In mathematics, the Heisenberg group H {\displaystyle H} , named after Werner Heisenberg, is the group of 3×3 upper triangular matrices of the form
( 1 a c 0 1 b 0 0 1 ) {\displaystyle {\begin{pmatrix}1&a&c\\0&1&b\\0&0&1\\\end{pmatrix}}}
under the operation of matrix multiplication. Elements a, b and c can be taken from any commutative ring with identity, often taken to be the ring of real numbers (resulting in the "continuous Heisenberg group") or the ring of integers (resulting in the "discrete Heisenberg group"). The continuous Heisenberg group arises in the description of one-dimensional quantum mechanical systems, especially in the context of the Stone–von Neumann theorem. More generally, one can consider Heisenberg groups associated to n-dimensional systems, and most generally, to any symplectic vector space.
Three-dimensional case In the three-dimensional case, the product of two Heisenberg matrices is given by
( 1 a c 0 1 b 0 0 1 ) ( 1 a ′ c ′ 0 1 b ′ 0 0 1 ) = ( 1 a + a ′ c + a b ′ + c ′ 0 1 b + b ′ 0 0 1 ) . {\displaystyle {\begin{pmatrix}1&a&c\\0&1&b\\0&0&1\\\end{pmatrix}}{\begin{pmatrix}1&a'&c'\\0&1&b'\\0&0&1\\\end{pmatrix}}={\begin{pmatrix}1&a+a'&c+ab'+c'\\0&1&b+b'\\0&0&1\\\end{pmatrix}}.}
As one can see from the term ab′, the group is non-abelian. The neutral element of the Heisenberg group is the identity matrix, and inverses are given by
( 1 a c 0 1 b 0 0 1 ) − 1 = ( 1 − a a b − c 0 1 − b 0 0 1 ) . {\displaystyle {\begin{pmatrix}1&a&c\\0&1&b\\0&0&1\\\end{pmatrix}}^{-1}={\begin{pmatrix}1&-a&ab-c\\0&1&-b\\0&0&1\\\end{pmatrix}}.}
The group is a subgroup of the 2-dimensional affine group Aff(2): the action of the element
( 1 a c 0 1 b 0 0 1 ) {\displaystyle {\begin{pmatrix}1&a&c\\0&1&b\\0&0&1\\\end{pmatrix}}} on the vector ( x → , 1 ) {\displaystyle ({\vec {x}},1)} corresponds to the affine transform ( 1 a 0 1 ) x → + ( c b ) . {\displaystyle {\begin{pmatrix}1&a\\0&1\end{pmatrix}}{\vec {x}}+{\begin{pmatrix}c\\b\end{pmatrix}}.}
There are several prominent examples of the three-dimensional case.
Continuous Heisenberg group If a, b, c, are real numbers (in the ring R), then one has the continuous Heisenberg group H3(R). It is a nilpotent real Lie group of dimension 3. In addition to the representation as real 3×3 matrices, the continuous Heisenberg group also has several different representations in terms of function spaces. By Stone–von Neumann theorem, there is, up to isomorphism, a unique irreducible unitary representation of H in which its centre acts by a given nontrivial character. This representation has several important realizations, or models. In the Schrödinger model, the Heisenberg group acts on the space of square integrable functions. In the theta, or holomorphic, model, the Heisenberg group acts on a Hilbert space of entire functions, with the model depending on a parameter in the upper half-plane.
