Fubini–Study metric
Fubini–Study metric
In mathematics, the Fubini–Study metric is a Kähler metric on projective Hilbert space, that is, on a complex projective space CPn endowed with a Hermitian form. This metric was originally described in 1904 and 1905 by Guido Fubini and Eduard Study.[1][2]
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 {CP}}^{n}=\left\{{\mathbf {Z}}=[Z_{0},Z_{1},\ldots ,Z_{n}]\in {{\mathbf C}}^{{n+1}}\setminus \{0\}\,\right\}/\{{\mathbf {Z}}\sim c{\mathbf {Z}},c\in {\mathbf {C}}^{*}\}.](https://wikimedia.org/api/rest_v1/media/math/render/svg/6392aca942341900088d1fde767243890aec42e4)
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.
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 ~ eiθZ.
As a metric quotient
The standard Hermitian metric on Cn+1 is given in the standard basis by
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.
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
![{\displaystyle [Z_{0}:\dots :Z_{n}]{\sim }[1,z_{1},\dots ,z_{n}],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/97d12f36eeaab46d7d21fdd09ee0ad17dabc9ed2)
where |z|2 = |z1|2+...+|z**n|2. That is, the Hermitian matrix of the Fubini–Study metric in this frame is
![{\displaystyle {\bigl [}g_{i{\bar {j}}}{\bigr ]}={\frac {1}{(1+|\mathbf {z} |^{2})^{2}}}\left[{\begin{array}{cccc}1+|\mathbf {z} |^{2}-|z_{1}|^{2}&-{\bar {z}}_{1}z_{2}&\cdots &-{\bar {z}}_{1}z_{n}\\-{\bar {z}}_{2}z_{1}&1+|\mathbf {z} |^{2}-|z_{2}|^{2}&\cdots &-{\bar {z}}_{2}z_{n}\\\vdots &\vdots &\ddots &\vdots \\-{\bar {z}}_{n}z_{1}&-{\bar {z}}_{n}z_{2}&\cdots &1+|\mathbf {z} |^{2}-|z_{n}|^{2}\end{array}}\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/288982f060242143ba99c7c18ee9c5493f017348)
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.
The metric can be derived from the following Kähler potential:[3]
as
Using homogeneous coordinates
An expression is also possible in the notation of homogeneous coordinates, commonly used to describe projective varieties of algebraic geometry: Z = [Z0:...:Z**n]. Formally, subject to suitably interpreting the expressions involved, one has
![{\begin{aligned}ds^{2}&={\frac {|{\mathbf {Z}}|^{2}|d{\mathbf {Z}}|^{2}-({\bar {{\mathbf {Z}}}}\cdot d{\mathbf {Z}})({\mathbf {Z}}\cdot d{\bar {{\mathbf {Z}}}})}{|{\mathbf {Z}}|^{4}}}\\&={\frac {Z_{\alpha }{\bar {Z}}^{\alpha }dZ_{\beta }d{\bar {Z}}^{\beta }-{\bar {Z}}^{\alpha }Z_{\beta }dZ_{\alpha }d{\bar {Z}}^{\beta }}{(Z_{\alpha }{\bar {Z}}^{\alpha })^{2}}}\\&={\frac {2Z_{{[\alpha }}dZ_{{\beta ]}}\overline {Z}^{{[\alpha }}\overline {dZ}^{{\beta ]}}}{\left(Z_{\alpha }\overline {Z}^{\alpha }\right)^{2}}}.\end{aligned}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b7f3030099ca326a90040641623bce030c63b2a6)
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}}\left(Z_{{\alpha }}W_{{\beta }}-Z_{{\beta }}W_{{\alpha }}\right).](https://wikimedia.org/api/rest_v1/media/math/render/svg/2ca1668ed5c8bdb979877b7c79091b3f9d53faf4)
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
In bra-ket coordinate notation
In quantum mechanics, the Fubini–Study metric is also known as the Bures metric.[4] 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.[4]
The Fubini–Study metric may be written using the bra–ket notation commonly used in quantum mechanics. To explicitly equate this notation to the homogeneous coordinates given above, let
![\vert \psi \rangle =\sum _{{k=0}}^{n}Z_{k}\vert e_{k}\rangle =[Z_{0}:Z_{1}:\ldots :Z_{n}]](https://wikimedia.org/api/rest_v1/media/math/render/svg/55a69fef50eeb2171ccb65bc161a7c4dfa5483c8)
![Z_{\alpha }=[Z_{0}:Z_{1}:\ldots :Z_{n}]](https://wikimedia.org/api/rest_v1/media/math/render/svg/fb3478a5f2a2547fe5ce0b4e9591befe5a67c59f)
or, equivalently, in projective variety notation,
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.
The n = 1 case
- on S
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
The Kahler form is
Applying the Hodge star to the Kahler form, one obtains
implying that K is harmonic.
The n = 2 case
The line element, starting with the previously given expression, is given by
The vierbeins can be immediately read off from the last expression:
That is, in the vierbein coordinate system, using roman-letter subscripts, the metric tensor is Euclidean:
and is covariantly constant, which, for spin connections, means that it is antisymmetric in the vierbein indexes:
The above is readily solved; one obtains
The curvature 2-form is defined as
and is constant:
The Ricci tensor in veirbein indexes is given by
where the curvature 2-form was expanded as a four-component tensor:
The resulting Ricci tensor is constant
so that the resulting Einstein equation
The Weyl tensor for Fubini-Study metrics in general is given by
For the n=2 case, the two-forms
are self-dual:
Curvature properties
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.
- .
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.
- .
Product metric
Connection and curvature
The fact that the metric can be derived from the Kahler potential means that the Christoffel symbols and the curvature tensors contain a lot of symmetries, and can be given a particulary simple form:[7] The Christoffel symbols, in the local affine coordinates, are given by
The Riemann tensor is also particularly simple:
The Ricci tensor is
Pronunciation
A common pronunciation mistake, made especially by native English speakers, is to assume that Study is pronounced the same as the verb to study. Since it is actually a German name, the correct way to pronounce the u in Study is the same as the u in Fubini. In terms of phonetics: ʃtuːdi.
See also
Non-linear sigma model
Kaluza–Klein theory
Arakelov height