In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Fubini–Study metric is a
Kähler metric Kähler may refer to:
;People
*Alexander Kähler (born 1960), German television journalist
*Birgit Kähler (born 1970), German high jumper
*Erich Kähler (1906–2000), German mathematician
*Heinz Kähler (1905–1974), German art historian and arc ...
on
projective Hilbert space In mathematics and the foundations of quantum mechanics, the projective Hilbert space P(H) of a complex Hilbert space H is the set of equivalence classes of non-zero vectors v in H, for the relation \sim on H given by
:w \sim v if and only if v = \ ...
, that is, on a
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 ...
CP
''n'' endowed with a
Hermitian form
In mathematics, a sesquilinear form is a generalization of a bilinear form that, in turn, is a generalization of the concept of the dot product of Euclidean space. A bilinear form is linear in each of its arguments, but a sesquilinear form allow ...
. This
metric
Metric or metrical may refer to:
* Metric system, an internationally adopted decimal system of measurement
* An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement
Mathematics
In mathem ...
was originally described in 1904 and 1905 by
Guido Fubini
Guido Fubini (19 January 1879 – 6 June 1943) was an Italian mathematician, known for Fubini's theorem and the Fubini–Study metric.
Life
Born in Venice, he was steered towards mathematics at an early age by his teachers and his father, wh ...
and
Eduard Study
Eduard Study ( ), more properly Christian Hugo Eduard Study (March 23, 1862 – January 6, 1930), was a German mathematician known for work on invariant theory of ternary forms (1889) and for the study of spherical trigonometry. He is also known f ...
.
A
Hermitian form
In mathematics, a sesquilinear form is a generalization of a bilinear form that, in turn, is a generalization of the concept of the dot product of Euclidean space. A bilinear form is linear in each of its arguments, but a sesquilinear form allow ...
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
Homogeneity and heterogeneity are concepts often used in the sciences and statistics relating to the uniformity of a substance or organism. A material or image that is homogeneous is uniform in composition or character (i.e. color, shape, siz ...
. Equipped with a Fubini–Study metric, CP
''n'' is a
symmetric space
In mathematics, a symmetric space is a Riemannian manifold (or more generally, a pseudo-Riemannian manifold) whose group of symmetries contains an inversion symmetry about every point. This can be studied with the tools of Riemannian geometry, l ...
. The particular normalization on the metric depends on the application. In
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 that varies smoothly from point to poin ...
, one uses a normalization so that the Fubini–Study metric simply relates to the standard metric on the
(2''n''+1)-sphere. In
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
, one uses a normalization making CP
''n'' a
Hodge manifold.
Construction
The Fubini–Study metric arises naturally in the
quotient space construction of
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 ...
.
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\ by the
equivalence relation
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation.
Each equivalence relation ...
relating all complex multiples of each point together. This agrees with the quotient by the diagonal
group action
In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism ...
of the multiplicative group C
* = C \ :
:
This quotient realizes C
''n''+1\ as a complex
line 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 organisin ...
over the base space CP
''n''. (In fact this is the so-called
tautological bundle In mathematics, the tautological bundle is a vector bundle occurring over a Grassmannian in a natural tautological way: for a Grassmannian of k- dimensional subspaces of V, given a point in the Grassmannian corresponding to a k-dimensional vector ...
over CP
''n''.) A point of CP
''n'' is thus identified with an equivalence class of (''n''+1)-tuples
0,...,''Z''''n''">'Z''0,...,''Z''''n''modulo nonzero complex rescaling; the ''Z''
''i'' are called
homogeneous coordinates
In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work , are a system of coordinates used in projective geometry, just as Cartesian coordinates are used in Euclidean geometry. T ...
of the point.
Furthermore, one may realize this quotient mapping 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 mapping C
''n''+1 → CP
''n'' splits into two pieces.
:
where step (a) is a quotient by the dilation Z ~ ''R''Z for ''R'' ∈ R
+, the multiplicative group of
positive real numbers
In mathematics, the set of positive real numbers, \R_ = \left\, is the subset of those real numbers that are greater than zero. The non-negative real numbers, \R_ = \left\, also include zero. Although the symbols \R_ and \R^ are ambiguously used fo ...
, and step (b) is a quotient by the rotations Z ~ ''e''
iθZ.
The result of the quotient in (a) is the real hypersphere ''S''
2''n''+1 defined by the equation , Z,
2 = , ''Z''
0,
2 + ... + , ''Z''
''n'',
2 = 1. The quotient in (b) realizes CP
''n'' = ''S''
2''n''+1/''S''
1, where ''S''
1 represents the group of rotations. This quotient is realized explicitly by the famous
Hopf fibration
In the mathematical field of differential topology, the Hopf fibration (also known as the Hopf bundle or Hopf map) describes a 3-sphere (a hypersphere in four-dimensional space) in terms of circles and an ordinary sphere. Discovered by Heinz H ...
''S''
1 → ''S''
2''n''+1 → CP
''n'', the fibers of which are among the
great circles
In mathematics, a great circle or orthodrome is the circle, circular Intersection (geometry), intersection of a sphere and a Plane (geometry), plane incidence (geometry), passing through the sphere's centre (geometry), center point.
