The Heisenberg Group

The relations, which define the Heisenberg group or its Lie algebra, are of a fundamental nature and appeared in very different areas. For example, the basic operators of differentiation and multiplication by an independent variable in analysis satisfy to the same commutation relations as observables of momentum and coordinate in quantum mechanics.

We shall start from the general properties the Heisenberg group and its representations. Many important applications will follow.

Remark 1 It is worth to mention that the Heisenberg group is also known as the Weyl (or Heisenberg–Weyl) group in physical literature. It is another illustration of its perception as an extraneous object: physicists call it by the name of a mathematician, and mathematicians by the name of a physicists.

To add more confusion the Lie algebra of the Heisenberg group is called Weyl algebra . However, the commutator relations [Q,P]=I in the Weyl algebra are called Heisenberg commutator relations , however the commutation relation sm=q ms in the representation the Heisenberg group are known as the Weyl commutation relation . The most obvious simplification would be to call every above object the Heisenberg–Weyl. However we will use the most common names in this work.

2.1 The Symplectic Form and the Heisenberg group

Let n≥ 1 be an integer. For two real n-vectors x, y∈ℝⁿ, we write xy for their inner product:

The following notion is the central for Hamiltonian formulation of classical mechanics [11]*§ 37.

Definition 2 The symplectic form ω on ℝ²ⁿ is a function of two vectors such that:

ω(x,y;x′,y′)=xy′−x′y, where (x,y), (x′,y′)∈ℝ²ⁿ. (3)

Exercise 3 Check the following properties:

ω is anti-symmetric ω(x,y;x′,y′)=− ω(x′,y′;x,y).

ω is bilinear:

ω( x, y;α x′,α y′)	=	α ω(x,y;x′,y′),
ω( x, y; x′+x″, y′+x″)	=	α ω(x,y;x′,y′)+ω(x,y;x″,y″).

Let z=x+i y and w=x′+i y′ then ω can be expressed through the complex inner product (2) as ω(x,y;x′,y′)=−ℑ (zw).
The symplectic form on ℝ² is equal to det (
x x′

y y′

). Consequently it vanishes if and only if (x,y) and (x′,y′) are collinear.

Let A∈SL₂(ℝ) be a real 2× 2 matrix with the unit determinant. Define:

⎛
⎜
⎝

x′

y′

⎞
⎟
⎠

⎛
⎜
⎝

⎞
⎟
⎠

and

⎛
⎜
⎝

x′′

y′′

⎞
⎟
⎠

⎛
⎜
⎝

x′

y′

⎞
⎟
⎠

. (4)

Then, ω(x′,y′;x′′,y′′)=ω(x,y;x′,y′). Moreover, the symplectic group [2]—the set of all linear transformations of ℝ² preserving ω—coincides with SL₂(ℝ).

Definition 4 An element of the n-dimensional Heisenberg group ℍⁿ [104, 139] is (s,x,y)∈ℝ²ⁿ⁺¹, where s∈ℝ and x, y∈ ℝⁿ. The group law on ℍⁿ is given as follows:

(s,x,y)·(s′,x′,y′)=(s+s′+

ω(x,y;x′,y′),x+x′,y+y′), (5)

where ω the symplectic form.

For the sake of simplicity, we will work with the one-dimensional Heisenberg group ℍ¹ on several occasions. This shall be mainly in the cases involving the symplectic group, since we want to the stay the basic case [2]∼ SL₂(ℝ). However, consideration of the general case of ℍⁿ is similar in most respects.

Exercise 5 For the Heisenberg group ℍⁿ, check that:

The unit is (0,0,0) and the inverse (s,x,y)⁻¹=(−s,−x,−y).
It is a non-commutative Lie group. Hint: Use the properties of the symplectic form from the Exercise 3. For example, the associativity follows from the linearity of ω. ⋄
It has the centre
Z={(s,0,0)∈ ℍⁿ, s ∈ ℝ}. (6)

Exercise 6

Introduce complexified coordinates (s,z) on ℍ¹ with z=x+i y. Then the group law can be written as:
(s,z)·(s′,z′)=(s+s′+
1

2

ℑ(z′z),z+z′).
Show that the set ℝ³ with the group law
(s,x,y)·(s′,x′,y′)=(s+s′+xy′,x+x′,y+y′) (7)

is isomorphic to the Heisenberg group ℍ¹. It is called polarised Heisenberg group [104]*§ 1.2. Hint: Use the explicit form of the homomorphism (s,x,y)↦ (s+1/2xy,x,y). ⋄
Define the map φ: ℍ¹ → M₃(ℝ) by
φ(s,x,y)=
⎛
⎜
⎜
⎜
⎜
⎜
⎝
1 x
s+
1

2

xy

0 1 y

0 0 1

⎞
⎟
⎟
⎟
⎟
⎟
⎠

. (8)

This is a group homomorphism from ℍ¹ to the group of 3× 3 matrices with the unit determinant and the matrix multiplication as the group operation. Write also a group homomorphism from the polarised Heisenberg group to M₃(ℝ).
Expand the above items from this Exercise to ℍⁿ.

