An alternative proof of Wigner theorem on quantum transformations based on elementary complex analysis
aa r X i v : . [ qu a n t - ph ] S e p An alternative proof of Wigner theorem on quantumtransformations based on elementary complex analysis
Amaury Mouchet
Laboratoire de Math´ematiques et de Physique Th´eoriqueUniversit´e Fran¸cois Rabelais de Tours — cnrs (umr 7350)
F´ed´eration Denis PoissonParc de Grandmont 37200Tours, France.
Abstract
According to Wigner theorem, transformations of quantum states which preserve theprobabilities are either unitary or antiunitary. This short communication presents anelementary proof of this theorem that significantly departs from the numerous ones al-ready existing in the literature. The main line of the argument remains valid even inquantum field theory where Hilbert spaces are non-separable.
Keywords:
Wigner theorem, symmetry, unitary representations
Highlights:
A new elementary proof of Wigner theorem on quantum transformations ispresented. It relies on substantially different hypothesis from the previous proofs. Itallows a straightforward generalisation to non-separable Hilbert spaces. pacs:
1. Preliminaries
Wigner theorem is a cornerstone of theoretical physics since it encapsulates all thelinear structure of quantum transformations, among which the evolution of quantumsystems (aside from the measurement process). More precisely, given a Hilbert space H endowed with a Hermitian product h χ | ψ i Wigner, in the early 30’s (Wigner, 1959, Appendix to chap. 20, pp 233–236, for theupdated English translation), acknowledged that any transformation T : | ψ i 7→ | ψ i T = T (cid:0) | ψ i (cid:1) such that ∀ (cid:0) | χ i , | ψ i (cid:1) ∈ H , (cid:12)(cid:12) T h χ | ψ i T (cid:12)(cid:12) = (cid:12)(cid:12) h χ | ψ i (cid:12)(cid:12) (1)is either(a) linear T (cid:0) c | ψ i + c | ψ i (cid:1) = (cid:0) c | ψ i T + c | ψ i T (cid:1) and unitary T − = T ∗ ;or Email address: [email protected] (Amaury Mouchet)
Preprint submitted to Elsevier October 31, 2018 b) antilinear T (cid:0) c | ψ i + c | ψ i (cid:1) = c ∗ | ψ i T + c ∗ | ψ i T and unitary T − = T ∗ (such amap is also called antiunitary).We shall systematically use the usual Dirac bra-ket notation and, in the above de-finitions, ( c , c ) stands for any pair of complex coefficients, (cid:0) | ψ i , | ψ i (cid:1) is any pair ofelements of H . The Hermitian conjugate will be denoted by ( ) ∗ and therefore, if z isjust one complex number, z ∗ stands for its complex conjugate.Since Wigner’s original work that pertained to the representation of the rotationgroup, many proofs and generalizations have been proposed whose levels of rigour arenot necessarily correlated to their length but, rather, vary depending on the concernof their author. See for example (Simon et al., 2008) and references therein to which Ishall add (Fonda and Ghirardi, 1970, § § XV-2). The present work is further added to this list because itappears to be almost a back-of-the-envelope presentation while keeping a level of rigourthat is acceptable by physicists (hopefully, little additionnal work on the main key ideasshould meet to the requirements of mathematicians as well). Moreover, the majorityof the previous proofs, if not all of them, can hardly be transposed to non-separableHilbert spaces, that is to spaces where a countable orthonormal basis does not exist.Yet, in quantum field theory, such non-separable Hilbert spaces are unavoidable: anycontinuous canonical transformation or rearrangement of the infinitely many degreesof freedom — that physically describe a renormalisation of the bare particles into thedressed ones like, for instance, when a condensation occurs — requires that the