Lie group

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https://en.wikipedia.org/wiki/Lie_group

In mathematics, a Lie group (pronounced /liː/ LEE) is a group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable.

A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additional properties it must have to be thought of as a "transformation" in the abstract sense, for instance multiplication and the taking of inverses (division), or equivalently, the concept of addition and the taking of inverses (subtraction). Combining these two ideas, one obtains a continuous group where multiplying points and their inverses is continuous. If the multiplication and taking of inverses are smooth (differentiable) as well, one obtains a Lie group.

Lie groups provide a natural model for the concept of continuous symmetry, a celebrated example of which is the circle group. Rotating a circle is an example of a continuous symmetry. For any rotation of the circle, there exists the same symmetry, and concatenation of such rotations makes them into the circle group, an archetypal example of a Lie group. Lie groups are widely used in many parts of modern mathematics and physics.

Lie groups were first found by studying matrix subgroups ${\displaystyle G}$ contained in ${\displaystyle {\text{GL}}_n(\mathbb{R})}$ or ⁠${\displaystyle {\text{GL}}_n(\mathbb{C})}$⁠, the groups of ${\displaystyle n\times n}$ invertible matrices over ${\displaystyle \mathbb{R}}$ or ⁠${\displaystyle \mathbb{C}}$⁠. These are now called the classical groups, as the concept has been extended far beyond these origins. Lie groups are named after Norwegian mathematician Sophus Lie (1842–1899), who laid the foundations of the theory of continuous transformation groups. Lie's original motivation for introducing Lie groups was to model the continuous symmetries of differential equations, in much the same way that finite groups are used in Galois theory to model the discrete symmetries of algebraic equations.

History

Sophus Lie considered the winter of 1873–1874 as the birth date of his theory of continuous groups. Thomas Hawkins, however, suggests that it was "Lie's prodigious research activity during the four-year period from the fall of 1869 to the fall of 1873" that led to the theory's creation. Some of Lie's early ideas were developed in close collaboration with Felix Klein. Lie met with Klein every day from October 1869 through 1872: in Berlin from the end of October 1869 to the end of February 1870, and in Paris, Göttingen and Erlangen in the subsequent two years. Lie stated that all of the principal results were obtained by 1884. But during the 1870s all his papers (except the very first note) were published in Norwegian journals, which impeded recognition of the work throughout the rest of Europe. In 1884 a young German mathematician, Friedrich Engel, came to work with Lie on a systematic treatise to expose his theory of continuous groups. From this effort resulted the three-volume Theorie der Transformationsgruppen, published in 1888, 1890, and 1893. The term groupes de Lie first appeared in French in 1893 in the thesis of Lie's student Arthur Tresse.

Lie's ideas did not stand in isolation from the rest of mathematics. In fact, his interest in the geometry of differential equations was first motivated by the work of Carl Gustav Jacobi, on the theory of partial differential equations of first order and on the equations of classical mechanics. Much of Jacobi's work was published posthumously in the 1860s, generating enormous interest in France and Germany. Lie's idée fixe was to develop a theory of symmetries of differential equations that would accomplish for them what Évariste Galois had done for algebraic equations: namely, to classify them in terms of group theory. Lie and other mathematicians showed that the most important equations for special functions and orthogonal polynomials tend to arise from group theoretical symmetries. In Lie's early work, the idea was to construct a theory of continuous groups, to complement the theory of discrete groups that had developed in the theory of modular forms, in the hands of Felix Klein and Henri Poincaré. The initial application that Lie had in mind was to the theory of differential equations. On the model of Galois theory and polynomial equations, the driving conception was of a theory capable of unifying, by the study of symmetry, the whole area of ordinary differential equations. However, the hope that Lie theory would unify the entire field of ordinary differential equations was not fulfilled. Symmetry methods for ODEs continue to be studied, but do not dominate the subject. There is a differential Galois theory, but it was developed by others, such as Picard and Vessiot, and it provides a theory of quadratures, the indefinite integrals required to express solutions.

Additional impetus to consider continuous groups came from ideas of Bernhard Riemann, on the foundations of geometry, and their further development in the hands of Klein. Thus three major themes in 19th century mathematics were combined by Lie in creating his new theory:

• The idea of symmetry, as exemplified by Galois through the algebraic notion of a group;
• Geometric theory and the explicit solutions of differential equations of mechanics, worked out by Poisson and Jacobi;
• The new understanding of geometry that emerged in the works of Plücker, Möbius, Grassmann and others, and culminated in Riemann's revolutionary vision of the subject.

Although today Sophus Lie is rightfully recognized as the creator of the theory of continuous groups, a major stride in the development of their structure theory, which was to have a profound influence on subsequent development of mathematics, was made by Wilhelm Killing, who in 1888 published the first paper in a series entitled Die Zusammensetzung der stetigen endlichen Transformationsgruppen (The composition of continuous finite transformation groups). The work of Killing, later refined and generalized by Élie Cartan, led to classification of semisimple Lie algebras, Cartan's theory of symmetric spaces, and Hermann Weyl's description of representations of compact and semisimple Lie groups using highest weights.

In 1900 David Hilbert challenged Lie theorists with his Fifth Problem presented at the International Congress of Mathematicians in Paris.

Weyl brought the early period of the development of the theory of Lie groups to fruition, for not only did he classify irreducible representations of semisimple Lie groups and connect the theory of groups with quantum mechanics, but he also put Lie's theory itself on firmer footing by clearly enunciating the distinction between Lie's infinitesimal groups (i.e., Lie algebras) and the Lie groups proper, and began investigations of topology of Lie groups. The theory of Lie groups was systematically reworked in modern mathematical language in a monograph by Claude Chevalley.

Overview

The set of all complex numbers with absolute value 1 (corresponding to points on the circle of center 0 and radius 1 in the complex plane) is a Lie group under complex multiplication: the circle group.

Lie groups are smooth differentiable manifolds and as such can be studied using differential calculus, in contrast with the case of more general topological groups. One of the key ideas in the theory of Lie groups is to replace the global object, the group, with its local or linearized version, which Lie himself called its "infinitesimal group" and which has since become known as its Lie algebra.

Lie groups play an enormous role in modern geometry, on several different levels. Felix Klein argued in his Erlangen program that one can consider various "geometries" by specifying an appropriate transformation group that leaves certain geometric properties invariant. Thus Euclidean geometry corresponds to the choice of the group E(3) of distance-preserving transformations of the Euclidean space ⁠${\displaystyle {\mathbb{R}}^3}$⁠, conformal geometry corresponds to enlarging the group to the conformal group, whereas in projective geometry one is interested in the properties invariant under the projective group. This idea later led to the notion of a G-structure, where G is a Lie group of "local" symmetries of a manifold.

Lie groups (and their associated Lie algebras) play a major role in modern physics, with the Lie group typically playing the role of a symmetry of a physical system. Here, the representations of the Lie group (or of its Lie algebra) are especially important. Representation theory is used extensively in particle physics. Groups whose representations are of particular importance include the rotation group SO(3) (or its double cover SU(2)), the special unitary group SU(3) and the Poincaré group.

On a "global" level, whenever a Lie group acts on a geometric object, such as a Riemannian or a symplectic manifold, this action provides a measure of rigidity and yields a rich algebraic structure. The presence of continuous symmetries expressed via a Lie group action on a manifold places strong constraints on its geometry and facilitates analysis on the manifold. Linear actions of Lie groups are especially important, and are studied in representation theory.

In the 1940s–1950s, Ellis Kolchin, Armand Borel, and Claude Chevalley realised that many foundational results concerning Lie groups can be developed completely algebraically, giving rise to the theory of algebraic groups defined over an arbitrary field. This insight opened new possibilities in pure algebra, by providing a uniform construction for most finite simple groups, as well as in algebraic geometry. The theory of automorphic forms, an important branch of modern number theory, deals extensively with analogues of Lie groups over adele rings; p-adic Lie groups play an important role, via their connections with Galois representations in number theory.

Definitions and examples

A real Lie group is a group that is also a finite-dimensional real smooth manifold, in which the group operations of multiplication and inversion are smooth maps. Smoothness of the group multiplication

${\displaystyle \mu :G\times G\to G\mu (x,y)=xy}$

means that μ is a smooth mapping of the product manifold G × G into G. The two requirements can be combined to the single requirement that the mapping

${\displaystyle (x,y)\mapsto x^{-1}y}$

be a smooth mapping of the product manifold into G.

First examples

• The 2×2 real invertible matrices form a group under multiplication, called general linear group of degree 2 and denoted by ${\displaystyle \mathrm{GL}(2,\mathbb{R})}$ or by ⁠${\displaystyle {\mathrm{GL}}_2(\mathbb{R})}$⁠: $${\displaystyle \mathrm{GL}(2,\mathbb{R})=\left\{A=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right):detA=ad-bc\ne 0\right\}.}$$ This is a four-dimensional noncompact real Lie group; it is an open subset of ⁠${\displaystyle {\mathbb{R}}^4}$⁠. This group is disconnected; it has two connected components corresponding to the positive and negative values of the determinant.
• The rotation matrices form a subgroup of ⁠${\displaystyle \mathrm{GL}(2,\mathbb{R})}$⁠, denoted by ⁠${\displaystyle \mathrm{SO}(2,\mathbb{R})}$⁠. It is a Lie group in its own right: specifically, a one-dimensional compact connected Lie group which is diffeomorphic to the circle. Using the rotation angle ${\displaystyle \phi }$ as a parameter, this group can be parametrized as follows: $${\displaystyle \mathrm{SO}(2,\mathbb{R})=\left\{\left(\begin{array}{cc}\cos \phi & -\sin \phi \\ \sin \phi & \cos \phi \end{array}\right):\phi \in \mathbb{R}/2\pi \mathbb{Z}\right\}.}$$ Addition of the angles corresponds to multiplication of the elements of ⁠${\displaystyle \mathrm{SO}(2,\mathbb{R})}$⁠, and taking the opposite angle corresponds to inversion. Thus both multiplication and inversion are differentiable maps.
• The affine group of one dimension is a two-dimensional matrix Lie group, consisting of ${\displaystyle 2\times 2}$ real, upper-triangular matrices, with the first diagonal entry being positive and the second diagonal entry being 1. Thus, the group consists of matrices of the form $${\displaystyle A=\left(\begin{array}{cc}a& b\\ 0& 1\end{array}\right),a>0,b\in \mathbb{R}.}$$

Non-example

Further information: Linear flow on the torus

We now present an example of a group with an uncountable number of elements that is not a Lie group under a certain topology. The group given by

${\displaystyle H=\left\{\left(\begin{array}{cc}{e}^{2\pi i\theta }& 0\\ 0& e^{2\pi ia\theta }\end{array}\right):\theta \in \mathbb{R}\right\}\subset {\mathbb{T}}^2=\left\{\left(\begin{array}{cc}{e}^{2\pi i\theta }& 0\\ 0& e^{2\pi i\varphi }\end{array}\right):\theta ,\varphi \in \mathbb{R}\right\},}$

with ${\displaystyle a\in \mathbb{R}\setminus \mathbb{Q}}$ a fixed irrational number, is a subgroup of the torus ${\displaystyle {\mathbb{T}}^2}$ that is not a Lie group when given the subspace topology. If we take any small neighborhood ${\displaystyle U}$ of a point ${\displaystyle h}$ in ⁠${\displaystyle H}$⁠, for example, the portion of ${\displaystyle H}$ in ${\displaystyle U}$ is disconnected. The group ${\displaystyle H}$ winds repeatedly around the torus without ever reaching a previous point of the spiral and thus forms a dense subgroup of ⁠${\displaystyle {\mathbb{T}}^2}$⁠.

