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Helvetica Physica Acta, Vol. 51 (1978), Birkhäuser Verlag, Basel
About the structure -preserving maps
of aquantum mechanical
propositional system
by Dirk Aerts1) and Ingrid Daubechies1)
Theoretische Natuurkunde, Vrije Universiteit Brüssel, Pleinlaan 2, B1050 Brussels
(2. II. 1978; rev. 26. VII. 1978)
Abstract. We study c-morphisms from one Hilbert space lattice (with dimension at least three) to
another one; we show that for ac-morphism conserving modular pairs, there exists alinear structure
underlying such amorphism, which enables us to construct explicitly afamily of linear maps generating this
morphism. As aspecial case we prove that aunitary c-morphism which preserves the atoms (i.e. maps one-
dimensional subspaces into one-dimensional subspaces) is necessarily an isomorphism. Counterexamples
are given when the Hilbert space has dimension 2.
1. Definition of apropositional system and Piron's representation theorem
According to Piron's axiomatic description of quantum mechanics [1], the
structure of the set of the propositions corresponding to 'yes-no' experiments on a
physical system is that of acomplete, orthocomplemented, weakly modular and
atomic lattice which satisfies the covering law. Such alattice is called apropositional
system. If the physical system has no super-selection rules, the propositional system is
irreducible. We will first give some definitions concerning propositional systems. For
more details we refer the reader to [1].
Let JP be the collection of all the propositions concerning aphysical system.
1.1 Definition. fP is called aCROC whenever SP satisfies the following conditions:
(i) SP, <is apartially ordered set with <as partial order relation (1.1)
(ii) SP is acomplete lattice, which means that for every family {bi}iel of
elements in f£ there exists agreatest lower bound /\il b{ and a
least upper bound \/i^jbi. (1.2)
(iii) SP is an orthocomplemented lattice, which means that there exists a
bijection ':Sf -> SP such that: Vb,ceSf: (1.3)
Wetenschappelijke medewerkers bij het Interuniversitair Instituut voor Kernwetenschappen (in het
kader van navorsingsprogramma 21 EN).
638 Dirk Aerts and Ingrid Daubechies H. P. A.
(bj b
bab' 0and bvb' /
b<c=> d<b'
where IV*er °and 0Abese b
(iv) SP is weakly modular, which means that if b<cthen ca(c' vb) b(LA)
In aCROC it may be interesting to consider pairs of propositions. In the following
definition we introduce some pair-properties (see [1], [2], [3]).
1.2 Definition. In aCROC two propositions band care said to be
-compatible if the sublattice generated by {b, b', c, d} is distributive,
notation: b<-^c
-amodular pair if for any d> c: (b vc) ad(b ad) vc
notation: (b, c)M2)
It is easy to see that if b<-> c, (b, c)M. The converse is not true. Moreover, we see
now that the condition for weak modularity (1.4) can be reinterpreted as:
b<c=> b*-*c.
1.3 Definition. Fhe 'center' of aCROC is the set of propositions compatible with all
other propositions.
The center of aCROC is also aCROC which is distributive.
1.4 Definition. If SP is aCROC and be SP we consider the set {x \x<b, xeSP}.
If we define on this set arelative orthocomplementation xr x' ab, then it is easy to
check that this set becomes aCROC; we will call it the segment [0, b~\.
1.5 Definition. ACROC SP will be called 'irreducible' if the center of SP contains
only 0and I.
1.6 Definition. Ifb, ce SP,b #candb <c, one says that c'covers' b, ifb <x<c
for some xe SS implies xborx c.An element of SP which covers 0is called an atom.
Now we have all the notions to define apropositional system. So let SP be again
the collection of all the propositions concerning aphysical system.
1.7 Definition. SP is apropositional system if
(i) SP is aCROC
(ii) SP is atomic, which means that for every beSP, b#0, there exists
an atom peSP such that 0<p<b. (1.5)
(iii) SP satisfies the covering law, which means that ifp is an atom of SS
and be SS and pab0, then pvbcovers b. (1-6)
In [2] Birkhoff defines two kinds of these pairs: modular pairs and dual modular pairs. In the
following we shall only need one of these two kinds ;since no confusion can arise, we shall call them
modular pairs. Moreover, one can prove that when the CROC is an irreducible propositional system
isomorphic to aP(3f (see further) every modular pair is adual modular pair in Birkhoff's
terminology, and vice versa. For more details concerning these pairs, see [3].
Vol. 51, 1978 Maps of aquantum mechanical propositional system 639
Example: If we take acomplex Hilbert space JP and if we define P(Jt) to be the
collection of all the closed subspaces of JP, then P(jP) becomes an irreducible
propositional system if we define the operations as follows :
(i) If G, Fe P(Jt), then G<Fiff. GcFset-theoretically. (1.7)
(ii) If Gì eP(3P) Vi e/, then Aiel Gt fì,t/ G(1.8)
(iii) and V,e, Gt is the closure of the linear subspace generated by all
the Gi's. (1.9)
(iv) If GeP(2fe) then G' GL which is the subspace orthogonal to
G. (1.10)
(v) The atoms are the one-dimensional subspaces. (Ill)
P(JP) is the propositional system one uses in ordinary quantum mechanics. As
expected two propositions of P(PC) are compatible iff the corresponding projection
operators commute.
Conversely, we can ask ourselves if it is always possible to represent an irreducible
propositional system by astructure which resembles the given example. This question
is answered by the representation theorem of Piron ([1] and [4]) which says that it is
always possible to realize an irreducible propositional system by the set P(V) of all
biorthogonal subspaces of avector space Vover some field K. The orthocomplemen¬
tation defines on Kan involutive anti-automorphism and on Fanon-degenerate
sesquilinear form; the weak modularity ensures that the whole space is linearly
generated by any element and the corresponding orthogonal subspace. Hence all
irreducible propositional systems are given by generalization of the P(3P") in the
example above. The representation theorem enables us to prove interesting results
about propositional systems. As an example we give the following easy characteri¬
zation of modular pairs in an irreducible propositional system.
1.8 Lemma. Let P(V) be the realization of an irreducible propositional system SS;
let a, be SP and let a,, è, be their realizations in P(V). Fhen
(a, Z?)M<=> at +bx a^ vbx.
Proof: see [3].
An immediate consequence of the lemma is the following property:
(a, b)M => (b, a)M. If we take the field in Piron's representation theorem to be Cand
the involutive anti-automorphism of <C to be the conjugation, then the vector space V
becomes an Hilbert space over <C ([4], [5]). Since the set of all biorthogonal subspaces
of aHilbert space is exactly P(Jf we are now reduced to the case considered in the
example. In the following we will restrict ourselves to the cases where the field is <C.
