Virtual Particles Explain Everything

This is preliminary. It's a work in progress.

It may take a minute to load
this page because of all the math symbols.

INTRODUCTION

The Spacetime Manifold and Quantum Theory

Energy is Virtual Particle
Trading

ABSTRACT

In my previous article I explained how the path integral for a free particle can be derived from the principles of logic alone. There it was discovered that the wave function represents the material implication of logic. This wave function is the transition amplitude for a particle to go from one place to another. And it seems intuitive to expect that all the properties and events in nature will be the result of some underlying principle of cause and effect, some premises having consequences that we observe as properties and events. Since implications gave rise to transition amplitudes that are so basic to the development of the path integral, I will take these as the most fundamental virtual process in all of physics. And I will refer to them as virtual particles. It's my intent here to show that these virtual particles are part of the definition of spacetime itself, and that they are fundamental in the definition of the wave function of quantum mechanics, and that spacetime, mass, energy, the speed of light and the gravitational field, entanglement, charge, the coupling constants of nature, dark energy, dark matter, the Higgs field and the inflaton field, can all be explained in terms of them. This is accomplished by recognizing that all processes must propagate through the collection of these virtual particles which defines space, and any propagation at all occurs through a series of "virtual particle trading" through this space of virtual particles. The properties of these virtual particles are the properties of space itself and get transferred to properties of particles and how they move. Obviously this is an ambitious project, and for now I have only heuristic arguments. But it seems to construct a consistent story that provides clues on how to develop a mathematical model from it. I think it is at least a path to a theory of everything.

Virtual particles are somewhat of a contentious issue in the physics community. They seem to be used as an explanatory device to create a visual image of what is going on at the quantum level. But there does not seem to be an independent mathematical description of their behavior by themselves. They exist as parts of an integral that are summed over in the larger calculation of observables. So they don't seem to have any independent existence outside the integral. And so they are not in and of themselves observable. But the accumulative effect they have can be summed up to give measurable effects like the Lamb shift and the Casimir effect. So the debate can rage on over whether they actually exist or not. Here I'm content to think of virtual particles as mathematical devices, as small parts which only have accumulative effects. But if it so happens that this perspective does in fact explain everything, then it will be hard to deny that they actually exist on some level.

Some of the hesitation to accept virtual particles may come from the fact that they seem to be an attempt to explain the fundamental, ontological underpinnings for quantum theory. And the consensus seems to be that asking questions about why the universe is quantum mechanical is immature. We have our mathematical framework which allows us to calculate observables which we can confirm by experiment. And so there is no need to explain it further. The advice seems to be "shut up and calculate". And any attempt to dig deeper is discouraged. However, in the previous article I believe I've explained from the first principles of reason why quantum theory is logically necessary. And with this as my encouragement, I believe it's possible to understanding everything else.

In classical physics, events are described in terms of differential things such as velocity, acceleration, and the force at various points on an object. Then these differential parts are summed up in integrals to give measurable things like distance and weight and other properties. And this summing up of differential parts is meant to be an explanation of the whole in terms of fundamental constituents. These differential parts form the ontological foundations on which the theory rests. And no body questions the legitimacy of doing that in classical physics. So it would be nice if we were able to do the same thing in quantum physics.

The problem is that in quantum theory we have the concept of superposition. The various ways an event can happen all occur at the same time and interfere with each other to produce the final effect. There is no one differential cause that can be integrated to give us an overall effect. And so it is difficult to see the foundations of quantum theory. In classical physics, the integrand involves one differential quantity that describes what is happening at each point in space. Then that is integrated along a path or in a volume to give overall effects. That seems intuitive enough. But in quantum theory, the integrand involves a function of many points, and there is an infinite number of integrations to take into account every possible combination of points that the function can have. The function being integrated loses any connection to what's happening at any particular time and place. The infinite integrations seem to be describing what's happening everywhere at the same time. And when one particular event is described by what's happening everywhere, then the importance of particular events gets blurred. The formulism loses explanatory power to describe exactly what cause had exactly what effect. And it becomes difficult to say what the foundational building blocks are on which the theory is built.

What I intend to show is that virtual particle pairs are simply an alternate description of the spacetime manifold. And I will show that they give rise to the differential parts that are necessary to construct the path integrals of quantum mechanics. And they are responsible for the emergence of particle properties and the curvature of space due to gravity as well as other phenomena. I will try to show some of the math that describes virtual particles. But in my explanation of various phenomena, I will only use the basic properties commonly attributed to virtual particles. Virtual particles come in pairs, particle and antiparticle, that pop into existence for a very brief moment and then pop out of existence by canceling each other out. And if they do not cancel out with their antiparticle, they accumulate mass and charge and become real particles.

