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The quantum probability theory of fuzzy event is suggested by using the idea and method of fuzzy mathematics, giving the form of fuzzy event path integral, membership degree amplitude, fuzzy field function, Green function, physical quantity and fuzzy diagram. This theory reforms quantum mechanics, making the later become its special case. This theory breaks unitarity, gauge invariance, probability conservation and information conservation, making these principles become approximate ones under certain conditions. This new theory, which needs no renormalization and can naturally give meaningful results which are in accordance with the experiments, is the proper theory to describe microscopic high-speed phenomenon, whereas quantum mechanics is only a proper theory to describe microscopic low-speed phenomenon. This theory is not divergent under the condition of there being no renormalization and infinitely many offsetting terms, thereby it can become the theoretical framework required for the quantization of gravity.

Since Hawking offered the problem of black hole information loss [

In my first work [

In this paper, the form of the path integral of fuzzy event is established in Section 2. The quantum field theory of fuzzy event is discussed in Section 3. In Section 4, the mathematical form of membership degree amplitude of fuzzy path is discussed. Fuzzy diagram is also introduced and discussed in this Section.

As we know, quantum probability is calculated by “probability amplitude”, whereas probability amplitude is calculated by “path integral”. Thereby, if one attempts to create a theory about the quantum probability of fuzzy event, it is necessary to introduce fuzziness into path integration. For this, the three different probability theories about paths should be first reviewed.

In

First, from the viewpoint of classical probability theory, if there exists a probability distribution for the momentum values of the emitted particles, the particle has a certain probability to move along the path r → ( t ) and are observed at pointB. This probability is denoted as P [ r → ( t ) ] . Then, the probability that a particle will be observed at point B is

P ( A → B ) = ∑ all r → ( t ) from A to B P [ r → ( t ) ] . (1)

Second, from the viewpoint of the probability theory of fuzzy event, considering the fuzziness of the position (coordinate) of the particle, the path of the particle also is unsharp. So, A → B is a fuzzy event, which can be denoted using a fuzzy set R ˜ . Every path belongs to this fuzzy set only in a certain degree of membership. For instance, the membership degree of the path r → ( t ) is expressed as μ R ˜ [ r → ( t ) ] ∈ [ 0 , 1 ] . According to the probability formula of fuzzy event [

P ( R ˜ = ( A → B ) ) = ∑ all r → ( t ) from A to B μ R ˜ [ r → ( t ) ] ⋅ P [ r → ( t ) ] . (2)

Third, from the viewpoint of quantum mechanics, considering the volatility of the particle, according to the principle of superposition, the probability amplitude for A → B is

K ( A → B ) = ∑ all r → ( t ) from A to B ψ [ r → ( t ) ] .

Here ψ [ r → ( t ) ] is the probability amplitude for the path r → ( t ) . According to the statistical interpretation of wave function, the total probability should be calculated using probability amplitude

P ( A → B ) = | K ( A → B ) | 2 . (3)

The logic vein among these three probability theories is shown in

The designed path summation form about fuzzy event is also exhibited in

1) When the interference terms

∑ all r → ( t ) from A to B ∑ all r → ′ ( t ) from A to B ( r → ′ ( t ) ≠ r → ( t ) ) λ R ˜ ∗ [ r → ( t ) ] λ R ˜ [ r → ′ ( t ) ] × ψ ∗ [ r → ( t ) ] ψ [ r → ′ ( t ) ] → 0 ,

i.e. wave effect disappears, the form of the quantum probability of fuzzy event changes into

P ( R ˜ = ( A → B ) ) = ∑ all r → ( t ) from A to B | λ R ˜ [ r → ( t ) ] | 2 × | ψ [ r → ( t ) ] | 2 .

This just accords with the form of the probability of fuzzy event.

2) When the membership degree amplitude

λ R ˜ [ r → ( t ) ] → 1 , ( forarbitrarypath )

i.e. fuzziness effect disappears, the form of the quantum probability of fuzzy event changes into

P ( R ˜ = ( A → B ) ) = ∑ all r → ( t ) from A to B | ψ [ r → ( t ) ] | 2 + ∑ all r → ( t ) from A to B ∑ all r → ′ ( t ) from A to B ( r → ′ ( t ) ≠ r → ( t ) ) ψ * [ r → ( t ) ] ψ [ r → ′ ( t ) ] .