2. m-morphisms
2.1 Definition. Ac-morphism from aCROC SS\ to aCROC SP2 is amapfof SS\ to
SS2 which preserves unions and compatible pairs, i.e.
te/
f\\lbi \Vf(bi) bteSS, Vie/ (2.1)
b~c=>f(b)~f(c) b,ceSS± (2.2)
640 Dirk Aerts and Ingrid Daubechies H. P. A.
Abijective c-morphism from SSX to SS2 will be called an isomorphism.
The terminology c-morphism is justified because we want (2.1) to be valid for
every non-empty family of bis. Weakening condition (2.1) we find the definitions of a
morphism ((2.1) is only required for finite families) and of a<x-morphism ((2.1) is only
required for countable families). For ageneral CROC however only c-morphisms
preserve the completeness. It is easy to prove [1] that when/is ac-morphism from SS ^
to SS2 then :
/(0) 0(2.3)
f(b')=f(b)' Af(I) be SS, (2.4)
/(a*.Ì A/(*i) he SS, Vie/ (2.5)
Vie/ /iel
2.2 Definition. Amap ffrom aCROC SS1 to aCROC SP2 is called an m-morphism
if it is ac-morphism which preserves modular pairs, i.e. (b, c)M => (f(b),f(c))M.
If we want to study c-morphisms between propositional systems, we can in general
restrict ourselves to c-morphisms between irreducible propositional systems. Indeed,
let SS xand SP2 be two propositional systems with centers Zx, Z2 respectively. One can
prove that Zj and Z2 are also propositional systems [1]. Let (zi)XEA, (z2)ßeBbe the sets
of atoms of Z1, Z2.
Iff is ac-morphism from SS1 to SS2, we can define
ULO,zn^[_o,z2Pi
X^-f(x) AZ2
These (fxp)XeA,peB are c-morphisms between irreducible propositional systems and the
set offxß determines completely/:
VxeSS1:f(x)= VVfxß(x az1)
aeA ßeB
If/is an w-morphism, it is easy to check that the/a/3 are w-morphisms too. We will
henceforth restrict ourselves to the study of c-morphisms (or m-morphisms) from one
irreducible propositional system to another one.
It is interesting to remark that any c-morphism from an irreducible CROC SS jto
aCROC SS2 is either injective or the zero-morphism (see [1], pp. 31, 33).
2.3 Definition. Aunitary c-morphism fof aCROC SS\ into aCROC SS2 is a
c-morphism such that f(Ix) I2. (2.6)
In the study of c-morphisms we can always restrict ourselves to unitary c-
morphisms because if
/: SP >SP2 is ac-morphism
Vol. 51, 1978 Maps of aquantum mechanical propositional system 641
then we can always study/by studying the morphism/defined by
/= JS?! [0,/(/)] suchthat f(b)=f(b) for beSPX
This /is aunitary c-morphism.
Taking together all these remarks, and remembering that we decided to consider
only irreducible propositional systems isomorphic to aP(Jf), we see now that we can
restrict ourselves to unitary c-morphisms from an irreducible propositional system
P(Pf) into an irreducible propositional system P(PC).
It is aremarkable fact that when the dimensions of the Hilbert spaces Pt° and 2P"
are at least equal to 3, one can prove that any isomorphism from P(Jf into P(PC) can
be generated by aunitary or anti-unitary map from Jif into PC. This is aconsequence
of the following theorem proved by Wigner [6].
2.4 Theorem. Let Jf and PC be two complex Hilbert spaces of dimension at least
3and f: P(3P)^> P(PC) an isomorphism, then it is always possible to find amap U:
JP >JP' which is unitary or anti-unitary and such that:
f(G) {U(x) ;xeG} VG eP( Jf
This is not only an amazing result but it is also very useful because it is always by
using this theorem that one is able to prove the deepest results about problems in
relation with isomorphisms of propositional systems.
Our aim in this paper is to prove asimilar result for amore general structure
preserving map of apropositional system, namely an m-morphism. In fact we shall
prove our main theorem with the help of apparently weaker conditions than the one
given in the definition of an m-morphism. These weaker conditions are specific for
atomic CROC's, where our definition of an m-morphism is valid for any CROC. The
following proposition states moreover that these weaker conditions are equivalent
with the fact that/is an m-morphism. To distinguish the individual elements of Kfrom
the linear subspaces they generate, and which are elements of P(V), we will henceforth
write xto denote the linear subspace generated by x.
2.5 Proposition. Let 3^, M" be two complex Hilbert spaces, let fbe ac-morphism
from P(JP) to P(jC). Fhen fis an m-morphism iff one of the following is true:
(1) Vx, ynon-zero vectors in JP:f(x y) <=f(x) +f(y) (2.7)
(2) Vx, y, znon-zero vectors in PC: z<xvy=>f(z) <=/(x) +f(y) (2.8)
(3) Vx, ynon-zero vectors in JC:f(x) +f(y) is aclosed subspace of PC (2.9)
The proof of this proposition is given in the Appendix.
If the/(x) are finite-dimensional, condition (2.9) is automatically satisfied, and/is
an m-morphism. In particular, if the/(x) are one-dimensional, /is an m-morphism.
One can even prove (see the beginning ofthe proof of corollary 4.2) the following :if for
one atom pin P(jP),f(p) is an atom in P(JC), then/is an m-morphism.
Our main theorem can now be stated as follows :
642 Dirk Aerts and Ingrid Daubechies H. P. A.
2.6 Theorem. Let Pf and PC be two complex Hilbert spaces with dimension greater
than or equal to three, and fa unitary m-morphism from P(Jf) to P(JC). Fhere exists a
family of maps (tp^^jfrom $f to PC such that:
-each <f>j is an isometry or an anti-isometry
-the different q)j(JP) are orthogonal subspaces of PC, and their direct sum is #C:
PP ©<t>ji^)
-(<Pj)jej generates fin the following sense:
\/GeP(Pf):f(G) V<t>}(G) 0{^(x); xeG}
je Jje J
We will prove this theorem and others in Section 3. In Section 4we restrict
ourselves to the important special case of c-morphisms mapping atoms onto atoms. It
is to be remarked that the theorem we obtain there is still amore general one than
Theorem 2.4 :our results are the same, but where Wigner supposed /to be an
isomorphism, we only use that/is aunitary c-morphism mapping one atom onto an
atom :no surjectivity is needed. Our different theorems hold only if dim Pf >3(the
same applies for Wigner's theorem) ;in Section 5we give some counter-examples for
dim Pe 1.
