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Symmetric transition probabilities in convex model of quantum mechanics
VoL 8 fl~'l~
x ~ v o m = o~v uAr,.tru.erte.,~ . , , ~
No. 3
SYMME'I~C 'T'RA~SYrlON PROBABIIdTW.S IN CONVEX MODEL
o~' Q U ~ r g t . ~ MF..CHA~C:S
lnsdumm of Tl~oreugal Physics. Warsaw U u l v ~ . (gegm',~ M,m:k ~S, l~S)
Warsa~o Polmsd
Ths ~ b ~ the S e o m ~ 7 ~ d the ~ a ~ of the q ~ t u m ~ k most e~4dkl Ja the conv~ modcls oF quagtum mechsnlcs. A d u s of ~ / s Im,est/l~ ~ tlm ms~foM $ of all s~t~ is • com,~ set with a smooth bmmc~7. It is shown ~ fa"
models tim slmple assumption about symmetry of the ~ pro~bt'Ut~ implkm an eJlJpsoldal shape of $, thus leading to the "sphericaJ" gemnetrlm degn'l~d in [71. L IutmdEtiom the development o f quantum theory the question ~rose whether the formalism o f quantum nua:hanics admits a descrimJou in terms of purely geometric concepts. The most consistent approach to this question was presented in [2], [5],'[61, [71, [9]. The tms~ structure in this approach is tha set $ of all pure and mixed states of a physical system. In S tl~re exists a natural operation of convex combinadon which corresponds to the physical mixture of states:
$~x,y-, ~tx÷~ye$
with
~÷ts -,, 1, 0 ~ 2 , ~ G R t.
(1.1)
This loads to a representation of $ as a convex set called a stutistfcalflgtue [9]. The extrem~ points of $ correspond to the pure states of the system, whereas the remaining points of $ stand for quantum mixtures. The siip~ficance of the statistical figure has beer. recently r~li,,,d ([5], [6], ['/]) leading to what might be called a "convex model" of quantum mechanics. This model provides the most complete insiigh£ into ~he statistical aspects of quantum theory: it determines tl~ strucuu~ o f the transition probabilities for the quantum systems of arbitrary geometry, without any additional re~ictions concerning the quantum logic (which are essential in the lattice theoretical approagh). The convex model is also adequate for studying an inverse dependence: how is the shape of $ conditioned by the structure of the probabilities? It was shown by Dahn [3] that an apparently weak assumption about symmetry of transition probabilities is related to the modularity of the lattice o£ quantumproposition: which are represented by the faces Of S. ps'q
388
S. w o J c l ~ C ~ o w s ~
In this note we take another step in that direction by considering a special model in which the statistical figure 5 has a smooth boundary. As will be shown below, in thz framework of the "convex form~i..'~m" the symmetry of the transition probabilities for such models leads uniquely to the elfipsoidal shape o f S discussed in [8].
7. Symaeu~ t r a ~ e e a Below, a subclass of quantum systems v-ill be considered for which the set $ o f all states of a physical system is a compact convex s,.-t ~ith no empty interior (convex body) aud a smooth boundary in a finite-dimensional affme space E (therefore, E is s~e,nned
byS). For an arbitrary statistical figure $ there exists, in general, a noatrivial structure o f walls (see, e.g-, [9]) determining the "quantumlogic'. Since it is assumed that $ is smooth, the only nontrivial walls (different from S and the empty set ~ ) are the extreme poineJ (pure states) which form the boundary o f $. Hence, the "quantum logic" d e g e n e ~ t ~ to a three-fold structure (0, I and pure suttes being the only three types of propositions admissible) and so the laxticc theoretical model tells nothing about the transition probabfiitles. There exists, however, a general definition of the transition probability exploring the convex structure [9]. It leads here to a simple geometric construction d ~ u g with the support hyperplanes of the stathtJt~ figure S and with the norma/functinnals on S. D~u~rnot~ 1. Let S b e the subset ofF., and let H be a closed real hyperphne in F. H is called a supportb:g ~ p e ~ l n n e of $ if: S ~ l ~ vt f$ and if S is contained in one o f the dosed semispaces determ;,,ed by H. . The boundary 05 of the dosed convex set S = £ is called ~Jooth if each point x o f the boundary S determines exactly one supporting hyperplane containing this po;-t_ D~vmrno~ 2. In a finite-dimensional atF~e space the ~2near functionals are those which fulfil the restricted condition of Unearity: y,(ex+~),) =- ~r-(X)+~F(y)
for
x, yeE;
~ ' , ~ e R z with ct+~ ,= 1.