Hilbertspace is made of a continuous family of orthogonal Fock spaces and one cannot contentone self with the unique Fock space that represents the physical (dressed) particles; seefor example (Emch, 1972, § (cid:0) | ϕ ν i (cid:1) ν ∈ I in H , h ϕ ν | ϕ µ i = δ νµ , (2)but the discreteness of the set I will be just a matter of convenience. The transpositionto a continuous (multi-)index is straightforward (we will not use any induction argumentsthat would prevent such generalization) and physicists are quite used to it. For instance,with an appropriate choice of normalization, the Kronecker symbol in (2) is replaced bya Dirac distribution, the discrete sequences z = ( z ν ) ν ∈ I labelled by ν become regularfunctions z : ν z ( ν ), functions of z become functionals, matrices are turned intooperators, etc.Unlike in many proofs, we shall not suppose a priori that T is bijective. The onlyadditional property will be that T is differentiable twice (just once may be sufficient butwe will not try no minimize the requirements on the regularity of the transformation).Let us just mention that such smoothness is a reasonable supposition based on physicalgrounds. Except during the measuring process, when the number of degrees of freedominvolved in the interaction of the system with a measuring device becomes infinite —and we know that the superposition principle as well as condition (1) are lost by thetransformation on the states induced by the measuring process —, we have never observedany discontinuity nor singularity with respect to the quantum state in a transformation.To see how this differentiable hypothesis simplifies considerably the proof of Wignertheorem by offering a simple strategy, let us see how this works in the Euclidean case2here the analogous of the Wigner theorem is known as Mazur-Ulam theorem(Mazur and Ulam,1932, this first version, concerning more generally isometries in real normed vector spaces,was published the following year after Wigner’s). Let us suppose that T is a differentiableapplication in an Euclidean space E preserving the scalar product “ · ” i.e. such that ∀ (cid:0) u, v (cid:1) ∈ E , T ( u ) · T ( v ) = u · v . (3)Then, differentiating (3) with respect to u and v leads to ∀ (cid:0) u, v (cid:1) ∈ E , (cid:0) T ′ ( u ) (cid:1) t T ′ ( v ) = 1 (4)(“ t ” denotes the transposition and “ ′ ” the derivative) which shows immediately thatthe Jacobian matrix T ′ ( v ) is invertible, independent of v and indeed orthogonal. Sincemoreover T (0) = 0 (the scalar product is non degenerate), we have proven that T isnecessary a linear orthogonal transformation (it is obviously a sufficient condition).Under the alternative hypothesis of being surjective, the Mazur-Ulam theorem isalso immediate since, then, the orthonormal basis ( e ν ) ν ∈ I is mapped to an orthonormalbasis ( e T ν ) ν ∈ I on which, for any v in E , we can expand v T and then, v T = X ν ( e T ν · v T ) e T ν = X ν ( e ν · v ) e T ν . (5)If there were a non-null vector w in E that were orthogonal to the subspace spannedby ( e T ν ) ν ∈ I , then its preimage T − ( w ), whose existence is guaranteed by the surjectivityhypothesis, would be the null vector, being a vector orthogonal to any e ν . This is incontradiction with T (0) = 0. Relation (5) can be written as T X ν ( e ν · v ) e ν ! = X ν ( e ν · v ) T ( e ν ) (6)which expresses the linearity of T whose orthogonality follows. However, in the Hermi-tian case, the possible phase factors that may appear in the relation between T h ϕ ν | ψ i T and h ϕ ν | ψ i make the proofs following the second line of thought much less straightfor-ward.
2. The core of the proof
The invariance property formulated in terms of complex analysis.
