Symplectic group

Last updated

In mathematics, the name symplectic group can refer to two different, but closely related, collections of mathematical groups, denoted Sp(2n, F) and Sp(n) for positive integer n and field F (usually C or R). The latter is called the compact symplectic group and is also denoted by . Many authors prefer slightly different notations, usually differing by factors of 2. The notation used here is consistent with the size of the most common matrices which represent the groups. In Cartan's classification of the simple Lie algebras, the Lie algebra of the complex group Sp(2n, C) is denoted Cn, and Sp(n) is the compact real form of Sp(2n, C). Note that when we refer to the (compact) symplectic group it is implied that we are talking about the collection of (compact) symplectic groups, indexed by their dimension n.

Contents

The name "symplectic group" was coined by Hermann Weyl as a replacement for the previous confusing names (line) complex group and Abelian linear group, and is the Greek analog of "complex".

The metaplectic group is a double cover of the symplectic group over R; it has analogues over other local fields, finite fields, and adele rings.

Sp(2n, F)

The symplectic group is a classical group defined as the set of linear transformations of a 2n-dimensional vector space over the field F which preserve a non-degenerate skew-symmetric bilinear form. Such a vector space is called a symplectic vector space, and the symplectic group of an abstract symplectic vector space V is denoted Sp(V). Upon fixing a basis for V, the symplectic group becomes the group of 2n × 2n symplectic matrices, with entries in F, under the operation of matrix multiplication. This group is denoted either Sp(2n, F) or Sp(n, F). If the bilinear form is represented by the nonsingular skew-symmetric matrix Ω, then

where MT is the transpose of M. Often Ω is defined to be

where In is the identity matrix. In this case, Sp(2n, F) can be expressed as those block matrices , where , satisfying the three equations:

Since all symplectic matrices have determinant 1, the symplectic group is a subgroup of the special linear group SL(2n, F). When n = 1, the symplectic condition on a matrix is satisfied if and only if the determinant is one, so that Sp(2, F) = SL(2, F). For n > 1, there are additional conditions, i.e. Sp(2n, F) is then a proper subgroup of SL(2n, F).

Typically, the field F is the field of real numbers R or complex numbers C. In these cases Sp(2n, F) is a real or complex Lie group of real or complex dimension n(2n + 1), respectively. These groups are connected but non-compact.

The center of Sp(2n, F) consists of the matrices I2n and I2n as long as the characteristic of the field is not 2. [1] Since the center of Sp(2n, F) is discrete and its quotient modulo the center is a simple group, Sp(2n, F) is considered a simple Lie group.

The real rank of the corresponding Lie algebra, and hence of the Lie group Sp(2n, F), is n.

The Lie algebra of Sp(2n, F) is the set

equipped with the commutator as its Lie bracket. [2] For the standard skew-symmetric bilinear form , this Lie algebra is the set of all block matrices subject to the conditions

Sp(2n, C)

The symplectic group over the field of complex numbers is a non-compact, simply connected, simple Lie group.

Sp(2n, R)

Sp(n, C) is the complexification of the real group Sp(2n, R). Sp(2n, R) is a real, non-compact, connected, simple Lie group. [3] It has a fundamental group isomorphic to the group of integers under addition. As the real form of a simple Lie group its Lie algebra is a splittable Lie algebra.

Some further properties of Sp(2n, R):

The matrix D is positive-definite and diagonal. The set of such Zs forms a non-compact subgroup of Sp(2n, R) whereas U(n) forms a compact subgroup. This decomposition is known as 'Euler' or 'Bloch–Messiah' decomposition. [5] Further symplectic matrix properties can be found on that Wikipedia page.

Infinitesimal generators

The members of the symplectic Lie algebra sp(2n, F) are the Hamiltonian matrices.

These are matrices, such that

where B and C are symmetric matrices. See classical group for a derivation.

Example of symplectic matrices

For Sp(2, R), the group of 2 × 2 matrices with determinant 1, the three symplectic (0, 1)-matrices are: [7]

Sp(2n, R)

It turns out that can have a fairly explicit description using generators. If we let denote the symmetric matrices, then is generated by where

are subgroups of [8] pg 173 [9] pg 2.

Relationship with symplectic geometry

Symplectic geometry is the study of symplectic manifolds. The tangent space at any point on a symplectic manifold is a symplectic vector space. [10] As noted earlier, structure preserving transformations of a symplectic vector space form a group and this group is Sp(2n, F), depending on the dimension of the space and the field over which it is defined.