This just accords with the form of the quantum probability of the event.

That is to say, under certain conditions, the quantum probability of fuzzy event can respectively transition to the probability of fuzzy event and the quantum probability of the event. So, the designed path summation form for fuzzy event in

According to the above thought and the explanation about

K ( ( r → 1 , t 1 ) → ( r → 2 , t 2 ) ) = C ∑ allpaths λ R ˜ [ r → ( t ) ] ⋅ exp { i S [ r → ( t ) ] / ℏ } , (4)

or

K ( ( r → 1 , t 1 ) → ( r → 2 , t 2 ) ) = ∫ λ R ˜ [ r → ( t ) ] ⋅ exp { i S [ r → ( t ) ] / ℏ } D [ r → ( t ) ] . (5)

Equation (5) also can be introduced in another way. Usually, it is thought that the position (coordinate) r → ( t ) of the particle is clear and certain at one moment. However, according to the idea of fuzzy theory [

K ˜ ( ( r → 1 , t 1 ) → ( r → 2 , t 2 ) ) = ∫ a ˜ b → exp { i S [ r → ( t ) ] / ℏ } D [ r → ( t ) ] = lim N → ∞ ε → 0 ∫ x ˜ 1 a x ˜ 1 b ∫ x ˜ 2 a x ˜ 2 b ⋯ ∫ x ˜ N − 1 , a x ˜ N − 1 , b C N exp { i S N [ r → ( t ) ] / ℏ } ∏ j = 1 N − 1 d 3 x j . (6)

This fuzzy integral can be computed in an alternate form

K ( ( r → 1 , t 1 ) → ( r → 2 , t 2 ) ) = lim N → ∞ ε → 0 ∫ − ∞ ∞ λ R ˜ ( r → 0 , r → 1 , ⋯ , r → N ) C N exp { i S N [ r → ( t ) ] / ℏ } ∏ j = 1 N − 1 d 3 x j = ∫ λ R ˜ [ r → ( t ) ] exp { i S [ r → ( t ) ] / ℏ } D [ r → ( t ) ] . (7)

Thus, (5) is obtained ((7) is just (5)). (7) can be regarded as the result of the non-fuzzification of (6).

Is the above design from (6) to (7) rational? An explanation can be made as follows:

From (6) and (7), it can be obtained respectively that

| K ˜ ( ( r → 1 , t 1 ) → ( r → 2 , t 2 ) ) | 2 = lim N → ∞ ε → 0 ∫ x ˜ 1 a x ˜ 1 b ∫ x ˜ 2 a x ˜ 2 b ⋯ ∫ x ˜ N − 1 , a x ˜ N − 1 , b | C N exp { i S N [ r → ( t ) ] / ℏ } | 2 ∏ j = 1 N − 1 d 3 x j + the interference terms ,

| K ( ( r → 1 , t 1 ) → ( r → 2 , t 2 ) ) | 2 = lim N → ∞ ε → 0 ∫ − ∞ ∞ | λ R ˜ ( r → 0 , r → 1 , ⋯ , r → N ) C N exp { i S N [ r → ( t ) ] / ℏ } | 2 ∏ j = 1 N − 1 d 3 x j + the interference terms .

When the interference terms tend to zero (i.e. wave effect disappears), the above two equations change respectively into

| K ˜ ( ( r → 1 , t 1 ) → ( r → 2 , t 2 ) ) | 2 = lim N → ∞ ε → 0 ∫ x ˜ 1 a x ˜ 1 b ∫ x ˜ 2 a x ˜ 2 b ⋯ ∫ x ˜ N − 1 , a x ˜ N − 1 , b | C N exp { i S N [ r → ( t ) ] / ℏ } | 2 ∏ j = 1 N − 1 d 3 x j , (8)

| K ( ( r → 1 , t 1 ) → ( r → 2 , t 2 ) ) | 2 = lim N → ∞ ε → 0 ∫ − ∞ ∞ | λ R ˜ ( r → 0 , r → 1 , ⋯ , r → N ) | 2 × | C N exp { i S N [ r → ( t ) ] / ℏ } | 2 ∏ j = 1 N − 1 d 3 x j . (9)

According to the integral theory of fuzzy mathematics [

For the non-relativistic particle system (microscopic low-speed phenomenon) whose particle-number is certain, probability conservation is satisfied. So, there is no fuzziness, λ R ˜ [ r → ( t ) ] = 1 . Thus, (5) becomes usual Feynman path integral.

