An elementary proof of Wigner's theorem on quantum mechanical symmetry transformations

An elementary proof of Wigner's theorem on quantum mechanical symmetry transformations University of Szeged, Bolyai Institute and MTA-DE "Lendület" Functional Analysis Research Group, University of Debrecen 5th June 2014, Ljubljana LAW'14

Introduction

Wigner's theorem Theorem Let H be a complex Hilbert space and let S denote the set of unit vectors of H. Let us consider an arbitrary mapping φ: S S such that the following holds: u, v = φ( u), φ( v) ( u = v = 1). (W's C) Then there exists a linear or a conjugatelinear isometry W : H H and a function f : S T := {z C: z = 1} such that we have φ( u) = f ( u) W u is satised for every unit vector u H. Remark: Originally Wigner assumed bijectivity of φ.

E. P. Wigner (1931) the proof was not complete. J. A. Lomont and P. Mendleson (1963) rst known proof for the classical (i.e. bijective) case. V. Bargmann (1964) His proof follows the thoughtline suggested by Wigner. Figure: Valentine Bargmann

U. Uhlhorn (1963) A nice generalization (dim H 3, φ is bijective and preserves orthogonality in both directions). Figure: Ulf Uhlhorn

L. Molnár (1996) algebraic approach, he managed to generalize Wigner's theorem in several ways. Figure: Molnár Lajos

P 1 := P 1 (H) the set of rank-one (self-adjoint) projections. If u = 1, then P[ u] the rank-one projection with precise range C u. u, v 2 = Tr P[ u]p[ v] } {{ } P[ u] P[ v] = transition probability 1 u, v 2..

Wigner's theorem, two reformulations Theorem Let us consider an arbitrary mapping φ: P 1 P 1 for which the following holds Tr P[ u]p[ v] = Tr f (P[ u])f (P[ v]) ( u = v = 1). (W's C) Then there is a linear or antilinear isometry W : H H such that Theorem φ(p[ u]) = WP[ u]w = P[W u] ( u = 1). For every isometry φ: P 1 P 1 there exists a linear or an antilinear isometry W : H H such that we have φ(p[ u]) = WP[ u]w = P[W u] ( u = 1).

Resolving sets

Denition (S. Bau and A. F. Beardon) Let (X, d) be a metric space and D, R X. We say that R is a resolving set of D if for every two points x 1, x 2 D whenever d(x 1, y) = d(x 2, y) is satised for all y R, we necessarily have x 1 = x 2.

Denition (S. Bau and A. F. Beardon) Let (X, d) be a metric space and D, R X. We say that R is a resolving set of D if for every two points x 1, x 2 D whenever d(x 1, y) = d(x 2, y) is satised for all y R, we necessarily have x 1 = x 2. dim H = N N {ℵ 0 }, N > 1. We x an orthonormal base: { e j } N j=1. Let v j := v, e j denote the jth coordinate of a unit vector v. The set D := {P[ v]: v j 0, j} P 1 is dense in P 1.

Lemma Let H be an arbitrary separable (nite or innite-dimensional) Hilbert space. Then the set R = {P[ e j ]} N { P[ 1 ( e j=1 2 j e j+1 )], P[ 1 2( e j + i e j+1 )] } 1 j<n resolves D.

Lemma Let H be an arbitrary separable (nite or innite-dimensional) Hilbert space. Then the set R = {P[ e j ]} N { P[ 1 ( e j=1 2 j e j+1 )], P[ 1 2( e j + i e j+1 )] } 1 j<n resolves D. Proof. An easy calculation.

Proof of Wigner's theorem (in the separable case)

Proof. Let P[ f j ] = φ(p[ e j ]), then { f j } N j=1 H := { f j } N j=1. is an ONS.

Proof. Let P[ f j ] = φ(p[ e j ]), then { f j } N j=1 H := { f j } N j=1. is an ONS. Set a unit vector v H and let φ(p[ v]) = P[ w]. (W's C) implies v j = w, f j ( j), and from Parseval's identity we get w H.

Proof. Let P[ f j ] = φ(p[ e j ]), then { f j } N j=1 H := { f j } N j=1. is an ONS. Set a unit vector v H and let φ(p[ v]) = P[ w]. (W's C) implies v j = w, f j ( j), and from Parseval's identity we get w H. We dene a linear isometry V : H H H, V e j = f j (j N N ). The mapping φ 1 ( ) := V φ( )V satisfy (W's C).