A portion of the group ${\displaystyle H}$ inside ⁠${\displaystyle {\mathbb{T}}^2}$⁠. Small neighborhoods of the element ${\displaystyle h\in H}$ are disconnected in the subset topology on ⁠${\displaystyle H}$⁠

The group ${\displaystyle H}$ can, however, be given a different topology, in which the distance between two points ${\displaystyle h_1,h_2\in H}$ is defined as the length of the shortest path in the group ${\displaystyle H}$ joining ${\displaystyle h_1}$ to ⁠${\displaystyle h_2}$⁠. In this topology, ${\displaystyle H}$ is identified homeomorphically with the real line by identifying each element with the number ${\displaystyle \theta }$ in the definition of ⁠${\displaystyle H}$⁠. With this topology, ${\displaystyle H}$ is just the group of real numbers under addition and is therefore a Lie group.

The group ${\displaystyle H}$ is an example of a "Lie subgroup" of a Lie group that is not closed. See the discussion below of Lie subgroups in the section on basic concepts.

Matrix Lie groups

Let ${\displaystyle \mathrm{GL}(n,\mathbb{C})}$ denote the group of ${\displaystyle n\times n}$ invertible matrices with entries in ⁠${\displaystyle \mathbb{C}}$⁠. Any closed subgroup of ${\displaystyle \mathrm{GL}(n,\mathbb{C})}$ is a Lie group; Lie groups of this sort are called matrix Lie groups. Since most of the interesting examples of Lie groups can be realized as matrix Lie groups, some textbooks restrict attention to this class, including those of Hall, Rossmann, and Stillwell. Restricting attention to matrix Lie groups simplifies the definition of the Lie algebra and the exponential map. The following are standard examples of matrix Lie groups.

• The special linear groups over ${\displaystyle \mathbb{R}}$ and ⁠${\displaystyle \mathbb{C}}$⁠, ${\displaystyle \mathrm{SL}(n,\mathbb{R})}$ and ⁠${\displaystyle \mathrm{SL}(n,\mathbb{C})}$⁠, consisting of ${\displaystyle n\times n}$ matrices with determinant one and entries in ${\displaystyle \mathbb{R}}$ or ${\displaystyle \mathbb{C}}$
• The unitary groups and special unitary groups, ${\displaystyle \mathrm{U}(n,\mathbb{C})}$ and ⁠${\displaystyle \mathrm{SU}(n,\mathbb{C})}$⁠, consisting of ${\displaystyle n\times n}$ complex matrices satisfying ${\displaystyle U^{\ast }=U^{-1}}$ (and also ${\displaystyle det(U)=1}$ in the case of ${\displaystyle \mathrm{SU}(n)}$)
• The orthogonal groups and special orthogonal groups, ${\displaystyle \mathrm{O}(n,\mathbb{R})}$ and ⁠${\displaystyle \mathrm{SO}(n,\mathbb{R})}$⁠, consisting of ${\displaystyle n\times n}$ real matrices satisfying ${\displaystyle R^{\mathrm{T}}=R^{-1}}$ (and also ${\displaystyle det(R)=1}$ in the case of ${\displaystyle \mathrm{SO}(n,\mathbb{R})}$)

All of the preceding examples fall under the heading of the classical groups.

Related concepts

A complex Lie group is defined in the same way using complex manifolds rather than real ones (example: ${\displaystyle \mathrm{SL}(2,\mathbb{C})}$), and holomorphic maps. Similarly, using an alternate metric completion of ⁠${\displaystyle \mathbb{Q}}$⁠, one can define a p-adic Lie group over the p-adic numbers, a topological group which is also an analytic p-adic manifold, such that the group operations are analytic. In particular, each point has a p-adic neighborhood.

Hilbert's fifth problem asked whether replacing differentiable manifolds with topological or analytic ones can yield new examples. The answer to this question turned out to be negative: in 1952, Gleason, Montgomery and Zippin showed that if G is a topological manifold with continuous group operations, then there exists exactly one analytic structure on G which turns it into a Lie group (see also Hilbert–Smith conjecture). If the underlying manifold is allowed to be infinite-dimensional (for example, a Hilbert manifold), then one arrives at the notion of an infinite-dimensional Lie group. It is possible to define analogues of many Lie groups over finite fields, and these give most of the examples of finite simple groups.

The language of category theory provides a concise definition for Lie groups: a Lie group is a group object in the category of smooth manifolds. This is important, because it allows generalization of the notion of a Lie group to Lie supergroups. This categorical point of view leads also to a different generalization of Lie groups, namely Lie groupoids, which are groupoid objects in the category of smooth manifolds with a further requirement.

Topological definition

A Lie group can be defined as a (Hausdorff) topological group that, near the identity element, looks like a transformation group, with no reference to differentiable manifolds. First, we define an immersely linear Lie group to be a subgroup G of the general linear group ${\displaystyle \mathrm{GL}(n,\mathbb{C})}$ such that

1. for some neighborhood V of the identity element e in G, the topology on V is the subspace topology of ${\displaystyle \mathrm{GL}(n,\mathbb{C})}$ and V is closed in ⁠${\displaystyle \mathrm{GL}(n,\mathbb{C})}$⁠.
2. G has at most countably many connected components.

(For example, a closed subgroup of ⁠${\displaystyle \mathrm{GL}(n,\mathbb{C})}$⁠; that is, a matrix Lie group satisfies the above conditions.)

Then a Lie group is defined as a topological group that (1) is locally isomorphic near the identities to an immersely linear Lie group and (2) has at most countably many connected components. Showing the topological definition is equivalent to the usual one is technical (and the beginning readers should skip the following) but is done roughly as follows:

1. Given a Lie group G in the usual manifold sense, the Lie group–Lie algebra correspondence (or a version of Lie's third theorem) constructs an immersed Lie subgroup ${\displaystyle G^{\prime }\subset \mathrm{GL}(n,\mathbb{C})}$ such that ${\displaystyle G,G^{\prime }}$ share the same Lie algebra; thus, they are locally isomorphic. Hence, ${\displaystyle G}$ satisfies the above topological definition.
2. Conversely, let ${\displaystyle G}$ be a topological group that is a Lie group in the above topological sense and choose an immersely linear Lie group ${\displaystyle G^{\prime }}$ that is locally isomorphic to ⁠${\displaystyle G}$⁠. Then, by a version of the closed subgroup theorem, ${\displaystyle G^{\prime }}$ is a real-analytic manifold and then, through the local isomorphism, G acquires a structure of a manifold near the identity element. One then shows that the group law on G can be given by formal power series; so the group operations are real-analytic and ${\displaystyle G}$ itself is a real-analytic manifold.

The topological definition implies the statement that if two Lie groups are isomorphic as topological groups, then they are isomorphic as Lie groups. In fact, it states the general principle that, to a large extent, the topology of a Lie group together with the group law determines the geometry of the group.

More examples of Lie groups

See also: Table of Lie groups and List of simple Lie groups

Lie groups occur in abundance throughout mathematics and physics. Matrix groups or algebraic groups are (roughly) groups of matrices (for example, orthogonal and symplectic groups), and these give most of the more common examples of Lie groups.

Dimensions one and two

The only connected Lie groups with dimension one are the real line ${\displaystyle \mathbb{R}}$ (with the group operation being addition) and the circle group ${\displaystyle S^1}$ of complex numbers with absolute value one (with the group operation being multiplication). The ${\displaystyle S^1}$ group is often denoted as ⁠${\displaystyle \mathrm{U}(1)}$⁠, the group of ${\displaystyle 1\times 1}$ unitary matrices.

In two dimensions, if we restrict attention to simply connected groups, then they are classified by their Lie algebras. There are (up to isomorphism) only two Lie algebras of dimension two. The associated simply connected Lie groups are ${\displaystyle {\mathbb{R}}^2}$ (with the group operation being vector addition) and the affine group in dimension one, described in the previous subsection under "first examples".

Additional examples

• The group SU(2) is the group of ${\displaystyle 2\times 2}$ unitary matrices with determinant ⁠${\displaystyle 1}$⁠. Topologically, ${\displaystyle \text{SU}(2)}$ is the ⁠${\displaystyle 3}$⁠-sphere ⁠${\displaystyle S^3}$⁠; as a group, it may be identified with the group of unit quaternions.
• The Heisenberg group is a connected nilpotent Lie group of dimension ⁠${\displaystyle 3}$⁠, playing a key role in quantum mechanics.
• The Lorentz group is a 6-dimensional Lie group of linear isometries of the Minkowski space.
• The Poincaré group is a 10-dimensional Lie group of affine isometries of the Minkowski space.
• The exceptional Lie groups of types G₂, F₄, E₆, E₇, E₈ have dimensions 14, 52, 78, 133, and 248. Along with the A–B–C–D series of simple Lie groups, the exceptional groups complete the list of simple Lie groups.
• The symplectic group ${\displaystyle \text{Sp}(2n,\mathbb{R})}$ consists of all ${\displaystyle 2n\times 2n}$ matrices preserving a symplectic form on ⁠${\displaystyle {\mathbb{R}}^{2n}}$⁠. It is a connected Lie group of dimension ⁠${\displaystyle 2n^2+n}$⁠.

