Von Neumann’s concept of quantum logic
1
Miklós Rédei
Department of History and Philosophy of Science
Eötvös University, Budapest
http://hps.elte.hu/∼ redei
Prepared for the talk
Brussels, April 24, 2004
Structure of talk:
• The Main Message
• Standard concept of quantum logic
2
• Why the standard concept is not
what von Neumann had in mind
• Von Neumann’s proposal
• Why von Neumann did not like his own concept
• Summary
Main message of talk:
By creating quantum logic von Neumann wanted to create a
non-commutative = quantum version of
the classical situation:
3
Classical propositional logic
||
Boolean algebra
||
Random event structure
Classical probability
||
normalized measure µ on Boolean algebra
and probability is interpreted as relative frequency
and he saw that he had not succeeded
The non-commutative version of the classical situation would be:
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quantum (propositional) logic
||
non-distributive lattice L
||
quantum (random) event structure
quantum probability
||
normalized additive measure φ on non-distributive lattice L
probability = relative frequency
Standard concept of quantum logic:
L = P(H) = set of projections on a Hilbert space H
represents the set of all quantum propositions
Details:
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Q=
Z
λdP Q (λ)
Q = sa operator on H
observable
P Q = spectral measure of Q
hψ, P Q (E)ψi
||
probability that Q takes its value in E ⊆ IR in state ψ ∈ H
ψ ∈ P Q (E)
⇒
hψ, P Q (E)ψi = 1
Quantum logic:
read
hψ, P Q (E)ψi = 1
as
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ψ makes true the proposition
“Q takes its value in E (with probability 1) =Prop(Q, E)
P (E) ⊆ H closed linear subspace
||
set of interpretations (ψ) making Prop(Q, E) true
||
Prop(Q, E)
(P(H), ≤, ∧, ∨, ⊥)
Hilbert lattice
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atomic
atomistic
orthomodular
modular if dim(H) < ∞
NOT modular if dim(H) = ∞
complete lattice
Hilbert lattice
||
Quantum Logic
Orthomodularity:
If A ≤ B and A⊥ ≤ C then A ∨ (B ∧ C) = (A ∨ B) ∧ (A ∨ C)
Distributivity:
A ∨ (B ∧ C) = (A ∨ B) ∧ (A ∨ C)
for all
A, B, C
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Modularity:
If A ≤ B then A ∨ (B ∧ C) = (A ∨ B) ∧ (A ∨ C)
distributivity ⇒ modularity ⇒ orthomodularity
distributivity ⇐
6 modularity 6⇐ orthomodularity
Birkhoff – von Neumann 1936:
“Hence we conclude that the propositional calculus of quantum
mechanics has the same structure as an
abstract projective geometry.”
||
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Orthocomplemented MODULAR lattice
P(H) (dim(H) = ∞) is NOT modular
??????
Answer:
Von Neumann wanted to create a non-commutative
quantum version of the classical situation:
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Classical propositional logic
||
Boolean algebra
||
Random event structure
Classical probability
||
normalized measure µ on Boolean algebra with
subadditivity property:
µ(A) + µ(B) = µ(A ∨ B) + µ(A ∧ B)
where probability is interpreted as relative frequency
Why isn’t this non-commutative version of the classical situation OK?:
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quantum (propositional) logic
||
non-distributive Hilbert lattice P(H)
||
quantum (random) event structure
quantum probability
||
normalized additive measure φ on non-distributive lattice P(H)
probability = relative frequency
Answer:
Because (P(H), φ) is not a “good” non-commutative probability
space
for two reasons:
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1. Subadditivity of a measure is necessary for the measure to be
probability in the sense of a relative frequency interpretation
(in the sense of R. von Mises)
2. φ cannot be subadditive
φ(A) = T r(ρA)
Relative frequency interpretation of probability
(von Mises)
(X, S, p) has a relative frequency interpretation if there exists a
fixed statistical ensemble {e1 , e2 , . . .} such that
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• For every attribute (event) A, presence/absence of A on every
element ei of the ensemble can be decided unambiguously
without changing ei /the ensemble
for every A ∈ S
• p(A) = limN →∞ #(A)
N
#(A) = number of occurrences of event A in {e1 , e2 , . . . eN }
Observation:
#(A)
N
is subadditive !
Von Mises: the ensemble is supposed to be random
Randomness is tricky and problematic but this is not the reason why a frequency interpretation of
quantum probability spaces is not possible.
φ cannot be subadditive
because
Theorem : There is no finite, subadditive probability measure on
the Hilbert lattice P(H) of an infinite dimensional Hilbert space
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because
Theorem : If there exists a faithful subadditive probability measure
on a lattice then the lattice is modular (which P(H) is not)
Von Neumann thus had these choices:
1. Give up the frequency interpretation of probability in quantum
mechanics
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2. Give up that the algebraic structure representing quantum
logic also represents the structure of (quantum) events
3. Give up Hilbert lattice (of an ∞-dimensional Hilbert space) as
quantum logic
None of the options is attractive
Von Neumann’s choice
in the 1936 Birkhoff-von Neumann paper:
Giving up Hilbert lattice as quantum logic!
