Derivation of the Born rule from manyworlds interpretation and probability theory
Home > Quantum mechanics > Derivation of Born rule of quantum mechanics
2019/02/20
Published 2012/09/26
K. Sugiyama^{[1]}
Born rule is a rule that a probability we observe a small particle like an electron is proportional to the square of the absolute value of the wave function. In this paper, we try to derive the Born rule from the manyworlds interpretation.
Figure 36: Derivation of Born rule by the network structure of the path integral
Many researchers have tried to derive the Born rule (also called Born’s rule, Born's law, or probability interpretation) from manyworlds interpretation (MWI). However, nobody succeeded. Thus, the derivation of Born rule had become an important issue for MWI. We try to derive Born rule by introducing an elementaryevent (also called an atomic event or simple event) of probability theory to the quantum theory as a new method.
The absolute value of a wave function is regarded as a surface area of a manifold. A position on the surface of the manifold is regarded as an elementaryposition. A particle existing at the elementaryposition is regarded as a stateelement. The state that the stateelement existing at the elementaryposition is regarded as an elementarystate. A transition from one elementarystate to the other elementarystate is regarded as an elementaryevent.
A probability is proportional to the number of elementaryevents, and the number of elementaryevents is the square of the number of elementarystates. The number of elementarystates is proportional to the surface area of the manifold, and the surface area of the manifold is an absolute value of a wave function. Therefore, the probability is proportional to a square of an absolute value of a wave function.
CONTENTS
1.2 The importance of the subject
1.4 New derivation method of this paper
2 Traditional method of deriving and the problem
2.2 Everett's manyworlds interpretation
2.2.1 The basis problem of manyworlds interpretation
2.2.2 The probability problem of manyworlds interpretation
3.1 Elementaryevent of manyworlds interpretation
3.2 Elementarystates of manyworlds interpretation
3.3 Introduction of a elementarystate to the quantum theory
3.4 Application of path integral to the field
3.5 Introduction of elementaryevent to the quantum theory
5.1 Supplement of the manyparticle wave function
5.2 Supplement of the method of deriving Born rule
5.3 Supplement of basis problem in manyworlds interpretation
5.4 Interpretation of time in manyworlds interpretation
5.5 Supplement of long distance transition
7 Consideration of the future issues
7.1 Consideration of principles
7.2 Consideration of formulation for the quantum field theory
8 Appendix: Review of existing ideas
8.1 Universal Wave function of Wheeler and DeWitt
8.2 Barbour's manyparticle wave function of the universe
8.3 Laplace's Probability Theory
8.6 Dirac's quantum field theory
9 Appendix: Review of existing ideas (part 2)
9.4 CauchyRiemannFueter equation
9.5 Cartan's differential form
10 Appendix: Hierarchical universe
10.1 Universe of Twodimensional spacetime
10.1.1 Closed path of Twodimensional spacetime
10.1.2 Introduction of the absolute value of wave function
10.1.3 Introduction of the phase of wave function
10.2 Universe of fourdimensional spacetime
10.2.1 Closed path of fourdimensional spacetime
10.2.2 Introduction of the absolute value of wave function
10.2.3 Introduction of the phase of wave function
According to Born rule, an observed probability of a particle is proportional to the square of the absolute value of the wave function. It is the subject of this paper to derive Born rule by counting the number of the events.
Wave function collapse and Born rule are the principle of the quantum mechanics. We can eliminate the wave function collapse from the quantum mechanics by manyworlds interpretation (MWI), but we cannot eliminate Born rule. For this reason, many researchers have tried to derive Born rule from MWI. However, it has not resulted in the success. Therefore, it has become an important subject to derive Born rule.
Hugh Everett III^{[2]} claimed that he derived Born rule from manyworlds interpretation (MWI) in 1957. After that, many researchers claimed that they derived Born rule from the method that is different from the method of Everett. James Hartle^{[3]} claimed in 1968, Bryce DeWitt^{[4]} claimed in 1970 and Neil Graham^{[5]} claimed in 1973 that they derived Born rule. However, Adrian Kent pointed out that their method of deriving Born rule was insufficient^{[6]} in 1990. David Deutsch^{[7]} tried to derive Born rule.in 1999. Sumio Wada^{[8]} also tried it in 2007. However, many researchers do not agree on the methods of deriving Born rule.
Max Born^{[9]} proposed the Born rule in 1926. It is also called Born’s rule, the Born law, the Born's law, or the probability interpretation. Born rule is a principle of quantum mechanics. We express the state of the particle by the wave function ψ (x) in quantum mechanics. We show an example of the wave function in the following figure.
Figure 21: An example of a wave function
The observed probability of a particle is proportional to the square of the absolute value of the wave function. We express the observed probability P (x) of a particle at the position x as follows.

(2.1) 
According to the Copenhagen interpretation that is a general interpretation of quantum mechanics, we cannot describe the state of the particle before the observation because the wave function does not exist physically. However, the wave function might exist physically. One of the interpretations based on the existence of a wave function is manyworlds interpretation.
Everett proposed manyworlds interpretation (MWI) in order to deal with the universal wave function. He tried to derive Born rule from the measure.
For example, we consider the SternGerlach experiment. We can express the state of spin in any direction by superposition of spinup state along zaxis and spindown state along zaxis. Therefore, we express the state vector of the spin of an electron by the eigenstate vector of the spin along the zaxis as follows.

(2.2) 
The coefficients a and a_{k} are complex number. Here, we have normalized ψ>, z+> and, z−> as follows.

(2.3) 

(2.4) 

(2.5) 
In order to derive the probability, Everett introduced a new concept, measure. He expressed the measure by a positive function m(a). He requested the following equation for the measure.

(2.6) 
He adduced a probability conservation law to justify the request. We write the function m(a) satisfying the above equation by using a positive constant c as follows.

(2.7) 

(2.8) 

(2.9) 
Andrew Gleason^{[10]} generally proved the above equation in 1957. His proof is called the Gleason's theorem. Everett considered the infinite time measurement, and concluded that the measure behaves like the probability. However, MWI of Everett has a basis problem and a probability problem. I will explain them in the following sections.
If we define the measure by using a particular basis, we need to show how to select a particular basis. However, Everett did not show how to select a particular basis in his paper.
We can express the wave function of an electron by the basis of the spin eigenstate along zaxis as follows.

(2.10) 
We can express the measure of the spin eigenstate along zaxis as follows.

