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| In [[mathematics]], the '''Heisenberg group''', named after [[Werner Heisenberg]], is the [[group (mathematics)|group]] of 3×3 [[triangular matrix|upper triangular matrices]] of the form
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| ::<math>\begin{pmatrix}
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| 1 & a & c\\
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| 0 & 1 & b\\
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| 0 & 0 & 1\\
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| \end{pmatrix}</math>
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| or its generalizations under the operation of matrix multiplication. Elements ''a, b'' and ''c'', can be taken from some ([[wiktionary:arbitrary|arbitrary]]) [[commutative ring]] with identity, often taken to be the ring of [[real number]]s or the ring of [[integer]]s.
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| The real Heisenberg group arises in the description of one-dimensional [[quantum mechanical]] systems. More generally, one can consider groups associated to n-dimensional systems, and most generally, to any [[symplectic vector space]].
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| ==The three-dimensional case==
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| In the three-dimensional case, the product of two Heisenberg matrices is given by:
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| :<math>\begin{pmatrix}
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| 1 & a & c\\
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| 0 & 1 & b\\
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| 0 & 0 & 1\\
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| \end{pmatrix}
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| \begin{pmatrix}
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| 1 & a' & c'\\
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| 0 & 1 & b'\\
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| 0 & 0 & 1\\
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| \end{pmatrix}=
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| \begin{pmatrix}
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| 1 & a+a' & c+c'+ab'\\
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| 0 & 1 & b+b'\\
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| 0 & 0 & 1\\
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| \end{pmatrix}\, .</math>
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| There are several prominent examples of the three-dimensional case.
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| ===Continuous Heisenberg group===
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| If {{math|''a, b, c''}}, are [[real number]]s (in the ring '''R''') then one has the '''continuous Heisenberg group''' H<sub>3</sub>('''R''').
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| It is a [[nilpotent group|nilpotent]] [[Lie group]].
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| In addition to the representation as real 3x3 matrices, the continuous Heisenberg group also has several different [[group representation|representations]] in terms of [[function space]]s. By [[Stone–von Neumann theorem]], there is a unique irreducible unitary representation of H in which its [[center of a group|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 function]]s on the [[upper half-plane]]; it is so named for its connection with the [[theta function]]s.
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| ===Discrete Heisenberg group===
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| [[Image:HeisenbergCayleyGraph.png|thumb|350px|right|right|A portion of the [[Cayley graph]] of the discrete Heisenberg group, with generators ''x,y,z'' as in the text. (The coloring is only for visual aid.)]]
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| If {{math|''a, b, c''}}, are integers (in the ring '''Z''') then one has the '''discrete Heisenberg group''' H<sub>3</sub>('''Z'''). It is a [[nonabelian group|non-abelian]] [[nilpotent group]]. It has two generators,
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| :<math>x=\begin{pmatrix}
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| 1 & 1 & 0\\
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| 0 & 1 & 0\\
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| 0 & 0 & 1\\
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| \end{pmatrix},\ \ y=\begin{pmatrix}
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| 1 & 0 & 0\\
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| 0 & 1 & 1\\
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| 0 & 0 & 1\\
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| \end{pmatrix}</math>
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| and relations
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| :<math> z^{}_{}=xyx^{-1}y^{-1},\ xz=zx,\ yz=zy </math>,
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| where
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| :<math>z=\begin{pmatrix}
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| 1 & 0 & 1\\
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| 0 & 1 & 0\\
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| 0 & 0 & 1\\
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| \end{pmatrix}</math>
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| is the generator of the [[center of a group|center]] of H<sub>3</sub>. (Note that the inverses of ''x'', ''y'', and ''z'' replace the 1 above the diagonal with −1.) | |
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| By [[Bass's theorem]], it has a polynomial [[Growth rate (group theory)|growth rate]] of order 4.
