Heisenberg group

In mathematics, the Heisenberg group formula_1, named after Werner Heisenberg, is the group of 3×3 upper triangular matrices of the form
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.
The three-dimensional case.
In the three-dimensional case, the product of two Heisenberg matrices is given by:
As one can see from the term , the group is non-abelian.
The neutral element of the Heisenberg group is the identity matrix, and inverses are given by
The group is a subgroup of the 2-dimensional affine group Aff(2): formula_2 acting on formula_6 corresponds to the affine transform formula_7.
There are several prominent examples of the three-dimensional case.
Continuous Heisenberg group.
If , 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 3x3 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 representation, it acts on the space of holomorphic functions on the upper half-plane; it is so named for its connection with the theta functions.
Discrete Heisenberg group.
If , 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,
and relations
where
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
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 "p"3 with generators "x,y" and relations:
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 from "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
Then
and
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 formula_16 of the Heisenberg group formula_1 (over the real numbers) is known as the Heisenberg algebra. It is represented using the space of formula_18 matrices of the form
with formula_20. The following three elements form a basis for formula_16:
The basis elements satisfy the commutation relations:
The name "Heisenberg group" is motivated by the preceding relations, which have the same form as the canonical commutation relations in quantum mechanics:
where formula_25 is the position operator, formula_26 is the momentum operator, and formula_27 is Planck's constant.
The Heisenberg group formula_1 has the special property that the exponential map is a one-to-one and onto map from the Lie algebra formula_16 to the group formula_1.
Higher dimensions.
More general Heisenberg groups formula_31 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 formula_32, for any integer formula_33. As a group of matrices, formula_31 (or formula_35 to indicate this is the Heisenberg group over the field formula_36 of real numbers) is defined as the group formula_37 matrices with entries in formula_36 and having the form:
where
Group structure.
This is indeed a group, as is shown by the multiplication:
and
Lie algebra.
The Heisenberg group is a simply-connected Lie group whose Lie algebra consists of matrices
where
By letting e1, ..., e"n" be the canonical basis of R"n", and setting
the associated Lie algebra can be characterized by the canonical commutation relations,
where "p"1, ..., "p""n", "q"1, ..., "q""n", "z" are the algebra generators.
In particular, "z" is a "central" element of the Heisenberg Lie algebra. Note that the Lie algebra of the Heisenberg group is nilpotent.
Exponential map.
Let 
which fulfills formula_47. The exponential map evaluates to
The exponential map of any nilpotent Lie algebra is a diffeomorphism between the Lie algebra and the unique associated connected, simply-connected Lie group.
This discussion (aside from statements referring to dimension and Lie group) further applies if we replace R by any commutative ring "A". The corresponding group is denoted "H""n"("A" ).
Under the additional assumption that the prime 2 is invertible in the ring "A", the exponential map is also defined, since it reduces to a finite sum and has the form above (i.e. "A" could be a ring Z/"p" Z with an odd prime "p" or any field of characteristic 0).
Representation theory.
The unitary representation theory of the Heisenberg group is fairly simple – later generalized by Mackey theory – and was the motivation for its introduction in quantum physics, as discussed below.
For each nonzero real number formula_27, we can define an irreducible unitary representation formula_50 of formula_31 acting on the Hilbert space formula_52 by the formula:
This representation is known as the Schrödinger representation. The motivation for this representation is the action of the exponentiated position and momentum operators in quantum mechanics. The parameter formula_54 describes translations in position space, the parameter formula_55 describes translations in momentum space, and the parameter formula_56 gives an overall phase factor. The phase factor is needed to obtain a group of operators, since translations in position space and translations in momentum space do not commute.
The key result is the Stone–von Neumann theorem, which states that every (strongly continuous) irreducible unitary representation of the Heisenberg group in which the center acts nontrivially is equivalent to formula_50 for some formula_27. Alternatively, that they are all equivalent to the Weyl algebra (or CCR algebra) on a symplectic space of dimension 2"n".