Discrete Heisenberg group
If a, b, c are integers (in the ring Z), then one has the discrete Heisenberg group H3(Z). It is a non-abelian nilpotent group. It has two generators:
x = ( 1 1 0 0 1 0 0 0 1 ) , y = ( 1 0 0 0 1 1 0 0 1 ) {\displaystyle x={\begin{pmatrix}1&1&0\\0&1&0\\0&0&1\end{pmatrix}},\quad y={\begin{pmatrix}1&0&0\\0&1&1\\0&0&1\end{pmatrix}}}
and relations
z = x y x − 1 y − 1 , x z = z x , y z = z y , {\displaystyle z=xyx^{-1}y^{-1},\quad xz=zx,\quad yz=zy,}
where
z = ( 1 0 1 0 1 0 0 0 1 ) {\displaystyle z={\begin{pmatrix}1&0&1\\0&1&0\\0&0&1\end{pmatrix}}}
is the generator of the center of H3. (Note that the inverses of x, y, and z replace the 1 above the diagonal with −1.) By Bass's theorem, it has a polynomial growth rate of order 4. One can generate any element through
( 1 a c 0 1 b 0 0 1 ) = y b z c x a . {\displaystyle {\begin{pmatrix}1&a&c\\0&1&b\\0&0&1\end{pmatrix}}=y^{b}z^{c}x^{a}.}
Heisenberg group modulo an odd prime p If one takes a, b, c in Z/p Z for an odd prime p, then one has the Heisenberg group modulo p. It is a group of order p3 with generators x, y and relations
z = x y x − 1 y − 1 , x p = y p = z p = 1 , x z = z x , y z = z y . {\displaystyle z=xyx^{-1}y^{-1},\quad x^{p}=y^{p}=z^{p}=1,\quad xz=zx,\quad yz=zy.}
Analogues of Heisenberg groups over finite fields of odd prime order p are called extra special groups, or more properly, extra special groups of exponent p. More generally, if the derived subgroup of a group G is contained in the center Z of G, then the map G/Z × G/Z → Z is a skew-symmetric bilinear operator on abelian groups. However, requiring that G/Z to be a finite vector space requires the Frattini subgroup of G to be contained in the center, and requiring that Z be a one-dimensional vector space over Z/p Z requires that Z have order p, so if G is not abelian, then G is extra special. If G is extra special but does not have exponent p, then the general construction below applied to the symplectic vector space G/Z does not yield a group isomorphic to G.
Heisenberg group modulo 2 The Heisenberg group modulo 2 is of order 8 and is isomorphic to the dihedral group D4 (the symmetries of a square). Observe that if
x = ( 1 1 0 0 1 0 0 0 1 ) , y = ( 1 0 0 0 1 1 0 0 1 ) , {\displaystyle x={\begin{pmatrix}1&1&0\\0&1&0\\0&0&1\end{pmatrix}},\quad y={\begin{pmatrix}1&0&0\\0&1&1\\0&0&1\end{pmatrix}},}
then
x y = ( 1 1 1 0 1 1 0 0 1 ) , {\displaystyle xy={\begin{pmatrix}1&1&1\\0&1&1\\0&0&1\end{pmatrix}},}
and
y x = ( 1 1 0 0 1 1 0 0 1 ) . {\displaystyle yx={\begin{pmatrix}1&1&0\\0&1&1\\0&0&1\end{pmatrix}}.}
The elements x and y correspond to reflections (with 45° between them), whereas xy and yx correspond to rotations by 90°. The other reflections are xyx and yxy, and rotation by 180° is xyxy (= yxyx).
Heisenberg algebra The Lie algebra h {\displaystyle {\mathfrak {h}}} of the Heisenberg group H {\displaystyle H} (over the real numbers) is known as the Heisenberg algebra. It may be represented using the space of 3×3 matrices of the form
( 0 a c 0 0 b 0 0 0 ) {\displaystyle {\begin{pmatrix}0&a&c\\0&0&b\\0&0&0\end{pmatrix}}}
with a , b , c ∈ R {\displaystyle a,b,c\in \mathbb {R} } . The following three elements form a basis for h {\displaystyle {\mathfrak {h}}} :
X = ( 0 1 0 0 0 0 0 0 0 ) , Y = ( 0 0 0 0 0 1 0 0 0 ) , Z = ( 0 0 1 0 0 0 0 0 0 ) . {\displaystyle X={\begin{pmatrix}0&1&0\\0&0&0\\0&0&0\end{pmatrix}},\quad Y={\begin{pmatrix}0&0&0\\0&0&1\\0&0&0\end{pmatrix}},\quad Z={\begin{pmatrix}0&0&1\\0&0&0\\0&0&0\end{pmatrix}}.}
These basis elements satisfy the commutation relations
[ X , Y ] = Z , [ X , Z ] = 0 , [ Y , Z ] = 0. {\displaystyle [X,Y]=Z,\quad [X,Z]=0,\quad [Y,Z]=0.}
The name "Heisenberg group" is motivated by the preceding relations, which have the same form as the canonical commutation relations in quantum mechanics:
[ x ^ , p ^ ] = i ℏ I , [ x ^ , i ℏ I ] = 0 , [ p ^ , i ℏ I ] = 0 , {\displaystyle [{\hat {x}},{\hat {p}}]=i\hbar I,\quad [{\hat {x}},i\hbar I]=0,\quad [{\hat {p}},i\hbar I]=0,}
where x ^ {\displaystyle {\hat {x}}} is the position operator, p ^ {\displaystyle {\hat {p}}} is the momentum operator, and ℏ {\displaystyle \hbar } is the Planck constant. The Heisenberg group H has the special property that the exponential map is a one-to-one and onto map from the Lie algebra h {\displaystyle {\mathfrak {h}}} to the group H:
exp ( 0 a c 0 0 b 0 0 0 ) = ( 1 a c + a b 2 0 1 b 0 0 1 ) . {\displaystyle \exp {\begin{pmatrix}0&a&c\\0&0&b\\0&0&0\end{pmatrix}}={\begin{pmatrix}1&a&c+{\frac {ab}{2}}\\0&1&b\\0&0&1\end{pmatrix}}.}
In conformal field theory In conformal field theory, the term Heisenberg algebra is used to refer to an infinite-dimensional generalization of the above algebra. It is spanned by elements a n , n ∈ Z {\displaystyle a_{n},n\in \mathbb {Z} } with commutation relations
[ a n , a m ] = δ n + m , 0 . {\displaystyle [a_{n},a_{m}]=\delta _{n+m,0}.}
Under a rescaling, this is simply a countably-infinite number of copies of the above algebra.
