### WIGNER`S THEOREM AND ITS GENERALIZATIONS

```HANDBOOK O F QUANTUM LOGIC AND QUANTUM STRUCTURES: QUANTUM STRUCTURES
Edited by K. Engesser, D. M. Gabbay and D. Lehmann
429
WIGNER'S THEOREM AND ITS
GENERALIZATIONS
Georges Chevalier
1 INTRODUCTION
Invariance principles play an important role in physics and, as Wigner noted in
[Houtappel et al., 19651, they serve as a guide in the search for new laws of nature
and as tools for obtaining properties of the solutions of equations provided by
the laws of nature. Newtonian mechanics and Einstein's theory of relativity are
remarkable illustrations of these ideas.
In his book "Gruppentheorie und ihre Anwendung auf die Quantenmechanzk der
Atomspektren" published in 1931, Wigner postulated that the transition probability between two states has an invariant physical sense. This assumption led him
to consider transformations of the states of a physical system which preserve the
transition probability associated to any pair of states. He proved that any such
transformation, called a symmetry transformation, is induced either by a unitary
or by an antiunitary operator on the Hilbert space associated to the physical system. This result is, today, known as Wigner's theorem. In the particular case of a
conservative physical system, Wigner's theorem together with certain reasonable
physical assumptions permits to deduce the Schrodinger equation [Simon, 19761.
From the mathematical point of view, the proof given by Wigner is incomplete.
Strangely enough, more than thirty years separate the publication of Wigner's
book and the first correct proof of his theorem by U. Uhlhorn [1962].
Uhlhorn's paper is of great interest in another respect, namely because he replaced the requirement of the invariance of the transition probabilities by the
requirement that any pair of orthogonal states, also called incoherent states in the
physical literature, be transformed into a pair of orthogonal states. Thus a symmetry transformation is a'mapping preserving the logical structure of quantum
mechanics. Uhlhorn's result highlights the connection between Wigner's theorem
and the First Fundamental Theorem of projective geometry. This connection does
not come as a surprise in view of the fact that Birkhoff and von Neumann [1936]
already proved in 1936 that the logic of quantum mechanics can be represented
by certain projective spaces and the work of Piron and the Geneva School [Piron,
19761 has developed a similar idea from the 1960s.
In the last decade, L. Moln6.r renewed the subject, especially with regard to
its mathematical aspects. He published numerous papers in which he used a new
430
Georges Chevalier
algebraic approach [ ~ o l n k 1996;
,
Molnir, 19981 which allowed him and other
,
Molnir,
authors to generalize Wigner 's theorem to other structures [ ~ o l n i r1999;
2000aI.
Let us briefly summarize the contents of this chapter.
In Section 2, we recall the Hilbert space formulation of quantum mechanics and
we introduce some notations. Section 3 deals with the origin of Wigner's theorem
and its various proofs.
In Section 4, Winer's theorem is proved in a way which uses only the definition
of an inner product space and elementary results of linear algebra and complex
numbers. All calculations are given in detail and a deeper mathematical result,
namely the existence of an adjoint for a bounded operator defined on a Hilbert
space, is needed only in the last step of the proof.
Section 5 is devoted to Uhlhorn's version of Wigner's theorem. A shortened
proof is given and its connection with projective geometry is discussed.
In Section 6, we describe the formalism of quantum mechanics introduced by
Piron and the Geneva School. The version of Wigner's theorem given by Piron
[I9761 seems to be a purely lattice theoretical result, close to the First Fundamental
Theorem of projective geometry, but its equivalence to the usual one is established.
Section 7, 8 and 9 deal with some generalizations of Wigner's theorem to indefinite inner product spaces, Hilbert modules, type I1 factors, quaternionic Hilbert
spaces, etc. Generalizations of the classical form of Wigner's theorem often have
an Uhlhorn version.
Motivated by the large number of versions of Wigner's theorem, we try, in
Section 10, to obtain a very general form of this result by replacing Hilbert spaces
by vector spaces equipped with linear topologies.. A Wigner type theorem in its
Uhlhorn form is proved with a difference: the orthogonality relation is not between
two lines of a vector space E but between a line of E and a line of its topological
dual. Wigner's theorem in its classical form is deduced from this general result.
In the last two sections, we outline two topics at the frontier of the subject of
this chapter. In Section 11, we introduce several automorphism groups having an
important physical meaning, and we prove that they are isomorphic to the symmetry group. Section 12 is devoted to the derivation of the Schrodinger equation
from Wigner's theorem in the case of a conservative physical system.
2 THE CLASSICAL HILBERT SPACE FORMULATION OF QUANTUM
MECHANICS.
In this Section, we recall the classical Hilbert space formulation of quantum m e
chanics. This is useful for understanding the meaning of Wigner's theorem. Moreover, we will fix some notations used throughout. A complete and rigorous exposition of this formulation can be found in [Beltrametti and Cassinelli, 1981, Chapters
1-51.
Wigner's Theorem and its Generalizations
43 1
To every physical system S is associated a separable complex Hilbert space
H. Real or quaternionic Hilbert spaces can also be used, see for example
[~eltramettiand Cassinelli, 1981, Chapter 221, [Finkelstein et al., 19621 or
[Uhlhorn, 19621.
In all Hilbert spaces and, more generally, in all inner product spaces considered in this paper, the inner product is taken t o be linear with respect to the
first argument. By an operator of H we mean a linear or antilinear bounded
mapping from H to H.
A state is a complete description of the physical system. To each state
corresponds a ray in H, i.e. an equivalence class of vectors that differ by
multiplication by a nonzero complex scalar. A ray can also be viewed as a
onedimensional subspace of H or a rank-one orthogonal projection. In the
framework of linear algebra or projective geometry, a ray is also called a line.
In general, if cp is a nonzero vector in H then the ray generated by cp will be
denoted by [cp] and the orthogonal projection on [cp] by P, or q,].We often
identify a state with its representative ray or its representative projection.
If cp is a unit vector generating the ray [cp], then cp is also called the wave
function of the physical system in the state represented by [cp]. The set of
all rays of H is denoted by [HI.
A real scalar product (., .) on the set of all rays is defined by
where cp and \$ are unit vectors generating [cp] and
[\$I.
If cp and \$ are unit vectors, we have
where tr denotes the trace functional (see [Beltrametti and Cassinelli, 1981,
Appendix A]).
For a proof of the second identity, note that P, o P\$(x) = (x,\$) (\$, cp) cp and
consider cp as an element of a Hilbertian basis of H. For the last identity,
if I1 x Il= 1 then I1 P, 0 P\$(x) 112= 1(x,\$)12 1(\$,cp>125 l(\$,cp)12 and, since
11 p p oP\$(\$)[I2= I ( \$ I ( P > I ~ ~ 11 p p oP\$]I2= I ( Y ! ' , ( P=>([(PI,
~~
Two states are said to be orthogonal or incoherent if the scalar product of
their representative rays is zero.
Actually, rays correspond to the secalled pure states of S, and general states
are associated to positive trace class operators of class one, also called in the
literature "density operators", "statistical operators", "density matrices".
This set of operators is a convex subset of the vector space of all trace class
operators: if S1and S2represent states then for any X E [0, 11,AS1 (1- X I S 2
+
Georges Chevalier
is also the representation of a state. As for pure states, we often identlfy a
general state with its operator representation.
Any pure state, considered as a rank-one projection, is an extremal point
of the convex set of all states and any state is a linear combination of pure
states: if S is a state then there exist a sequence of complex numbers (wi)
and a sequence (w) of unit vectors such that
wi = 1and S =
wiPVi.
x
x
i
i
a An observable of S is a property that, in principle, can be measured. To
each observable corresponds a not necessarily bounded self-adjoint operator
defined on H.
a In the measurement of an observable represented by a self-adjoint operator
A, the spectrum a(A) of A defined by
a(A)
=
{A E C I A - X . l H not invertible)
plays a crucial role. In particular, the possible values of a measurement of
the physical quantity represented by A are all elements of a(A).
Consider the simplest case of an observable represented by a self-adjoint
operator A with a pure point spectrum. Let X be an eigenvalue of A and PA
the orthogonal projection on the corresponding eigenspace. If, just prior to
the measurement, the physical system S is in the pure state [cp], 11 cp )I= 1,
then the outcome X is obtained with probability
If the outcome X is effectively obtained then the state after the measurement
is [Px(cp)]. Note that if the measurement is immediately repeated, then the
outcome X is attained with probability 1.
For an observable with a spectrum containing a continuous part the situation is, roughly speaking, very similar but its rigorous mathematical treatment is more complicated and it uses the projection-valued form of the spectral theorem (see [~eltramettiand Cassinelli, 1981, Chapters 1 and 31 or
[~vure~enskij,
1993, Chapter 11)
Any state is also an observable and thus can be measured. In particular the
possible measures of a pure state are 0 or 1. If the physical system is in
the pure state [cp], then the probability of the outcome 1if an experiment is
performed to determine if the system is in the pure state [\$] is:
Prob([\$l = 1) =II Pdcp) /I2= ([\$I, PI)^ (I1 cp II=II ?/1 II= 1).
It is the square of the scalar product of the rays [cp] and [\$I. If the outcome
1 is effectively attained then the state of the system immediately after the
the transition
measurement is [\$I. This is the origin of the name of ( [cp] ,
probability between the pure states [cp] and [\$I.
Wigner's Theorem and its Generalizations
433
Now a natural question is: Is any self-adjoint operator in correspondence with a
physical quantity and does any ray represent a pure state of the physical system?
In general the answer is no and the answer yes corresponds to strict quantum
systems, i.e. systems without nonquantum (i.e. classical) features. In the presence
of nonquantum features so called superselection rules must be introduced and
the Hilbert space formulation becomes more complicated (see [~eltramettiand
Cassinelli, 1981, Chapter 51).
3 THE ORIGIN OF WIGNER'S THEOREM AND A SHORT HISTORY OF
ITS PROOFS
The result known today as Wigner's theorem originally appeared in a book [Winer,
19311 written by E. Wigner in 1931. An English translation [Wigner, 19591 of this
book, with three new chapters, was published in 1959. In the sequel, we briefly
summarize the part of the book dealing with Wigner's theorem. we use the same
notation and vocabulary as Wigner.
In Chapter 6, entitled "Transformation Theory and the Bases for the Statistical
Interpretation of Quantum Mechanics"', Wigner considers observables represented
by operators G, G', G" ,. . .and wave functions cpl, cp2,. . . and he proves that the
same results are obtained from this system of operators and wave functions as
from the system in which the operators are replaced by
-
-
-
G = U G U - ~ ; G' = UG'U-~; GI' = ~
~"u-l
I . . . ,
and the state functions by
-
-
pz = Ucpz; (P3 = Up3. . . ,
where U is an arbitrary unitary operator. First of all, the eigenvalues which define
the possible results of the measurements of G and E = UGU-I are identical: if
Xk is an eigenvalue of G, and qk the corresponding eigenfunction, then X k is also
an eigenvalue of E and the corresponding eigenfunction is UQk. Moreover, the
probability of the eigenvalue X k for the quantity corresponding to G in the first
"coordinate system" and the one corresponding to ?? in the second are equal since
cpl = Up1;
cp2
Similarly, the transition probabilities between pairs of corresponding states cpl,
and Ucpl, Up2 are also the same in the two coordinate systems, since
Wigner calls such a transformation of the coordinate system by a similarity
transformation of the operators and simultaneous replacement of the wave functions cp by U p a canonical transformation. Two descriptions which result from
one another by a canonical transformation are equivalent.
The converse of this is the topic of the Appendix of Chapter 20, which is devoted
to the study of the equation
Georges Chevalier
where and 5 are the wave functions ascribed by the second observer of a physical
system to the states which the first observer describes by Q and a.
Wigner proves that, up to a phase factor, = U(cP) with U being a unitary or
antiunitary operator. By considering two stationary states with diierent energies
he shows that the operator U is actually unitary. Wigner notes that for the
exclusion of the antiunitary case he used the time dependence of the states. More
precisely, he postulated that if state is transformed into state cP' in the course
of the time interval t, then state is transformed into
during the same time
interval.
In Chapter 26, "Time Inversion", added in the English translation of his book
[Wigner, 19591, Wigner considers a physical system with zero linear momentum
and the transformation t -t -t which transforms a state cp into a state 8(p) in
which all velocities have opposite directions to those of cp. He proves that 8 satisfies
Equation (2) and corresponds to an antiunitary operator. The chapter continues
with a general study of time inversion and antiunitary operators. A good reference
concerning time inversion, also called time reversal, and antiunitary operators is
[ ~ o u t a ~et~ al.,
e l 1965, expecially Section 4.101. See also [Henley,20011, [Uhlhorn,
1962, Section 91.
Wigner's original proof is not complete. Nevertheless it represents "an excellent
basis for an elementary and straightforward proof' [Bargmann, 19641. In his 1962
paper [Uhlhorn, 19621, U. Uhlhorn gave a proof using new ideas and, moreover,
discussed the proofs that had been given until 1962 by E. Wigner, R. Hagedorn,
J . M. Jauch and G. Ludwig. All these proofs are incomplete or incorrect and thus
it seems that hi paper contains the first proof of Wigner's theorem. This proof is
analysed in Section 5.
After Uhlhorn's proof numerous proofs of Wigner's theorem appeared in the
literature. We can mention the following authors:
J . S. Lomont and P. Mendelson [1963]: an elementary proof apart the use
of the following result of analysis: a bounded subset of a Hilbert space is
weakly sequentially compact. Thii result can be obtained as a consequence
of the reflexivity of Hilbert spaces.
G. Emch and C. Piron 119631: a proof in the framework of lattice theory and
projective geometry. See also iron, 19761, Theorems 3.23 and 3.24. This
proof is summarized in Section 6.
V. Bargman 119641: an elementary proof in the spirit of the original work of
Wigner. Contrary to Uhlhorn, Bargman uses the same proof if dim H = 2
and if dim H >_ 3. For this reason, his proof is often considered as the first
correct proof of Wigner's theorem.
Wigner's Theorem and its Generalizations
435
S. C. Sharma and D. F. Almeida [1990b]. Their proof forms the subject of
our Section 4.