Constructions

There are several standard ways to form new Lie groups from old ones:

• The product of two Lie groups is a Lie group.
• Any topologically closed subgroup of a Lie group is a Lie group. This is known as the closed subgroup theorem or Cartan's theorem.
• The quotient of a Lie group by a closed normal subgroup is a Lie group.
• The universal cover of a connected Lie group is a Lie group. For example, the group ${\displaystyle \mathbb{R}}$ is the universal cover of the circle group ⁠${\displaystyle S^1}$⁠. In fact any covering of a differentiable manifold is also a differentiable manifold, but by specifying universal cover, one guarantees a group structure (compatible with its other structures).

Related notions

Some examples of groups that are not Lie groups (except in the trivial sense that any group having at most countably many elements can be viewed as a 0-dimensional Lie group, with the discrete topology), are:

• Infinite-dimensional groups, such as the additive group of an infinite-dimensional real vector space, or the space of smooth functions from a manifold ${\displaystyle X}$ to a Lie group ⁠${\displaystyle G}$⁠, ⁠${\displaystyle C^{\mathrm{\infty }}(X,G)}$⁠. These are not Lie groups as they are not finite-dimensional manifolds.
• Some totally disconnected groups, such as the Galois group of an infinite extension of fields, or the additive group of the p-adic numbers. These are not Lie groups because their underlying spaces are not real manifolds. (Some of these groups are "p-adic Lie groups".) In general, only topological groups having similar local properties to Rⁿ for some positive integer n can be Lie groups (of course they must also have a differentiable structure).

Basic concepts

The Lie algebra associated with a Lie group

Main article: Lie group–Lie algebra correspondence

To every Lie group we can associate a Lie algebra whose underlying vector space is the tangent space of the Lie group at the identity element and which completely captures the local structure of the group. Informally we can think of elements of the Lie algebra as elements of the group that are "infinitesimally close" to the identity, and the Lie bracket of the Lie algebra is related to the commutator of two such infinitesimal elements. Before giving the abstract definition we give a few examples:

• The Lie algebra of the vector space Rⁿ is just Rⁿ with the Lie bracket given by
      [A, B] = 0.
  (In general the Lie bracket of a connected Lie group is always 0 if and only if the Lie group is abelian.)
• The Lie algebra of the general linear group GL(n, C) of invertible matrices is the vector space M(n, C) of square matrices with the Lie bracket given by
      [A, B] = AB − BA.
• If G is a closed subgroup of GL(n, C) then the Lie algebra of G can be thought of informally as the matrices m of M(n, C) such that 1 + εm is in G, where ε is an infinitesimal positive number with ε² = 0 (of course, no such real number ε exists). For example, the orthogonal group O(n, R) consists of matrices A with AA^T = 1, so the Lie algebra consists of the matrices m with (1 + εm)(1 + εm)^T = 1, which is equivalent to m + m^T = 0 because ε² = 0.
• The preceding description can be made more rigorous as follows. The Lie algebra of a closed subgroup G of GL(n, C), may be computed as

${\displaystyle \mathrm{Lie}(G)=\{X\in M(n;\mathbb{C})|\exp (tX)\in G\text{ for all }t\text{ in }\mathbb{R}\},}$ where exp(tX) is defined using the matrix exponential. It can then be shown that the Lie algebra of G is a real vector space that is closed under the bracket operation, ⁠${\displaystyle [X,Y]=XY-YX}$⁠.

The concrete definition given above for matrix groups is easy to work with, but has some minor problems: to use it we first need to represent a Lie group as a group of matrices, but not all Lie groups can be represented in this way, and it is not even obvious that the Lie algebra is independent of the representation we use. To get around these problems we give the general definition of the Lie algebra of a Lie group (in 4 steps):

1. Vector fields on any smooth manifold M can be thought of as derivations X of the ring of smooth functions on the manifold, and therefore form a Lie algebra under the Lie bracket [X, Y] = XY − YX, because the Lie bracket of any two derivations is a derivation.
2. If G is any group acting smoothly on the manifold M, then it acts on the vector fields, and the vector space of vector fields fixed by the group is closed under the Lie bracket and therefore also forms a Lie algebra.
3. We apply this construction to the case when the manifold M is the underlying space of a Lie group G, with G acting on G = M by left translations L_g(h) = gh. This shows that the space of left invariant vector fields (vector fields satisfying L_g*X_h = X_gh for every h in G, where L_g* denotes the differential of L_g) on a Lie group is a Lie algebra under the Lie bracket of vector fields.
4. Any tangent vector at the identity of a Lie group can be extended to a left invariant vector field by left translating the tangent vector to other points of the manifold. Specifically, the left invariant extension of an element v of the tangent space at the identity is the vector field defined by v^_g = L_g*v. This identifies the tangent space T_eG at the identity with the space of left invariant vector fields, and therefore makes the tangent space at the identity into a Lie algebra, called the Lie algebra of G, usually denoted by a Fraktur ${\displaystyle \mathfrak{g}.}$ Thus the Lie bracket on ${\displaystyle \mathfrak{g}}$ is given explicitly by [v, w] = [v^, w^]_e.

This Lie algebra ${\displaystyle \mathfrak{g}}$ is finite-dimensional and it has the same dimension as the manifold G. The Lie algebra of G determines G up to "local isomorphism", where two Lie groups are called locally isomorphic if they look the same near the identity element. Problems about Lie groups are often solved by first solving the corresponding problem for the Lie algebras, and the result for groups then usually follows easily. For example, simple Lie groups are usually classified by first classifying the corresponding Lie algebras.

We could also define a Lie algebra structure on T_e using right invariant vector fields instead of left invariant vector fields. This leads to the same Lie algebra, because the inverse map on G can be used to identify left invariant vector fields with right invariant vector fields, and acts as −1 on the tangent space T_e.

The Lie algebra structure on T_e can also be described as follows: the commutator operation

(x, y) → xyx⁻¹y⁻¹

on G × G sends (e, e) to e, so its derivative yields a bilinear operation on T_eG. This bilinear operation is actually the zero map, but the second derivative, under the proper identification of tangent spaces, yields an operation that satisfies the axioms of a Lie bracket, and it is equal to twice the one defined through left-invariant vector fields.

Homomorphisms and isomorphisms

If G and H are Lie groups, then a Lie group homomorphism f : G → H is a smooth group homomorphism. In the case of complex Lie groups, such a homomorphism is required to be a holomorphic map. However, these requirements are a bit stringent; every continuous homomorphism between real Lie groups turns out to be (real) analytic.

The composition of two Lie homomorphisms is again a homomorphism, and the class of all Lie groups, together with these morphisms, forms a category. Moreover, every Lie group homomorphism induces a homomorphism between the corresponding Lie algebras. Let ${\displaystyle \varphi :G\to H}$ be a Lie group homomorphism and let ${\displaystyle {\varphi }_{\ast }}$ be its derivative at the identity. If we identify the Lie algebras of G and H with their tangent spaces at the identity elements, then ${\displaystyle {\varphi }_{\ast }}$ is a map between the corresponding Lie algebras:

${\displaystyle {\varphi }_{\ast }:\mathfrak{g}\to \mathfrak{h},}$

which turns out to be a Lie algebra homomorphism (meaning that it is a linear map which preserves the Lie bracket). In the language of category theory, we then have a covariant functor from the category of Lie groups to the category of Lie algebras which sends a Lie group to its Lie algebra and a Lie group homomorphism to its derivative at the identity.

Two Lie groups are called isomorphic if there exists a bijective homomorphism between them whose inverse is also a Lie group homomorphism. Equivalently, it is a diffeomorphism which is also a group homomorphism. Observe that, by the above, a continuous homomorphism from a Lie group ${\displaystyle G}$ to a Lie group ${\displaystyle H}$ is an isomorphism of Lie groups if and only if it is bijective.

Lie group versus Lie algebra isomorphisms

Isomorphic Lie groups necessarily have isomorphic Lie algebras; it is then reasonable to ask how isomorphism classes of Lie groups relate to isomorphism classes of Lie algebras.

The first result in this direction is Lie's third theorem, which states that every finite-dimensional, real Lie algebra is the Lie algebra of some (linear) Lie group. One way to prove Lie's third theorem is to use Ado's theorem, which says every finite-dimensional real Lie algebra is isomorphic to a matrix Lie algebra. Meanwhile, for every finite-dimensional matrix Lie algebra, there is a linear group (matrix Lie group) with this algebra as its Lie algebra.

On the other hand, Lie groups with isomorphic Lie algebras need not be isomorphic. Furthermore, this result remains true even if we assume the groups are connected. To put it differently, the global structure of a Lie group is not determined by its Lie algebra; for example, if Z is any discrete subgroup of the center of G then G and G/Z have the same Lie algebra (see the table of Lie groups for examples). An example of importance in physics are the groups SU(2) and SO(3). These two groups have isomorphic Lie algebras, but the groups themselves are not isomorphic, because SU(2) is simply connected but SO(3) is not.

On the other hand, if we require that the Lie group be simply connected, then the global structure is determined by its Lie algebra: two simply connected Lie groups with isomorphic Lie algebras are isomorphic. (See the next subsection for more information about simply connected Lie groups.) In light of Lie's third theorem, we may therefore say that there is a one-to-one correspondence between isomorphism classes of finite-dimensional real Lie algebras and isomorphism classes of simply connected Lie groups.

Simply connected Lie groups

See also: Lie group–Lie algebra correspondence and Fundamental group § Lie groups

A Lie group ${\displaystyle G}$ is said to be simply connected if every loop in ${\displaystyle G}$ can be shrunk continuously to a point in ⁠${\displaystyle G}$⁠. This notion is important because of the following result that has simple connectedness as a hypothesis:

Theorem: Suppose ${\displaystyle G}$ and ${\displaystyle H}$ are Lie groups with Lie algebras ${\displaystyle \mathfrak{g}}$ and ${\displaystyle \mathfrak{h}}$ and that ${\displaystyle f:\mathfrak{g}\to \mathfrak{h}}$ is a Lie algebra homomorphism. If ${\displaystyle G}$ is simply connected, then there is a unique Lie group homomorphism ${\displaystyle \varphi :G\to H}$ such that ⁠${\displaystyle {\varphi }_{\ast }=f}$⁠, where ${\displaystyle {\varphi }_{\ast }}$ is the differential of ${\displaystyle \varphi }$ at the identity.

Lie's third theorem says that every finite-dimensional real Lie algebra is the Lie algebra of a Lie group. It follows from Lie's third theorem and the preceding result that every finite-dimensional real Lie algebra is the Lie algebra of a unique simply connected Lie group.