But, for field theory (microscopic high-speed phenomenon), it is difficult to accurately determine the particle-number of every momentum value because of the generation and annihilation of the virtual particle pairs in vacuum. According to the fuzzy quantity theory of fuzzy mathematics [

Here it should be pointed out that if (6) and (7) are used to describe the field theory, (6) is based on the fuzziness of the field function and fuzzy field function should be written as

φ ˜ ( x ) = 1 ( 2 π ) 3 ∫ − ∞ ∞ q ˜ ( k → , t ) e i k → ⋅ x → d 3 k .

Where q ˜ ( k → , t ) indicates the fuzziness of microscopic state of the field. Whereas in (7), since (7) is the result of the non-fuzzification of (6), the field function φ ( x ) also should be regarded as the result of the non-fuzzification of φ ˜ ( x ) with the form

φ ( x ) = ( 1 / ( 2 π ) 3 ) ∫ − ∞ ∞ λ ( k → ) q ( k → , t ) e i k → ⋅ x → d 3 k . (10)

Where q ( k → , t ) is a possible value of q ˜ ( k → , t ) , whose membership degree is maximum. λ ( k → ) is the relative membership degree amplitude of q ( k → , t ) , and μ ( k → ) = | λ ( k → ) | 2 represents the relative membership degree for q ( k → , t ) . For the non-vacuum field (the field of excited state), there is no fuzziness, μ ( k → ) = 1 ; whereas for vacuum field (the field of ground state), there is fuzziness, 0 ≤ μ ( k → ) ≤ 1 . Here it is might as well supposed temporarily that

λ ( k → ) = Λ 2 n / ( k → 2 + ω k → 2 + Λ 2 ) n . (11)

This is a form satisfying the requirement of theoretical relativistic invariance. Λ ( Λ ∈ R ) and n ( n ∈ N ) are two undetermined parameters (constants). Of course, besides of (11), λ ( k → ) or μ ( k → ) = | λ ( k → ) | 2 also can take other forms (see

The introduce of λ R ˜ [ φ ( x ) ] into path integral of field theory will lead to the change of the forms of Green function and physical quantity. It is easy to prove that the expressions of the n-point Green function and physical quantity, which are corresponding to the fuzzy path integral, respectively are

G ( x 1 , ⋯ , x n ) = 〈 0 | T { λ R ˜ [ φ ( x 1 ) ] φ ^ ( x 1 ) ⋯ λ R ˜ [ φ ( x n ) ] φ ^ ( x n ) } | 0 〉 = i − n δ n W F [ J ] δ J ( x 1 ) ⋯ δ J ( x n ) | J = 0 , (12)

〈 O 〉 = 1 Z ∫ D φ O [ φ ( x ) ] λ R ˜ [ φ ( x ) ] exp { i ℏ S [ φ ( x ) ] } . (13)

Where W F [ J ] is the generating functional based on the fuzzy path integral, O [ φ ( x ) ] is the physical quantity under study, and Z is the fuzzy path integral about φ ( x ) and its form is like (5).

It is worth being pointed out that in the quantum probability theory of fuzzy event, since probability conservation is broke, unitarity does not hold (because unitarity means probability conservation). Because unitarity also implies gauge invariance and information conservation, the breaking of unitarity will mean the breaking of gauge invariance and information (entropy) conservation. The unconservation of information (entropy) further means irreversibility. The breaking degree of unitarity can be depicted by the rate of the change of information (entropy): the change rate being large (corresponding to irreversible process) means large breaking degree, the change rate being small implies less breaking degree, and the change rate tending to zero (corresponding to reversible process) means that unitarity is not broken. Traditional quantum field theory, which argues unitarity, is applicable to the reversible process; whereas quantum field theory of fuzzy event in this paper, which negates unitarity, is suitable for the irreversible process.