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| One can generate any element through
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| ::<math>\begin{pmatrix}
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| 1 & a & c\\
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| 0 & 1 & b\\
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| 0 & 0 & 1\\
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| \end{pmatrix}=y^bz^cx^a\, .</math>
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| ===Heisenberg group modulo an odd prime ''p''===
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| 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 (group theory)|order]] ''p''<sup>3</sup> with generators ''x,y'' and relations:
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| :<math> z^{}_{}=xyx^{-1}y^{-1},\ x^p=y^p=z^p=1,\ xz=zx,\ yz=zy. </math>
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| Analogues of Heisenberg groups over finite fields of odd prime order ''p'' are called [[extra special group]]s, or more properly, extra special groups of [[exponent (group theory)|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'' 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''.
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| ===Heisenberg group modulo 2===
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| The Heisenberg group modulo 2 is of order 8 and is isomorphic to the [[dihedral group]] D<sub>4</sub> (the symmetries of a square). Observe that if
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| :<math>x=\begin{pmatrix}
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| 1 & 1 & 0\\
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| 0 & 1 & 0\\
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| 0 & 0 & 1\\
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| \end{pmatrix},\ \ y=\begin{pmatrix}
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| 1 & 0 & 0\\
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| 0 & 1 & 1\\
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| 0 & 0 & 1\\
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| \end{pmatrix}</math>.
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| Then
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| :<math>xy=\begin{pmatrix}
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| 1 & 1 & 1\\
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| 0 & 1 & 1\\
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| 0 & 0 & 1\\
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| \end{pmatrix}, </math>
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| and
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| :<math>yx=\begin{pmatrix}
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| 1 & 1 & 0\\
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| 0 & 1 & 1\\
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| 0 & 0 & 1\\
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| \end{pmatrix}. </math>
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| The elements ''x'' and ''y'' correspond to reflexions (with 45° between them), whereas ''xy'' and ''yx'' correspond to rotations by 90°. The other reflexions are ''xyx'' and ''yxy'', and rotation by 180° is ''xyxy'' (=''yxyx'').
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| ==Higher dimensions==
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| More general Heisenberg groups H<sub>''n''</sub> may be defined for higher dimensions in Euclidean space, and more generally on [[symplectic vector space]]s. The simplest general case is the real Heisenberg group of dimension 2''n''+1, for any integer ''n'' ≥ 1. As a group of matrices, H<sub>''n''</sub> (or H<sub>''n''</sub>('''R''') to indicate this is the Heisenberg group over the ring '''R''' or real numbers) is defined as the group of square matrices of size ''n''+2 with entries in '''R''':
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| :<math> \begin{bmatrix} 1 & \mathbf a & c \\ 0 & I_n & \mathbf b \\ 0 & 0 & 1 \end{bmatrix} </math>
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| where
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| : '''a''' is a [[row vector]] of length ''n'',
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| : '''b''' is a [[column vector]] of length ''n'',
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| : ''I''<sub>''n''</sub> is the [[identity matrix]] of size ''n''.
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| ===Group Structure===
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| This is indeed a group, as is shown by the multiplication:
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| :<math> \begin{bmatrix} 1 & \mathbf a & c \\ 0 & I_n & \mathbf b \\ 0 & 0 & 1 \end{bmatrix} \cdot \begin{bmatrix}1 & \mathbf a' & c' \\ 0 & I_n & \mathbf b' \\ 0 & 0 & 1 \end{bmatrix} = \begin{bmatrix} 1 & \mathbf a+ \mathbf a' & c+c' +\mathbf a \cdot \mathbf b' \\ 0 & I_n & \mathbf b+\mathbf b' \\ 0 & 0 & 1 \end{bmatrix} </math>
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| and
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| :<math> \begin{bmatrix} 1 & \mathbf a & c \\ 0 & I_n & \mathbf b \\ 0 & 0 & 1 \end{bmatrix} \cdot \begin{bmatrix}1 & -\mathbf a & -c +\mathbf a \cdot \mathbf b\\ 0 & I_n & -\mathbf b \\ 0 & 0 & 1 \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & I_n & 0 \\ 0 & 0 & 1 \end{bmatrix}. </math>
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| The Heisenberg group is a [[connected space|connected]], [[simply-connected]] Lie group whose [[Lie algebra]] consists of matrices
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| :<math> \begin{bmatrix} 0 & \mathbf a & c \\ 0 & 0_n & \mathbf b \\ 0 & 0 & 0 \end{bmatrix}, </math>
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| where
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| : '''a''' is a row vector of length ''n'',
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| : '''b''' is a column vector of length ''n'',
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| : ''0''<sub>''n''</sub> is the [[zero matrix]] of size ''n''.
