Example:
Two-valued logic as used for example in computer circuits
"Quantum theory is simply the replacement in standard probability theory of event-as-subset-of-a-set (abelian, distributive) by event-as-subspace-of-a-Hilbert-space (non-abelian, non-distributive)"Read this introduction by Martin Ziegler to understand what he means by that.
| a) | a ^ b = b ^ a | a v b = b v a | commutativity |
|---|---|---|---|
| b) | (a ^ b) ^ c = a ^ (b ^ c) | (a v b) v c = a v (b v c) | associativity |
| c) | (a v b) ^ a = a | (a ^ b) v c = a) | absorption |
| d) | a ^ (b v c) = (a ^ b) v (a ^ c) | distributive | |
| e) | a v ¬a = 1 a ^ ¬a = 0 ¬¬a = a | invertibility | |
Theorem: Elements 0 and 1 are rightly called neutral, since they fulfill
Example:
Two-valued logic as used for example in computer circuits
Example:
If Li are Boolean algebras for each
i, then their direct product
L:=×Li
again constitutes a Boolean algebra via
In fact every finite (!) Boolean algebra L can be shown to be a direct product of 2¹, see [3]:
Example: L := { A subset of N : A finite or N\A finite } forms a Boolean algebra via
Example: The graph to the right represents a poset:
Idefix and Obelix are connected to indicate
that they are comparable; Idefix is lower, so
A poset is called complete if for any nonvoid subset
M of P there exist sup M and inf M.
Example:
The set N0 of natural numbers
including 0 constitutes a partially ordered set. It is not
complete, since sup N0 does not exist.
Example: Every Boolean algebra L can be turned into a poset by letting
Note that the Boolean algebra's orthocomplementation ¬ : L -> L is compatible with the relation « in the following sense:
| d') | a « b => ¬a » ¬b |
|---|
Consider orthogonal unit vectors x and y in H and for v:=2-1/2(x+y) the projectors to the corresponding onedimensional subspacesYet P can be turned into a poset the same way as before. This poset turns out to be even complete.p:=|x><x|, q:=|y><y|, r:=|v><v| Then we haver ^ (p v q) = r <> 0 = 0 v 0 = (r ^ p) v (r ^ q)
Strategy:
To learn about the differences and similarities
between classical and Quantum mechanics let
us investigate their logics' common categorial
substructure(s).
One such substructure is that of posets. But there are others:
| a) | a ^ b = b ^ a | a v b = b v a | commutativity |
|---|---|---|---|
| b) | (a ^ b) ^ c = a ^ (b ^ c) | (a v b) v c = a v (b v c) | associativity |
| c) | (a v b) ^ a = a | (a ^ b) v c = a | absorption |
Examples: Every Boolean algebra is trivially a lattice.
The Standard Quantum Logic P forms a lattice.
Indeed axiom c) is fulfilled, since for
closed subspaces A and B of H we have
Every lattice can be turned into a poset by the method already mentioned. Conversely every poset, for which sup(a,b) and inf(a,b) always exist, is the so constructed poset of a lattice. This lattice is completely determined by the poset :
Theorem (Iseki): Let (L, ^ , v ) and (K, ^ , v ) be lattices. A mapping f : L -> K is a lattice-isomorphism if and only if f is an order-isomorphism of the posets (L, « ) and (K, « ). Or stated similarly:
Examples: The following directed graphs represent lattices:
| ~d) | a « c => a v (b ^ c) = (a v b) ^ c | modularity |
|---|
Example: Every lattice fulfilling the distributive law, and in particular every Boolean algebra, is obviously modular :
But pentagon and benzene are not :
| pentagon: a « c | but | a v (b ^ c) = a v 0 = a <> c = 1 ^ c = (a v b) ^ c |
|---|---|---|
| benzene: a « b | but | a v ( ¬ a ^ b) = a v 0 = a <> b = 1 ^ b = (a v ¬ a) ^ b |
Theorem: A lattice is modular if and only if it does not contain the pentagon as an isomorphic sublattice.
| a) | a ^ b = b ^ a | a v b = b v a | commutativity |
|---|---|---|---|
| b) | (a ^ b) ^ c = a ^ (b ^ c) | (a v b) v c = a v (b v c) | associativity |
| c) | (a v b) ^ a = a | (a ^ b) v c = a) | absorption |
| d') | a « b => ¬ a » ¬ b | compatibility | |
| e) | a v ¬a = 1 a ^ ¬a = 0 ¬¬a = a | invertibility | |
Theorem: If a), b), c), and e) hold, then d') is equivalent to the de Morgan laws
Examples:
Every Boolean algebra is an ortholattice.
M02 is a (modular) ortholattice,
benzene and P are (non-modular) ortholattices.
