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A Few Good Examples why to Refute
Classical Logic in Quantum Mechanics
Lovre Grisogono
Mateo Paulišić
University of Zagreb
Faculty of Science
Department of Physics
Split, July 7-8 2014
Lovre Grisogono & Mateo Paulišić
A Few Good Examples why to Refute Classical Logic in QM
Classical Logic
Classical logic is a type of formal logic which is most
intensively studied and most widely used on all levels of
education.
Classical logic is defined by a set of properties.
Classical logic generally includes propositional logic and
first-order logic.
Lovre Grisogono & Mateo Paulišić
A Few Good Examples why to Refute Classical Logic in QM
Definition of CL
We define classical logic with the following set of properties:
Binarity:
Set of truth values - V(P) ∈ {0, 1}
Commutativity:
P ∧ Q ≡ Q ∧ P, P ∨ Q ≡ Q ∨ P
Distributivity:
P ∧ (Q ∨ R) ≡ (P ∧ Q) ∨ (P ∧ R)
P ∨ (Q ∧ R) ≡ (P ∨ Q) ∧ (P ∨ R)
Principle of excluded middle:
V(P ∨ ¬P) = 1
Principle of non-contradiction:
V(¬(P ∧ ¬P)) = 1, alternatively V(P ∧ ¬P) = 0
Lovre Grisogono & Mateo Paulišić
A Few Good Examples why to Refute Classical Logic in QM
Quantum Mechanics
Branch of physics
Atomar and subatomar scale
Commutative and non-commutative operators
Uncertainty principle
Lovre Grisogono & Mateo Paulišić
A Few Good Examples why to Refute Classical Logic in QM
But the Question Is?
In 1936 Birkoff and von Neumann wrote the article “The
Logic of Quantum Mechanics”.
Birkoff and von Neumann wanted to find the logical
structure in quantum mechanics which did not conform to
classical logic.
Lovre Grisogono & Mateo Paulišić
A Few Good Examples why to Refute Classical Logic in QM
But the Question Is?
In 1936 Birkoff and von Neumann wrote the article “The
Logic of Quantum Mechanics”.
Birkoff and von Neumann wanted to find the logical
structure in quantum mechanics which did not conform to
classical logic.
Should we refute classical logic for QM?
Lovre Grisogono & Mateo Paulišić
A Few Good Examples why to Refute Classical Logic in QM
Definitions
Quantum mechanical proposition is a projection operator
(generally observable) P
- experiment outcome prediction
Px = x = 1 · x for x ∈ LP
Py = 0 = 0 · y for y ∈ L⊥
P
P ∧ Q - operator projecting on LP∧Q = LP ∩ LQ
P ∨ Q - operator projecting on LP + LQ
¬P = 1 − P
P ∧ Q = lim (PQ)n
n→∞
P ∨ Q = ¬[¬P ∧ ¬Q]
Lovre Grisogono & Mateo Paulišić
A Few Good Examples why to Refute Classical Logic in QM
Examples
Example 1: Sz = ( ~2 )P1 − ( ~2 )P2
P1 = |↑i h↑| P2 = |↓i h↓|
1
0
Matrix notation: |↑i =
|↓i =
0
1
P1 =
1 0
0 0
0 0
P2 =
0 1
For Sz :
Sz =
~
2
1 0
0 −1
Lovre Grisogono & Mateo Paulišić
A Few Good Examples why to Refute Classical Logic in QM
Using previous definitions:
P1 ∧ P2 = lim (P1 P2 )n
n→+∞
n
1 0
0 0
= lim
·
0 0
0 1
n→+∞
n
0 0
= lim
=0
n→+∞ 0 0
P1 ∨ P2 = 1 − [(1 − P1 ) ∧ (1 − P2 )]
= 1 − [P2 ∧ P1 ]
=1
Lovre Grisogono & Mateo Paulišić
A Few Good Examples why to Refute Classical Logic in QM
Using previous definitions:
P1 ∧ P2 = lim (P1 P2 )n
n→+∞
n
1 0
0 0
= lim
·
0 0
0 1
n→+∞
n
0 0
= lim
=0
n→+∞ 0 0
P1 ∨ P2 = 1 − [(1 − P1 ) ∧ (1 − P2 )]
= 1 − [P2 ∧ P1 ]
=1
Calculations were simplified due to [P1 , P2 ] = 0
Lovre Grisogono & Mateo Paulišić
A Few Good Examples why to Refute Classical Logic in QM
Example of non-commuting observables:
Q1 − ~2 Q2 , it is easily seen that:
1 1
1 −1
1
1
Q1 = 2
Q2 = 2
1 1
−1 1
1
1
are projectors projecting on subspaces
,
1
−1
Operators Q1 and Q2 do not commute with P1 and P2
P1 ∧ Q1 = lim (P1 Q1 )n
n→+∞
1 n 1 1
= lim ( 2 )
=0
0 0
n→+∞
Sx =
~
2
P1 ∨ Q1 = 1 − ((1 − P1 ) ∧ (1 − Q1 ))
= 1 − (P2 ∧ Q2 )
=1−0=1
Lovre Grisogono & Mateo Paulišić
A Few Good Examples why to Refute Classical Logic in QM
Non-commutativity 6= zero conjunction
Example: particle with spin 1


1 0 0
B = 2~12 [S 2 − Sz2 ] + 2~12 Sz = 0 1 0
0 0 0
1

 
1
0 2
0
2
1 2



D = ~2 Sz = 0 1 0 , [B, D] 6= 0 but 1 is a
1
0 12
0
2
common eigenvector.