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| ===Exponential Map===
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| The [[exponential map]] is given by the following expression
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| :<math> \exp \begin{bmatrix} 0 & \mathbf a & c \\ 0 & 0_n & \mathbf b \\ 0 & 0 & 0 \end{bmatrix} = \sum_{k=0}^\infty \frac{1}{k!}\begin{bmatrix} 0 & \mathbf a & c \\ 0 & 0_n & \mathbf b \\ 0 & 0 & 0 \end{bmatrix}^k = \begin{bmatrix} 1 & \mathbf a & c + {1\over 2}\mathbf a \cdot \mathbf b\\ 0 & I_n & \mathbf b \\ 0 & 0 & 1 \end{bmatrix}. </math>
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| By letting e<sub>1</sub>, ..., e<sub>''n''</sub> be the canonical basis of '''R'''<sup>''n''</sup>, and setting
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| :<math> p_i = \begin{bmatrix} 0 & \operatorname{e}_i^{\mathrm{T}} & 0 \\ 0 & 0_n & 0 \\ 0 & 0 & 0 \end{bmatrix}, </math>
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| :<math> q_j = \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0_n & \operatorname{e}_j \\ 0 & 0 & 0 \end{bmatrix}, </math>
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| :<math> z = \begin{bmatrix} 0 & 0 & 1\\ 0 & 0_n & 0 \\ 0 & 0 & 0 \end{bmatrix}, </math>
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| the associated [[Lie algebra]] can be characterized by the [[canonical commutation relations]],
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| :<math> \begin{bmatrix} p_i, q_j \end{bmatrix} = \delta_{ij}z, \qquad \qquad
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| \begin{bmatrix} p_i, z \end{bmatrix} = 0, \qquad \begin{bmatrix} q_j, z \end{bmatrix} = 0~,</math>
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| where ''p''<sub>1</sub>, ..., ''p''<sub>''n''</sub>, ''q''<sub>1</sub>, ..., ''q''<sub>''n''</sub>, ''z'' are the algebra generators.
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| In particular, ''z'' is a ''central'' element of the Heisenberg Lie algebra. Note that the Lie algebra of the Heisenberg group is nilpotent. The [[exponential map]] of a nilpotent Lie algebra is a [[diffeomorphism]] between the Lie algebra and the unique associated [[connected space|connected]], [[simply-connected]] Lie group.
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| This discussion (aside from statements referring to dimension and Lie group) applies if we replace '''R''' by any commutative ring ''A''. The corresponding group is denoted ''H''<sub>''n''</sub>(''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 (mathematics)|field]] of [[characteristic (algebra)|characteristic]] 0).
