Unitary-antiunitary symmetries

Transcription

1 Unitary-antiunitary symmetries György Pál Gehér University of Reading, UK Preservers: Modern Aspects and New Directions 18 June 2018, Belfast

2 Some classical results

3 H a complex (or real) Hilbert space, P 1 (H) the projective space over H: P 1 (H) = {P B sa (H): P 2 = P, dim(imp) = 1} (natural identification with one-dimensional subspaces), (P, Q) angle between P and Q: (P, Q) = arccos u, v [ 0, π ] 2, where u = v = 1, u ImP, v ImQ, TrPQ transition probability between P and Q. By an easy calculation we get TrPQ = u, v 2 = cos 2 ( (P, Q)) and P Q = 1 TrPQ = sin( (P, Q)).

4 The bijective and non-bijective Wigner theorem Theorem (bijective version (1932!), Lomont Mendelson (1963)) Let φ: P 1 (H) P 1 (H) be a bijective map that satisfies Trφ(P)φ(Q) = TrPQ (P, Q P 1 (H)) ( ) (and nothing else is assumed!). Then there exists a unitary or an antiunitary operator U : H H such that φ(p) = UPU (P P 1 (H)). Theorem (non-bijective version, Bargmann (1964)) Let φ: P 1 (H) P 1 (H) be a map that satisfies ( ). Then there exists a linear or antilinear isometry V : H H such that φ(p) = VPV (P P 1 (H)). Note that ImVPV = V (ImP).

5 Proof for H = C 2 Let us consider Bloch s vectorspace isomorphism: ρ: B sa (C 2 ) R 4, ρ(a) = ρ(x 0 σ 0 +x 1 σ 1 +x 2 σ 2 +x 3 σ 3 ) = (x 0, x 1, x 2, x 3 ), where σ 0 = Note that [ ] 1 0, σ = [ ] 0 1, σ = [ ] [ ] 0 i 1 0, σ i 0 3 =. 0 1 TrA = ρ(a) (2, 0, 0, 0) = 2 ρ(a) s first coordinate, thus ρ(p 1 (H)) {( 1 2,,, ): R}.

6 More precisely, ρ(p 1 (H)) is a sphere in the 3-dimensional affine subspace {( 1 2,,, ): R} with centre ρ( 1 2 I ) = ( 1 2, 0, 0, 0): ρ(q) ρ(p) ρ( 1 2 I ) 2 (P, Q) The beauty of Bloch s transformation: (ρ(p), ρ(q)) = 2 (P, Q). So Wigner maps are angle preserving maps on the sphere. We only have to note that sphere-transformations from SO(3) correspond to unitary maps on P 1 (H), and that the reflection through a particular two-dimensional affine plane corresponds to the coordinate-wise conjugation operator on P 1 (H).

7 Uhlhorn s remarkable improvement orthogonality of rank-one projections: P Q ImP ImQ PQ = 0 TrPQ = 0. Theorem (Uhlhorn (1963)) Assume dim H 3, φ: P 1 (H) P 1 (H) is a bijective map and Trφ(P)φ(Q) = 0 TrPQ = 0 (P, Q P 1 (H)) ( ) Then there exists a unitary/antiunitary operator U : H H so that φ(p) = UPU (P P 1 (H)). This is a consequence of the fundamental theorem of projective geometry. This result implies the bijective Wigner theorem, but NOT the non-bijective version! Unlike Wigner s theorem, Uhlhorn s result cannot be extended for non-bijective maps.

8 Some recent improvements (in Hilbert spaces)

9 P n (H) the Grassmann space over H: P n (H) = {P B sa (H): P 2 = P, dim(imp) = n} (again, natural identification with n-dimensional subspaces). Theorem (Halmos two projections theorem) Let dim H 2n and P, Q P n (H). Then up to a unitary similarity H can be written as an orthogonal direct sum decomposition H = K K L such that dim K = n and the projections P and Q have the corresponding matrix representations: I 0 0 C 2 CS 0 P = 0 0 0, Q = CS S 2 0, (1) where S, C : K K are self-adjoint operators satisfying 0 S, C I and S 2 + C 2 = I (hence CS = SC). Note that C and S can be diagonalized in a joint ONB of K.