M. Gyory [2004]. This is the last known proof in 2005. It uses Zorn's lemma.
See also: [Simon, 1976; Varadarajan, 1985; Zhmud, 1992; Weinberg, 1995; Ratz,
1996; Cassinelli et al., 19971.
Of course, proofs of generalized versions of Wigner's theorem can also be considered prooh of the original theorem [Wright, 1977; Molnir, 1996; Molnir, 1998;
Molnk, 1999; BakiC and Guljas, 2002; Chevalier, 2005bl in chronological order).
4
AN ELEMENTARY PROOF OF WIGNER'S THEOREM
The purpose of this section is to give a simple proof of Wigner's theorem in full
detail and, as V. Bargmann noted in [1964], to emphasize its "quite elementary
nature". Our proof is close to the proof given in [ ~ h a r m aand Almeida, 1990bl.
This latter proof is itself similar to Bargmann's [~argmann,19641 which is, two
years after the proof of Uhlhorn, one of the first complete and rigorous proofs of
the theorem.
Before giving a precise statement of Wigner's theorem we need some definitions.
Let H be an inner product space over K = B or C.
1. A mapping U : H + H is an isometry if (U(x), U(y)) = (x, y) for all x,
y E H . Any isometry is an injective operator. If U is onto and H a Hilbert
space then U is invertible with U-l = U*, the adjoint of U. In this case, U
is called a unitary operator.
2. If K = C, an additive mapping U : H + H is said to be antilinear or
conjugate linear if U(Xx) = XU(Z) , x E H, X E C. If, for any x, y E
H, (U(x), U(y)) = (y, x) then U is an anti-isometry. In a Hilbert space,
a surjective anti-isometry is called an antiunitary operator. Unitary and
antiunitary operators satisfy U-I = U*.
More generally, let El and E2be two vector spaces over the fields K1 and K2
and a : Kl + K 2 be a field isomorphism. An additive mapping f : El + E2
is said to be semilinear or a-linear if, for any X E Kl, any x E El,
3. Two mappings T : H -t H and TI : H -t H differ by a phase factor or are
phase factor equivalent if there exists cp : H + K such that for any x E H,
T(x) = cp(x)T1(x) with Icp(x)I = 1. Remark that the binary relation "to be
phase factor equivalent" is an equivalence relation.
Georges Chevalier
436
4. Let f be a mapping from a set X into a set Y and let D be a subset of
P(X), the power set of X. A mapping F : D + P(Y) is said to be induced
or generated by f if for every M E D l F ( M ) = { f (m) I m E M}. As usual,
this last set is also noted f (M) .
5. A bijection T : H 4 H is called a symmetry if T preserves the modulus of
the inner product:
Every symmetry T generates a bijection f' of [HI,defined by f'([cp]) = [T(cp)],
cp E H, which preserves the scalar product of rays. A proof of this claim
uses the forthcoming Lemma 1.
A bijection of [HI preserving the scalar product of rays is also called a symmetry
in the literature. Let ? be such a symmetry and choose in every ray X a unit
vector u(X). Define T : H + H by T(0) = 0 and if x # 0 then x = Xu(X)
for a unique ray X and T(x) = Xu(T(X)). The mapping T is a bijection of H
preserving the modulus of the scalar product and generating the same bijection of
[HI than ?. When it is useful to avoid ambiguity, this second kind of symmetry
will be called, as in [~aradarajan,19851, a physical symmetry.
(T(x),T(y)) = 0 for a bijection T of H then T is called an
If (x, y) = 0
I-symmetry (ortho-symmetry). As for symmetries, any I-symmetry generates a
bijection of [HI preserving the orthogonality of rays in both directions (use the
forthcomming Lemma 7 for a proof) and, conversely, a bijection of [HI preserving
orthogonality in both directions allows to define a I-symmetry. In the literature, a
bijection of [HI preserving the orthogonality of rays is also called an I-symmetry.
Recall that a mapping which differs by a phase factor from a symmetry or an
I-symmetry is also a symmetry or a I-symmetry respectively.
A first lemma [Sharma and Almeida, 1990b, Lemma 51 (see also [Wigner, 1959,
page 2341) will show that a mapping defined on an inner product space and preserving the modulus of the inner product has a property close to linearity.
LEMMA 1. Let H be a complex inner product space and consider a mapping f
from a subset D of H into H which preserves the modulus of the inner product.
For any pair x and y of mutually nonzero orthogonal vectors of D and any pair of
complex numbers a and p such that a x py E D there exist a', p' E C such that
+
f (ax
Moreover, Ial = la'l and
+ Py) = ff'f(XI+ P'f (Y),
Ipl = IP'I.
Proof. Assume P = 0 and a # 0. The Schwarz equality states that two vectors u and v in an inner product space are linearly dependent if and only if
11 u 11 11 u I[= 1 (u, v )1. Thus a mapping preserving the modulus of the inner product
437
Wigner's Theorem and its Generalizations
also preserves the linear dependence of two vectors. Since x and a x are linearly
dependent, there exits a' E C such that f ( a x ) = a'f ( x ) . We have
and thus la1 = ldl.
Now if x and y are orthogonal unit vectors then f ( x ) and f ( y ) are also orthogonal unit vectors and, by the Fourier expansion, the claim of the lemma is true if
and only if
By a straightforward calculation
( f ( a x PY) - ( f ( a x PY), f ( x ) ) f(\$1 ( f ( a x + f l y ) ,f ( Y ) ) f ( Y ) ,
f ( a x PY) - ( f ( a x PY), f ( x ) ) f(XI + ( f ( a x + PY),f ( Y ) ) ~ ( Y ) =
) 0 and thus
Equation (3) is satisfied.
We have
+
+
+
+
+
l(f(ax+Py),f(x))l = I(ffx+P~,x)I=lalIIxIl~
= I(alf(4 P'f ( Y ) , f ().I
= la'l
+
II f (4112=
la'l
and thus Ial = la'1 . Similarly, IPI = IP'I .
The general case is obtained by combining these two particular cases.
II 112
rn
REMARKS.
1. If f is defined on the whole of H, then the forthcoming Lemma 7 allows one
to obtain a shorter proof.
2. Iff preserves the modulus of the inner product then, by the previous lemma,
the image of the line spanned by a nonzero vector x is contained in the
line spanned by f ( x ) and the image of the plane spanned by two mutually
orthogonal vectors x and y is contained in the plane generated by f ( x ) and
f (y) but, in general, the image by f of a line or a plane is not a line or a
plane respectively. For example, choose a unit vector u ( D ) on any line D
and let f : H + H be the mapping defined as follows:
f (0)= 0,
if x f 0 then let D be the line generated by x and if x = Xu(D) set
f ( X I =I
l u(D).
Clearly, f preserves the modulus of the inner product and the image by f of the line
Cu(D)is B+u(D)which is not a line. Recall that f is not linear or antilinear since
f ( x ) = f ( i x ) and thus a mapping preserving the modulus of the inner product is
not necessarily linear or antilinear.
Wigner's theorem can now be stated as follows.
Georges Chevalier
438
PROPOSITION 2. Let H be a complex inner product space with dim H 2 2 and
T : H 4 H a mapping presenring the modulus of the inner product. There exists an
isometry or a n anti-isometry A on H which diflers from T only by a phase factor.
T w o isometries A and A' or two anti-isometrics B and B' satisfy the requirement
of the preceding sentence if and only if they difler by a constant phase factor. The
mapping T cannot be phase equivalent to an isometry and to a n anti-isometry.
If T is a symmetry (i.e. if T i s surjective) and H a Hilbert space then A is a
unitary or an antiunitary operator.
Proof. The proof splits into six steps.
Step 1: Definition of a mapping A which differs from T by a phase factor.
Let xo be a unit vector in H chosen once and for all, X = @so the line spanned
by xo and let D = (xo X I ) U X I . Note that ( x o X I ) n X I = 0.
First we will try to define a mapping A : D + H which differs from T by a
phase factor and such that:
+
+
If A exists then A preserves the modulus of the inner product. Thus for any
y E X I and any a E @, A ( a y ) = a t A ( y ) with la]= la'[.
For any mapping f : D -+ H which preserves the modulus of the inner product
and any y E X I , ( f ( s o Y ) - f ( s o ) - f ( Y ) ,f ( s o Y ) - f ( s o ) - f ( Y ) ) =
1+ II Y 112 +1+ II Y 112 - ( f ( 2 0 + Y ) , f(xo>>- ( f (xo)1f (xo + Y ) ) - ( f (xo + Y ) , f ( Y ) )( f ( y ) ,f ( s o y ) ) . Therefore, for the satisfaction of ( 4 ) by f , it suffices that ( f (xo
y ) , f (so))
= 1 and ( f (so+ y ) , f ( y ) ) =I] y 112. We will prove that such a mapping
exists.
+
+
+
+
If y E xo
+ X ' - define
We have A ( x o ) = T ( x o ) , ( A ( y ) , A ( x o ) )= 1 and
I(T(Y),T(xo)>l= I(Y,xo)I
If y E X I then xo
=
I(x0
+
1
I
)I
T(xo)
(Y - xo),xo)I = 1(xo,xo)I = 1.
= 1 since
+ y E xo + X I and, by the previous definition,
and so (A(xo+y ) , A ( x o ) ) = 1. Thus for the satisfaction of (4) by A it suffices
to find X E @ such that 1X1 = 1 , to define A ( y ) = A T ( y ) and to check that
( 4 x 0 Y ) ,A(Y))=11 Y 112.
- ( T ( x 0 Y ) ,T ( Y ) )
If A ( y ) = A T ( y ) we have ( A ( x 0 y), A ( y ) ) = X
(Tho +I, T ( x 0 ) )'
+
+
+
+
Wigner's Theorem and its Generalizations
Thus X
=[I
y
(I2
(T(~0)lT(xo + y)) is convenient since
(T(Y),T(xo + Y))
We have succeeded in defining A : D + H which differs from T only by a phase
factor and which fulfills (4).
Step 2: The restriction of A to a line of
XIis
linear or antilinear.
Let y E x'- a unit vector. By Lemma 1,for any a E C there exists a' E C such
that A(ay) = a1A(y) with la1 = la'l.
Using Equation (4), we have
On the other hand,
I(A(x0
+ ffy),A(xo+y))l = I(xo + a y , x o +Y)I = I1 + al
+
and so 11+ a1 = 11 all.
Therefore 1+ a + E a E = 1+ a' + 2 0'2and, since la1 = la'l, a E =
a' + 2. Thus a and a' have the same real part. Since they have also the same
modulus, their imaginary parts
- are equal or opposite and finally, a = a' (Referred
as Condition (PI)) or a = a' (Referred as Condition (Pz)). Note that the end of
the previous proof shows that:
+
+
+
LEMMA 3. Let a and a' be two complex
- numbers with the'same modulus. If
Il+al = Il+alI then a = a' or a = a'.
(This Lemma is used by Wigner in his proof, see [Wigner, 1959, page 2341.)
Now we will prove that for any unit vector y E XI,Condition (PI) holds for
any a E C or Condition (P2) holds for any a E C.
Let a, /? E C*. By Lemma 1, there exists a', P1 E cC such that la1 = la'[,
IPI = IP'I and A(ay) = a1A(y), A(Py) = PtA(y). Using Equation (4), A(xo+ay) =
A(x0) A(ay) = A(x0) a1A(y) and, similarly, A(x0 Py) = A(x0) PIA(y).
Thus ,
+
+
+
I(x0 + PY,XO
+ ~ Y )=I I(A(xo),A(xo)>+ (PIA(y),a1A(y))I
I1 +Wl=
ImI
+
I1+P1a'l
=
But
= [@'TI
and, using Lemma 3,
or ,BE = Fa'.
Assume that a satisfies (PI) and p satisfies (P2) then
= @ or @ = Pa.
In the first case, ,8= and @ also satisfies ( 4 ) . In the second case a = E and a
also satisfies (P2). In both cases, a and P satisfy the same condition.
p
m
440
Georges Chevalier
Finally, the restriction of A to a line of X I is linear or antilinear (In the
definition of a linear mapping f on a Hilbert space, it suffices to assume that
f (Ax) = Af (x) for any unit vector x).
Step 3: The restriction of A to X I is additive.
A proof is necessary only if dim H > 2.
Let y and z be two orthogonal unit vectors in X I . For all a and p in C,
a y + p z E XI.Using Equation (4) and Lemma 1, A(xo + (ay + pz)) = A(xo) +
A(ay + pz) = A(xo) + al'A(y) + pl'A(z), Ial = Ia1'I and IpI = IP1'I. Let a' and \$
be such that A(ay) = alA(y) and A(@z) = PIA(z) with la1 = \all and
= I/?'I.
Since IaEl = 1aU21,we can use Lemma 3 and we get aE = aN2.Since a"7 E W+,
a' and a" we get by the same argument that Ia'l = la"/implies a' = a". By similar
reasoning we get p1 = P" and
Let u and v be two nonzero vectors in x'- and let w be a unit vector orthogonal
t o u in the subspace spanned by u and u. There exist a and ,B in C such that
In the following calculation, if the restriction of A to Cu is linear then & = cr and
if this restriction is antiliiear then & = E
We conclude that the restriction of A to X I is additive.
Step 4: The restriction of A to XIis linear or antilinear and A is extended to H.
Wigner's Theorem and its Generalizations
44 1
This result is already proved if dim H = 2. Thus we assume dim H > 2.
Let yl and y2 be two nonzero and noncollinear vectors in X I and assume that
the restriction of A to (Cyl is linear while its restriction to Cy2 is antilinear. If yl
is orthogonal to y2 then
which is absurd. Now if yl and y2 are not orthogonal then let z be a vector
orthogonal to yl in the subspace spanned by yl and y2. There exist a and P E C
such that yz = a y l pz. By the beginning of the proof, the restriction of A to
Cz is linear and for all X E (C
+
In particular, A(iy2) = iA(y2) and also A(iy2) = -iA(yz) since the restriction of
A to Cy2 is antilinear. Therefore A(y2) = 0, which contradicts the hypothesis of
preservation of the modulus of the inner product. The restriction of A to X I is
linear or antilinear. If this restriction is linear, we extend it to H by linearity and,
if this restriction is antilinear, the extension is by antilinearity. Now, A is defined
in the whole of H and, since A(x0) = T(xo), is phase equivalent to T.