3. Construction of the underlying linear structure
We start by stating our main theorem. The formulation as presented here is rather
compact :we will relate it to the former one at the end of this section.
3.1 Theorem. Let Pf and #C be two complex Hilbert spaces, with dimension greater
than or equal to 3. Let fbe an m-morphism mapping P(Pf) into P(PC).
Fhen for every couple (x, y) of non-zero elements of Pf, there exists abijective
bounded linear map Fyx mapping f(x) ontof(y), such that the set of maps {Fyx ;x, yePf}
has the following properties:
Pxx l/(x)
F=(F V1
xxy x* yx/
FFF
rzy* yx *¦ zx
Fj. F+F
*y+z.x *y. x**¦ z. x
Pxy.Xx= Py.x Xe<C,XpO
Fyx is an isomorphism if \\x\\ \\y\\
For every non-zero xin Pf, there exist moreover two orthogonal projections P\ and P\,
elements of SS(f(x)), such that
P\-P\ 0
Pi +PÌ *m
py, FP?F i=\ 2
1iAyx1- l*xy lx>^
and FXXiX XPi +XP\
Vol. 51, 1978 Maps of aquantum mechanical propositional system 643
The proof of this theorem is aquite extensive one; this is why it has been split into
different lemmas.
If/is zero, the/(x) are zero for every xand the theorem is trivial. We therefore will
restrict ourselves to the cases where/is different from zero.
3.2 Lemma. Let JP, PC,f be as in Fheorem 3.1, with fdifferent from the null-
morphism. For every two non-zero elements x, yin Jf one can define alinear map Fyx,
element of SP(f(x),f(y)), such that
(a) Fyx is bijective (3.1)
(b) Fxy (Fyx)'1 (3.2)
(c) FyzFzx Fyx for every non-zero zin Pf (3.3)
(d) Fxx lfixl (3.4)
(e) FXy Xx Fyx for every non-zero Xin C(3.5)
Proof. Since/is different from zero, it is injective. This implies that for any xin
JP, different from zero, the space/(x) is asubspace of PC different from the null-space.
Let x, ybe two linearly independent elements in Jf.
We have xaj0, xaxy=0 and yaxy=0, hence
f(x)Af(y) 0,f(x)Af(x-y) 0and f(y) a/(x -y) 0(3.6)
Moreover/(x) c/(j) +f(x -y).
If now x1 is an element of/(x), the previous remarks imply that there exist unique
y', u', elements off(y) and/(x y) respectively, such that
x' =y' +if (3.7)
This correspondence defines alinear map Fyx from f(x) to f(y) :
Fyx(x')=y
This map is bijective ;moreover :
*xy \*yx)
We will now prove that these Fxy are bounded linear maps. From the already proven
results we see that Fxy is an everywhere defined linear map from one F-space to another
(both/(x) and/(j) are closed subspaces). We will prove that Fxy is closed. Indeed, let
iy'n)n be aconverging sequence in f(y) with limit /, such that the sequence (x'n
Pxy(y'n))n converges too: x'n-*x'. Then there exists asequence u' in/(x y) such
that
un Vn ~X
Since both sequences (P) and (/) converge, the sequence (t4) is aCauchy sequence
with limit u'. Since moreover the spaces /(x), f(y), f(x y) are closed, we have
x' e/(x), y' ef(y), u' ef(x y). But y' x' +u', hence x' Fxy(y'), which implies
that Fxy is closed. Using the closed graph theorem ([7], p. 57), we see that Fxy is
bounded.
644 Dirk Aerts and Ingrid Daubechies H. P. A.
Up till now, we have proven statements (a) and (b) for x, ylinearly independent.
We will proceed further, and prove statement (c) for x, y, zlinearly independent, after
which we will re-examine the different cases for linearly dependent vectors.
Take x, y, zlinearly independent in #C We have
y-z<yvzand yz=(y x) +(x z)<y-x vxz
which implies
yz<(yvP)A(y xvx r)
Because of the linear independency of x, y, zwe can conclude that
y-z=(yvz)A(y xv x z) (3.8)
which implies
f(y -z) -(f(y) v/(f)) a(f(x -y)vf(x- z)) (3.9)
Take now x' in f(x), and define /, z' by
y'=Fyx(x'), z' Fzx(P) (3.10)
From the construction of Fyx, Fzx we know that there exist u', v' with u' ef(x y),
v' ef(x z), such that
x' y' +u' z' +v'
This implies that /z' t/ u', hence /z' ef(x y) vf(x z). Since/ z' is
obviously an element of/(j) v/(z), we conclude from (3.9) that /z' ef(y z).
This implies y' Fyz(z').
Because ofthe definitions (3.10) of/, z', this implies
F°FF
yz zx *yx 5
which proves statement (c) for three linearly independent vectors.
Whenever xand yare linearly dependent, i.e. yex, we define Fyx by
Fyx Fyt oFtx where tePf\x. (3.11)
The fact that this definition of Fyx does not depend on the choice of tis an almost trivial
application of the just proved chain rule for linearly independent vectors. The
bijectivity of Fyx follows immediately from its definition (3.11). It is also easy to check
that statements (b) and (c) hold even when the vectors are not linearly independent.
Statement (d) is now atrivial consequence of (b) and (3.11). Statement (e) is atrivial
consequence ofthe construction ofthe Fxy. I
Remark. It is acrucial point in this proof that dim Pf >3. If dim JP 2, one can
still construct the Fxy in the same way as was done in the lemma, but it is then
impossible to prove the chain rule. We give acounter-example in the last section.
3.3 Lemma. Let Pf, PC ,fbe as in Lemma 3.2. Let {Fxy; x, yeJf, x#0py) be
the set of maps constructed in the proof of Lemma 3.2, and let x, y, zbe three non-zero
vectors in Pf with y+z^0. Fhen the following holds:
Vol. 51, 1978 Maps of aquantum mechanical propositional system 645
for every x/ in f(x): Fyx(x!) +FZX(P) ef(y +z)
and Fyx +Fzx Fy+Z<x (3.12)
Proof. Suppose first that x, y, zare linearly independent. We rewrite (3.8) in two
different forms :
y+z=(yvz)A(xvx yz) (3.13)
and
xyz(x yvz) a(x zvy) (3.14)
Take Pin f(x), and define /, z' by
/F,x(xO, z' Fzx(P) (3.15)
Since x' y' ef(x y) and x' z' e/(x z), we have
x' -/-z' e(/(x -j) v/(z)) a(/(x -z) v/(.y)) =/(x -j-z)
(because of (3.14))
Hence
/+z' x' -(x' -/-z') e/(x) v/(x -y-z)
which implies
y' +z'e (f(y) vf(z)) a(f(x) vf(x -y-z)) =f(y +z) (because of (3.13))
We have now
x' y+z' +x' yp
with
y' +z' ef(y +z), x' -y' -z' e/(x -y-z),
hence
Fy +z,x(x')=y' +P.