We shall take advan-tage of the isomorphism that allows us to identify the Hilbert space H to the set ofsequences of complex numbers z = ( z ν ) ν ∈ I . More precisely, once given an orthonomalbasis (cid:0) | ϕ ν i (cid:1) ν ∈ I , we can map any | ψ i in H to z with z ν = h ϕ ν | ψ i . The Hermitian productin H reads h w | z i def = w ∗ z = X ν w ∗ ν z ν = h χ | ψ i (7)with z = (cid:0) h ϕ ν | ψ i (cid:1) ν ∈ I and w = (cid:0) h ϕ ν | χ i (cid:1) ν ∈ I . Any application T in H can be seenas a either a function T ( x, y ) of the sequence of the real part x = Re z and imaginary3art y = Im z of the components of the state, or rather as a function of two independent complex sequences z and ¯ zT ( z, ¯ z ) def = T (cid:0) (¯ z + z ) / , i(¯ z − z ) /
2) (8)evaluated at ¯ z = z ∗ . When T can be differentiated, we can define the derivatives withrespect to the complex variables by (Dubrovine et al., 1984, § ∂ z T def = 12 ( ∂ x − i ∂ y ) T ; (9a) ∂ ¯ z T def = 12 ( ∂ x + i ∂ y ) T . (9b)In particular, T is analytic if and only if ∂ ¯ z T = 0. From the differential of T , we get ∂ z ∗ (cid:0) T ( z, z ∗ ) (cid:1) = ( ∂ ¯ z T )( z, z ∗ ) . (10)In the following we will drop the distinction between z ∗ and the variable ¯ z that may varyindependently of z because the continuation of any application T ( z, ¯ z ) defined by (8) inthe domain where ¯ z = z ∗ is unique.In this language, the invariance condition (1) can be reformulated as follows, ∀ ( w, z ) ∈ H , (cid:12)(cid:12)(cid:0) T ( w, w ∗ ) (cid:1) ∗ T ( z, z ∗ ) (cid:12)(cid:12) = | w ∗ z | . (11)Therefore we have two possibilities : there exists a real function θ of the complex vari-ables ( w, w ∗ , z, z ∗ ) such that(a) either (cid:0) T ( w, w ∗ ) (cid:1) ∗ T ( z, z ∗ ) = e i θ ( w,w ∗ ,z,z ∗ ) w ∗ z ; (12a)(b) or (cid:0) T ( z, z ∗ ) (cid:1) ∗ T ( w, w ∗ ) = e − i θ ( w,w ∗ ,z,z ∗ ) w ∗ z . (12b)To understand where these two conditions come from, first divide the both sides of theequality in (11) by | w ∗ z | when non vanishing (then eventually include these cases bycontinuity of the transformation T ). Thus, we are led to an equation of the form | Z | = 1where Z is a complex number. Then, the argument η of Z can take the two solutions ± arccos(Re Z ) in [ − π, π ] of the equation cos η = Re Z . The two options (a) and (b)correspond to the two forms e ± i arccos(Re Z ) that is Z = e i arccos(Re Z ) or Z ∗ = e i arccos(Re Z ) .Permuting ( w, w ∗ ) and ( z, z ∗ ), then by conjugation we have necessarily θ ( z, z ∗ , w, w ∗ ) = − θ ( w, w ∗ , z, z ∗ ) (13)which incidentally shows that θ (0 , , ,
0) = 0.
Adjusting the phase.
The first step of our proof is to redefine the phase of the transformedstates by considering e T ( z, z ∗ ) = e i α ( z,z ∗ ) T ( z, z ∗ ) (14)where α is a real function. Conditions (12) can be written(a) either (cid:0) e T ( w, w ∗ ) (cid:1) ∗ e T ( z, z ∗ ) = e i˜ θ ( w,w ∗ ,z,z ∗ ) w ∗ z ; (15a)(b) or (cid:0) e T ( z, z ∗ ) (cid:1) ∗ e T ( w, w ∗ ) = e − i˜ θ ( w,w ∗ ,z,z ∗ ) w ∗ z . (15b)4ith ˜ θ ( w, w ∗ , z, z ∗ ) = θ ( w, w ∗ , z, z ∗ ) + α ( z, z ∗ ) − α ( w, w ∗ ) . (16)The latter identity can be used to cancel ˜ θ (0 , , z, z ∗ ) with the choice α ( z, z ∗ ) = − θ (0 , , z, z ∗ ) = θ ( z, z ∗ , , . (17)Therefore without loss of generality, in the following we can consider that for all zθ (0 , , z, z ∗ ) = 0 . (18) Differentiation.