Step 5: A and T differ by a phase factor, A is an isometry or an anti-isometry
and A is uniaue UD to a constant ~ h a s efactor.
Let ax0 + p y E H = [email protected]'-.
Since A(py) differs from T(py) by a phase factor,
we can assume a # 0 and write in the linear case:
In the antilinear case, E-' replaces a-I and we have 171 = 1 and 101 = lall.
Therefore laI7a-l = Id7~-l= 1and thus T(axo py) and A(axo +fly) differs
by a phase factor. Note that, since T is norm preserving, A is norm preserving
too and, since A is linear or antilinear, A is bounded and therefore continuous.
1
+
1
Let y, z E H. We have
(A(Y - z), A(Y - 2))
=II
A(Y) 112
+ II A(%) 112
- M y ) , A(%))- (A(%),A(Y)
and also
(A(y - ~ ) , A ( Y
- 2)) = (Y- 2, Y - 4
and therefore,
=11
Y
112
+ II z 112
-(y, 2) - (2, y)
Georges Chevalier
442
(5)
(Y,z)
+ (z, Y) = (A(Y),
+ (A(%),A(Y))
Replacing z by iz in (5) allows one to obtain (y, z)
case and (y, z) = (A(z), A(y)) in the antilinear one.
=
(A(y), A(z)) in the linear
In order to make precise the uniqueness of A, we separate out a lemma which
is of interst in its own right.
LEMMA 4. Let H be a complex inner product space with dim H 2 2 and T : H +
H a mapping preserving the modulus of the inner product.
I. Let A be an isometry or an anti-isometry phase equivalent to T. Then the
nature of A can be predicted fi-om T.
2. An isometry cannot be phase equivalent to an anti-isometry.
3. If two isometries or two anti-isometrics are phase equivalent then they difler
by a constant factor.
Proof. As in [~argmann,19641, consider three rays [pl], [pa]and [p3]generated
by unit vectors p l , p2 and p3. The expression (pl, p2) (p2, p3) (p3,pl) is independent of the choice of the unit vectors generating the rays and is a function
A([cpl], [pz], [p3]) of the rays [pi].Note that
if T is phase equivalent to TI then
Therefore, if there exist three unit vectors el, e2 and e3 such that A([el], [ea],[es])
is not real, an isometry cannot be phase equivalent to an anti-isometry and T
is phase equivalent to an isometry A if and only if A(T([el]),T([ez]),T([e3]))
=
A([el], [ez],[e31>
Since dimH 2 2, let e and f be two orthogonal unit vectors and consider the
1
1
unit vectors el = e, e2 = -(e
a:
\$
- f ) and e3 = -(e
6
+ (1 - i)f) [Bargmann, 19641.
We have A([el], [el], [e3]) = and the proof of 1. and 2. is complete.
(For another approach to the distinction between unitary and antiunitary operators, see [Wigner, 19601.)
If A and B are two isometries or two anti-isometrics phase equivalent then, for
every line p in H, A(p) = B ( p ) . A classical exercise in elementary linear algebra
states that if f and g are two linear mappings defined on a vector space E over
a field K and if, for every line p, f (p) = g(p) then there exists X E K such that
Wigner's Theorem and its Generalizations
443
f = Xg. The result is true as well for antilinear mappings and if E is a Hilbert
space and f and g are norm preserving then IXI = 1. Therefore two isometries
or two anti-isometries which are phase equivalent to T d 8 e r by a constant of
modulus 1.
To sum up, we have defined an isometry or an anti-isometry A which is phase
factor equivalent to T and the result is valid for any inner product space of dimension greater than 2.
S t e p 6: If T is a symmetry and H a Hilbert space then A is a unitary or an antiunitary operator. Let 0 # z E H. If T is surjective, there exists x E H such that
1
T ( x ) = z and A(%) = XT(x) with X E C*. Thus A(-x) = z if A is linear and
X
1
A(:%) = z in the antilinear case. The mapping A is surjective.
X
If H is a Hilbert space, then the adjoint A* of A is a mapping from H to H
which is linear or antilinear if A is linear or antilinear respectively. Let y, z E H.
If A is linear
and, in the antilinear case,
and thus, in the two cases, A*A = l H .
(Information about adjoints of semilinear mappings can be found in Section 10)
Since A is bijective, A-' exists and
If the mapping A is linear then A is a unitary operator and, otherwise, A is an
antiunitary operator.
The following corollary is useful for the understanding of generalizations of
Wigner's Theorem or its formulation in the framework of projections (for example,
see [Wright, 19771).
COROLLARY 5. Let T be a bijection of the set of all rank-one projections of
a Hilbert space H . If t r ( P Q ) = t r ( T ( P ) T ( Q ) )then there exists a unitary or
antiunitary operator U : H + H such that
T ( P )= UPU*,
444
Georges Chevalier
We begin the proof by a remark: if U is a unitary or antiunitary operator then,
)
for any closed subspace M of H, P U ( ~=) UPMU*where PM and P U ( ~denote
the orthogonal projections on M and U ( M ) . Indeed,
and clearly UPMU*is an orthogonal projection.
Using Equation (1)in Section 2 we see that the bijection T induces a bijection of
[HI preserving the scalar product of rays. The later bijection generates a bijection
of H preserving the modulus of the inner product. Let us denote these three
bijections by the same symbol T . By Wigner's theorem, there exists a unitary or
antiunitary operator U : H + H such that, for any unit vector cp E H, T(cp) =
X(cp)U(cp)with IX(cp) 1 = 1. We have [T(cp)]= [U(cp)] and thus, using the hypothesis
concerning the symbol T ,
COROLLARY 6. The set of all physical symmetries of H forms a group isomorphic t o the quotient group of the group of all unitary or antiunitary operators by
= 1).
i t s subgroup {XIH I
The set of all physical symmetries is clearly a subgroup of the group of all
bijections of [HI. Let f be the mapping £rom the group of all unitary or antiunitary
operators into this subgroup which associates to each U the physical symmetry
induced by U . The mapping f is a group homomorphism and Wigner's theorem
means that f is onto. Its kernel is the set of all unitary or antiunitary operators
U satisfying, for any line p of H , U(p) = p. By using Lemma 4, kerf = {XIH I
1x1 = 1).
The group of all physical symmetries of H is called the symmetry group of H or
of the physical system associated to H . The symmetries, considered a s mapping
from H to H, form a group as well.
By Wigner's theorem the binary relation "to be phase equivalent" is a congruence relation and the quotient group is isomorphic to the symmetry group of H.
Some other groups isomorphic to the symmetry group will be studied in Section 11
REMARKS.
1) Let H be an inner product space over K = W or C and f : H + H . Consider
the two properties:
Wigner's Theorem and its Generalizations
445
+
I f f satisfies 1. or 2. then a straightforward calculation yields to ( f ( x Y) f (x) - f ( Y ) ,f ( x + y ) - f ( x ) - f ( y ) ) = 0 and therefore f is additive. If f
satisfies 1. then, for any X E K, ( f ( X x ) - X f ( x ) ,f ( X x ) - X f ( x ) ) =Oand f is
an isometry and, similarly, iff satisfies 2. and K = C, f is an anti-isometry.
Moreover, if f is onto and H a Hilbert space, then f is a unitary operator
when f fulfills 1. and an antiunitary operator if K = C and f satisfies 2..
2) A linear mapping which preserves the inner product is not necessarily onto
in the infinitedimensional case. For example, let (en)nENbe an orthonormal
basis of a separable Hilbert space H. The shift operator x =
Enen 3
nEM
is a non surjective operator preserving the inner product.
S ( x )=
nEN
In the finite dimensional case, any operator preserving the inner product is
a bijection since it is a one-bone operator and the dimension of the vector
space is finite.
3) In [Sharma and Almeida, 1990b]the authors give a simple proof that an additive mapping preserving the modulus of the inner product is linear or antilinear. They use a result of [pian and Sharma, 19831: An additive bounded
mapping f is a sum of linear operator and an antilinear operator. By using
the first claim, the step 4 in the proof of Wigner's theorem is unnecessary.
4 ) ) Let T be a symmetry on a one-dimensional Hilbert space and let xo be
a unit vector. Define two mappings A : H + H and B : H + H by
A(Xx0) = XT(xo) and B(Xx0) = ~ T ( X ~for) any X E C. It is easy to check
that A is an isometry, B is an anti-isometry and that T, A and B differs by
a phase factor. A part of Lemma 4 is no longer true if dimH = 1.
5) The original proof of W i n e r [1959, Appendix of Ch. 201 and the proof of
Bargmann [1964]use a Hilbertian basis and thus cannot be done in an inner
product space.
5 UHLHORN'S VERSION OF WIGNER'S THEOREM
The main idea and the great significance of the paper [Uhlhorn, 19621 is well
summarized by the following part of its Introduction:
W e shall replace the requirement of the invariance of transition probabilities by the requirement that orthogonal vector rays are transformed
into orthogonal vector rays, that is, incoherent states are transformed
into incoherent states. B y this definition, a symmetry transformation
i s a mapping preserving the logical structure of quantum mechanics,
whereas the definition stated above corresponds to a mapping preserving the probabilistic structure of quantum mechanics.
Georges Chevalier
446
In his proof of Wigner's theorem, Uhlhorn considers bijections of lines preserving
linear independence and a lemma is useful to understand the connection between
this notion and the notion of bijections of lines preserving orthogonality. By
definition, n lines in a vector space E, 11, . . . ,In, are said to be linearly independent
n
li is a
i=l
direct sum of subspaces. This also means that 11,. . . ,ln considered as elements of
the modular lattice of all subspaces of E are independent [Birkhoff, 1967, Ch. IV,
941
LEMMA 7.
if they are generated by linearly independent vectors or equivalently if
+
1. I n a n inner product space H, n 1 vectors XI, . . .,xn+l are linearly independent exactly if the n first vectors are linearly independent and there exists
a vector y which is orthogonal to the n first vectors and non-orthogonal to
xn+ 12. Every mapping j : H 4 H preserving orthogonality of vectors in both directions also preserves linear independence of vectors in both directions and its
extension to the set of all lines of H preserves independence of lines in both
directions.
Proof. 1. Assume that X I , . . . ,xn+l are linearly independent. Let X I , y2,. . . ,yn+l
be the n 1 vectors obtained from X I , . . . ,xn+l by using the Gram-Schmidt orthonormalization process. It is easy to check that y = yn+l is convenient. Con-
+
versely, assume that the n first vectors are linearly independent and there exists a vector y which is orthogonal to the n first vectors and non-orthogonal to
%,+I.
If Xlxl . . . Xn+lxn+l = 0 then ( y , Xlxl . . . Xn+lxn+l) = 0 implies
An+l = 0. Since the n first vectors are linearly independent, X1 = . . . = An = 0
and X I , . . . ,xn+l are linearly independent.
The claim 2. is an easy consequence of 1.
+ +
+ +
In Euclidean spaces and, more generally, in inner product spaces, the preservation of linear independence of lines is an hypothesis strictly weaker than the
preservation of orthogonality of lines. For example, let j be an isometry of the
Euclidean space Rn equipped with its canonical scalar product. If one changes
the scalar product then f always preserves linear independence of lines but not,
in general, orthogonality of lines with respect to the new product.
We can now state the main result of [Uhlhorn, 19621 as follows.
PROPOSITION 8 ([Uhlhorn, 1962, Lemma 3.4, Theorems 4.1 and 4.21). Let H
and H' be two complex Hilbert spaces with d i m H 2 3. If T : [HI + [H'] i s
a n I-symmetry then there exists a unitary or antiunitary operator O : H -+ H'
which induces T. The operator O is determined by T up to a complex factor of
modulus 1.
Wigner's Theorem and its Generalizations
447
Shortened proof.
Using the Axiom of Choice, there exists r : C + C such that T(C x) = C r(x)
if x # 0 and r(0) = 0. Let x and y be two linearly independent vectors. Since x,
y and x y are linearly dependent, the same holds for r(x), r(y) and r ( x y).
There exists a mapping w defined for any pairs of linearly independent vectors and
with values in C such that
+
+
x) instead of w(y, x) is useful for the simplicity of the forthcoming
(To write W(X+Y,
Equation (7)).
Using (6) and r((x y) z) = r ( x (y z)), we have for arbitrary linearly
independent vectors x, y and z (Remark we use dim H 3)
+ +
+ +
>
Another relation for linearly independent vectors x, y is w(x, y)w(y, x) = 1.
The definition of w extends to pairs of linearly dependent vectors in such a way
that (7) becomes valid. This part of the proof is based on the following fact: if z
is linearly independent of the linearly dependent vectors x, y then w(x, z)w(z, y)
is independent of z and allows to define w(x, y).
Now choose 0 # xo E H and define 8 : H 4 HI by 8(x) = w(xo,x)r(x) if x # 0
and 8(0) = 0. This mapping is bijective, additive and, since C 8(x) = C r(x) =
T(C x), 8 induces T.
For 0 # a E C and 0 # x E H , C x = C(ax) and therefore 8(x) and 8(ax)
generate the same ray in HI. So we can consider cp : C* x ( H - (0)) + C defined
by 8(ax) = cp(a, X ) ~ ( X )For
. any fixed 0 # x E H , a + cp(a, x) is a mapping
cp, : C + C.
Using
it is easy to see that all the mappings cp, coincide. Let +(a)= cp(a,x) for an arbitrary nonzero x H with +(O) = 0. The mapping 4 is a nonzero homomorphiim
of the complex field.
Let x and y be two orthogonal vectors in H with the same norm. For a # 0,
x - ~ - l y and x + a y are orthogonal vectors and therefore 8(z-E-ly) and B(x+ay)
are also orthogonal (It is the first time we use the preservation of orthogonality of
lines. So far, we have only used the preservation of linear independence of lines).
This yields to
11 8(x) ]I2 - + ( a ) + ( ~ - l ) 11 8(y) 112=
-
0 and +(a)+(-E-l) =
11 e(x>[I2
II 8(Y) 112
+
is independent of a. By choosing a = 1, we have +(E) = +(a). Thus is the
identity or the conjugation and 8 is linear or conjugate linear.