n 

0 12
0 0 0
D ∧ B = lim  0 1 0  = 0 1 0
n→+∞
1
0 12
0 0 0
2
1
2
Lovre Grisogono & Mateo Paulišić
A Few Good Examples why to Refute Classical Logic in QM
Failure of distributivity:
Easily calculated as before:
P1 ∨ P2 = 1 Q1 ∧ P1 = 0 Q1 ∧ P2 = 0
Lovre Grisogono & Mateo Paulišić
A Few Good Examples why to Refute Classical Logic in QM
Failure of distributivity:
Easily calculated as before:
P1 ∨ P2 = 1 Q1 ∧ P1 = 0 Q1 ∧ P2 = 0
Q1 ∧ (P1 ∨ P2 ) ≡ (Q1 ∧ P1 ) ∨ (Q1 ∧ P2 )
Lovre Grisogono & Mateo Paulišić
A Few Good Examples why to Refute Classical Logic in QM
Failure of distributivity:
Easily calculated as before:
P1 ∨ P2 = 1 Q1 ∧ P1 = 0 Q1 ∧ P2 = 0
Q1 ∧ (P1 ∨ P2 ) ≡ (Q1 ∧ P1 ) ∨ (Q1 ∧ P2 )
Q1 ∧ 1 ≡ 0 ∨ 0
Lovre Grisogono & Mateo Paulišić
A Few Good Examples why to Refute Classical Logic in QM
Failure of distributivity:
Easily calculated as before:
P1 ∨ P2 = 1 Q1 ∧ P1 = 0 Q1 ∧ P2 = 0
Q1 ∧ (P1 ∨ P2 ) ≡ (Q1 ∧ P1 ) ∨ (Q1 ∧ P2 )
Q1 ∧ 1 ≡ 0 ∨ 0
Q1 = 0
Lovre Grisogono & Mateo Paulišić
A Few Good Examples why to Refute Classical Logic in QM
Failure of distributivity:
Easily calculated as before:
P1 ∨ P2 = 1 Q1 ∧ P1 = 0 Q1 ∧ P2 = 0
Q1 ∧ (P1 ∨ P2 ) ≡ (Q1 ∧ P1 ) ∨ (Q1 ∧ P2 )
Q1 ∧ 1 ≡ 0 ∨ 0
Q1 = 0
Comment: Modular law holds:
C →A
A ∧ (B ∨ C ) ≡ (A ∧ B) ∨ (A ∧ C ) ≡ (A ∧ B) ∨ C
Lovre Grisogono & Mateo Paulišić
A Few Good Examples why to Refute Classical Logic in QM
Failure of continuity - example using polarization
let P =
cos2 cos sin cos sin sin2 Lovre Grisogono & Mateo Paulišić
A Few Good Examples why to Refute Classical Logic in QM
Failure of continuity - example using polarization
let P =
cos2 cos sin cos sin sin2 lim P = P1
→0
Lovre Grisogono & Mateo Paulišić
A Few Good Examples why to Refute Classical Logic in QM
Failure of continuity - example using polarization
let P =
cos2 cos sin cos sin sin2 lim P = P1
→0
for 6= 0, P ∧ P1 = 0
Lovre Grisogono & Mateo Paulišić
A Few Good Examples why to Refute Classical Logic in QM
Failure of continuity - example using polarization
let P =
cos2 cos sin cos sin sin2 lim P = P1
→0
for 6= 0, P ∧ P1 = 0
lim (P ∧ P1 ) = 0
→0
Lovre Grisogono & Mateo Paulišić
A Few Good Examples why to Refute Classical Logic in QM
Failure of continuity - example using polarization
let P =
cos2 cos sin cos sin sin2 lim P = P1
→0
for 6= 0, P ∧ P1 = 0
lim (P ∧ P1 ) = 0
→0
(lim P ) ∧ P1 = P1 ∧ P1 = P1
→0
Lovre Grisogono & Mateo Paulišić
A Few Good Examples why to Refute Classical Logic in QM
Polarization
Figure: Polarization example
P1
light bulb
unpolarized light vertical polarizing
filter
P2
filter polarizing
at direction b
Polarizing filter = Mechanical analogue to the projection
operator
Lovre Grisogono & Mateo Paulišić
A Few Good Examples why to Refute Classical Logic in QM
Polarization