The Spacetime Manifold and Quantum Theory

Previously, I showed how a conjunction of points necessarily means that one point logically implies another. Implication was implemented using the Dirac measure. The Dirac measure assigns a numeric value of 1 if the element of the Dirac measure is included inside the set of the Dirac measure, and it assigns a value of 0 if the element is not in the set. This was how numeric values were introduced into propositional logic. This is also how a primitive probability distribution was introduce into the theory. The set of the Dirac measure can be thought of as a bin in a sorting operation; the element can be thought of as a sample that belongs in that bin or another. And if the element of the Dirac measure is found inside the set of the Dirac measure, then we add 1 to the number of samples in that bin. This is the basic operation necessary in forming a probability distribution since it is derived by counting the relative number of samples in the various possible bins.

The material implication of logic has a premise and a conclusion. The set of the Dirac measure can be thought of as the premise of the implication, and the element of the Dirac measure can be thought of as the conclusion. A set can be thought of as a single proposition as well when we equated it to the conjunction of all the propositions it contains that are then thought of as elements in that set. Then each element represents a proposition and is considered true if it is included in the set. An implication being true then means that the conjunction of all the propositions that make up the premise are true, including the proposition representing the element of the Dirac measure. So if the set-proposition representing the premise is true, then the element-proposition representing the conclusion is true simply because that element is in the set. And in this way the Dirac measure is made to represent the logic of material implication.

Yet, we think of the spacetime manifold as consisting of a conjunction of all the various points in it. For we can say this point is part of it and that point is part of it. And we can think of each point as a proposition. For we can say that it is true that this point exists in space, and it is true that that point exists in space, etc. Then we can consider how this conjunction of points means that each point implies another. And it turns out that the implication between points and the implication of implications between points forms the quantum fields of the Standard Model.

So, we want to assign unique coordinates to each point in spacetime. These points sometimes serve as a premise and sometimes serve as a conclusion in the various implications in the theory. But the set of the Dirac measure is usually described as encompassing a region of space; it covers a whole range of coordinates that can't be assigned a unique point in coordinate space. So it is difficult to see how the set of the Dirac measure, which is a region, can interchangeably play the role of a premise, which is a point. This is fixed by letting the set of the Dirac measure shrink down in size until it only surrounds one point. Then it can be assigned the unique coordinates of that point and serve as either a particular premise or conclusion as needed. And functions can be constructed from the two coordinates, and the coordinates can be interchanged if the roles are reversed.

In the language of point set topology, the set of the Dirac measure can be seen as a neighborhood of the point element. And the premise-set shrinking in size to include only the conclusion-point describes the Hausdorff property of a manifold. So my prescription specifies a manifold. And if we are allowed to consider whether any neighborhood contains or not any point and assign a function the value of 1 if it does and 0 otherwise, then we can use this to represent the implication of logic and derive the quantum fields of quantum field theory. So I believe my construction implies both a manifold and quantum theory. And quantum field theory is seen as a construction necessarily implied by the spacetime manifold. The fields of quantum field theory are not added constructions onto spacetime; they are derived directly from the fact that the points of the spacetime manifold exist in conjunction and so imply each other, and those implications give us the quantum fields.

(As an aside, I wonder if these differential neighborhoods can be added up in some way to give the usual union and intersection of larger sets as required in the definition of a topology or a probability space. I wonder also if the function of 1 or 0 depending on whether a point is an element or not of a set is part of the definition of a completely regular space. If so then my construction describes a topological space necessary in the definition of a manifold. I ask these questions here.)

A coordinate system labels points of a frame of reference, and these points exist in conjunction with each other. And the implications that follow from this conjunction form the quantum fields that give rise to virtual particles everywhere in that reference frame. So each person's reference frame refers to his own set of virtual particles. Space itself is equivalent to virtual particles. Everything that occurs in that coordinate system has to be described in terms of the virtual particles of that system, e.g. how virtual particles become real, and how real particles get their properties, how they move through space, and how they interact with each other. Any change or propagation whatsoever, must be a process by which the virtual particles at one place exchange information with the virtual particles at another place. There simply is no other way to describe things except in terms of the set of virtual particles of that reference frame.

Though many physicists may talk of virtual particles in a heuristic sense, the math of virtual particles usually appears in the perturbation methods of quantum field theory (QFT). They show up in the Feynman diagrams showing how one set of particles can combine or decay into another set of particles. Virtual particles don't show up in regular quantum mechanics because no wave function for them has ever been written down. However, if they exist at all, as they do in QFT, then they should have a wave function as well. The wave function for a virtual particle can only propagate for a very short time before disappearing. Otherwise, we could calculate an expectation value for observable effects everywhere in space, but this we don't see. One way to get a wave function to become zero is if it encounters an infinite potential, but this we don't see at every point in space. Another way to get a wave function to disappear is to have it cancel with another wave function in superposition with it. This is why virtual particles are produced in pairs, so they can cancel each other out and leave no observable effects. Virtual particles always come in pairs, a particle and its antiparticle. They come into being at the same time and place, they travel some distance for a very brief moment, then they cancel each other out and annihilate each other at the same time and place. But how are both virtual particle and antiparticle guaranteed by the spacetime manifold?