G?v~n a close convex set Sin F. a normalfuc.ctionalon S i s any continuous linear functional ~: E--. R ' such that 0 ~ y(x) ~ 1 for each x e S . Since S is assumed to be smooth, for every state x e $ and every pure state y ¢ a S there exists a unique transition probabifity p(x, y) which can be constructed as follows. Let ¥ G aS and x ~ S. There exists strictly one supporting hyperplane H , tangent m $ in y. This hype~lane determines a parallel congruence C ofhyperplanes in E, one o f which is H~, (.see Fig. 1). Because S is a convex compact set, the congruence C contains the unique support hyperplan¢ FIr, tangent to S in the "complementary point" y'. The pair of hyperplanes: Ht. I t r defines precisely one normal functional ~, with values 1 at H, and 0 at H r . This functional d ~ e r ~ n e s the required "transition probab'diW':
p(x..1,) - r,(x).
(2.D
SYMMETRIC l"RA~srrloN PRO3AB/LEF11~IN CONVEX MODEL
389
Note that thz above quantity is well d e c a l for quite arbitrary x ,~~ and y • 05. Fo¢ pure stat~ x, y (x, y ,~ S) the probability thus defined is, in general, not sy~,,,mic p(x, .V) p0v, z). Some special "spherical" models with symmetricp(x, y) have b a n considen:d by Haag and Miekdk [8]. As will bc shown now, for 0~ smooth, these are the only models
,¢
lqs. l let S be a ~at~ical f i ~ e w~m a ~zooth bozmdarp In a
spacr and
tr
ion probab O p(x,y) be
ven by ('2.1).
p(x, 7)
~ o f : It sufHces t.o construct a posk/ve q u n d n ~ form in E ~ on e ~ The quan~/ty p(x, ~) is well defined for eve~ pair of poinu ~ e ~ , x G ~ S~ncc $ has a nonempty interior,every point of 2~ ~ be dccompmed into an a/finecomb/n~on of two boundary points of $ (s~ figure below). Now for any x, 3' • £ de,re th~ form qOr, y) by
~s. 2
where the points Ys,.Vz e&S provide th~ unique decomposition of 3,, that is, y -- as.v+ +azyz, as, ~z being real numbers and or, + ~ : - I. To check the uniqueness assume that there is some other represcnta~oa of 7 of the form y - - p,v, +~zv=, whm'e oz, vz • aS; P~, 0: e R s, Ps +Pz =" J, and choose any representation x = 2sx1+2zx.
(xl,x2¢O$, ~:,;~2eR l, ~Is+~lz - l).
390
$. W O ~ C ~ H O W S U
The linearity of p(x, y) ia the first variable x and the assumed symmetry of p ( x , y) for both x, y taken from ~$ subu:quently imply: p~p(x, ~,)+p~p(x, ~
-
~
.
~ "
;.,O~,(vz, x ~ + z,p~,,o(e~, x~)+ Z,p~,(e~, x J + ~ p ~ ( v ~ , z . )
- z,~,
xJ+~zffifo, x ~
..,
- ~,p(x,~,)+a~p(x,y~).