Let 6 : ( H - (0)) + cC be the mapping defined by 11 8(x) I]= 6(x) 11 x 11 if x # 0.
If x and y are two non-orthogonal vectors then z = (y, y)x - (x, y)y is orthogonal
to y and thus we have
448
Georges Chevalier
By interchanging x and y and taking the complex conjugate
and combining (8) and (9) yields b ( x ) = 6 ( y ) . The same is true if s and y
are orthogonal. Let 6 = 6 ( x ) for a non-zero x. Using equation ( 8 ) we have
1
( e ( x )8(y))
,
= h 2 4 ( ( x ,y ) ) . Finally Q = 1 0 is an isometry or an anti-isometry. If T
6
is onto and H a Hilbert space then Q is a unitary or anti-unitary operator which
induces T.
REMARKS. 1) A careful study of the proof of Theorem 4.1 in [Uhlhorn, 19621
shows, as Uhlhorn noted himself, that it gives rise to a proof of the following
version of the First Fundamental Theorem of projective geometry.
PROPOSITION 9. Let El and E2 be two vector spaces over the commutative fields
K1 and K2 with dim El 2 3. If there exists a bijection T from the set of all the
lines of El onto the set of all the lines of Ez which preserves linear independence
of lines then in both directions:
1. the two fields K 1 and K2 are isomorphic;
2. there exists a semilinear bijection s : El -+ E2 inducing T
In [ ~ h l h o r n19621
,
the surjectivity of the homomorphism 4 of C is not proved
since the last part of the proof of Proposition 8 implies that 4 is onto if T preserves
orthogonality of lines. With the weaker hypothesis saying that T preserves linear
independence of lines the proof of its surjectivity is not difficult and it is similar
to a part Baer's proof of the First Fundamental Theorem of projective geometry
[ ~ a e r1952,
,
page 501.
from the following Lemma we see that Proposition 9 is actually not an improvement of the First Fundamental Theorem of projective geometry.
LEMMA 10. A bijection cp of the set of all lines of a vector space E over the field
K preserves linearly independence of lines in both directions if a n only if cp extends
t o a n isomorphism @ of the lattice of all subspaces of E.
Proof. An isomorphism of the lattice of all subspaces of E preserves, in both
directions, the dimension of subspaces and therefore the linear independence of
lines.
Now let cp be a bijection of the set of all lines of E which preserves linear indepencp(1). If
dence in both directions. Define for any subspace F of E , @ ( F )=
V
1 line,1CF
F c G then @ ( F )c @ ( G )and conversely assume that @ ( F )c @ ( G )for two subp(1) =
spaces F and G . Let m C F be a line. We have p ( m ) C @ ( G )=
V
1 line,1CG
Wigner's Theorem and its Generalizations
449
~ ( 1 ) If
. p(m) = K a, a E E, there exists n linearly independent vectors,
1 line, 1CG
a l , . . . ,a, such that a = a1 + . . . + an with ai E cp(li), li a line of G. We have
cp(m) c cp(ll) . . . cp(ln) and, as {cp(ll), . . . ,cp(1,)) is a linearly independent
set of lines and {cp(m), cp(ll), . . . ,cp(1,)) is not linearly independent, (11,. . . ,I,)
is linearly independent and {m, 11, . . . , 1,) is not linearly independent. Therefore
m C l1 @ . . . @ 1, and m is a line of G. Thus F c G and @ is an isomorphism of
the lattice of all the subspaces of E.
2) Uhlhorn's version of Wigner's theorem is invalid for twedimensional Hilbert
spaces. A first example of a ray transformation defined on a two-dimensional
Hilbert space which preserves orthogonality of lines but not all transition probabilities is given by Uhlhorn in [1962] and a simplified version of this example may
be found in [Cassinelli et al., 19971.
In [1962, Theorem 5.11, Uhlhorn states Wigner's theorem in the two dimensional
case for a ray transformation preserving transition probabilities. He has thus
obtained the first complete proof of the usual version of Wigner's theorem.
6
WIGNER'S THEOREM VIEWED BY THE GENEVA SCHOOL
In the early 1960s, in Geneva, J. M. Jauch and especially C. Piron developed,
jointly with some other mathematicians and physicists, a new formulation of quantum mechanics [Jauch, 1968; Piron, 19761. Some other mathematicians (D. Aerts,
I. Daubechie, B. Coecke,. . .) continued this work in Brussels. This approach
to the foundations of quantum mechanics is known as the Geneva School or the
Geneva-Brussels School. We will outline only that part of this. formulation which
is relevant to the understanding of Wigner's theorem.
Every measurement on a physical system can be reduced, at least in principle,
to the measurements of a series of yes-no experiments which, in a sense, represent propositions. These experiment are observations permitting only one of two
alternatives as an answer [Jauch, 1968; Piron, 19761. The set of all such propositions of a physical system is a complete atomic orthomodular lattice satisfying the
exchange axiom p iron, 1976, Chapter 21 and [Kalmbach, 19831 or [ ~ a e d aand
Maeda, 19701 for information about this kind of lattices). It is called a propositional system.
Any propositional system is isomorphic to a product of irreducible propositional
systems. The following result gives a representation theorem for such structures. It
also motivates the use of Hilbert spaces in the formalization of quantum mechanics.
We need a definition.
Let E be a vector space over a field K equipped with an involutorial antiautomorphism X + A* and let (., .) be a mapping from E x E into K. If, for x, y,
ZEE,AEK,
Georges Chevalier
3. (x,x) = 0 implies x
= 0.
then (., .) is called a definite Hermitian form. A subspace F of E is said to be
closed or I-closed if F = FLL.
PROPOSITION 11 p iron, 1976, Theorems 3.23 and 3.241).
I . Let L be a n irreducible propositional system of height 2 4. The orthocomplemented lattice L is isomorphic to the lattice of all closed subspaces of a
vector space E equipped with a definite Hermitian form.
2. Conversely, the lattice of all closed subspaces of a vector space E equipped
with a definite Hermitian f o m is a n irreducible propositional system if and
only if, for any closed subspace F,
A vector space E equipped with a definite Hermitian form satisfying Equation
(10) is said to be a generalized Hilbert space. Any classical Hilbert space over R,
C or IHI is a generalized Hilbert space but there exist non classical Hilbert spaces
[Keller, 19801.
In iron, 19761, the following result is called Wigner's theorem.
PROPOSITION 12 s iron, 1976, Theorem 3.281; see also [ ~ m c and
h Piron, 19631
and [Jauch, 1968, 9.41). Let (El, (., .)l) and (E2, (., .)2) be two generalized Hilbert
spaces of dimensions at least equal to 3 and let Li, i = 1, 2, be the irreducible
propositional systems of all closed subspaces of Ei. A n y isomorphism Q, of L1
onto L2 i s induced by a semilinear bijection of El onto E2. Conversely, a a-linear
bijection f : El + E2 induces a n isomorphism of L1 onto L2 i f and only i f there
exists a n element a in the scalar field K1 of El such that
Idea of the proof. The isomorphism Q, extends to the lattice of all subspaces of El
by
Q,(F) =
V @(PI
p line, pCF
for all subspaces F of El. By the First Fundamental Theorem of projective g e
ometry [Baer, 19521 there exists a semilinear bijection f : El + E2 inducing Q,.
Conversely, if a semilinear bijection satisfies (11) then
and thus, for any subspace F c El, f (FL) = f ( F ) ~ . The image of a closed
subspace of El under f is a closed subspace of E2 by Equation (11) and Equation
Wigner's Theorem and its Generalizations
45 1
(11) is sufficient for the preservation of the structure L1 and L2 as orthocomplemented lattices. As the two forms (.,
and (x, y ) + a-'((f (x), f ( y ) ) ~define
)
the
same orthocomplementation on L1, a classical result of Birkhoff and von Neumann
[I9361 or [Varadarajan, 1985, Theorem 2.61 shows the necessity of Equation (11).
Let * and be the involutorial antiautomorphisms of K1 and K2 associated to
and (., .)2. Replacing y by Xy in Equation (11) yields
(.,
(12) a(X)* = a(a-lX*a).
If Kl and K2 are both the field of complex numbers and * and agree with
the usual conjugation then, by Equation (12), a(X) = a(1) and thus a(R) = R.
The automorphim a is either the identity or the conjugation. Moreover x = y in
Equation (11) implies that a is a positive real number. Define u =
-,f
6
then u
induces the same isomorphism than f and the following corollary is proved.
COROLLARY 13 ([Piron, 1976, Corollary 3.311).
I f H is a complex Hilbert space of dimension at least 3, then every isomorphism
of the propositional system of all closed subspaces of H is induced by a unitary or
antiunitary operator.
The resemblance with Uhlhorn's version of Wigner's Theorem is highlighted by:
PROPOSITION 14 ([Piron, 1976, Theorem 2.461). A bijective mapping of the set
of all atoms of a propositional system L1 onto the set of all atoms of a propositional
system L2 which preserves orthogonality of atoms in both directions can be uniquely
extended to a n isomorphism of Ll onto La.
7 GENERALIZATIONS TO INDEFINITE INNER PRODUCT SPACES
7.1 Generalizations of the classical version of Wigner's theorem
Let H be a vector space over K = R or @ together with a form (., .) : H x H 4 K
which is bilinear and symmetric in the real case and sesquilinear and hermitian in
the complex case. An element x E H is called positive if (x, x) > 0 and negative
if (x, x) < 0. If H contains positive as well as negative elements, we say that H is
an indefinite inner product space or an indefinite metric space, see [~ogntir,19741
for information about general indefinite inner product spaces. Any indefinite inner
product space always contains nonzero isotropic elements, i.e. elements x # 0 such
that (x, x) = 0.
This subject appeared in a paper by Dirac dealing with quantum field theory.
in recent decades indefinite inner product spaces proved useful for certain physical
problems as well as mathematical questions, see for example the Introduction in
[Bracci et al., 19751.
If H is a Hilbert space with the product (., .) then any linear mapping a : H 4 H
induces a new inner product on H denoted by (., .), and defined by the formula
452
Georges Chevalier
This product allows, in general, to define a structure of indefinite inner product
space on H. The operator a is called the metric operator and certain hypotheses
on a such as bounded, self-adjoint, invertible ,... are required in order to obtain
interesting results about the new product. Many proofs in indefinite inner product
spaces obtained in the tradition1 way don't make use of specific properties of this
structure, as for example the existence of nonzero isotropic elements, and thus are
also valid in Hilbert spaces by choosing a = lH.
The definition of a ray in an indefinite inner product space is the same as that in
a Hilbert space but the scalar product and orthogonality of rays are now defined by
means of the indefinite inner product (., .),. A bijective transformation of the set
[HI of all rays which preserve the new scalar product of rays or the orthogonality of
rays in both directions will be called a a-symmetry or a I,-symmetry respectively.
The first paper [Bracci et al., 19751 dealing with the generalization of Wigner's
theorem to indefinite inner product spaces is one of the most interesting on the
subject. In its remarkable introduction the authors give numerous references highlighting the imoportance of indefinite inner product spaces in physics, in particular
in quantum field theory, and on the other hand references concerning the purely
mathematical development of the theory. They also justify their mathematical
hypotheses by physical arguments.
i al., 19751. In this paper, the metric operator a is
Let's summarize [ ~ r a c cet
assumed to be a self-adjoint operator with a bounded inverse and Proposition 1
states that, without loss of generality, one can assumes that a2= 1. A particularity
of the paper, justified by physical arguments, is that a-symmetries are not defined
on the whole set of rays, and therefore a precise definition of some classes of
operators with an analogous property is necessary.
i al., 19751). Let H be an indefinite inner product
DEFINITION 15 ( [ ~ r a c cet
space defined by means of a Hilbert space and linear operator a : H --+ H .
An operator U with dense domain Du and dense range such that
is called a a-unitary operator.
i al., 1975, Proposition 21). A a-unitary operator is
PROPOSITION 16 ( [ ~ r a c c et
linear having inverse which is a a-unitary operator and it is closable.
If in the previous definition, Equation (13) is replaced by (U(x), U(y)), =
(Y,x)U, (U(x);U(y))u = - ( x , Y ) ~or (U(x),U(y))u = - ( Y , X ) ~one obtains the
definition of, respectively, a-antiunitary operators, a-pseudo unitary operators
and a-pseudo antiunitary operators. Since (U(x), U(x)) = -(x, a ) for x # 0 is impossible in a Hilbert space, a-pseudo unitary operators and a-pseudo antiunitary
operators do not exist in Hilbert spaces. In indefinite inner product spaces, they
only exist if the eigenvalues 1 and -1 of the metric operator a have the same multiplicity. For a-antiunitary operators, a-pseudo unitary operators and a-pseudo
antiunitary operators a result similar to Proposition 16 exists, with the difference
that a-antiunitary and a-pseudo antiunitary operators are antilinear.
Wigner's Theorem and its Generalizations
453
As we shall see below the big difference between the definite and the indefinite case is the presence of a-pseudo unitary operators and a-pseudo antiunitary
operators in the indefinite case in the statement of Wigner's theorem.
i al., 19751). Let H be an indefinite inner product
PROPOSITION 17 ( [ ~ r a c cet
space defined by means of a Hilbert space H and a linear operator a : H -+ H
which i s self-adjoint with a bounded inverse. Let T be a bijective mapping defined
o n a set R of rays of H onto a set R' of rays such that
1. the set D of vectors belonging to the rays of R and the set D' of vectors
belonging to the rays of R' are dense linear subspaces in H;
2. For any rays
[XI,
[y] E R,
There exists an operator U : D --t D' such that, for any x E D , U(x) E T([x]). The
operator U i s either a-unitary or a-antiunitary or a-pseudo unitary or a-pseudo
antiunitary.
In [2000a],Molnk proved a similar result without the hypothesis of self-adjointness
for a , see Section 8.2.
7.2
Genemlixations of Uhlhorn's version of Wigner's theorem
The first generalization of Uhlhorn's version of Wigner's theorem to indefinite
inner product space appeared in [van der Broek, 1984al. In this paper, the author
considers an n-dimensional complex Hilbert space H , n 2 3, and a metric operator
a which is self-adjoint and bijective. If T is a I,-symmetry then, as in [Bracci et
al., 19751, T is induced by an operator U which is either a-unitary or a-antiunitary
or a-pseudo unitary or a-pseudo antiunitary ([van der Broek, 1984a, Theorem 11
or [van der Broek, 1984bl). The main differences which distinguishes this result
i al., 19751 are that T is a 1,-symmetry defined on the whole of [HI
from [ ~ r a c cet
and 3 5 dimH < ca.