Because of (3.15) this implies
Py+z,x Pyx +Pzx f°r x, j, zlinearly independent
Suppose now that zand yare linearly dependent, i.e. yezwith y+z#0, and suppose
x$y. There exists a;such that {x, t,y +z] and {x, t+y, z} are both linearly
independent.
We have
Ft +y+z,x ^t +y,x +*'zx ^tx +"yx +^zx
^tx +-T' y+z,x
which implies
Fi F+F
1y+z,x *yx <xzx-
We have only one more case to check :suppose xey vzand y+z#0(y and z
646 Dirk Aerts and Ingrid Daubechies H. P. A.
may be linearly dependent). We can choose atsuch that t$y vz. We have
Fy +z,x "y +z.fftx ~^yt^tx +''zd'tx
*yx '"zx m
3.4 Lemma. Let Jtf, JtP,f, {Fxy} be as in Lemma 3.3. Let x, ybe two non-zero
vectors in JP with \\x\\ \\y\\. Fhen Fyx is an isomorphism.
Proof. Consider first the case where x±.y. Since ||x|| \\y\\, this implies
xy1x+yand thus/(x -y) 1f(x +y). Take x', x" in/(x) and define y', y" in f(y)
by
y' Fyx(x') y" Fyx(x") (3.16)
Because of Lemma 3.3 we know that
xf +Yef(x +y) x" -y"ef(x-y) (3.17)
From (3.16), (3.17) and the fact that f(x) l/(y), we infer that
0(x' +/, x" -y") (x-, x") -(/, /')
But this leads to
(Fyx(P), Fyx(x")) (x-, x")
which implies that Fyx is an isomorphism.
If xand yare not orthogonal, we can choose avector zsuch that
\\A\ IMI IIjII and z±x,z±y
Because ofthe previous result, we know that Fzx and Fyz are isomorphisms. From the
chain rule (3.3) it follows now immediately that Fyx Fyz °Fzx is an isomorphism too.
I
We have now proven the first part of Theorem 3.1. To prove the second part,
which gives in fact aspectral decomposition of the operators FXx x, we need the
following two remarks.
3.5 Remarks.
1. Let JP be aHilbert space, and let G, Hbe two commuting bounded self-adjoint
operators on M' with G>H. Let a(G) be the spectrum of the operator Gand
g(H) be the spectrum ofthe operator H. Fhen inf a(G) ^inf o(H) and sup a(G)
>sup a(H).
2. Let Pf be aHilbert space, Aabounded normal operator such that A2 1.
Fhen o(A) <= {;', /}.
(These remarks are easy to prove if one uses the Gel'fand isomorphism for
commutative C*-algebras: see [8].)
3.6 Lemma. Let Pf, PC',/, {Fyx} be as in Lemma 3.3. For every xin Pf, x#0, there
Vol. 51, 1978 Maps of aquantum mechanical propositional system 647
exist two orthogonal projectors Pf, Pf in SS(f(x)), such that the following holds:
(a) Pf-Pf 0
(b) Pf +Pf lm
(c) Vx, ynon-zero vectors in Jf: PJ FyxPfFxy /=1,2 (3.18)
(d) VAeC, X#0: FAXjX XPf +XPf (3.19)
Proof. To alleviate somewhat our notations, we will write A(X;x) for FXXix. From
(3.5) we see that
A(X; px) A(X; x) (3.20)
On the other hand, (3.3) implies
A(X; y) FXy>y FXy.XxFXx>xFxy FyxA(X\ x)Fxy (3.21)
These two relations (3.20) and (3.21) imply that the same structure will be found for all
A(X ;x), since they are all equal up to aunitary transformation :
AiX; y) (Fy^l^x^Xiy)~1A(X; x)F\\y\\i\\x\\x,, (3.22)
From (3.12) we see that
A(X +p; x) Fa+IÂ)XyX FXXtX +FXiX A(X; x) +A(p; x) (3.23)
while (3.3) and (3.5) imply
A(Xp; x) FXllXtX FXliX^xF^x FXXtXFßx<x A(X; x)A(p; x) (3.24)
From (3.23) and (3.24) we infer that the map
A(-;x):^SS(f(x))
A> A(X; x)
is in fact arepresentation of the (commutative) field Cin SS(f(x)). But we can prove
more.
Let x, ybe two non-zero vectors in Jf, with ||x|| ||j|| and xLy. Choose
XeC\{0}. We have Xx +y1x-Xy, hence
f(Xx +y)±f(x-Xy) (3.25)
Let x', x" be elements of/(x), and /Fyx(P), y" Fyx(x"). Then:
Fxx+y,x(x')'= A(X; x)x' +y' _
Px-Iy,x(x") -x" -FXyiyFyxx" x" -A(X; y)y"
From (3.25) and f(x) -Lf(y) we see that
0(A(X; x)x' +/, x" -A(Y y)y")
(A(X;x)x',x")-(y',A(X-y)f)
(A(X; x)x', x") -(Fyx(x') A(X; y)Fyx(x"))
Using (3.21) and the fact that Fyx is an isomorphism (see Lemma 3.5), this leads to:
(A(X; x)x', x") (x', A(X; x)x"),
648 Dirk Aerts and Ingrid Daubechies H.P.A.
hence
(A(X; x))* A(T; x) (3.26)
We will now use the three relations (3.23), (3.24) and (3.26) to prove the lemma.