An example of a simply connected group is the special unitary group SU(2), which as a manifold is the 3-sphere. The rotation group SO(3), on the other hand, is not simply connected. (See Topology of SO(3).) The failure of SO(3) to be simply connected is intimately connected to the distinction between integer spin and half-integer spin in quantum mechanics. Other examples of simply connected Lie groups include the special unitary group SU(n), the spin group (double cover of rotation group) Spin(n) for ⁠${\displaystyle n\ge 3}$⁠, and the compact symplectic group Sp(n).

Methods for determining whether a Lie group is simply connected or not are discussed in the article on fundamental groups of Lie groups.

Exponential map

Main article: Exponential map (Lie theory)

See also: derivative of the exponential map and normal coordinates

The exponential map from the Lie algebra ${\displaystyle \mathrm{M}(n;\mathbb{C})}$ of the general linear group ${\displaystyle \mathrm{G}\mathrm{L}(n;\mathbb{C})}$ to ${\displaystyle \mathrm{G}\mathrm{L}(n;\mathbb{C})}$ is defined by the matrix exponential, given by the usual power series:

${\displaystyle \exp (X)=1+X+\frac{X^2}{2!}+\frac{X^3}{3!}+\cdots }$

for matrices ⁠${\displaystyle X}$⁠. If ${\displaystyle G}$ is a closed subgroup of ⁠${\displaystyle \mathrm{G}\mathrm{L}(n;\mathbb{C})}$⁠, then the exponential map takes the Lie algebra of ${\displaystyle G}$ into ⁠${\displaystyle G}$⁠; thus, we have an exponential map for all matrix groups. Every element of ${\displaystyle G}$ that is sufficiently close to the identity is the exponential of a matrix in the Lie algebra.

The definition above is easy to use, but it is not defined for Lie groups that are not matrix groups, and it is not clear that the exponential map of a Lie group does not depend on its representation as a matrix group. We can solve both problems using a more abstract definition of the exponential map that works for all Lie groups, as follows.

For each vector ${\displaystyle X}$ in the Lie algebra ${\displaystyle \mathfrak{g}}$ of ${\displaystyle G}$ (i.e., the tangent space to ${\displaystyle G}$ at the identity), one proves that there is a unique one-parameter subgroup ${\displaystyle c:\mathbb{R}\to G}$ such that ⁠${\displaystyle c^{\prime }(0)=X}$⁠. Saying that ${\displaystyle c}$ is a one-parameter subgroup means simply that ${\displaystyle c}$ is a smooth map into ${\displaystyle G}$ and that

${\displaystyle c(s+t)=c(s)c(t)}$

for all ${\displaystyle s}$ and ⁠${\displaystyle t}$⁠. The operation on the right hand side is the group multiplication in ⁠${\displaystyle G}$⁠. The formal similarity of this formula with the one valid for the exponential function justifies the definition

${\displaystyle \exp (X)=c(1).}$

This is called the exponential map, and it maps the Lie algebra ${\displaystyle \mathfrak{g}}$ into the Lie group ⁠${\displaystyle G}$⁠. It provides a diffeomorphism between a neighborhood of 0 in ${\displaystyle \mathfrak{g}}$ and a neighborhood of ${\displaystyle e}$ in ⁠${\displaystyle G}$⁠. This exponential map is a generalization of the exponential function for real numbers (because ${\displaystyle \mathbb{R}}$ is the Lie algebra of the Lie group of positive real numbers with multiplication), for complex numbers (because ${\displaystyle \mathbb{C}}$ is the Lie algebra of the Lie group of non-zero complex numbers with multiplication) and for matrices (because ${\displaystyle M(n,\mathbb{R})}$ with the regular commutator is the Lie algebra of the Lie group ${\displaystyle \mathrm{G}\mathrm{L}(n,\mathbb{R})}$ of all invertible matrices).

Because the exponential map is surjective on some neighbourhood ${\displaystyle N}$ of ⁠${\displaystyle e}$⁠, it is common to call elements of the Lie algebra infinitesimal generators of the group ⁠${\displaystyle G}$⁠. The subgroup of ${\displaystyle G}$ generated by ${\displaystyle N}$ is the identity component of ⁠${\displaystyle G}$⁠.

The exponential map and the Lie algebra determine the local group structure of every connected Lie group, because of the Baker–Campbell–Hausdorff formula: there exists a neighborhood ${\displaystyle U}$ of the zero element of ⁠${\displaystyle \mathfrak{g}}$⁠, such that for ${\displaystyle X,Y\in U}$ we have

${\displaystyle \exp (X)\exp (Y)=\exp \left(X+Y+\frac{1}{2}[X,Y]+\frac{1}{12}[[X,Y],Y]-\frac{1}{12}[[X,Y],X]-\cdots \right),}$

where the omitted terms are known and involve Lie brackets of four or more elements. In case ${\displaystyle X}$ and ${\displaystyle Y}$ commute, this formula reduces to the familiar exponential law ⁠${\displaystyle \exp (X)\exp (Y)=\exp (X+Y)}$⁠.

The exponential map relates Lie group homomorphisms. That is, if ${\displaystyle \varphi :G\to H}$ is a Lie group homomorphism and ${\displaystyle {\varphi }_{\ast }:\mathfrak{g}\to \mathfrak{h}}$ the induced map on the corresponding Lie algebras, then for all ${\displaystyle x\in \mathfrak{g}}$ we have

${\displaystyle \varphi (\exp (x))=\exp ({\varphi }_{\ast }(x)).}$

In other words, the following diagram commutes,

(In short, exp is a natural transformation from the functor Lie to the identity functor on the category of Lie groups.)

The exponential map from the Lie algebra to the Lie group is not always onto, even if the group is connected (though it does map onto the Lie group for connected groups that are either compact or nilpotent). For example, the exponential map of SL(2, R) is not surjective. Also, the exponential map is neither surjective nor injective for infinite-dimensional (see below) Lie groups modelled on C^∞ Fréchet space, even from arbitrary small neighborhood of 0 to corresponding neighborhood of 1.

Lie subgroup

A Lie subgroup ${\displaystyle H}$ of a Lie group ${\displaystyle G}$ is a Lie group that is a subset of ${\displaystyle G}$ and such that the inclusion map from ${\displaystyle H}$ to ${\displaystyle G}$ is an injective immersion and group homomorphism. According to Cartan's theorem, a closed subgroup of ${\displaystyle G}$ admits a unique smooth structure which makes it an embedded Lie subgroup of ${\displaystyle G}$—i.e. a Lie subgroup such that the inclusion map is a smooth embedding.

Examples of non-closed subgroups are plentiful; for example take ${\displaystyle G}$ to be a torus of dimension 2 or greater, and let ${\displaystyle H}$ be a one-parameter subgroup of irrational slope, i.e. one that winds around in G. Then there is a Lie group homomorphism ${\displaystyle \phi :\mathbb{R}\to G}$ with ⁠${\displaystyle \mathrm{i}\mathrm{m}(\phi )=H}$⁠. The closure of ${\displaystyle H}$ will be a sub-torus in ⁠${\displaystyle G}$⁠.

The exponential map gives a one-to-one correspondence between the connected Lie subgroups of a connected Lie group ${\displaystyle G}$ and the subalgebras of the Lie algebra of ⁠${\displaystyle G}$⁠. Typically, the subgroup corresponding to a subalgebra is not a closed subgroup. There is no criterion solely based on the structure of ${\displaystyle G}$ which determines which subalgebras correspond to closed subgroups.

Representations

Main article: Representation of a Lie group

See also: Compact group § Representation theory of a connected compact Lie group, and Lie algebra representation

One important aspect of the study of Lie groups is their representations, that is, the way they can act (linearly) on vector spaces. In physics, Lie groups often encode the symmetries of a physical system. The way one makes use of this symmetry to help analyze the system is often through representation theory. Consider, for example, the time-independent Schrödinger equation in quantum mechanics, ⁠${\displaystyle \hat{H}\psi =E\psi }$⁠. Assume the system in question has the rotation group SO(3) as a symmetry, meaning that the Hamiltonian operator ${\displaystyle \hat{H}}$ commutes with the action of SO(3) on the wave function ⁠${\displaystyle \psi }$⁠. (One important example of such a system is the hydrogen atom, which has a spherically symmetric potential.) This assumption does not necessarily mean that the solutions ${\displaystyle \psi }$ are rotationally invariant functions. Rather, it means that the space of solutions to ${\displaystyle \hat{H}\psi =E\psi }$ is invariant under rotations (for each fixed value of ${\displaystyle E}$). This space, therefore, constitutes a representation of SO(3). These representations have been classified and the classification leads to a substantial simplification of the problem, essentially converting a three-dimensional partial differential equation to a one-dimensional ordinary differential equation.

The case of a connected compact Lie group K (including the just-mentioned case of SO(3)) is particularly tractable. In that case, every finite-dimensional representation of K decomposes as a direct sum of irreducible representations. The irreducible representations, in turn, were classified by Hermann Weyl. The classification is in terms of the "highest weight" of the representation. The classification is closely related to the classification of representations of a semisimple Lie algebra.

One can also study (in general infinite-dimensional) unitary representations of an arbitrary Lie group (not necessarily compact). For example, it is possible to give a relatively simple explicit description of the representations of the group SL(2, R) and the representations of the Poincaré group.

Classification

Lie groups may be thought of as smoothly varying families of symmetries. Examples of symmetries include rotation about an axis. What must be understood is the nature of 'small' transformations, for example, rotations through tiny angles, that link nearby transformations. The mathematical object capturing this structure is called a Lie algebra (Lie himself called them "infinitesimal groups"). It can be defined because Lie groups are smooth manifolds, so have tangent spaces at each point.

The Lie algebra of any compact Lie group (very roughly: one for which the symmetries form a bounded set) can be decomposed as a direct sum of an abelian Lie algebra and some number of simple ones. The structure of an abelian Lie algebra is mathematically uninteresting (since the Lie bracket is identically zero); the interest is in the simple summands. Hence the question arises: what are the simple Lie algebras of compact groups? It turns out that they mostly fall into four infinite families, the "classical Lie algebras" A_n, B_n, C_n and D_n, which have simple descriptions in terms of symmetries of Euclidean space. But there are also just five "exceptional Lie algebras" that do not fall into any of these families. E₈ is the largest of these.

Lie groups are classified according to their algebraic properties (simple, semisimple, solvable, nilpotent, abelian), their connectedness (connected or simply connected) and their compactness.

A first key result is the Levi decomposition, which says that every simply connected Lie group is the semidirect product of a solvable normal subgroup and a semisimple subgroup.

