Integrable system

In mathematics, the Fubini–Study metric is a Kähler metric on projective Hilbert space, that is, complex projective space CPⁿ endowed with a Hermitian form. This metric was originally described in 1904 and 1905 by Guido Fubini and Eduard Study.

A Hermitian form in (the vector space) Cⁿ⁺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, CPⁿ is a symmetric space. 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 CPⁿ a Hodge manifold.

Construction

The Fubini–Study metric arises naturally in the quotient space construction of complex projective space.

Specifically, one may define CPⁿ to be the space consisting of all complex lines in Cⁿ⁺¹, i.e., the quotient of Cⁿ⁺¹\{0} by the equivalence relation relating all complex multiples of each point together. This agrees with the quotient by the diagonal group action of the multiplicative group C^* = C \ {0}:

${\mathbf{C}\mathbf{P}}^n=\{\mathbf{Z}=[Z_0,Z_1,\dots ,Z_n]\in {\mathbf{C}}^{n+1}\setminus \{0\}\}/\{\mathbf{Z}\sim c\mathbf{Z},c\in {\mathbf{C}}^{*}\}.$

This quotient realizes Cⁿ⁺¹\{0} as a complex line bundle over the base space CPⁿ. (In fact this is the so-called tautological bundle over CPⁿ.) A point of CPⁿ is thus identified with an equivalence class of (n+1)-tuples [Z₀,...,Z_n] modulo nonzero complex rescaling; the Z_i are called homogeneous coordinates 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 $\theta$, the quotient Cⁿ⁺¹ → CPⁿ splits into two pieces.

${\mathbf{C}}^{n+1}\setminus \{0\}\stackrel{(a)}{⟶}{S}^{2n+1}\stackrel{(b)}{⟶}{\mathbf{C}\mathbf{P}}^n$

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^iθZ.

The result of the quotient in (a) is the real hypersphere S²ⁿ⁺¹ defined by the equation |Z|² = |Z₀|² + ... + |Z_n|² = 1. The quotient in (b) realizes CPⁿ = S²ⁿ⁺¹/S¹, where S¹ represents the group of rotations. This quotient is realized explicitly by the famous Hopf fibration S¹ → S²ⁿ⁺¹ → CPⁿ, the fibers of which are among the great circles of $S^{2n+1}$.

As a metric quotient

When a quotient is taken of a Riemannian manifold (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, $g$ must be constant along G-orbits in the sense that for any element h ∈ G and pair of vector fields $X,Y$ we must have g(Xh,Yh) = g(X,Y).

The standard Hermitian metric on Cⁿ⁺¹ is given in the standard basis by

$d{s}^2=d\mathbf{Z}\otimes d\bar{\mathbf{Z}}=d{Z}_0\otimes d\overline{Z_0}+\cdots +d{Z}_n\otimes d\overline{Z_n}$

whose realification is the standard Euclidean metric on R²ⁿ⁺². This metric is not invariant under the diagonal action of C^*, so we are unable to directly push it down to CPⁿ in the quotient. However, this metric is invariant under the diagonal action of S¹ = 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 CPⁿ = S²ⁿ⁺¹/S¹, where $S^{2n+1}$ 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 CPⁿ with homogeneous coordinates (Z₀,...,Z_n), there is a unique set of n coordinates (z₁,…,z_n) such that

$[Z_0,\dots ,Z_n]\sim [1,z_1,\dots ,z_n],$

provided Z₀ ≠ 0; specifically, z_j = Z_j/Z₀. The (z₁,…,z_n) form an affine coordinate system for CPⁿ in the coordinate patch U₀ = {Z₀ ≠ 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 CPⁿ, and it is possible to give the metric explicitly in terms of the affine coordinates (z₁,…,z_n) on U_i. The coordinate derivatives define a frame $\{{\partial }_1,\dots ,{\partial }_n\}$ of the holomorphic tangent bundle of CPⁿ, in terms of which the Fubini–Study metric has Hermitian components

$h_{i\overline{j}}=h({\partial }_i,{\overline{\partial }}_j)=\frac{(1+|\mathbf{z}{|}^2){\delta }_{i\overline{j}}-{\overline{z}}_i{z}_j}{(1+|\mathbf{z}{|}^2{)}^2}.$

where |z|² = z₁²+...+z_n². That is, the Hermitian matrix of the Fubini–Study metric in this frame is

