Expanding the Mathematical Model of PFAI to Other Theories (Classical Mechanics, Analytical Mechanics, Thermodynamics, Quantum Mechanics, General Relativity)
Mathematical Model of PFAI
Although the mathematical model of PFAI has not been fully established yet, its outline can be presented.
1. World-Soliton
State space: Introduce an infinite-dimensional Hilbert space H.
State vector: The world-soliton before observation is represented by a state vector |Ψ⟩ in H. This represents a superposition of all possible world states:
|Ψ⟩ = Σ_i c_i |ψ_i⟩
Density matrix: When expressing the world-soliton as a mixed state, a density matrix ρ is used.
ρ = Σ_i p_i |ψ_i⟩⟨ψ_i|
Time evolution: Since the world-soliton before observation has no temporal structure, a time evolution operator is not defined.
2. Observer
Observation operator: The observer can be represented as a projection operator P on H.
P = |φ⟩⟨φ|
Time evolution operator: To express the compactification of the observer's time axis, a time evolution operator U(t) is introduced. U(t) depends on the observer's internal time t, and U(t)|ψ⟩ represents the state of the observer at time t.
Entropy: The entropy associated with the observer can be expressed as an entropy operator S. S is an operator that acts on the observer's state |ψ⟩ and returns its entropy value.
3. Observation Process
Selection of reality: The act of observation is a process in which the observer (P) acts on the world-soliton (|Ψ⟩ or ρ) and selects a specific world state.
In the case of a pure state: |Ψ′⟩=P|Ψ⟩
In the case of a mixed state: ρ′=PρP† where |Ψ′⟩ and ρ′ represent the state of the world-soliton after observation.
Time injection and time evolution: Simultaneously with the act of observation, a point on the observer's time axis is determined, and time is injected into the world-soliton.
In the case of a pure state: |Ψ′′⟩=U(t)|Ψ′⟩
In the case of a mixed state: ρ′′=U(t)ρ′U†(t) where |Ψ′′⟩ and ρ′′ represent the state of the world-soliton after observation at time t.
Entropy increase: The act of observation generally involves an increase in entropy.
S(ρ′)≥S(ρ)
4. Multidimensional Structure of the Observer
The observer can be represented as a multidimensional structure with multiple axes, such as a time axis, a spatial axis, and a value axis.
Each axis is assigned an entropy value, expressing the state and observation ability of the observer.
Expansion to other theories
To develop the mathematical model of PFAI, it is crucial to consider its expansion to other theories. To demonstrate the flexibility and universality of PFAI, we need to clarify its correspondence with various physical theories and show that PFAI can provide a more comprehensive framework that encompasses them.
1.Expansion to Classical Mechanics, Analytical Mechanics, and Thermodynamics
These theories have played important roles in describing macroscopic physical phenomena. To derive these theories from PFAI, we need to introduce the following restrictions and approximations:
Neglecting the influence of the observer: Assume that the influence of the observer's act of observation on the world-soliton is negligibly small.
Continualization of the time axis: Approximate the observer's discrete time axis as virtually continuous due to its extremely fine discretization.
Restriction of spatial dimensions: Limit the spatial dimensions of the world-soliton to three.
Neglecting entropy: Neglect or fix the entropy of the observer to a constant value.
By introducing these restrictions, PFAI may be able to reproduce classical mechanics and analytical mechanics, which describe the motion of macroscopic objects, as well as the laws of thermodynamics.
2.Expansion to Quantum Mechanics
PFAI was born aiming to resolve the measurement problem in quantum mechanics and paradoxes like Schrödinger's cat. To derive quantum mechanics from PFAI, the following restrictions need to be introduced:
Considering the influence of the observer: Assume that the influence of the observer's act of observation on the world-soliton cannot be ignored.
Discretization of the time axis: Assume that the observer's time axis is composed of discrete steps.