(quotation follows shortly)
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This choice is surprising and counterintuitive because it means
abandoning Hilbert space QM!
Von Neumann’s choice leads to the question:
What to replace Hilbert space QM by?
Von Neumann’s choice was possible
because he had known the following
Theorem:
(Murray-von Neumann, 1936):
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There exists a non-distributive, modular lattice P(N ) of non-finite
linear dimensional projections on an ∞ dimensional Hilbert space
such that there exists a τ normalized, subadditive probability
measure on P(N ).
“Projection lattice of a
type II1 von Neumann algebra”
||
Quantum logic
in the Birkhoff-Neumann (1936) sense
“I would like to make a confession which may seem immoral: I do not
believe absolutely in Hilbert space any more. After all Hilbert-space (as
far as quantum-mechanical things are concerned) was obtained by
generalizing Euclidean space, footing on the principle of “conserving the
validity of all formal rules”. This is very clear, if you consider the
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axiomatic-geometric definition of Hilbert-space, where one simply takes
Weyl’s axioms for a unitary-Euclidean-space, drops the condition on the
existence of a finite linear basis, and replaces it by a minimum of
topological assumptions (completeness + separability). Thus
Hilbert-space is the straightforward generalization of Euclidean space, if
one considers the vectors as the essential notions.
Now we [with F.J. Murray, von Neumann’s coauthor] begin to believe,
that it is not the vectors which matter but the lattice of all linear
(closed) subspaces. [...]
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But if we wish to generalize the lattice of all linear closed subspaces from
a Euclidean space to infinitely many dimensions, then one does not
obtain Hilbert space, but that configuration, which Murray and I called
“case II1 .” (The lattice of all linear closed subspaces of Hilbert-space is
our “case I∞ ”.) And this is chiefly due to the presence of the rule
a ≤ c → a ∪ (b ∩ c) = (a ∪ b) ∩ c
[ modularity !]
This “formal rule” would be lost, by passing to Hilbert space!”
Von Neumann to Birkhoff, November 13, 1935
What is a “type II1 von Neumann algebra”?
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Von Neumann algebra theory is the non-commutative
generalization of Hilbert space measure/probability theory that
yields the non-commutative versions of all the typical types of
classical measure/probability theories
A “type II1 von Neumann algebra” corresponds to the classical
probability measure space [0, 1] with the Lebesgue measure on [0, 1]
Definition : N ⊆ B(H) is a von Neumann algebra if
• I ∈ N (I= identity operator )
• If Q ∈ N then Q∗ ∈ N
(*-closed)
• If Q1 , Q2 ∈ N then (λ1 Q1 + λ2 Q2 ) ∈ N and Q1 Q2 ∈ N
(N algebra)
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• N is closed in the sense that if for some Q ∈ B(H) and Qn ∈ N
(n = 1, . . .) we have φ(Qn ) → φ(Q) for all states φ then Q ∈ N
B(H) = set of all bounded operators
is (obviously) a von Neumann algebra
Are there any other examples?
(Other = non-isomorphic to B(H))
m
Classification problem
Classification of von Neumann algebras
The set of projections P(N ) of a von Neumann algebra is an
orthomodular lattice (= sublattice of the Hilbert lattice P(H))
Definition : d: N → IR+ ∪ ∞ is a dimension function if
d(A) + d(B) = d(A ∪ B) + d(A ∩ B)
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m
subadditivity !
Definition N is a factor von Neumann algebra if N ∩ N 0 = {λI}
with
N 0 = {Q ∈ B(H) : QA = AQ
∀A ∈ N }
i.e. if there are no non-trivial elements in N that commute with
every element in N
Proposition (Murray-von Neumann, 1935) There exists a unique
(up to multiplication by a constant) dimension function on every
factor von Neumann algebra
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The classification of von Neumann algebras is in terms of the type
of the range of the dimension function defined on the projection
lattice P(N ): the type of the range of the dimension function
coincides with the notion of type used in classifying the classical
probability spaces. These types of classical measure/probability
spaces are shown on the next slide.
The typical classical measure/probability spaces
X = {x1 , x2 , . . . xN }
discrete
p(xi ) = 1
finite
(i = 1, . . . N )
X = {x1 , x2 , . . . xN , . . .}
discrete
p(xi ) = 1
infinite
(i = 1, . . . N . . .)
