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Research
, Volume: 10( 8) DOI: 10.37532/2320-6756.2022.10(8).292

Time Evolution of Particle in Quantized Gravitational Field

*Correspondence:
Angshuman Roy Physical Science, DAV Siliguri, India
E-mail:
[email protected]

Received date: June 25, 2022, Manuscript No. tspa-22-67675; Editor assigned: June 27, 2022, PreQC No. tspa-22-67675 (PQ); Reviewed: July11, 2022, QC No. tspa-22-67675 (Q); Revised: July 15, 2022, Manuscript No. tspa-22-67675 (R); Published: July 20, 2022, DOI: 10.37532/2320-6756.2022.10(8).292

Citation: Angshuman Roy. Time Evolution of Particle in Quantized Gravitational Field. J Phys Astron.2022;10(8):292.

Abstract

Through this article, an attempt has been made to propose a heuristic method to determine wave evolution in a quantized gravitational field. In this paper, propagation of particles in a quantized gravitational field is studied. Instead of quantizing the gravitational field into superposition of plane wave solutions, here at a certain point in spacetime, the all-possible values of the field due uncertainty of the gravitational field at that point due to uncertainty in momentum and position of the particle curving spacetime is assigned a probability using the derived rules in the paper. The interaction between other fields and the gravitational field is determined by superimposing all possible solutions what the field would be like in every possible geometry of the curved spacetime with uncertainty of the gravitational field at that point due to uncertainty in momentum and position of the particle curving spacetime.

Keywords

Physics of gravity; Gyroscopic forces of electron rotation; Temperature dependence

Introduction

Renormalization is needed for integrating over internal x in Feynman diagrams, But it can avoid it in finding a quantum theory of gravity, if we use general relativity's original view of curvature in spacetime cause gravity instead of gravitons. Here, we study evolution of a field in each of the geometries for each mass equivalence eigen-sates of the particle curving spacetime's at every spacetime point separated by light like intervals from a given point. This process is repeated for all point in spacetime for an observer. Due to uncertainty in position and momentum of the particle curving spacetime, there will be uncertainty in the geometry of spacetime at every point.

Now by using conditional probability theory we can get the particle evolution in each of the possible geometries but in probability, not probability density, ie, we need phase retrieval. So, now an iterative algorithm can retrieve its phase if we have the probability of all possible eigenstates of its momenta.

Field evolution in quantized gravitational field

Using Schwarzschild metric, we get x1 = r

Then, the geodesics equation becomes [1,2]

Where,

We know that,

where g is the proper acceleration seen by the observer.

We know,

Therefore,

A change in curvature of spacetime at a point in spacetime due to change in position of particles should be causally connected by light like spacetime interval. The change in curvature of spacetime at a point , xt in spacetime due to wave packet must be causally connected by light like spacetime interval, i.e, . Therefore, the wave packet at to induce the change in curvature of spacetime at x0, t0.

The absolute value of time interval is taken as negative value of time interval will imply effect precedes the cause.

Let’s assume there is a particle A at x0, t0 and another particle B at ,the particle at (x0, t0) which will be accelerated due to gravity by a value , whose value is dependent on the distance of separation.

If the particle B’s probability to exist at rather than some probability less than 1 to exist at , let’s say probability for the particle B to exist at then the probability of particle A being accelerated by as we know depends on mass equivalence of the other particle and distance of separation x . Since, here Mrel is constant, probability of will only depend on the probability of the distance to separation between the particles equal to x [3,4].

By Born rule, we know

Since proper length in change with distance of separation, the term is multiplied.

Since we know sum of probability of all possibilities should be 1, therefore a condition must be set

To get,

We can set the normalizing constant

Let’s say there are N number of spacetime co-ordinates is non-zero, i.e., the probability of the particle B described by has a non-zero chance of existing at . Let the position co-ordinates that satisfy the above criterion be . Besides it, on taking into account the uncertainty in momentum of the particle B at time coordinates we can say there are number of momentum eigenvalues at time respectively [5].

Since there is an uncertainty in momentum of the particle B, the mass equivalence of particle B be uncertain. The probability of a mass equivalence eigenstate is of the particle B.

Thus, probability of the particle A being accelerated by is equal to

Let there be a particles particle A at x0 , t0 . As mentioned earlier there are N number of spacetime co-ordinates where is non zero. Since there is uncertainty in momentum, at time let’s say there are number of momentum eigenvalues at time respectively.

Therefore, there are possible values of which will have a non-zero probability by which the particle can be accelerated towards x0+x in time interval . Therefore, particle A will have a non-zero probability of existing in all possible coordinates after the time interval .

If particle A described by has uncertainty in position such that it could be at two positions r and s at time t0 simultaneously such that In such a case, particle A will be accelerated by all values of at r and by at s bearing a non-zero probability. Since there are possible values of which particle A at r and s respectively will be accelerated by, in time interval . Thus, there are possible particle evolution since at r, there are ways the particle can evolve in time and at r, there are ways the particle can evolve in time and maximum number of possible locations where particle A can have non-zero probability of existing after time interval [6,7].

To determine wave evolution in 1 dimension, first we divide the X axis into infinite infinitesimally small segments such that the mean position of the ith segment is termed as and points from be labelled as and let be one of the mass equivalence eigenvalue of [7].

Let’s say using the above-mentioned rules, we get to know that the part of the wave in the 1,2,3,4….ith segment can evolve in time interval number of ways respectively, i.e., the part of the wave in the ith segment can be can be accelerated by all values of towards where for distance of separation from the ith segment , the condition is non-zero is satisfied [1].

