# Definition of Complete

April 12, 2022

A complete theory must corresponding to each element of reality. It can predict reality with certainty, without disturbing the system.

In quantum mechanics, in the case of two physical quantities described by non-commuting operators, the knowledge of one precludes the knowledge of the other.

This means that either:

1. The description of reality given by the quantum wave function is not complete or
2. These two quantities cannot have simultaneous reality

Quantum mechanics makes predictions in one system based of measurements made on another system that had previously interacted with it. It leads to the result that if (1) is false then (2) is also false.

Thus, the description of reality as given by a wave function is not complete theory.

Any serious physical theory must take into account the distinction between=

• the reality which is independent of any theory
• the concepts in the theory
• These concepts must correspond wit reality and give us a picture of this reality to ourselves.

A successful physical theory is one that is=

• correct
• its descriptions are complete

“Complete” means that every element of reality must have a counterpart in the theory.

What are the elements of a complete theory?

These cannot be determined by a priori philosophical considerations. Instead, it must be based on experiments and measurements. Only real, measureable experiences can be used in such a theory.

We do not need a comprehensive deﬁnition of reality. Our criterion for reality is simply if, without disturbing a system, we can predict with certainty (i.e., with probability equal to unity) a physical value, then that value has physical reality.

This criterion provides us with one way of recognizing physical reality. This agrees with both=

• quantum mechanics
• classical mechanics

For example, there is a particle with a single degree of freedom.

The fundamental concept of quantum theory is state, completely characterized by the wave function .. which is a function of the variables chosen to describe the particle’s behavior. Each physical value A has an operator, which may be designated by the same letter.

If `Ψ` is an eigenfunction of the operator `A`, that is if:

```````Ψ' = AΨ = aΨ`
``````

where `a` is a number, then the physical value `A` has with certainty `a` whenever the particle is in the state given by `Ψ`.

In accordance with our criterion of reality, for a particle in the state given by `Ψ` for which Eq. (1) holds, there is an element of physical reality corresponding to the physical value `A`.

For example=

``````Ψ = e^(2πi/h)p0x             (2)
``````
• h is Planck’s constant
• p0 is some constant number
• x the independent variable

Since the operator corresponding to the momentum of the particle is=

``````p = (h/2πi)d/dx                (3)
``````

we obtain

``````Ψ’= pΨ = (h/2πi)dΨ/dx = p0Ψ          (4)
``````

Thus, in Eq. 2’s state, the momentum has certainly `p0`. The particle’s momentum in the state given by Eq. (2) is real.

On the other hand, if Eq. (1) does not hold, thn `A` no longer has a particular value. This is the case, for example, with the coordinate of the particle.

The operator corresponding to it, such as `q`, is the operator of multiplication by the independent variable. Thus,

``````qΨ = xΨ ≠ aΨ                     (5)
``````

In line with quantum mechanics, the relative probability that a measurement of the coordinate will give a result lying between `a` and `b` is=

``````P (a, b) = ∫ab ΨΨdx = ∫ dx = b - a
``````

## Einstein Doesn’t Want a Universe That He Cannot Predict

Since this probability is independent of `a`, but depends only on the difference `b—a`, we see that all values of the coordinate are equally probable.

A deﬁnite value of the coordinate, for a particle in the state given by Eq. (2), is thus not predictable. It may be obtained only by a direct measurement. Such a measurement however disturbs the particle and thus alters its state. After the coordinate is determined, the particle will no longer be in the state given by Eq. (2).

From this, quantum mechanics concludes that when a particle’s momentum is known, its coordinate has no physical reality.

Quantum mechanics shows that if the operators corresponding to two physical quantities, say `A` and `B`, do not commute AB=BA, then the precise knowledge of one of them precludes such a knowledge of the other.

Any attempt to determine the latter experimentally will alter the state of the system in such a way as to destroy the knowledge of the ﬁrst.

This means that either:

1. The quantum-mechanical description of reality given by the wave function is incomplete or
2. When the operators corresponding to two physical quantities do not commute, the two quantities cannot have simultaneous reality. If both of them had simultaneous reality—and thus deﬁnite values—these values would enter into the complete description, according to the condition of completeness.

If then the wave function provided such a complete description of reality, it would contain these values. These would then be predictable. This not being the case, we are left with the alternatives stated.

Quantum mechanics assumes that the wave function does contain a complete description of the physical reality of the system in the state to which it corresponds.

At ﬁrst sight, this assumption is entirely reasonable. The information in a wave function seems to correspond exactly to what can be measured, without altering the state of the system.

But this contradicts with my criterion of reality [perfect predictability].