A symplectic vector space is itself a symplectic manifold. A transformation under an action of the symplectic group is thus, in a sense, a linearised version of a symplectomorphism which is a more general structure preserving transformation on a symplectic manifold.

Sp(n)

The compact symplectic group [11] Sp(n) is the intersection of Sp(2n, C) with the unitary group:

It is sometimes written as USp(2n). Alternatively, Sp(n) can be described as the subgroup of GL(n, H) (invertible quaternionic matrices) that preserves the standard hermitian form on Hn:

That is, Sp(n) is just the quaternionic unitary group, U(n, H). [12] Indeed, it is sometimes called the hyperunitary group. Also Sp(1) is the group of quaternions of norm 1, equivalent to SU(2) and topologically a 3-sphere S3.

Note that Sp(n) is not a symplectic group in the sense of the previous sectionit does not preserve a non-degenerate skew-symmetric H-bilinear form on Hn: there is no such form except the zero form. Rather, it is isomorphic to a subgroup of Sp(2n, C), and so does preserve a complex symplectic form in a vector space of twice the dimension. As explained below, the Lie algebra of Sp(n) is the compact real form of the complex symplectic Lie algebra sp(2n, C).

Sp(n) is a real Lie group with (real) dimension n(2n + 1). It is compact and simply connected. [13]

The Lie algebra of Sp(n) is given by the quaternionic skew-Hermitian matrices, the set of n-by-n quaternionic matrices that satisfy

where A is the conjugate transpose of A (here one takes the quaternionic conjugate). The Lie bracket is given by the commutator.

Important subgroups

Some main subgroups are:

Conversely it is itself a subgroup of some other groups:

There are also the isomorphisms of the Lie algebras sp(2) = so(5) and sp(1) = so(3) = su(2).

Relationship between the symplectic groups

Every complex, semisimple Lie algebra has a split real form and a compact real form; the former is called a complexification of the latter two.

The Lie algebra of Sp(2n, C) is semisimple and is denoted sp(2n, C). Its split real form is sp(2n, R) and its compact real form is sp(n). These correspond to the Lie groups Sp(2n, R) and Sp(n) respectively.

The algebras, sp(p, np), which are the Lie algebras of Sp(p, np), are the indefinite signature equivalent to the compact form.

Physical significance

Classical mechanics

The non-compact symplectic group Sp(2n, R) comes up in classical physics as the symmetries of canonical coordinates preserving the Poisson bracket.

Consider a system of n particles, evolving under Hamilton's equations whose position in phase space at a given time is denoted by the vector of canonical coordinates,

The elements of the group Sp(2n, R) are, in a certain sense, canonical transformations on this vector, i.e. they preserve the form of Hamilton's equations. [14] [15] If

are new canonical coordinates, then, with a dot denoting time derivative,

where

for all t and all z in phase space. [16]

For the special case of a Riemannian manifold, Hamilton's equations describe the geodesics on that manifold. The coordinates live on the underlying manifold, and the momenta live in the cotangent bundle. This is the reason why these are conventionally written with upper and lower indexes; it is to distinguish their locations. The corresponding Hamiltonian consists purely of the kinetic energy: it is where is the inverse of the metric tensor on the Riemannian manifold. [17] [15] In fact, the cotangent bundle of any smooth manifold can be a given a symplectic structure in a canonical way, with the symplectic form defined as the exterior derivative of the tautological one-form. [18]

Quantum mechanics

Consider a system of n particles whose quantum state encodes its position and momentum. These coordinates are continuous variables and hence the Hilbert space, in which the state lives, is infinite-dimensional. This often makes the analysis of this situation tricky. An alternative approach is to consider the evolution of the position and momentum operators under the Heisenberg equation in phase space.

Construct a vector of canonical coordinates,

The canonical commutation relation can be expressed simply as

where

and In is the n × n identity matrix.

Many physical situations only require quadratic Hamiltonians, i.e. Hamiltonians of the form

where K is a 2n × 2n real, symmetric matrix. This turns out to be a useful restriction and allows us to rewrite the Heisenberg equation as

The solution to this equation must preserve the canonical commutation relation. It can be shown that the time evolution of this system is equivalent to an action of the real symplectic group, Sp(2n, R), on the phase space.