First of all, it follows from (3.23) and (3.24) that for any rational number qone has
A(q;x) qtnx) (qe<S>) (3.27)
If r,, r2 are real numbers with rt >r2, then we have
A(r,; x) -A(r2;x) A(r^ -r2;x) A2(y/r1 -r2; x)
A*(Jri-r2; x)A(^r[-r2; x) >0(3.28)
which implies that the restriction to IR ofthe map A( ¦;x) preserves the order. Let rbe a
real number, and qx, q2 two rational numbers such that
q,<r<q2. (3.29)
From (3.27) and (3.28) we see that
qitnx)<A(r;x)<q2tnx)
Since A(r; x) is self-adjoint, we can apply the first remark in 3.5 to obtain
inf o(A(r; x)) >qx, sup a(A(r; x)) <q2
This holds for any two rational numbers satisfying (3.29), which implies
A(r;x) rif(xl (3.30)
Since rwas arbitrarily chosen, it is obvious that (3.30) holds for any real number. On
the other hand we have that (A(i;x))* A( i;x), and (A(i;x))2 A( l;x)
tf(x), which implies that A(i; x) is anormal operator satisfying the conditions in the
second remark in 3.6. Applying remark 3.5 leads to
a(A(i;x)) c{/, -/}
This implies the existence of two orthogonal projections Pf, P2 in SS(f(x)) such that
PfPf 0(3.31)
Pf +Pf lfix) (3.32)
A(i; x) iPf -iPf (3.33)
Using (3.28), (3.30) and (3.33), we conclude that
A(X;x) XPf +XPf
From (3.22) we see that
A(X;y) XP\ +XP\
where the P\ Fy.\\y\\i\\x\\xPiP\\y\\i\\x\\x.y are st^n orthogonal projections satisfying,
mutatis mutandum, the relations (3.31) and (3.33) (we use the fact that Fy< y,^nxnx is an
Vol. 51, 1978 Maps of aquantum mechanical propositional system 649
isomorphism). We have moreover that
P' -F>>.llyll/!l*||x-p?/ii)>||/||x||x,.v'
FAfi^-x)p?a(^-x)f
FP?F
*yx* i* xy
which was the last relation we had to prove. I
Theorem 3.1 is now completely proven: if we gather the results of Lemmas 3.2,
3.3,3.4 and 3.6, we get exactly Theorem 3.1. We introduce now the following definition
which is motivated by the results of our theorem.
3.7 Definition. Let Pf, PC be two Hilbert spaces, with dim 2ff >3. Letf be an m-
morphism different from the null-morphism, mapping P(Pf) into P(3P").
-f is called alinear m-morphism ;/ the Fyx constructed in Lemma 3.2 have the
property:
P).x,x ^f(x-)
-f is called an anti-linear m-morphism if they have the property
Pkx,x ^f(x)
-f is called amixed m-morphism if it is neither linear, nor anti-linear.
In the following theorem we show that any mixed m-morphism can be written as a
combination of alinear one and an anti-linear one. This decomposition turns out to be
unique if/ is unitary. We formulate the theorem only for the mixed case :the same
techniques as used in the proof yield trivial results if the m-morphism is linear or anti-
linear, which implies that the theorem works also in these cases. One should however
drop then the condition that the Jtt are non-trivial since either 3/fx or Pf 2would be
zero.
3.8 Theorem. Let Jf, PC be two Hilbert spaces, with dim Pf >3. Letf be aunitary
mixed m-morphism mapping P(Pf) into P(PC). Fhen there exist two non-trivial
orthogonalsubspaces Jf\, 3ff2 offPC, aunitary linear m-morphism f\ mapping P(Pf) into
P(Pf\), and aunitary anti-linear m-morphism f2 mapping P(Pf) into P(Pf2) such that
PC Pfx®Pf2 (3.34)
Va eP(Pf):f(a) =ft(a) vf2(a) (3.35)
Phis decomposition is unique.
Proof. We first remark that/is different from zero (it is unitary), hence injective.
For any xin Pf, put
f(x) Pf(f(x)) /=1,2 (3.36)
We see immediately that/(x) =/t(x) v/2(x). We define Pf 1; Jf 2by
Mi VMx). (3.37)
xeje
650 Dirk Aerts and Ingrid Daubechies H. P. A.
Let x, ybe arbitrary non-zero elements in Pf. We shall prove that/^x) Pf2(y). If yex,
then/2(j) =/2(x) and so /i(x) 1f2(y). If y£x, then there exists az_I_x such that j
l/\\x\\2(x,y)x +z. Take x'e/^x), y"ef2(y). We have
(x', /') tx', -^ F(je.,,,(/) +Fzy(y")
x\\
1Ff* >P(x,y)x,y(y
\ï\x',P\
l|x||2V '^«GO.
Applying (3.18), and using the fact that F? (/') 0, we see that the right-hand side
is reduced to zero, which proves /i(x)l/2(j). Because ofthe definition (3.37) ofthe
Pfi, this implies:
Jt^lJfa (3.38)
On the other hand we have
*"«/(¦*) V/(*)=¦ VC/i(*) v/2(x))
Vk(x) vV/2(*) Jf ivPf 2(3.39)
\i6Jf /\xe.je /
Combining (3.39) with (3.38), we have
jf '^fj ©jf 2
For any arbitrary element aof F(^f we define
Ma) =f(a) a.Pfi (3.40)
Since/is injective, the restriction of this definition to the set of atoms coincides with
(3.36). On the other hand (3.40) defines an m-morphism/ mapping P($f) to P(Pf A.
We have indeed
Md) \yf(x)\APf
V/i(*))v( \/f2(x)
x<a
-VMx)
AJfi
/f Afl* =/ Aak ajf, Afiak) ajf,
\teX /\keK f\keK
A(/(a*) ajr,) AMak)
keK keK
Ma)' {zf ePfi-, z'lf(a)} =/(a)x ajf,
/(«') a*», =/(a')
Vol. 51, 1978 Maps of aquantum mechanical propositional system 651
We have moreover
Aia) vf2(a) VA(x)) vV/2(*)
\x<a /\x<fl y
VC/i(*) v/2(x)) Vf(x) =f(a)
x<a x<a
Suppose now that Pf tis the zero-subspace of PC. Then/^x) P\f(x) 0for
every xin Pf, which implies F2 tnx) for every x. This is however equivalent to saying
that/is antilinear, while it was supposed to be mixed. This proves that Pf xcannot be
the null-space in PC. In the same manner one proves that Pf 2is different from the null-
space, which implies that both Pf jand Pf2 are non-trivial.
On the other hand we have
fitjt) =f(Pf) aJfi pf 'aPfi Pfi,
which proves that the/ are unitary.
As an immediate consequence of (3.18) and of the construction of the/ we have
%x Fyx\fM (3.41)
where {lFyx} is the set of {Fyx} corresponding to/. From (3.41) we see that
I'Xy.x F».y,x\ft(x)
Fyx\fl(x)o(XPf +XPf)
^-Fyxlf^x) A. Fyx,
which implies that/ is aunitary linear m-morphism. The fact that/2 is anti-linear is
proven in the same way.
The proof of the existence of the decomposition is now complete.
