Fubini-Study metric

Encyclopedia

In mathematics

, the Fubini–Study metric is a Kähler metric on projective Hilbert space

, that is, complex projective space

CP

was originally described in 1904 and 1905 by Guido Fubini

and Eduard Study

.

A Hermitian form in (the vector space) C

. The particular normalization on the metric depends on the application. In Riemannian geometry

, one uses a normalization so that the Fubini-Study metric simply relates to the standard metric on the (2n+1)-sphere. In algebraic geometry

, one uses a normalization making CP

construction of complex projective space

.

Specifically, one may define CP

relating all complex multiples of each point together. This agrees with the quotient by the diagonal group action

of the multiplicative group C

This quotient realizes C

over the base space CP

over CP

of the point.

Furthermore, one may realize this quotient in two steps: since multiplication by a nonzero complex scalar z = R e

where step (a) is a quotient by the dilation Z ~ RZ for R ∈ R

The result of the quotient in (a) is the real hypersphere S

(or metric space

in general), care must be taken to ensure that the quotient space is endowed with a metric that is well-defined. For instance, if a group G acts on a Riemannian manifold (X,g), then in order for the orbit space X/G to possess an induced metric, must be constant along G-orbits in the sense that for any element h ∈ G and pair of vector fields we must have g(Xh,Yh) = g(X,Y).

The standard Hermitian metric on C

whose realification is the standard Euclidean metric on R

The Fubini-Study metric is the metric induced on the quotient CP

provided Z

where |z|

Note that each matrix element is unitary-invariant: the diagonal action will leave this matrix unchanged.

Accordingly, the line element is given by

In this last expression, the summation convention is used to sum over Latin indices i,j that range from 1 to n.

Here the summation convention is used to sum over Greek indices α β ranging from 0 to n, and in the last equality the standard notation for the skew part of a tensor is used:

Now, this expression for ds

The Kähler form of this metric is, up to an overall constant normalization,

the pullback of which is clearly independent of the choice of holomorphic section. The quantity log|Z|

. This leads to the "special" Hopf fibration S

4) on S

Namely, if z = x + iy is the standard affine coordinate chart on the Riemann sphere

CP

where is the round metric on the unit 2-sphere. Here φ, θ are "mathematician's spherical coordinates" on S

where is an orthonormal basis of the 2-plane σ, J : TCP

on CP

A consequence of this formula is that the sectional curvature satisfies for all 2-planes . The maximum sectional curvature (4) is attained at a holomorphic 2-plane — one for which J(σ) ⊂ σ — while the minimum sectional curvature (1) is attained at a 2-plane for which J(σ) is orthogonal to σ. For this reason, the Fubini-Study metric is often said to have "constant holomorphic sectional curvature" equal to 4.

This makes CP

The Fubini-Study metric is also an Einstein metric in that it is proportional to its own Ricci tensor: there exists a constant λ such that for all i,j we have

.

This implies, among other things, that the Fubini-Study metric remains unchanged up to a scalar multiple under the Ricci flow

. It also makes CP

, where it serves as a nontrivial solution to the vacuum Einstein equations.

commonly used in quantum mechanics

, or the notation of projective varieties of algebraic geometry

. To explicitly equate these two languages, let

where is a set of orthonormal basis vectors for Hilbert space

, the are complex numbers, and is the standard notation for a point in the projective space in homogeneous coordinates

. Then, given two points and in the space, the distance between them is

or, equivalently, in projective variety notation,

Here, is the complex conjugate

of . The appearance of in the denominator is a reminder that and likewise were not normalized to unit length; thus the normalization is made explicit here. In Hilbert space, the metric can be rather trivially interpreted as the angle between two vectors; thus it is occasionally called the quantum angle. The angle is real-valued, and runs from zero to .

The infinitesimal form of this metric may be quickly obtained by taking , or equivalently, to obtain

In the context of quantum mechanics

, CP

; the Fubini–Study metric is the natural metric

for the geometrization of quantum mechanics. Much of the peculiar behaviour of quantum mechanics, including quantum entanglement

and the Berry phase effect, can be attributed to the peculiarities of the Fubini–Study metric.

. That is, if is a separable state, so that it can be written as , then the metric is the sum of the metric on the subspaces:

where and are the metrics, respectively, on the subspaces A and B.

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the Fubini–Study metric is a Kähler metric on projective Hilbert space

Projective Hilbert space

In mathematics and the foundations of quantum mechanics, the projective Hilbert space P of a complex Hilbert space H is the set of equivalence classes of vectors v in H, with v ≠ 0, for the relation given by...

, that is, complex projective space

Complex projective space

In mathematics, complex projective space is the projective space with respect to the field of complex numbers. By analogy, whereas the points of a real projective space label the lines through the origin of a real Euclidean space, the points of a complex projective space label the complex lines...