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| ==On symplectic vector spaces==
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| The general abstraction of a Heisenberg group is constructed from any [[symplectic vector space]].<ref>Hans Tilgner, "[http://www.numdam.org/numdam-bin/fitem?id=AIHPA_1970__13_2_103_0 A class of solvable Lie groups and their relation to the canonical formalism]", ''Annales de l'institut Henri Poincaré (A) Physique théorique'', '''13 ''' no. 2 (1970), pp. 103-127.</ref> For example, let (''V'',ω) be a finite dimensional real symplectic vector space (so ω is a [[nondegenerate form|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
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| :<math>(v,t)\cdot(v',t') =\left (v+v',t+t'+\tfrac{1}{2}\omega(v,v')\right).</math>
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| The Heisenberg group is a [[Group extension%23Central extension|central extension]] of the additive group ''V''. Thus there is an [[exact sequence]]
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| :<math>0\to\mathbf{R}\to H(V)\to V\to 0.</math>
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| Any symplectic vector space admits a [[Symplectic vector space#Standard symplectic space|Darboux basis]] {'''e'''<sub>''j''</sub>,'''f'''<sup>''k''</sup>}<sub>1 ≤ ''j'',''k'' ≤ ''n''</sub> satisfying ω('''e'''<sub>''j''</sub>,'''f'''<sup>''k''</sup>) = δ<sub>''j''</sub><sup>''k''</sup> 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
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| :<math>v=q^a\mathbf{e}_a+p_a\mathbf{f}^a.</math>
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| The ''q''<sup>''a''</sup> and ''p''<sub>''a''</sub> are [[canonically conjugate coordinates]].
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| If {'''e'''<sub>''j''</sub>, '''f'''<sup>''k''</sup>}<sub>1 ≤ ''j'',''k'' ≤ ''n''</sub> is a Darboux basis for ''V'', then let {''E''} be a basis for '''R''', and {'''e'''<sub>''j''</sub>, '''f'''<sup>''k''</sup>, ''E''}<sub>1 ≤ ''j'',''k'' ≤ ''n''</sub> is the corresponding basis for ''V''×'''R'''. A vector in H(''V'') is then given by
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| :<math>v=q^a\mathbf{e}_a+p_a\mathbf{f}^a+tE</math>
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| and the group law becomes
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| :<math>(p,q,t)\cdot(p',q',t') =\left (p+p',q+q',t+t'+\frac{1}{2}(p q'-p' q)\right).</math>
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| 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
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| :<math>\begin{bmatrix} (v_1,t_1),(v_2,t_2) \end{bmatrix} =\omega(v_1,v_2)</math>
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| or written in terms of the Darboux basis
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| :<math>[\mathbf{e}_a,\mathbf{f}^b]=\delta_a^b</math>
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| and all other commutators vanish.
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| 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
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| :<math>v=q^a\mathbf{e}_a+p_a\mathbf{f}^a+uE</math>
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| and the group law is
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| :<math>(p,q,u)\cdot(p',q',u')=(p+p',q+q',u+u'+p q').</math>
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| An element of the group
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| :<math>v=q^a\mathbf{e}_a+p_a\mathbf{f}^a+uE</math>
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| can then be expressed as a matrix
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| :<math>
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| \begin{bmatrix}
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| 1 & p & u\\
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| 0 & I_n & q\\
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| 0 & 0 & 1
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| \end{bmatrix}</math> ,
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| which gives a faithful [[matrix representation]] of H(''V''). The ''u'' in this formulation is related to ''t'' in our previous formulation by <math>u=t+\tfrac{1}{2}pq</math>, so that the ''t'' value for the product comes to
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| :<math>u+u'+p q'-\tfrac{1}{2}(p+p')(q+q')</math>
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| :<math>=t+\tfrac{1}{2}p q+t'+\tfrac{1}{2}p' q'+p q'-\tfrac{1}{2}(p+p')(q+q')</math>
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| :<math>=t+t'+\tfrac{1}{2}(p q'-p' q)</math> ,
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| as before.
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| 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 (abelian variety)|polarization]]).
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| To any Lie algebra, there is a unique [[connected space|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.