Since the Heisenberg group is a one-dimensional central extension of formula_59, its irreducible unitary representations can be viewed as irreducible unitary projective representations of formula_59. Conceptually, the representation given above constitutes the quantum mechanical counterpart to the group of translational symmetries on the classical phase space, formula_59. The fact that the quantum version is only a "projective" representation of formula_62 is suggested already at the classical level. The Hamiltonian generators of translations in phase space are the position and momentum functions. The span of these functions do not form a Lie algebra under the Poisson bracket however, because formula_63 Rather, the span of the position and momentum functions "and the constants" forms a Lie algebra under the Poisson bracket. This Lie algebra is a one-dimensional central extension of the commutative Lie algebra formula_59, isomorphic to the Lie algebra of the Heisenberg group.
On symplectic vector spaces.
The general abstraction of a Heisenberg group is constructed from any symplectic vector space. For example, let ("V",ω) be a finite-dimensional real symplectic vector space (so ω is a nondegenerate skew symmetric bilinear form on "V"). The Heisenberg group H("V") on ("V",ω) (or simply "V" for brevity) is the set "V"×R endowed with the group law
The Heisenberg group is a central extension of the additive group "V". Thus there is an exact sequence
Any symplectic vector space admits a Darboux basis {e"j",f"k"}1 ≤ "j","k" ≤ "n" satisfying ω(e"j",f"k") = δ"j""k" and where 2"n" is the dimension of "V" (the dimension of "V" is necessarily even). In terms of this basis, every vector decomposes as
The "q""a" and "p""a" are canonically conjugate coordinates.
If {e"j", f"k"}1 ≤ "j","k" ≤ "n" is a Darboux basis for "V", then let {"E"} be a basis for R, and {e"j", f"k", "E"}1 ≤ "j","k" ≤ "n" is the corresponding basis for "V"×R. A vector in H("V") is then given by
and the group law becomes
Because the underlying manifold of the Heisenberg group is a linear space, vectors in the Lie algebra can be canonically identified with vectors in the group. The Lie algebra of the Heisenberg group is given by the commutation relation
or written in terms of the Darboux basis
and all other commutators vanish.
It is also possible to define the group law in a different way but which yields a group isomorphic to the group we have just defined. To avoid confusion, we will use "u" instead of "t", so a vector is given by
and the group law is
An element of the group 
can then be expressed as a matrix
which gives a faithful matrix representation of H("V"). The "u" in this formulation is related to "t" in our previous formulation by formula_76, so that the "t" value for the product comes to
as before.
The isomorphism to the group using upper triangular matrices relies on the decomposition of "V" into a Darboux basis, which amounts to a choice of isomorphism "V" ≅ "U" ⊕ "U"*. Although the new group law yields a group isomorphic to the one given higher up, the group with this law is sometimes referred to as the polarized Heisenberg group as a reminder that this group law relies on a choice of basis (a choice of a Lagrangian subspace of "V" is a polarization).
To any Lie algebra, there is a unique connected, simply connected Lie group "G". All other connected Lie groups with the same Lie algebra as "G" are of the form "G"/"N" where "N" is a central discrete group in "G". In this case, the center of H("V") is R and the only discrete subgroups are isomorphic to "Z". Thus H("V")/Z is another Lie group which shares this Lie algebra. Of note about this Lie group is that it admits no faithful finite-dimensional representations; it is not isomorphic to any matrix group. It does however have a well-known family of infinite-dimensional unitary representations.
The connection with the Weyl algebra.
The Lie algebra formula_80 of the Heisenberg group was described above, (1), as a Lie algebra of matrices. The Poincaré–Birkhoff–Witt theorem applies to determine the universal enveloping algebra formula_81. Among other properties, the universal enveloping algebra is an associative algebra into which formula_80 injectively imbeds.
By the Poincaré–Birkhoff–Witt theorem, it is thus the free vector space generated by the monomials
where the exponents are all non-negative.
Consequently, formula_81 consists of real polynomials 
with the commutation relations
The algebra formula_81 is closely related to the algebra of differential operators on ℝ"n" with polynomial coefficients, since any such operator has a unique representation in the form
This algebra is called the Weyl algebra. It follows from abstract nonsense that the Weyl algebra "Wn" is a quotient of formula_81. However, this is also easy to see directly from the above representations; viz. by the mapping
Applications.