CP

^{n}endowed with a Hermitian form. This metricMetric (mathematics)

In mathematics, a metric or distance function is a function which defines a distance between elements of a set. A set with a metric is called a metric space. A metric induces a topology on a set but not all topologies can be generated by a metric...

was originally described in 1904 and 1905 by Guido Fubini

Guido Fubini

Guido Fubini was an Italian mathematician, known for Fubini's theorem and the Fubini–Study metric.Born in Venice, he was steered towards mathematics at an early age by his teachers and his father, who was himself a teacher of mathematics...

and Eduard Study

Eduard Study

Eduard Study was a German mathematician known for work on invariant theory of ternary forms and for the study of spherical trigonometry. He is also known for contributions to space geometry, hypercomplex numbers, and criticism of early physical chemistry.Study was born in Coburg in the Duchy of...

.

A Hermitian form in (the vector space) C

^{n+1}defines a unitary subgroup U(n+1) in GL(n+1,C). A Fubini–Study metric is determined up to homothety (overall scaling) by invariance under such a U(n+1) action; thus it is homogeneous. Equipped with a Fubini–Study metric, CP^{n}is a symmetric spaceSymmetric space

A symmetric space is, in differential geometry and representation theory, a smooth manifold whose group of symmetries contains an "inversion symmetry" about every point...

. The particular normalization on the metric depends on the application. In Riemannian geometry

Riemannian geometry

Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric, i.e. with an inner product on the tangent space at each point which varies smoothly from point to point. This gives, in particular, local notions of angle, length...

, one uses a normalization so that the Fubini-Study metric simply relates to the standard metric on the (2n+1)-sphere. In algebraic geometry

Algebraic geometry

Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...

, one uses a normalization making CP

^{n}a Hodge manifold.## Construction

The Fubini-Study metric arises naturally in the quotient spaceQuotient space

In topology and related areas of mathematics, a quotient space is, intuitively speaking, the result of identifying or "gluing together" certain points of a given space. The points to be identified are specified by an equivalence relation...

construction of complex projective space

Complex projective space

In mathematics, complex projective space is the projective space with respect to the field of complex numbers. By analogy, whereas the points of a real projective space label the lines through the origin of a real Euclidean space, the points of a complex projective space label the complex lines...

.

Specifically, one may define CP

^{n}to be the space consisting of all complex lines in C^{n+1}, i.e., the quotient of C^{n+1}\{0} by the equivalence relationEquivalence relation

In mathematics, an equivalence relation is a relation that, loosely speaking, partitions a set so that every element of the set is a member of one and only one cell of the partition. Two elements of the set are considered equivalent if and only if they are elements of the same cell...

relating all complex multiples of each point together. This agrees with the quotient by the diagonal group action

Group action

In algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...

of the multiplicative group C

^{*}= C \ {0}:This quotient realizes C

^{n+1}\{0} as a complex line bundleLine bundle

In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example a curve in the plane having a tangent line at each point determines a varying line: the tangent bundle is a way of organising these...

over the base space CP

^{n}. (In fact this is the so-called tautological bundleTautological bundle

In mathematics, tautological bundle is a term for a particularly natural vector bundle occurring over a Grassmannian, and more specially over projective space...

over CP

^{n}.) A point of CP^{n}is thus identified with an equivalence class of (n+1)-tuples [Z_{0},...,Z_{n}] modulo nonzero complex rescaling; the Z_{i}are called homogeneous coordinatesHomogeneous coordinates

In mathematics, homogeneous coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrische Calcül, are a system of coordinates used in projective geometry much as Cartesian coordinates are used in Euclidean geometry. They have the advantage that the coordinates of points,...

of the point.

Furthermore, one may realize this quotient in two steps: since multiplication by a nonzero complex scalar z = R e

^{iθ}can be uniquely thought of as the composition of a dilation by the modulus R followed by a counterclockwise rotation about the origin by an angle , the quotient C^{n+1}→ CP^{n}splits into two pieces.where step (a) is a quotient by the dilation Z ~ RZ for R ∈ R

^{+}, the multiplicative group of positive real numbers, and step (b) is a quotient by the rotations Z ~ e^{iθ}Z.The result of the quotient in (a) is the real hypersphere S

^{2n+1}defined by the equation |Z|^{2}= |Z_{0}|^{2}+ ... + |Z_{n}|^{2}= 1. The quotient in (b) realizes CP^{n}= S^{2n+1}/S^{1}, where S^{1}represents the group of rotations. This quotient is realized explicitly by the famous Hopf fibration S^{1}→ S^{2n+1}→ CP^{n}, the fibers of which are among the great circles of .### As a metric quotient

When a quotient is taken of a Riemannian manifoldRiemannian manifold

In Riemannian geometry and the differential geometry of surfaces, a Riemannian manifold or Riemannian space is a real differentiable manifold M in which each tangent space is equipped with an inner product g, a Riemannian metric, which varies smoothly from point to point...