Figure: Polarization example
a
P1
P2
I1
b
(a)
a
c
a
P3
P1
φ
P2
c
I1
b I1cos2φsin2φ
(b)
Lovre Grisogono & Mateo Paulišić
A Few Good Examples why to Refute Classical Logic in QM
Polarization
H - horizontal filter, V - vertical filter, D - diagonal filter
H n = H, V n = V , HV = 0
Lovre Grisogono & Mateo Paulišić
A Few Good Examples why to Refute Classical Logic in QM
Polarization
H - horizontal filter, V - vertical filter, D - diagonal filter
H n = H, V n = V , HV = 0
VHD 6= VDH
Lovre Grisogono & Mateo Paulišić
A Few Good Examples why to Refute Classical Logic in QM
Polarization
H - horizontal filter, V - vertical filter, D - diagonal filter
H n = H, V n = V , HV = 0
VHD 6= VDH
H ∧ V = lim (HV )n = 0
n→∞
V ∧ D = lim (VD)n = 0
n→∞
D ∨ V = ¬[¬D ∧ ¬V ]
= ¬[Danti ∧ H] = ¬0 = 1
Lovre Grisogono & Mateo Paulišić
A Few Good Examples why to Refute Classical Logic in QM
Polarization
H - horizontal filter, V - vertical filter, D - diagonal filter
H n = H, V n = V , HV = 0
VHD 6= VDH
H ∧ V = lim (HV )n = 0
n→∞
V ∧ D = lim (VD)n = 0
n→∞
D ∨ V = ¬[¬D ∧ ¬V ]
= ¬[Danti ∧ H] = ¬0 = 1
Failure of distributivity
H ∧ (D ∨ V ) ≡ H ∧ 1 ≡ H
Lovre Grisogono & Mateo Paulišić
A Few Good Examples why to Refute Classical Logic in QM
Polarization
H - horizontal filter, V - vertical filter, D - diagonal filter
H n = H, V n = V , HV = 0
VHD 6= VDH
H ∧ V = lim (HV )n = 0
n→∞
V ∧ D = lim (VD)n = 0
n→∞
D ∨ V = ¬[¬D ∧ ¬V ]
= ¬[Danti ∧ H] = ¬0 = 1
Failure of distributivity
H ∧ (D ∨ V ) ≡ H ∧ 1 ≡ H
(H ∧ D) ∨ (H ∧ V ) ≡ 0 ∨ 0 ≡ 0
Lovre Grisogono & Mateo Paulišić
A Few Good Examples why to Refute Classical Logic in QM
Non-continuity of conjunction
D - polarizing filter inclined at to the horizontal
lim D = H
→0
Lovre Grisogono & Mateo Paulišić
A Few Good Examples why to Refute Classical Logic in QM
Non-continuity of conjunction
D - polarizing filter inclined at to the horizontal
lim D = H
→0
D ∧ H = 0
Lovre Grisogono & Mateo Paulišić
A Few Good Examples why to Refute Classical Logic in QM
Non-continuity of conjunction
D - polarizing filter inclined at to the horizontal
lim D = H
→0
D ∧ H = 0
lim (D ∧ H) = 0
→0
Lovre Grisogono & Mateo Paulišić
A Few Good Examples why to Refute Classical Logic in QM
Non-continuity of conjunction
D - polarizing filter inclined at to the horizontal
lim D = H
→0
D ∧ H = 0
lim (D ∧ H) = 0
→0
(lim D ) ∧ H = H
→0
Lovre Grisogono & Mateo Paulišić
A Few Good Examples why to Refute Classical Logic in QM
Two slit experiment - failure of distributivity
A1 - electron passes through slit 1
A2 - electron passes through slit 2
P(A1 , R) - probablity that electron hits region R, after A1
P(A2 , R) - probablity that electron hits region R, after A2
Classically, one would expect
P(R) = 12 P(A1 , R) + 21 P(A1 , R)
