Lecture 3 Representations of the Heisenberg Group

The Heisenberg group, as many other things, is worth to be seen in action. We are going to describe various form of its actions on linear spaces, that is, representations of ℍⁿ.

3.1 Left Regular Representations and Its Subrepresentations

As usual, we can extend geometrical action of ℍ¹ on itself by left shift to a linear representation

on a certain linear space of functions.

Thus the (left invariant) Haar measure on ℍ¹ coincides with the Lebesgue measure. The same measure is also invariant under the right shifts, thus ℍ¹ is unimodular. Consequently, the action (1) on L₂(ℍ,dg) is unitary, it is called the left regular representation. Moreover, the action (1) on the Banach space L_p(ℍ,dg) for any 1≤ p < ∞ is also an isometry.

The left regular representation (1) is reducible on L₂(ℍ,dg). To find its (possibly irreducible) subrepresentations, we can construct linear representations of ℍ¹ by induction from a character χ of the centre Z, see Section 1.4 and [159]*§ 13. There are several equivalent forms for the construction, here we prefer the following one, cf. § 3.2 and [159]*§ 13 [321]*Ch. 5.

Let χ be a character of Z, that is, a group homomorphism of Z to the group of unimodular complex numbers. Let F₂^χ(ℍ¹) be the space of functions on ℍ¹ having the properties:

and

Then F₂^χ(ℍ¹) is invariant under the left shifts (1) and those shifts restricted to F₂^χ(ℍ¹) make a representation χ of ℍ¹ induced by χ.

However, the representation χ is not necessarily irreducible. Indeed, left shifts are commuting with the right action of the group. Thus, any subspace of null-solutions of a linear combination aS+∑_j=1ⁿ (b_jX_j+c_jY_j) of left-invariant vector fields is left-invariant and we can restrict χ to this subspace. The left-invariant differential operators define analytic condition for functions, cf. Cor. 6.

3.2 Induced Representations on Homogeneous Spaces

We can also build representations on the homogeneous spaces using the formula (5). We briefly remind this alternative construction of induced representations here [159]*§ 13.2. Consider a subgroup H of a group G. Let a smooth section s:G/H→ G be a right inverse of the natural projection p:G→ G/H. Thus any element g∈ G can be uniquely decomposed as g=s(p(g))*r(g) where the map r:G→ H is defined by the previous identity. For a character χ of H we can define a lifting L_χ: L₂(G/H) → L₂^χ(G) as follows:

The image space of the lifting L_χ satisfies to (2) and is invariant under left shifts. We also define the pulling P:L₂^χ(G) → L₂(G/H), which is a left inverse of the lifting and explicitly can be given, for example, by [PF](x)=F(s(x)). Then the induced representation on L₂(G/H) is generated by the formula

It implies the formula (5) for induced representations on the homogeneous spaces:

The corresponding forms of the induced representations are:

1. For H=Z the map r: ℍ¹→ Z is r(s,x,y)=(s,0,0). For the character χ_ℏ(s,0,0)=e^{2π i ℏ s}, the representation of ℍ¹ on L₂(ℝ²) is, cf. (21):

   [ℏ(s,x,y)f] (x′,y′) = e

   2π i ℏ  (−s− 1 2 ω(x,y;x′y′))

   f(x′−x,y′−y). (7)

   The Fourier transform maps this representation to the Fock–Segal–Bargmann (FSB) representation (13), see Exercise 5.
2. For H=H_x the map r(s,x,y)=(0,x,0). For the character χ(0,x,0)=e^{2π i ℏ x}, the representation ℍ¹ on L₂(ℝ²) is, cf. (22):

   [ℏ(s,x,y)f] (s′,y′) = e−2π i ℏ x f(s′−s−xy′+

   1 2

   xy,y′−y). (8)

3. For H=H′_x={(s,0,y)∈ℍ¹} the map r: ℍ¹→ H′_x is r(s,x,y)=(s−1/2xy,0,y). For the character χ_ℏ(s,0,y)=e^{2π i (ℏ s+qy)}, the representation of ℍ¹ on L₂(ℝ¹) is, cf. (24):

   [ℏ(s,x,y)f](x′)= exp(2π i (ℏ (−s+yx′−

   1 2

   xy)−qy))  f(x′−x). (9)

   For q=0, this a key to the Schrödinger representation of the Heisenberg group, which is obtaned by the Fourier transform.