Weyl's parameterization of quantum mechanics.
The application that led Hermann Weyl to an explicit realization of the Heisenberg group was the question of why the Schrödinger picture and Heisenberg picture are physically equivalent. Abstractly, the reason is the Stone–von Neumann theorem: there is a unique unitary representation with given action of the central Lie algebra element "z", up to a unitary equivalence: the nontrivial elements of the algebra are all equivalent to the usual position and momentum operators.
Thus, the Schrödinger picture and Heisenberg picture are equivalent – they are just different ways of realizing this essentially unique representation.
Theta representation.
The same uniqueness result was used by David Mumford for discrete Heisenberg groups, in his theory of equations defining abelian varieties. This is a large generalization of the approach used in Jacobi's elliptic functions, which is the case of the modulo 2 Heisenberg group, of order 8. The simplest case is the theta representation of the Heisenberg group, of which the discrete case gives the theta function.
Fourier analysis.
The Heisenberg group also occurs in Fourier analysis, where it is used in some formulations of the Stone–von Neumann theorem. In this case, the Heisenberg group can be understood to act on the space of square integrable functions; the result is a representation of the Heisenberg groups sometimes called the Weyl representation.
As a sub-Riemannian manifold.
The three-dimensional Heisenberg group "H"3(R) on the reals can also be understood to be a smooth manifold, and specifically, a simple example of a sub-Riemannian manifold. Given a point "p"=("x","y","z") in R3, define a differential 1-form Θ at this point as
This one-form belongs to the cotangent bundle of R3; that is,
is a map on the tangent bundle. Let
It can be seen that "H" is a subbundle of the tangent bundle TR3. A cometric on "H" is given by projecting vectors to the two-dimensional space spanned by vectors in the "x" and "y" direction. That is, given vectors formula_94 and formula_95 in TR3, the inner product is given by
The resulting structure turns "H" into the manifold of the Heisenberg group. An orthonormal frame on the manifold is given by the Lie vector fields
which obey the relations ["X","Y"]="Z" and ["X","Z"]=["Y","Z"]=0. Being Lie vector fields, these form a left-invariant basis for the group action. The geodesics on the manifold are spirals, projecting down to circles in two dimensions. That is, if
is a geodesic curve, then the curve formula_101 is an arc of a circle, and
with the integral limited to the two-dimensional plane. That is, the height of the curve is proportional to the area of the circle subtended by the circular arc, which follows by Stokes' theorem.
Heisenberg group of a locally compact abelian group.
It is more generally possible to define the Heisenberg group of a locally compact abelian group "K", equipped with a Haar measure. Such a group has a Pontrjagin dual formula_103, consisting of all continuous formula_104-valued characters on "K", which is also a locally compact abelian group if endowed with the compact-open topology. The Heisenberg group associated with the locally compact abelian group "K" is the subgroup of the unitary group of formula_105 generated by translations from "K" and multiplications by elements of formula_103.
In more detail, the Hilbert space formula_105 consists of square-integrable complex-valued functions formula_108 on "K". The translations in "K" form a unitary representation of "K" as operators on formula_105:
for formula_111. So too do the multiplications by characters:
for formula_113. These operators do not commute, and instead satisfy
multiplication by a fixed unit modulus complex number.
So the Heisenberg group formula_115 associated with "K" is a type of central extension of formula_116, via an exact sequence of groups:
More general Heisenberg groups are described by 2-cocyles in the cohomology group formula_118. The existence of a duality between formula_119 and formula_103 gives rise to a canonical cocycle, but there are generally others.
The Heisenberg group acts irreducibly on formula_105. Indeed, the continuous characters separate points so any unitary operator of formula_105 that commutes with them is an formula_123 multiplier. But commuting with translations implies that the multiplier is constant.
A version of the Stone–von Neumann theorem, proved by George Mackey, holds for the Heisenberg group formula_115. The Fourier transform is the unique intertwiner between the representations of formula_105 and formula_126. See the discussion at Stone–von Neumann theorem#Relation to the Fourier transform for details.