(or metric space

Metric space

In mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...

in general), care must be taken to ensure that the quotient space is endowed with a metric that is well-defined. For instance, if a group G acts on a Riemannian manifold (X,g), then in order for the orbit space X/G to possess an induced metric, must be constant along G-orbits in the sense that for any element h ∈ G and pair of vector fields we must have g(Xh,Yh) = g(X,Y).

The standard Hermitian metric on C

^{n+1}is given in the standard basis bywhose realification is the standard Euclidean metric on R

^{2n+2}. This metric is not invariant under the diagonal action of C^{*}, so we are unable to directly push it down to CP^{n}in the quotient. However, this metric is invariant under the diagonal action of S^{1}= U(1), the group of rotations. Therefore, step (b) in the above construction is possible once step (a) is accomplished.The Fubini-Study metric is the metric induced on the quotient CP

^{n}= S^{2n+1}/S^{1}, where carries the so-called "round metric" endowed upon it by restriction of the standard Euclidean metric to the unit hypersphere.### In local affine coordinates

Corresponding to a point in CP^{n}with homogeneous coordinates (Z_{0},...,Z_{n}), there is a unique set of n coordinates (z_{1},…,z_{n}) such thatprovided Z

_{0}≠ 0; specifically, z_{j}= Z_{j}/Z_{0}. The (z_{1},…,z_{n}) form an affine coordinate system for CP^{n}in the coordinate patch U_{0}= {Z_{0}≠ 0}. One can develop an affine coordinate system in any of the coordinate patches U_{i}= {Z_{i}≠ 0} by dividing instead by Z_{i}in the obvious manner. The n+1 coordinate patches U_{i}cover CP^{n}, and it is possible to give the metric explicitly in terms of the affine coordinates (z_{1},…,z_{n}) on U_{i}. The coordinate derivatives define a frame of the holomorphic tangent bundle of CP^{n}, in terms of which the Fubini–Study metric has Hermitian componentswhere |z|

^{2}= z_{1}^{2}+...+z_{n}^{2}. That is, the Hermitian matrix of the Fubini-Study metric in this frame isNote that each matrix element is unitary-invariant: the diagonal action will leave this matrix unchanged.

Accordingly, the line element is given by

In this last expression, the summation convention is used to sum over Latin indices i,j that range from 1 to n.

### Homogeneous coordinates

An expression is also possible in the homogeneous coordinates Z = [Z_{0},...,Z_{n}]. Formally, subject to suitably interpreting the expressions involved, one hasHere the summation convention is used to sum over Greek indices α β ranging from 0 to n, and in the last equality the standard notation for the skew part of a tensor is used:

Now, this expression for ds

^{2}apparently defines a tensor on the total space of the tautological bundle C^{n+1}\{0}. It is to be understood properly as a tensor on CP^{n}by pulling it back along a holomorphic section σ of the tautological bundle of CP^{n}. It remains then to verify that the value of the pullback is independent of the choice of section: this can be done by a direct calculation.The Kähler form of this metric is, up to an overall constant normalization,

the pullback of which is clearly independent of the choice of holomorphic section. The quantity log|Z|

^{2}is the Kähler scalar of CP^{n}.### The n = 1 case

When n = 1, there is a diffeomorphism given by stereographic projectionStereographic projection

The stereographic projection, in geometry, is a particular mapping that projects a sphere onto a plane. The projection is defined on the entire sphere, except at one point — the projection point. Where it is defined, the mapping is smooth and bijective. It is conformal, meaning that it...

. This leads to the "special" Hopf fibration S

^{1}→ S^{3}→ S^{2}. When the Fubini-Study metric is written in coordinates on CP^{1}, its restriction to the real tangent bundle yields an expression of the ordinary "round metric" of radius 1/2 (and Gaussian curvatureGaussian curvature

In differential geometry, the Gaussian curvature or Gauss curvature of a point on a surface is the product of the principal curvatures, κ1 and κ2, of the given point. It is an intrinsic measure of curvature, i.e., its value depends only on how distances are measured on the surface, not on the way...

4) on S

^{2}.Namely, if z = x + iy is the standard affine coordinate chart on the Riemann sphere

Riemann sphere

In mathematics, the Riemann sphere , named after the 19th century mathematician Bernhard Riemann, is the sphere obtained from the complex plane by adding a point at infinity...