Let’s define a scenario as a possible particle time evolution in time interval in which the part of the wave in each of the segments is translated from its initial position by is any of the values by which the part of the wave in the ith can be accelerated towards can be.

This can be written in form of an equation as:

Where,

Here the translates only the part of the wave solutions of the field in ith segment by any of the possible values of .

Probability of a possible wave evolution, let’s say scenario Q:

Let be the field after a time interval in scenario Q and be normalized by a normalizing co-efficient as due to uncertainty in acceleration in each of the segments; in some scenarios the part of the wave solutions of the field in, let’s say the jth segment accelerates faster than the part of the wave solutions of the field in any of the segments prior to it and end up out of phase with each other at, say x, which arises the need for normalization [7].

Probability of the particle A to possess momentum p due to this scenario contributing to overall probability of particle A have momentum after time interval

of the particle A to exist in the co-ordinate x due to this scenario contributing to overall probability of particle A being at x after time interval

Therefore,

III Creation and Annihilation Operators in Quantized Gravitational Field

In case of any free field, we can directly apply the same principle to get its time evolution in a quantized gravitational since quantum fields are superposition of wave solutions.

But in case of an interacting field theory, we need to take into note redshift of each frequency node of the field in every possible scenario. Using the above derived rules, we can retrieve what would the frequency node look like in a quantized gravitational field. After superimposing all the nodes, we can get to know how the field would look like in a quantized gravitational field.

In case of any free field, we can directly apply the same principle to get its time evolution in a quantized gravitational since quantum fields are superposition of wave solutions.

But in case of an interacting field theory, we need to take into note redshift of each frequency node of the field in every possible scenario. Using the above derived rules, we can retrieve what would the frequency node look like in a quantized gravitational field. After superimposing all the nodes, we can get to know how the field would look like in a quantized gravitational field.

For example, let’s take the interacting electric field

We can see E(x) depends on and A . Since we know let us see what the field A induced by the fermion is like:

The contribution of the frequency node k of the field to the field is

As we know in a gravitational field, waves are redshifted by gravity,

where and are wavelength at infinitely far away from the particle and at distance r away from the particle.

Therefore, momentum of the wave at distance r away from the particle will be

Since is the momentum is absence of a gravitational field, therefore,

Let

Substituting value of r in the above equation,

Therefore, in a gravitational field,

As we know the uncertainty in the geometry of the space time as mentioned earlier due to uncertainty in momentum and position of the particle as mentioned in the second section of the article. Let us say, at the point at time t, there are spacetime geometries, each with a certain probability of it.

After that, we need to find the modulus of the amplitude of that frequency node of the field in both momentum and position spaces of each of the scenarios using the earlier derived rules and sum up all the modulus of the amplitude of that frequency node over all possible scenarios in both spaces to get the resultant amplitude of that frequency node of the field in both spaces. Then using the Gerchberg-Saxton algorithm, we can retrieve the field in the momentum space for both the creation and the annihilation part which can be used to get the field in position space [8].

Summary of the Above Derived Rules

Let me explain the above-mentioned method, suppose the range over which the excitation in the field, i.e., particle A is non-zero be divided into 3 segments such that these segments have negligible width FIG.1.

physics-astronomy-field

Figure 1: The range over which the excitation in the field

Using the above-mentioned rules, one can find the possible values of acceleration due to gravity with their associated probability for a particle in segment 1, 2 and 3. Suppose there was n, m and o number of values by which a particle could be accelerated, if present in any of the segments respectively. Similarly, part of the field in each of the segments is accelerated by any of the possible value of acceleration it could undergo in that segment for time interval . To get the probability of this field evolution, the probability of the value of acceleration the part of the field in each of the segments is multiplied. Then, the field in this field evolution after time interval is normalized and its contribution to the overall probability of the particle existing at a point x is the sum of probability of the field evolution and the squared modulus of the field in that scenario.

Since, there are n, m and o number of values by which a particle could be accelerated, if present in any of the segments respectively, there are number of possible field evolutions. The overall probability of the particle existing at a point x is the sum of probabilities of all number of possible field evolutions. Similarly, the overall probability of the particle having momentum p is also determined. Then the Gerchberg Saxton algorithm is used to retrieve the field. To increase accuracy of the method, one needs to increase the number of segments the range over which the excitation in the field, i.e., particle A is non-zero.

Similarly, for a particle self-interacting with itself in the gravitational field, we space into infinite infinitesimally small segments. The part of the field of the particle A in the ith segment at will be accelerated by the above mentioned rules, but values of will depend on The part of the field in ith segment will be accelerated by whose values depend on x (the distance and separation) and the mass equivalence eigen values at time .

When x=0,

We know that the part of the field in ith segment will interact with itself, i.e., it will be accelerated towards itself, Thus can be ignored . Therefore, it will be equal to when x=0 for all self-interactions FIG.2.

physics-astronomy-vacuum

Figure 2: To study energy conservation due to self-interaction, let’s imagine an excitation in the field in vacuum

We can see that the wave function is non-zero over the range (-4,0) at t=0. The part of the wave function at -4 will be accelerated in forward direction due to self-interaction. The scenario in which the part of the wave function will be accelerated the least is when x=0, in which it will be translated by . The part of the wave function at 0 gets translated the most is when x=0, as in other scenario we can see it will be accelerated in the direction opposite to its motion [9].

Therefore, the particle after time interval will have a probability to exist over the range . Since is non-zero over the range , there will also be non-zero over it. Since the wavelength of the particle does not change, we can say conservation will not be violated due to self-interactions since

References