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| ==The connection with the Weyl algebra==
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| {{main|Weyl algebra}}
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| The Lie algebra <math>\mathfrak{h}_n</math> of the Heisenberg group was described above as a Lie algebra of matrices. The [[Poincaré–Birkhoff–Witt theorem]] applies to determine the [[universal enveloping algebra]] <math>U(\mathfrak{h}_n)</math>. Among other properties, the universal enveloping algebra is an [[associative algebra]] into which <math>\mathfrak{h}_n</math> injectively imbeds. By Poincaré–Birkhoff–Witt, it is the [[free module|free vector space]] generated by the monomials
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| :<math> z^j p_1^{k_1} p_2^{k_2} \cdots p_n^{k_n} q_1^{\ell_1} q_2^{\ell_2} \cdots q_n^{\ell_n}</math>
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| where the exponents are all non-negative. Thus <math>U(\mathfrak{h}_n)</math> consists of real polynomials
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| :<math> \sum_{j, \vec{k}, \vec{\ell}} c_{j \vec{k} \vec{\ell}} \,\, z^j p_1^{k_1} p_2^{k_2} \cdots p_n^{k_n} q_1^{\ell_1} q_2^{\ell_2} \cdots q_n^{\ell_n}</math>
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| with the commutation relations
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| :<math> p_k p_\ell = p_\ell p_k, \quad q_k q_\ell = q_\ell q_k, \quad p_k q_\ell - q_\ell p_k = \delta_{k \ell} z, \quad z p_k - p_k z =0, \quad z q_k - q_k z =0.</math>
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| The algebra <math>U(\mathfrak{h}_n)</math> is closely related to the algebra of differential operators on '''R'''<sup>''n''</sup> with polynomial coefficients, since any such operator has a unique representation in the form
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| :<math>P=\sum_{\vec{k}, \vec{\ell}} c_{\vec{k} \vec{\ell}} \,\, \partial_{x_1}^{k_1} \partial_{x_2}^{k_2} \cdots \partial_{x_n}^{k_n} x_1^{\ell_1} x_2^{\ell_2} \cdots x_n^{\ell_n}</math>
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| This algebra is called the [[Weyl algebra]]. It follows from [[abstract nonsense]] that the [[Weyl algebra]] ''W<sub>n''</sub> is a quotient of <math>U(\mathfrak{h}_n)</math>. However, this is also easy to see directly from the above representations; [[viz.]] by the mapping
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| :<math> z^j p_1^{k_1} p_2^{k_2} \cdots p_n^{k_n} q_1^{\ell_1} q_2^{\ell_2} \cdots q_n^{\ell_n} \, \mapsto \, \partial_{x_1}^{k_1} \partial_{x_2}^{k_2} \cdots \partial_{x_n}^{k_n} x_1^{\ell_1} x_2^{\ell_2} \cdots x_n^{\ell_n}.</math>
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| == Representation theory ==
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| {{main|Stone–von Neumann theorem}}
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| The [[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.
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| The key result is the [[Stone–von Neumann theorem]], which, informally stated, says that (with certain technical assumptions) every representation of the Heisenberg group H<sub>2''n''+1</sub> is equivalent to the position operators and momentum operators on '''R'''<sup>''n''</sup>. Alternatively, that they are all equivalent to the [[Weyl algebra]] (or [[CCR algebra]]) on a symplectic space of dimension 2''n''.
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| More formally, there is a unique (up to scale) non-trivial central strongly continuous unitary representation.
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| Further, as the Heisenberg group is a [[semidirect product]], its representation theory can be studied in terms of [[ergodic theory]], via ergodic actions of the group, as in the work of [[George Mackey]].
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| == Applications ==
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| === Weyl's parameterization of quantum mechanics ===
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| {{main|Weyl quantization}}
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| 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.
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| Thus, the Schrödinger picture and Heisenberg picture are equivalent – they are just different ways of realizing this essentially unique representation.
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| === Theta representation ===
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| {{main|theta representation}}
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| 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]].
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| === Fourier analysis ===
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| 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.