See also

Notes

  1. "Symplectic group", Encyclopedia of Mathematics Retrieved on 13 December 2014.
  2. Hall 2015 Prop. 3.25
  3. "Is the symplectic group Sp(2n, R) simple?", Stack Exchange Retrieved on 14 December 2014.
  4. "Is the exponential map for Sp(2n, R) surjective?", Stack Exchange Retrieved on 5 December 2014.
  5. "Standard forms and entanglement engineering of multimode Gaussian states under local operations – Serafini and Adesso", Retrieved on 30 January 2015.
  6. "Symplectic Geometry – Arnol'd and Givental", Retrieved on 30 January 2015.
  7. Symplectic Group, (source: Wolfram MathWorld), downloaded February 14, 2012
  8. Gerald B. Folland. (2016). Harmonic analysis in phase space. Princeton: Princeton Univ Press. p. 173. ISBN   978-1-4008-8242-7. OCLC   945482850.
  9. Habermann, Katharina, 1966- (2006). Introduction to symplectic Dirac operators. Springer. ISBN   978-3-540-33421-7. OCLC   262692314.{{cite book}}: CS1 maint: multiple names: authors list (link) CS1 maint: numeric names: authors list (link)
  10. "Lecture Notes – Lecture 2: Symplectic reduction", Retrieved on 30 January 2015.
  11. Hall 2015 Section 1.2.8
  12. Hall 2015 p. 14
  13. Hall 2015 Prop. 13.12
  14. Arnold 1989 gives an extensive mathematical overview of classical mechanics. See chapter 8 for symplectic manifolds.
  15. 1 2 Ralph Abraham and Jerrold E. Marsden, Foundations of Mechanics, (1978) Benjamin-Cummings, London ISBN   0-8053-0102-X
  16. Goldstein 1980 , Section 9.3
  17. Jurgen Jost, (1992) Riemannian Geometry and Geometric Analysis, Springer.
  18. da Silva, Ana Cannas (2008). Lectures on Symplectic Geometry. Lecture Notes in Mathematics. Vol. 1764. Berlin, Heidelberg: Springer Berlin Heidelberg. p. 9. doi:10.1007/978-3-540-45330-7. ISBN   978-3-540-42195-5.

Related Research Articles

<span class="mw-page-title-main">Lie group</span> Group that is also a differentiable manifold with group operations that are smooth

In mathematics, a Lie group is a group that is also a differentiable manifold.

In differential geometry, a subject of mathematics, a symplectic manifold is a smooth manifold, , equipped with a closed nondegenerate differential 2-form , called the symplectic form. The study of symplectic manifolds is called symplectic geometry or symplectic topology. Symplectic manifolds arise naturally in abstract formulations of classical mechanics and analytical mechanics as the cotangent bundles of manifolds. For example, in the Hamiltonian formulation of classical mechanics, which provides one of the major motivations for the field, the set of all possible configurations of a system is modeled as a manifold, and this manifold's cotangent bundle describes the phase space of the system.

<span class="mw-page-title-main">Orthogonal group</span> Type of group in mathematics

In mathematics, the orthogonal group in dimension n, denoted O(n), is the group of distance-preserving transformations of a Euclidean space of dimension n that preserve a fixed point, where the group operation is given by composing transformations. The orthogonal group is sometimes called the general orthogonal group, by analogy with the general linear group. Equivalently, it is the group of n × n orthogonal matrices, where the group operation is given by matrix multiplication (an orthogonal matrix is a real matrix whose inverse equals its transpose). The orthogonal group is an algebraic group and a Lie group. It is compact.

In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space under the operation of composition.

In mathematics, a symplectic matrix is a matrix with real entries that satisfies the condition

<span class="mw-page-title-main">Unitary group</span> Group of unitary matrices

In mathematics, the unitary group of degree n, denoted U(n), is the group of n × n unitary matrices, with the group operation of matrix multiplication. The unitary group is a subgroup of the general linear group GL(n, C). Hyperorthogonal group is an archaic name for the unitary group, especially over finite fields. For the group of unitary matrices with determinant 1, see Special unitary group.

<span class="mw-page-title-main">Special unitary group</span> Group of unitary matrices with determinant of 1

In mathematics, the special unitary group of degree n, denoted SU(n), is the Lie group of n × n unitary matrices with determinant 1.

In mathematics, a congruence subgroup of a matrix group with integer entries is a subgroup defined by congruence conditions on the entries. A very simple example is the subgroup of invertible 2 × 2 integer matrices of determinant 1 in which the off-diagonal entries are even. More generally, the notion of congruence subgroup can be defined for arithmetic subgroups of algebraic groups; that is, those for which we have a notion of 'integral structure' and can define reduction maps modulo an integer.

