Transcript Document

Complementarity and
Bistable Perception
Thomas Filk
Institute for Frontier Areas in Psychology, Freiburg
Parmenides Foundation for the Study of Thinking, Munich,
Department of Physics, University of Freiburg
Monte Verità – May, 23rd 2007
Can we apply ideas and part of
the mathematical formalism of
(quantum) physics to describe
phenomena of consciousness?
Can we apply ideas and part of
the mathematical formalism of
(quantum) physics to describe
phenomena of consciousness?
Not: consciousness as an
immediate quantum phenomenon
Content
• Bistable Perception
• Weak Quantum Theory
• The Necker-Zeno Model for Bistable
Perception
• Tests for Non-classicality
Bistable Perception
Bistable perception - cup or faces
Bistable perception – mother or daughter
The Necker cube
Louis Albert Necker (1786-1861)
The mental states
state 1
state 2
Rates of perceptive shifts
2
1
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 t (sec)
T=t
J.W.Brascamp et.al, Journal of Vision (2005) 5, 287-298
Weak Quantum Theory
“Observation”
• An observation not only changes the state of
the observing system but also the state of the
observed system. It is an interaction between
these two systems.
• The algebraic formalism of quantum
mechanics grew out of the necessity that
observations may have an influence on the
observed system.
“Observation”
• No discussion of
– the role of consciousness
– the relevance of the partition
– the pointer basis problem
– the problem of state reduction
Observables and States
• Measurable quantity (measuring recipe): A
detailed prescription for the performance of an
experiment yielding a definite result.
• Observable: A mathematical object
„representing“ a measurable quantity.
• State: A functional (mapping) which
associates to each observable a number
(expectation value).
Mathematical formalization of classical
and quantum mechanics
Classical mechanics
Quantum mechanics
• Observables:
• Observables:
– Commutative C*Algebra
– Distributive
proposition calculus
– Boolean lattice
• States: positive, linear
functionals on the set
of observables
(expectation values)
– Non-commutative C*Algebra
– Non-distributive
proposition calculus
– Non-boolean lattice
• States: positive, linear
functionals on the set
of observables
(expectation values)
Mathematical formalization of classical
and quantum mechanics
Classical mechanics
Quantum mechanics
• Observables:
• Observables:
– Commutative C*Algebra
– Distributive
proposition calculus
– Boolean lattice
• States: positive, linear
functionals on the
observables
(expectation values)
– Non-commutative C*Algebra
– Non-distributive
proposition calculus
– Non-boolean lattice
• States: positive, linear
functionals on the
observables
(expectation values)
Weak quantum mechanics
H. Atmanspacher, H. Römer, H. Wallach (2001)
• Generalization of the algebraic description of
classical and quantum physics
• A framework for a theory of observables
(propositions) for any system which “has
enough internal structure to be a possible
object of a meaningful study”.
• No Hilbert-space of states, no a priori
probability interpretation, no Schrödinger
equation, no Born rule, ….
Sketch of the axioms of weak QT
• The exist states {z} and observables {A}.
Observables act on states (change states).
• Observables can be multiplied (related to
successive observations).
• Observables have a “spectrum”, i.e.,
measurements yield definite results.
• There exists an “identity” observable: the trivial
“measurement” giving always the same result.
Complementarity
• Two observables A and B are complementary if
they do not commute AB  BA .
• Two (sets of) observables A and B are
complementary, if they do not commute and if
they generate the observable algebra .
• Two (sets of) observables A and B are
complementary, if they do not commute on states
AB z  BA z.
• Two (sets of) observables A and B are
complementary, if the eigenstates (dispersion-free
states) have a maximal distance.
The Necker-Zeno Model for
Bistable Perception
The quantum Zeno effect
B. Misra and E.C.G. Sudarshan (1977)
Dynamics:
0 1

H  g 
 1 0
U (t )  e
1 0 

Observation:  3  
 0  1
States:
1
   
 0
iHt
 cos gt i sin gt 

 
 i sin gt cos gt 
Dynamics and observation
are complementary
 0
   
1
Results of
observations
The quantum Zeno effect
B. Misra and E.C.G. Sudarshan (1977)
Dynamics:
0 1

H  g 
 1 0
U (t )  e
1 0 

Observation:  3  
 0  1
States:
1
   
 0
iHt
 cos gt i sin gt 

 
 i sin gt cos gt 
Dynamics and observation
are complementary
 0
   
1
Results of
observations
The quantum Zeno effect
The probability that the system is in state |+ at t=0 and still
in state |+ at time t is:
w(t) = |+|U(t)|+|2 = cos2gt .
t0~1/g is the time-scale of unperturbed time evolution.
The probability that the system is in state |+ at t=0 and is
measured to be in state |+ N times in intervals Δt and still
in state |+ at time t=N·Δt is given by:
wΔt(t) := w(Δt)N = [cos2gΔt]N
Decay time:

 Ng t 
 g t t
e
e
t02
1
T t  2 
g t t
2
2
2
Quantum Zeno effect
w(t)
Δt
t0
T
t 0  t  T
The Necker-Zeno model
H. Atmanspacher, T. Filk, H. Römer (2004)
Mental state 1:
Mental state 2:
dynamics 
„decay“ (continuous change) of a mental state
observation 
„update“ of one of the mental states
Internal dynamics and internal observation are
complementary.
Time scales in the Necker Zeno model
• Δt : internal „update“ time. Temporal separability
of stimuli  25-70 ms
• t0 : time scale without updates (“P300”)  300 ms
• T : average duration of a mental state  2-3 s.
Prediction of the Necker-Zeno model:
t 0  t  T
A first test of the Necker-Zeno model
Assumption: for long off-times t0  off-time
t0  T
Necker-Zeno model predictions for the
distribution functions
probability
density
Cum. probability
J.W.Brascamp et.al, Journal of Vision (2005) 5, 287-298
Refined model
g(t), t(t)
Modification of
- g  g(t)
t
the „decay“-parameter is smaller in the beginning:
- t  t(t)
the update-intervals are shorter in the beginning
Increased attention?
Tests for Non-Classicality
Bell‘s inequalities
J. Bell (1964)
Let Q1, Q2, Q3, Q4 be observables with possible
results +1 and –1.
Let E(i,j)=QiQj
Then the assumption of “local realism” leads to
–2  E(1,2) + E(2,3) + E(3,4) – E(4,1)  +2
Temporal Bell’s inequalities
A.J. Leggett, A. Garg (1985)
1
t
-1
Let K(ti,tj)=σ3(ti)σ3(tj) be the 2-point correlation
function for a measurement of the state, averaged over a
classical ensemble of “histories”.
Then the following inequality holds:
|K(t1,t2) + K(t2,t3) + K(t3,t4) – K(t1,t4)|  2 .
This inequality can be violated in quantum mechanics,
e.g., in the quantum Zeno model.
Caveat
• The derivation of temporal Bell‘s inequalities
requires the assumption of „non invasive“
measurements.
(This corresponds to locality in the standard
case: the first measurement has no influence
on the second measurement.)
Summary and Challenges
• The Necker-Zeno model makes predictions for
time scales which can be tested.
• The temporal Bell’s inequalities can be tested.
• Complementarity between the dynamics and
observations of mental states is presumably easier
to find than complementary observables for
mental states.
• If Bell’s inequalities are violated (an noninvasiveness has been checked), what are the
„non-classical“ states in the Necker-Zeno model?
(acategorical mental states?)