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Projective Hilbert space
Projective Hilbert space
Projective Hilbert space
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In mathematics and the foundations of quantum mechanics, the projective Hilbert space or ray space of a complex Hilbert space is the set of equivalence classes of non-zero vectors , for the equivalence relation on given by

if and only if for some non-zero complex number .

This is the usual construction of projectivization, applied to a complex Hilbert space.[1] In quantum mechanics, the equivalence classes are also referred to as rays or projective rays. Each such projective ray is a copy of the nonzero complex numbers, which is topologically a two-dimensional plane after one point has been removed.

Overview

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See also: Wigner's theorem § Rays and ray space

The physical significance of the projective Hilbert space is that in quantum theory, the wave functions and represent the same physical state, for any . The Born rule demands that if the system is physical and measurable, its wave function has unit norm, , in which case it is called a normalized wave function. The unit norm constraint does not completely determine within the ray, since could be multiplied by any with absolute value 1 (the circle group action) and retain its normalization. Such a can be written as with called the global phase.

Rays that differ by such a correspond to the same state (cf. quantum state (algebraic definition), given a C*-algebra of observables and a representation on ). No measurement can recover the phase of a ray; it is not observable. One says that is a gauge group of the first kind.

If is an irreducible representation of the algebra of observables then the rays induce pure states. Convex linear combinations of rays naturally give rise to density matrix which (still in case of an irreducible representation) correspond to mixed states.

In the case is finite-dimensional, i.e., , the Hilbert space reduces to a finite-dimensional inner product space and the set of projective rays may be treated as a complex projective space; it is a homogeneous space for a unitary group . That is,

,

which carries a Kähler metric, called the Fubini–Study metric, derived from the Hilbert space's norm.[2][3]

As such, the projectivization of, e.g., two-dimensional complex Hilbert space (the space describing one qubit) is the complex projective line . This is known as the Bloch sphere or, equivalently, the Riemann sphere. See Hopf fibration for details of the projectivization construction in this case.

Product

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The Cartesian product of projective Hilbert spaces is not a projective space. The Segre mapping is an embedding of the Cartesian product of two projective spaces into the projective space associated to the tensor product of the two Hilbert spaces, given by . In quantum theory, it describes how to make states of the composite system from states of its constituents. It is only an embedding, not a surjection; most of the tensor product space does not lie in its image and represents entangled states.

See also

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Notes

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References

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Projective Hilbert space

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In mathematics and physics, the projective Hilbert space associated with a complex Hilbert space HH is the set of equivalence classes of non-zero vectors in HH, where two vectors ψ\psi and ϕ\phi are considered equivalent if ψ=cϕ\psi = c \phi for some non-zero complex scalar c∈C∗c \in \mathbb{C}^*. These equivalence classes, often called rays or one-dimensional subspaces, form the projective Hilbert space P(H)\mathbb{P}(H), which generalizes the finite-dimensional complex projective space CPn\mathbb{CP}^n to infinite dimensions. This structure arises naturally in the foundations of , where pure quantum states are represented not by vectors in HH but by rays in P(H)\mathbb{P}(H), as overall normalization and global phase factors have no physical significance and do not affect measurement probabilities via the . The space P(H)\mathbb{P}(H) inherits a natural geometry from HH, including a Fubini-Study metric that induces a Kähler structure, enabling the formulation of as Hamiltonian flows on this manifold. In infinite dimensions, P(H)\mathbb{P}(H) requires careful definition of topologies—such as the quotient topology or weak topologies—to ensure measurability and continuity of quantum operations, addressing challenges absent in finite-dimensional cases. Key applications include , where P(H)\mathbb{P}(H) serves as a for classical-like descriptions of quantum evolution, and the study of projective unitary representations of symmetry groups in quantum theory.

Mathematical Foundations

Hilbert Spaces and Rays

A is defined as a complete over the complex numbers, meaning it is a equipped with an inner product that induces a norm, and every converges within the space. In the context of , Hilbert spaces are typically taken to be separable, ensuring a countable exists, which facilitates the representation of physical systems with discrete and continuous spectra. In , the state of a is represented by a vector ψ in the , subject to the normalization condition ‖ψ‖ = 1, where the norm is derived from the inner product ⟨ψ|ψ⟩ = 1. This normalization ensures the vector lies on the unit sphere in the , but physical observables depend only on directions rather than specific vector representatives. A ray in the is the one-dimensional subspace generated by a non-zero vector ψ, consisting of all scalar multiples of the form { c \psi \mid c \in \mathbb{C} \setminus {0} }. In , these rays capture the physically equivalent states, as overall scaling and global phase factors have no physical significance. For example, in the finite-dimensional ℂ^n, rays correspond to lines through the origin, with each ray intersecting the unit sphere S^{2n-1} in a circle, representing distinct directions modulo phase and scaling. were formalized by in the 1920s and 1930s as the mathematical foundation for , building on earlier work by and others to rigorously describe infinite-dimensional systems. The , to be detailed later, emerges as the of these rays under equivalence.