Lovre Grisogono & Mateo Paulišić
A Few Good Examples why to Refute Classical Logic in QM
Two slit experiment - failure of distributivity
A1 - electron passes through slit 1
A2 - electron passes through slit 2
P(A1 , R) - probablity that electron hits region R, after A1
P(A2 , R) - probablity that electron hits region R, after A2
Classically, one would expect
P(R) = 12 P(A1 , R) + 21 P(A1 , R)
Quantum formula P(R) = p
1
P(A1 , R) + 12 P(A1 , R) + P(A1 , R)P(A2 , R) cos δ
2
Lovre Grisogono & Mateo Paulišić
A Few Good Examples why to Refute Classical Logic in QM
P((A1 ∨ A2 ) · R)
P(A1 ∨ A2 )
P((A1 · R) ∨ (A2 ∨ R))
=
P(A1 ∨ A2 )
P(A1 · R)
P(A2 · R)
=
+
P(A1 ∨ A2 ) P(A1 ∨ A2 )
P(A1 ∨ A2 , R) =
Lovre Grisogono & Mateo Paulišić
A Few Good Examples why to Refute Classical Logic in QM
P((A1 ∨ A2 ) · R)
P(A1 ∨ A2 )
P((A1 · R) ∨ (A2 ∨ R))
=
P(A1 ∨ A2 )
P(A1 · R)
P(A2 · R)
=
+
P(A1 ∨ A2 ) P(A1 ∨ A2 )
P(A1 ∨ A2 , R) =
P(A1 ) = P(A2 )
P(A1 ∨ A2 ) = 2P(A1 ) = 2P(A2 )
Lovre Grisogono & Mateo Paulišić
A Few Good Examples why to Refute Classical Logic in QM
P((A1 ∨ A2 ) · R)
P(A1 ∨ A2 )
P((A1 · R) ∨ (A2 ∨ R))
=
P(A1 ∨ A2 )
P(A1 · R)
P(A2 · R)
=
+
P(A1 ∨ A2 ) P(A1 ∨ A2 )
P(A1 ∨ A2 , R) =
P(A1 ) = P(A2 )
P(A1 ∨ A2 ) = 2P(A1 ) = 2P(A2 )
P(A1 · R)
P(A1 · R)
=
=
P(A1 ∨ A2 )
2P(A1 )
P(A2 · R)
P(A2 · R)
=
=
P(A1 ∨ A2 )
2P(A2 )
Lovre Grisogono & Mateo Paulišić
1
P(A1 , R) ;
2
1
P(A2 , R)
2
A Few Good Examples why to Refute Classical Logic in QM
P((A1 ∨ A2 ) · R)
P(A1 ∨ A2 )
P((A1 · R) ∨ (A2 ∨ R))
=
P(A1 ∨ A2 )
P(A1 · R)
P(A2 · R)
=
+
P(A1 ∨ A2 ) P(A1 ∨ A2 )
P(A1 ∨ A2 , R) =
P(A1 ) = P(A2 )
P(A1 ∨ A2 ) = 2P(A1 ) = 2P(A2 )
P(A1 · R)
P(A1 · R)
=
=
P(A1 ∨ A2 )
2P(A1 )
P(A2 · R)
P(A2 · R)
=
=
P(A1 ∨ A2 )
2P(A2 )
1
P(A1 , R) ;
2
1
P(A2 , R)
2
P(R) = 12 P(A1 , R) + 21 P(A1 , R)
Lovre Grisogono & Mateo Paulišić
A Few Good Examples why to Refute Classical Logic in QM
Conclusion
The question was:
Should we refute classical logic for QM?
And the anwser is:
Lovre Grisogono & Mateo Paulišić
A Few Good Examples why to Refute Classical Logic in QM
Conclusion
The question was:
Should we refute classical logic for QM?
And the anwser is:
We should.
Lovre Grisogono & Mateo Paulišić
A Few Good Examples why to Refute Classical Logic in QM
Literature
’Quantum mechanics’ [Auletta, G., Fortunato, M., Parisi,
G. Quantum mechanics, Cambridge University Press,
2009]
’Quantum logic’ [Adler, C. G., Wirth, J. F. Quantum
Logic. In American Journal of Physics 51(5): 1983, pp.
412-417.]
’Is logic empirical’ [Putnam, H. Is Logic Empirical? In
Boston Studies in the Philosophy of Science 5, R. S.
Cohen, M. W. Wartofsky, ur. Dodrecht: D. Reidel, 1968,
pp. 174-197.]
Lovre Grisogono & Mateo Paulišić
A Few Good Examples why to Refute Classical Logic in QM