(2.11) 

(2.12) 
On the other hand, we can also express the wave function of an electron by the basis of the spin eigenstate along xaxis as follows.

(2.13) 
We can express the measure of the spin eigenstate along xaxis as follows.

(2.14) 

(2.15) 
If the measure of Everett is a quantity that exists physically, it should not change by choice of a basis of eigenstate, because a quantity that exists physically does not depend on the method to observe it. Therefore, we need a method to choose a specific basis. However, Everett did not show the method.
Everett tried to derive Born rule from the measure theory. Then, Everett did not give the physical meaning to the measure. However, to request the conservation law of the probability for the equation of the measure is equivalent to defining the measure as the probability. Therefore, it is circular reasoning to show that measure acts like a probability for infinite time measurement.
If the number of worlds is proportional to the measure, it is necessary to clarify the mechanism that the number of worlds is proportional to the measure. If the number of worlds is not proportional to the measure, it is necessary to explain how the probability of occurrence of a world is proportional to the measure.
We introduce elementaryevents to quantum mechanics by embracing the manyworlds interpretation (MWI) in this paper.
If we interpret an event of quantum theory as a set of elementaryevents, we can derive the probability of an event by counting the number of elementaryevents that belong to the event. If the event R or event B occurs in some observations, a history branches to the history that event R occurs and the other history that event B occurs in the manyworlds interpretation.
For example, if the number of elementaryevents of the event R is three, and the number of elementaryevents of the event B is two, the probability of occurrence of the event R is 3/5.
Figure 31: A history is a set of elementary histories in manyworlds interpretation
The concrete implementation method of elementaryevents is described in the following sections.
In the wave function of a manyparticle system in configuration space (manyparticle wave function), we call the state (position eigenstate) that positions of all particles are decided a position certainstate, or certainstate.
In the actual experiment, a particle spreads in the narrow range. Therefore, actual state diffuses in the narrow range in configuration space. We can regard the state as the set of the certainstates. We call the state a localizedstate.
Figure 32: Elementarystates in configuration space
The absolute value of a wave function is regarded as a surface area of a manifold. A position on the surface of the manifold is regarded as an elementaryposition. A particle existing at the elementaryposition is regarded as a stateelement. The state that the stateelement existing at the elementaryposition is regarded as an elementarystate.
In the discussion of this paper, there is no difference between the discussion using the manyparticle wave function and the discussion using the wave function of one particle. Therefore, in the discussion of this paper, we do not use the manyparticle wave function but the wave function of one particle.
We express the wave function ψ(x) by Dirac delta function as follows.

(3.1) 
We interpret the state ψ(x')δ(x − x') as the state that the position x of the particle is fixed at the position x', "Certainstate." The state δ(x − x') is regarded as an elementarystate. K. Sugiyama^{[11]} introduced the elementarystate in 1999.
We divided the virtual highdimensional Euclidean space by using "elementary domain" and we suppose that lattice points are elementarystates. The certainstate is 1sphere or 3sphere and the lattice points on the sphere are elementarystates of the certainstate.
Figure 33: Elementarystates of manyworlds interpretation
Since the surface area S of the manifold is the absolute value ψ (y, t) of the wave function, we describe the number M (y, t) of the elementarystates by using Planck length as follows.

(3.2) 
In quantum field theory, we apply pathintegral to field. On the other hand, the stateelements construct the field. Therefore, we apply to the stateelements.
We introduce a new wave function Ψ that has an argument of a wave function ψ.

(3.3) 
There is a network structure of the path integral for the position x that is "positional physical quantity." (See section 8.5.) Therefore, we apply a network structure of the path integral for the field ψ that is "positional physical quantity" like the following figure.
Figure 34: Application of network structure of path integral to field itself
In the above figure, we apply the "network structure of the path integral" to the region that is smaller than Δψ.
We introduce a new concept, an elementaryevent to the quantum theory in this paper.
We express an event as a transition from one state to the other state in quantum theory. Therefore, we express an elementaryevent as a transition from one elementarystate to the other elementarystate.
If the arrow from the point A to the point B exists, the arrow from the point B to the point A also exists conversely. If the number of points is M, the number of arrows becomes M^{ }^{2}. In other words, if the number of elementarystates is M, the number of elementaryevents becomes M^{ }^{2}.
Though there is no clear evidence of the existence of an elementaryevent, we deduce it by the following reasons.
Figure 35: Elementarystates and elementaryevents of manyworlds interpretation
We assume that the elementaryevent of this paper have the same properties of elementaryevents (also called an atomic event or simple event) of probability theory. In other words, the probability of occurrence of an event is proportional to the number of elementaryevents those are included in the event.
In addition, we define the event that is a transition from any Certainstate to any Certainstate "Certain event." The Certain event is a set of elementaryevents.
Actual states are localized by the uncertainty principle. We call the states localizedstates. We call the event from any localizedstate to any localizedstate a "Localizedevent."
If we apply the path integral to the discrete spacetime, the long distance transition from Certainstate occurs. However, "long distance transition" is suppressed due to the localizedstates. It means that the number of elementaryevents of Localizedevent that is Long distance transition is very rare.
The existence of elementarystates and elementaryevents suggests that an existence probability and a probability of occurrence are different concepts. If the number of elementarystates of a state is m, the state existence probability is proportional to m. If the number of elementaryevents of an event is n, the event's existence probability is proportional to n.
This section describes how to derive this probability.
We express the observation probability P(x, t) of the particle by the wave function ψ(x, t) as follows.

(3.4) 
On the other hand, we express the probability P based on the Laplace's calculation method of the probability as follows.

(3.5) 
Here N_{a} is the number of all elementaryevents and N is the number of elementaryevents those are expected. If N_{a} is sufficiently larger than N, the probability P is proportional to N.

(3.6) 
Actual state is localizedstate. We apply the "network structure of path integral" to the localizedstate. Since "long distance transition" does not occur for localizedstate, the length of transition is small after a minimum time t_{P}.
The number M (x', t') of elementaryevents of the localizedstate ψ (x', t') Δx is proportional to the surface area of the manifold. We apply the "network structure of path integral" to the position on the surface area of the manifold.
Since the manifold after the minimum time almost same as the original manifold, we approximate it by the same manifold. We express the number M (x, t) of elementarystates on the surface area of the manifold as follows.

(3.7) 
Therefore, the number of the elementarystates is proportional to the absolute value of the wave function if the Δx is almost constant.