Equivalence Relations and Quotient Construction

The projective Hilbert space arises from the HH through an that identifies vectors differing only by a non-zero complex scalar, reflecting the physical indistinguishability of quantum states under C∗\mathbb{C}^* transformations. Specifically, two nonzero vectors ψ,ϕ∈H∖{0}\psi, \phi \in H \setminus \{0\} are equivalent, denoted ψ∼ϕ\psi \sim \phi, if there exists λ∈C∗\lambda \in \mathbb{C}^* such that ϕ=λψ\phi = \lambda \psi. This relation partitions H∖{0}H \setminus \{0\} into equivalence classes, each corresponding to a one-dimensional subspace or "ray" in HH. The projective , denoted PH(H)\mathrm{PH}(H), is then defined as the set PH(H)=(H∖{0})/∼\mathrm{PH}(H) = (H \setminus \{0\}) / \sim, where elements are equivalence classes [ψ]={λψ∣λ∈C∗}[\psi] = \{\lambda \psi \mid \lambda \in \mathbb{C}^*\}. An equivalent construction focuses on the unit sphere S(H)={ψ∈H∣∥ψ∥=1}S(H) = \{\psi \in H \mid \|\psi\| = 1\} in HH, which is a complete metric space under the norm topology. The group U(1)U(1) acts freely and continuously on S(H)S(H) via multiplication by unit complex scalars, and PH(H)\mathrm{PH}(H) is the orbit space S(H)/U(1)S(H) / U(1), inheriting the quotient topology. In this topology, a set U⊂PH(H)U \subset \mathrm{PH}(H) is open if its preimage under the quotient map π:S(H)→PH(H)\pi: S(H) \to \mathrm{PH}(H), defined by π(ψ)=[ψ]\pi(\psi) = [\psi], is open in S(H)S(H). This ensures that PH(H)\mathrm{PH}(H) is a topological space where convergence of sequences [ψn]→[ψ][\psi_n] \to [\psi] corresponds to the existence of phases λn∈U(1)\lambda_n \in U(1) such that λnψn→ψ\lambda_n \psi_n \to \psi in the norm of HH. For a separable infinite-dimensional Hilbert space HH, the quotient PH(H)\mathrm{PH}(H) is Hausdorff and well-behaved topologically. The Hausdorff property follows from the compactness of U(1)U(1), which implies that the quotient map π\pi separates distinct orbits: for distinct [ψ][\psi] and [ϕ][\phi], there exist disjoint open neighborhoods in S(H)S(H) that are saturated under the U(1)U(1)-action, projecting to disjoint opens in PH(H)\mathrm{PH}(H). Moreover, since S(H)S(H) is second-countable (as HH is separable), the quotient topology on PH(H)\mathrm{PH}(H) is also second-countable, making it a metrizable space without pathologies like non-separated points. This construction yields a locally compact Hausdorff space that serves as the state space in quantum mechanics. Unlike the real projective space RPn\mathbb{RP}^n, which quotients Rn+1∖{0}\mathbb{R}^{n+1} \setminus \{0\} by positive R+\mathbb{R}^+, the complex structure of PH(H)\mathrm{PH}(H) involves the C∗\mathbb{C}^* rather than R∗\mathbb{R}^*, leading to a with distinct geometric properties such as complex dimension and Kähler potential (though the metric is addressed elsewhere). This emphasis on non-zero complex scalars ensures that PH(H)\mathrm{PH}(H) captures the phase invariance inherent to quantum superpositions, distinguishing it from real projective geometries.