In the case of P we have
Definition: Let L be some ortholattice. Two elements a,b of L are said to be orthogonal if a « ¬ b. The so defined relation "Orthogonality" is symmetric.
Theorem: For q,r in P the condition q·r=r·q is equivalent to
Definition:
Let L be an ortholattice and a,b in L. We say
that a commutes with b if
a = (a ^ b) v (a ^ ¬ b).
Unfortunately, on some ortholattices this is no equivalence relation:
In benzene, a commutes with b, but b does not
commute with a.
So P posesses additional properties, which are not reflected
by the mathematical structure "Ortholattice".
Theorem: For an ortholattice L, the following are equivalent
| ~d') | a « b => a v ( ¬ a ^ b) = b | orthomodularity |
|---|
Example: Every Boolean algebra and, more generally, every modular ortholattice is an orthomodular lattice. But there are orthomodular lattices which are not modular, like for example P, the projectors on a Hilbert space.
Theorem: An orthomodular lattice is modular if and only if it does not contain the pentagon with common 0 and 1 as a sublattice.
Theorem: For an orthomodular lattice L the
following are equivalent :
Theorem (Foulis-Holland): Let L be an orthomodular lattice and a,b,c in L such that any one of them commutes with the other two. In this particular case the distributive laws hold :
Theorem (Gudder-Schelp): If x,y,z are elements of an
orthomodular lattice L and y commutes with z
and x commutes with y ^ z.
Then each of x ^ y, ¬ x ^ y, ¬ x ^ y,
¬ x ^ ¬ y commutes with z
and each of x ^ z,
¬ x ^ z, x ^ ¬ z, ¬ x ^ ¬ z commutes with y.
| d') | a « b => ¬ a » ¬ b | compatibility |
|---|---|---|
| e') | ¬ ¬ a = a, a v ¬ a exists and =1 | weak invertibility |
a ^ ¬ a = 0 needs not be postulated, it follows automatically.
An orthomodular poset is an orthocomplemented poset fulfilling
| ~d') | a « b => a v ( ¬ a ^ b) = b |
|---|
Example:
Every ortholattice is an orthocomplemented poset.
Every orthomodular lattice is an orthomodular poset.
A probability measure on Q is a mapping s : Q -> [0,1] with s(Omega)=1 and s(A union B)=s(A)+s(B) whenever A and B are disjoint.
Q constitutes a Boolean algebra via
Definition: A state on a Boolean algebra Q is a mapping s from Q to the real interval [0,1] with s(1)=1 and s(A v B) = s(A)+s(B) whenever A and B are orthogonal.
Theorem: It follows s(0)=0, and s is order-preserving:
Theorem: Every Boolean algebra may be represented by some set algebra Q over an appropriate base set Omega.
So, what we achieved is an equivalent (!) reformulation of classical probability theory as a Boolean algebra Q and a set of states on it
Theorem (Gleason [10]): If H has dimension greater than two, then every state is obtained in such a way.
This time we got an equivalent (!) reformulation of Quantum probability theory as an orthomodular lattice Q := P and a set of states on it
Definition: A Quantum Logic is a tuple (Q,S) with Q some orthomodular lattice (called questions or propositions) and S some (separating) set of states on Q.
"Quantum theory is simply the replacement in standard probability theory of event-as-subset-of-a-set (abelian, distributive) by event-as-subspace-of-a-Hilbert-space (non-abelian, non-distributive)"
A.R.Marlow in [4]
| a) | a ^ b = b ^ a | a v b = b v a | commutativity |
|---|---|---|---|
| b) | (a ^ b) ^ c = a ^ (b ^ c) | (a v b) v c = a v (b v c) | associativity |
| c) | (a v b) ^ a = a | (a ^ b) v c = a) | absorption |
| d) | a ^ (b v c) = (a ^ b) v (a ^ c) | distributive | |
| e) | a v ¬a = 1 a ^ ¬a = 0 ¬¬a = a | invertibility | |
| d') | a « b => ¬ a » ¬ b | compatibility | |
| ~d) | a « c => a v (b ^ c) = (a v b) ^ c | modularity | |
| ~d') | a « b => a v ( ¬ a ^ b) = b | orthomodularity | |
| e') | ¬ ¬ a = a, a v ¬ a exists and =1 | weak invertibility | |
The following sketch depicts the different categorial structures
considered above. Read as follows:
An orthomodular lattice has to fulfill axioms
a), b), c), d'), ~d') and e). It is connected downwards
to the category of lattices, so every orthomodular lattice is
a lattice. It is not connected to the modular lattice,
meaning both: There exist orthomodular lattices which are not
modular and there are modular lattices that cannot be
turned into an orthomodular lattice by a suitable
orthocomplementation.