$\left(h_{i\overline{j}}\right)=\frac{1}{(1+|\mathbf{z}{|}^2{)}^2}\left[\begin{array}{cccc}1+|\mathbf{z}{|}^2-|z_1{|}^2& -{\overline{z}}_1{z}_2& \cdots & -{\overline{z}}_1{z}_n\\ -{\overline{z}}_2{z}_1& 1+|\mathbf{z}{|}^2-|z_2{|}^2& \cdots & -{\overline{z}}_2{z}_n\\ ⋮& ⋮& \ddots & ⋮\\ -{\overline{z}}_n{z}_1& -{\overline{z}}_n{z}_2& \cdots & 1+|\mathbf{z}{|}^2-|z_n{|}^2\end{array}\right]$

Note that each matrix element is unitary-invariant: the diagonal action $\mathbf{z}\mapsto e^{i\theta }\mathbf{z}$ will leave this matrix unchanged.

Accordingly, the line element is given by

$\begin{array}{rl}d{s}^2& =\frac{(1+|\mathbf{z}{|}^2)|d\mathbf{z}{|}^2-(\overline{\mathbf{z}}\cdot d\mathbf{z})(\mathbf{z}\cdot d\overline{\mathbf{z}})}{(1+|\mathbf{z}{|}^2{)}^2}\\ & =\frac{(1+z_i{\overline{z}}^i)d{z}_{j}d{\overline{z}}^j-{\overline{z}}^j{z}_{i}d{z}_{j}d{\overline{z}}^i}{(1+z_i{\overline{z}}^i{)}^2}.\end{array}$

In this last expression, the summation convention 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:

$K=\ln (1+{\delta }_{i{j}^{*}}{z}^i{\overline{z}}^{j^{*}})$

as

$g_{i{j}^{*}}=K_{i{j}^{*}}=\frac{{\partial }^2}{\partial z^i\partial {\overline{z}}^{j^{*}}}K$

Homogeneous coordinates

An expression is also possible in the homogeneous coordinates Z = [Z₀,...,Z_n]. Formally, subject to suitably interpreting the expressions involved, one has

$\begin{array}{rl}d{s}^2& =\frac{\left|\mathbf{Z}{|}^2\right|d\mathbf{Z}{|}^2-(\overline{\mathbf{Z}}\cdot d\mathbf{Z})(\mathbf{Z}\cdot d\overline{\mathbf{Z}})}{|\mathbf{Z}{|}^4}\\ & =\frac{Z_{\alpha }{\overline{Z}}^{\alpha }d{Z}_{\beta }d{\overline{Z}}^{\beta }-{\overline{Z}}^{\alpha }{Z}_{\beta }d{Z}_{\alpha }d{\overline{Z}}^{\beta }}{(Z_{\alpha }{\overline{Z}}^{\alpha }{)}^2}\\ & =\frac{2Z_{[\alpha }d{Z}_{\beta ]}\stackrel{[\alpha }{\bar{Z}}\stackrel{\beta ]}{\overline{dZ}}}{{\left(Z_{\alpha }\stackrel{\alpha }{\bar{Z}}\right)}^2}.\end{array}$

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:

$Z_{[\alpha }{W}_{\beta ]}=\frac{1}{2}(Z_{\alpha }{W}_{\beta }-Z_{\beta }{W}_{\alpha }).$

Now, this expression for ds² apparently defines a tensor on the total space of the tautological bundle Cⁿ⁺¹\{0}. It is to be understood properly as a tensor on CPⁿ by pulling it back along a holomorphic section σ of the tautological bundle of CPⁿ. 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,

$\omega =i\partial \bar{\partial }\log |\mathbf{Z}{|}^2$

the pullback of which is clearly independent of the choice of holomorphic section. The quantity log|Z|² is the Kähler scalar of CPⁿ.

The n = 1 case

When n = 1, there is a diffeomorphism $S^2\cong {\mathbb{ℂ}\mathbb{\mathbb{P}}}^1$ given by stereographic projection. This leads to the "special" Hopf fibration S¹ → S³ → S². When the Fubini–Study metric is written in coordinates on CP¹, its restriction to the real tangent bundle yields an expression of the ordinary "round metric" of radius 1/2 (and Gaussian curvature 4) on S².