Geometric and Topological Structure

Projective Space as a Manifold

In the finite-dimensional case, where the Hilbert space H=CnH = \mathbb{C}^n is an nn-dimensional complex vector space, the projective Hilbert space PH(H)\mathrm{PH}(H) is isomorphic to the CPn−1\mathbb{C}\mathbb{P}^{n-1}, which has complex dimension n−1n-1. This space consists of equivalence classes of nonzero vectors in HH, where two vectors z,z′∈H∖{0}z, z' \in H \setminus \{0\} are identified if z′=λzz' = \lambda z for some λ∈C∗=C∖{0}\lambda \in \mathbb{C}^* = \mathbb{C} \setminus \{0\}. Points in CPn−1\mathbb{C}\mathbb{P}^{n-1} are represented using homogeneous coordinates [z1:z2:⋯:zn][z_1 : z_2 : \dots : z_n], where (z1,…,zn)∈Cn∖{0}(z_1, \dots, z_n) \in \mathbb{C}^n \setminus \{0\} and the notation indicates equivalence under C∗\mathbb{C}^*-scaling. To endow CPn−1\mathbb{C}\mathbb{P}^{n-1} with a manifold structure, it is covered by nn standard affine charts Uk={∈CPn−1∣zk≠0}U_k = \{ \in \mathbb{C}\mathbb{P}^{n-1} \mid z_k \neq 0 \} for k=1,…,nk = 1, \dots, n, each diffeomorphic to Cn−1\mathbb{C}^{n-1}. On UkU_k, the holomorphic coordinates are given by wi(k)=zi/zkw_i^{(k)} = z_i / z_k for i≠ki \neq k, providing a local identification with Cn−1\mathbb{C}^{n-1}. The overlaps between charts ensure compatibility through transition functions that are holomorphic. For instance, on Uj∩UkU_j \cap U_k with j≠kj \neq k, the transition from coordinates on UjU_j to those on UkU_k is wi(k)=wi(j)/wj(j)w_i^{(k)} = w_i^{(j)} / w_j^{(j)} for i≠k,ji \neq k, j, while the coordinate corresponding to the jj-th direction in the UkU_k-system is wj(k)=1/wj(j)w_j^{(k)} = 1 / w_j^{(j)}. These transition maps are rational functions that are holomorphic on the overlaps, confirming that the atlas {(Uk,ϕk)∣k=1,…,n}\{ (U_k, \phi_k) \mid k = 1, \dots, n \}, where ϕk\phi_k denotes the coordinate map, defines a structure on CPn−1\mathbb{C}\mathbb{P}^{n-1}. In the infinite-dimensional setting, for a separable infinite-dimensional complex HH, the PH(H)\mathrm{PH}(H) inherits a smooth manifold structure as the of the unit in HH by the S1S^1-action, resulting in a non-compact infinite-dimensional . Unlike the finite-dimensional case, it lacks a finite atlas of charts analogous to the affine coverings, but it supports a differentiable structure compatible with the .