(3.8) 
According to the uncertainty principle, the deviation Δp of momentum is almost constant if Δx is almost constant. Therefore, the number M of the elementarystates approximately does not change for Planck time t_{P}.

(3.9) 
The number N of elementaryevents is the number of transitions from all elementarystates at time 1 to all elementarystates at time 2 after Planck time t_{P}. Therefore, the number N of elementaryevents is the square of the number M of elementarystates.

(3.10) 
We express those elementaryevents in the following figure.
Figure 36: Derivation of Born rule by the network structure of the path integral
The probability P is proportional to the number of the elementaryevents N.

(3.11) 
On the other hand, the number N of the elementaryevents is equal to the square of the number M of elementarystates.

(3.12) 
The number M of elementaryevents is proportional to the absolute value of the wave function.

(3.13) 
Therefore, the probability P is proportional to the square of the absolute value of the wave function.

(3.14) 
We derived the Born rule by the above method.
^{ }
We explained the method to derive Born rule from manyworlds interpretation and probability theory.
Probability is proportional to the number of elementaryevents. The number of elementaryevents is the square of the number of elementarystates because we apply the "network structure of path integral" to the elementarystate. The number of elementarystates is proportional to the absolute value of the wave function. Therefore, the probability is proportional to the absolute value of the wave function.
We call an elementarystate, a certainstate and a localizedstate for the universe "worldelement," "certainworld" and "localizedworld" respectively.
In addition, we call an elementaryevent, a certainevent and a localizedevent for the universe "elementaryhistory", "certain history" and "localizedhistory" respectively.
Figure 51: Elementaryhistory and worldelement of manyworlds interpretation
We interpret one point of the configuration space of the manyparticle wave function as the state that the positions of all particles are determined. The state is "certainworld."
In the view of classical mechanics, the point is our world. In the view of the quantum mechanics, localizedworld is our world.
I guess that the absolute value of the manyparticle wave function of the universe is most nearly zero in the almost area. The domain that the absolute value is large is localized like a network structure.
The simplest way to derive Born rule from manyworlds interpretation (MWI) is that we connect the number of worlds to the probability.
If the probability of occurrence of event A is higher than the probability of occurrence of event B, we deduce that the number of worlds those event A occurred is greater than the number of worlds those event B occurred.
For example, we suppose that we make the 100 planets those are exactly same as Earth. If the event A occurred on 80 planets and the event B occurred on 20 planets, then we interpret that the probability of the occurrence of the event A is 80%.
However, it is not clear how to count the world. Therefore, we count the number of worldelements of the localizedworld that event A occurred.
We express the number M of worldelements of the localizedworld by the wave function ψ that event A occurred as follows.

(5.1) 
Δx is the position deviation, and n is the number of all particles. The number of worldelements is proportional to the absolute value of the wave function.

(5.2) 
On the other hand, the probability is proportional to the square of the absolute value of the wave function. Therefore, we cannot explain the probability by using the number of the worldelements.
To solve this problem, we explain the probability by using the number of histories. We express the number N of elementary histories of the localizedhistory that event A occurred as follows.

(5.3) 
The probability is proportional to the number of elementary histories.

(5.4) 
The number of histories is the square of the number of worldelements.

(5.5) 
On the other hand, the number of the worldelements is proportional to the absolute values of the wave function.

(5.6) 
Therefore, the probability is proportional to the square of the absolute value of the wave function.

(5.7) 
In manyworlds interpretation, the absence of a particular basis of the wave function is a problem.
For example, we consider the SternGerlach experiment of the spin of electrons. In this experiment, we measure the spin by using a magnetic field gradient. Since the basis of the spin is determined by the direction of the gradient magnetic field, there is no particular basis for the spin.
In this paper, we chose position as the particular basis. We could also choose the momentum as the particular basis, but we did not do so, because we express the basis of the momentum by using a set of elementarystates those the positions are basis.
For spin, there is no way to select a particular basis. Therefore, we encounter the problem that the measure changes by choosing the basis of spin. The cause of the problem is in an interpretation that one elementary state construct a state that has a particle of spinup.
I showed that we can derive the spin from the rotation of 3sphere in the following paper.
・Derivation of twovaluedness and angular momentum of spin1/2 from rotation of 3sphere [2013/5]
https://xseekqm.net/Spin_e.htm
If the spin is rotation 3sphere, we can interpret that many elementary states construct a state that has a particle of spinup. Choosing the basis of spin is equivalent to changing the observation axis of the 3sphere. The observation decide a set of elementary states those belong to spinup state and a set of elementary states those belong to spindown state. Therefore, we can solve the basis problem of MWI by the rotation of 3sphere.
The position of all particles is different for a point in the configuration space of manyparticle wave function. Therefore, we define the time for a point in the configuration space. Since a point corresponds to a certainworld, we interpret the time as a parameter to classify the certainworlds.
A certainworld transits the minimum length continuously in the configuration space. I guess that we feel the transition as a time.
Figure 52: Manyworlds interpretation and arrow of
time
If a transition of a direction exists, the transition of the opposite direction also exists. However, since there are many "worldelements" of future more overwhelmingly than the number of worldelements of past, we feel that our worldelement always transits to worldelement of the future. In this way, manyworlds interpretation explains the arrow of time by.
In this paper, we have been thinking about one particle is localized in one place. Here we consider the wave function of one particle that was localized in one place at a time. We suppose that the wave function was separated and localized in two places. We call the state "many localizedstates." In this case, what would happen?
Elementaryevent exists between any two elementarystates. The world does not become disorder because long distance transition is suppressed due to the "localizedstate." We determine the number of elementaryevents between two localizedstates only by the number of elementarystates of the two localizedstates.
Therefore, if there are "many localizedstates," the transition between two localizedstates will occur.
Figure 53: Long distance transition between localizedstates
I call the phenomenon "localized long distance transition" or "localized shift."
Then, will localized shift between localizedstates those have different time occur?
In this case, since the elementaryevent exists between any two elementarystates, the localized shift occurs, too.
I do not deduce that the localized teleport send information, because we cannot send any information by using EPR correlation.
Observation is an operation to transform uncertain information entropy Q to certain information entropy H as follows.
We define certain information entropy H for the observed information as follows.
(Certain information entropy)

(5.8) 
We define the uncertain information entropy Q for the unobserved wave function ψ (x) as follows.
(Uncertain information entropy)

(5.9) 