Proof. Let P[ f j ] = φ(p[ e j ]), then { f j } N j=1 H := { f j } N j=1. is an ONS. Set a unit vector v H and let φ(p[ v]) = P[ w]. (W's C) implies v j = w, f j ( j), and from Parseval's identity we get w H. We dene a linear isometry V : H H H, V e j = f j (j N N ). The mapping φ 1 ( ) := V φ( )V satisfy (W's C). Moreover φ 1 (P[ e j ]) = V φ(p[ e j ])V = V P[ f j ]V = P[V fj ] = P[ e j ] ( j).

Again from (W's C) ( φ 1 P[1/ 2 ( ej e j+1 )] ) = P[1/ 2 ( e j δ j+1 e j+1 )] ( φ 1 P[1/ 2 ( ej + i e j+1 )] ) = P[1/ 2 ( e j ε j+1 e j+1 )] where δ j+1 = ε j+1 = 1 (1 j < N).

Again from (W's C) φ 1 ( P[1/ 2 ( ej e j+1 )] ) = P[1/ 2 ( e j δ j+1 e j+1 )] φ 1 ( P[1/ 2 ( ej + i e j+1 )] ) = P[1/ 2 ( e j ε j+1 e j+1 )] where δ j+1 = ε j+1 = 1 (1 j < N). Therefore 2 = 1 + δj+1 ε j+1, and consequently δ j+1 = ±iε j+1 (j < N).

Again from (W's C) φ 1 ( P[1/ 2 ( ej e j+1 )] ) = P[1/ 2 ( e j δ j+1 e j+1 )] φ 1 ( P[1/ 2 ( ej + i e j+1 )] ) = P[1/ 2 ( e j ε j+1 e j+1 )] where δ j+1 = ε j+1 = 1 (1 j < N). Therefore 2 = 1 + δj+1 ε j+1, and consequently δ j+1 = ±iε j+1 (j < N). We dene φ 2 ( ) := U φ 1 ( )U, where 1 If ε 2 = iδ 2, then let U be the unitary operator such that U e 1 = e 1, k U e k = e k (k > 1). 2 If ε 2 = iδ 2, then let U be the antiunitary operator dened by the equations above. j=2 δ j

Moreover, φ 2 (P[ e j ]) = P[ e j ] ( j), φ 2 (P[1/ 2 ( e j e j+1 )]) = P[1/ 2 ( e j e j+1 )] ( j), φ 2 (P[1/ 2 ( e 1 + i e 2 )]) = P[1/ 2 ( e 1 + i e 2 )].

Moreover, φ 2 (P[ e j ]) = P[ e j ] ( j), φ 2 (P[1/ 2 ( e j e j+1 )]) = P[1/ 2 ( e j e j+1 )] ( j), φ 2 (P[1/ 2 ( e 1 + i e 2 )]) = P[1/ 2 ( e 1 + i e 2 )]. We also have φ 2 (P[1/ 2 ( e j + i e j+1 )]) = P[1/ 2 ( e j ± i e j+1 )] (1 < j < N).

Moreover, φ 2 (P[ e j ]) = P[ e j ] ( j), φ 2 (P[1/ 2 ( e j e j+1 )]) = P[1/ 2 ( e j e j+1 )] ( j), φ 2 (P[1/ 2 ( e 1 + i e 2 )]) = P[1/ 2 ( e 1 + i e 2 )]. We also have φ 2 (P[1/ 2 ( e j + i e j+1 )]) = P[1/ 2 ( e j ± i e j+1 )] (1 < j < N). Assume that there exists an index j > 1 for which φ 2 (P[1/ 2 ( e j + i e j+1 )]) = P[1/ 2 ( e j i e j+1 )] holds. We may assume that this j is the rst such index.

Claim Then φ 2 (P[v j 1 e j 1 + t e j + v j+1 e j+1 ]) = P[v j 1 e j 1 + t e j + v j+1 e j+1 ] holds for every t > 0, v j 1 0, v j+1 0, v j 1 2 + t 2 + v j+1 2 = 1.

Claim Then φ 2 (P[v j 1 e j 1 + t e j + v j+1 e j+1 ]) = P[v j 1 e j 1 + t e j + v j+1 e j+1 ] holds for every t > 0, v j 1 0, v j+1 0, v j 1 2 + t 2 + v j+1 2 = 1. Proof of Claim An easy calculation.