In the previous article,
I described every point in space as a proposition, since it is true that each
point is on the manifold. So all these points exist in conjunction with each
other. And I show in Eq[4] that
for any two propositions in conjunction, we have, * q_{1}q_{2}* (

But exactly how does cancellation of amplitudes work? How do these virtual particles annihilate? Cancellation of amplitudes occur because the wave functions are of equal magnitude but 180° out of phase. When added together in superposition, one is in the opposite direction from the other so that they sum to zero. The phases come from the complex exponent in the gaussian representation of a transition amplitude, and the summation comes from using addition when dealing with alternatives. But since a complex number is not always 180° out of phase with its complex conjugate, the question is under what conditions can it be made to be 180° out of phase. Let's do the math.

The transition amplitude for a
particle to go from the position *x'* to *x* is denoted as
$<x|U(t)|x\text{'}>$, and is usually represented in quantum mechanical text books as,
$$<x|U(t)|x\text{'}>\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\left(\frac{m}{2\pi \hslash it}\right)}^{{\scriptscriptstyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{e}^{im{(x-x\text{'})}^{2}/2\hslash t}$$
where
*U*(*t*) is the propagator. I take this to be true even for
virtual particles for the time that they exist. Its antiparticle partner would be the complex conjugate. So it can be written,
$$<x|U(t)|x\text{'}{>}^{*}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\left(\frac{m}{2\pi \hslash (-i)t}\right)}^{{\scriptscriptstyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{e}^{-im{(x-x\text{'})}^{2}/2\hslash t}$$

And the minus sign under
the square root can be taken outside the parenthesis to get,
$$<x|U(t)|x\text{'}{>}^{*}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}i{\left(\frac{m}{2\pi \hslash it}\right)}^{{\scriptscriptstyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{e}^{-im{(x-x\text{'})}^{2}/2\hslash t}$$
with a leading factor of *i*. Both the virtual particle and
its virtual antiparticle partner both go from *x'* to *x* at the
same time. And they perfectly interfere and cancel out at the end point of *
x*. So they must exist in superposition with each other in order to
interfere and cancel out. Beside. virtual particle pairs are said to be
entangled with each other, and there must be a superposition in order for there
to be entanglement.

So in order to get the transition amplitudes for the virtual particle and antiparticle to cancel out, we need to have $$<x|U(t)|x\text{'}>\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}<x|U(t)|x\text{'}{>}^{*}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\text{\hspace{0.17em}}\text{\hspace{0.17em}}.$$ Or, expressing this with the exponential functions we get, $${\left(\frac{m}{2\pi \hslash it}\right)}^{{\scriptscriptstyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{e}^{im{(x-x\text{'})}^{2}/2\hslash t}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}i\text{\hspace{0.17em}}{\left(\frac{m}{2\pi \hslash it}\right)}^{{\scriptscriptstyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{e}^{-im{(x-x\text{'})}^{2}/2\hslash t}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}.$$ In order to simplify things a bit, let $$A={\left(\frac{m}{2\pi \hslash i}\right)}^{{\scriptscriptstyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}$$ and $$B=m{(x-x\text{'})}^{2}/2\hslash \text{\hspace{0.17em}}\text{\hspace{0.17em}}.$$ Then the above expression becomes, $$A\left(\frac{{e}^{iB/t}}{{t}^{{\scriptscriptstyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}+i\frac{{e}^{-iB/t}}{{t}^{{\scriptscriptstyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}\right)=0\text{\hspace{0.17em}}\text{\hspace{0.17em}}.$$ And then since $i={e}^{\frac{i\pi}{2}}$, we get $$A\frac{1}{{t}^{{\scriptscriptstyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}\left({e}^{iB/t}+{e}^{-iB/t+i\pi /2}\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\text{\hspace{0.17em}}\text{\hspace{0.17em}}.$$ It turns out that only for certain values of $t$ will this expression be true. This is shown by letting $$\frac{iB}{t}=\frac{i\pi}{4}+\frac{i\pi}{2}+i2\pi n$$ where $n$ is any integer. Then we have $$t=\frac{B}{\frac{\pi}{4}+\frac{\pi}{2}+2\pi n}$$ And substituting these into our expression gives $$A{\left(\frac{\frac{\pi}{4}+\frac{\pi}{2}+2\pi n}{B}\right)}^{\frac{1}{2}}\left({e}^{\frac{i\pi}{4}+\frac{i\pi}{2}+i2\pi n}+{e}^{-\frac{i\pi}{4}-\frac{i\pi}{2}-i2\pi n+\frac{i\pi}{2}}\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\text{\hspace{0.17em}}\text{\hspace{0.17em}}.$$ The first and last terms in the right exponential add up to $\frac{i\pi}{4}$ so that the expression becomes $$A{\left(\frac{\frac{\pi}{4}+\frac{\pi}{2}+2\pi n}{B}\right)}^{\frac{1}{2}}\left({e}^{\frac{i\pi}{4}+\frac{i\pi}{2}+i2\pi n}+{e}^{\frac{i\pi}{4}-\frac{i\pi}{2}-i2\pi n}\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\text{\hspace{0.17em}}\text{\hspace{0.17em}}.$$ The factor of ${e}^{\frac{i\pi}{4}}$ can be pulled out of both exponentials so that we get $$A{\left(\frac{\frac{\pi}{4}+\frac{\pi}{2}+2\pi n}{B}\right)}^{\frac{1}{2}}{e}^{\frac{i\pi}{4}}\left({e}^{\frac{i\pi}{2}+i2\pi n}+{e}^{-\frac{i\pi}{2}-i2\pi n}\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\text{\hspace{0.17em}}\text{\hspace{0.17em}}.$$ And since ${e}^{ix}={e}^{ix\pm i2\pi n}$ , this becomes $$A{\left(\frac{\frac{\pi}{4}+\frac{\pi}{2}+2\pi n}{B}\right)}^{\frac{1}{2}}{e}^{\frac{i\pi}{4}}\left({e}^{\frac{i\pi}{2}}+{e}^{-\frac{i\pi}{2}}\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\text{\hspace{0.17em}}0\text{\hspace{0.17em}}\text{\hspace{0.17em}}.$$ And finally, since ${e}^{\frac{i\pi}{2}}=i$ and ${e}^{-\frac{i\pi}{2}}=-i$ this becomes $$A{\left(\frac{\frac{\pi}{4}+\frac{\pi}{2}+2\pi n}{B}\right)}^{\frac{1}{2}}{e}^{\frac{i\pi}{4}}\left(i-i\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0$$ as desired. Notice that this is true for any $n$ whatsoever.