Unicity can be proven as following. Suppose that J^lt $f2,A>7i satisfy all the
conditions. Combining (3.34) and (3.35) we get
fix) =/i(x) ®/2(x) with /l(x) -L/2(x) for every xin Jf
It is now easy to check that
Fyx 1Fyx +2Fyx for any two non-zero vectors x, yin Pf
which implies
A(X\ x) FXx xFXXtX +FXxx
Al/lW +Xlf2(xi (3.42)
From (3.42) we conclude Pff(x) =/(x), which implies
fi(x)=fi(x) VxeJf, x#0
Since both f, fi are c-morphisms, we see that/ =/, and
fi(jf)=fi(jf) jtfi /=1,2
This implies that Pf{ =f(JP) is contained in #f. Since the #; are orthogonal
subspaces, and J>f 2ffx ®Pf2, this leads to
yf iyf iI1, Z
652 Dirk Aerts and Ingrid Daubechies H. P. A.
This proves the unicity of the decomposition. |
We have now achieved our goal :Theorem 3.1 gives us acomplete characterization
of the linear structure underlying aunitary m-morphism, and permits us to define two
special types of m-morphisms, i.e. the linear and the anti-linear ones. In Theorem 3.8
we proved that any unitary m-morphism can be written as a'direct union' of at most
two of these special m-morphisms. Applying the remarks made in Section 2, one can
extend these results to general (i.e. non-unitary) m-morphisms. Before passing on to
the next section, We want to show the connection between Theorem 3.1 and the
statement made at the end of Section 2:we will construct explicitly afamily of maps
satisfying all the conditions in Theorem 2.6.
Let #f, JP" be two Hilbert spaces,/a unitary m-morphism mapping P(Jff) into
P(Pf '). Let xbe anormalized vector in Jf :||x|| 1.
Since/(x) can bte written as the direct sum ofthe orthogonal subspaces P\f(x) and
Pi fix), we can choose an orthonormal basis (Xj)jeJ in/(x), and apartition {/,, J2) of
Jsuch that
jeJi<^P}ePff(x) /=1,2
We define now afamily of maps {<pj\ :
\/jeJ: (pf.Pf^PC
0->0
y^Fyx(x'j) ifyPO
It follows from (3.12) and (3.19) that for Je Jy the 4>j are linear maps, while for yeJ2
the 4>j are anti-linear. Applying Lemma 3.4 we get the following result:
\\y\\ i=> Uj(y)\\ lV/e/,
which implies that the tpj are isometric. Moreover, one can prove that the <pj(Pf) are
orthogonal subspaces. Indeed, let y, zbe two non-zero vectors in Pf. There exists a
vector u(which may be zero) such that :
zYYY (y' z^y +uwith u-^ y
We have
<t>j(P> Fzx(x-)
FzyFyx(Pj) Fzy(Yj)
y^2 F(y,z)y,y(y'j) +Ày(yj)
iPijT <Piiiy' z))(^') +Fuy(y'j)
where tpj is amap from Cto Cwhich is the identity if/ eJ±, and the usual conjugation if
jeJ2. We have now
(4>k(y), 4>j(z)) |-|2 cpjiiy, P))(y'k, y'j)
<Pjii y, z))(x'k,x'j)
Vol. 51, 1978 Maps of aquantum mechanical propositional system 653
where we have used that l/\\y\\ Fyx is an isomorphism. From this result we infer that for
different kandj vectors (j)k(y) and c/)j(z) are orthogonal. This implies that the different
<j>j(X) are orthogonal. On the other hand, the unitarity of /implies that the
(tj)j(x);j e/, xeJP) form atotal set, hence
X' ©<PjiX)
jej
We define now amap of P(3f) to P(PC) by
/: F(Jf)-> F(Jf')
a-V^(«)
ieJ
It is easily seen that for each non-zero vector jin jf, we have :
Ry) V^(JO VHy) \/y'j =/O0
/e7 je/ yeJ
Since each <^ is unitary or anti-unitary, each (pj generates ac-morphism, and the
following holds:
/(fl) V4>;(fl) VV</>/(*) VV4>j(x)
je JjeJ x<a x<a jeJ
Vfix) fia)
x<a
which proves that/and/are identical.
We can now sum up all these remarks, and state the results :we have constructed a
family of maps ((pj)jeJ mapping Jf into 2ff. Each of these maps is an isometry or an
anti-isometry ;their images are orthogonal subspaces <j>j(X) of 3tf" such that
X' ©(PjiJtf)
jeJ
This family generates the unitary m-morphism/in the following sense:
VaeF(^):/(a)=V<,j(fl)
je J
This proves Theorem 2.6.
It is to be remarked however that this family (4>j)jeJ is not unique.
4. Aspecial case:/maps atoms into atoms
In this section we shall consider the special case where the f(x) are one-
dimensional subspaces of #C, i.e. where/maps the atoms of P(JP) into the atoms of
P(jC). The physical meaning of this condition is that states are transformed into
states. In this case we can prove that the c-morphism/is automatically generated by an
isometric or anti-isometric map. More specifically:
654 Dirk Aerts and Ingrid Daubechies H.P.A.
4.1 Theorem. Let Pf, 3/f" be complex Hilbert spaces with dim JP >3; let fbe ac-
morphism mapping P(Pf) into P(PC) such that for any atomp in P(Pf),f(p) is an atom in
P(PC). Fhen fis an m-morphism, and the following holds: For any non-zero xin Pf, and
any non-zero x' infix) there exists aunique closed bounded linear or anti-linear map </>
from Pf into JC such that
(p(x) x'
<f) generates the c-morphism f.
Phis <p is equal to an isometric or an anti-isometric operator multiplied by aconstant.
Proof. We define tp :Jf -» Pf 'by :
<K0) 0
<p(y) Fyx(x') forj#0.
We have trivially
4>(x) Fxx(x') xf.
Since all the/(j) are one-dimensional,/is either alinear or an anti-linear m-morphism.
Suppose that/is linear. Then <p is linear:
<p(Xy +pz) FXy +flZ:X(x')
*-Fyx(xf) +pFzx(x')
X<p(y) +p<p(z)
If H^ll ||x||, then Fyx is an isomorphism; hence
\\<t>(y)\\ \\Fyx(x')\\ llx'H !|0(X)||
This proves that ||x||/||x'||0 is an isometry.
Since Hxll/llx'll qb is an isometry, we know that <p generates ac-morphism mapping
P(Pf)\n\o P(PC) ;since <p(y) f(y) for any yin Pf, this c-morphism is/. Suppose now
that ^is alinear or anti-linear map satisfying the conditions in the theorem. Take
yeJf, y$x. Put y" <p~(y). We have
y" Fxy(y") +Fy.x,y(y") (AA)
On the other hand
y" $(y) $(x) +$(y -x) (4.2)
Since <p generates/ we know that <p(x) is contained in/(x), and tp(y x) \nf(y x).