• Connected compact Lie groups are all known: they are finite central quotients of a product of copies of the circle group S¹ and simple compact Lie groups (which correspond to connected Dynkin diagrams).
• Any simply connected solvable Lie group is isomorphic to a closed subgroup of the group of invertible upper triangular matrices of some rank, and any finite-dimensional irreducible representation of such a group is 1-dimensional. Solvable groups are too messy to classify except in a few small dimensions.
• Any simply connected nilpotent Lie group is isomorphic to a closed subgroup of the group of invertible upper triangular matrices with 1s on the diagonal of some rank, and any finite-dimensional irreducible representation of such a group is 1-dimensional. Like solvable groups, nilpotent groups are too messy to classify except in a few small dimensions.
• Simple Lie groups are sometimes defined to be those that are simple as abstract groups, and sometimes defined to be connected Lie groups with a simple Lie algebra. For example, SL(2, R) is simple according to the second definition but not according to the first. They have all been classified (for either definition).
• Semisimple Lie groups are Lie groups whose Lie algebra is a product of simple Lie algebras. They are central extensions of products of simple Lie groups.

The identity component of any Lie group is an open normal subgroup, and the quotient group is a discrete group. The universal cover of any connected Lie group is a simply connected Lie group, and conversely any connected Lie group is a quotient of a simply connected Lie group by a discrete normal subgroup of the center. Any Lie group G can be decomposed into discrete, simple, and abelian groups in a canonical way as follows. Write

G_con for the connected component of the identity

G_sol for the largest connected normal solvable subgroup

G_nil for the largest connected normal nilpotent subgroup

so that we have a sequence of normal subgroups

1 ⊆ G_nil ⊆ G_sol ⊆ G_con ⊆ G.

Then

G/G_con is discrete

G_con/G_sol is a central extension of a product of simple connected Lie groups.

G_sol/G_nil is abelian. A connected abelian Lie group is isomorphic to a product of copies of R and the circle group S¹.

G_nil/1 is nilpotent, and therefore its ascending central series has all quotients abelian.

This can be used to reduce some problems about Lie groups (such as finding their unitary representations) to the same problems for connected simple groups and nilpotent and solvable subgroups of smaller dimension.

• The diffeomorphism group of a Lie group acts transitively on the Lie group
• Every Lie group is parallelizable, and hence an orientable manifold (there is a bundle isomorphism between its tangent bundle and the product of itself with the tangent space at the identity)

Infinite-dimensional Lie groups

Lie groups are often defined to be finite-dimensional, but there are many groups that resemble Lie groups, except for being infinite-dimensional. The simplest way to define infinite-dimensional Lie groups is to model them locally on Banach spaces (as opposed to Euclidean space in the finite-dimensional case), and in this case much of the basic theory is similar to that of finite-dimensional Lie groups. However this is inadequate for many applications, because many natural examples of infinite-dimensional Lie groups are not Banach manifolds. Instead one needs to define Lie groups modeled on more general locally convex topological vector spaces. In this case the relation between the Lie algebra and the Lie group becomes rather subtle, and several results about finite-dimensional Lie groups no longer hold.

The literature is not entirely uniform in its terminology as to exactly which properties of infinite-dimensional groups qualify the group for the prefix Lie in Lie group. On the Lie algebra side of affairs, things are simpler since the qualifying criteria for the prefix Lie in Lie algebra are purely algebraic. For example, an infinite-dimensional Lie algebra may or may not have a corresponding Lie group. That is, there may be a group corresponding to the Lie algebra, but it might not be nice enough to be called a Lie group, or the connection between the group and the Lie algebra might not be nice enough (for example, failure of the exponential map to be onto a neighborhood of the identity). It is the "nice enough" that is not universally defined.

Some of the examples that have been studied include:

• The group of diffeomorphisms of a manifold. Quite a lot is known about the group of diffeomorphisms of the circle. Its Lie algebra is (more or less) the Witt algebra, whose central extension the Virasoro algebra (see Virasoro algebra from Witt algebra for a derivation of this fact) is the symmetry algebra of two-dimensional conformal field theory. Diffeomorphism groups of compact manifolds of larger dimension are regular Fréchet Lie groups; very little about their structure is known.
• The diffeomorphism group of spacetime sometimes appears in attempts to quantize gravity.
• The group of smooth maps from a manifold to a finite-dimensional Lie group is an example of a gauge group (with operation of pointwise multiplication), and is used in quantum field theory and Donaldson theory. If the manifold is a circle these are called loop groups, and have central extensions whose Lie algebras are (more or less) Kac–Moody algebras.
• There are infinite-dimensional analogues of general linear groups, orthogonal groups, and so on. One important aspect is that these may have simpler topological properties: see for example Kuiper's theorem. In M-theory, for example, a 10-dimensional SU(N) gauge theory becomes an 11-dimensional theory when N becomes infinite.

Integrable system

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Integrable_system

In mathematics, integrability is a property of certain dynamical systems. While there are several distinct formal definitions, informally speaking, an integrable system is a dynamical system with sufficiently many conserved quantities, or first integrals, that its motion is confined to a submanifold of much smaller dimensionality than that of its phase space.

Three features are often referred to as characterizing integrable systems:

• the existence of a maximal set of conserved quantities (the usual defining property of complete integrability)
• the existence of algebraic invariants, having a basis in algebraic geometry (a property known sometimes as algebraic integrability)
• the explicit determination of solutions in an explicit functional form (not an intrinsic property, but something often referred to as solvability)

Integrable systems may be seen as very different in qualitative character from more generic dynamical systems, which are more typically chaotic systems. The latter generally have no conserved quantities, and are asymptotically intractable, since an arbitrarily small perturbation in initial conditions may lead to arbitrarily large deviations in their trajectories over a sufficiently large time.

Many systems studied in physics are completely integrable, in particular, in the Hamiltonian sense, the key example being multi-dimensional harmonic oscillators. Another standard example is planetary motion about either one fixed center (e.g., the sun) or two. Other elementary examples include the motion of a rigid body about its center of mass (the Euler top) and the motion of an axially symmetric rigid body about a point in its axis of symmetry (the Lagrange top).

In the late 1960s, it was realized that there are completely integrable systems in physics having an infinite number of degrees of freedom, such as some models of shallow water waves (Korteweg–de Vries equation), the Kerr effect in optical fibres, described by the nonlinear Schrödinger equation, and certain integrable many-body systems, such as the Toda lattice. The modern theory of integrable systems was revived with the numerical discovery of solitons by Martin Kruskal and Norman Zabusky in 1965, which led to the inverse scattering transform method in 1967.

In the special case of Hamiltonian systems, if there are enough independent Poisson commuting first integrals for the flow parameters to be able to serve as a coordinate system on the invariant level sets (the leaves of the Lagrangian foliation), and if the flows are complete and the energy level set is compact, this implies the Liouville–Arnold theorem; i.e., the existence of action-angle variables. General dynamical systems have no such conserved quantities; in the case of autonomous Hamiltonian systems, the energy is generally the only one, and on the energy level sets, the flows are typically chaotic.

A key ingredient in characterizing integrable systems is the Frobenius theorem, which states that a system is Frobenius integrable (i.e., is generated by an integrable distribution) if, locally, it has a foliation by maximal integral manifolds. But integrability, in the sense of dynamical systems, is a global property, not a local one, since it requires that the foliation be a regular one, with the leaves embedded submanifolds.

Integrability does not necessarily imply that generic solutions can be explicitly expressed in terms of some known set of special functions; it is an intrinsic property of the geometry and topology of the system, and the nature of the dynamics.

General dynamical systems

In the context of differentiable dynamical systems, the notion of integrability refers to the existence of invariant, regular foliations; i.e., ones whose leaves are embedded submanifolds of the smallest possible dimension that are invariant under the flow. There is thus a variable notion of the degree of integrability, depending on the dimension of the leaves of the invariant foliation. This concept has a refinement in the case of Hamiltonian systems, known as complete integrability in the sense of Liouville (see below), which is what is most frequently referred to in this context.

An extension of the notion of integrability is also applicable to discrete systems such as lattices. This definition can be adapted to describe evolution equations that either are systems of differential equations or finite difference equations.

The distinction between integrable and nonintegrable dynamical systems has the qualitative implication of regular motion vs. chaotic motion and hence is an intrinsic property, not just a matter of whether a system can be explicitly integrated in an exact form.

Hamiltonian systems and Liouville integrability

In the special setting of Hamiltonian systems, we have the notion of integrability in the Liouville sense. (See the Liouville–Arnold theorem.) Liouville integrability means that there exists a regular foliation of the phase space by invariant manifolds such that the Hamiltonian vector fields associated with the invariants of the foliation span the tangent distribution. Another way to state this is that there exists a maximal set of functionally independent Poisson commuting invariants (i.e., independent functions on the phase space whose Poisson brackets with the Hamiltonian of the system, and with each other, vanish).

In finite dimensions, if the phase space is symplectic (i.e., the center of the Poisson algebra consists only of constants), it must have even dimension ${\displaystyle 2n,}$ and the maximal number of independent Poisson commuting invariants (including the Hamiltonian itself) is ${\displaystyle n}$. The leaves of the foliation are totally isotropic with respect to the symplectic form and such a maximal isotropic foliation is called Lagrangian. All autonomous Hamiltonian systems (i.e. those for which the Hamiltonian and Poisson brackets are not explicitly time-dependent) have at least one invariant; namely, the Hamiltonian itself, whose value along the flow is the energy. If the energy level sets are compact, the leaves of the Lagrangian foliation are tori, and the natural linear coordinates on these are called "angle" variables. The cycles of the canonical ${\displaystyle 1}$-form are called the action variables, and the resulting canonical coordinates are called action-angle variables (see below).

There is also a distinction between complete integrability, in the Liouville sense, and partial integrability, as well as a notion of superintegrability and maximal superintegrability. Essentially, these distinctions correspond to the dimensions of the leaves of the foliation. When the number of independent Poisson commuting invariants is less than maximal (but, in the case of autonomous systems, more than one), we say the system is partially integrable. When there exist further functionally independent invariants, beyond the maximal number that can be Poisson commuting, and hence the dimension of the leaves of the invariant foliation is less than n, we say the system is superintegrable. If there is a regular foliation with one-dimensional leaves (curves), this is called maximally superintegrable.

Action-angle variables

When a finite-dimensional Hamiltonian system is completely integrable in the Liouville sense, and the energy level sets are compact, the flows are complete, and the leaves of the invariant foliation are tori. There then exist, as mentioned above, special sets of canonical coordinates on the phase space known as action-angle variables, such that the invariant tori are the joint level sets of the action variables. These thus provide a complete set of invariants of the Hamiltonian flow (constants of motion), and the angle variables are the natural periodic coordinates on the tori. The motion on the invariant tori, expressed in terms of these canonical coordinates, is linear in the angle variables.