I know of no other way to get these amplitudes to cancel except to make $t$ discrete. And since $t=B/(\frac{\pi}{4}+\frac{\pi}{2}+2\pi n)$ and $B=m{(x-x\text{'})}^{2}/2\hslash $ , then $$\frac{B}{t}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{m{(x-x\text{'})}^{2}\text{\hspace{0.17em}}}{2\hslash t}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{m{(x-x\text{'})}^{2}\text{\hspace{0.17em}}}{2{t}^{2}}\frac{t}{\hslash}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{\pi}{4}+\frac{\pi}{2}+2\pi n\text{\hspace{0.17em}}\text{\hspace{0.17em}}.$$ We can recognize the $m{(x-x\text{'})}^{2}/2{t}^{2}$ as the kinetic energy $m{v}^{2}/2\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{E}_{k}$ . And the $\hslash $ can be brought over to the other side of the equal sign, and we can use the fact that $\hslash =h/2\pi $ to show that the above is equal to, $${E}_{k}t=h\left(\frac{3}{4}+n\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}.$$ This says that virtual particles have a discretized action. Notice also that this says that the virtual antiparticle is 180° out of phase with its virtual particle. But this says nothing about the relation of one virtual pair to another virtual pair. This makes me wonder if one virtual pair can interact with a separate pair and under what condition they do. That would probably mean that the phases no longer cancel and there would be some sort of permanent particle or other effect.

And there are also momentum states associated with each of the transition amplitudes, $<x|U(t)|{x}^{\prime}>$ . And momentum can be derived from a function of position by Fourier transforming the function of position to a function of momentum. See this wikipedia.org page. Since the transition amplitude is a function of the position, $x$ and ${x}^{\prime}$, we can find the momentum by Fourier transforming it. In the previous article material implication was mapped to a gaussian with a complex exponent. So let us start with a generic virtual particle having this kind of transition amplitude. I am using the transition amplitude in position eigenspace to represent a virtual particle going from ${x}^{\prime}$ to $x$. So a virtual particle going from one place to another would be represented as, $$<x|U(t)|{x}^{\prime}>={\left(\frac{m}{2\pi \hslash it}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{e}^{im{(x-x\text{'})}^{2}/2\hslash t}\text{\hspace{0.17em}}\text{\hspace{0.17em}}.$$

The Fourier transform used in quantum mechanics is,

$$F(p)=\frac{1}{{(2\pi \hslash )}^{{\scriptscriptstyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}{\displaystyle {\int}_{-\infty}^{+\infty}f(x){e}^{\raisebox{1ex}{$-ipx$}\!\left/ \!\raisebox{-1ex}{$\hslash $}\right.}\text{\hspace{0.17em}}\text{\hspace{0.17em}}dx}\text{\hspace{0.17em}}\text{\hspace{0.17em}}.$$

