Multi-mix Path Algorithm

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Multi-mix Path Algorithm
In this primary investigation, we ignore the actions of the symmetry related potential. These actions are far less vigilant than the direct results of the embedding of individual locations of elementary modules.

The name “multi-mix algorithm” stands for an alternative of the algorithm that is known as “path integral.”

For the Multi-mix Path Algorithm, the name “path integral” is, in fact, a misnomer. The algorithm concerns a sequence of multiplications that can be approached by a sum of the terms.

Since during the regeneration of the considered object the displacement of the object is rather stable, will part of the multiplication factors reduce to unity. The other factors are close to unity.

The result is that the sequence reduces to a sequence of additions of many small contributions. These contributions are the actions of the individual hops of an elementary module.

Elementary modules reside on an individual symmetry center. A dedicated mechanism recurrently generates the landing locations of the hops of the elementary module in a stochastic fashion. The embedded object hops along the elements of the generated hop landing location swarm. A continuum embeds the hop landing locations. The path of the symmetry center is the averaged path of the embedded object. The mechanism applies a stochastic process that owns a characteristic function. The characteristic function implements a displacement generator. Consequently, at first approximation, the swarm moves as one coherent unit. The location density distribution of the hop landing location swarm owns a Fourier transform that equals the characteristic function that equals the characteristic function of the stochastic process.

This Fourier transform enables the description of the path of the swarm by a “multi-mix algorithm.”  A sequence of factors that after multiplication represent the whole path describe the hopping of the embedded object. Each factor represents three subfactors. The procedure that underlies the multi-mix algorithm depends on the fact that a summation can replace the multiplication of factors that are all very close to unity.

The first subfactor represents the jump from configuration space to momentum space. The inner product of the Hilbert vector that represents the current location and the Hilbert vector that represents the momentum of the swarm provides this subfactor. The algorithm assumes that during the current regeneration of the location swarm the momentum is constant.

1.      The second subfactor represents the effect of the hop in momentum space.

2.      The third subfactor represents the jump back from momentum space to configuration space.

The procedure runs over the complete hopping path. In the sequence of factors, the third subfactor of the current term compensates the effect of the first subfactor of next term. Their product equals unity.

What results is a sequence of factors that are very close to unity, and that represent the effects of the hops in momentum space. Because the momentum is considered constant, the logarithms of the terms can be taken and added to an overall sum. In this way, the multiplication is equal to the sum of the logarithms of the factors.

In more detail, the procedure describes as follows.

We suppose that during the particle generation cycle in which the controlling mechanism produces the swarm $$\{a_i\}$$, the momentum $$\vec{p}_n$$ is constant.

Every hop gives a contribution to the path. These contributions can be divided into three steps per contributing hop:

1.     Change to Fourier space. This act involves as subfactor the inner product $$\langle \vec{a}_i,\vec{p}_n \rangle$$.

2.       Evolve during an infinitesimal progression step into the future.

a.       Multiply with the corresponding displacement generator $$\vec{p}_n$$.

b.       The generated step in configuration space is $$(\vec{a}_{i+1}-\vec{a}_i)$$.

c.       The action contribution factor in Fourier space is $$\langle \vec{p}_n,\vec{a}_{i+1}-\vec{a}_i \rangle$$.

3.       Change back to configuration space. This act involves as subfactor the inner product $$\langle \vec{p}_n,\vec{a}_{i+1} \rangle$$.

The combined term contributes a factor $$\langle \vec{a}_i,\vec{p}_n\rangle\exp(\langle\vec{p}_n,\vec{a}_{i+1}-\vec{a}_i\rangle)\langle\vec{p}_n,\vec{a}_{i+1}\rangle$$.

Two subsequent steps give:

$$\langle \vec{a}_i,\vec{p}_n\rangle\exp(\langle\vec{p}_n,\vec{a}_{i+1}-\vec{a}_i\rangle) {\color{Red}\langle\vec{p}_n,\vec{a}_{i+1}\rangle \langle \vec{a}_{i+1},\vec{p}_n\rangle} \exp(\langle\vec{p}_n,\vec{a}_{i+2}-\vec{a}_{i+1}\rangle) \langle\vec{p}_n,\vec{a}_{i+2}\rangle$$

$$=\langle \vec{a}_i,\vec{p}_n\rangle\exp(\langle\vec{p}_n,\vec{a}_{i+1}-\vec{a}_i\rangle)

\exp(\langle\vec{p}_n,\vec{a}_{i+2}-\vec{a}_{i+1}\rangle) \langle\vec{p},\vec{a}_{i+2}\rangle$$

The $${\color{Red}red}$$ terms in the middle reduce to unity. The other terms also join.

Over a full particle generation cycle with $$N$$ steps this results in:

$$\prod_{i=1}^N\langle \vec{a}_i,\vec{p}_n\rangle\exp(\langle\vec{p}_n,\vec{a}_{i+1}-\vec{a}_i\rangle)\langle\vec{p}_n,\vec{a}_{i+1}\rangle$$

$$=\langle \vec{a}_1,\vec{p}_n\rangle\exp(\langle\vec{p}_n,\vec{a}_{N}-\vec{a}_1\rangle)\langle\vec{p}_n,\vec{a}_{N}\rangle$$

$$=\langle \vec{a}_1,\vec{p}_n\rangle\exp\biggl(\sum_{i=2}^N\langle\vec{p}_n,\vec{a}_{i+1}-\vec{a}_i\rangle\biggr)\langle\vec{p}_n,\vec{a}_{N}\rangle$$

$$=\langle \vec{a}_1,\vec{p}_n\rangle\exp(L\, d \tau)\langle\vec{p}_n,\vec{a}_{N}\rangle$$

$$L\,d\tau=\exp\biggl(\sum_{i=2}^N\langle\vec{p}_n,\vec{a}_{i+1}-\vec{a}_i\rangle\biggr)$$

$$L=\langle\vec{p}_n,\dot{\vec{q}}\rangle$$

Here, $$L$$ is known as the Lagrangian.

Equation (4) holds for the special condition in which $$\vec{p}_n$$ is constant. If $$\vec{p}_n$$ is not constant, then the Hamiltonian $$H$$ varies with location. In the next equations, we ignore subscript $$_n$$.

$$\frac{\partial H}{\partial q_i}=-\dot{p}_i$$

$$\frac{\partial H}{\partial p_i}=\dot{q}_i$$

$$\frac{\partial L}{\partial q_i}=\dot{p}_i$$

$$\frac{\partial L}{\partial \dot{q}_i}=p_i$$

$$\frac{\partial H}{\partial\tau}=-\frac{\partial L}{\partial\tau}$$

$$\frac{d}{\partial\tau}\frac{\partial L}{\partial \dot{q}_i}=\frac{\partial L}{\partial q_i}$$

$$H+L=\sum_{i=1}^3\dot{q}_ip_i$$

In these equations, we used proper time $$\tau$$ rather than coordinate time $$t$$.

The effect of the hopping path is that the geometric center of the symmetry center moves over a resulting small distance $$\vec{a}_n-\vec{a}_1$$. Together with “charge” $$(N\,Q_n)$$ this move determines the next version of momentum $$\vec{p}_n$$.

The result is that both the symmetry related fields $$\mathfrak{A}^x$$ and the embedding field $$\mathfrak{C}$$ influence the location of the geometric center of the symmetry center $$\mathfrak{S}^x_n$$.

In this investigation, we ignored the influence of the symmetry related field $$\mathfrak{A}$$. This field influences momentum $$\vec{p}_n$$ and the corresponding eigenvector $$|p\rangle$$. This means that the product of the red colored middle terms is no longer equal to unity. Instead the product differs slightly from unity and the effect can be included in the path integral. In this way, a small slowly varying extra contribution is added to each subsequent term in the summation. This extra contribution is a smooth function of progression and thus, it is a smooth function of the index of the term.

The result of the “multi-mix algorithm” is expectable. The “step” of the swarm equals the sum of the steps of the hops. The “multi-mix algorithm” is introduced to show the similarity with the “path integral.” The “path integral” is taken over all possible paths. The multi-mix algorithm only takes the actual hopping path. Starting from the Lagrangian usually introduces the “path integral” algorithm. Here we started the “multi-mix algorithm” from the hopping path and the “multi-mix algorithm” results in the Lagrangian.