(5.10) 
We define uncertain information entropy Q for the unobserved angular momentum of the kth particle as follows.
(Uncertain information entropy)

(5.11) 

(5.12) 

(5.13) 

(5.14) 
We define the general information entropy G.
(General information entropy)

(5.15) 
This general information entropy conserves.
(Law of general information entropy conservation)

(5.16) 
Certain information entropy always increases by a thermodynamics second law.
(Law of entropy increase)

(5.17) 
However, certain information entropy has the following upper limit because of uncertainty principle.
(Upper limit of certain information entropy)

(5.18) 
All the particle’s angular momentums of ydirection and zdirection become uncertain, when all the particle’s angular momentums of xdirection are observed.
Certain information entropy increases when the wave function collapses.
Uncertain information entropy increases when the wave function diffuses (anticollapses).
If we make a status uncertain status, uncertain information entropy increases.
Future issues are shown as follows.
(1) Consideration of the principle
(2) Formulation for the quantum field theory
(3) Consideration of the discrete space
(4) Formulation for the relativistic mechanics
(5) Formulation for the gravity theory
We consider some of these issues in the following chapters.
We consider the hierarchical principle and the event principle.
I propose the following hierarchical principle.
(1) Wave functions are quaternionic functions.
(2) The direct product of the closed path of a particle and the wave function is the other universe.
(3) Wave functions in the other universe are also quaternionic functions.
We call the theory based on the hierarchy principle the hierarchy theory.
I propose the following event principle.
(1) An elementaryevent is the transition from an elementarystate to the other elementarystate.
(2) Event probability of an event is proportional to the number of elementaryevents those the event includes.
We call the theory based on the event principle the event theory.
A certainstate has a phase and an absolute value of the wave function. Therefore, it is possible to use “suppression of long distance transition due to localizedstates” for the certainstate. On the other hand, an elementarystate does not have a phase and an absolute value of the wave function. Therefore, it is impossible to use “suppression of long distance transition due to localizedstates” for the elementarystate. In order to solve the problem, we consider the quantum field theory.
In the quantum mechanics, the position and the momentum of a particle have a commutation relation. It means that the position of the particle is distributed. On the other hand, in the quantum field theory the amplitude and the general momentum of the wave function have a commutation relation. It means that the amplitude of the wave function is distributed.
Then I propose the following new function.

(7.1) 
We call the function the second wave function because we get the wave function by the second quantization of the field. The second wave function exists in the second universe. The elementarystate of the first universe is the certainstate of the second universe. Therefore, it is possible to use “suppression of long distance transition due to localizedstates” for the elementarystate.
John Wheeler and Bryce DeWitt^{[12]} proposed the universal wave function in 1967. We have the wave function by the Hamiltonian operator H and the ket vector ψ> as follows.

(8.1) 
This ket vector ψ> is not a normal function but a functional.
A functional is mathematically almost equivalent to a function of many variables. Since the discussion based on the functional is difficult, we use a function of many variables for discussion in this paper. The following sections describe the manyparticle wave function, which is a function of many variables.
Julian Barbour^{[13]} expressed the universe by using the manyparticle wave function in his book The End of Time in 1999.
We suppose that the number of the particles in the universe is n, and the kth particle's position is r_{k} = (x_{k}, y_{k}, z_{k}). Then we express the manyparticle wave function ψ as follows.

(8.2) 
The many dimensional space expressing the positions of all the particles is called configuration space.
Figure 81: Manyparticle wave function
The configuration space expresses all the possible worlds those exist physically in the past, the present, and the future, because a point in the configuration expresses the positions of all the particles. In other words, manyparticle wave function expresses all the possible worlds in manyworlds interpretation.
If the combination of the positions of all particles of a world is decided, the state of the clock of the world will be decided. If the state of the clock of the world is decided, the time of the clock of the world is decided. Therefore, manyparticle wave function does not need time as the argument of the function.
It is possible to choose a position or a momentum as a basis of a wave function. This paper chooses the position as a basic basis, since we always observe a position finally by an experiment.
The number of particles changes in the quantum field theory. Therefore, it is impossible to express the quantum field by the manyparticle wave function. We need a functional in order to express the quantum field. On the other hand, it is possible to express the functional by manyvariable function approximately. Then, we use manyvariable function, manyparticle wave function in order to argue easily in this paper.
We express the probability P that we observe a world in the configuration space as follows.

(8.3) 
In order to consider the reason why we express the probability by this equation, we will review the probability theory in the following section.
PierreSimon Laplace^{[14]} summarized the classical probability theory in 1814. He described the following calculation method of the probability.
The probability of an event is the ratio of the number of cases favorable to it, to the number of all cases possible when nothing leads us to except that any one of these cases should occur more than any other, which renders them, for us, equally possible.
This "equally possible" case is an elementaryevent in probability theory. All elementaryevents have a same probability of occurrence.
An elementaryevent is also called an atomic event. In this paper, we call "equally possible" case an elementaryevent.
We suppose that the number of all elementaryevents is N_{a}, and the number of elementaryevents of an event is N. Then, we express the probability P of occurrence of the event as follows.

(8.4) 

(8.5) 
For example, we suppose that the five balls are in the bag. Three of five balls are red and two balls are blue. We suppose that the probability of the event that we take out the red ball is P. Then, the probability is 3/5.
Figure 82: Event is a set of elementaryevents
We explain the reason by the concept of an elementaryevent. According to the probability theory, we interpret the event that we take out a ball as an elementaryevent. We interpret an event as a set of elementaryevents.
In order to derive Born rule, we need to find elementaryevents of quantum theory. An elementaryevent of probability theory generally we cannot divide anymore, so it is expected that an elementaryevent of quantum theory also cannot be divided anymore.
Roger Penrose^{[15]} proposed spin networks in 1971. According to the spin networks, we express the space as a graph with a line that connects a point and the other point. This graph is called spin network. Since the spacetime is discrete, the spacetime has a minimum length and minimum time.
Figure 83: Penrose's spin network
In this paper, though we do not use a spin network, we assume that spacetime is discrete as well as by this theory and the space is a graph that connects the points. In this paper, we assume that the minimum length is Planck length and the minimum time is Planck time t_{P}.

(8.6) 

(8.7) 
We call the minimum domain that is constructed by the Planck length elementary domain.
If the spacetime is discrete, we need to review the theory that has been constructed based on the continuous spacetime. Therefore, in the next section, we review what happens in the path integral in the case of discrete spacetime.
Richard Feynman^{[16]} proposed path integral in 1948. It provides a new quantization method. In the path integral, we need to take the sum of all the possible paths of the particle.
We express the probability amplitude K (b, a) from the position a to the position b as follows.