This differs from the usual transform that uses $k$ in the exponent with no $\hslash $, but here we are letting $k=p/\hslash $. See for example, here and here. Then the inverse Fourier transform is, $$f(x)=\frac{1}{{(2\pi \hslash )}^{{\scriptscriptstyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}{\displaystyle {\int}_{-\infty}^{+\infty}F(p){e}^{\raisebox{1ex}{$ipx$}\!\left/ \!\raisebox{-1ex}{$\hslash $}\right.}\text{\hspace{0.17em}}\text{\hspace{0.17em}}dp}\text{\hspace{0.17em}}\text{\hspace{0.17em}}.$$

But when we apply the forward Fourier transform to the gaussian representation of $<x|U(t)|{x}^{\prime}>$, we will get its Fourier transform with respect to $x$, which is $<p|U(t)|{x}^{\prime}>$. So to that end, $$<p|U(t)|{x}^{\prime}>=\frac{1}{{(2\pi \hslash )}^{{\scriptscriptstyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}{\displaystyle {\int}_{-\infty}^{+\infty}\left(\frac{{m}^{{\scriptscriptstyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}{{(2\pi \hslash it)}^{{\scriptscriptstyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{e}^{\frac{im{(x-x\text{'})}^{2}}{2\hslash t}}\right){e}^{\frac{-ipx}{\hslash}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}dx}\text{\hspace{0.17em}}\text{\hspace{0.17em}}.$$ which equals, $$\frac{1}{{(2\pi \hslash )}^{{\scriptscriptstyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}\frac{{m}^{{\scriptscriptstyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}{{(2\pi \hslash it)}^{{\scriptscriptstyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}{\displaystyle {\int}_{-\infty}^{+\infty}{e}^{\frac{im{(x-x\text{'})}^{2}}{2\hslash t}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\frac{ipx}{\hslash}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}dx}\text{\hspace{0.17em}}\text{\hspace{0.17em}}.$$

Now, it is interesting to note that, $${{\displaystyle {\int}_{-\infty}^{+\infty}{e}^{A{u}^{2}+Bu}\text{\hspace{0.17em}}du=\left(\frac{\pi}{-A}\right)}}^{{\scriptscriptstyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{e}^{\frac{-{B}^{2}}{4A}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}.$$

As is listed in the integration table here and here.

We can use this in the above by letting $u=x-x\text{'}$. Then $x=u+x\text{'}$, and $dx=du$. Then the Fourier transform becomes, $$\frac{1}{{(2\pi \hslash )}^{{\scriptscriptstyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}\frac{{m}^{{\scriptscriptstyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}{{(2\pi \hslash it)}^{{\scriptscriptstyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}{\displaystyle {\int}_{-\infty}^{+\infty}{e}^{\frac{im{u}^{2}}{2\hslash t}\text{\hspace{0.17em}}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{ip(u+x\text{'})}{\hslash}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}du}\text{\hspace{0.17em}}\text{\hspace{0.17em}},$$ or, $$\frac{1}{{(2\pi \hslash )}^{{\scriptscriptstyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}\frac{{m}^{{\scriptscriptstyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}{{(2\pi \hslash it)}^{{\scriptscriptstyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}\text{\hspace{0.17em}}{e}^{-\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{ipx\text{'}}{\hslash}}{\displaystyle {\int}_{-\infty}^{+\infty}{e}^{\frac{im{u}^{2}}{2\hslash t}\text{\hspace{0.17em}}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{ipu}{\hslash}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}du}\text{\hspace{0.17em}}\text{\hspace{0.17em}}.$$

If we let $A=im/2\hslash t$, and $B=-ip/\hslash $, then Fourier transform becomes, $$\frac{1}{{(2\pi \hslash )}^{{\scriptscriptstyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}\frac{{m}^{{\scriptscriptstyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}{{(2\pi \hslash it)}^{{\scriptscriptstyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}\text{\hspace{0.17em}}{e}^{-\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{ipx\text{'}}{\hslash}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\cdot \text{\hspace{0.17em}}\text{\hspace{0.17em}}\left({\left(\frac{\pi}{-A}\right)}^{{\scriptscriptstyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{e}^{\frac{-{B}^{2}}{4A}}\text{\hspace{0.17em}}\right),$$ or, $$\frac{1}{{(2\pi \hslash )}^{{\scriptscriptstyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}\frac{{m}^{{\scriptscriptstyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}{{(2\pi \hslash it)}^{{\scriptscriptstyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}\text{\hspace{0.17em}}{e}^{-\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{ipx\text{'}}{\hslash}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\cdot \text{\hspace{0.17em}}\text{\hspace{0.17em}}{\left(\frac{\pi}{-\left(\frac{im}{2\hslash t}\right)}\right)}^{{\scriptscriptstyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}{e}^{\frac{-{\left(\frac{-ip}{\hslash}\right)}^{2}}{4\left(\frac{im}{2\hslash t}\right)}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}.$$ And after some cancellation, this becomes the end result of, $$<p|U(t)|{x}^{\prime}>\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{1}{{\left(2\pi \hslash \right)}^{{\scriptscriptstyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}{e}^{-\text{\hspace{0.17em}}\frac{ipx\text{'}}{\hslash}\text{\hspace{0.17em}}-\text{\hspace{0.17em}}\frac{i{p}^{2}t}{2m\text{\hspace{0.17em}}\hslash}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}.$$