The author of [van der Broek, 1984a] applies this result to representations of
groups in indefinite inner product spaces in [van der Broek, 1984bl
In his generalization of the classical version of Wigner's theorem to indefinite inner product space Molnk removes self-adjointness of the metric operator from his
hypotheses and proves in [Molnir, 20021 the following generalization of Uhlhorn's
version of Wigner's theorem to indefinite inner product spaces.
PROPOSITION 18 ([Molnir, 2002, Corollary 21). Let H be a (real or complex)
Hilbert space of dimension not less than 3 and let a : H -+ H be a n invertible
operator. Suppose that T : [HI + [HI i s a 1,-symmetry.
If H i s real then T i s induced by an invertible linear operator U o n H. Similarly,
if H i s complex then T i s induced by an invertible linear or antilinear operator U
on H . The operator U inducing T i s unique up to multiplication by a scalar.
454
George Chevalier
The invertible linear operator U : H + H induces an I,-symmetry if an only
if
(U(X),U(Y))U= c(x, Y ) U
(x, Y E H )
holds for some constant c E R in the real case and c E C in the complez case.
If H is complex, then the antilinear operator U : H + H induces an I,symmetry if and only if
holds for some constant d E (C. Here, a* denotes the adjoint of a
This result is a corollary of the main theorem of [ ~ o l n ~20021,
r,
which describes the form of all bijective transformations @ of the set Il(X) of all rank-one
idempotents on a Banach space X such that
Its proof is based on a result of Ovchinnikov [1993]describing the automorphisms
of the poset of all idempotents on a separable Hilbert H space by means of automorphisms and anti-automorphisms of the lattices of all closed subspaces of H.
REMARK. If a is a self-adjoint operator then, for any z E H, (z, z), = (o(z), z)
is real and thus the constant d in the previous proposition as well. Let w =
1
or w =
and define V : H --+ H by V(x) = -U(x). The operator V generates
W
the same I,-symmetry as U, is linear or antilinear if U is linear or antilinear
respectively and
(V(x), V ( Y ) ) =
~ f(x, Y),.
8 SOME OTHER GENERALIZATIONS
8.1
The case of Hilbert modules
Roughly speaking, a Hilbert module is a Hilbert space in which a C*-algebra
replaces the complex field. More precisely, let A be a C*-algebra and let 7-1 be
a left A-module with a product [.,.] : 'FI x 'FI + A. The module 7-1 is called an
A-Hilbert module or a Hilbert C*-module over A if:
2. [af, 91 = a[f, gl ;
3. [g,f l = [f,gl*;
4. [f,f ] 2 0 and [f,f ] = 0
* f = 0;
Wigner's Theorem and its Generalizations
5. 7-1 is complete with respect to the norm f
[f,f
11 4
The concept of Hilbert module is due to I. Kaplansky and in its full generality
to Paschke [1973].
In [Molnbr, 19981 and [Molnk, 19991, L. Molnbr proves some generalizations
of Wigner's theorem to Hilbert modules using a new algebraic approach and, in
particular, results from ring theory. If, as usual in C*-algebras, 1x1 denotes the
,
can be stated
unique positive square root of xx*, the main result of [ ~ o l n b r19991
as follows.
PROPOSITION 19 ( [ ~ o l n b r19991).
,
Let 7-1 be a Hilbert C*-module over the C*algebra A = Mn(C) of all n x n complex matrices with n > 1. Let T : 7-i + 7-1 be
a mapping with the property that:
There exists a linear mapping U, called an A-isometry, and cp : 7-1 + C such that:
The absence of A-anti-isometries is a consequence of the noncommutativity of A.
In [Bakit and Guljas, 20021, the C*-algebra A = Mn(C) of all n x n complex
matrices is replaced by the C*-algebra K ( H ) of all compact operators on a Hilbert
space and a result similar to the above proposition is proved. Since the C*-algebra
A = Mn(C) of all n x n complex matrices can be considered as the C*-algebra of
all compact operators on Cn, this result generalizes Proposition 19.
8.2
Genemlization to pairs of symmetries
Using, as in [Molnbr, 19981 and [Molnk, 19991, a new algebraic approach Molnbr,
in [2000a], first proved a generalization of Wigner's Theorem for pairs of symmetries.
PROPOSITION 20 ([Molnbr, 2000a, Theorem I]). Let H be a complex Hilbert
space of dimension at least 3 and let S and T be two bijections of [HI with the
property that:
Then there are invertible either both linear or both antilinear operators U, V :
H -t H such that V = U*-I and
Georges Chevalier
456
The proof is not very simple. It uses Gleason's theorem and one of its variations
due to A. Dvureeenskij [1993, Chapter 31 and also a result of Jacobson and Rickart
stating that any Jordan homomorphiim of a local matrix ring is the sum of a
homomorphism and an anti-homomorphism. As a corollary, the author obtains
the following generalization of Wigner's theorem to a-symmetries in indefinite
inner product spaces. Note that in this proposition, the metric operator a is not
assumed to be self-adjoint.
PROPOSITION 21 ([Molndr, 2000a, Corollary 21). Let H be a complex Hilbert
space with dimH 2 3 and let a : H + H be a n invertible operator. For any
a-symmetry T on H there exists a n invertible either linear or anti-linear operator
U o n H with U*aU = €afor some scalar E E (C of modulus 1 such that
We remark that if E = 1 then U is either a a-unitary or an a-anti-unitary
operator and if 6 = -1, U is either a a-pseudo unitary or a a-pseudo antiunitary
operator since in the latter case
8.3 Preservation of angles
Let [q5] and [cp] be two rays generated by the unit vectors q5 and cp. The scalar
product ([\$I, [cp]) = I(q5, cp)l can be interpreted as the cosine of the angle between
the rays [I\$] and [cp] and so Wigner's theorem states that any bijection of rays which
preserves angles of rays in both directions is generated by a unitary or antiunitary
operator. This interpretation has led L. Molndr to investigate bijections between
,
A first problem is
subspaces of a Hilbert space preserving angles ( [ ~ o l n d r20011).
to make a choice among several possible definitions of the angle of two subspaces
in a Hilbert space. Molndr holds that the adequate concept is that of principal
angles defined as follows.
Let F and G be two finite dimensional subspaces of a Hilbert space with 1 5
7r
p = d i m F 5 dimG. T h e p principal angles of F and G, 0 5 el 5 ... 5 0 < -,
P-2
are defined recursively by means of their cosines:
a
For 1 < k 5 p,
where [xl, . . . ,x , ] ~denotes the subspace orthogonal to the subspace generated by
{XI,.
- .,
57%).
Wigner's Theorem and its Generalizations
457
The definition is not easy. Molnir, however, asserts that a characterization of
the cos Ole'sis simple: they are the square roots of the eigenvalues of the positive
counted by their multiplicity.
If L(F,G) denotes the system of principal angles between F and G, then
L(Fl, GI) = L(F2,G2) if and only if the operators PFlPGIPF,and PF2PG2
PF2
are unitarily equivalent, that is there exists a unitary operator U such that
PF~
P GPF~
~ = UPFP
~ GPF~
~ u*.
For infinite dimensional subspaces the definition of the principal angles is more
difficult. But we still have that the unitary equivalence of PFlPGlPFland PF2PG2
PF2
characterizes the system of principal angles, i.e. L(Fl, GI) = L(F2,G2).
PROPOSITION 22 ( [ ~ o l n i r2001,
,
Main he or em]). Let H be a real or complex
Hilbert space with dim H 2 n, n E N. Let Pn(H) and P,(H) if dimH = ca be
the sets of all rank-n projections and all infinite rank projections.
Suppose that 4 : Pn(H) 4 Pn(H) i s a transformation with the property that
dim H
Ifn=l orn#then there exists a linear or conjugate-linear isometry
2
V : H + H such that
4(P) = VPV* ( P € Pn(H)).
If H i s infinite dimensional and if the surjective transformation 4 : P,
satisfies
L(4(P), 4(Q)) = L(P, Q)
then there exists a unitary or antiunitary operator U : H 4 H , such that
4(P)
4
P,
UPU* ( P € Pw(H)).
dimH
I f n = -then the transformation P + 1 -P preserves angles but cannot be
2
written in the form of the above proposition. Thus this case is really exceptional,
and Molnar hopes that in this case the only transformations preserving angles are
P + P and P -t 1 - P. This problem is open.
In [Aerts and Daubechies, 19831, another notion of angle is considered, corresponding to the first principal angle in the previous one. It is proved that a morphism from the orthocomplemented lattice of all subspaces of a complex Hilbert
space H (dim H 2 3) to another preserves the angles of lines.
8.4
Generalization to type 11factors and Banach spaces
Let H be a Hilbert space. The usual trace is a mapping t r defined on the cone of
positive operators B+ (a)
of the algebra B,(H) of all self-adjoint operators of H
with values in the extended positive reals [0,+m]. This mapping satisfies:
Georges Chevalier
Traces on a von Neumann algebra are, as above, defined by the properties I.,
2., 3.' and a trace r on the cone M+ of all positive elements of a von Neumann
algebra M is said to be
a
faithful if x
> 0 implies r ( x ) > 0;
semifinite if for any nonzero x E M+ there exists a nonzero y 5 x such that
r ( y ) < +m:
normal if
SUP xi) = SUP r ( x i ) for any bounded increasing net
{xi) E M+.
Type I and type I1 factors have a common characterization in terms of traces: a
factor is type I or type I1 if and only if it admits a faithful semifinite normal trace.
With the help of Corollary 5, Wigner's theorem can be stated as follows: if T is
a bijection of the set of all rank-one projection defined on a Hilbert space H such
that
t r ( T ( P ) T ( Q ) )= t r ( P Q )
then T can be extended to a *-automorphism or a *-antiautomorphism of B ( H ) .
In [Moln&r,2000b], the corresponding result for Type I1 factors is stated as
follows.
PROPOSITION 23 ( [ ~ o l n k2000b,
,
he or em]). Let p be a faithful normal semifinite trace on a type II factor M. If T is a bijective transformation of the set of
all nonzero finite projections for which
then there is either a linear *-automorphism or a linear *-antiautomorphism
M which extends T .
of
The proof is not easy using deep results from algebra and functional analysis.
In [MolnLr, 2000c], Wigner's theorem is generalized to Banach spaces. This
generalization is based on formulation of Wigner's theorem given in Corollary 5.
There, the Hilbert space H is replaced by a real or complex Banach space X and
projections are replaced by rank-one idempotents.
Let us denote by X' the topological dual of X and by A' : X' + X' the adjoint
of the bounded linear operator A : X + X and by I l ( X ) the set of all rank-one
idempotents on X . Wigner's theorem for Banach spaces reads as follows.
Wigner's Theorem and its Generalizations
PROPOSITION 24 ( [ ~ o l n b r2000c,
,
Theorem 11). Let
bijective mapping for which
a1 : Il(X) + Il(X)
459
be a
There exists a bijective bounded operator A : X + X such that
or there exits a bounded biiective operator B : X' + X such that
For the proof Molnar extends to the operator algebra F ( X ) of all finite rank
operators and applies a result of [Omladie and ~ e m r l ,19931 related to additive
mappings on F(X).
The usual version of Wigner's theorem is deduced as a corollary and, in the
second part of the paper, a Wigner-type result for matrix algebras over fields of
,
Theorem 21.
characteristic different from 2 is proved [ ~ o l n L r2000c,
8.5 Continuous solutions of 1 (x,y ) 1 = (IS(x),S ( y )1)
In the last section of [ ~ a t z 1996],
,
J. Ratz studies the continuous solutions of
the equation I (x, y) 1 = I (S(x), S(y)) 1 where S is a mapping defined on a real or
complex inner product space. Since in the usual version of Wigner's theorem the
unitary or antiunitary mapping U is defined up to a phase factor, it would seem
that the continuous solutions of the equation l(x, y) I = I (S(x), S(y))I are of the
form S = WU for a continuous phase factor a. The precise result, however, reads
as follows.
,
Theorem 13, (b) and (c)]). Let H and H' two
PROPOSITION 25 ( [ ~ a t z1996,
2 and S : H 4 H' a
inner product spaces over IK = W or cG with dimH
continuous solution of 1(x,y) 1 = I (S(x), S(y)) 1.
>
I . If
IK = R then S is a linear isometry;
2. If K = cG then there exists an isometry or an anti-isometry U : H 4 H' and
f : H + R such that cos of and sin of are continuous on H - (0) and
S(x) = eif( x ) ~ ( x ) (x E H).
Conversely, all these mappings are continuous solutions of I(x, y)l
I(s(x)ls(~>>l-
=
Georges Chevalier
9 QUATERNIONIC HILBERT SPACES
9.1
T h e definite case
The field of quaternions, denoted by W, is a non-commutative field which is also
a four dimensional vector space over R. Any quaternion q may be represented in
the form
q = a bil ciz dig (a, b, c) E R3,
+ + +
with
:i
= -1 and
iris
..
= -zsz,
= it
for any even permutation (r,s, t) of (1,2,3). The set of all quaternions q with
b = c = d = 0 is a subfield of W isomorphic to R.
The conjugate i j of q is defined by i j = a - bil - cia - dig and the modulus of q
b2 c2 d2 = (qq)+.
is 191 = Ja2
The center of W, that is the set of quaternions which commute with any quaternion, is W and for any automorphism f of W there exists qo E W,
= 1, such
that, for any q E W,f (q) = qoqq;l. In other words, any automorphism of W is
inner. We remark that for any x E W and any automorphism f , f (x) = x.
Let H be is a vector space over W endowed with a definite sesquilinear Hermitian
form with respect to conjugation. The pair (H, (., .)) is called a quaternionic
Hilbert space if, for any x E H, (x, x) 2 0 and if H is complete for the topology
defined by the norm 11 x !I= (x, x);.