(8.8) 
The probability amplitude K (b, a) is called propagator. The symbol Dx (t) represents the sum of the probability amplitudes for all paths. We express the wave functions by the propagator as follows.

(8.9) 
In the path integral, an event that a particle moves from a position a to the other position b is made to correspond to the propagator K (b, a). We get the wave function of time t_{b} by multiplying the propagator K (b, a) to a wave function of time t_{a} and integrating it.
As shown in the following figure, there is not only a normal path of α but also the other path of β to travel long distance in a short period. Such path might have a speed that is greater than the speed of light. Since the path is contrary to the special relativity, the path is not allowed. In this paper, we call such a movement of the path long distance transition.
Figure 84: Feynman's path integral
Generally, the textbook of a path integral explains as follows.
The sum of a minutely different path near a path α becomes large. On the other hand, the sum of a minutely different path near a path β becomes small. For this reason, long distance transition is suppressed and the path β does not remain.
Then, what happens after the minimum time t_{P}? We express the propagator from a position a to a position b after a minimum time t_{P} as follows.

(8.10) 
In this paper, we assume the discrete time. Since we cannot divide minimum time any more, when the departure point and the point of arrival are decided, it cannot take a minutely different path near a path β. For this reason, we cannot suppress long distance transition and the path β remains.
Therefore, if we apply the path integral to the discrete spacetime and the position of a particle is determined like a delta function of the Dirac, long distance transition occurs after the minimum time t_{P}.

(8.11) 
Figure 85: Long distance transition in the path integral
However, we do not observe the long distance transition. We deduce the reason is that the position of the particle is distributed with a normal distribution like the following figure.
Figure 86: A wave function of a localizedstate
Therefore, position x is distributed
with deviation Δx, momentum p is also distributed with deviation Δp.
According to the Uncertainty Principle, the product of Δx and Δp
is close to the Planck constant

(8.12) 
We call the state of the wave function with a normal distribution localizedstate.
We express a wave function of a particle with momentum p as follows.

(8.13) 
We suppose that this particle has a mass m and the velocity v. The momentum is shown below.

(8.14) 
We get the following formula by substituting this formula to the wave function.

(8.15) 
We express the velocity v by the moving distance x and the Planck length t_{P}.

(8.16) 
We get the following formula by substituting this formula to the wave function.

(8.17) 
From the above formula, the wavelength of the wave function is long at the short range. On the other hand, the wavelength of the wave function is short at the long range.
In the short distance, the sum of the path integral of localizedstate becomes large. On the other hand, in long distance, the sum of the path integral of localizedstate becomes small. We call this phenomenon “suppression of long distance transition due to localizedstates.”
If the state is localizedstate, the long distance transition does not occur after the minimum time t_{P}. Therefore, the localizedstate is localized near the place after the time t_{P}. For this reason, we deduce that network structure of the path integral is realized, as shown in the following figure.
Figure 87: Network structure of the path integral
In this paper, we call the network structure of “path network structure of the path integral.”
We suppose that there is an event AB that is a transition from a state A to a state B. If the state A has three positions and the state B has three positions, the event AB has 3 × 3 = 9 paths.
In "network structure of the integral path," the number of paths is the square of the number of positions. On the other hand, according to Born rule, the probability becomes the square of the absolute value of the wave function. In this paper, we discuss the similarities of these squares.
Paul Dirac^{[17]} proposed the quantum field theory to explain the emission and absorption of electromagnetic waves in 1927. We express the fundamental commutation relation^{[18]} of the quantum field theory in the case of onedimensional space as follows.

(8.18) 
Then ψ is the field and π is the conjugate operator of the field ψ. The variable x and y are positions. The function δ is Dirac's delta function.
This commutation relation is similar to the following commutation relation between position x and momentum p.

(8.19) 
This indicates that field ψ is a physical quantity that has a property similar to the position x. In this paper, we call the physical quantity “positional physical quantity.”
We got a field ψ(x) by the first quantization for the position x. On the other hand, the field ψ(x) is "positional physical quantity" like the position x. Therefore, we get a new field Ψ(x, ψ(x)) by the second quantization for the field ψ. We call the field Ψ(x, ψ(x)) the second wave function. We express the second wave function Ψ(x, ψ(x)) in the following figure.
Figure 88: The second wave function
It is possible to interpret the second wave function Ψ(x, ψ(x)) as a functional Φ [ψ(x)]. We express the functional Φ [ψ(x)] by the manyparticle wave function ψ (x_{1}, x_{2}, x_{3}, …, x_{n}) approximately. To argue a point easily, we use manyparticle wave functions by this paper.
Theodor Kaluza^{[19]} proposed in 1921 and Oskar Klein^{[20]} proposed in 1926 the extra space like a onedimensional circle, in order to unify the electromagnetic field and gravity. This theory is called KaluzaKlein theory.
We express a new space M^{4}×S^{1 }by using a normal fourdimensional spacetime M^{4} and an extra space S^{1} like a onedimensional circle as follows.

(8.20) 
Figure 89: KaluzaKlein theory
Euler published the following formula in 1748.
(Euler’s formula)

(9.1) 
Imaginary number i satisfies the following equation.

(9.2) 
We express the complex number as follows.

(9.3) 

(9.4) 
We express the complex conjugate as follows.

(9.5) 
We express the complex function as follows.

(9.6) 
We express the square of the absolute value of the complex number as follows.

(9.7) 
We use the following symbols as follows.

(9.8) 

(9.9) 
Augustin Louis Cauchy[21] introduced the following equation in 1814 for complex analysis. Riemann[22] used the following equation in 1851.
(CauchyRiemann equation)

(9.10) 
We express the above equation shortly as follows.
(CauchyRiemann equation)

(9.11) 
Cauchy introduced the following formula.
(Cauchy's integral formula)

(9.12) 
S is the contour path.
William Rowan Hamilton[23] proposed the quaternion in 1843.

(9.13) 
We express the quaternion as follows.

(9.14) 

(9.15) 
We express the quaternion conjugate as follows.

(9.16) 
We express the quaternion function as follows.

(9.17) 
We express the square of the absolute value of the quaternion as follows.

(9.18) 
We use the following symbols as follows.

