ϲ Ͳ ί ò

( 19 2005 ..)

 

..,

 

. ( P, Q, p, q,) - Y S. QQY, pD L1-L3, L^, Lr, OML.

 

, , .

QQY ( QQY) - , : , , - . - , p, q , P, Q,, Qi y (: ) Q - Y=y1+y2+...+y+... S, MÍH H.

Q, M, (, , ) Q , , QQ=Q Q = 1, . , QQ = Q, , , Q(1-Q)=0 Q1=1 Q2=0 . , QQ=Q , (1-Q)Q=Q Q MÍH. - -Q=ØQ , , (1-Q) : ØQ (1-Q) H H-MÍH.

Q .

, , QÙP, QÙP=PÙQ , QP, , QÙP , , , QP¹PQ. . . () - y, , ()

r = (Qy,y) = ||Qy||2

()

r = (1-Q)y,y)) = ||(1Q)y||2

y.

Y=ånyn, (S)

||ånyn||2 r(Q) .

, , , , , - (poset) , , .

ϳ - , - , , , -. (. ., 1956/60, .101) :

, = B, B=D(A) , Ø = (pD)

(. )

= , D() = D() (pD=)

Í , D() Í D() (pDÍ)

. , . , , . , , , , , , . - , pD.

(, , unité, Einheit, unity)

q = åk qkQk +dQ(l), (L1)

q q qk

Qk Qk - Q(l) () () . , L1 , - , - . Գ , , . ᒺ ( Q) ( q). , L1 q Q .

, Q, pD, ØQ = P P,

p = ål plPl +dP(m) D(L1)

Q P QP¹PQ - , , , , q p. qp ¹ pq qp-pq @ ik, q, p ( ) q p = mv , DqDp³ , , k=h, .

DD ³h, (WH)

, . D D, (NB), .

(1927.) ͳ - . (1905.), . - (1924.) . . . , , , , . . , WH , º- ( Ù-) . , WH : q, -p (qÍØp <0111>) p, -q (pÍØq <0111>), qÍØp = pÍØq <1111> qÍØp Ù pÍØq <0111> .

WH : -q, p, -p, q <1110> <1111> (Øq£p)=(Øp£q).

ϳ , DqDp³h ,

(Q P P Q) ÞDqDp ³h.

˳ (1963, .15-16)

((QP = PQ) £ (QP PQ)) = (0 = (QP = PQ)) = (P^Q)

, WH (!) qp ¹ pq WH (!) q^p=p^q. , WH (qÍØp) = ((q ÙØp)= q)= ((q (1-p)= q)= (qp=0)= q^p. p^q=q^p pÍØq. ,

DqD ³ h, q^p L^

( qÙp =<0000> q=1 p=1!).

pD, qÙp, <0000>-, Ø(qÙp) = <1111> , , L// = D(L^) L^, L^ L// ØL^. pD, L// ØL^ .

<0000> pD D(qÙp) = qÚp. , WH : -q, p, -p, q. (ØqÍp) = ((ØqÙp) = Øq) = (((1-q)p) = (1-q)) = ((p+q-qp) = 1) (ØpÍq) = ((ØpÚq) = q) = (((1-p)Úq) = q) = ((1-p)+q-(1-p)q = q) = (1-p-q+pq = 0), pq = qp , p^q, p+q = 1 p q, p = Øq q = Øp .

. : -(q = p) , (q ¹ p) , , -(q ¹ p), (q = p). ᒺ º- Ù- (Q ¹ P) (Q = P) ØQ £ P Q £ Ø P (Q ¹ P) = ((ØQ £ P) Ù (Q £ Ø P)) (Q = P) = Ø((ØQ £ P) Ù (Q £ Ø P). ͳ ᒺ - (WH) - Q P,

Ø(((Q ¹ P) Ù (Q = P)) = (((ØQ £ P) Ù (Q £ Ø P)) = (Q ¹ P)) (NB)

(1111)- (NB) (1111)-

(NB) = (L2)

Ù -, º - - , ,

((Q ¹ P) Ù (Q = P)) = (((ØQ £ P) = (Q £ Ø P)) Ù (Q = P)) (L2)

- , , : () ,

= Ø D(), (L3)

D() - ().

. , , (a £ b) = (Ùb = a) (a £ b) = (aÚb = b), - . , (. structura, . lattice, . treillis, . Verband, . ), .

³ q Ù p, . , , q Ú p, - Ø (Ø q Ù Øp) = (q Ú p). - (Ø q Ù Øp) = Ø (q Ú p) Ø (q Ù p) = q // p, . (WH) q Ù p . - - , - . .

-

[r(Q)y = (Qy,y)]£ [(r(Qy) = 1) = ((□)QQM =

= (Øà)1Q(ØQH-M))] £ [(((r(QH) = 1) = ◘QH) = بQ(0))] (Lr)

r(Q)y y, 䳺 Q, Q yMÍH, 䳺 Q, Q , 1- Q H-M Ø Q, Ø Ø Q = Q. H H , , . - □=1 □Q = QØQ = ØQ, ◘=1 , ◘QH = بQ(0) , , QH. Q(0) H , y.