Fubini-Study Metric and Kähler Geometry

The Fubini-Study metric on the projective P(H)\mathbb{P}(\mathcal{H}), where H\mathcal{H} is a finite-dimensional complex of dimension n+1n+1, arises as the quotient metric induced from the round metric on the unit sphere S2n+1⊂HS^{2n+1} \subset \mathcal{H} via the π:S2n+1→CPn\pi: S^{2n+1} \to \mathbb{C}\mathbb{P}^n. Specifically, for tangent vectors H1,H2H_1, H_2 at a line L∈CPnL \in \mathbb{C}\mathbb{P}^n, the metric is given by gFS,L(H1,H2)=gSx(H1x,H2x)g_{\mathrm{FS},L}(H_1, H_2) = g_{S_x}(H_1 x, H_2 x), where x∈L∩S2n+1x \in L \cap S^{2n+1} and gSg_S is the standard round metric on the sphere, ensuring the projection is a Riemannian submersion. In affine charts, such as the standard chart on Cn\mathbb{C}^n where [z0:z1:⋯:zn]=[1:w1:⋯:wn][z_0 : z_1 : \cdots : z_n] = [1 : w_1 : \cdots : w_n] with w=(w1,…,wn)w = (w_1, \dots, w_n), the Fubini-Study metric takes the local form ds2=∑i=1ndwidwˉi(1+∥w∥2)2,ds^2 = \frac{\sum_{i=1}^n dw_i d\bar{w}_i}{(1 + \|w\|^2)^2}, which is the Hermitian metric gijˉ=δij(1+∥w∥2)−wˉiwj(1+∥w∥2)2g_{i\bar{j}} = \frac{\delta_{ij} (1 + \|w\|^2) - \bar{w}_i w_j}{(1 + \|w\|^2)^2}. This expression derives from the Kähler potential K(w)=log⁡(1+∥w∥2)K(w) = \log(1 + \|w\|^2) in these coordinates, where the metric components are the mixed partial derivatives gijˉ=∂i∂jˉKg_{i\bar{j}} = \partial_i \partial_{\bar{j}} K. In homogeneous coordinates [Z]=[Z0:⋯:Zn][Z] = [Z_0 : \cdots : Z_n] with Z∈H∖{0}Z \in \mathcal{H} \setminus \{0\}, the Kähler potential is K(Z)=log⁡∑i=0n∣Zi∣2K(Z) = \log \sum_{i=0}^n |Z_i|^2, providing a gauge-invariant description that descends to the quotient. The associated Kähler form is the fundamental 2-form ω=i2∂∂ˉlog⁡(1+∥w∥2)\omega = \frac{i}{2} \partial \bar{\partial} \log(1 + \|w\|^2) in affine coordinates, or more generally ω=i2∂∂ˉK\omega = \frac{i}{2} \partial \bar{\partial} K, making CPn\mathbb{C}\mathbb{P}^n a . This form is closed (dω=0d\omega = 0) and compatible with the complex structure, endowing the space with a natural symplectic structure where ω\omega serves as the symplectic 2-form. Consequently, the Fubini-Study geometry supports , with geodesics corresponding to Hamiltonian flows generated by the metric's symplectic potential. The Fubini-Study metric exhibits constant holomorphic sectional curvature equal to 4 (in the standard normalization), with sectional curvatures ranging from 1 to 4, achieving the maximum on holomorphic planes. It is an Einstein metric, satisfying Ric(gFS)=2(n+1) gFS\mathrm{Ric}(g_{\mathrm{FS}}) = 2(n+1) \, g_{\mathrm{FS}} for dim⁡CCPn=n\dim_{\mathbb{C}} \mathbb{C}\mathbb{P}^n = n, and thus Kähler-Einstein, reflecting its role as a of rank 1. In the infinite-dimensional case, where H\mathcal{H} is a separable complex , the Fubini-Study metric extends to P(H)\mathbb{P}(\mathcal{H}) via the same quotient construction from the unit sphere in H\mathcal{H}, yielding a complete Kähler metric σ2\sigma^2 that generalizes the finite-dimensional version. However, challenges arise in ensuring well-definedness and completeness, often requiring weak topologies on H\mathcal{H} to handle convergence of rays and to embed bounded domains isometrically into P(H)\mathbb{P}(\mathcal{H}) using Bergman kernels.

Properties and Theorems

Dimension and Homogeneity

The projective Hilbert space PH(H) associated to a finite-dimensional complex Hilbert space H of dimension nn is diffeomorphic to the CPn−1\mathbb{CP}^{n-1}, a smooth manifold of real dimension 2(n−1)2(n-1). In the infinite-dimensional case, where dim⁡H=∞\dim H = \infty, PH(H) has infinite real dimension and is non-compact. For finite nn, CPn−1\mathbb{CP}^{n-1} is compact as a quotient of the compact unit sphere in HH by the S1S^1-action. Its singular homology groups satisfy Hk(CPn−1;Z)=ZH_k(\mathbb{CP}^{n-1}; \mathbb{Z}) = \mathbb{Z} for k=0,2,…,2(n−1)k = 0, 2, \dots, 2(n-1) and Hk(CPn−1;Z)=0H_k(\mathbb{CP}^{n-1}; \mathbb{Z}) = 0 otherwise, reflecting its CW-complex structure with one cell in each even dimension from 0 to 2(n−1)2(n-1). The Euler characteristic is χ(CPn−1)=n\chi(\mathbb{CP}^{n-1}) = n, obtained as the alternating sum of Betti numbers or the number of CW-cells. As a homogeneous space, CPn−1≅U(n)/(U(1)×U(n−1))\mathbb{CP}^{n-1} \cong U(n) / (U(1) \times U(n-1)), where the U(n)U(n) acts transitively by transforming lines in HH into each other. The integral cohomology ring is H∗(CPn−1;Z)≅Z/(xn)H^*(\mathbb{CP}^{n-1}; \mathbb{Z}) \cong \mathbb{Z} / (x^n), where xx is the generator in degree 2 represented by the Kähler class [ω][\omega] of the Fubini-Study form.