CP

^{1}and x = r cosθ, y = r sinθ are polar coordinates on C, then a routine computation showswhere is the round metric on the unit 2-sphere. Here φ, θ are "mathematician's spherical coordinates" on S

^{2}coming from the stereographic projection r tan(φ/2) = 1, tanθ = y/x. (Many physics references interchange the roles of φ and θ.)## Curvature properties

In the n = 1 special case, the Fubini-Study metric has constant scalar curvature identically equal to 4, according to the equivalence with the 2-sphere's round metric (which given a radius R has scalar curvature ). However, for n > 1, the Fubini-Study metric does not have constant curvature. Its sectional curvature is instead by the equationwhere is an orthonormal basis of the 2-plane σ, J : TCP

^{n}→ TCP^{n}is the complex structureLinear complex structure

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

on CP

^{n}, and is the Fubini-Study metric.A consequence of this formula is that the sectional curvature satisfies for all 2-planes . The maximum sectional curvature (4) is attained at a holomorphic 2-plane — one for which J(σ) ⊂ σ — while the minimum sectional curvature (1) is attained at a 2-plane for which J(σ) is orthogonal to σ. For this reason, the Fubini-Study metric is often said to have "constant holomorphic sectional curvature" equal to 4.

This makes CP

^{n}a (non-strict) quarter pinched manifold; a celebrated theorem shows that a strictly quarter-pinched simply connected n-manifold must be homeomorphic to a sphere.The Fubini-Study metric is also an Einstein metric in that it is proportional to its own Ricci tensor: there exists a constant λ such that for all i,j we have

.

This implies, among other things, that the Fubini-Study metric remains unchanged up to a scalar multiple under the Ricci flow

Ricci flow

In differential geometry, the Ricci flow is an intrinsic geometric flow. It is a process that deforms the metric of a Riemannian manifold in a way formally analogous to the diffusion of heat, smoothing out irregularities in the metric....

. It also makes CP

^{n}indispensable to the theory of general relativityGeneral relativity

General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...

, where it serves as a nontrivial solution to the vacuum Einstein equations.

## In quantum mechanics

The Fubini-Study metric may be defined either using the bra-ket notationBra-ket notation

Bra-ket notation is a standard notation for describing quantum states in the theory of quantum mechanics composed of angle brackets and vertical bars. It can also be used to denote abstract vectors and linear functionals in mathematics...

commonly used in quantum mechanics

Quantum mechanics

Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

, or the notation of projective varieties of algebraic geometry

Algebraic geometry

Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...

. To explicitly equate these two languages, let

where is a set of orthonormal basis vectors for Hilbert space

Hilbert space

The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...

, the are complex numbers, and is the standard notation for a point in the projective space in homogeneous coordinates

Homogeneous coordinates

In mathematics, homogeneous coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrische Calcül, are a system of coordinates used in projective geometry much as Cartesian coordinates are used in Euclidean geometry. They have the advantage that the coordinates of points,...

. Then, given two points and in the space, the distance between them is

or, equivalently, in projective variety notation,

Here, is the complex conjugate

Complex conjugate

In mathematics, complex conjugates are a pair of complex numbers, both having the same real part, but with imaginary parts of equal magnitude and opposite signs...

of . The appearance of in the denominator is a reminder that and likewise were not normalized to unit length; thus the normalization is made explicit here. In Hilbert space, the metric can be rather trivially interpreted as the angle between two vectors; thus it is occasionally called the quantum angle. The angle is real-valued, and runs from zero to .

The infinitesimal form of this metric may be quickly obtained by taking , or equivalently, to obtain

In the context of quantum mechanics

Quantum mechanics

Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...

, CP

^{1}is called the Bloch sphereBloch sphere

In quantum mechanics, the Bloch sphere is a geometrical representation of the pure state space of a two-level quantum mechanical system , named after the physicist Felix Bloch....

; the Fubini–Study metric is the natural metric

Metric (mathematics)

In mathematics, a metric or distance function is a function which defines a distance between elements of a set. A set with a metric is called a metric space. A metric induces a topology on a set but not all topologies can be generated by a metric...

for the geometrization of quantum mechanics. Much of the peculiar behaviour of quantum mechanics, including quantum entanglement

Quantum entanglement

Quantum entanglement occurs when electrons, molecules even as large as "buckyballs", photons, etc., interact physically and then become separated; the type of interaction is such that each resulting member of a pair is properly described by the same quantum mechanical description , which is...

and the Berry phase effect, can be attributed to the peculiarities of the Fubini–Study metric.

## Product metric

The common notions of separability apply for the Fubini–Study metric. More precisely, the metric is separable on the natural product of projective spaces, the Segre embeddingSegre embedding

In mathematics, the Segre embedding is used in projective geometry to consider the cartesian product of two or more projective spaces as a projective variety...

. That is, if is a separable state, so that it can be written as , then the metric is the sum of the metric on the subspaces:

where and are the metrics, respectively, on the subspaces A and B.