10 (P, Q) system of principal angles between P and Q: (P, Q) = (θ 1,..., θ n ) = (arccos σ 1,..., arccos σ n ), where σ 1,..., σ n are the n largest singular values of PQ so that σ 1 σ n. So π 2 θ 1 θ n 0. Orthogonoality of rank-n projections: ImP ImQ. Note that θ 1 = arcsin P Q. We also have TrPQ = n i=1 cos2 θ i. P Q PQ = 0 TrPQ = 0 (P, Q) = ( π 2,..., π 2 ).

11 Generalized non-bijective Wigner theorem on P n (H) Theorem (Molnár, 2001 and 2008) Let dim H > n 1 and φ: P n (H) P n (H) be a transformation that satisfies (φ(p), φ(q)) = (P, Q) (P, Q P n (H)). Then either φ is induced by a linear/conjugatelinear isometry V : H H φ(p) = VPV (P P n (H)), (2) or we have dim H = 2n, n > 1 and φ(p) = I VPV (P P n (H)). (3) If n = 1 and H = C 2, then P I P is implemented the following antiunitary operator: U(v, w) = (w, v). If n > 1 and dim H = 2n, then (3) cannot be written in the form (2) (see Molnár s article).

12 Generalized Uhlhorn theorem on P n (H) Theorem (Győry Šemrl, ; G. Šemrl, 2016) Assume that dim H > 2n, and that the surjective map φ : P n (H) P n (H) preserves orthogonality in both directions: P Q φ(p) φ(q). Then there exists a unitary/antiunitary operator U : H H such that φ(p) = UPU (P P n (H)). The main idea of the proof is to show that φ preserves adjacency: P and Q P n (H) are said to be adjacent, in notation P Q, if dim(imp ImQ) = n 1. And then we apply Chow s theorem (if dim H < ).

13 Lemma For P, Q P n (H), P Q, the following are equivalent: P Q; R P n (H) \ {P, Q} we have #({R} {P, Q} ) 1. Theorem (Chow, 1949) Assume that 2n dim H < and that the bijection φ: P n (H) P n (H) satisfies P Q φ(p) φ(q) (P, Q P n (H)). Then there exists a bijective semi-linear map A: H H such that either Imφ(P) = A(ImP) (P P n (H)), or dim H = 2n and Imφ(P) = (A(ImP)) (P P n (H)).

14 Generalized bijective Wigner theorem on P n (H) Theorem (Botelho Jamison Molnár, 2013; G. Šemrl, 2016) Assume that n < dim H, and that the surjective map φ : P n (H) P n (H) satisfies φ(p) φ(q) = P Q (P, Q P n (H)). Then there exists a unitary/antiunitary operator U such that either φ(p) = UPU (P P n (H)), or dim H = 2n and φ(p) = U(I P)U (P P n (H)). Remark: Mori generalized this theorem further (Tuesday, last talk). The main idea of the proof: P Q if and only if P Q = 1 and M(P, Q) := {R P n (H): P R = Q R = sin π 4 } is a compact manifold.

15 Generalized non-bijective Wigner on P n (H) Part 2 The following result generalizes Molnár s theorem: Theorem (G., 2017) Let dim H > n 2 and φ: P n (H) P n (H) be a transformation that satisfies Trφ(P)φ(Q) = TrPQ (P, Q P n (H)). Then either φ is induced by a linear/conjugatelinear isometry V φ(p) = VPV (P P n (H)), or we have dim H = 2n and φ(p) = I VPV (P P n (H)). Molnár s idea (from his 2001 paper): such a map can be uniquely extended linearly to F sa (H) so that we still have Trφ(A)φ(B) = TrAB (A, B F sa (H)).