The second step of the proof is to differentiate the conditions writtenin (12) with respect to w ∗ and z . From (12a), using ∂ w ∗ (cid:0) T ( w, w ∗ ) (cid:1) ∗ = (cid:0) ∂ w T ( w, w ∗ ) (cid:1) ∗ ,we get (cid:0) ∂ w T ( w, w ∗ ) (cid:1) ∗ ∂ z T ( z, z ∗ ) = e i θ ( w,w ∗ ,z,z ∗ ) n w ∗ ∂ w ∗ θ ( w, w ∗ , z, z ∗ ) + i ∂ z θ ( w, w ∗ , z, z ∗ ) z + (cid:2) i ∂ z,w ∗ θ ( w, w ∗ , z, z ∗ ) − ∂ z θ ( w, w ∗ , z, z ∗ ) ∂ w ∗ θ ( w, w ∗ , z, z ∗ ) (cid:3) w ∗ z o . (19)Because of this last computation, and in particular because of the presence of the Hes-sian of θ , we assume that the transformation is differentiable twice. When evaluatedfor ( w, w ∗ ) = (0 , (cid:0) ∂ z T (0 , (cid:1) ∗ ∂ z T ( z, z ∗ ) = 1 (20)since (18) implies ∂ z θ (0 , , z, z ∗ ) = 0. Then (cid:0) ∂ z T ( z, z ∗ ) (cid:1) − = (cid:0) ∂ z T (0 , (cid:1) ∗ appears to beindependent of ( z, z ∗ ) and unitary. Moreover, because T (0 ,
0) = 0, we have, for all z , T ( z, z ∗ ) = ∂ z T (0 , z (21)and therefore z z T = T ( z, z ∗ ) is linear. The linear operator ˆ U defined by its matrixelements h φ ν | ˆ U | φ ν ′ i def = ( ∂ z T (0 , νν ′ is such that ˆ U − = ˆ U ∗ and the expression (21) canbe written as h ϕ ν | ψ i T = P ν ′ h ϕ ν | ˆ U | ϕ ν ′ i h ϕ ν ′ | ψ i = h ϕ ν | ˆ U | ψ i , hence | ψ i T = ˆ U | ψ i which isthe first option offered by Wigner theorem.The second option comes when differentiating (12b) which just modifies the left-handside of (19) and the irrelevant sign in front of θ in the right-hand side. Then we obtain (cid:0) ∂ z ∗ T (0 , (cid:1) ∗ ∂ z ∗ T ( z, z ∗ ) = 1 (22)and eventually T ( z, z ∗ ) = ∂ z ∗ T (0 , z ∗ . (23)The transformation z z T = T ( z, z ∗ ) is antilinear. An antilinear operator ˆ A can be de-fined from its matrix elements h φ ν | (cid:0) ˆ A | φ ν ′ i (cid:1) def = ( ∂ z ∗ T (0 , νν ′ and we have h φ ν | ψ i T = P ν ′ h φ ν | (cid:16) ˆ A | φ ν ′ i (cid:17) h φ ν ′ | ψ i ∗ = h φ ν | (cid:16) ˆ A P ν ′ | φ ν ′ i h φ ν ′ | ψ i (cid:17) = h φ ν | (cid:16) ˆ A | ψ i (cid:17) thatis | ψ i T = ˆ A | ψ i . The relation (cid:0) ∂ z ∗ T (0 , (cid:1) − = (cid:0) ∂ z ∗ T (0 , (cid:1) ∗ reads ˆ A − = ˆ A ∗ . Thereforecase (b) implies the second alternative of Wigner theorem.5 ummary. Looking back the proof over one’shoulder, once keeping in mind the simpleline of thought used in the real case — the double differentiation of (3) that leadsto (4)—, the backbone of the argument in the Hilbert case do not appear much morecomplicated and can be captured in the following few lines : from the key equations (12),it is straightforward by a double differentiation to obtain (19). Then having redefinedthe phases of the transformed states in order to get (18), we are immediately led to (20)or (22) then to (21) or (23) respectively.
Acknowledgements
It is a pleasure to thanks Xavier Bekaert for the substantial improvements he sug-gested after his careful reading of the first version of this note.
References
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