The decomposition (4.1) is however unique (see the proof of Lemma 3.2) which implies
4>ix) Fxy(y")
hence
$(y) y" Fyx($(x)) Fyx(P) <p(y)
If yex, then we can choose t$xand apply the same reasoning. This yields
$(y) Fyt($(t)) FytFtx(xf) Fyx(xf) 4>(y).
This proves the unicity. |
This theorem has the following interesting consequence.
Vol. 51, 1978 Maps of aquantum mechanical propositional system 655
4.2 Corollary. Letf be aunitary c-morphism mapping P(Pf) into P(3P") such that
for one atomp in P(jP),f(p) is an atom in P(PC). Fhen fis an isomorphism, and every
isometry (or anti-isometry) generating fis aunitary (or anti-unitary) operator.
Proof. The injectivity of/is aconsequence ofthe fact that/is different from zero.
Let xbe anon-zero vector such that xp. For any non-zero yin JP we have that
fiy) +f(x) is closed (f(y) is closed and f(x) is one-dimensional), hence f(y)
cf(x) vf(x y) f(x) +f(x y). Using the same arguments as in the proof of
Lemma 3.2 we see that/(x) and/(j) are isomorphic, hence that all the/(j) are one-
dimensional, which implies that we can apply Theorem 4.1.
Suppose/to be linear, and let cj) be an isometry generating/. Let (e;)>e/ be an
orthonormal basis in JP. Since Pf \ZieJëi we have:
V4>(ed yRed =f(3tf) PC
iel iel
which implies that (^>(e,))ie/ is an orthonormal basis in PC.
Since tp is an isometry, this implies that <p(X) PC, hence that (p is unitary. For
each atom qin P(JP") there exists ayin PC such that yeq. For this ythere exists an x
4>~ l(y) in Xsuch that <p(x) y, hence/(x) (f>(x) yq. Since any element of
P(JP") can be written as aunion of atoms, this proves the surjectivity of/. |
All the results we have obtained were only proven for dim Pf >3. The proof of
Lemma 3.2 for instance relies rather heavily on this condition. One would thus expect
counter-examples to occur for dim jf 2(the case where dim Pf 1is trivial). They
do indeed exist :some of them are given in the next section.
5. Counter-examples in the case where Pf has dimension 2
We first construct acounter-example against Theorem 3.1, more specifically
against Lemma 3.2 :we define aunitary c-morphism of P(<£2) into F(C4) for which the
corresponding Fyx do not satisfy the chain rule.
5.1 Counter-example. Take Pf (C2, and let {ex, e2) be the standard basis in C2.
Then we can write P(<F2) as:
P(<F2) {0,11} y{Fe,; F, C¦(cos Ge, +e* sin 6e2) with
06 tt
°'2 ,<pe[0,27t[}.
It is easy to check that the orthocomplementation on P(<C2) is given by
P^lPe^o-i +9' yand \q> -q>'\ nif 0#0#ff
+0'=| if0 Oor0' O
656 Dirk Aerts and Ingrid Daubechies H. P. A.
Define
<x: 0, 0, by a(0) -(1 -cos 20)
This function has the property that a(7t/2 0) n/2 a(0), hence
0+0'=^Ua(0) +a(0') |
Take now JP' <C4 with standard basis {fi,f2,A,fi}> and define
fe,<p cos6f1 +é"sinef2
n
(5.1)
(5.2)
0, cp e[0, ln[
ge,9 cos a(0)/3 +<?'* sin a(0)/4 for 0e
Qe,<p Lin (/,,,#)
It is easy to check that for (0', cp') P(0, cp) we have
Qe.v aG>,. 0
For 0+0' re/2 and |<p <p'| nwe have moreover that
Qe-,q>' ße,<p (this is aconsequence of (5.2)).
It follows immediately that the map/from P(<F2) to F(<C4) defined by
/(0) 0
/(1) 1
n
f(Pe,v) Qe,v for 0e 0, <p e[0, 2tc[
is aunitary m-morphism.
One can now construct the Ffl>-^ Fe9V,e9s) where e9^ cos 0^! +e"" sin 0e2.
Arather lengthy but straightforward calculation yields
''ie <p+ n.Bip °re<i>,o<i>(9o<p)
cos (a(20) -a(0/2)) cos a(0/2)
cos (a(0) +a(0/2)) cos (a(0)
cos a(0)
(do«)
2$ <p +n, o<p\fdo<p.
«(0/2))
926 q> +n
9ie y-t
cos (a(20) -a(0)) '
Since ais astrictly convex function on ]0, n/A[, we have for 0<0<7t/8 :
F26 (p +n, 6(p (9oV) XF.28 (p+ n,o<p (gov) withA<l. (5.3)
We see immediately from (5.3) that the chain rule (3.3) does not hold in this case.
In Section 4we proved some theorems about c-morphisms mapping atoms into
atoms. The first one stated that any such c-morphism was generated by an isometry or
an anti-isometry. This theorem uses explicitly Theorem 3.1, which can only be proven
when ,?f has dimension greater than two (see Counter-example 5.1). It might however
happen that pathologies such as the one in this counter-example drop out if the/(x) are
one-dimensional. The following counter-example shows that this is afalse hope.
Vol. 51, 1978 Maps of aquantum mechanical propositional system 657
5.2 Counter-example. Take Pf PC C2, and define the map/from P(2) to
P((C2) by
/(0) 0
/(1) 1
j(Pe,<i>) °a(9),c>
where we use the same notation Fe, as in Counter-example 5.1, and where ais the
function from [0, n/2'] to [0, n/2] defined by (5.1). Suppose now this/to be generated
by alinear map (p. Since this linear map conserves the orthogonality, it conserves the
angles. For <p different from 0, n/A or n/2 we have however
<p(ee, o) e<?«<»), o
Since (p(ex) eex, this implies
(<t>(et), <p(e6r0)) cos a(0) pcos 0-1' e'°
:l^t)ll \\<p(ee,o) e.oll
This is acontradiction, which implies that no linear map generating /exists. In a
completely analogous manner we can prove that/can not be generated by an anti-
linear map.
In Corollary 5.2 we proved that aunitary c-morphism mapping P(Pf) into P(PC)
with the additional condition that it maps atoms into atoms has to be an isomorphism
if dim Pf >3. In the proof of this corollary we used the fact that such ac-morphism is
generated by an isometry or an anti-isometry, but it might be that the surjectivity-
statement follows already from much weaker conditions :the c-morphism in Counter¬
example 5.2 is not generated by an (anti)isometry, and yet it is onto. In the following
counter-example however we construct anon-surjective c-morphism from P(<£2) to
F(C2) satisfying all the conditions, which implies that even the first statement in
Corollary 4.2 does not hold for dimension 2.