+ + +
Quantum mechanics using quaternionic Hilbert spaces has been introduced in
et a l , 19621 and also in [E'inkelstein et al., 19591. The first quaternionic version of Wigner's Theorem appeared in '[Uhlhorn, 19621 in its Uhlhorn
formulation and a generalization of the classical version of Wigner's theorem to
quaternionic Hilbert spaces is established in [Sharma and Almeida, 1990al. There
are only two differences with the complex case.
inkel el stein
1. Symmetries and I-symmetries are induced by unitary operators only. This
result is linked to the form of the automorphisms of W and can be compared
to a result of quaternionic projective geometry: any isomorphism of the
lattice of all subspaces of a quaternionic vector space is generated by a linear
bijection,
2. Wigner's theorem is false in the two-dimensional case even for symmetries.
A counterexample is given in [Bargmann, 19641 and the general form of
counterexamples is in [Sharma and Almeida, 1990a].
9.2
Theindefinitecase
In [Molnbr, 20021, the author suggested to generalize his main result to quaternionic Hilbert spaces. This was done in [~emrl,20031 in the definite case and as
Wigner's Theorem and its Generalizations
46 1
well as the indefinite case. As in [ ~ o l n k20021,
,
the first result of [~emrl,20031
is concerned with bijective transformations of the set of all rank-one idempotent
operators which preserve zero products.
PROPOSITION 26 ([~emrl,2003, Theorem I]). Let I ( H ) be the set of all rankone idempotent operators of a quaternionic Hilbert space H with dimH 3. If
: I ( H ) + I ( H ) is a bijective transformation satisfying
>
TS = 0 if and only if +(T)+(S) = 0 (T, S E I(H)).
Then
@(T)= ATA-I
( T E I(H))
where A : H + H i s a invertible semilinear operator.
This proposition allows to prove the following generalization of Wigner's theorem to indefinite quaternionic inner product spaces.
PROPOSITION 27 ([~emrl,2003, Theorem 21). Let H be a quaternionic Hilbert
space with dim H 2 3 and let a : H -+ H be an invertible operator. If T i s a
I,-symmetry then there exist a nonzero c E W and a bounded semilinear bijective
operator U : H + H such that T([x]) = [U(x)]for every nonzero x E W and
where a : W + ]HI i s the automorphism of the field W corresponding t o the semilinear
operator U.
REMARK. Assume that a = l~ and let d be the quaternion such that a(%) =
d-lxd with Id1 = 1. The real number c in (14) is positive by 11 U(x) [I2= c 11 x 112
and if V : H + H is defined by V(x)
=
d
-U(x)
fi
then V is an operator satisfying
We have got the quaternionic version of Uhlhorn's generalization of Wigner's theorem as in [Sharma and Almeida, 1990al.
10 A TOPOLOGICAL AND LATTICE APPROACH
In order to obtain a very general form of Wigner's theorem in its Uhlhorn version,
we will, in this Section, replace the Hilbert space H by a topological vector space
E over a field K.
The first problem is to define an orthogonality relation on the set of all lines
of E . In general, orthogonality relations on a lattice of subspaces are defined by
means of non degenerate bilinear forms and usually no natural bilinear form is
available on E x E . On the other hand, let E* be the algebraic dual space of E
462
Georges Chevalier
formed by all linear functionals on E. There always exists a natural non degenerate
bilinear form on E x E*, namely the mapping B : E x E* t K defined by
B(x, y) = y(x), x E E, y E E*.
Since E is a topological space, closed subspaces seem more convenient than general
subspaces and this condition forces us to replace E* by E', the topological dual
of E formed by all continuous liiear functionals on E. But now the restriction
of the bilinear form B to E x E' is not necessarily non degenerate. We therefore
consider only pairs (E, E') which are pairs of dual spaces in the sense of Mackey
[I9451 or DieudonnC [1942].
If (E, F ) is a pair of dual spaces then the lattice of all closed subspaces of E is an
irreducible complete DAC-lattice and such lattices appear as the natural setting
of the lattice part of this study. In the first part of this Section, we will specify
the definitions and the main properties of pairs of dual spaces and DAC-lattices.
The second part is devoted to the lattice tools necessary to the generalization
of Wigner's theorem. In particular we will prove a generalization of the First
Fundamental Theorem of projective geometry to lattices of closed subspaces.
As a consequence of the previous results, different Wigner-type theorems are
proved in the last part.
In the following, we will assume that the dimensions of all vector spaces is not
less than 3 and the heights of all the lattices are not less than 4.
1 0 1 DAC-lattices and pairs of dual spaces
An AC-lattice is an atomistic lattice with the covering property: if p is an atom
a n d a A p = O t h e n a a a v p , that i s a < x < a V p i m p l i e s a = x o r a V p = x . In
general, At(L) will denote the set of all atoms of lattice L and if L* is the dual
lattice of L then At(L*) is also the set of all coatoms of L.
If L and and its dual lattice L* are AC-lattices, L is called a DAC-lattice
[Maeda and Maeda, 19701. Irreducible complete DAC-lattices of heights 2 4 are
representable by lattices of closed subspaces and many lattices of subspaces are
DAC-lattices. Let us specify this last assertion.
Let K be a field, E a left vector space over K , F a right vector space over K.
If there exists a non degenerate bilinear form B on E x F, we say that (E, F ) is
a pair of dual spaces [DieudonnC, 19421. Since the form is non degenerate, F can
be interpreted as a subspace of the algebraic dual E* of E and E as a subspace
of F*. This interpretation allows one to write, for any x E E and any y E F ,
X(Y)= Y(Z)= B(x, Y).
For example, if E is a locally convex space and E' its topological dual space
then (E,E') is naturally a pair of dual spaces with B(x, y) = y (x) [Kothe, 1969,
page 2341.
For a subspace A of E, we put
A~ = {y E F I B(x, y) = 0 for every x E A).
Wigner's Theorem and its Generalizations
Similarly, let
B I = {x E E
I B(x, y) = 0 for every y E B )
for every subspace B of F . A subspace A of E is called F-closed if A = ALL and
the set of all F-closed subspaces, denoted by LF(E) and ordered by set-inclusion,
is a complete irreducible DAC-lattice [ ~ a e d aand Maeda, 1970, Theorem 33.41.
Conversely, for any irreducible complete DAC-lattice L of height 2 4, there exists
a pair (E, F) of dual spaces such that L is isomorphic to the lattice of all F-closed
subspaces of E [Maeda and Maeda, 1970, Theorem 33.71, [Kothe, 1969, §10.3].
The set LE(F) of all E-closed subspaces of F is similarly defined and is also
a DAC-lattice. The two DAC-lattices LF(E) and LE(F) are dual isomorphic by
the mapping A + AL [ ~ a e d aand Maeda, 1970, Theorem 33.41 and an element
X of LF(E) and an element Y of LE(F) are said to be orthogonal if X C YI
(Equivalently, Y C x L ) and we write X IY.
Let (E, F ) be a pair of dual spaces. The linear weak topology on E, denoted
by a ( E , F ) , is the linear topology defined by taking {GL I G C F, dim G < m)
as a basis of neighborhoods of 0. If F is interpreted as a subspace of the algebraic
dual of E then a subbasis of neighborhoods of 0 consists of the kernels of elements
of F .
The linear weak topology on F, denoted by a(F,E), is defined in the same
way. The space F can be interpreted as the topological dual of E for the a(E,F )
topology and E as the topological dual of F for the a(F,E) topology. Equipped
with their linear weak topologies, E and F are topological vector spaces [Kothe,
1969, ~10.31if the topology on K is discrete.
Moreover, for a subspace G c E, we have ?? = GIIand thus to be a closed
subspace in E is an unambiguous notion. If K = W or C, this result generalizes to
any pair (E, F ) and any locally convex topology over E when F is the dual of E
for this topology [Kothe, 1969, \$20.31.
10.2
The adjoint of a semi-linear map
For a future generalization of the First Fundamental Theorem of projective geometry, it is necessary to make precise the concept of the adjoint of a semilinear
mapping.
Let (E, F ) be a pair of dual spaces and f : E + E a r-linear mapping with
respect to an automorphism r of K.
If y is an element of E* then the mapping x E E 4 ~ - l ( ~ (x)))
( f belongs to
E*. Let us define f * : E* --t E* by f*(y)(x) = ~ - l ( ~ ( f ( x ) for
) ) any y E E* and
any x E E. The mapping f * is r-l-linear and will be called the adjoint of f .
Assume that f is weakly continuous. If y E F c E* then the mapping x E E +
~ - ' ( ~ ((x)))
f is weakly continuous and therefore f*(y) E F. The restriction of f *
to F is weakly continuous [Chevalier, 2005b] and in what follows, i f f : E t E is
a weakly continuous r-linear mapping then f * will always mean the restriction
of f* to F C E* and thus f ** is a mapping from E to E.
464
Georges Chevalier
Now, let us consider a clmed subspace X of E. For any x E X and any
y E F , y ( f ( x ) ) = 0 is equivalent to f * ( Y )( x ) = 0 and so, as for linear mappings,
f *-l(xL)
= f ( X ) I for any X E L F ( E ) . Others results about adjoints are:
f** = f for a weakly continuous semilinear mapping and f*-l = f-l* if f is a
weakly continuous semilinear bijection with a weakly continuous inverse [Chevalier,
2005bl.
10.3 The lattice tools for a lattice approach to Wigner's theorem
The first propmition is comparable to Proposition 14.
PROPOSITION 28 ([chevalier, 2005b, Proposition 11). Let L be a complete D A C
lattice. I f f is an automorphism of the poset A t ( L ) U A t ( L * ) then f extends to a n
automorphism B of the lattice L.
Idea of the proof. For two families of atoms (pi)iEr and ( q j )j E
of L
and an extension B of f to L can be defined by B ( 0 ) = 0 and ,(z) =
V f (pi) if
iEI
0 # x = V p i , . Using (l5), it is easy to check that B is an automorphism of the
iEI
DAC-lattice L.
The following result generalizes the First Fundamental Theorem of projective
,
to lattices of closed subspaces. In its statement a bicontingeometry [ ~ a e r19521
uous bijection is meant to be a continuous bijection with a continuous inverse.
PROPOSITION 29 (Chevalier [2005a; 2005bl). Let ( E l ,F l ) and (E2,F 2 ) be two
pairs of dual spaces over the fields Kl and Kz.
+
1. If there exists a n isomorphism
of the lattice LFl(E1)onto the lattice
LFz (E2)then Kl and Kz are isomorphic fields and there exists a bicontinuous semilinear bijection s : El H E2 such that, for every Fl-closed subspace
M of El, + ( M ) = s ( M ) . If a bicontinuous r-linear bijection s and a bacontinuous TI-linear bijection st generate the same automorphism then there
exists k E K 2 such that T I = k r k k l and s f = ks.
+
2. If that K1 and Kz are isomorphic fields then, for every semilinear bijection
s : El H E2, the following statements are equivalent:
(a) The bijection s i s bicontinuous.
(b) H E LF,( E l )I+ s ( H ) is a bijection from the set of all Fl-closed hyperplanes of El onto the set of all F2-closed hyperplanes of E2.
(c) M E LFl ( E l )H s ( M ) is a n isomorphism from the lattice LFl( E l ) into
-
L F 2 (E2)
Wigner's Theorem and its Generalizations
465
+
Idea of the proof. The mapping is an order isomorphism of the poset of all finite
dimensional subspaces of El into the poset of all finite dimensional subspaces of
Ez. This isomorphism extends to an isomorphism cp of the lattice of all subspaces
of El into the lattice of all subspaces of E2by
p(N) = U{\$(M)
I M c N,
dim M
< m)
and cp extends \$. The First Fundamental Theorem of projective geometry then
concludes the proof of 1.
The bijection s maps the set of all hyperplanes of El bijectively into the set of all
hyperplanes of E2 and , if s is bicontinuous, closed hyperplanes of El correspond
with closed hyperplanes of E2. Thus (a) + (b) holds true. The proof of (b) +
(c) uses the fact that in a DAC-lattice LF(E), any element is an intersection of
closed hyperplanes. Since the family of all closed hyperplanes is a O-neighborhood
subbasis for the linear weak topology (c) +- (a) is clear.
10.4
Wigner-type theorems
PROPOSITION 30 (A Wigner-type theorem for DAC-lattices). Let L be an irreducible complete DAC-lattice and f an automorphism of the poset At(L) UAt(L*).
If L is representable as the lattice LF(E) of all F-closed subspaces of a pair of
dual spaces (E, F ) then f extends to an automorphism q5 of LF(E) and there exists a bicontinuous semilinear bijection s : E -+ E such that @(M)= s(M) for all
M E LF(E).
Proof. Use Propositions 28 and 29.
REMARK. Let L be the lattice of all subspaces of a vector space E. If f is
an automorphism of At(L) U At(L*) (informally speaking, f preserves in both
directions inclusion of lines in hyperplanes) then f extends to an automorphism
of L and there exists a semilinear bijection s : E -t E such that, for any subspace
x,f (XI = 4x1.
Let (E, F ) be a pair of dual spaces. If f : At (LF (E)) u A ~ ( L E
(F)) + At (LF (E)) U
At(LE(F)) is at the same time a bijection of A~(LF(E))and a bijection of
At(LE(F)) such that, for any p E At(LF(E)) and any q E At(LE(F)),
then f is called an I-symmetry over (E, F ) .
PROPOSITION 31 (A Wigner-type theorem for a pair of dual spaces). Let f be
an I-symmetry over a pair (E, F ) of dual spaces. There exists a bicontinuous
semilinear bijection s : E -+ E such that:
Georges Chevalier
2. for any q E A ~ ( L E ( F )f) (q)
, = s*-'(q).
Proof. As M E L F ( E )-+ M-'- E L E ( F )is an anti-isomorphism of lattices, we can
define a bijection fl of At(LF(E)*)by fl(P) = f ( p L ) l for any P E At(LF(E)*).
Let g be the extension of fl to A t ( L F ( E ) U
) At(LF(E)*)which agrees with f on
At(LF( E ) ) .If p E At(LF( E ) )and P E A ~ ( L F ( E ) *we
) , have p 5 P if and only if
p IPL which is also equivalent to f (p) I f ( p L )= g ( ~ ) that
L
is g(p) 5 g(P).