2.2 Lie algebra of the Heisenberg group

The Lie algebra of the Heisenberg group h₁ is also called Weyl algebra. From the general theory we know, that h₁ is a three-dimensional real vector space, thus, it can be identified as a set with the group ℍ¹∼ℝ³ itself.

There are several standard possibilities to realise h₁, cf. Sect. 2.3. Firstly, we link h₁ with one-parameter subgroups as in Sect. 2.3.1.

Exercise 7

Show that 3× 3 matrices from (8) are created by the following exponential map:

exp

⎛
⎜
⎜
⎝

0	x	s
0	0	y
0	0	0

⎞
⎟
⎟
⎠

⎛
⎜
⎜
⎜
⎜
⎜
⎝

⎞
⎟
⎟
⎟
⎟
⎟
⎠

. (9)

Thus h₁ isomorphic to the vector space of matrices in the left-hand side. We can define the explicit identification exp: h₁ → ℍ¹ by (9), which is also known as the exponential coordinates on ℍ¹.

Define the basis of h₁:

⎛
⎜
⎜
⎝

0	0	1
0	0	0
0	0	0

⎞
⎟
⎟
⎠

, X=

⎛
⎜
⎜
⎝

0	1	0
0	0	0
0	0	0

⎞
⎟
⎟
⎠

, Y=

⎛
⎜
⎜
⎝

0	0	0
0	0	1
0	0	0

⎞
⎟
⎟
⎠

. (10)

Write the one-parameter subgroups of ℍ¹ generated by S, X and Y.

Another possibility is a description of h₁ as the collection of invariant vector fields, see Sect. 2.3.2.

Exercise 8

Check that the following vector fields on ℍ¹ are left (right) invariant:
S^l(r)=±∂_s, X^l(r)=±∂_x−
1

2

y∂_s, Y^l(r)=±∂_y+
1

2

x∂_s. (11)

Show also that they are linearly independent and, thus, are bases the Lie algebra h₁ in two different realisations.
Calculate one-parameter groups of right (left) shifts on ℍ¹ generated by these vector fields.

The principal operation on a Lie algebra, besides the linear structure, is the commutator [A,B]=AB−BA, see Sect. 2.3.3. In the above exercises we can define the commutator for matrices and vector fields through the corresponding algebraic operations in these algebras.

Exercise 9

Check that bases from (10) and (10) satisfy the Heisenberg commutator relation
[X^l(r),Y^l(r)]=S^l(r) (12)

and all other commutators vanishing. More generally:
[A,A′]=ω(x,y;x′,y′) S, where A^(′)=s^(′)S+x^(′)X+y^(′)Y, (13)

and ω is the symplectic form.
Show that any second (and, thus, any higher) commutator [[A,B],C] on h₁ vanishes. This property can be stated as “the Heisenberg group is a step 2 nilpotent Lie group”.
Check the formula
exp(A)exp(B)=exp(A+B+
1

2

[A,B]), where A,B∈h₁. (14)

The formula is also true for any step 2 nilpotent Lie group and is a particular case of the Baker–Campbell–Hausdorff formula. Hint: In the case of ℍ¹ you can use the explicit form of the exponential map (9). ⋄
Define the vector space decomposition
h₁=V₀⊕ V₁, such that V₀=[V₁,V₁]. (15)

Consequently, we can start from definition of the Lie algebra h_n through the commutation relations (12). Thereafter, ℍⁿ and the group law (5) can be derived from the exponentiation of h_n.

We note that any element A∈h₁ defines an adjoint map ad(A): B↦ [A,B] on h₁.

Exercise 10 Write matrices corresponding to transformations ad(S), ad(X), ad(Y) of h₁ in the basis S, X, Y.

2.3 Automorphisms of the Heisenberg group

Erlangen programme suggest investigate invariants under group action. This recipe can be applied recursively to groups themselves. Transformations of a group which preserve its structure are called group automorphisms.

Exercise 11 Check that the following are automorphisms of ℍ¹:

Inner automorphisms or conjugation with (s,x,y)∈ℍ¹:

(s′,x′,y′)↦(s,x,y)· (s′,x′,y′) · (s,x,y)⁻¹	=	(s′+ω(x,y;x′,y′),x′,y′)
	=	(s′+xy′−x′y,x′,y′).	(16)

Symplectic maps (s,x,y)=(s,x′,y′), where (
x′

y′

)=A (
x

y

) with A from the symplectic group [2] ∼SL₂(ℝ), see Exercise 5.
Dilations: (s,x,y)↦(r²s,rx,ry) for a positive real r.
Inversion: (s,x,y)↦(−s,y,x).