The relations among these several conceptions are shown in

In the path integral of fuzzy event, what is the mathematical form of the membership degree amplitude λ R ˜ [ φ ( x ) ] ? This is a process of exploration.

For instance, for the scattering problem of the electron-photon interaction system, it can be ordered that

λ R ˜ 1 [ ψ ¯ ( x ) , ψ ( x ) , A μ ( x ) ] = F 1 exp { i ℏ ∫ d 4 x [ δ m ψ ¯ ( x ) ψ ( x ) + 1 2 δ μ A 2 A μ ( x ) A μ ( x ) ] } . (14)

Where d 4 x = d x 1 d x 2 d x 3 d x 4 , x 4 = i t is virtual time. F 1 is a constant satisfying

0 ≤ | λ R ˜ 1 [ ψ ¯ ( x ) , ψ ( x ) , A μ ( x ) ] | 2 ≤ 1.

δ m = ∑ * ( p ) | γ ν p ν = i m . δ μ A 2 satisfies

− i ( 2 π ) 4 δ μ ν δ μ A 2 = Π μ ν * ( k → 0 ) .

Since gauge invariance is broken in quantum field theory of fuzzy event, the square term of gauge field is introduced into (14). This square term breaks unitarity and probability conservation. The breaking degree is decided by the integral of this square term. When the integral is relatively large, the breaking degree of unitarity also is relatively large; when the integral is relatively small, the breaking degree of unitarity also is relatively small. When the integral tends to zero, the square term of the gauge field can be ignored, and gauge invariance, unitarity, probability conservation and information conservation come into true.

By using (10) and (14), a S matrix element without divergence, from which the theoretical results can be computed in accordance with the experimental data, can be obtained from (12). The propagators in the S matrix element have the following forms:

S F ( p ) = μ ( p ) × i ( 2 π ) 4 i p ^ − m p 2 + m 2 − i ε , ( for electron )

δ μ ν D F ( k ) = μ ( k ) × − i ( 2 π ) 4 δ μ ν k 2 − i ε . ( for photon )

Here μ ( p ) and μ ( k ) are the membership degree factors of the propagators. Several Feynman diagrams are drawn in

These diagrams in

Since there is no divergence in the quantum field theory of fuzzy events, the renormalization is not need (Some people think that the renormalization theory cannot be regarded as a completely correct theory even though it’s supported by experiments. We should also seek a deeper understanding to explain the intrinsic nature of the agreement between the experimental results and the calculated results [

It should be pointed out that under the condition of Λ 1 = 6 ∼ 10 m and Λ 2 < 1 × 10 − 4 m , the above fuzzy diagrams give

δ μ A 2 ≈ 0 a n d k μ [ Π μ ν * ( k ) − Π μ ν * ( 0 ) ] ≈ 0 → k μ Π μ ν * ( k ) ≈ − i ( 2 π ) 4 k μ δ μ v δ μ A 2 ≈ 0.

Thus, gauge invariance, unitarity and the conservation of probability and information come into true in an approximate sense (it is mentioned above that these principles are only approximate ones).

Similarly, for the quark-gluon system, it can be ordered that

λ R ˜ 2 [ ψ ¯ ( x ) , ψ ( x ) , B μ ρ ( x ) , η ¯ ( x ) , η ( x ) ] = F 2 exp { i ℏ ∫ d 4 x [ δ m ψ ⋅ ψ ¯ ( x ) ψ ( x ) + 1 2 δ μ B 2 ⋅ B μ ρ ( x ) B μ ρ ( x ) + δ μ η 2 ⋅ η ¯ c ( x ) η c ( x ) ] } . (15)

Where η ( x ) represents ghost field. But, in the case of lattice gauge field, there is no ghost field. According to the above thought, under the condition of large momentum exchange, the perturbation S matrix element without divergence can be obtained whose theoretical results accord with the experimental data when