16 Idea of the proof in 2n-dimensions (the core of the proof) Note that P n (H) is a compact and connected manifold, thus by Brouwer s domain invariance theorem every injective and continuous map is actually a homeomorphism. As TrPQ = n P Q 2 HS, φ is an isometry w.r.t. the Hilbert-Schmidt norm, thus continuous and injective. For P, Q P n (H) we define the set A P,Q = {R P n (H): P + Q R P n (H)}. Lemma A P,Q is a 1-dimensional (real) manifold iff P Q and PQ QP. Then by continuity one can show that φ preserves adjacency in both directions, and use Chow s theorem. Remark: Qian, Wang, Wua and Yuan have recently generalized this theorem further.

17 A linear generalization of Wigner s theorem Theorem (Stromer, 2017; Sarbicki Chruscinski Mozrzymas, 2016; Aniello Chruscinski, 2017; Pankov, 2018) Let dim H > n and assume that L: F sa (H) F sa is an injective linear map such that L(P n (H)) P n (H). Then either L is induced by a linear/conjugatelinear isometry V or dim H = 2n and L(A) = VAV L(A) = TrA n I VAV (A F sa (H)), (A F sa (H)). Open Problem: What happens for non-injective maps? A trivial example: L(A) = TrA n P, where P P n(h) is fixed.

18 Uhlhorn s theorem for angles 0 < α < π 2 Theorem (Li Plevnik Šemrl, 2012; G., 2018) Assume that H is a real Hilbert space with dim H 3, 0 < α < π 2, and φ: P 1 (H) P 1 (H) is a bijective map which satisfies TrPQ = cos 2 α Trφ(P)φ(Q) = cos 2 α (P, Q P 1 (H)). Then φ is induced by a bijective linear isometry O : H H, φ(p) = OPO (P P 1 (H)). The proof first establishes a similar result on the unit sphere of H a generalization of Everling s theorem. When α < π 3, then the set {R P 1(H): TrPR = cos 2 α} is basically a sphere. The restriction of φ to this sphere preserves a spherical angle (determined by α), so the generalized Everling theorem can be applied. For α π 3, the story is much more complicated and technical, but we can still succeed using the generalized Everling theorem.

19 Theorem (G., 2018) Suppose that 0 < α < π 2 and that the bijective map φ: P 1 (C 2 ) P 1 (C 2 ) satisfies TrPQ = cos 2 α Trφ(P)φ(Q) = cos 2 α (P, Q P 1 (C 2 )). Then (i) either φ is induced by a unitary/antiunitary operator: (ii) or α = π 4 and φ(p) = UPU (P P 1 (C 2 )), φ(p) {I UPU, UPU } (P P 1 (C 2 )). We only have to use Bloch s transformation which translates the problem into a problem on the unit sphere of R 3. Thus we can apply the aforementioned generalized Everling s theorem.

20 Theorem (G., 2018) Suppose that H is a complex Hilbert space, 3 dim H and 0 < α π 4. Assume that φ: P 1(H) P 1 (H) is a bijective map which satisfies TrPQ = cos 2 α Trφ(P)φ(Q) = cos 2 α (P, Q P 1 (H)). Then φ is induced by a unitary/antiunitary operator U, φ(p) = UPU (P P 1 (H)). Open Problem: What happens for α > π 4 in the complex case? The main idea: For any P, Q P 1 (H) we consider the set P α Q α = {R P 1 (H): TrPR = TrQR = cos 2 α}. We can prove that cos 2 α = min{trr 1 R 2 : R 1, R 2 P α Q α } if and only if TrPQ = cos 2 γ, where γ is uniquely determined by α. Then we show that φ preserves the angle γ too. Then obtain more angles which are preserved.