Any Circula ...
of
.
As a metric quotient
When a quotient is taken of a
Riemannian manifold
In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real manifold, real, smooth manifold ''M'' equipped with a positive-definite Inner product space, inner product ...
(or
metric space
In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
in general), care must be taken to ensure that the quotient space is endowed with a
metric
Metric or metrical may refer to:
* Metric system, an internationally adopted decimal system of measurement
* An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement
Mathematics
In mathem ...
that is well-defined. For instance, if a group ''G'' acts on a Riemannian manifold (''X'',''g''), then in order for the
orbit space
In celestial mechanics, an orbit is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an object or position in space such as a ...
''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
In mathematics, and more specifically in differential geometry, a Hermitian manifold is the complex analogue of a Riemannian manifold. More precisely, a Hermitian manifold is a complex manifold with a smoothly varying Hermitian inner product on ea ...
on C
''n''+1 is given in the standard basis by
:
whose realification is the standard
Euclidean metric
In mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two points.
It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, therefore oc ...
on R
2''n''+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''
2''n''+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
0:...:''Z''''n''">'Z''0:...:''Z''''n'' there is a unique set of ''n'' coordinates (''z''
1,...,''z''
''n'') such that
:
provided ''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 = . One can develop an affine coordinate system in any of the coordinate patches ''U''
''i'' = 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 components
:
where , z,
2 = , ''z''
1,
2+...+, ''z''
''n'',
2. That is, the
Hermitian matrix
In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose—that is, the element in the -th row and -th column is equal to the complex conjugate of the element in the -th ...
of the Fubini–Study metric in this frame is
:
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
In mathematics, especially the usage of linear algebra in Mathematical physics, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational convention that implies summation over a set of i ...
is used to sum over Latin indices ''i'',''j'' that range from 1 to ''n''.
The metric can be derived from the following
Kähler potential Kähler may refer to:
;People
* Alexander Kähler (born 1960), German television journalist
* Birgit Kähler (born 1970), German high jumper
*Erich Kähler (1906–2000), German mathematician
*Heinz Kähler (1905–1974), German art historian and a ...
:
:
as
:
Using homogeneous coordinates
An expression is also possible in the notation of
homogeneous coordinates
In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work , are a system of coordinates used in projective geometry, just as Cartesian coordinates are used in Euclidean geometry. T ...
, commonly used to describe
projective varieties
In algebraic geometry, a projective variety over an algebraically closed field ''k'' is a subset of some projective ''n''-space \mathbb^n over ''k'' that is the zero-locus of some finite family of homogeneous polynomials of ''n'' + 1 variables wi ...
of
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
: Z =
0:...:''Z''''n''">'Z''0:...:''Z''''n'' Formally, subject to suitably interpreting the expressions involved, one has
:
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 d''s''
2 apparently defines a tensor on the total space of the tautological bundle C
''n''+1\. 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 Kähler may refer to:
;People
* Alexander Kähler (born 1960), German television journalist
* Birgit Kähler (born 1970), German high jumper
*Erich Kähler (1906–2000), German mathematician
*Heinz Kähler (1905–1974), German art historian and a ...
of this metric is
:
where the
are the
Dolbeault operators
In mathematics, a complex differential form is a differential form on a manifold (usually a complex manifold) which is permitted to have complex coefficients.
Complex forms have broad applications in differential geometry. On complex manifolds, t ...
.
The pullback of this is clearly independent of the choice of holomorphic section. The quantity log, Z,
2 is the
Kähler potential Kähler may refer to:
;People
* Alexander Kähler (born 1960), German television journalist
* Birgit Kähler (born 1970), German high jumper
*Erich Kähler (1906–2000), German mathematician
*Heinz Kähler (1905–1974), German art historian and a ...
(sometimes called the Kähler scalar) of CP
''n''.
In bra-ket coordinate notation
In
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
, the Fubini–Study metric is also known as the
Bures metric
In mathematics, in the area of quantum information geometry, the Bures metric (named after Donald Bures) or Helstrom metric (named after Carl W. Helstrom) defines an infinitesimal distance between density matrix operators defining quantum states. ...
.
[Paolo Facchi, Ravi Kulkarni, V. I. Man'ko, Giuseppe Marmo, E. C. G. Sudarshan, Franco Ventriglia]
Classical and Quantum Fisher Information in the Geometrical Formulation of Quantum Mechanics
(2010), ''Physics Letters'' A 374 pp. 4801. However, the Bures metric is typically defined in the notation of
mixed states, whereas the exposition below is written in terms of a
pure state
In quantum physics, a quantum state is a mathematical entity that provides a probability distribution for the outcomes of each possible measurement on a system. Knowledge of the quantum state together with the rules for the system's evolution in t ...
. The real part of the metric is (four times) the
Fisher information metric In information geometry, the Fisher information metric is a particular Riemannian metric which can be defined on a smooth statistical manifold, ''i.e.'', a smooth manifold whose points are probability measures defined on a common probability space ...
.
[
The Fubini–Study metric may be written using the bra–ket notation commonly used in ]quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
. To explicitly equate this notation to the homogeneous coordinates given above, let
: