Quantum probability from a geometrical interpretation of a wave function
Home > Quantum mechanics > Event theory
2019/02/17
Published 1999/11/21^{[1]}
K. Sugiyama^{[2]}
The probabilistic prediction of quantum theory is mystery. I solved the mystery by a geometrical interpretation of a wave function.
Figure 41: The number of elementary events is the square of the number of elementary states.
This suggests the unification between quantum theory and the theory of relativity. This suggests ManyWorlds Interpretation is true, too.
CONTENTS
2.1 An assumption of "no boundary"
2.2 An assumption of "a minimum unit"
2.3 An assumption of "a wave space"
2.4 An assumption of "a phase space"
2.5 An assumption of "an elementary state"
2.6 An assumption of "an elementary event"
There was no theory for ManyWorlds Interpretation, which can provide the probabilistic prediction of quantum theory^{[3]}. To obtain the probabilistic prediction of quantum theory, I suppose some assumptions and construct a theory for ManyWorlds Interpretation. I call the theory ManyEvents Theory.
There are some physical quantities, position x, time t or phase θ, etc. I call these physical quantities "Position quantity." I suppose that all position quantities have no boundaries and are compact.
I suppose that all position quantities have a common minimum unit u.
I suppose new another position quantity w. I call a space of the quantity a wave space. This quantity has something to do with a wave function.
I suppose a phase space θ have a structure like a Mobius strip. This is analogous with a spin of an electron. Thus, I define a wave function

(2.1) 
where R_{w} is the radius of curvature of a wave space and R_{θ} is the radius of curvature of a phase space. R_{w}, R_{θ} and θ are functions of θ, x and t.
I suppose that there is a minimum unit of a state. I call this an elementary state. For example in the quantum theory one particle’s elementary state is

(2.2) 
We can point the elementary state using a combination of position quantities. I suppose that same elementary state is only one. Thus, we can interpret it as a point in a space. A state is connected to a set of points.
I call the transition from one elementary state to another elementary state an elementary event. I suppose that same elementary event is only one. Thus, we can interpret an elementary event as one line in a space. An event is connected to a set of lines.
All possible elementary events exist. This is analogous with path integrals.
To make the problem easy I think one particle. And, I use w, θ, x and t. I construct a torus,

(3.1) 
And, I define a point f on the torus,

(3.2) 
R_{w} is linear like a wave function approximately. If there is a wave on the torus, an effect of one loop in a phase space and an effect of two loops cancel out. Because the phase space has a structure like a Mobius strip. Thus, R_{w} has properties like a wave function.
The shape of the torus is decided by the Hamiltonian H of the system. To describe many particles, we need a new position quantity. This is analogous with second quantization.
A point on the torus is an elementary state f (w, θ, x, t). We can count elementary states since a position quantity has a minimum unit. The number of elementary states N (θ, x, t) is

(3.3) 
where u is a minimum unit and f (w, θ, x, t) = 1.
The probability of quantum theory is the probability of an event. We consider only the number of different elementary events since we measure only one elementary event of many elementary events.
I suppose that

(4.1) 
and

(4.2) 
Then, we can obtain the number of different combinations of the elementary states, m^{2} since the time has minimum unit. The combinations are elementary events. If there are only m different elementary states, there are m^{2} different elementary events.
Figure 41: The number of elementary events is the square of the number of elementary states.
If there is a state , the probability of finding a particle at the position x, P (x) is

(4.3) 
Thus, we can obtain the probability from this theory. Each elementary event exists. Thus, each observer measuring each elementary event exists, too. This is ManyWorlds Interpretation.
This theory provides the probability of quantum theory.
[1] K. Sugiyama (1999), "Quantum probability from a geometrical interpretation of a wave function", (http://arxiv.org/abs/quantph/9910108).
[2] Mail: sugiyama_xs@yahoo.co.jp
[3] A. Kent, "Against ManyWorlds Interpretations", Int. J. Mod. Phys. A5 (1990) 1745, (http://arxiv.org/abs/grqc/9703089).