Namely, if z = x + iy is the standard affine coordinate chart on the Riemann sphere CP¹ and x = r cosθ, y = r sinθ are polar coordinates on C, then a routine computation shows

$d{s}^2=\frac{\mathrm{\Re }(dz\otimes d\bar{z})}{{(1+|z{|}^2)}^2}=\frac{d{x}^2+d{y}^2}{{(1+r^2)}^2}=\frac{1}{4}(d{\varphi }^2+{\sin }^2\varphi d{\theta }^2)=\frac{1}{4}d{s}_{us}^2$

where $d{s}_{us}^2$ is the round metric on the unit 2-sphere. Here φ, θ are "mathematician's spherical coordinates" on S² 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 $1/R^2$). However, for n > 1, the Fubini–Study metric does not have constant curvature. Its sectional curvature is instead given by the equation

$K(\sigma )=1+3⟨JX,Y{⟩}^2$

where $\{X,Y\}\in T_p{\mathbf{C}\mathbf{P}}^n$ is an orthonormal basis of the 2-plane σ, J : TCPⁿ → TCPⁿ is the complex structure on CPⁿ, and $⟨\cdot ,\cdot ⟩$ is the Fubini–Study metric.

A consequence of this formula is that the sectional curvature satisfies $1\le K(\sigma )\le 4$ for all 2-planes $\sigma$. 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 CPⁿ 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

$Ri{c}_{ij}=\lambda g_{ij}$.

This implies, among other things, that the Fubini–Study metric remains unchanged up to a scalar multiple under the Ricci flow. It also makes CPⁿ indispensable to the theory of general relativity, where it serves as a nontrivial solution to the vacuum Einstein field equations.

In quantum mechanics

In quantum mechanics, the Fubini–Study metric is also known as the Bures metric. 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. The real part of the metric is (four times) the Fisher information metric.

The Fubini–Study metric may be written either using the bra-ket notation commonly used in quantum mechanics, or the notation of projective varieties of algebraic geometry. To explicitly equate these two languages, let

$|\psi ⟩=\sum _{k=0}^n{Z}_k|e_k⟩=[Z_0:Z_1:\dots :Z_n]$

where $\{|e_k⟩\}$ is a set of orthonormal basis vectors for Hilbert space, the $Z_k$ are complex numbers, and $Z_{\alpha }=[Z_0:Z_1:\dots :Z_n]$ is the standard notation for a point in the projective space $\mathbb{ℂ}{P}^n$ in homogeneous coordinates. Then, given two points $|\psi ⟩=Z_{\alpha }$ and $|\varphi ⟩=W_{\alpha }$ in the space, the distance between them is

$\gamma (\psi ,\varphi )=\arccos \sqrt{\frac{⟨\psi |\varphi ⟩⟨\varphi |\psi ⟩}{⟨\psi |\psi ⟩⟨\varphi |\varphi ⟩}}$

or, equivalently, in projective variety notation,

$\gamma (\psi ,\varphi )=\gamma (Z,W)=\arccos \sqrt{\frac{Z_{\alpha }\stackrel{\alpha }{\bar{W}}{W}_{\beta }\stackrel{\beta }{\bar{Z}}}{Z_{\alpha }\stackrel{\alpha }{\bar{Z}}{W}_{\beta }\stackrel{\beta }{\bar{W}}}}.$

Here, $\stackrel{\alpha }{\bar{Z}}$ is the complex conjugate of $Z_{\alpha }$. The appearance of $⟨\psi |\psi ⟩$ in the denominator is a reminder that $|\psi ⟩$ and likewise $|\varphi ⟩$ 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 $\pi /2$.

The infinitesimal form of this metric may be quickly obtained by taking $\varphi =\psi +\delta \psi$, or equivalently, $W_{\alpha }=Z_{\alpha }+d{Z}_{\alpha }$ to obtain

$d{s}^2=\frac{⟨\delta \psi |\delta \psi ⟩}{⟨\psi |\psi ⟩}-\frac{⟨\delta \psi |\psi ⟩⟨\psi |\delta \psi ⟩}{{⟨\psi |\psi ⟩}^2}.$

In the context of quantum mechanics, CP¹ is called the Bloch sphere; 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.

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 embedding. That is, if $|\psi ⟩$ is a separable state, so that it can be written as $|\psi ⟩=|{\psi }_A⟩\otimes |{\psi }_B⟩$, then the metric is the sum of the metric on the subspaces:

$d{s}^2={d{s}_A}^2+{d{s}_B}^2$

where ${d{s}_A}^2$ and ${d{s}_B}^2$ are the metrics, respectively, on the subspaces A and B.

See also

• Non-linear sigma model
• Kaluza–Klein theory
• Arakelov height

References

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