This proves the uniquencss of the q,,*-dcy (2.2). The biliueadxy and symmetry o f q(x, y) are obviously due to the same dceompo~i~on. The quadratic form q(z, z) corresponding to the bilinear form q(x, y) is well d,.f~ned and positive on FThe form q(z, z) can be used to desc~oe the boundary of $: Indeed, for an arbitrary reprmentafioa g , , p l g t + p z z 2 ( : I , Z Z G ~ $ and p s + P 3 - - I) the decomposition o f g ( : , : ) takes the form: q(:. ~) - j,=,+ ~ + 2 ~ , , , , , ~ - ~ : . : 0 . .. Since the point z •ffi P~ zz + P: zz belongs to $ i £ a n d only if both P~ , Pz ~ O, therefore the quadnttic form ¢(:, : ) - - * ; , , - the values: 0 < ¢(z, z) < 1 on $, and q(z, z) > 1 for all z ~ S. Since q is po~tire, S is an ellipsoid in the affme space E. Consequently, the rcsnltinl| structm'e of the transition probabilities p(x. y) for x , y e ~$ must coincide with one o f the "spherical" stx,,zctures described in [8]. Thus, the spherical probability geome~es considered by Haag and Mielnik are fully characterized by the only conditions of the smoothness of $ and ~ c r/rnmcu~ of chc t~_-~fion probabilities. In the above considexatious the finite dimeasionality of E h a s been assumed for simplicity but it dc,cs not seem to play an essential role. The result obt~;ned can be thus generaIized to th: c a ~ oflnfinlte dimension too, if adequate assumptions concerning the existence of support hyper~L~-,~ of S are made. KEFEKENC5 Ill Birkho~, (3., and J. voa l'qetmzaan: Ann. Math. 37 (1936), 823. [21 Daha, G.: ~ M,,,h Phys. 9 (1968), 192. [ 2 ] - - : ~ . 2S (1972), 123. [41 Gedder, S.: ~ 29 (]~r3), 249. [5] Guinea, L: ~t~. • (1967), 2'~. [6] Ludwig, G.: 7- ~ ISl (1964), 233. [7] - - : Comm~. Math. Phys. 4 (1967), 331:9 (1968), 1. [81 Mie.Lqik,B.: ~ 9 (t96s), ss.
[91 --: ~d. 15 0969), L [10] --: Gnwa//:ed ~ams~m mzr3smzk~ (prcprlm). [11] Piton, G.: Holy. Phys. Acta 37 (I964), 439.
x ~ v o m = o~v uAr,.tru.erte.,~ . , , ~
No. 3
SYMME'I~C 'T'RA~SYrlON PROBABIIdTW.S IN CONVEX MODEL
o~' Q U ~ r g t . ~ MF..CHA~C:S
lnsdumm of Tl~oreugal Physics. Warsaw U u l v ~ . (gegm',~ M,m:k ~S, l~S)
Warsa~o Polmsd
Ths ~ b ~ the S e o m ~ 7 ~ d the ~ a ~ of the q ~ t u m ~ k most e~4dkl Ja the conv~ modcls oF quagtum mechsnlcs. A d u s of ~ / s Im,est/l~ ~ tlm ms~foM $ of all s~t~ is • com,~ set with a smooth bmmc~7. It is shown ~ fa"
models tim slmple assumption about symmetry of the ~ pro~bt'Ut~ implkm an eJlJpsoldal shape of $, thus leading to the "sphericaJ" gemnetrlm degn'l~d in [71. L IutmdEtiom the development o f quantum theory the question ~rose whether the formalism o f quantum nua:hanics admits a descrimJou in terms of purely geometric concepts. The most consistent approach to this question was presented in [2], [5],'[61, [71, [9]. The tms~ structure in this approach is tha set $ of all pure and mixed states of a physical system. In S tl~re exists a natural operation of convex combinadon which corresponds to the physical mixture of states:
$~x,y-, ~tx÷~ye$
with
~÷ts -,, 1, 0 ~ 2 , ~ G R t.