By Proposition 28, g extends to an automorphism G of the lattice L F ( E )and by
using Proposition 30 there exists a bicontinuous semilinear bijection s such that,
for every F-closed subspace M , G ( M )= s ( M ) . In particular, for every atom p of
LF( E ) ,S ( P ) = G(P)= g ( p ) = f b).
Let q E At(LE(F)).We have :
10.5 Examples
= W or C is
weakly continuous if and only if f is continuous with respect to the linear weak
topology u ( E ,El) [Kothe, 1969, 20.41. If K = W then a semi- linear mapping is
linear since the identity is the only automorphism of W and we have the following
version of Wigner's theorem.
A linear mapping f , defined on a locally convex space E over K
COROLLARY 32. Let E be a real locally convex space and E' its dual. I f f is
an I-symmetry over the dual pair ( E ,E') then there exists a weakly bicontinuozls
linear bijection s : E + E such that
for any p E A ~ ( L E ~ ( Ef )b)
), =sb),
for any q E A ~ ( L E ( E ' )f) (q)
, = s*-'(q).
If E is metrizable then s is continuous.
For the last claim of this corollary we have used the fact that weakly continuous
linear mappings between metrizable spaces are continuous [Schaefer, 1964, Chapter
IV, 3.4 and 7.41.
If K = C then the automorphism r associated to the semi-linear bijection s of
Proposition 31 cannot be continuous (In a locally convex space over a field K , the
topology on K is not the discrete one but is defined by means of the modulus)
and an extra hypothesis seems necessary to obtain a version of Wigner's theorem
close to the classical one.
Wigper's Theorem and its Generalizations
467
COROLLARY 33. Let E be a n infinite-dimensional complex n o w e d space and
f a n I-symmetry over the dual pair ( E lE'). There exists a linear or conjugate
linear bijection s : E + E which is bicontinuous for the n o r m topology and such
that:
for any q E A ~ ( L E ( E ' f)(, q ) = s*-'(q).
Proof. Let s be the semi-linear bijection obtained by using Proposition 31. Since
s is continuous for the weak linear topology, s carries orthogonally closed hyperplanes to orthogonally closed hyperplanes. But orthogonally closed subspaces of
E agree with topologically closed subspaces he, 1969, 520, 3 (2)] and by using
a result of [Kakutani and Mackey, 19461 or illmo more and Longstaff, 19841, Lemma
2, s is either linear or conjugate linear. A linear mapping on a metrizable space E
is continuous if and only if this mapping is continuous for the linear weak topology
a ( E ,E') and the generalization of this result to a conjugate linear mapping is easy.
Thus, s is continuous and, by using a similar argument, s-l is also continuous.
REMARKS.
1. In [2002], L. MolnLr proved the same result for complex Banach spaces.
2. If E is a finite-dimensional complex normed space and s : E + E is
any T-linear bijection then one can define an I-symmetry on ( E lE') by
f (p) = s(p) if p E &(LEI ( E ) ) and f ( q ) = s * - ' ( ~ ) if q E,A ~ ( L E ( E ' )If. the
automorphism T of C is neither the identity nor the conjugation then s is
not continuous.
The following corollary is the classical version of Wigner's Theorem in its
Uhlhorn version. What is of interest here is only its proof, which uses the previous
results and especially the Wigner-type theorem for pairs of dual spaces.
COROLLARY 34. Let H be a Hzlbert space over K = B or cG (dim H 2 3) and T
an I - s y m m e t r y o n [HI. There exists a semilinear mapping r : H + H such that:
1. for any ray p E [HI, r(p) = T ( p ) ,
2. if K = B,r i s a unitary operator,
3. If K = C, r is either a unitary or an antiunitary operator.
T h e mapping T extends to a n automorphism cP of the orthomodular lattice of all
closed subspaces of H
Georges Chevalier
468
Proof. Since H is a Hilbert space, the correspondence 6 which associates to every
y E H the continuous functional 6(y) : x + (x, y) is an isomorphism from H onto
its dual H' in the real case and an anti-isomorphism in the complex case. This
mapping generates an isomorphism, denoted by the same symbol, from the lattice
of all closed subspaces of H onto the lattice of all closed subspaces of its dual.
with two different
Remark that for (p,q) E [HI x [H'], p Iq ++ p I
meanings for the ~ r t h o g o n ~ relations.
ty
Let S be the I-symmetry on (H, H')
defined for (p, q) E [HI x [H'] by S(p) = T(p) and S(q) = 6(T(B-'(q))).
Assume that H is infinite-dimensional in the complex case.
If B is always the canonical bilinear form on the pair (H, H') then for any
(5, y) E H x H', B(x, y) = (x, 6-'(y)) and if f f l : H t H denotes the adjoint,
defined by means of the inner product of the linear or antilinear operator f , then
f f l = 6-' f*6 where f* : H' t H' is the adjoint of f defined in 10.2.
By using Corollary 32 or Corollary 33, there exists a linear or antilinear bijective
operator s : H + H such that for any p E [HI, s(p) = S(p) = T(p) and for any
q E [H'], s*-'(9) = S(q) = O(T(6-'(q))). Therefore, for any p E [HI, T(p) =
8-'(s*-'(6(p))) = SU-'(~). Thus s and su-' generate the same automorphism of
the lattice of all subspaces of H and there exists X E K such that s = Xsfl-'. We
have ssfl= XIK and therefore X > 0.
1
If r = -s
fi
then, for any ray p, r(p) = T(p) and r-'
= rfl. If
K
=
R, r is a
unitary operator and, in the infinitedimensional complex case, r is a unitary or
antiunitary operator.
Now, we assume that H is a finite-dimensional complex Hilbert space.
If f is a T-linear mapping then, for any y E H, the mapping x E H t
7-I ((f (x), y)) is a linear form and there exists an element f fl(y) E H such that
~ - ' ( ( f (x), y)) = (x, ffl(y)). It is easy to check that the mapping f fl : x E H +
ffl(z) E H satisfies f#(y y') = ffl(y) fU(yr) and ffl(Xy) = ~-l(X)f(y) if Y,
y' E H, X E C. Moreover, f f l = O-'f*6.
As in the infinite-dimensional case, let S be the I-symmetry on (H, H') defined
for (p, q) E [HI x [H'] by S(p) = T(p) and S(q) = 6 ( ~ ( 6 - ~ ( q ) ) )If. s : H t H
is the T-linear bijection obtained by using Proposition 31 then, as in the infinitedimensional case, there exits a bijection r such that r-' = rfl and r(p) = T(p) for
any P E [HI.
For any p E C and 0 # x E H, r-'(px) = 7-'(p)r-'(2)
and rfl(px) =
T-l(jZ)rfl(x). Therefore ~ - ' ( p ) = ~-l(jZ) and thus r-'(IW) = R. The automorphism T is the identity or the conjugation and r a unitary or antiunitary operator.
Let Q, be the automorphism of the lattice of all closed subspace generated by
the unitary or antiunitary operator r. This automorphism extends T and by using
= r , we
the relations rfl-'(XI) = T ( x ) ~for a closed subspace X of H and TU-'
.
mapping Q, is an automorphism of the orthomodular
have @ ( X I )= @ ( x ) ~ The
lattice of all closed subspaces of H.
This last result can be also obtained by using Proposition 14.
+
+
Wigner's Theorem and its Generalizations
11 SOME OTHER SYMMETRY GROUPS
In his books (Wigner [1931; 19591) Wigner postulated the invariance of the transition probabilities between pure states and this hypothesis leads to the symmetry
group of a physical system. As in [Cassinelli e t al., 19971 or [Cassinelli e t al., 2004,
Chapter 21, one can consider several other objects playing a fundamental role in
the Hilbert space formulation of quantum mechanics. Among these there are:
1. The convex set S of all states. This set is represented by the set of all positive
trace class operators of trace 1 on H.
2. The orthomodular lattice L of all closed subspaces of H. This lattice r e p
resents the lattice of all yesno experiments or propositions on the physical
system.
3. The effect algebra E ([~reechieand Foulis, 19951) of all positive operators
bounded by the unit operator.
4. The Jordan algebra B, of all bounded self-adjoint operators. In this algebra,
AB+BA
the product is the Jordan product defined by A o B =
and an
2
element of this algebra can be interpreted as a bounded observable.
5. The C*-algebra B of all bounded operators on H.
To each of these objects there corresponds a group of automorphisms, i.e. bijective mappings preserving the relevant structures. Let us give three examples.
1. . An automorphism of the orthomodular lattice L is any bijective mapping
f : L + L such that for any closed subspaces M and N of H
Any automorphism preserves all joins and meets.
2. If A and B are two positive operators bounded by the unit operator I of H
then, in the effect algebra E, the sum A+ B is defined only if A+ B 5 I. If it
is the case, it agrees with the usual sum of two operators. An automorphism
of E is a bijective mapping f : E + E such that and in this case, f (A+ B) =
f (A) f (B). Any automorphism preserves the unit operator and the order
([Cassinelli e t al., 19971).
+
3. An element of Aut(B) is a linear or antilinear bijection satisfying @(AB)=
@(A)@(B)and @(A*)= @(A)*,A, B E B. This is not the usual definition
of a C*-algebra automorphism.
Georges Chevalier
If Aut(X) is one of the above groups of automorphisms then it can be considered
as the group of all transformations of the physical system preserving the physical
features associated to X . Thus its elements are the symmetries of the system
when the formulation of quantum mechanics is based on the set X of objects.
For example, Piron defines a symmetry as an isomorphism of the orthomodular
lattices L of all propositions iron, 1976, page 371.
The following proposition shows that all these automorphism groups are essentially the same. Roughly speaking, if a transformation of a physical system leaves
invariant some convenient physically significant features then all the physically
significant features are also invariant.
PROPOSITION 35 ([Cassinelli et al., 19971).
1. If dim H 2 2 then all the automorphism groups Aut(P), Aut(B), Aut(B,),
Aut(E) and Aut(S) are isomorphic.
2. If dim H 2 3, then the above groups are also isomorphic to Aut(L) and to
the group Autl(P) of all bijections of P preserving orthogonality.
Shortened proof.
In [Cassinelli et al., 19971, the authors prove the existence of five injective group
homomorphisms
Aut(P)
% Aut(B) % Aut(B,) % Aut(E) 2 Aut(S) 5 Aut(P)
defined as follows.
1. Any automorphism of P (identified here to the set of all rank-one projections)
is of the form P + UPU* with U unitary or antiunitary (Corollary 5) and
a1 associates to this automorphism the automorphism A E B + UAU* of
B.
2. Since E c B, c B the definitions of Q2 and a3are easy: it suffices for Qa to
check that the restriction of an element of Aut(B) to B, belongs to Aut(B,)
and similarly for a3.
3. The definition of a4is more difficult. In [Cassinelli et al., 19971, it is proved
that, for'any f E Aut(E) and any state p, there exists a state p' such that,
for any E E E,
tr[p o f -'(E)] = t ~ [o ~E)]
'
and
a4is defined by @4(f)(p) = p'
4. Since P c S, it suffices for defining a5to check that the restriction to pure
states of a mapping preserving the convex structure of S also preserves transition probabilities between pure states [Cassinelli et al., 1997, Proposition
4.61.
Wigner's Theorem and its Generalizations
471
All the previous homomorphisms are isomorphisms since by a straightforward
computation, using the properties of the trace for @4,
If dim H 2 3 then, by using Proposition 14, Aut(L) is isomorphic to Autl(P)
and Corollary 13 implies that Aut(L) and Aut(P) are isomorphic.
REMARKS.
1. If dim H = 2 then Aut(P) is a proper subgroup of Autl(P) [~hlhorn,1962,
5.21, [~assinelliet al., 1997, Example 4.11.
This result can be also obtained by a comparison of the cardinals of Aut(P)
and Autl(P). Indeed, Card(Aut(P)) = Card(C) and Card(Autl(P)) is the
cardinal of the set of all bijections from C to C, that is 2Card". Thus, if
dim H = 2, Aut(P) and Autl(P) are dserent and non-isomorphic groups.
Without any assumption on the dimension of H, the groups Autl(P) and
Aut(L) are isomorphic [Cassinelli et al., 1997, Proposition 4.8 and Corollary
4.31.
2. In [Cassinelli et al., 19971, the previous automorphism groups are all endowed
with natural initial topologies and it is proved that there are all homeomorphic topological groups. Each of them is a second countable metrizable
topological group where second countable means that the open sets have a
denumerable basis.
3. In [1962], Uhlhorn associated to any automorphism f of P a mapping Q(f) :
B + B defined by
If f is generated by a unitary operator U then Q(f)(A) = UAU*,
If f is generated by an antiunitary operator U then Q(f )(A) = UA*U*,
AEB.
If U is unitary then Q(f) is an automorphism of the C*-algebra B and otherwise, Q(f) is an anti automorphism, that is Q(f)(AB) = Q(f)(B)Q(f)(A).
The mapping Q is an isomorphism of the group Aut(S) into the group of all
automorphisms or antiautomorphisms of the C*-algebra B.
If, as in [~assinelliet al., 19971, one consider Q1(f) defined in the same way
for U unitary or antiunitary by Q1(f)(A) = UAU* then Q1is an isomorphism
of the group Aut(P) into the group Aut(B) where an element 4 of Aut(B) is
defined as a linear or antilinear bijection satisfying 4(AB) = 4(A)+(B) and
4(A*) = +(A)*. In the antiunitary case, such automorphism is neither an
automorphism nor an antiautomorphism of the C*-algebra B in the usual
sense.
4. The isomorphism of Aut(P), Aut(S) and Aut(B,) is also a consequence of
Theorems 2.1, 2.2 and 2.3 in [Simon, 19761. A more elementary study of
these groups may be found in [Hunziker, 19721.
Georges Chevalier
12 FROM AUTOMORPHISMS TO THE HAMILTONIAN
In this concluding Section we will show how Wigner's theorem permits us to derive
the Schrdinger equation for a conservative physical system. While the proof of
Wiper's theorem is quite elementary, this derivation requires deep results from
functional analysis, so we will outline only its main steps. For more information,
see [Jauch, 1968, Chapter 101, [Beltrametti and Cassinelli, 1981, Chapters 6 and
231. For a rigorous derivation and to go deeper in the question see [~argmann,
1970; Simon, 19761 or [Varadarajan, 19851; [Jordan, 1991] is also an interesting
reference.