{ Λ 3 ≤ 10 − 7 m e , ( for the membership degree factor of the vacuum propagator of gluon ) Λ 4 ≤ 10 − 8 m e , ( for the membership degree factor of the vacuum propagator of ghost particle ) Λ 5 ≤ 10 − 9 m e . ( for the membership degree factor of the vacuum propagator of quark )

What is more, gauge invariance, unitarity, probability conservation and information conservation are all satisfied because

δ μ B 2 ≈ 0 and k λ [ ( Π * ) λ λ ′ a a ′ ( k ) − ( Π * ) λ λ ′ a a ′ ( 0 ) ] ≈ 0 → k λ ( Π * ) λ λ ′ a a ′ ( k ) ≈ − i ( 2 π ) 4 k λ δ λ λ ′ f a b c f a ′ b c δ μ B 2 ≈ 0.

However, under the condition of extremely strong gluon field, the integral of the square term of gluon field cannot be ignored, and the breaking of unitarity also has to be considered.

Since the quantum field theory of fuzzy event need no renormalization and infinitely many offsetting terms and can naturally give meaningful results without divergence, it is the theoretical framework required for the quantization of gravity. For the system of gravity and scalar field, it is ordered that

λ R ˜ 3 [ h μ ν ( x ) , θ ¯ α ( x ) , θ β ( x ) , ϕ ( x ) ] = F 3 exp { ( i / ℏ ) ∫ d 4 x [ ( 1 / 2 ) δ μ h 2 h μ ν ( x ) h μ ν ( x ) + δ μ θ 2 θ ¯ c ( x ) θ c ( x ) + 1 2 δ μ ϕ 2 ϕ ( x ) ϕ ( x ) ] } . (16)

Where θ ( x ) is the ghost field, h μ ν = ( g g μ ν − η μ ν ) / κ is the gravitational field. δ μ h 2 is a very small value (this is similar to δ μ A 2 and δ μ B 2 above). For weak gravity system, the gravitational square term in (16) is very small and thereby can be ignored, unitarity is tenable. But, for the black hole or the universe of very early stage, due to extremely strong gravity, the gravitational square term in (16) can no longer be ignored. So, the breaking of unitarity also can no longer be ignored, and information is no longer conserved.

In this paper the second kind of fuzzy quantum probability theory (quantum probability of fuzzy event) is proposed. This theory can be regarded as the reformation and redevelopment of usual quantum theory, making fuzzy events also be able to be described. Under certain conditions, the quantum probability theory of fuzzy event will transition to usual quantum theory. Thereby, traditional quantum theory is only the special case of the quantum probability theory of fuzzy event. Due to the introduction of fuzziness, this new theory breaks unitarity, gauge invariance, probability conservation and information conservation, making these principles become approximate ones under certain conditions. This new theory is a theory which can describe irreversible process, whereas quantum mechanics is only a theory to describe reversible process. This new theory, which needs no renormalization and can naturally give meaningful results in accordance with the experiments, is the proper theory to describe microscopic high-speed phenomenon, whereas quantum mechanics is only a proper theory to describe microscopic low-speed phenomenon.

The novelty of the work of this paper lies in: 1) bringing forward the conept and idea of “fuzzy vacuum” or “fuzziness of vacuum”; 2) putting forward that under the condition of extremely strong gauge field or gravitational field, usual quantum theory and unitarity will fail.

At present, seemly there are six directions for the quantum theory of fuzzy event to play roles:

1) Being helpful to meaningfully quantize gravity.

2) Being helpful to construct the unified theory of four basic forces.

3) Being helpful to solve the puzzle of cosmological constants.

4) Being helpful to research quantitatively the origin of the Big Bang.

5) Being helpful to understand the problem of black hole information loss.

6) Possibly providing a new idea and solution to the “abnormal” decay probability of B mesons in LHCb.

The author declares no conflicts of interest regarding the publication of this paper.

Qiu, W.B. (2020) Fuzzification of Feynman Path Integral and Its Effect on Field Theory and Quantum Gravity—Reformation and Redevelopment of Quantum Theory. Journal of Modern Physics, 11, 2053-2065. https://doi.org/10.4236/jmp.2020.1112129