In order to prevent losing the meaning of contrary or reference, in **4D space wave function**, we engage that space and momentum, one is 4D space physical quantity, then the other is 3D space physical quantity. This is called the **1/2 Rule or the Special Engagement**.

Blow is not something new. It's just a previous post. I move it here and make minor amendment to make the concept of the "hidden variables" a bit more explicit.

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The position space wave function for a free particle in 4D space is:

Ψ(X, t) = A exp( i(P ·X – Et )) Amplitude A is a constant. (omit “ℏ” and vector marks for convenience of observation)

X = X4x

Then, the probability density｜A｜² is a constant. It means the probability of appearance of the particle in any point of 4D position space is the same. In turn, it means the probability of the X4 state the free particle be in is the same. The situation is the same in any position x in the 3D space.

Because it’s a free particle, no interaction is concerned, the above analysis reflects the property of the particle itself regardless it is moving with any momentum P in the 3D space or not (It has nothing to do with wavelength λ = h / P ).

The momentum space wave function for a free particle in 4D space is:

Ψ(P, t) = A exp( i(P·x – Et )) Amplitude A is a constant. (omit “ℏ” and vector marks for convenience of observation)

P = X4 P

Then, the probability density｜A｜² is a constant. It means the probability of appearance of the particle in any point of 4D momentum space is the same. In turn, it means the probability of the X4 state the free particle be in is the same. The situation is the same in any position x in the 3D space. Because it’s a free particle, no interaction is concerned, the above analysis reflects the property of the particle itself regardless it is moving with any momentum P in the 3D space or not (It has nothing to do with wavelength λ = h / P ).

Look at the 3D space wave function for free particle in QM again.

Ψ = A exp( i(p•x– Et )) omit “ℏ” and vector marks for convenience of observation.

When position x and momentum p are both determined at the same time, it will no longer be a wave function, instead, an oscillation function. It seems that the “wave character” should disappear either.

What interesting is it could be considered as a special situation and the calculation of probability is still applicable. “A” is a constant. ｜A｜² is a constant. It means the probability of appearance of the particle in any space point is the same. In turn, it means the position of the free particle can not be determined.

Contradiction…

Next, resort to 4D space wave function. In the item P ·X or P·x, among space and momentum, if one physical quantity is 3D space quantity, then the other is 4D space quantity. It reflects that they two can not be determined simultaneously and avoid the dispute.

Classical physics describes the certainty in 3D space while QM describes the uncertainty in 4D space.