Robert's Mistakes

The Heisenberg-Weyl group

Apr 30, 2022

Let's have some fun with the Heisenberg group HH! It is the group of upper triangular matrices of the form:

(1ac01b001) \begin{pmatrix} 1 & a & c \\ 0 & 1 & b \\ 0 & 0 & 1 \end{pmatrix}

Muliplying two such matrices you can easily see it's a monoid, because the result is again an upper triangular matrix:

(1a+Ac+aB+C01b+B001) \begin{pmatrix} 1 & a+A & c + aB + C \\ 0 & 1 & b+B \\ 0 & 0 & 1 \end{pmatrix}

Of course, the identity matrix Id\textbf{Id} is also the identity of the monoid. This also makes clear that the group is non-commutative. The inverse is also rather easy to find, and that makes it a group:

(1aabc01b001) \begin{pmatrix} 1 & -a & ab - c \\ 0 & 1 & -b \\ 0 & 0 & 1 \end{pmatrix}

Two ways to embed an affine transform

If we let such a matrix act on 2-vectors embedded like this

(xy1) \begin{pmatrix} x \\ y \\ 1 \end{pmatrix}

it gives an affine transform, that is, a shear plus a translation:

(1a01)(xy)+(cb) \begin{pmatrix} 1 & a \\ 0 & 1 \end{pmatrix} \begin{pmatrix} x \\ y \\ \end{pmatrix} + \begin{pmatrix} c \\ b \end{pmatrix}

Notice that there are two ways such a 2-dimensional affine transform can be embedded in the Heisenberg group!

Lie algebra

The way I described it, you're probably thinking of real numbers for aa, bb, and cc, and that makes it a continuous- or Lie group! Let's stick with that (the discrete Heisenberg groups are also fun to explore, have a look on Wikipedia's take for some more on this).

As such, it is nilpotent, and non-abelian, which I had mentioned above already. There is also an associated Lie algebra, which I find to be a nice example to illustrate the relation between Lie groups and Lie algebras. I think so, because it's small, and because the exponential map, which always translates a Lie algebra to a Lie group, is bijective in this case! Which is quite obvious if you look at it:

exp(0ac00b000)=(1aab+c01b001) exp \begin{pmatrix} 0 & a & c \\ 0 & 0 & b \\ 0 & 0 & 0 \end{pmatrix} = \begin{pmatrix} 1 & a & ab + c \\ 0 & 1 & b \\ 0 & 0 & 1 \end{pmatrix}

There's a fun way to generalize all this to higher dimensions using a row vector a\textbf{a} a column vector b\textbf{b} and a scalar cc:

H2n+1=(1ac0Idnb001) H_{2n+1} = \begin{pmatrix} 1 & \textbf{a} & c \\ 0 & \textbf{Id}_n & \textbf{b} \\ 0 & 0 & 1 \end{pmatrix}

This almost looks just like what I wrote earlier, but with the types of some entries changed! All the other things also works out just as nicely!

Polarized Heisenberg group

If you look at this the right way, you get an even general picture of the Heisenberg group, for symplectic space!

Remember, symplectic space is this freak 2n2n-dimensional space where, instead of our dot product, we get a product which yields zero when its argument vectors are parallel! (This is then called a nondegenerate skew-symmetric bilinear form). Now, if we pick a (Darboux-) basis for it, we can get the corresponding Heisenberg group to look almost like before again!

Picking a basis for a symplectic space V amounts to choosing a Lagrangian, and that is sometimes often called the polarization of V, which leads to the beautiful name polarized Heisenberg group for a representation of this form. For some more details on this see here:

Heisenberg Group on Wikipedia - https://en.wikipedia.org/wiki/Heisenberg_group

Nilgeometry

William Thurston solved the geometrization conjecture for 3-space. That is, he found eight homogeneous geometries (all points look the same) for 3-space, in analogy to spherical, euclidean, and hyperbolic geometry in two dimensions. One of these is called Nil.

Thurston's space sort of looks like a plane, but if you move in a circle, you also move perpendicular to the plane! It is similar to an example for an effect called holonomy: when you roll a ball in a small circle on a plane, the ball ends up twisted along an axis pointing away from the plane. While moving in two dimensions on a plane, we have also moved using a third degree of freedom. The difference is that in nilgeometry that third direction does not repeat every full turn.

The notion of Nilgeometry was introduced by Anatoly Mal'cev in 1951 as the quotient N/HN/H of a nilpotent Lie group NN modulo a closed subgroup HH. Just so you know where the funny name comes from. For solvable Lie groups there is an analog, and there's a corresponding Thurston geometry beautifully named Sol!

In any case, there's a representation of the Nilmanifold which, again, looks almost like the things we did earlier! Of course these ideas make sense in dimensions other than three, see here for a more complete account:

Nil manifold on Wikipedia - https://en.wikipedia.org/wiki/Nil_manifold

Nilgeometry explained - [youtube]

Quantum mechanics

The Heisenberg group turns up in quantum mechanics.

One of the earliest attempts at a natural quantization was Weyl quantization, proposed by Hermann Weyl in 1927.[3] Here, an attempt is made to associate a quantum-mechanical observable (a self-adjoint operator on a Hilbert space) with a real-valued function on classical phase space. The position and momentum in this phase space are mapped to the generators of the Heisenberg group, and the Hilbert space appears as a group representation of the Heisenberg group. - https://en.wikipedia.org/wiki/Quantization_%28physics%29

We can fix generators for the small Heisenberg group like this:

x=(110010001)y=(100011001) x = \begin{pmatrix} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \\ y = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{pmatrix}

That leads to the center generated by z=xyx1y1z=xyx^{-1}y^{-1}:

z=(101010001) z = \begin{pmatrix} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}

If that doesn't mean much to you, at least notice that the center is the part of a group which is commuative. Look what happens if we substitute the identity for zz and right-multiply first by yy and then by xx:

Id=xyx1y1yx=xy Id = xyx^{-1}y^{-1} \\ yx = xy

End

I secretly wanted to make this post about Weyl algebra, which can also be called symplectic Clifford algebra, another name I find irresistably attractive. Let me say just this much: Herrmann Weyl wanted to understand why the Schrödinger picture of quantum mechanics is equivalent to Heisenberg's, and that's why he came up with this stuff.

This post is already too long, so I will stop here.

Thanks to ZenoRogue for showing me the relation between Nil-geometry and the Heisenberg group H3H_3, which got me started on my journey!

Links

Heisenberg Group on Wikipedia - https://en.wikipedia.org/wiki/Heisenberg_group

Nil manifold on Wikipedia - https://en.wikipedia.org/wiki/Nil_manifold

Videos

Short visual introduction to nilgeometry narrated by Tehora Rogue - [youtube]

Here's a seminar with two speakers talking about the Heisenberg group: History for Physics - "The Weyl-Heisenberg group: from quantum mechanics to quantum information" - [youtube] The first part delves into a specific problem set, the second half gives a more bird's eye account on the history of Heisenberg group.

Edward Teller "Understanding Group theory with Heisenberg" - [youtube]

Further links

Weyl algebra - https://en.wikipedia.org/wiki/Weyl_algebra

Group center - https://en.wikipedia.org/wiki/Center_%28group_theory%29

Quantizaton - https://en.wikipedia.org/wiki/Quantization_%28physics%29

Schrödinger picture - https://en.wikipedia.org/wiki/Schr%C3%B6dinger_picture

Heisenberg picture - https://en.wikipedia.org/wiki/Heisenberg_picture

Wigner-Weyl transform - https://en.wikipedia.org/wiki/Wigner%e2%80%93Weyl_transform