The last three types of transformations are outer automorphisms.

Exercise 12 Show that

Automorphism groups of ℍ¹ and h₁ coincide as groups of maps of ℝ³ onto itself. Hint: Use the exponent map and the relation (14). The crucial step is a demonstration that any automorphism of ℍ¹ is a linear map of ℝ³. See details in [104]*Ch. 1, (1.21). ⋄
All transforms from Exercise 11 viewed as automorphisms of h₁ preserve the decomposition (15).
Every automorphism of ℍ¹ can be written uniquely as composition of a symplectic map, an inner automorphism, a dilation and a power (mod 2) of the inversion from Exercise 11. Hint: Any automorphism is a linear map (by the previous item) of ℝ³ which maps the centre Z to itself. Thus it shall have the form (s,x,y)↦ (cs+ax+by, T(x,y)), where a, b and c are real and T is a linear map of ℝ², see [104]*Ch. 1, (1.22). ⋄

The symplectic automorphisms from Exercise 2 can be characterised as the group of outer automorphisms of ℍ¹, which trivially acts on the centre of ℍ¹. It is the group of symmetries of the symplectic form ω in (5) [104]*Thm. 1.22 [138]*p. 830. The symplectic group is isomorphic to SL₂(ℝ) considered in the first half of this work, see Exercise 5. For future use we will need Sp′(2) which is the double cover of [2].

We can build the semidirect product G=ℍ¹⋊Sp′(2) with the standard group law for semidirect products:

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 (4). 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 [263]. This group is of interest not only in quantum mechanics but also in optics [323, 322].

Consider the Lie algebra sp₂ of the group [2]. We again use the basis A, B, Z (13) with commutators (13). Vectors Z, B−Z/2 and B are generators of the one-parameter subgroups K, N′ and (3–5) respectively. Furthermore we can consider the basis {S, X, Y, A, B, Z} of the Lie algebra g of the Schrödinger group G=ℍ¹⋊Sp′(2). All non-zero commutators besides those already listed in (12) and (13) are:

2.4 Subgroups of ℍⁿ and Homogeneous Spaces

We want to classify up to certain equivalences all possible ℍ¹–homogeneous spaces. According to Sect. 2.2.2 we will look for continuous subgroups of ℍ¹.

One-dimensional continuous subgroups of ℍ¹ can be classified up to group automorphism. Two one-dimensional subgroups of ℍ¹ are the centre Z (6) and

Exercise 13 Show that

There is no an automorphism of ℍ¹ which maps Z to H_x.
For any one-parameter continuous subgroup H of ℍ¹ there is an automorphism of ℍ¹ which maps H either to Z or H_x.

Next, for a subgroup H we wish to describe the respective homogeneous space X=ℍ¹/H and actions of ℍ¹ on X. See, Section 2.2.2 for the background general theory and definitions of maps p: ℍ¹→ X and s: X → ℍ¹.

Exercise 14 Check that:

Both homogeneous spaces ℍ¹/Z and ℍ¹/H_x are parametrised by ℝ²;
The ℍ¹-action on ℍ¹/Z is:
(s,x,y): (x′,y′) ↦ (x+x′,y+y′). (21)

Hint: Use the following maps: p: (s′,x′,y′)↦ (x′,y′), s:(x′,y′)↦(0,x′,y′). ⋄
The ℍ¹-action on ℍ¹/H_x is:
(s,x,y): (s′,y′) ↦ (s+s′+
1

2

xy′,y+y′). (22)

Hint: Use the following maps: p: (s′,x′,y′)↦ x′, s: x′↦(0,x′,0). ⋄

Exercise 15 Show that

For any two-dimensional continuous subgroup of ℍ¹ there is an automorphism of ℍ¹ which maps the subgroup to
H′_x={(s,0,y)∈ℍ¹, s,y∈ℝ}. (23)
The homogeneous space ℍ¹/H′_x is parametrised by ℝ.
ℍ¹-action on ℍ¹/H′_x is
(s,x,y): x′ ↦ x+x′ (24)

Hint: Use the maps p: (s′,x′,y′)↦ x′ and s: x′ ↦ (0,x′,0). ⋄

Actions (21) and (24) are Euclidean shifts and much more simple than the Möbius action of the group SL₂(ℝ) (1). Nevertheless, the associated representations of the Heisenberg group will be far from trivial.

Lecture 2 The Heisenberg Group

2.1 The Symplectic Form and the Heisenberg group

2.2 Lie algebra of the Heisenberg group

2.3 Automorphisms of the Heisenberg group

2.4 Subgroups of ℍn and Homogeneous Spaces

2.4 Subgroups of ℍⁿ and Homogeneous Spaces