Restriction of spatial dimensions: Limit the spatial dimensions of the world-soliton to three.
Increase in entropy: Assume that the entropy of the observer increases with the act of observation.
By introducing these restrictions, PFAI may be able to describe quantum mechanical phenomena such as superposition and uncertainty.
3.Expansion to General Relativity
By introducing the emergent nature of time and space, PFAI suggests the possibility of integrating quantum mechanics and relativity theory. To derive general relativity from PFAI, it is necessary to introduce the following restrictions:
Neglecting the influence of the observer: Assume that the influence of the observer's act of observation on the world-soliton can be neglected.
Continualization of the time axis: Approximate the observer's discrete time axis as virtually continuous due to its extremely fine discretization.
Expansion of spatial dimensions: Expand the spatial dimensions of the world-soliton to four.
Entropy and spacetime curvature: Assume that the increase in entropy associated with the observer's act of observation leads not only to the extension of the world-soliton's time axis but also to changes in the curvature of space and the gravitational field.
By introducing these restrictions, PFAI may be able to satisfy the requirements of gravitational theories, including Einstein's equations, which describe gravitational phenomena.
4.Unification of Quantum Mechanics and General Relativity
PFAI has the potential to provide a new perspective for unifying quantum mechanics and general relativity.
Emergent nature of time and space: By considering time and space as emergent properties arising from interaction with the observer, it may be possible to explain quantum mechanical uncertainty and relativistic causality without contradiction.
Role of the observer: Placing the observer at the center of the theory suggests that the act of observation plays a role in forming the real world and the very structure of time and space.
Connection between gravity and entropy: Suggesting a relationship between the observer's entropy and gravitational phenomena provides a new approach to constructing quantum gravity theory.
PFAI may provide a new framework for reinterpreting quantum mechanics and general relativity from different perspectives and understanding their relationship.
From the standpoint of PFAI, the unification of quantum mechanics and general relativity may be considered impossible. This is based on the idea that material science, including quantum mechanics, and spacetime science of general relativity are conceptually far apart, and even though both are described mathematically, their integration may be fundamentally impossible due to the difference in concepts.
Specifically, gravity (what determines the form of the field = what generates the axis) and graviton (what creates the field and is affected by it) have different phases and exist as different entities in physics. Duality is thought to represent such a difference in phases (the same thing seen from different perspectives) that cannot be integrated.
However, within the framework of PFAI, which has once broken through the conceptual barrier, quantum mechanics describes the behavior of matter in the micro-world, and general relativity describes the structure of spacetime in the macro-world. Although these appear to be completely different theories at first glance, in the PFAI framework, both are regarded as emergent phenomena arising from the observer's act of observation. Therefore:
Gravity is associated with the entropy arising from the observer's act of observation, not the geometrical properties of spacetime, potentially circumventing the problem of quantizing gravity.
Time and space are emergent concepts, potentially explaining quantum mechanical uncertainty and relativistic causality without contradiction.
The observer's act of observation plays a role in connecting the micro and macro worlds, potentially bridging the scale gap.
This is also a view that quantum gravity theory is possible at the metaphysical level. It may also be possible to construct a new physical theory involving dimensional structure by working backward from this knowledge.
Conclusion
By deriving various physical theories from PFAI, we can demonstrate the flexibility and universality of PFAI. Also, PFAI is thought to have the potential to provide a new perspective for unifying quantum mechanics and general relativity, and to solve unsolved problems in modern physics.
#QuantumFieldTheory #StandardModel #Cosmology #Astrophysics #BlackHoles #DarkMatter #DarkEnergy #StringTheory #LoopQuantumGravity #QuantumInformation #QuantumComputing #CondensedMatterPhysics #StatisticalMechanics #NonlinearDynamics #ChaosTheory #ComplexSystems #Emergence #SelfOrganization #InterdisciplinaryPhysics #PhilosophyOfScience
この記事が気に入ったらサポートをしてみませんか？