First, we will assume that the state of the physical system at time t is completely
determined by its state at any time to < t, such a system is called a conservative
system. Let us denote by [cpt] the state at time t and by Vt,t~the two-parameter
family of transformations of states such that [cptt] = Vt,t~([cpt])if t < t'. If to <
tl < tz then
and thus
Now if we assume that Vt,t~depends only on the difference of times r = t' - t
then (16) becomes
This hypothesis corresponds to those situations in which there is homogeneity
of time and, in the terminology of Markov process, to a stationary Markovian
evolution.
Another requirement that has natural physical motivations is the continuity of
the real function
-,(&([PI)>[+I)
for every pure states [cp] and [+I. Intuitively, this hypothesis means that small
changes in time produce small changes in probabilities.
We also assume the reversibiity of the evolution of the system which forces V,
t o be invertible with Vyl = V-,.
Finally, r 4 V, is a homomorphism from the group (R, +) into a group of
bijective transformations of the set of all states.
In order to use Wigner's theorem a last hypothesis is necessary: r + V, preserves the transition probabilities, i.e.
for every pure states [cp] and
[\$I.
Wigner's Theorem and its Generalizations
473
Now we are in position to apply Wigner's theorem: V, is phase equivalent to a
unitary or antiunitary operator U, defined up to a phase factor and considering
that U, = U+o U; , U, is in fact a unitary operator. Using Equation (17), we have
and r -+ U, is a projective representation of the additive group of W into the group
of all unitary operators of H. By a result of Wigner [1939],(see also [Simon, 1976]),
w(r, 7') can be chosen in such a way that w(r, 7') = 1 and Equation (18) becomes
Thus T + U, is a strongly continuous one-parameter unitary group [Reed and
Simon, 1972, page 2651. Using Stone's theorem [Reed and Simon, 1972, page 2641,
[Simon, 1976, Theorem VIII], we get that there exists a self-adjoint operator A on
H such that
- -irA.
,-e
In mathematics, the self-adjoint operator A is called the infinitesimal generator
of U, and, in physics, the Hamiltonian of the system. It represents the physical
quantity called the energy of the system.
Now the Schrodinger equation is at hand: if the state of the system at time t is
\$(t) then +(t)= eVirA4(0)and satisfies the Schrodinger differential equation
+
BIBLIOGRAPHY
[Aerts and Daubechies, 19831 D. Aerts and I. Daubechies. Simple proof that the structurepreserving maps between quantum-mechanical propositional systems conserve the angles, Helvetica Physica Acta 56, 1187-1190, 1983.
[Baer, 19521 R. Baer. Linear algebra and projective geometry, Academic Press Inc. 1952.
[BakiC and GuljaB, 20021 D. BakiC and B. GuljS. Wigner's Theorem i n Hilbert C*-modules
over C-algebras of compact operators, Proc. Amer. Math. Soc. 130 ( 8 ) , 2343-2349, 2002.
[Bargmann, 19641 V. Bargmann. Note on Wigner's T h w r e m o n symmetry operations, J . Math.
Phys. 5 , 862-868, 1964.
[Bargmann, 19701 V. Bargmann. O n unitary ray representations of continuous groups, Annals
of Math., 59 ( I ) , 1-46, 1954. 1970.
[Beltrametti and Cassinelli, 19811 E. Beltrametti and G. Cassinelli. The logic of quantum mechanics, Addison- Wesley Publishing Company, 1981.
[Birkhoff, 19671 G. Birkhoff. Lattice theory, Amer. Math. Soc. Colloquium Publications,Volume
XXV, 1967.
[Birkhoff and von Neumann, 19361 G . Birkhoff and J. von Neumann. The logic of quantum mechanics, Ann. Math., 37, 823-843, 1936.
[Bognk, 19741 J . Bognkr. Indefinite Inner product spaces, Springer-Verlag 1974.
[Bracci et al., 19751 L. Bracci, G. Morchio and F. Stocchi. Wigner's theorem o n symmetries i n
indefinite metric spaces, Cornmun. Math. Phys. 41, 289-299, 1975.
[Cassinelli et al., 19971 G. Cassinelli, E. de Vito, P. Lahti and A. Levrero. Symmetry groups
i n quantum mechanics and the theorem of Wigner o n the symmetry transformations, Rar.
Math. Phys. 9 , 921-941, 1997.
474
Georges Chevalier
[Cassinelli et al., 20041 G. Cassinelli, E. de Vito, P. Lahti and A. Levrero. The Theory of Symmetry Actions i n Quantum Mechanics, Springer, 2004.
[Chevalier, 2005al G. Chevalier. Automorphisms of a n orthomodular poset of projections, Int.
J. Theor. Phys. 44 (7), 985-998, 2005.
[Chevalier, 2005b] G. Chevalier. Lattice approach to Wigner-type thwrenw, Int. J. Theor. Phys.
44 ( l l ) , 1903-1913, 2005.
[Di?udonnk, 19421 J. DieudonnB. La dualit6 d a m les espaces vectoriels topologiques, Ann. Sci.
Ecole Norm. Sup., 59, 107-139, 1942.
[Dvure~enskij,19931 A. DvureEenskij. Gleason's Theorem and its applications, Kluwer Academic Publisher, 1993.
[Emch and Piron, 19631 G. Emch and C. Piron. Symmetry i n quantum theory, J . Math. Phys.
4 (4), 469-473, 1963.
[Fillmore and Longstaff, 19841 P. A. Fillmore and W. E. Longstaff. Zsomorphisnzs of lattices of
closed subspaces, Can. J. Math. Vol XXXVI (5), 820-829, 1984.
inkel el stein et al., 19591 D. Finkelstein, J. M. Jauch and D. Speiser. Notes on quaternion quant u m mechanics, Cern Theoretical Study Division No 59-7 1-17, No 59-9 1-36, No 59-17 1-28,
1959.
[Finkelstein et al., 19621 D. Finkelstein, J. M. Jauch, S. Schiminovich and D. Speiser. Foundations of quaternion quantum mechanics, J. Math. Phys. 3, 207-220, 1962.
[Greechie and Foulis, 19951 R. Greechie and D. Foulis. 'Pansition to effect algebras, Int. J .
Theor. Phys. 34 (8), 1369-1382, 1995.
[Gyory, 20041 M. Gyory. A new proof of Wigner's Theorem, Rep. Math. Phys. 54 (2) 159-167,
2004.
[Henley, 20011 E. M. Henley. What do we know about time reversal invariance, Fisika B 10 (3),
161-165, 2001.
[Houtappel et al., 19651 R. M. F. Houtappel, H. van Dam and P. Wigner. The conceptual basis
and w e of the geometric invariance principles, h v . Mod. Phys. 37, 595-632, 1965.
[Hunziker, 19721 W. Hunziker. A note o n symmetry operations in quantum mechanics, Helvetica
Physica Acta 45, 233-236, 1972.
[Kakutani and Mackey, 19461 S. Kakutani and G. Mackey. Ring and lattice characterizations of
wnzplex Hilbert spaces, Bull. Amer. Soc. 52, . 727-733, 1946.
[Kalmbach, 19831 G. Kalmbach. OrUlonzodular lattices, Academic Press, 1983.
[Keller, 19801 H. A. Keller. Ein nicht-klassischer Hilbertscher m u m , Mathemathische
Zeitschrift, 172, 41-49, 1980.
[Kothe, 19691 G. Kothe. Topological vector spaces I, Springer-Verlag, 1969.
[Jauch, 19681 J. Jauch. Foundations of quantum mechanics, Addison-Wesley Publishing Company, 1968.
[Jordan, 19911 T . F. Jordan. Assumptions implying the Schrodinger equation, Amer. J. Phys.
59 (7), 606-608, 1991.
[Lomont and Mendelson, 19631 J. S. Lomont and P. Mendelson. The Wigner unitaryantiunitary theorem, Annals of Math. 78 (3), 548-559, 1963.
[Mackey, 19451 G. Mackey. O n infinite-dimensional linear spaces, Trans. Amer. Math. Soc. 57,
155-207, 1945.
[Maeda and Maeda, 19701 F. Maeda and S. Maeda. Theory of symmetric lattices, SpringerVerlag, 1970.
[Molniir, 19961 L. Molnk. Wigner's unitary-antiunitary theorem via Herstein's theorem on Jordan homomorphisnm, J. Nat. Geom. 10, 137-148, 1996.
[ ~ o l n r i r19981
,
L. Molnir. A n algebraic approach to Wigner's unitary-antiunitary theorem, J .
Austral. Math. Soc. 65, 354-369, 1998.
[Molniir, 19991 L. Molniir. A generalization of Wigner's unitary-antiunitary theorem to Hilbert
modules, J . Math. Phys. 40 ( l l ) , 55445554, 1999.
[MolnAr, 2000aI L. Molniir. Genemlization of Wigner's unitary-antiunitary theorem for indejinite inner product spaces, Commun. Math. Phys. 210, p. 785-791, 2000.
[Molnrir, 2000bI L. Molnk. Wigner-type thwrem o n symmetry tmnsfomaations in type II factors, Int. J. Thwr. Phys. 39 (6), 1463-1466, 2000.
[Molniir, 2000c] L. Molnir. Wigner-type thwrem o n synznletry tmnsfornzations i n Banach
spaces, Publ. Math. (Debrecen) 58 , 231-239, 2000.
Wigner's Theorem and its Generalizations
475
[Molnir, 20011 L. M o l n k . !lkansfornaations o n the set of all n-dimensional subspaces of a Hilbert
space preserving principal angles, Commun. Math. Phys. 217, 409-421, 2001.
[ M o l d r , 20021 L. Molnir. Orthogonality preserving transformations o n indefinite inner product spaces: Generalization of Uhlhorn's version of Wigner's theorem, Journal o f Functional
Analysis 194, 248-262, 2002.
[OmladiE and Semrl, 19931 M. OmladiE and P. Semrl. Additive mappings preserving operators
of rank-one, Linear Algebra Appl. 182, 239-256, 1993.
[Ovchinnikov, 19931 P. Ovchinnikov. Automorphwms of the poset of skew projections, Journal
o f functional analysis 115, 184189, 1993.
[Paschke, 19731 W . Paschke. Inner pmduct spaces over B*-algebras, 'lkans. Amer. Math. Soc.
182, 443-468, 1973.
[Pian and Sharma, 19831 J. Pian and C . S. Sharma. Calculw o n wmplez Banach spaces, Inter.
J . Theo. Phys. 22, 107-130, 1983.
[Piron, 19761 C . Piron. Foundations of quantuna physics, W . A. Benjamin, Reading, Massachussetts, 1976.
[Poschadel, 20001 N . Poschadel. A note o n families of observables and a generalization of
Wigner's theorem i n c', J . Math. Phys. 41 ( l l ) , 7832-7838, 2000.
[Rlitz, 19961 J . Rlitz. O n Wigner's theorem: Remarks, wnaplements, comments, and corollaries,
Aequationes Mathematicae 52, 1-9, 1996.
[Reed and Simon, 19721 M. Reed and B. Simon. Methods of modern mathemetical physics I,
[Roberts and Roepstorff, 19691 J . E. Roberts and G. Roepstorff. Some basic concepts of algebraic quantum theory, Commun. math. Phys. 11, 321-338, 1969.
[Sharrna and Almeida, 1990al C. S. Sharma and D. F. Almeida. Additive isonzetries o n a quaternionic Hilbert space, J. Math. Phys. 31 ( 5 ) 1035-1041, 1990.
[ ~ h a r m and
a Almeida, 1990bl C. S. Sharma and D. F. Almeida. A direct proof of Wigner's theorem o n maps which preserve transition probabilities between pure states of quantum systems,
Ann. Phys. 197, 300-309, 1990.
[Simon, 19761 B. Simon. Quantum dynamics: from autonaorphism to hamiltonian i n Studies i n
Mathematical Physics. Essays i n Honor of Valentine Bargmann, eds E. H. Lieb, B. Simon,
A. S. Wightman, Princeton Series i n Physics, Princeton University Press, Princeton, 327-349,
1976.
[Schaefer,19641 H . Schaefer. Topological vector spaces, T h e Macmillan Company, 1964.
[Semrl, 20031 P. Semrl. Generalized symmetry transfomaations on quaternionic indefinite inner
product spaces: A n extension of quaternionic version of Wigner's theorem, Comrn. Math.
Phys. 242, 579-584, 2003.
[Uhlhorn, 19621 U . Uhlhorn. Representation of symmetry tmnsfornzations i n quantum mechanics, Arkiv for Fysik 23 (30), 307-340, 1962.
[van der Broek, 1984a] P. M. van der Broek. Symmetry transformations in indefinite metric
spaces: A generalization of Wigner's theorem, Physica A 127, 599-612, 1984.
[van der Broek, 1984b] P. M. van der Broek. Group representations in indefininite metric
spaces, J . Math. Phys. 25 ( 5 ) , 1205-1210, 1984.
[Varadarajan, 19851 V . S. Varadarajan. Geometry of quantuna theory (2nd edition), SpringerVerlag, 1985.
[Weinberg, 19951 S. Weinberg. The Quantum Theory of Fields, Cambridge University Press,
1995.
[Wigner, 19311 E. Wigner. Gruppentheorie und ihre Anwendung auf die Quantenmechanik der
Atomspektren, Vieweg, Braunscheig, 1931.
[wigner, 19391 E. Wigner. O n unitary representations of the inhomogeneow Lorentz Groups,
Annals o f Math. 40 ( I ) , 149-204, 1939.
[Wigner, 19591 E. Wigner. Group theory and its applications to quantum mechanics of atomic
spectra, Academic Press Inc., New York, 1959.
[Wigner, 19601 P. Wigner. Phenomenological distinction between unitary and antiunitary symmetry operators ,J. Math. Phys. 1, 414-416, 1960.
[Wright, 19771 R. Wright. The structure of projection-valued states: A generalization of the
Wigner's theorem, Int. Journ. Theor. Physics 16 ( a ) , 567-573, 1977.
[Zhmud, 19921 E. M. Zhmud. O n a Theorem of Wigner, Selecta Mathernatica Sovietica 11 ( 2 ) ,
71-82, 1992.
```