This is the momentum of the particle at ${x}^{\prime}$. It is a gaussian distribution centered at $p=-2mx\text{'}/t$.

If we want the change of momentum of the particle as it moves from
${x}^{\prime}$
to
$x$,
this would

And the Fourier transform of this is, $$<p\text{'}|U(t)|p>=\frac{1}{{(2\pi \hslash )}^{{\scriptscriptstyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}{\displaystyle {\int}_{-\infty}^{+\infty}\left(\frac{1}{{\left(2\pi \hslash \right)}^{{\scriptscriptstyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}}}{e}^{+\text{\hspace{0.17em}}\frac{ipx\text{'}}{\hslash}\text{\hspace{0.17em}}+\frac{i{p}^{2}t}{2m\text{\hspace{0.17em}}\hslash}}\right){e}^{-\frac{ip\text{'}x\text{'}}{\hslash}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}dx}\text{'}\text{\hspace{0.17em}}\text{\hspace{0.17em}},$$ or, $$\frac{1}{\hslash}\text{\hspace{0.17em}}{e}^{\frac{i{p}^{2}t}{2m\text{\hspace{0.17em}}\hslash}}{\displaystyle {\int}_{-\infty}^{+\infty}\frac{1}{2\pi}{e}^{ix\text{'}(\frac{p}{\hslash}-\frac{p\text{'}}{\hslash})\text{\hspace{0.17em}}}\text{\hspace{0.17em}}dx}\text{'}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}.$$

But we have $${\int}_{-\infty}^{+\infty}\frac{1}{2\pi}{e}^{ix\text{'}(\frac{p}{\hslash}-\frac{p\text{'}}{\hslash})\text{\hspace{0.17em}}}\text{\hspace{0.17em}}dx}\text{'}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\text{\hspace{0.17em}}\delta (\frac{p}{\hslash}-\frac{p\text{'}}{\hslash})\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\text{\hspace{0.17em}}\hslash \delta (p-p\text{'})\text{\hspace{0.17em}}\text{\hspace{0.17em}}.$$ See the article here. And this gives us, $$<p\text{'}|U(t)|p>\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\text{\hspace{0.17em}}{e}^{\frac{i{p}^{2}t}{2m\text{\hspace{0.17em}}\hslash}}\delta (p-p\text{'})\text{\hspace{0.17em}}\text{\hspace{0.17em}}.$$ And if we take the complex conjugate we finally get, $$<p|U(t)|p\text{'}>\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\text{\hspace{0.17em}}{e}^{-\frac{i{p}^{2}t}{2m\text{\hspace{0.17em}}\hslash}}\delta (p-p\text{'})\text{\hspace{0.17em}}\text{\hspace{0.17em}}.$$

The only non-zero value this has is when $p=p\text{'}$. This means that the virtual particle described by $<x|U(t)|{x}^{\prime}>$ does not change its momentum as it travels from ${x}^{\prime}$ to $x$.

And of course, we could have guessed this would be the momentum of a transition amplitude. For we have that if the amplitude is,

$$<x|U(t)|{x}^{\prime}>={\left(\frac{m}{2\pi \hslash it}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}{e}^{im{(x-x\text{'})}^{2}/2\hslash t}\text{\hspace{0.17em}}\text{\hspace{0.17em}},$$

then the exponent can be manipulated to get, $$\frac{im{(x-x\text{'})}^{2}}{2\hslash t}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{i{m}^{2}}{2m\hslash}\frac{{(x-x\text{'})}^{2}}{{t}^{2}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}t\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{i{(mv)}^{2}t}{2m\hslash}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{i{p}^{2}t}{2m\hslash}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}.$$

Notice that if we know $x\text{'}$ precisely, then this could be part of any transistion amplitude, $<x|U(t)|{x}^{\prime}>$. In other words, the $x$ of the transition amplitude could be anywhere. And since the momentum was derived from knowing both ${x}^{\prime}$ and $x$, and we don't know $x$, then we can't know anything about the moment. So if we know a particle's position exactly, then we can't know anything about its momentum.