The Hamilton–Jacobi approach

In canonical transformation theory, there is the Hamilton–Jacobi method, in which solutions to Hamilton's equations are sought by first finding a complete solution of the associated Hamilton–Jacobi equation. In classical terminology, this is described as determining a transformation to a canonical set of coordinates consisting of completely ignorable variables; i.e., those in which there is no dependence of the Hamiltonian on a complete set of canonical "position" coordinates, and hence the corresponding canonically conjugate momenta are all conserved quantities. In the case of compact energy level sets, this is the first step towards determining the action-angle variables. In the general theory of partial differential equations of Hamilton–Jacobi type, a complete solution (i.e. one that depends on n independent constants of integration, where n is the dimension of the configuration space), exists in very general cases, but only in the local sense. Therefore, the existence of a complete solution of the Hamilton–Jacobi equation is by no means a characterization of complete integrability in the Liouville sense. Most cases that can be "explicitly integrated" involve a complete separation of variables, in which the separation constants provide the complete set of integration constants that are required. Only when these constants can be reinterpreted, within the full phase space setting, as the values of a complete set of Poisson commuting functions restricted to the leaves of a Lagrangian foliation, can the system be regarded as completely integrable in the Liouville sense.

Solitons and inverse spectral methods

A resurgence of interest in classical integrable systems came with the discovery, in the late 1960s, that solitons, which are strongly stable, localized solutions of partial differential equations like the Korteweg–de Vries equation (which describes 1-dimensional non-dissipative fluid dynamics in shallow basins), could be understood by viewing these equations as infinite-dimensional integrable Hamiltonian systems. Their study leads to a very fruitful approach for "integrating" such systems, the inverse scattering transform and more general inverse spectral methods (often reducible to Riemann–Hilbert problems), which generalize local linear methods like Fourier analysis to nonlocal linearization, through the solution of associated integral equations.

The basic idea of this method is to introduce a linear operator that is determined by the position in phase space and which evolves under the dynamics of the system in question in such a way that its "spectrum" (in a suitably generalized sense) is invariant under the evolution, cf. Lax pair. This provides, in certain cases, enough invariants, or "integrals of motion" to make the system completely integrable. In the case of systems having an infinite number of degrees of freedom, such as the KdV equation, this is not sufficient to make precise the property of Liouville integrability. However, for suitably defined boundary conditions, the spectral transform can, in fact, be interpreted as a transformation to completely ignorable coordinates, in which the conserved quantities form half of a doubly infinite set of canonical coordinates, and the flow linearizes in these. In some cases, this may even be seen as a transformation to action-angle variables, although typically only a finite number of the "position" variables are actually angle coordinates, and the rest are noncompact.

Hirota bilinear equations and τ-functions

Another viewpoint that arose in the modern theory of integrable systems originated in a calculational approach pioneered by Ryogo Hirota, which involved replacing the original nonlinear dynamical system with a bilinear system of constant coefficient equations for an auxiliary quantity, which later came to be known as the τ-function. These are now referred to as the Hirota equations. Although originally appearing just as a calculational device, without any clear relation to the inverse scattering approach, or the Hamiltonian structure, this nevertheless gave a very direct method from which important classes of solutions such as solitons could be derived.

Subsequently, this was interpreted by Mikio Sato and his students, at first for the case of integrable hierarchies of PDEs, such as the Kadomtsev–Petviashvili hierarchy, but then for much more general classes of integrable hierarchies, as a sort of universal phase space approach, in which, typically, the commuting dynamics were viewed simply as determined by a fixed (finite or infinite) abelian group action on a (finite or infinite) Grassmann manifold. The τ-function was viewed as the determinant of a projection operator from elements of the group orbit to some origin within the Grassmannian, and the Hirota equations as expressing the Plücker relations, characterizing the Plücker embedding of the Grassmannian in the projectivization of a suitably defined (infinite) exterior space, viewed as a fermionic Fock space.

Quantum integrable systems

There is also a notion of quantum integrable systems.

In the quantum setting, functions on phase space must be replaced by self-adjoint operators on a Hilbert space, and the notion of Poisson commuting functions replaced by commuting operators. The notion of conservation laws must be specialized to local conservation laws. Every Hamiltonian has an infinite set of conserved quantities given by projectors to its energy eigenstates. However, this does not imply any special dynamical structure.

To explain quantum integrability, it is helpful to consider the free particle setting. Here all dynamics are one-body reducible. A quantum system is said to be integrable if the dynamics are two-body reducible. The Yang–Baxter equation is a consequence of this reducibility and leads to trace identities which provide an infinite set of conserved quantities. All of these ideas are incorporated into the quantum inverse scattering method where the algebraic Bethe ansatz can be used to obtain explicit solutions. Examples of quantum integrable models are the Lieb–Liniger model, the Hubbard model and several variations on the Heisenberg model. Some other types of quantum integrability are known in explicitly time-dependent quantum problems, such as the driven Tavis-Cummings model.

Exactly solvable models

In physics, completely integrable systems, especially in the infinite-dimensional setting, are often referred to as exactly solvable models. This obscures the distinction between integrability, in the Hamiltonian sense, and the more general dynamical systems sense.

There are also exactly solvable models in statistical mechanics, which are more closely related to quantum integrable systems than classical ones. Two closely related methods: the Bethe ansatz approach, in its modern sense, based on the Yang–Baxter equations and the quantum inverse scattering method, provide quantum analogs of the inverse spectral methods. These are equally important in the study of solvable models in statistical mechanics.

An imprecise notion of "exact solvability" as meaning: "The solutions can be expressed explicitly in terms of some previously known functions" is also sometimes used, as though this were an intrinsic property of the system itself, rather than the purely calculational feature that we happen to have some "known" functions available, in terms of which the solutions may be expressed. This notion has no intrinsic meaning, since what is meant by "known" functions very often is defined precisely by the fact that they satisfy certain given equations, and the list of such "known functions" is constantly growing. Although such a characterization of "integrability" has no intrinsic validity, it often implies the sort of regularity that is to be expected in integrable systems.

List of some well-known integrable systems

Classical mechanical systems

• Calogero–Moser–Sutherland model
• Central force motion (exact solutions of classical central-force problems)
• Geodesic motion on ellipsoids
• Harmonic oscillator
• Integrable Clebsch and Steklov systems in fluids
• Lagrange, Euler, and Kovalevskaya tops
• Neumann oscillator
• Two center Newtonian gravitational motion

Integrable lattice models

• Ablowitz–Ladik lattice
• Toda lattice
• Volterra lattice

Integrable systems in 1 + 1 dimensions

• AKNS system
• Benjamin–Ono equation
• Boussinesq equation (water waves)
• Camassa–Holm equation
• Classical Heisenberg ferromagnet model (spin chain)
• Degasperis–Procesi equation
• Dym equation
• Garnier integrable system
• Kaup–Kupershmidt equation
• Krichever–Novikov equation
• Korteweg–de Vries equation
• Landau–Lifshitz equation (continuous spin field)
• Nonlinear Schrödinger equation
• Nonlinear sigma models
• Sine–Gordon equation
• Thirring model
• Three-wave equation

Integrable PDEs in 2 + 1 dimensions

• Davey–Stewartson equation
• Ishimori equation
• Kadomtsev–Petviashvili equation
• Novikov–Veselov equation

Integrable PDEs in 3 + 1 dimensions

• The Belinski–Zakharov transform generates a Lax pair for the Einstein field equations; general solutions are termed gravitational solitons, of which the Schwarzschild metric, the Kerr metric and some gravitational wave solutions are examples.

Exactly solvable statistical lattice models

• 8-vertex model
• Gaudin model
• Ising model in 1- and 2-dimensions
• Ice-type model of Lieb
• Quantum Heisenberg model

Spacetime symmetries

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Spacetime symmetries are features of spacetime that can be described as exhibiting some form of symmetry. The role of symmetry in physics is important in simplifying solutions to many problems. Spacetime symmetries are used in the study of exact solutions of Einstein's field equations of general relativity. Spacetime symmetries are distinguished from internal symmetries.

Physical motivation

Physical problems are often investigated and solved by noticing features which have some form of symmetry. For example, in the Schwarzschild solution, the role of spherical symmetry is important in deriving the Schwarzschild solution and deducing the physical consequences of this symmetry (such as the nonexistence of gravitational radiation in a spherically pulsating star). In cosmological problems, symmetry plays a role in the cosmological principle, which restricts the type of universes that are consistent with large-scale observations (e.g. the Friedmann–Lemaître–Robertson–Walker (FLRW) metric). Symmetries usually require some form of preserving property, the most important of which in general relativity include the following:

• preserving geodesics of the spacetime
• preserving the metric tensor
• preserving the curvature tensor

These and other symmetries will be discussed below in more detail. This preservation property which symmetries usually possess (alluded to above) can be used to motivate a useful definition of these symmetries themselves.

Mathematical definition

A rigorous definition of symmetries in general relativity has been given by Hall (2004). In this approach, the idea is to use (smooth) vector fields whose local flow diffeomorphisms preserve some property of the spacetime. (Note that one should emphasize in one's thinking this is a diffeomorphism—a transformation on a differential element. The implication is that the behavior of objects with extent may not be as manifestly symmetric.) This preserving property of the diffeomorphisms is made precise as follows. A smooth vector field X on a spacetime M is said to preserve a smooth tensor T on M (or T is invariant under X) if, for each smooth local flow diffeomorphism ϕ_t associated with X, the tensors T and ϕ^∗
_t(T) are equal on the domain of ϕ_t. This statement is equivalent to the more usable condition that the Lie derivative of the tensor under the vector field vanishes: $${\displaystyle {\mathcal{L}}_{X}T=0}$$ on M. This has the consequence that, given any two points p and q on M, the coordinates of T in a coordinate system around p are equal to the coordinates of T in a coordinate system around q. A symmetry on the spacetime is a smooth vector field whose local flow diffeomorphisms preserve some (usually geometrical) feature of the spacetime. The (geometrical) feature may refer to specific tensors (such as the metric, or the energy–momentum tensor) or to other aspects of the spacetime such as its geodesic structure. The vector fields are sometimes referred to as collineations, symmetry vector fields or just symmetries. The set of all symmetry vector fields on M forms a Lie algebra under the Lie bracket operation as can be seen from the identity: $${\displaystyle {\mathcal{L}}_{[X,Y]}T={\mathcal{L}}_X({\mathcal{L}}_{Y}T)-{\mathcal{L}}_Y({\mathcal{L}}_{X}T)}$$ the term on the right usually being written, with an abuse of notation, as $${\displaystyle [{\mathcal{L}}_X,{\mathcal{L}}_Y]T.}$$

Killing symmetry

Main article: Killing vector field

A Killing vector field is one of the most important types of symmetries and is defined to be a smooth vector field X that preserves the metric tensor g: $${\displaystyle {\mathcal{L}}_{X}g=0.}$$

This is usually written in the expanded form as: $${\displaystyle X_{a;b}+X_{b;a}=0.}$$

Killing vector fields find extensive applications (including in classical mechanics) and are related to conservation laws.

