quantum logic

quantum logic

Introduction

In classical physics, results regarding experiments performed on physical

propositions

obey classical

propositional logic

. When a ball is tossed, one can, at any given time

t

,

measure

its position

x

⁢

(

t

)

and velocity

v

⁢

(

t

)

with great precision almost “simultaneously”. If we know where the ball is, then we know, at that location

x

⁢

(

t

)

, whether it is moving at

v

⁢

(

t

)

or not.

Let’s

translate

the above

sentence

into a logical statement. Suppose

p

is the statement that “the ball is at

x

⁢

(

t

)

”, and

q

is the statement that “the ball is traveling at velocity

v

⁢

(

t

)

”. Then last sentence in the previous paragraph becomes

if

p

is true, then either

p

∧

q

is true or

p

∧

q

′

is true.

Put it more succinctly, we have the

tautology

p

→

(

p

∧

q

)

∨

(

p

∧

q

′

)

.

Converted to the

language

of

Boolean algebra

(via Lindenbaum’s construction (

http://planetmath.org/LindenbaumAlgebra

)), it says

p

≤

(

p

∧

q

)

∨

(

p

∧

q

′

)

which is clearly true, since the right hand side is equal to

p

(the

distributive property

of a Boolean algebra).

However, when we downsize the scale of the system (the ball) to something at the subatomic level, say, an electron, we enter the realm of non-classical physics (quantum mechanics), the story is much different. Because of the uncertainty principle, the nature of being able to measure everything simultaneously is gone here. Even if we know the location

x

⁢

(

t

)

, we will not be able to know its velocity

v

⁢

(

t

)

. In other words, the above inequality no longer holds. A new logic is needed for reasoning in non-classical physics, in particular,

quantum theory

.

In the 1930’s, Birkhoff and von Neumann introduced a new kind of logic in dealing with quantum mechanics in their paper ”The logic of quantum mechanics”. In that paper, physical constructs were converted into mathematical ones. For example, a physical observable is nothing more than a

Hermitian operator

acting on an

infinite dimensional

Hilbert space

over the

complex numbers

. And it turns out that many of the

concepts

quantum mechanics can be explained in terms of doing math on

operators

in a Hilbert space. Furthermore, the logic of quantum mechanics corresponds to that of the set of closed subspaces of an infinite dimensional Hilbert space. This set is a

lattice

and it is not a Boolean algebra, because its lack of distributivity, and therefore does not correspond to classical propositional logic. They tried to axiomatize this lattice as a

complete

complemented

modular lattice

(with atoms). It was soon realized (by Piron) that this lattice is not even

modular

.

Definition

Today, there are several inequivalent

definitions

of

quantum logic

, but all of which are in one form or another a

generalization

of a Boolean algebra (the algebraic form of classical logic). Initially, a quantum logic is none other than the

lattice of projection

operators over an

infinite-dimensional

separable Hilbert space. Since its

introduction

, the definition has been generalized, stripping away unnecessary details (atomicity for example) while keeping the essentials.

One of the more general definitions says that a

quantum logic

is an

ordered pair

(

L

,

F

)

, where

L

is an

orthomodular poset

L

, and

F

is a set of

functions

on

L

, called

states

, taking values in the unit inverval

[

0

,

1

]

, satisfying the following conditions:

1.

f

⁢

(

0

)

=

0

for each

f

∈

F

,

2.

f

⁢

(

1

)

=

1

for each

f

∈

F

,

3.

if

a

≤

b

⟂

, then

f

⁢

(

a

∨

b

)

=

f

⁢

(

a

)

+

f

⁢

(

b

)

for each

f

∈

F

4.

if

f

⁢

(

a

)

≤

f

⁢

(

b

)

for each

f

∈

F

, then

a

≤

b

. We say that

F

is

full

.

One can think of

L

as a physical system (an electron, for example), and elements of

L

are

propositions

concerning the system.

F

can be regarded as a set of probability measures on the propositions. Conditions 1 and 2 say that

1

and

0

are propositions of

L

that are always true or always false, regardless of its states. Condition 3 is

equivalent

to the well-known law of finite additivity of a probability measure.

Remark

. One can define the notion of

compatibility

of a pair of

elements

(propositions) in a quantum logic:

a

,

b

∈

L

are

compatible

if they

orthogonally commute

, that is, there exist pairwise

orthogonal

elements

c

,

d

,

e

∈

L

such that

a

=

c

∨

e

and

b

=

d

∨

e

. It turns out that compatible propositions exactly

parallel

those physical propositions which can be experimentally tested “simultaneously”. A quantum logic

L

in which every two propositions are compatible is said to be a

classical logic

. It is not hard to show that a classical logic is a Boolean algebra.

References

1

L. Beran,

Orthomodular Lattices, Algebraic Approach

, Mathematics and Its Applications (East European Series), D. Reidel Publishing Company, Dordrecht, Holland (1985).

2

G. Birkhoff,

Lattice Theory

, 3rd Edition, AMS Volume XXV, (1967).

3

G. Birkhoff, J. von Neumann,

The logic of quantum mechanics

, Ann. of Math. 37, pp. 823-843, (1936).

4

D. W. Cohen,

An Introduction to Hilbert Space and Quantum Logic

, Springer, (1989).

5

P. Mittelstaedt,

Quantum Logic

, D. Reidel Publishing Company, (1978).

Title

quantum logic

Canonical

name

QuantumLogic

Date of creation

2013-03-22 16:49:09

Last modified on

2013-03-22 16:49:09

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

10

Author

CWoo (3771)

Entry type

Definition

Classification

msc 03G12

Synonym

compatible propositions

Related topic

LatticeOfProjections

Related topic

QuantumLogicsTopoi

Related topic

TopicEntryOnFoundationsOfMathematics

Related topic

ETAS

Related topic

QuantumTopos

Related topic

QuantumGroupsAndVonNeumannAlgebras

Defines

state

Defines

full set of states

Defines

compatible

Defines

classical logic

Generated on Thu Feb 8 20:27:04 2018 by

LaTeXML