(!) 0 £ r(y) £1,

r(Q) y = (Qy,y) ()

(: ) U=SrQy, ( Sp.U = Sr . Spur - ) r(Q)r(y) = (Sp.UQy,y) r(Ø Q)r(y)=(Sp.U(1- Q)y,y) . . U £ 1 U < ¥ , , U = ¥ - . , ᒺ () , ᒺ .

,

[Q £ P] Þ [Q Ú (Q^Ù P)] (OML)

.

(Modulargesetz) :

QÍP, QÚ(XÙP) = (QÚX)ÙP, (LD)

(1936.) , OML (1954.). , , .

, , , , . ( ) . .

, Q £ P . Q P , Q£X£P. , , Q Q y ||MÍN||, Q £ P,

XÚÙ: X = Q Ú (X Ù P), XÙÚ: X = (Q Ú X) Ù P

, Q£P XÚÙ £ XÙÚ, Q£P

(P Ù X) Ú Q £ P Ù (X Ú Q) (.)

Mutatis mutandis, , Q ³ X ³ P,

sÚn iÙm dij £ iÙm sÚn dij (mM)

.

(.) Q£P . , (.) ,

(Q Ú (X Ù P) = (Q Ú X) Ù (Q Ú P)) £ (Q Ú X) Ù P

((QÚ X) Ù P = (Q Ù X) Ú (Q Ù P)) £ Q Ú (X Ù P),

(Q Ú X) Ù P £ Q Ú (X Ù P) ()

() (.), . (.) (), .

. , () (.)

Q £ P, Q Ú (Ù P) = (Q Ú ) Ù P (M)

, . , - Q£ØX£P (Q£ØX)£P Q, Q £(ØX£ P) P.

Q £ P X^ Q X Q^ ()

P = Q Ú (Q^ Ù P) (w^M)

(), Q=QÙP Q£P

= (QÙP)Ú( Q^ÙP) QP

, Q£P, QP.

P£Q X P^

Q = (QÙP)Ú (QÙ P^) PQ

, £Q, QP.

, , Q1, P1 R. ,

Q = Q1Ú R P=P1Ú R.

QP QÛP, 򳺿 Q = P

QÚ (Q^ÙP) = PÚ (P^ÙQ) QÛP

ҳ , . , , Q£ P P £ Q .

P £ Q, P Ú (P^Ù Q)

Q £ P (OML)

Q £ P, Q Ú (Q^Ù P)

(OML).

Q£ØP Q£P , Ø(Q£P), , 1, , . . Q£(X£ØP) X^P

Q^P, Q=(QÚX)ÙØP (qz)

Q £ P Þ P = Q Ú (Q^ Ù P) 1954. . , , Q Ú (Q^Ù P) Q Ù (Q^Ú P), ,

Q £ P Þ Q^ Ú (Q Ù P)

, (, QL ) . a priori . , , .

(OML) . , , . . . , 1989., 1991. .

. , Q=Q KL QL q = åkqkQk +dQ(l). () Ø(QÙØQ) tertium non datur (QÚØQ) KL QL L^ QÙP = 0 QÚP = 1 D(L^). , Q=ØØQ KL QL Q=ØP L3, P=D(Q) Q.

, , , - , , , , .

, 19 . ., . , , .

. . , , , Y ïçYïç , , , , , , . , .

(QÙP)=0=(Q^P), QÄR=0, QÅR=1 . {, Å, 0, 1} 0 1 Å , : 1. xÅy , (xÅy)Åz ( , z (xÄy)Äz ..) xÅy=yÅz; 2. xÅy (xÅy)Åz , yÅz xÅ(yÅz) xÅ(yÅz) = (xÅy)Åz; 3. - xP x¢P , xÅx¢=1 ( , tertium non datur, xÄx¢=0 ..); 4. xÅ1 , =0.

, , Å , Ä, 쳺. , Å . . , .

 

.. //, 1984, 39:2, 3-52

.., .. . ., 1988. 680 . (228-260: )

. . ., 1961. - 151.; . . . ., 1971. .2, . .413

.., .. . ., 1963

., . (1931/32). ., 1964. 367 . (184-189: )

.., .. . ., 2000. 312 .

. . 1 (1956). ., 1960, . 101.

Birkhoff G., Neumann J., von. The logic of quantum mechanics //Annal. Math., 1936, 37:4, 823-843

Gudder S. Examples, problems and result in effect algebras (1995) //Int. J. Theor. Phys., 1996, 35:11, 2365-2376

Maurin K. Duality (polarity) in mathematics, physics and philosophy //Repts Math. Phys., 1988, 25:3, 357-388. See p.360-377

Pták P., Pulmannová S. Kvantove logiky. Bratislava, 1989. - 222 c. See engl. tranl.: Orthomodular structures as quantum logics. Dordrecht1991

Sasaki U. Orthocomplemented lattices satisfying the exchange axiom //J. Sci. Hiroshima Univ., 1954, A17:3, 293-302 (see p.293)