Homothetic symmetry

Main article: Homothetic vector field

A homothetic vector field is one which satisfies: $${\displaystyle {\mathcal{L}}_{X}g=2cg.}$$ where c is a real constant. Homothetic vector fields find application in the study of singularities in general relativity.

Affine symmetry

Main article: Affine vector field

An affine vector field is one that satisfies: $${\displaystyle ({\mathcal{L}}_{X}g{)}_{ab;c}=0}$$

An affine vector field preserves geodesics and preserves the affine parameter.

The above three vector field types are special cases of projective vector fields which preserve geodesics without necessarily preserving the affine parameter.

Conformal symmetry

Main article: Conformal vector field

A conformal vector field is one which satisfies: $${\displaystyle {\mathcal{L}}_{X}g=\varphi g}$$ where ϕ is a smooth real-valued function on M.

Curvature symmetry

Main article: Curvature collineation

A curvature collineation is a vector field which preserves the Riemann tensor: $${\displaystyle {\mathcal{L}}_X{R^a}_{bcd}=0}$$

where R^a_bcd are the components of the Riemann tensor. The set of all smooth curvature collineations forms a Lie algebra under the Lie bracket operation (if the smoothness condition is dropped, the set of all curvature collineations need not form a Lie algebra). The Lie algebra is denoted by CC(M) and may be infinite-dimensional. Every affine vector field is a curvature collineation.

Matter symmetry

Main article: Matter collineation

A less well-known form of symmetry concerns vector fields that preserve the energy–momentum tensor. These are variously referred to as matter collineations or matter symmetries and are defined by: $${\displaystyle {\mathcal{L}}_{X}T=0}$$ where T is the covariant energy–momentum tensor. The intimate relation between geometry and physics may be highlighted here, as the vector field X is regarded as preserving certain physical quantities along the flow lines of X, this being true for any two observers. In connection with this, it may be shown that every Killing vector field is a matter collineation (by the Einstein field equations, with or without cosmological constant). Thus, given a solution of the EFE, a vector field that preserves the metric necessarily preserves the corresponding energy–momentum tensor. When the energy–momentum tensor represents a perfect fluid, every Killing vector field preserves the energy density, pressure and the fluid flow vector field. When the energy–momentum tensor represents an electromagnetic field, a Killing vector field does not necessarily preserve the electric and magnetic fields.

Local and global symmetries

Main articles: Local symmetry and Global symmetry

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Applications

As mentioned at the start of this article, the main application of these symmetries occur in general relativity, where solutions of Einstein's equations may be classified by imposing some certain symmetries on the spacetime.

Spacetime classifications

Classifying solutions of the EFE constitutes a large part of general relativity research. Various approaches to classifying spacetimes, including using the Segre classification of the energy–momentum tensor or the Petrov classification of the Weyl tensor have been studied extensively by many researchers, most notably Stephani et al. (2003). They also classify spacetimes using symmetry vector fields (especially Killing and homothetic symmetries). For example, Killing vector fields may be used to classify spacetimes, as there is a limit to the number of global, smooth Killing vector fields that a spacetime may possess (the maximum being ten for four-dimensional spacetimes). Generally speaking, the higher the dimension of the algebra of symmetry vector fields on a spacetime, the more symmetry the spacetime admits. For example, the Schwarzschild solution has a Killing algebra of dimension four (three spatial rotational vector fields and a time translation), whereas the Friedmann–Lemaître–Robertson–Walker metric (excluding the Einstein static subcase) has a Killing algebra of dimension six (three translations and three rotations). The Einstein static metric has a Killing algebra of dimension seven (the previous six plus a time translation).

The assumption of a spacetime admitting a certain symmetry vector field can place restrictions on the spacetime.

List of symmetric spacetimes

See also: List of spacetimes

The following spacetimes have their own distinct articles in Wikipedia:

• Static spacetime
• Stationary spacetime
• Spherically symmetric spacetime
• Minkowski Space
• de Sitter space
• Anti-de Sitter space

Delayed-choice quantum eraser

From Wikipedia, the free encyclopedia
https://en.wikipedia.org/wiki/Delayed-choice_quantum_eraser

A delayed-choice quantum eraser experiment, first performed by Yoon-Ho Kim, R. Yu, S. P. Kulik, Y. H. Shih and Marlan O. Scully, and reported in early 1998, is an elaboration on the quantum eraser experiment that incorporates concepts considered in John Archibald Wheeler's delayed-choice experiment. The experiment was designed to investigate peculiar consequences of the well-known double-slit experiment in quantum mechanics, as well as the consequences of quantum entanglement.

The delayed-choice quantum eraser experiment investigates a paradox. If a photon manifests itself as though it had come by a single path to the detector, then "common sense" (which Wheeler and others challenge) says that it must have entered the double-slit device as a particle. If a photon manifests itself as though it had come by two indistinguishable paths, then it must have entered the double-slit device as a wave. Accordingly, if the experimental apparatus is changed while the photon is in mid‑flight, the photon may have to revise its prior "commitment" as to whether to be a wave or a particle. Wheeler pointed out that when these assumptions are applied to a device of interstellar dimensions, a last-minute decision made on Earth on how to observe a photon could alter a situation established millions or even billions of years earlier.

While delayed-choice experiments might seem to allow measurements made in the present to alter events that occurred in the past, this conclusion requires assuming a non-standard view of quantum mechanics. If a photon in flight is instead interpreted as being in a so-called "superposition of states"—that is, if it is allowed the potentiality of manifesting as a particle or wave, but during its time in flight is neither—then there is no causation paradox. This notion of superposition reflects the standard interpretation of quantum mechanics.

Introduction

In the basic double-slit experiment, a beam of light (usually from a laser) is directed perpendicularly towards a wall pierced by two parallel slit apertures. If a detection screen (anything from a sheet of white paper to a CCD) is put on the other side of the double-slit wall (far enough for light from both slits to overlap), a pattern of light and dark fringes will be observed, a pattern that is called an interference pattern. Other atomic-scale entities such as electrons are found to exhibit the same behavior when fired toward a double slit. By decreasing the brightness of the source sufficiently, individual particles that form the interference pattern are detectable. The emergence of an interference pattern suggests that each particle passing through the slits interferes with itself, and that therefore in some sense the particles are going through both slits at once. This is an idea that contradicts our everyday experience of discrete objects.

A well-known thought experiment, which played a vital role in the history of quantum mechanics (for example, see the discussion on Einstein's version of this experiment), demonstrated that if particle detectors are positioned at the slits, showing through which slit a photon goes, the interference pattern will disappear. This which-way experiment illustrates the complementarity principle that photons can behave as either particles or as waves, but cannot be simultaneously observed to be both a particle and a wave. However, technically feasible realizations of this experiment were not proposed until the 1970s.

Which-path information and the visibility of interference fringes are complementary quantities, meaning that information about a photon's path can be observed, or interference fringes can be observed, but they cannot both be observed in the same trial. In the double-slit experiment, conventional wisdom held that observing the particles' path inevitably disturbed them enough to destroy the interference pattern as a result of the Heisenberg uncertainty principle.

In 1982, Scully and Drühl pointed out a workaround alternative to this interpretation. They proposed to save the information about which slit the photon went through - or, in their setup, from which atom the photon was re-emitted - in the excited state of that atom. At this point the which-path information is known and no interference is observed. However, one can "erase" this information by making the atom to emit another photon and fall to the ground state. That on its own will not bring the interference pattern back, the which-path information can still be extracted from an appropriate measurement of the second photon. However, if the second photon is measured at a place where it could get to equally likely from any of the atoms, that successfully "erases" the which-path information. The original photon would now show the interference pattern (the position of its fringes depends on where exactly the second photon was observed, so that in the total statistics they average out and no fringes are seen). Since 1982, multiple experiments have demonstrated the validity of this so-called quantum "eraser".

A simple quantum-eraser experiment

A simple version of the quantum eraser can be described as follows: Rather than splitting one photon or its probability wave between two slits, the photon is subjected to a beam splitter. If one thinks in terms of a stream of photons being randomly directed by such a beam splitter to go down two paths that are kept from interaction, it would seem that no photon can then interfere with any other or with itself.

If the rate of photon production is reduced so that only one photon enters the apparatus at any one time, it becomes impossible to understand the photon as only moving through one path, because when the path outputs are redirected so that they coincide on a common detector or detectors, interference phenomena appear. This is similar to envisioning one photon in a two-slit apparatus: even though it is one photon, it still somehow interacts with both slits.

Figure 1. Experiment that shows delayed determination of photon path

In the two diagrams in Fig. 1, photons are emitted one at a time from a laser symbolized by a yellow star. They pass through a 50% beam splitter (green block) that reflects or transmits 1/2 of the photons. The reflected or transmitted photons travel along two possible paths depicted by the red or blue lines.

In the top diagram, it seems as though the trajectories of the photons are known: If a photon emerges from the top of the apparatus, it seems as though it had to have come by way of the blue path, and if it emerges from the side of the apparatus, it seems as though it had to have come by way of the red path. However, it is important to keep in mind that the photon is in a superposition of the paths until it is detected. The assumption above—that it 'had to have come by way of' either path—is a form of the 'separation fallacy'.

In the bottom diagram, a second beam splitter is introduced at the top right. It recombines the beams corresponding to the red and blue paths. By introducing the second beam splitter, the usual way of thinking is that the path information has been "erased." However, we have to be careful, because the photon cannot be assumed to have 'really' gone along one or the other path. Recombining the beams results in interference phenomena at detection screens positioned just beyond each exit port. What issues to the right side displays reinforcement, and what issues toward the top displays cancellation. It is important to keep in mind however that the illustrated interferometer effects apply only to a single photon in a pure state. When dealing with a pair of entangled photons, the photon encountering the interferometer will be in a mixed state, and there will be no visible interference pattern without coincidence counting to select appropriate subsets of the data.