Unitary Group Action

The unitary group U(H)U(\mathcal{H}) acts on the Hilbert space H\mathcal{H} by unitary operators, which preserve the inner product and thus map rays to rays, inducing an action on the projective Hilbert space PH\mathbb{P}\mathcal{H} defined by [Uψ]=[Uψ][U\psi] = [U \psi], where rays are preserved up to a global phase factor eiθe^{i\theta} with θ∈R\theta \in \mathbb{R}. This action is well-defined because unitary operators maintain the equivalence relation on nonzero vectors, ensuring that the projective space, consisting of one-dimensional subspaces, is invariant under the transformation. The kernel of this induced action is the center U(1)U(1), consisting of phase multiplications, so the effective group acting on PH\mathbb{P}\mathcal{H} is the projective unitary group PU(H)=U(H)/U(1)\mathrm{PU}(\mathcal{H}) = U(\mathcal{H}) / U(1). For finite-dimensional H=Cn\mathcal{H} = \mathbb{C}^n, this is PU(n)\mathrm{PU}(n), which acts transitively and effectively on PCn≅CPn−1\mathbb{P}\mathbb{C}^n \cong \mathbb{CP}^{n-1}, meaning any two rays can be mapped to each other by some element of PU(n)\mathrm{PU}(n), and the action is faithful. The stabilizer of a specific point, such as the ray [e1][e_1] spanned by the first standard basis vector, is the subgroup isomorphic to U(n−1)U(n-1), consisting of unitaries that fix this ray up to phase. Under the action of PU(n)\mathrm{PU}(n), the entire space CPn−1\mathbb{CP}^{n-1} forms a single , confirming its homogeneity as a manifold where the realizes the transitive . In the infinite-dimensional case for separable H\mathcal{H}, PU(H)\mathrm{PU}(\mathcal{H}) similarly acts transitively on PH\mathbb{P}\mathcal{H}, with the projective space being the set of all one-dimensional subspaces. The infinitesimal action arises from the u(n)\mathfrak{u}(n), consisting of skew-Hermitian matrices, which generates derivations on CPn−1\mathbb{CP}^{n-1} via the fundamental vector fields; these correspond to Killing vector fields with respect to the Fubini-Study metric, where the is proportional to the negative restricted to the . As a consequence, PH\mathbb{P}\mathcal{H} is a symmetric space under the action of PU(H)\mathrm{PU}(\mathcal{H}), characterized by an involution at each point whose fixed-point set is the isotropy subgroup, yielding a Riemannian symmetric structure invariant under the group.

Applications in Physics

Quantum Mechanics State Space

In quantum mechanics, the physical states of a system are represented by rays in the Hilbert space rather than individual vectors, owing to the invariance of observables under global phase transformations. Specifically, for a normalized state vector ∣ψ⟩∈H|\psi\rangle \in \mathcal{H}, the expectation value of an operator AA is ⟨ψ∣A∣ψ⟩\langle \psi | A | \psi \rangle, which remains unchanged if ∣ψ⟩|\psi\rangle is replaced by eiθ∣ψ⟩e^{i\theta} |\psi\rangle for any real θ\theta. This phase invariance implies that states differing only by a phase factor describe the same physical situation, leading to the identification of equivalence classes known as rays [ψ][\psi], which form the projective Hilbert space P(H)\mathbb{P}(\mathcal{H}). Pure quantum states correspond bijectively to rank-1 orthogonal projectors Pψ=∣ψ⟩⟨ψ∣P_\psi = |\psi\rangle\langle\psi|, where the density operator for the pure state is ρ=Pψ\rho = P_\psi, and the expectation value of an AA is given by Tr⁡(ρA)\operatorname{Tr}(\rho A). This representation ensures that the projective structure captures the physically distinguishable states without redundancy from phase factors. Superpositions in the H\mathcal{H} map to nonlinear combinations in P(H)\mathbb{P}(\mathcal{H}), as the projection operation normalizes linear combinations, which underpins quantum interference phenomena essential to the theory. A canonical example is the state space of a single , which is the CP1≅S2\mathbb{CP}^1 \cong S^2, known as the . Basis states such as ∣0⟩|0\rangle correspond to the , while equatorial points represent equal superpositions like 12(∣0⟩+∣1⟩)\frac{1}{\sqrt{2}}(|0\rangle + |1\rangle)
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