21 Generalized bijective Wigner s theorem on P (H) A crucial difference here: φ does NOT necessarily preserve orthogonality, unlike in all of the previous results! P (H) the infinite Grassmannian: P (H) = {P B sa (H): P 2 = P, dim(imp) = dim(ker P) = ℵ 0 }. Theorem (G. Šemrl, 2018) Assume that dim H = ℵ 0 and φ: P (H) P (H) is a surjective map which satisfies φ(p) φ(q) = P Q (P, Q P (H)). Then there exists a unitary/antiunitary operator U such that either or φ(p) = UPU φ(p) = U(I P)U (P P (H)), (P P (H)).

22 The main idea of the proof We investigate the geodesic structure of P (H): Lemma Let P, Q P (H) with P Q = 1. The following are equivalent: P Q or I P I Q; for every R P (H) with R P = R Q = sin π 4 = 1 2 there exists exactly one mapping γ : [0, π 2 ] P (H) such that γ(0) = P, γ ( π 2 ) = Q, and γ ( π 4 ) = R, and γ(θ 1 ) γ(θ 2 ) = sin θ 1 θ 2 (θ 1, θ 2 [0, π 2 ]).

23 References E.P. Wigner, Gruppentheorie und ihre Anwendung auf die Quantenmechanik der Atomspektrum, Fredrik Vieweg und Sohn, J.S. Lomont and P. Mendelson, The Wigner unitary-antiunitary theorem, Ann. Math. 78 (1963), V. Bargmann, Note on Wigner s theorem on symmetry operations, J. Math. Phys. 5 (1964), U. Uhlhorn, Representation of symmetry transformations in quantum mechanics, Ark. Fysik 23 (1963), C.-A. Faure, An elementary proof of the fundamental theorem of projective geometry, Geometriae Dedicata 90 (2002), A. Böttcher and I.M. Spitkovsky, A gentle guide to the basics of two projections theory, Linear Algebra Appl. 432 (2010), L. Molnár, Transformations on the set of all n-dimensional subspaces of a Hilbert space preserving principal angles, Comm. Math. Phys. 217 (2001), L. Molnár, Maps on the n-dimensional subspaces of a Hilbert space preserving principal angles, Proc. Amer. Math. Soc. 136 (2008),

24 References P. Šemrl, Orthogonality preserving transformations on the set of n-dimensional subspaces of a Hilbert space, Illinois J. Math. 48 (2004), M. Győry, Transformations on the set of all n-dimensional subspaces of a Hilbert space preserving orthogonality, Publ. Math. Debrecen 65 (2004), G.P. Gehér and P. Šemrl, Isometries of Grassmann spaces, J. Funct. Anal. 270 (2016), W.-L. Chow, On the geometry of algebraic homogeneous spaces, Ann. Math. (2) 50 (1949), F. Botelho, J. Jamison, and L. Molnár, Surjective isometries on Grassmann spaces, J. Funct. Anal. 265 (2013), M. Mori, Isometries between projection lattices of von Neumann algebras, arxiv: , preprint. G.P. Gehér, Wigner s theorem on Grassmann spaces, J. Funct. Anal. 273 (2017),

25 References W. Qian, L. Wang, W. Wu and W. Yuan, Wigner-Type Theorem on transition probability preserving maps in semifinite factors, J. Funct. Anal., in press. E. Stormer, Positive maps which map the set of rank k projections onto itself, Positivity 21 (2017), G. Sarbicki, D. Chruscinski, M. Mozrzymas, Generalising Wigner s theorem, J. Phys. A: Math. Theor. 49 (2016), 7 p. P. Aniello, D. Chruscinski, Symmetry witnesses, J. Phys. A: Math. Theor. 50 (2017), 16 p. M. Pankov, Linear transformations preserving projections of fixed finite rank, arxiv: , preprint C.-K. Li, L. Plevnik, P. Šemrl, Preservers of matrix pairs with a fixed inner product value, Oper. Matrices 6 (2012), G.P. Gehér, Symmetries of projective spaces and spheres, Int. Math. Res. Not. IMRN, accepted for publication. G.P. Gehér and P. Šemrl, Isometries of Grassmann spaces II, Adv. Math. 332 (2018)

26 Thank you