(9.19) 

(9.20) 
Fueter [24] introduced the following equation in 1934 for quaternionic analysis.
(CauchyRiemannFueter equation)

(9.21) 
We express the above equation shortly as follows.
(CauchyRiemannFueter equation)

(9.22) 

(9.23) 
Fueter introduced the following formula.
(CauchyFueter integral formula)

(9.24) 
Here, S^{3 }is a threedimensional closed surface. The detail of the quaternionic analysis was described in the Anthony Sudbery’s paper[25] in 1979.
We use the following formula.
(Integral formula of quaternion)

(9.25) 
Here, u^{ }is a unit quaternion.
Please refer to the following paper for the integral formula of quaternion.
・Derivation of the reflection integral equation of the zeta function by the quaternionic analysis [2014/5/18]
https://xseekqm.net/Quaternion_e.htm
Elie Cartan^{[26]} defined differential form in 1899 in order to describe manifold by the method that is independent to the coordinates.
Though the differential form dω is infinitesimal, we use difference form δω of finitesimal
We express the surface area A of the manifold S as follows.

(9.26) 
We express the difference form δS of the surface area of the manifold S as follows.

(9.27) 
Figure 91: Manifold
Here, we express the difference form δS_{1} of the surface area of the manifold S_{1} as follows.

(9.28) 
Then, we express the difference form δS_{2} of the surface area of the manifold S_{2} as follows.

(9.29) 
We get the following manifold S as the superposition of the manifold S_{1} and S_{2}.

(9.30) 
We sum the complex numbers of wave functions every position for the superposition of a wave function. Therefore, we deduce that we sum the surface areas of manifolds at every solid angle for the superposition of manifolds.
Then, we express the difference form δS of the manifold S as follows.

(9.31) 
Therefore, we have the following formula for the spherical harmonics.

(9.32) 
We define the superposition of the manifolds by the above formula.
We consider the universe U of twodimensional spacetime.
We express the world lines of the particle by a complex number.

(10.1) 

(10.2) 
We express the wave function of this particle as follows.

(10.3) 
We suppose that particles are generated by pair production and destroyed by pair annihilation.
Figure 101: Pair production and pair annihilation
We express the closed path C by the circle C of radius R as follows.
Figure 102: Closed path C
We express this closed path as follows.

(10.4) 
We express the circumference a of this circle C as follows.

(10.5) 
Here we introduce the complex solid angle ω.

(10.6) 

(10.7) 
We express the closed path as follows.

(10.8) 
Then we express the circumference a of this circle C as follows.

(10.9) 
We express the difference form of the circumference a as follows.

(10.10) 
We introduce a circle S as an extra space like KaluzaKlein theory.
We express the point on the circle C by matrix representation of a complex number as follows.

(10.11) 

(10.12) 

(10.13) 

(10.14) 
We call the circle an amplitude circle or an amplitude 1sphere.
Figure 103: Amplitude 1sphere S
We express the circumference A of the amplitude 1sphere S by the radius R and the solid angle Ω as follows.

(10.15) 
If the radius R is the function of r and ω, the circumference A becomes the function of r and ω.

(10.16) 
We express the difference form δS of the sphere S.

(10.17) 
We interpret the circumference A as the absolute value of the wave function.

(10.18) 
In order to introduce the phase of wave function, we rotate the sphere S. by the rotational transform angle Φ. We transform the sphere S to the new sphere S’ by the rotational transform which depends on ω.

(10.19) 
We define the rotational transform of the rotational transform angle Φ that depends on ω as follows.

(10.20) 
We express the function Φ (ω) by the natural number n as follows.

(10.21) 
We interpret the rotational angle Φ as the phase of the wave function.

(10.22) 
We transform the sphere S by the rotational transform as follows.

(10.23) 
Figure 104: Rotation of the amplitude 1sphere S
We define the superposition of a sphere S_{1} and a sphere S_{2} as follows.

(10.24) 
The superposition of the sphere and the sphere that is rotated by the angle 180 degrees is zero.

(10.25) 

(10.26) 
The direct product of the closed path C of the particle and the sphere S becomes a torus T.

(10.27) 

(10.28) 

(10.29) 
We call this torus T a torus worldsheet.
Figure 105: The torus worldsheet
Here, we introduce the following new solid angle ν.

(10.30) 
Here we introduce the following new solid radius ρ.

(10.31) 
Here we introduce the following new function f (ρ, ν).

(10.32) 
We express the torus worldsheet T by the function f (ρ, ν) as follows.

(10.33) 
The torus worldsheet is twisted like a helical torus as the following figure.
Figure 106: The torus worldsheet is twisted like a helical torus.
This dimension of the torus worldsheet is same as the dimension of the universe because the universe is twodimensional spacetime in this section.
Here we use a surprising idea.
We interpret the torus worldsheet as a new spacetime. We call the spacetime toric spacetime.
We interpret the toric spacetime an independent universe. We call the universe the second universe.
It is possible to construct the third and the forth universe in the same way that we construct the second universe. We construct many universes by repeating the same way. We call these universes hierarchical universe.
We show the hierarchical universe as follows.
Figure 107: Hierarchical universe
We express the above hierarchical universe by the following symbol.

(10.34) 
We express the position s by the complex number as follows.

(10.35) 
Then the wave function becomes a complex function.

(10.36) 
We assume that the complex function is an analytic function.
Analytic functions satisfy the CauchyRiemann equation.
(CauchyRiemann equation)

(10.37) 
We call the equation path differential equation.
We define the complex conjugate as follows.

(10.38) 
Then we express the path differential equation shortly as follows.

(10.39) 
Analytic function satisfy the Cauchy's integral formula.
(Cauchy's integral formula)

(10.40) 
S is the contour path.
We interpret the Cauchy's integral formula as the path integral equation of Feynman’s path integral.
Figure 108: The path integral equation of Feynman’s path integral
We interpret that the particle on the circle S transit from the position t to the position s for the long distance directly.
We call the new interpretation the path integral of spacetime view that is different from the traditional Feynman’s path integral.
It is possible to use these equations for wave functions of each universe because the wave functions of each universe are complex functions.
We consider the universe U of fourdimensional spacetime.
We express the world lines of the particle by quaternion.

(10.41) 

(10.42) 
We express the wave function of this particle as follows.

(10.43) 
We suppose that particles are generated by pair production and destroyed by pair annihilation.
Figure 109: Pair production and pair annihilation
We express the closed path s by the circle C of radius r as follows.
Figure 1010: Closed path C
We express this closed path as follows.

