one, cf. § 3.1 and [129, 190]. From this we obtain the parabolic version of the commutator (18):

for the partial parabolic Fourier-type transform k_s of the kernels. Thus the parabolic representation of the dynamical equation (4) becomes:

Although there is no possibility to divide by ε (since it is a zero divisor) we can obviously eliminate ε h from the both sides if the rest of the expressions are real. Moreover this can be done “in advance” through a kind of the antiderivative operator considered in [181]*(4.1). This will prevent “imaginary parts” of the remaining expressions (which contain the factor ε) from vanishing.

Using the duality between the Lie algebra of ℍⁿ and the phase space we can find an adjoint equation for observables on the phase space. To this end we apply the usual Fourier transform (x,y)↦(q,p). It turn to be the Hamilton equation [181]*(4.7). However the transition to the phase space is more a custom rather than a necessity and in many cases we can efficiently work on the Heisenberg group itself.

An alternative derivation of classical dynamics from the Heisenberg group is given in the recent paper [247].

4.5.3 Classical probabilities

It is worth to notice that dual numbers are not only helpful in reproducing classical Hamiltonian dynamics, they also provide the classic rule for addition of probabilities. We use the same formula (10) to calculate kernels of the states. The important difference now that the representation (46) does not change the support of functions. Thus if we calculate the correlation term ⟨ v₁,(g)v₂ ⟩ in (11), then it will be zero for every two vectors v₁ and v₂ which have disjoint supports in the phase space. Thus no interference similar to quantum or hyperbolic cases (Section 4.2.3) is possible.

4.5.4 Ladder Operator for the Nilpotent Subgroup

Finally we look for ladder operators for the Hamiltonian B′+Z′/2 or, equivalently, −B′+Z′/2. It can be identified with a free particle [338]*§ 3.8.

We can look for ladder operators in the representation (15–16) within the Lie algebra h₁ in the form L_ε^±=aX′+bY′. This is possible if and only if

The compatibility condition λ²=0 implies λ=0 within complex numbers. However such a “ladder” operator produces only the zero shift on the eigenvectors, cf. (??).

Another possibility appears if we consider the representation of the Heisenberg group induced by dual-valued characters. On the configuration space such a representation is [199]*(4.11):

The corresponding derived representation of h₁ is

However the Shale–Weil extension generated by this representation is inconvenient. It is better to consider the FSB–type parabolic representation (46) on the phase space induced by the same dual-valued character. Then the derived representation of h₁ is:

An advantage of the FSB representation is that the derived form of the parabolic Shale–Weil representation coincides with the elliptic one (22).

Eigenfunctions with the eigenvalue µ of the parabolic Hamiltonian B′+Z′/2=q∂_p have the form

The linear equations defining the corresponding ladder operator L_ε^±=aX′+bY′ in the algebra h₁ are (48). 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

For the eigenvalue µ=µ₀+εµ₁ with µ₀, µ₁∈ℂ the eigenfunction (52) can be rewritten as:

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, cf. § ??. There is the only operator L_p^±=B′+Z′/2 corresponding to complex coefficients, which does not affect the eigenvalues. However the dual numbers lead to the operators

These operator act on eigenvalues in a non-trivial way.

4.5.5 Similarity and Correspondence

We wish to summarise our findings. Firstly, the appearance of hypercomplex numbers in ladder operators for h₁ follows exactly the same pattern as was already noted for sp₂, see Rem. 9:

• 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 5 we have the following extension of Prop. 10:

It is worth continuing this investigation and describing in details hyperbolic and parabolic versions of FSB spaces.