<span class="mw-page-title-main">Spin group</span> Double cover Lie group of the special orthogonal group

In mathematics the spin group, denoted Spin(n), is a Lie group whose underlying manifold is the double cover of the special orthogonal group SO(n) = SO(n, R), such that there exists a short exact sequence of Lie groups (when n ≠ 2)

In mathematics, the Heisenberg group, named after Werner Heisenberg, is the group of 3×3 upper triangular matrices of the form

<span class="mw-page-title-main">Holonomy</span> Concept in differential geometry

In differential geometry, the holonomy of a connection on a smooth manifold is a general geometrical consequence of the curvature of the connection measuring the extent to which parallel transport around closed loops fails to preserve the geometrical data being transported. For flat connections, the associated holonomy is a type of monodromy and is an inherently global notion. For curved connections, holonomy has nontrivial local and global features.

In group theory, restriction forms a representation of a subgroup using a known representation of the whole group. Restriction is a fundamental construction in representation theory of groups. Often the restricted representation is simpler to understand. Rules for decomposing the restriction of an irreducible representation into irreducible representations of the subgroup are called branching rules, and have important applications in physics. For example, in case of explicit symmetry breaking, the symmetry group of the problem is reduced from the whole group to one of its subgroups. In quantum mechanics, this reduction in symmetry appears as a splitting of degenerate energy levels into multiplets, as in the Stark or Zeeman effect.

In mathematics, the Iwasawa decomposition of a semisimple Lie group generalises the way a square real matrix can be written as a product of an orthogonal matrix and an upper triangular matrix. It is named after Kenkichi Iwasawa, the Japanese mathematician who developed this method.

In mathematics, a complex structure on a real vector space V is an automorphism of V that squares to the minus identity, −IdV. Such a structure on V allows one to define multiplication by complex scalars in a canonical fashion so as to regard V as a complex vector space.

<span class="mw-page-title-main">Hermitian symmetric space</span> Manifold with inversion symmetry

In mathematics, a Hermitian symmetric space is a Hermitian manifold which at every point has an inversion symmetry preserving the Hermitian structure. First studied by Élie Cartan, they form a natural generalization of the notion of Riemannian symmetric space from real manifolds to complex manifolds.

<span class="mw-page-title-main">Classical group</span>

In mathematics, the classical groups are defined as the special linear groups over the reals R, the complex numbers C and the quaternions H together with special automorphism groups of symmetric or skew-symmetric bilinear forms and Hermitian or skew-Hermitian sesquilinear forms defined on real, complex and quaternionic finite-dimensional vector spaces. Of these, the complex classical Lie groups are four infinite families of Lie groups that together with the exceptional groups exhaust the classification of simple Lie groups. The compact classical groups are compact real forms of the complex classical groups. The finite analogues of the classical groups are the classical groups of Lie type. The term "classical group" was coined by Hermann Weyl, it being the title of his 1939 monograph The Classical Groups.

SL<sub>2</sub>(<b>R</b>) Group of real 2×2 matrices with unit determinant

In mathematics, the special linear group SL(2, R) or SL2(R) is the group of 2 × 2 real matrices with determinant one:

In differential geometry, a metaplectic structure is the symplectic analog of spin structure on orientable Riemannian manifolds. A metaplectic structure on a symplectic manifold allows one to define the symplectic spinor bundle, which is the Hilbert space bundle associated to the metaplectic structure via the metaplectic representation, giving rise to the notion of a symplectic spinor field in differential geometry.

In mathematics, the oscillator representation is a projective unitary representation of the symplectic group, first investigated by Irving Segal, David Shale, and André Weil. A natural extension of the representation leads to a semigroup of contraction operators, introduced as the oscillator semigroup by Roger Howe in 1988. The semigroup had previously been studied by other mathematicians and physicists, most notably Felix Berezin in the 1960s. The simplest example in one dimension is given by SU(1,1). It acts as Möbius transformations on the extended complex plane, leaving the unit circle invariant. In that case the oscillator representation is a unitary representation of a double cover of SU(1,1) and the oscillator semigroup corresponds to a representation by contraction operators of the semigroup in SL(2,C) corresponding to Möbius transformations that take the unit disk into itself.

In mathematics, an invariant convex cone is a closed convex cone in a Lie algebra of a connected Lie group that is invariant under inner automorphisms. The study of such cones was initiated by Ernest Vinberg and Bertram Kostant.

References