Also, if we know the momentum exactly, then notice that we could have derived it from just about any pair of ${x}^{\prime}$ and $x$, as long as their difference is the same. So if we know a particle's momentum exactly, then we can't know anything about its position.

Given an initial wave function, $\psi (x\text{'},0)$, at $t=0$, the wave function at a later time, $t$, is $$\psi (x,t)={\displaystyle {\int}_{-\infty}^{+\infty}U(x,t;x\text{'})}\text{\hspace{0.17em}}\psi (x\text{'},0)\text{\hspace{0.17em}}dx\text{'}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}},$$where $U(x,t;x\text{'})=<x\left|{e}^{-iHt/\hslash}\right|x\text{'}>$ is called the propagator. It is also called the transition amplitude to go from $x\text{'}$ to $x$. But we can break up $H$ in to $n$ equal pieces of width $\epsilon $ so that $n\epsilon =1$ to get, $H=n\epsilon H=\epsilon H+\epsilon H+\epsilon H+\cdot \cdot \cdot +\epsilon H$. And we can insert $n$ copies of the resolution of identity, $${\int}_{-\infty}^{+\infty}|{x}_{i}><{x}_{i}|}\text{\hspace{0.17em}}d{x}_{i}=1\text{\hspace{0.17em}}\text{\hspace{0.17em}},$$ since it is equal to one. Then the propagator, $U(x,t;x\text{'})$, is equal to

${\int}_{-\infty}^{+\infty}{\displaystyle {\int}_{-\infty}^{+\infty}\cdot \cdot \cdot {\displaystyle {\int}_{-\infty}^{+\infty}<x\left|{e}^{-i\epsilon H/\hslash}\right|{x}_{1}><{x}_{1}\left|{e}^{-i\epsilon H/\hslash}\right|{x}_{2}><{x}_{2}\left|{e}^{-i\epsilon H/\hslash}\right|{x}_{3}>\cdot \cdot \cdot <{x}_{n}\left|{e}^{-i\epsilon H/\hslash}\right|x\text{'}>}}}\text{\hspace{0.17em}}d{x}_{1}d{x}_{2}d{x}_{3}\cdot \cdot \cdot d{x}_{n}\text{\hspace{0.17em}}\text{\hspace{0.17em}},$

and the wave function, $\psi (x,t)$, is equal to

${\int}_{-\infty}^{+\infty}{\displaystyle {\int}_{-\infty}^{+\infty}\cdot \cdot \cdot {\displaystyle {\int}_{-\infty}^{+\infty}<x\left|{e}^{-i\epsilon H/\hslash}\right|{x}_{1}><{x}_{1}\left|{e}^{-i\epsilon H/\hslash}\right|{x}_{2}><{x}_{2}\left|{e}^{-i\epsilon H/\hslash}\right|{x}_{3}>\cdot \cdot \cdot <{x}_{n}\left|{e}^{-i\epsilon H/\hslash}\right|x\text{'}>}}}\text{\hspace{0.17em}}\psi (x\text{'},0)\text{\hspace{0.17em}}d{x}_{1}d{x}_{2}d{x}_{3}\cdot \cdot \cdot d{x}_{n}\text{\hspace{0.17em}}\text{\hspace{0.17em}}.$

What this shows is that the way a particle
propagates is through a series of different
transition amplitudes from the initial to the final point. Since these $<{x}_{j}\left|{e}^{-i\epsilon H/\hslash}\right|{x}_{i}>$
inside the integrals are only a
small piece of the calculation of some observable, they are not considered
observables by themselves. In this sense they are virtual particle transition
amplitudes. A particle gets from one point to another by jumping from the
starting point to some other arbitrary point and then to the next arbitrary point
and then to the
next, until it reaches the final point. This is one possible way to get from
start to finish. But there is nothing special about this particular erratic path
from start to finish, so each possible path must be considered. The transition
amplitudes of a virtual particle and its complex conjugate virtual antiparticle
can be seen as paths of one step each from some* x** _{i}*
to some

But why should a particle make these jumps from one point to another along a path from start to finish? That seems a bit arbitrary and contrived. Yet what if these arbitrary jumps already existed everywhere anyway as virtual particles that continually pop into and out of existence? Then a path is accomplished by a real particle interacting with a series of virtual particles so that the transition amplitudes of the virtual particles are given to the real particle one after another until it makes its way from start to finish. If we assume that a real, bare particle can cancel with a virtual particle, then a real particle propagating through space is accomplished as follows: There already exists virtual particle pairs everywhere, including near a real particle. If a virtual particle-antiparticle pair appears near a real particle, then the real particle can annihilate with the antiparticle of the virtual pair. This leaves real the virtual particle that did not annihilate with its original partner; it becomes real. Thus the real particle uses the transition amplitude of the virtual particle to jump from one position to the next. And now that the "real-ness" has been handed off to the unannihilated virtual particle, it is now subject to another jump by annihilating with yet another virtual pair that pops up nearby, and so on through the path normally described with the path integral. So the particle propagates through space by trading off its real-ness property through a succession of unannihilated virtual partners. Since the distance and direction of the virtual pairs is unpredictable, this mechanism of virtual-particle-trading appears to be a random walk from the starting point to the ending point of the particle's motion.