We will see below that all these (and many others) representations with the same value ℏ are unitary equivalent.

3.3 Co-adjoint Representation and Method of Orbits

Let h₁^* be the dual space to h₁, that is the space of all linear functional on h₁∼ℝ³. For physical reasons, we use letter (h ,q,p) to denote bi-orthonormal coordinates to the exponential ones (s,x,y) on hⁿ.

The inner automorphism from the Exercise 1 map the unit (0,0,0) of ℍ¹ to itself. Thus inner automorphisms generate a transformation of the tangent space at (0,0,0) to itself, which is a linear map given by the same formula (16).

The above map is called the co-adjoint representation [159, § 15.1] Ad^*: h_n^* → h_n^* of ℍⁿ. There are two types of orbits in (10) for Ad^*, i.e. Euclidean spaces ℝ²ⁿ and single points:

All complex representations are induced [159, § 13] by a character χ_h(s,0,0)=e^{i h s} of the centre of ℍⁿ generated by (h,0,0)∈h_n^* and shifts (10) from the left on orbits (11). The explicit formula respecting physical units [181] is:

This is the Fock–Segal–Bargmann (FSB) type representation

The Stone–von Neumann Theorem 6 describes all unitary irreducible representations of ℍⁿ parametrised up to equivalence by two classes of orbits (11) and (12):

• The infinite dimensional representations by transformation χ (13) for h ≠ 0 in Fock [104, 139] space F₂(O_h)⊂L₂(O_h) of null solutions of Cauchy–Riemann type operators [181].
• The one-dimensional representations as multiplication by a constant on ℂ=L₂(O_(q,p)) which drops out from (13) for h =0:

  (q,p)(s,x,y): c ↦ e−i(qx+py)c. (14)

3.4 Stone–von Neumann Theorem

The following result [159, § 18.4], [104, Chap. 1, § 5] reduces

Proof.

3.5 Shale–Weil Representation

The Shale–Weil theorem [104]*§ 4.2 [138]*p. 830 states that any representation ℏ of the Heisenberg groups generates a unitary oscillator (or metaplectic) representation SWℏ of the Sp′(2), the two-fold cover of the symplectic group [104]*Thm. 4.58. The Shale–Weil theorem allows us also to expand any representation ℏ of the Heisenberg group to the representation 2ℏ=ℏ⊕SWℏ of the group Schrödinger group G.

Of course, there is the derived form of the Shale–Weil representation for g. It can often be explicitly written in contrast to the Shale–Weil representation.

Note that SWℏ(Z) is the Hamiltonian of the harmonic oscillator (up to a factor). Then we can consider SWℏ(B) as the Hamiltonian of a repulsive (hyperbolic) oscillator. The operator SWℏ(B−Z/2)=ℏi/4πd²/dq² is the parabolic analog. A graphical representation of all three transformations defined by those Hamiltonian is given in Fig. 11.1 and a further discussion of these Hamiltonians can be found in [338]*§ 3.8.

An important observation, which is often missed, is that the three linear symplectic transformations are unitary rotations in the corresponding hypercomplex algebra, cf. [194]*§ 3. This means, that the symplectomorphisms generated by operators Z, B−Z/2, B within time t coincide with the multiplication of hypercomplex number q+ι p by e^{ι t}, see Section 3.1 and Fig. 11.1, which is just another illustration of the Similarity and Correspondence Principle 5.