Delayed choice

Elementary precursors to current quantum-eraser experiments such as the "simple quantum eraser" described above have straightforward classical-wave explanations. Indeed, it could be argued that there is nothing particularly quantum about this experiment. Nevertheless, Jordan has argued on the basis of the correspondence principle, that despite the existence of classical explanations, first-order interference experiments such as the above can be interpreted as true quantum erasers.

These precursors use single-photon interference. Versions of the quantum eraser using entangled photons, however, are intrinsically non-classical. Because of that, in order to avoid any possible ambiguity concerning the quantum versus classical interpretation, most experimenters have opted to use nonclassical entangled-photon light sources to demonstrate quantum erasers with no classical analog.

Furthermore, the use of entangled photons enables the design and implementation of versions of the quantum eraser that are impossible to achieve with single-photon interference, such as the delayed-choice quantum eraser, which is the topic of this article.

The experiment of Kim et al. (1999)

Figure 2. Setup of the delayed-choice quantum-eraser experiment of Kim et al. Detector D₀ is movable

The experimental setup, described in detail in Kim et al., is illustrated in Fig 2. An argon laser generates individual 351.1 nm photons that pass through a double-slit apparatus (vertical black line in the upper left corner of the diagram).

An individual photon goes through one (or both) of the two slits. In the illustration, the photon paths are color-coded as red or light blue lines to indicate which slit the photon came through (red indicates slit A, light blue indicates slit B).

So far, the experiment is like a conventional two-slit experiment. However, after the slits, spontaneous parametric down-conversion (SPDC) is used to prepare an entangled two-photon state. This is done by a nonlinear optical crystal BBO (beta barium borate) that converts the photon (from either slit) into two identical, orthogonally polarized entangled photons with 1/2 the frequency of the original photon. The paths followed by these orthogonally polarized photons are caused to diverge by the Glan–Thompson prism.

One of these 702.2 nm photons, referred to as the "signal" photon (look at the red and light-blue lines going upwards from the Glan–Thompson prism) continues to the target detector called D₀. During an experiment, detector D₀ is scanned along its x axis, its motions controlled by a step motor. A plot of "signal" photon counts detected by D₀ versus x can be examined to discover whether the cumulative signal forms an interference pattern.

The other entangled photon, referred to as the "idler" photon (look at the red and light-blue lines going downwards from the Glan–Thompson prism), is deflected by prism PS that sends it along divergent paths depending on whether it came from slit A or slit B.

Somewhat beyond the path split, the idler photons encounter beam splitters BS_a, BS_b, and BS_c that each have a 50% chance of allowing the idler photon to pass through and a 50% chance of causing it to be reflected. M_a and M_b are mirrors.

Figure 3. x axis: position of D₀. y axis: joint detection rates between D₀ and D₁, D₂, D₃, D₄ (R₀₁, R₀₂, R₀₃, R₀₄). R₀₄ is not provided in the Kim article and is supplied according to their verbal description.

Figure 4. Simulated recordings of photons jointly detected between D₀ and D₁, D₂, D₃, D₄ (R₀₁, R₀₂, R₀₃, R₀₄)

The beam splitters and mirrors direct the idler photons towards detectors labeled D₁, D₂, D₃ and D₄. Note that:

• If an idler photon is recorded at detector D₃, it can only have come from slit B.
• If an idler photon is recorded at detector D₄, it can only have come from slit A.
• If an idler photon is detected at detector D₁ or D₂, it might have come from slit A or slit B.
• The optical path length measured from slit to D₁, D₂, D₃, and D₄ is 2.5 m longer than the optical path length from slit to D₀. This means that any information that one can learn from an idler photon must be approximately 8 ns later than what one can learn from its entangled signal photon.

Detection of the idler photon by D₃ or D₄ provides delayed "which-path information" indicating whether the signal photon with which it is entangled had gone through slit A or B. On the other hand, detection of the idler photon by D₁ or D₂ provides a delayed indication that such information is not available for its entangled signal photon. Insofar as which-path information had earlier potentially been available from the idler photon, it is said that the information has been subjected to a "delayed erasure".

By using a coincidence counter, the experimenters were able to isolate the entangled signal from photo-noise, recording only events where both signal and idler photons were detected (after compensating for the 8 ns delay). Refer to Figs 3 and 4.

• When the experimenters looked at the signal photons whose entangled idlers were detected at D₁ or D₂, they detected interference patterns.
• However, when they looked at the signal photons whose entangled idlers were detected at D₃ or D₄, they detected simple diffraction patterns with no interference.

Significance

This result is similar to that of the double-slit experiment since interference is observed when it is extracted according to phase value (R₀₁ or R₀₂). Note that the phase cannot be measured if the photon's path (the slit through which it passes) is known.

Figure 5. Distribution of signal photons at D₀ can be compared with distribution of bulbs on digital billboard. When all the bulbs are lit, billboard does not reveal any pattern of image, which can be "recovered" only by switching off some bulbs. Likewise interference pattern or no-interference pattern among signal photons at D₀ can be recovered only after "switching off" (or ignoring) some signal photons and which signal photons should be ignored to recover pattern, this information can be gained only by looking at corresponding entangled idler photons in detectors D₁ to D₄.

However, what makes this experiment possibly astonishing is that, unlike in the classic double-slit experiment, the choice of whether to preserve or erase the which-path information of the idler was not made until 8 ns after the position of the signal photon had already been measured by D₀.

Detection of signal photons at D₀ does not directly yield any which-path information. Detection of idler photons at D₃ or D₄, which provide which-path information, means that no interference pattern can be observed in the jointly detected subset of signal photons at D₀. Likewise, detection of idler photons at D₁ or D₂, which do not provide which-path information, means that interference patterns can be observed in the jointly detected subset of signal photons at D₀.

In other words, even though an idler photon is not observed until long after its entangled signal photon arrives at D₀ due to the shorter optical path for the latter, interference at D₀ is determined by whether a signal photon's entangled idler photon is detected at a detector that preserves its which-path information (D₃ or D₄), or at a detector that erases its which-path information (D₁ or D₂).

Some have interpreted this result to mean that the delayed choice to observe or not observe the path of the idler photon changes the outcome of an event in the past. Note in particular that an interference pattern may only be pulled out for observation after the idlers have been detected (i.e., at D₁ or D₂).

The total pattern of all signal photons at D₀, whose entangled idlers went to multiple different detectors, will never show interference regardless of what happens to the idler photons. One can get an idea of how this works by looking at the graphs of R₀₁, R₀₂, R₀₃, and R₀₄, and observing that the peaks of R₀₁ line up with the troughs of R₀₂ (i.e. a π phase shift exists between the two interference fringes). R₀₃ shows a single maximum, and R₀₄, which is experimentally identical to R₀₃ will show equivalent results. The entangled photons, as filtered with the help of the coincidence counter, are simulated in Fig. 5 to give a visual impression of the evidence available from the experiment. In D₀, the sum of all the correlated counts will not show interference. If all the photons that arrive at D₀ were to be plotted on one graph, one would see only a bright central band.

Implications

Retrocausality

Delayed-choice experiments raise questions about the causal connections between the events. If events at D₁, D₂, D₃, D₄ determine outcomes at D₀, then the effects might seem to precede their causes in time.

Consensus: no retrocausality

However, the interference pattern can only be seen retroactively once the idler photons have been detected and the detection information used to select subsets of signal photons.

Moreover, it's observed that the apparent retroactive action vanishes if the effects of observations on the state of the entangled signal and idler photons are considered in their historic order. Specifically, in the case when detection/deletion of which-way information happens before the detection on D₀, the standard simplistic explanation says "The detector D_i, at which the idler photon is detected, determines the probability distribution at D₀ for the signal photon". Similarly, in the case when D₀ precedes detection of the idler photon, the following description is just as accurate: "The position at D₀ of the detected signal photon determines the probabilities for the idler photon to hit either of D₁, D₂, D₃ or D₄". These are just equivalent ways of formulating the correlations of entangled photons' observables in an intuitive causal way, so one may choose any of those (in particular, that one where the cause precedes the consequence and no retrograde action appears in the explanation).

The total pattern of signal photons at the primary detector never shows interference (see Fig. 5), so it is not possible to deduce what will happen to the idler photons by observing the signal photons alone. In a paper by Johannes Fankhauser, it is shown that the delayed choice quantum eraser experiment resembles a Bell-type scenario in which the paradox's resolution is rather trivial, and so there really is no mystery. Moreover, it gives a detailed account of the experiment in the de Broglie-Bohm picture with definite trajectories arriving at the conclusion that there is no "backwards in time influence" present. The delayed-choice quantum eraser does not communicate information in a retro-causal manner because it takes another signal, one which must arrive by a process that can go no faster than the speed of light, to sort the superimposed data in the signal photons into four streams that reflect the states of the idler photons at their four distinct detection screens.

A theorem proven by Phillippe Eberhard shows that if the accepted equations of relativistic quantum field theory are correct, faster than light communications is impossible.

Other delayed-choice quantum-eraser experiments

Many refinements and extensions of Kim et al. delayed-choice quantum eraser have been performed or proposed. Only a small sampling of reports and proposals are given here:

Scarcelli et al. (2007) reported on a delayed-choice quantum-eraser experiment based on a two-photon imaging scheme. After detecting a photon passed through a double-slit, a random delayed choice was made to erase or not erase the which-path information by the measurement of its distant entangled twin; the particle-like and wave-like behavior of the photon were then recorded simultaneously and respectively by only one set of joint detectors.

Peruzzo et al. (2012) have reported on a quantum delayed-choice experiment based on a quantum-controlled beam splitter, in which particle and wave behaviors were investigated simultaneously. The quantum nature of the photon's behavior was tested with a Bell inequality, which replaced the delayed choice of the observer.

Rezai et al. (2018) have combined the Hong-Ou-Mandel interference with a delayed choice quantum eraser. They impose two incompatible photons onto a beam-splitter, such that no interference pattern could be observed. When the output ports are monitored in an integrated fashion (i.e. counting all the clicks), no interference occurs. Only when the outcoming photons are polarization analysed and the right subset is selected, quantum interference in the form of a Hong-Ou-Mandel dip occurs.

The construction of solid-state electronic Mach–Zehnder interferometers (MZI) has led to proposals to use them in electronic versions of quantum-eraser experiments. This would be achieved by Coulomb coupling to a second electronic MZI acting as a detector.

Entangled pairs of neutral kaons have also been examined and found suitable for investigations using quantum marking and quantum-erasure techniques.

A quantum eraser has been proposed using a modified Stern-Gerlach setup. In this proposal, no coincident counting is required, and quantum erasure is accomplished by applying an additional Stern-Gerlach magnetic field.