(10.44) 
We express the circumference a of this circle C as follows.

(10.45) 
Here we introduce the quaternionic solid angle ω.

(10.46) 

(10.47) 
We express the closed path as follows.

(10.48) 
Then we express the circumference a of this circle C as follows.

(10.49) 
We express the difference form of the circumference a as follows.

(10.50) 
We introduce a threedimensional sphere (3sphere) S as an extra space like KaluzaKlein theory.
We express the point S on the 3sphere by matrix representation of quaternion.

(10.51) 

(10.52) 

(10.53) 

(10.54) 

(10.55) 

(10.56) 
We call the circle amplitude 3sphere.
Figure 1011: Amplitude 3sphere S
We express the circumference A of the amplitude 3sphere S by the radius R and the solid angle Ω as follows.

(10.57) 
If the radius R is the function of r and ω, the circumference A becomes the function of r and ω.

(10.58) 
We express the difference form δS of the sphere S.

(10.59) 
We interpret the circumference A as the absolute value of the wave function.

(10.60) 
In order to introduce the phase of wave function, we rotate the sphere S. by the rotational transform angle Φ. We transform the sphere S to the new sphere S’ by the rotational transform which depends on ω.

(10.61) 
We define the rotational transform of the rotational transform angle Φ that depends on ω as follows.

(10.62) 
We express the function Φ (ω) by the natural number n as follows.

(10.63) 
We interpret the rotational angle Φ as the phase of the wave function.

(10.64) 
We transform the sphere S by the rotational transform as follows.

(10.65) 
Figure 1012: Rotation of the amplitude 3sphere S
We define the superposition of a sphere S_{1} and a sphere S_{2} as follows.

(10.66) 
The superposition of the sphere and the sphere that is rotated by the angle 180 degrees is zero.

(10.67) 

(10.68) 
The direct product of the closed path C of the particle and the sphere S becomes a manifold like a torus T.

(10.69) 

(10.70) 

(10.71) 
We call this manifold T a torus worldsheet.
Figure 1013: The torus worldsheet
Here we introduce the following new solid angle ν.

(10.72) 
Here we introduce the following new solid radius ρ.

(10.73) 
Here we introduce the following new function f (ρ, ν).

(10.74) 
We express the torus T by the function f (ρ, ν) as follows.

(10.75) 
The torus worldsheet is twisted like a helical torus as the following figure.
Figure 1014: The torus worldsheet is twisted like a helical torus.
This dimension of the torus worldsheet is same as the dimension of the universe because the universe is fourdimensional spacetime in this section.
Here we use a surprising idea.
We interpret the torus worldsheet as a new spacetime. We call the spacetime toric spacetime.
We interpret the toric spacetime an independent universe. We call the universe the second universe.
It is possible to construct the third and the forth universe in the same way that we construct the second universe. We construct many universes by repeating the same way. We call these universes hierarchical universe.
We show the hierarchical universe as follows.
Figure 1015: Hierarchical universe
We express the above hierarchical universe by the following symbol.

(10.76) 
We express the position s by the quaternion number as follows.

(10.77) 
Then, the wave function becomes a quaternionic function.

(10.78) 
We assume that quaternionic functions are analytic functions.
Analytic functions satisfy the CauchyRiemannFueter equation.
(CauchyRiemannFueter equation)

(10.79) 
We call the above equation path differential equation in this paper.
We define the quaternionic conjugate as follows.

(10.80) 
Then we express the path differential equation shortly as follows.

(10.81) 
Analytic functions satisfy the integral formula of quaternion.
(Integral formula of quaternion)

(10.82) 
Here, u^{ }is a unit quaternion.
We interpret the integral formula of quaternion as the path integral equation of Feynman’s path integral.
Figure 1016: Path integral equation of Feynman’s path integral
We interpret that the particle on the circle S^{1} transit from the position t to the position s for the long distance directly.
We call the new interpretation a spacetime view path integral that is different from the traditional Feynman’s path integral.
It is possible to use these equations for the wave function of each universe because the wave functions of each universe are quaternionic functions.
We express the surface area a of the normal space U as follows.

(10.83) 
We express the difference form of the normal space U as follows.

(10.84) 
Here we replace the r^{3} to the function f (ω).

(10.85) 
Then we express the following formula.

(10.86) 
We interpret the above formula like the following figure.
We call the interpretation manifold view.
Figure 1017: Manifold view of the normal space U
Here we rewrite the formula as follows by a new function f (r, ω).

(10.87) 
We interpret the above formula like the following figure.
We call the interpretation spherical harmonics view.
Figure 1018: Spherical harmonics view of the normal space U
In the spherical harmonics view, we interpret the function f (r, ω) as the spherical harmonics.
The spherical harmonics f (r, ω) is the solution of the Laplace equation of the spheric polar coordinates.
Therefore, spherical harmonics f (r, ω) satisfies the following Laplace equation.

(10.88) 
Here, we used the following symbols.

(10.89) 

(10.90) 
We express the normal spacetime U by the radius r and the solid angle ω as follows.

(10.91) 
We express the amplitude 3sphere S by the radius R and the solid angle Ω as follows.

(10.92) 
We define the wave spacetime W as the direct product of the normal spacetime U and the amplitude 3sphere S as follows.

(10.93) 

(10.94) 
Here we introduce the new solid angle.

(10.95) 
Here we introduce the new radius.

(10.96) 
Here we introduce the new function.

(10.97) 
Then we express the wave spacetime shortly.

(10.98) 
Figure 1019: Wave spacetime
The spherical harmonics g (ρ, ν) is the solution of Laplace equation of the spheric polar coordinates.
Therefore, the spherical harmonics g (ρ, ν) satisfies the following harmonic equation.

(10.99) 
Here, we used the following symbols.