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| ==As a sub-Riemannian manifold==
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| The three-dimensional Heisenberg group ''H''<sub>3</sub>('''R''') on the reals can also be understood to be a smooth [[manifold]], and specifically, a simple example of a [[sub-Riemannian manifold]].<ref>Richard Montgomery, ''A Tour of Subriemannian Geometries, Their Geodesics and Applications (Mathematical Surveys and Monographs, Volume 91)'', (2002) American Mathematical Society, ISBN 0-8218-1391-9.</ref> Given a point ''p''=(''x'',''y'',''z'') in '''R'''<sup>3</sup>, define a differential [[1-form]] Θ at this point as
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| :<math>\Theta_p=dz -\frac{1}{2}\left(xdy - ydx\right).</math>
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| This [[one-form]] belongs to the [[cotangent bundle]] of '''R'''<sup>3</sup>; that is,
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| :<math>\Theta_p:T_p\mathbf{R}^3\to\mathbf{R}</math> | |
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| is a map on the [[tangent bundle]]. Let
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| :<math>H_p = \{ v\in T_p\mathbf{R}^3 \mid \Theta_p(v) = 0 \}.</math> | |
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| It can be seen that ''H'' is a [[subbundle]] of the tangent bundle T'''R'''<sup>3</sup>. 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 <math>v=(v_1,v_2,v_3)</math> and <math>w=(w_1,w_2,w_3)</math> in T'''R'''<sup>3</sup>, the inner product is given by
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| :<math>\langle v,w\rangle = v_1w_1+v_2w_2.</math>
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| The resulting structure turns ''H'' into the manifold of the Heisenberg group. An orthonormal frame on the manifold is given by the Lie [[vector field]]s
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| :<math>X=\frac{\partial}{\partial x} - \frac{1}{2} y\frac{\partial}{\partial z},</math>
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| :<math>Y=\frac{\partial}{\partial y} + \frac{1}{2} x\frac{\partial}{\partial z},</math>
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| :<math>Z=\frac{\partial}{\partial z},</math>
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| 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 [[geodesic]]s on the manifold are spirals, projecting down to circles in two dimensions. That is, if
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| :<math>\gamma(t)=(x(t),y(t),z(t))</math> | |
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| is a geodesic curve, then the curve <math>c(t)=(x(t),y(t))</math> is an arc of a circle, and
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| :<math>z(t)=\frac{1}{2}\int_c xdy-ydx</math>
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| 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]].
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| ==See also==
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| * [[Weyl quantization]]
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| * [[Stone–von Neumann theorem]]
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| ==Notes==
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| <references/>
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| ==References==
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| *Ernst Binz & Sonja Pods (2008) ''Geometry of Heisenberg Groups'' [[American Mathematical Society]],ISBN 978-0-8218-4495-3.
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| *{{cite book | last = Hall | first = Brian C. | title = Lie Groups, Lie Algebras, and Representations: An Elementary Introduction | year = 2004 | publisher = Springer | location = Berlin | isbn = 0-387-40122-9}}
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| *[[Roger Evans Howe]] (1980) "On the role of the Heisenberg group in harmonic analysis", [[Bulletin of the American Mathematical Society]] 3(2):821.
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| *A.A. Kirilov (2004) ''Lectures on the Orbit Method'', Chapter 2: "Representations and Orbits of the Heisenberg Group", American Mathematical Society, ISBN 0-8218-3530-0 .
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| *G Mackey (1976) ''The theory of Unitary Group Representations (Chicago Lectures in Mathematics)'' University Of Chicago Press (1976) ISBN 978-0226500522 .
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| ==External links ==
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| * Groupprops, The Group Properties Wiki [http://groupprops.subwiki.org/w/index.php?title=Unitriangular_matrix_group:UT(3,p) Unitriangular matrix group UT(3,p)]
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| {{DEFAULTSORT:Heisenberg Group}}
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| [[Category:Group theory]]
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| [[Category:Lie groups]]
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| [[Category:Mathematical quantization]]
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| [[Category:Mathematical physics]]
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