SUPERPOSITIONS AND

PROBABILITIES – QUANTUM THEORY

WITHOUT IDEOLOGY

“The fact that an adequate philosophical

presentation has been so long delayed is no

doubt caused by the fact that Niels Bohr

brainwashed a whole generation of theorists

into thinking Ref. 82 that the job was done fifty years

ago. Murray Gell-Mann”

Why is this famous physical issue arousing such strong emotions? In particular,

ho is brainwashed, Gell-Mann, the discoverer of the quarks, or most of the

orld’s physicists working on quantum theory who follow Niels Bohr’s opinion?

In the twentieth century, quantum mechanics has thrown many in disarray. Indeed, it

radically changed the two most basic concepts of classical physics: state and system.The

state is not described any more by the specific values taken by position and momentum,

but by the specific wave function ‘taken’ by the position and momentum operators.* In

addition, in classical physics a system was described as a set of permanent aspects of nature;

permanence was defined as negligible interaction with the environment. Quantum

mechanics shows that this definition has to be modified as well.

The description of nature with quantum theory is unfamiliar for two reasons: it allows

superpositions and it leads to probabilities. Let us clarify these issues. A clarification is

essential if we want to avoid getting lost on our way to the top ofMotionMountain, as

happened to quite a number of people since quantum theory appeared, including Gell-

Mann.

Why are people either dead or alive?

The evolution equation of quantum mechanics is linear in the wave function; linearity

implies the existence of superpositions. Therefore we can imagine and try to construct

systems where the state ψ is a superposition of two radically distinct situations, such as

those of a dead and of a living cat.This famous fictional animal is called Schrödinger’s

cat after the originator of the example. Is it possible to produce it? And how would it

evolve in time?We can ask the same two questions in other situations. For example, can

we produce a superposition of a state where a car is inside a closed garage with a state

where the car is outside?What happens then?

Such strange situations are not usually observed in everyday life.The reason for this

rareness is an important aspect of what is often called the ‘interpretation’ of quantum

mechanics. In fact, such strange situations are possible, and the superposition of macroscopically

distinct states has actually been observed in a few cases, though not for cats,

people or cars. To get an idea of the constraints, let us specify the situation in more detail.*

Th

e object of discussion are linear superpositions of the type ψ = aψa + bψb, where

ψa and ψb are macroscopically distinct states of the system under discussion, and where

a and b are some complex coefficients. States are called macroscopically distinct when

each state corresponds to a different macroscopic situation, i.e., when the two states can

be distinguished using the concepts ormeasurement methods of classical physics. In particular,

thismeans that the physical action necessary to transform one state into the other

must be much larger than ħ. For example, two different positions of any body composed

of a large number of molecules are macroscopically distinct.

A ‘strange’ situation is thus a superposition of macroscopically distinct states. Let us

work out the essence of macroscopic superpositionsmore clearly. Given two macroscopically

distinct states ψa and ψb, a superposition of the type ψ = aψa + bψb is called a pure

state. Since the states ψa and ψb can interfere, one also talks about a (phase) coherent superposition.

In the case of a superposition of macroscopically distinct states, the scalar

product ψ†

aψb is obviously vanishing. In case of a coherent superposition, the coefficient

product a∗b is different from zero. This fact can also be expressed with the help of the

density matrix ρ of the system