(10.100) 

(10.101) 

(10.102) 

(10.103) 
We define terms in the following table.
Table 111: Normal space and wave space
Term 
Definition 
Normal space 
Threedimensional normal space 
Wave space 
Surface whose surface area is a square of an absolute value of a wave function 
Table 112: Elementarystate and so on
Category 
Term 
Definition 
Certain 
Certainposition 
Position in a threedimensional normal space 
Certain 
Certainstate 
State that one particle existing at a certainposition 
Certain 
certainworld 
State that all particles existing at a certainposition 
Certain 
Certainstate particle 
One particle that is in a certainstate 
Certain 
certainworld particle 
All particle those are in certainstates 
Certain 
Certainpath 
Transition from one certainposition to the other one 
Certain 
Certainevent 
Transition from one certainstate to the other one state 
Certain 
Certainhistory 
Transition from one certainworld to the other one 
Elementary 
Elementaryposition 
Position in a threedimensional normal space 
Elementary 
Elementarystate 
State that one particle existing at an elementaryposition 
Elementary 
Elementaryworld 
State that all particles existing at an elementaryposition 
Elementary 
Elementarystate particle 
One particle that is an elementarystate 
Elementary 
Elementaryworld particle 
All particles those are in elementarystates 
Elementary 
Elementarypath 
Transition from one elementaryposition to the other one 
Elementary 
Elementaryevent 
Transition from one elementarystate to the other one 
Elementary 
Elementaryhistory 
Transition from one elementaryworld to the other one 
Localized 
Localizedposition 
Position in a threedimensional normal space 
Localized 
Localizedstate 
State that one particle existing at a localizedposition 
Localized 
Localizedworld 
State that all particles existing at a localizedposition 
Localized 
Localizedstate particle 
One particle that is in a localizedstate 
Localized 
Localizedworld particle 
All particles those are in localizedstates 
Localized 
Localizedpath 
Transition from one localizedposition to the other one 
Localized 
Localizedevent 
Transition from one localizedstate to the other one 
Localized 
Localizedhistory 
Transition from one localizedworld to the other one 
We abbreviate an elementarystate particle to a stateelement.
We abbreviate an elementaryworld particle to a worldelement.
We arrange terms in the following table.
Table 113: Elementarystate and so on
Category 
Certain 
Elementary 
Localized 
Position 
Certainposition 
Elementaryposition 
Localizedposition 
State 
Certainstate 
Elementarystate 
Localizedstate 
World 
certainworld 
Elementaryworld 
Localizedworld 
State particle 
Certainstate particle 
Elementarystate particle 
Localizedstate particle 
World particle 
certainworld particle 
Elementaryworld particle 
Localizedworld particle 
Path 
Certainpath 
Elementarypath 
Localizedpath 
Event 
Certainevent 
Elementaryevent 
Localizedevent 
History 
Certainhistory 
Elementaryhistory 
Localizedhistory 
In writing this paper, I thank from my heart to TY and NS who gave valuable advice to me.
[1] Mail: sugiyama_xs@yahoo.co.jp
[2] Hugh Everett III, "Relative state" formulation of quantum mechanics, Reviews of Modern Physics, 29 (3):454462, (1957).
[3] James B. Hartle, Quantum Mechanics of Individual Systems, American Journal of Physics, 36, 704712, (1968).
[4] Bryce S. DeWitt, Quantum mechanics and Reality, Physics Today 23, No. 9, pp. 3035, (1970).
[5] Neil Graham, The Measurement of Relative Frequency, in De Witt and N. Graham (eds.) The ManyWords Interpretation of Quantum Mechanics, Princeton NJ: Princeton University Press, (1973).
[6] Adrian Kent, Against Manyworlds interpretations, Int. J. Mod. Phys. A5 (1990) 1745, (http://arxiv.org/abs/grqc/9703089).
[7] David Deutsch, Quantum Theory of Probability and Decisions, Proceedings of the Royal Society of London A 455, 31293137, (1999), (http://arxiv.org/abs/quantph/9906015).
[8] Sumio Wada, Derivation of the Quantum Probability Rule without the Frequency Operator, J. Phys. Soc. Jpn, Vol.76, 094004 (2007), (http://arxiv.org/abs/0707.1623).
[9] Max Born, On the quantum mechanics of collisions (Zur Quantenmechanik der Stoßvorgänge), Zeitschrift für Physik, 37, #12, pp. 863867, (1926).
[10] Andrew M. Gleason, Measures on the Closed Subspaces of Hilbert Space, Journal of Mathematics and Mechanics 6, 885894, (1957).
[11] K. Sugiyama, Quantum probability from a geometrical interpretation of a wave function, (1999), (http://arxiv.org/abs/quantph/9910108).
[12] Bryce S. DeWitt, Quantum Theory of Gravity. I. The Canonical Theory, Phys. Rev. 160 (5): 11131148, (1967).
[13] Julian Barbour, The End of Time, W&N., (1999).
[14] PierreSimon de Laplace, Théorie analytique des probabilités, Paris: Courcier Imprimeur, (1812).
[15] Roger Penrose, Angular momentum: An approach to combinatorial spacetime, Cambridge University Press, Cambridge, (1971).
[16] Richard P. Feynman. The SpaceTime Approach to NonRelativistic Quantum Mechanics, Reviews of Modern Physics 20 (2): 367387, (1948).
[17] Paul A.M. Dirac, The Quantum Theory of the Emission and Absorption of Radiation, Proceedings of the Royal Society of London A 114: 243265, (1927).
[18] I. J. R. Aitchison, A. J. G. Hey, Gauge Theories in Particle Physics, Second Edition, Taylor & Francis; 2 edition, (1989).
[19] Kaluza T., On the Unification Problem of Physics (Zum Unitätsproblem in der Physik), Sitzungsber. Preuss. Akad. Wiss. Berlin. (Math. Phys.), 96, 6972, (1921).
[20] Klein O., Quantum theory and five dimensional theory of relativity (Quantentheorie und fünfdimensionale Relativitätstheorie), Zeitschrift für Physik, A37(12), 895906, (1926).
[21] Cauchy, A.L., Mémoire sur les intégrales définies, Oeuvres complètes Ser. 1 1, Paris (published 1882), pp. 319506, (1814).
[22] Riemann, B., Grundlagen für eine allgemeine Theorie der Funktionen einer veränderlichen komplexen Grösse, (1851).
[23] Hamilton, William Rowan, On quaternions, or on a new system of imaginaries in algebra, Philosophical Magazine. Vol. 25, n 3. p. 489495, (1844).
[24] Fueter R., Die Funktionentheorie der Differentialgleichungen Δu=0 und ΔΔu=0 mit vier reellen Variablen.Comment, Math. Helv. 7, 307330, (1934).
[25] A. Sudbery, Quaternionic Analysis, Mathematical Proceedings of the Cambridge Philosophical Society 85:199225, (1979).
[26] Elie Cartan, Sur certaines expressions differentielles et le probleme de Pfaff, Annales scientifiques de l'Ecole Normale Superieure: 239332, (1899).