Classical Mechanics as a Fundamental Law of Quantum Mechanics

n our previous paper, we showed that the so-called quantum entanglement also exists in classical mechanics. The inability to measure this classical entanglement was rationalized with the definition of a classical observer which collapses all entanglement into distinguishable states. It was shown that evidence for this primary coherence is Newton’s third law. However, in reformulating a "classical entanglement theory" we assumed the existence of Newton’s second law as an operator form where a force operator was introduced through a Hilbert space of force states. In this paper, we derive all related physical quantities and laws from basic quantum principles. We not only define a force operator but also derive the classical mechanic's laws and prove the necessity of entanglement to obtain Newton’s third law.

labels the particles, breaks this coherence to provide the local classical mechanics laws. Although the 48 quantum terminology was implemented in the paper of ref [5], the main formalism was fundamental 49 in the sense that operators such as force, were introduced fundamentally through a series of projecting 50 operators with no quantum justification. In addition, Newton's second law was assumed to be fulfilled 51 in its operator form. In this paper, we reformulate classical mechanics beginning with pure quantum 52 principles. We then prove that the consistency of our theory must be accompanied with classical 53 entanglement.

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This is a pure mathematical work. We implemented techniques of Hilbert and mostly Folk 56 spaces to introduce classical mechanics in a quantum-like terminology. No experiment or computer 57 proceeding were implemented.
where in the Heisenberg picture we have We implement a conventional momentum operator, that is, P = P (t) which causes the last term of eq. 2 to be canceled. This yield: Using the space representation P = −ih ∇ it can also be shown that, with, where we assumed that H = H (t) .

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Thus, if we substitute eqs. 4 and 5 into eq. 1, we obtain the classical law ∇H = −˙ P as an output 63 of a trivial mathematical identity. Clearly, this is wrong as both operators are different in picture and 64 representation. For example, the momentum operator in eq. 4 is presented in the energy form while in eq. 5 it is shown in the conventional form. Nevertheless, this paper will show that despite their 66 different pictures, if we project both commutators of eqs. 4 and 5 onto a common representation, both 67 sides of eq. 1 will be equatable, again, to the classical laws of mechanics. In our paper we implement 68 the labeling basis of states as the common representation. In addition, we show that the phases that 69 are accompanied by the exponential operators U E and U p generate the system dynamics. An observer terminology describes the observable representation. By selecting a basis of states, 72 the observer defines the concepts to be used in describing its measurement results [28]. Because the 73 operators P E and H p in eqs. 4 and 5 are represented differently, it make no sense to equate them.

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Mathematically, this means that either variable can appear simultaneously at different sides of the same 75 equation provided that they possess the same dimensions, or that physical operators must appear with 76 the same representation. One method of resolving this scenario is by projecting operators to the same 77 representation. In this paper we suggest that this common representation becomes the labeling basis 78 of states that distinguishes between particles [5], provided that each particle is associated solely with  Experimentally, at any moment, forces can be detected on each isolated particle. This is because these 88 particles are already in a collapsed state, having adjusted to describe each isolated particle.

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To define mutual forces, we consider that under the labeling basis, the force operator is non-diagonal.

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Thus, the off-diagonal terms will correlate with different labeled states. It will be shown that this 91 non-diagonal representation gives rise to the pre-collapsed entangled states.
[5], we introduce the force state F as an eigenstate of the force operator where the operator 94 was defined through a general formalism of projecting operators. In this section we implement eq. 5 to 95 obtain an explicit expression for the force operator. 96 We first use the identities: to obtain: Using the momentum operator, which is expressed though projecting operators, we find that, Similarly, it can be shown that yielding, Consider a Hamiltonian with the form H = P 2 2m + V. Note that the factor p − p eliminates the diagonal term of the kinetic energy which gives, Defining the potential energy variable (its meaning, not as an operator) we obtain, Using eq. 5 and by identifying P, H p in the classical equation F = −∇H, we obtain the force operator, We notice the following:

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• F p is an off-diagonal operator. This means that when applied to a momentum state it changes 98 from p to p as a force operator.

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• The gradient operation that describes a physical law, is generated by the phases exp i • The time variable t is missing in eq. 17. It appears later when we present momentum in the 101 energy representation as described in section VI. Further discussion is provided in section VII.