5.3 Counter-example. Take Pf Pf 'C2, and define the map/from P(<F2) to
P(<C2) by
/(0) 0
/(1) 1
f(Pe.J Pm if(pe[0,n[_
Pe,(<p+n)/2 if (p e[«, Irti
For tp <nwe have
f(P'ocp) =f(Pnl2-e,y +iA
Pnl2-6,nl2 +(<t> +li)/2 Pn/2-e,<i>/2+n
P'e,<p/2 =f(Pe,q>)'
The same property can be proven for tp >n.
»From the definition and the properties of/it is now easy to see that/is aunitary
c-morphism mapping atoms into atoms, although it is obviously not surjective.
It is amusing to remark that cases like this one, i.e. non-surjective injective
658 Dirk Aerts and Ingrid Daubechies H. P. A.
c-morphisms from P(<F2) into F(C2) can be exluded if/is required to be continuous
with respect to the topology induced on F(C2) by the usual norm-topology on SS(<C2).
6. Conclusions
We have proven that any map/from aquantum-mechanical propositional system
P(P') to aquantum-mechanical propositional system P(JP'), which preserves the
complete orthocomplemented lattice structure of P(Pf), and which maps modular
pairs to modular pairs, is generated (in the sense of Theorem 2.6) by afamily of
isometries or anti-isometries from Jf to 2P". As aconsequence of this main theorem we
can prove that any map/from P(jf) to P(PC) which preserves not only the complete
orthocomplemented lattice structure of P(Pf), but also the property of aproposition to
be astate of the quantum system, is automatically an isomorphism of P(Pf) onto the
segment \_0,f(Pf)] of P(PC). This implies that we are able to consider Wigner's
theorem as aspecial case of our theorem ;moreover it turns out that Wigner's theorem
holds even under weaker conditions than originally.
Our main theorem, as well as Wigner's theorem, is only valid if dim JP >3. That
this is avital restriction is illustrated by several counter-examples showing that both
the main theorem and its weaker corollaries can be violated if Pf has dimension 2.
Appendix
We prove Proposition 2.5.
Proposition. Let Pf, PC be two complex Hilbert spaces with dim Pf >3; let fbe a
c-morphism from P(Pf) to P(PC). Fhen the following are equivalent:
(1) fis an m-morphism
(1) Vx, ynon-zero vectors in Pf:f(x y) <=/(x) +f(y)
(3) Vx, ynon-zero vectors in Pf: z<xvy^-f(z) c/(x) +f(y)
(A) Vx, ynon-zero vectors in f/f:f(x) +f(y) is aclosed subspace of PC.
Proof. We prove (1) => (4) => (3) => (2) => (1). The implication (1)=>(4) follows
immediately from the fact that xvyx+y, which implies (x, y)M. Hence
(f(x),f(y))M or f(x) +f(y) =/(x) vf(y) is aclosed subspace. The implication
(4) => (3) is trivial if one remarks that (4) implies f(x) +f(y) f(x) vf(y). The
implication (3) => (2) is immediate.
To prove the implication (2)=>(1) we use the results of Theorem 2.6. Since
Theorem 2.6 is aconsequence of Theorem 3.1, and since we used only condition (2) to
construct the Fxy and to prove their properties, we are allowed to do so.
Let (4>j)jsj be afamily isometric maps generating/. Let a, bbe amodular pair in
Vol. 51, 1978 Maps of aquantum mechanical propositional system 659
P(Pf). Because of Lemma 1.8 we have avba+b. Hence
f(a)vf(b)=f(avb)
©<pj(a vb)
JeJ
©<pj(a +b)
jet
©</>,((« A(fl Ab)') +b)
jeJ
(we have split a+binto two disjoint parts which form again amodular pair: see [3]).
Take
xEXj e©<pj((a a(a ab)') +b)
jeJ jeJ
Then for any 7e/, there exist unique y,, z, in <pj(a a(a ab)'), (pj(b) such that Xj y,
+Zj. One can prove3) that £jey||x;||2 <oo implies
Xiloli2 <and Xiloli2 <°o
jeJ jeJ
Hence
x=TJyj+TJzje® cpj(a a(a aby) +©cpj(b)
jeJ jeJ jeJ jeJ
This holds for any xin (BjeJ (pj(a +b), which implies
f(a)vf(b) ®cPj(a +b)
jeJ
CZ ©<p}(a A(fl Ab)') +©<t>j(b)
jeJ jeJ
<= ©<Pjia) +©<Pjib) =/(«) +f(b)
jeJ jeJ
Applying again Lemma 1.8, we see that this implies (f(a), f(b))M, which completes the
proof of (2) => (1). I
Acknowledgments
It is apleasure for us to thank professor C. Piron for his constant interest and
professor J. Reignier for his encouragement. Moreover we are very grateful to various
In [3] it is proven that (a a(a ab)', b)M is equivalent to
\(x,y)\
sup lv '"' «<1
»s« at« a»r 11*11 II yII
i'e/j
Since every $, is an isomorphism, the same holds for tj>j(a a(a ab)') and (j>j(b). It is then easy to see
that ||>>j||2 +Hz,-!!2 <1/(1 -cOllxJ2, which gives the desired result.
660 Dirk Aerts and Ingrid Daubechies
members of the Department of Mathematics of the V.U.B. for interesting remarks.
One of us (D. Aerts) would like to thank professor C. Piron for his kind hospitality at
the Department of Theoretical Physics of the University of Geneva.
Note. We want to thank professor C. Piron for pointing out an omission in the
proof of the proposition in the Appendix.
REFERENCES
[1] C. Piron, Foundations of Quantum Physics (W. A. Benjamin Inc., 1976).
[2] G. Birkhoff, Lattice Theory (Amer. Math. Soc, Colloquium Publications, Vol. 25, New York,
1940).
[3] D. Aerts and C. Piron, In preparation.
[4] C. Piron, Helv. Phys. Acta, 37, 440 (1964).
[5] I. Amemiya and H. Araki, Pubi. Research Inst. Math. Sci. Kyoto Univ., A2 423 (1967).
[6] E. P. Wigner, Group Theory (Academic Press, New York, 1959).
[7] N. Dunford and J. Schwartz, Linear Operators, Part I(Interscience Publishers, Inc., New York,
1957).
[8] J. Dixmier, Les C*-algèbres el leurs représentations (Gauthier-Villars, Paris, 1964).