Higher dimensions More general Heisenberg groups H 2 n + 1 {\displaystyle H_{2n+1}} may be defined for higher dimensions in Euclidean space, and more generally on symplectic vector spaces. The simplest general case is the real Heisenberg group of dimension 2 n + 1 {\displaystyle 2n+1} , for any integer n ≥ 1 {\displaystyle n\geq 1} . As a group of matrices, H 2 n + 1 {\displaystyle H_{2n+1}} (or H 2 n + 1 ( R ) {\displaystyle H_{2n+1}(\mathbb {R} )} to indicate that this is the Heisenberg group over the field R {\displaystyle \mathbb {R} } of real numbers) is defined as the group ( n + 2 ) × ( n + 2 ) {\displaystyle (n+2)\times (n+2)} matrices with entries in R {\displaystyle \mathbb {R} } and having the form
[ 1 a c 0 I n b 0 0 1 ] , {\displaystyle {\begin{bmatrix}1&\mathbf {a} &c\\\mathbf {0} &I_{n}&\mathbf {b} \\0&\mathbf {0} &1\end{bmatrix}},}
where
a is a row vector of length n, b is a column vector of length n, In is the identity matrix of size n.
Group structure This is indeed a group, as is shown by the multiplication:
[ 1 a c 0 I n b 0 0 1 ] ⋅ [ 1 a ′ c ′ 0 I n b ′ 0 0 1 ] = [ 1 a + a ′ c + c ′ + a ⋅ b ′ 0 I n b + b ′ 0 0 1 ] {\displaystyle {\begin{bmatrix}1&\mathbf {a} &c\\\mathbf {0} &I_{n}&\mathbf {b} \\0&\mathbf {0} &1\end{bmatrix}}\cdot {\begin{bmatrix}1&\mathbf {a} '&c'\\\mathbf {0} &I_{n}&\mathbf {b} '\\0&\mathbf {0} &1\end{bmatrix}}={\begin{bmatrix}1&\mathbf {a} +\mathbf {a} '&c+c'+\mathbf {a} \cdot \mathbf {b} '\\\mathbf {0} &I_{n}&\mathbf {b} +\mathbf {b} '\\0&\mathbf {0} &1\end{bmatrix}}}
and
[ 1 a c 0 I n b 0 0 1 ] ⋅ [ 1 − a − c + a ⋅ b 0 I n − b 0 0 1 ] = [ 1 0 0 0 I n 0 0 0 1 ] . {\displaystyle {\begin{bmatrix}1&\mathbf {a} &c\\\mathbf {0} &I_{n}&\mathbf {b} \\0&\mathbf {0} &1\end{bmatrix}}\cdot {\begin{bmatrix}1&-\mathbf {a} &-c+\mathbf {a} \cdot \mathbf {b} \\\mathbf {0} &I_{n}&-\mathbf {b} \\0&\mathbf {0} &1\end{bmatrix}}={\begin{bmatrix}1&\mathbf {0} &0\\\mathbf {0} &I_{n}&\mathbf {0} \\0&\mathbf {0} &1\end{bmatrix}}.}
Lie algebra The Heisenberg group is a simply-connected Lie group whose Lie algebra consists of matrices
[ 0 a c 0 0 n b 0 0 0 ] , {\displaystyle {\begin{bmatrix}0&\mathbf {a} &c\\\mathbf {0} &0_{n}&\mathbf {b} \\0&\mathbf {0} &0\end{bmatrix}},}
where
a is a row vector of length n, b is a column vector of length n, 0n is the zero matrix of size n. By letting e1, ..., en be the canonical basis of Rn and setting
p i = [ 0 e i T 0 0 0 n