(1.1)
This loads to a representation of $ as a convex set called a stutistfcalflgtue [9]. The extrem~ points of $ correspond to the pure states of the system, whereas the remaining points of $ stand for quantum mixtures. The siip~ficance of the statistical figure has beer. recently r~li,,,d ([5], [6], ['/]) leading to what might be called a "convex model" of quantum mechanics. This model provides the most complete insiigh£ into ~he statistical aspects of quantum theory: it determines tl~ strucuu~ o f the transition probabilities for the quantum systems of arbitrary geometry, without any additional re~ictions concerning the quantum logic (which are essential in the lattice theoretical approagh). The convex model is also adequate for studying an inverse dependence: how is the shape of $ conditioned by the structure of the probabilities? It was shown by Dahn [3] that an apparently weak assumption about symmetry of transition probabilities is related to the modularity of the lattice o£ quantumproposition: which are represented by the faces Of S. ps'q
388
S. w o J c l ~ C ~ o w s ~
In this note we take another step in that direction by considering a special model in which the statistical figure 5 has a smooth boundary. As will be shown below, in thz framework of the "convex form~i..'~m" the symmetry of the transition probabilities for such models leads uniquely to the elfipsoidal shape o f S discussed in [8].
7. Symaeu~ t r a ~ e e a Below, a subclass of quantum systems v-ill be considered for which the set $ o f all states of a physical system is a compact convex s,.-t ~ith no empty interior (convex body) aud a smooth boundary in a finite-dimensional affme space E (therefore, E is s~e,nned
byS). For an arbitrary statistical figure $ there exists, in general, a noatrivial structure o f walls (see, e.g-, [9]) determining the "quantumlogic'. Since it is assumed that $ is smooth, the only nontrivial walls (different from S and the empty set ~ ) are the extreme poineJ (pure states) which form the boundary o f $. Hence, the "quantum logic" d e g e n e ~ t ~ to a three-fold structure (0, I and pure suttes being the only three types of propositions admissible) and so the laxticc theoretical model tells nothing about the transition probabfiitles. There exists, however, a general definition of the transition probability exploring the convex structure [9]. It leads here to a simple geometric construction d ~ u g with the support hyperplanes of the stathtJt~ figure S and with the norma/functinnals on S. D~u~rnot~ 1. Let S b e the subset ofF., and let H be a closed real hyperphne in F. H is called a supportb:g ~ p e ~ l n n e of $ if: S ~ l ~ vt f$ and if S is contained in one o f the dosed semispaces determ;,,ed by H. . The boundary 05 of the dosed convex set S = £ is called ~Jooth if each point x o f the boundary S determines exactly one supporting hyperplane containing this po;-t_ D~vmrno~ 2. In a finite-dimensional atF~e space the ~2near functionals are those which fulfil the restricted condition of Unearity: y,(ex+~),) =- ~r-(X)+~F(y)
for
x, yeE;
~ ' , ~ e R z with ct+~ ,= 1.
G?v~n a close convex set Sin F. a normalfuc.ctionalon S i s any continuous linear functional ~: E--. R ' such that 0 ~ y(x) ~ 1 for each x e S . Since S is assumed to be smooth, for every state x e $ and every pure state y ¢ a S there exists a unique transition probabifity p(x, y) which can be constructed as follows. Let ¥ G aS and x ~ S. There exists strictly one supporting hyperplane H , tangent m $ in y. This hype~lane determines a parallel congruence C ofhyperplanes in E, one o f which is H~, (.see Fig. 1). Because S is a convex compact set, the congruence C contains the unique support hyperplan¢ FIr, tangent to S in the "complementary point" y'. The pair of hyperplanes: Ht. I t r defines precisely one normal functional ~, with values 1 at H, and 0 at H r . This functional d ~ e r ~ n e s the required "transition probab'diW':
p(x..1,) - r,(x).
(2.D
SYMMETRIC l"RA~srrloN PRO3AB/LEF11~IN CONVEX MODEL
389
Note that thz above quantity is well d e c a l for quite arbitrary x ,~~ and y • 05. Fo¢ pure stat~ x, y (x, y ,~ S) the probability thus defined is, in general, not sy~,,,mic p(x, .V) p0v, z). Some special "spherical" models with symmetricp(x, y) have b a n considen:d by Haag and Miekdk [8]. As will bc shown now, for 0~ smooth, these are the only models
,¢
lqs. l let S be a ~at~ical f i ~ e w~m a ~zooth bozmdarp In a
spacr and
tr
ion probab O p(x,y) be
ven by ('2.1).