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Note that if p| V | p = p | V | p and is real, the force operator is Hermitian; meaning that in V. Force Operator F F expressed in the Diagonal Force States 106 The purpose of this section is to show that the force operator can be represented with the standard tools of quantum mechanics. That is, as a series of projecting operators accompanied with eigenvalues.
Assuming a Hermitian F , its eigenstates F generate a complete orthogonal basis of states that can be represented through the identity operator: Thus, using the relation we obtain, As the force operator is diagonal in this representation, we obtain the relations that identify the state F : Eqs. 21 enables us to write the force operator as a series of projecting operators:

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The momentum operator, shown in the Heisenberg picture is: In a similar manner to the previous calculations it can be shown that: Or, Defining the parameter we obtain that H, Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 11 August 2020 doi:10.20944/preprints202008.0252.v1 In this representation t appears explicitly in the momentum operator 108

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Let us observe the classical equation˙ On the right side of eq. 28, we represent the classical Hamiltonian as a function of coordinates and Implementing eq. 1 for the common representation we obtain the operator equation: This equation that was considered as a postulate in ref.
[5] is now derived from basic quantum 121 principles.

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With regards to individual particles, classical mechanics allows particles to share forces (through mutual interactions) while each particle can still be associated with a distinguishable momentum. In quantum terminology, this means that the momentum operator is diagonal under the labeling basis of states. Assuming: we obtain from eq. 31 the momentum operator's diagonal form: Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 11 August 2020 doi:10.20944/preprints202008.0252.v1

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For simplicity we define the following variable: to obtain from eq. 30 the force operator, To distinguish between solitary and mutual interaction forces, we separate F s into the diagonal terms |s s| = δ s,s and the off-diagonal terms. This yields: where F s , the net force operator that acts upon a s-particle is: with F s being the s-diagonal term and F s =s is the off-diagonal term. Focusing on a single particle with 125 a label s, we identify the diagonal term F s |s s| with an external force operator while the off-diagonal 126 sum, ∑ s F s =s |s s| represents the interaction of the s particles with other particles.

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With the purpose of reconstructing Newton's third law, we obtain the relation F s =s = − F s =s Note that on both sides, s and s were switched.
Defining the interaction force magnitude as F s,s = F s =s we obtain: Following the convention of an Hermit observable and noticing that F s of eq. 40 is not such, we suggest the replacement, F s ,s (|s s| − |s s |) → i F s ,s (|s s| − |s s |) to obtain the Hermit operator: Note that ∑ s F s = ∑ s F s |s s|.

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Classical mechanics includes particles interactions and is therefore useful to introduce our system with multi-particle algebra. Now the states belong to the Fock space. Following ref.
[5] we use the transformation |s s| → a † s a s |s s| → a † s a s |s s | → a † s a s , Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 11 August 2020 doi:10.20944/preprints202008.0252.v1 where a and a † behave like spin 1 2 fermionic annihilation and creation operators, The momentum operator becomes: and Eq. 41 is now modified to, whereas, the total force operator was:

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Here, we introduce two states that serve as the eigenstates of F s :

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Vacuum state The state in which no particle is observed: For that state we obtain F s |∅ = 0. (48) Close system state The state in which all s-states are occupied with a single fermionic particle: Substituting in eq. 46 we obtain, approach cannot represent internal forces and consequently Newton's third law. In section XIII, we 137 resolve this difficulty by introducing entangled states.

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We implement single particle states of the particles pair s 1 and s 2 to generate doubles of entangled states: with the single particle states: The observer detects only the particle which is labeled as s 1 The observer detects only the particle which is labeled as s 2 (55) According to eq. 45, each s 1 or s 2 particle is associated with a force operators F s 1 or F s 2 , as follows: The f orce acting on particle s 1 The f orce acting on particle s 2 where we replaced the dummy index s with s.
We then obtain: and in eq. 55's symbol terminology, and in the same manner we obtain: To retrieve classical results, we consider an observer that by performing measurements, collapses the entanglement into the distinguishable (labeled) particles. Consequently, they detect only average quantities. Implementing eqs. 54, 58 and 59 we obtain: The net f orce acting on particle 1 : ⇒ The net f orce acting on particle 2 : Newton's third law has been reformulated. The third law is what is left from a primary entanglement 141 of classical particles.

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In the previous section we associated measurement outputs with averages. This means that the 144 so-called single observation in fact includes a multitude of measurements forming an ensemble that 145 allows the detection of average quantities.

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In this paper, we reformulated classical mechanics laws from pure quantum principles. We 148 showed that the identity H, P = − P, H can be implemented to reformulate the laws of classical 149 mechanics provided that the following two-step procedure is to be followed: First, the right side onto a common basis. Then, can we equate both commutators to obtain a physical law. We suggest 154 that by following this procedure and by implementing other operators, it may be possible to extend 155 this approach to find or reconstruct other physical laws.

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As shown earlier, using this approach to reconstruct classical mechanics laws, Newton's third 157 law corresponds with entanglement. That is to say, prior to a classical mechanical measurement, the 158 labeling states were defined as quantum states that cannot be factored as a product of the states of its 159 local constituents. We showed that the remaining evidence for this coherence is the internal forces.

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Philosophically speaking, we realize that beyond classical mechanics, there is another reality in which 161 even classical particles are entangled. As the classical observer can measure only a limited basis of 162 states, entangled states become impossible to measure.