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X = [0, 1]
continuous
p = Lebesgue measure on [0, 1]
finite
X = IR
continuos
p = Lebesgue measure on IR
infinite
HN , dim(HN ) = N
finite
N = B(HN ), P(N ) = P(HN )
type IN
dimensional
range of d(= T r) = {1, 2, . . . N }
finite, discrete
QM
H, dim(H) = ∞
standard
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N = B(H), P(N ) = P(H)
type I∞
Hilbert space
range of d(= T r) = {1, 2, . . .}
non-finite, discrete
QM
N , P(N )
type II1
Quantum
range of d = [0, 1]
finite, continuous
stat.phys.
N , P(N )
type II∞
Quantum
range of d = IR
non-finite, continuous
stat.phys.
N , P(N )
type III
Quantum
range of d = {0, ∞}
very non-finite
field theory
Is (N , P(N ), τ ) REALLY a NON-COMMUTATIVE probability
space whose probabilities can be interpreted as relative frequencies?
NO
because
τ is a trace :
τ (XY ) = τ (Y X)
for all
X, Y ∈ N
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“τ is insensitive for the non-commutativity”
Theorem:
A linear functional on a von Neumann algebra is subadditive
if and only if it is a trace
Consequently
If one wants to have a genuinely non-commutative probability
space then the frequency interpretation has to go!
Von Neumann gave up the frequency view in 1937:
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“This view, the so-called ‘frequency theory of probability’ has been
very brilliantly upheld and expounded by R. von Mises. This view,
however, is not acceptable to us, at least not in the present ‘logical’
context.”
“Quantum logic (strict- and probability logics)” unfinished, unpublished manuscript from 1937
How to interpret non-commutative probability
if not by relative frequency?
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Von Neumann had not been able to give an answer that would
satisfy him. He tried but confessed that he had not succeeded in
working out quantum logic in a satisfactory manner:
“Dear Doctor Silsbee,
It is with great regret that I am writing these lines to you, but I simply
cannot help myself. In spite of very serious attempts to write the article
on the “Logics of quantum mechanics” I find it completely impossible to
do it at this time.
As you may know, I wrote a paper on this subject with Garrett Birkhoff
in 1936 ([reference]), and I have thought a good deal on the subject
since. My work on continuous geometries, on which I gave the
Amer.Math.Soc. Colloqium lectures in 1937, comes to a considerable
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extent from this source. Also a good deal concerning the relationship
between strict and probability logics (upon which I touched briefly in the
Henry Joseph Lecture) and the extension of this “Propositional calculus”
work to “logics with quantifiers” (which I never so far discussed in
public). All these things should be presented as a connected whole ...
When I offered to give the Henry Joseph Lecture on this subject, I
thought (and I hope that I was not too far wrong in this) that I could
give a reasonable general survey of at least part of the subject in a talk,
which might have some interest to the audience. I did not realize the
importance nor the difficulties of reducing this to writing.
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I have now learned – after a considerable number of serious but very
unsuccesful efforts – that they are exceedingly great. I must, of course,
accept a good part of the responsibility for my method of writing – I
write rather freely and fast if a subject is “mature” in my mind, but
develop the worst traits of pedantism and inefficiency if I attempt to give
a preliminary account of a subject which I do not have yet in what I can
believe in its final form.
I have tried to live up to my promise and to force myself to write this
article, and spent much more time on it than on many comparable ones
which I wrote with no difficulty at all – and it just didn’t work.”
Von Neumann to Dr. Silsbee, July 2, 1945
Why did it not work?
Because what von Neumann wanted to have
could not be had:
Summary
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In the classical world Logic, Measure Theory and Probability Theory
(understood as relative frequency) are in great harmony. While it is
possible to create non-classical, Quantum Logic and an impeccable
Non-commutative Measure Theory, the conceptual relation of the latter
ones cannot be the same as in the classical case; specifically, non-Boolean
lattice structures cannot be interpreted as random event structures and
non-classical probabilities cannot be interpreted as relative frequencies.
The question of whether non-commutative measure theory can be
interpreted as probability in any way that does not cut this notion off
the empirical world is especially important and von Neumann had
realized it as such. It is remarkable that he remained frustrated by the
fact that he was unable to give a satisfactory answer.
References
M. Rédei: Why John von Neumann did not like the Hilbert space
formalism of quantum mechanics (and what he liked instead)
Studies in the History and Philosophy of Modern Physics 27 (1996)
493-510
M. Rédei: Quantum Logic in Algebraic Approach (Kluwer Academic
Publishers, 1998)
32
M. Rédei: “Unsolved problems in mathematics” J. von Neumann’s
address to the International Congress of Mathematicians, Amsterdam,
August 1954
The Mathematical Intelligencer 21 (1999) 7-12
M. Rédei: Von Neumann’s concept of quantum logic
in M. Rédei, M. Stöltzner: John von Neumann and the Foundations of
Quantum Physics (Kluwer Academic Publishers, 2001)