p(x, 7)
~ o f : It sufHces t.o construct a posk/ve q u n d n ~ form in E ~ on e ~ The quan~/ty p(x, ~) is well defined for eve~ pair of poinu ~ e ~ , x G ~ S~ncc $ has a nonempty interior,every point of 2~ ~ be dccompmed into an a/finecomb/n~on of two boundary points of $ (s~ figure below). Now for any x, 3' • £ de,re th~ form qOr, y) by
~s. 2
where the points Ys,.Vz e&S provide th~ unique decomposition of 3,, that is, y -- as.v+ +azyz, as, ~z being real numbers and or, + ~ : - I. To check the uniqueness assume that there is some other represcnta~oa of 7 of the form y - - p,v, +~zv=, whm'e oz, vz • aS; P~, 0: e R s, Ps +Pz =" J, and choose any representation x = 2sx1+2zx.
(xl,x2¢O$, ~:,;~2eR l, ~Is+~lz - l).
390
$. W O ~ C ~ H O W S U
The linearity of p(x, y) ia the first variable x and the assumed symmetry of p ( x , y) for both x, y taken from ~$ subu:quently imply: p~p(x, ~,)+p~p(x, ~
-
~
.
~ "
;.,O~,(vz, x ~ + z,p~,,o(e~, x~)+ Z,p~,(e~, x J + ~ p ~ ( v ~ , z . )
- z,~,
xJ+~zffifo, x ~
..,
- ~,p(x,~,)+a~p(x,y~).
This proves the uniquencss of the q,,*-dcy (2.2). The biliueadxy and symmetry o f q(x, y) are obviously due to the same dceompo~i~on. The quadratic form q(z, z) corresponding to the bilinear form q(x, y) is well d,.f~ned and positive on FThe form q(z, z) can be used to desc~oe the boundary of $: Indeed, for an arbitrary reprmentafioa g , , p l g t + p z z 2 ( : I , Z Z G ~ $ and p s + P 3 - - I) the decomposition o f g ( : , : ) takes the form: q(:. ~) - j,=,+ ~ + 2 ~ , , , , , ~ - ~ : . : 0 . .. Since the point z •ffi P~ zz + P: zz belongs to $ i £ a n d only if both P~ , Pz ~ O, therefore the quadnttic form ¢(:, : ) - - * ; , , - the values: 0 < ¢(z, z) < 1 on $, and q(z, z) > 1 for all z ~ S. Since q is po~tire, S is an ellipsoid in the affme space E. Consequently, the rcsnltinl| structm'e of the transition probabilities p(x. y) for x , y e ~$ must coincide with one o f the "spherical" stx,,zctures described in [8]. Thus, the spherical probability geome~es considered by Haag and Mielnik are fully characterized by the only conditions of the smoothness of $ and ~ c r/rnmcu~ of chc t~_-~fion probabilities. In the above considexatious the finite dimeasionality of E h a s been assumed for simplicity but it dc,cs not seem to play an essential role. The result obt~;ned can be thus generaIized to th: c a ~ oflnfinlte dimension too, if adequate assumptions concerning the existence of support hyper~L~-,~ of S are made. KEFEKENC5 Ill Birkho~, (3., and J. voa l'qetmzaan: Ann. Math. 37 (1936), 823. [21 Daha, G.: ~ M,,,h Phys. 9 (1968), 192. [ 2 ] - - : ~ . 2S (1972), 123. [41 Gedder, S.: ~ 29 (]~r3), 249. [5] Guinea, L: ~t~. • (1967), 2'~. [6] Ludwig, G.: 7- ~ ISl (1964), 233. [7] - - : Comm~. Math. Phys. 4 (1967), 331:9 (1968), 1. [81 Mie.Lqik,B.: ~ 9 (t96s), ss.
[91 --: ~d. 15 0969), L [10] --: Gnwa//:ed ~ams~m mzr3smzk~ (prcprlm). [11] Piton, G.: Holy. Phys. Acta 37 (I964), 439.