Claim Then φ 2 (P[v j 1 e j 1 + t e j + v j+1 e j+1 ]) = P[v j 1 e j 1 + t e j + v j+1 e j+1 ] holds for every t > 0, v j 1 0, v j+1 0, v j 1 2 + t 2 + v j+1 2 = 1. Proof of Claim An easy calculation. Now, let x = 1 2 e j 1 + 1 2 e j + 1 2 e j+1, y = i 2 e j 1 + 1 2 e j + i 2 e j+1.

Claim Then φ 2 (P[v j 1 e j 1 + t e j + v j+1 e j+1 ]) = P[v j 1 e j 1 + t e j + v j+1 e j+1 ] holds for every t > 0, v j 1 0, v j+1 0, v j 1 2 + t 2 + v j+1 2 = 1. Proof of Claim An easy calculation. Now, let x = 1 2 e j 1 + 1 2 e j + 1 2 e j+1, y = i 2 e j 1 + 1 2 e j + i 2 e j+1. Claim and (W's C) implies 2/4 = i/4 + 1/4 i/2 = i/4 + 1/4 + i/2 = 10/4, which is a contradiction.

Claim Then φ 2 (P[v j 1 e j 1 + t e j + v j+1 e j+1 ]) = P[v j 1 e j 1 + t e j + v j+1 e j+1 ] holds for every t > 0, v j 1 0, v j+1 0, v j 1 2 + t 2 + v j+1 2 = 1. Proof of Claim An easy calculation. Now, let x = 1 2 e j 1 + 1 2 e j + 1 2 e j+1, y = i 2 e j 1 + 1 2 e j + i 2 e j+1. Claim and (W's C) implies 2/4 = i/4 + 1/4 i/2 = i/4 + 1/4 + i/2 = 10/4, which is a contradiction. Therefore φ 2 is the identity mapping on P 1, and φ(p[ u]) = WP[ u]w with W = VU.

Remarks The proof is short and elementary, it does not contain any serious computations.

Remarks The proof is short and elementary, it does not contain any serious computations. The non-separable case can be proven as a consequence of the separable case (technical).

Remarks The proof is short and elementary, it does not contain any serious computations. The non-separable case can be proven as a consequence of the separable case (technical). It works for the general case (bijcetivity of φ or separability of H is not assumed).

Remarks The proof is short and elementary, it does not contain any serious computations. The non-separable case can be proven as a consequence of the separable case (technical). It works for the general case (bijcetivity of φ or separability of H is not assumed). Note that we could consider φ: P 1 (H) P 1 (K) with some Hilbert spaces H and K.

Remarks The proof is short and elementary, it does not contain any serious computations. The non-separable case can be proven as a consequence of the separable case (technical). It works for the general case (bijcetivity of φ or separability of H is not assumed). Note that we could consider φ: P 1 (H) P 1 (K) with some Hilbert spaces H and K. The proof is similar (but easier) in real Hilbert spaces.

References 1/2 E. P. Wigner, Gruppentheorie und ihre Anwendung auf die Quantenmechanik der Atomspektrum, Fredrik Vieweg und Sohn, 1931. J. S. Lomont and P. Mendelson, The Wigner unitary-antiunitary theorem, Ann. Math. 78 (1963), 548559. V. Bargmann, Note on Wigner's theorem on symmetry operations, J. Math. Phys. 5 (1964), 862868. U. Uhlhorn, Representation of symmetry transformations in quantum mechanics, Ark. Fysik 23 (1963), 307340.

References 2/2 L. Molnár, Wigner's unitary-antiunitary theorem via Herstein's theorem on Jordan homeomorphisms, J. Nat. Geom. 10 (1996), 137148. L. Molnár, An algebraic approach to Wigner's unitary-antiunitary theorem, J. Austral. Math. Soc. 65 (1998), 354369. Gy. P. Gehér, An elementary proof for the non-bijective version of Wigner's theorem, to appear in Phys. Lett. A.

This research was also supported by the European Union and the State of Hungary, co-nanced by the European Social Fund in the framework of TÁMOP-4.2.4.A/2-11/1-2012-0001 'National Excellence Program'

Thank You for Your Kind Attention