And all this virtual-particle-trading only adds uncertainty in the particle's position as time increases. Since it is completely random which virtual particles interact with the real particle to determine its path, the distribution of the particle's position must be a gaussian curve that gets ever wider with time. But this is exactly what is predicted from the wave function for the position of a free particle not influenced by any potential. This indicates that the wave function can be described in terms of virtual particles. This is exactly what the last equation above is telling us. There are many different ways that this virtual-particle-trading could carry the real-ness of a particle from one place to another. But at the moment a measurement is made, the real-ness property is caught in the last virtual particle trade that occurred. This is the way a wave function collapses with measurement.

ENERGY IS VIRTUAL PARTICLE TRADING

In the last section, the coordinate frame of the spacetime manifold was shown to consist of a set of virtual particle pairs. Each coordinate frame is equivalent to its own set of virtual particle pairs that continually pop into and out of existence. And the propagation of real particles was described as a process of virtual particle trading whereby a real particle annihilates with a virtual anitiparticle which leaves real its virtual particle partner. In this way the transition amplitude of the virtual particle is transferred to the real particle, and so the real particle moves from place to place in this way.

The virtual particle pairs are locked in position because they are linked to the coordinate points of a reference frame. And nothing moves or changes except that they navigate through this field of virtual particles. So it must be that the faster objects move through space, then the more virtual particle trading is needed to accomplish this motion. Also, the faster things move, the more kinetic energy they have. So it would appear that kinetic energy can be defined in terms of the rate of virtual particle trading. This is why the kinetic energy of an object in one frame has a different kinetic energy in a different coordinate frames. There is not as much virtual particle trading going on in one frame verses the other. This might be obvious enough for kinetic energy, but what about other forms of energy like mass and potential?

In the next section I describe how real particles can come from virtual particles. This usually takes some form of acceleration. But since real particles come out of virtual particle pairs, which have no observable properties or effects, it would seem that the observable properties of a real particle are established by how it interacts with the surrounding field of virtual particles. For the observable properties of particles can only be discerned by how they interact with other things. And this interaction only occurs through the field of virtual particles. So the particle properties themselves are established by interaction with the field of virtual particles. And it may take some time for these properties to take effect.

For example, mass would be explained as the rate of virtual particle trading going on in the center of mass frame of reference that occurs simply because there are virtual particle pairs popping into existence near the bare particle. The more virtual particle trading that occurs just to set up the mass, then the less virtual particles are available to set up motion. So it becomes difficult to get massive objects moving. And if a particle is less massive, it would seem the more virtual particle pairs are available to start motion. This assumes, of course, that there is a limit on the rate of virtual particle trading that can occur in a given space in a given time. This limited rate also means that it takes time to set up a particle's properties, like mass and charge, once a virtual particle becomes real. And this limited rate of virtual particle trading may also be responsible for the limited speed of light. The question that remains here is why the rate of virtual particle trading would be limited.

I have to wonder if this way of getting mass through virtual particle trading may be another way of describing the Higgs mechanism of the Standard Model. Perhaps this trading of virtual particles are other words for a particle coupling to the Higgs field. Coupling usually refers to the strength of interaction. And we would have an interaction at the basic level of virtual particle trading as the real, bare particle cancels with the virtual antiparticle to leave real the virtual particle. And the strength of coupling that a particle has to the Higgs field may be describing how often virtual particles trade for that kind of particle.

The question becomes, why is energy conserved, and how does one particle transfer energy to another? This gets to the heart of an interaction between particles. What exactly happens when one particle with a given mass and velocity encounters another particle of a given mass and velocity? There are interactions that repel each other, and there are interactions that attract each other. There are glancing blows that change things very little, and there are head on collisions that change trajectories a lot. How can interaction be described in terms of virtual particles?

For a single real particle at rest, since every virtual particle trade is a jump to a random place, on the average all jumps tend to cancel so that a particle at rest tends to stay at rest. Of course, in a reference frame that is moving with respect to a particle, it would be the particle that would seem to be moving in that frame. So a particle in motion tends to stay in motion with the same velocity. It must be that there is a higher probability for particles to annihilate with antiparticles of the same momentum.

Now suppose there are two particles approaching each other. Let's suppose for the moment that they are both two real particles (not antiparticles). As the first real particle approaches the second, it tends to cancel with a virtual antiparticles of the opposite momentum so that it leaves real the virtual particle of the pair such that it continues the momentum of the first real particle. This means that the second real particle no longer has the option available to cancel out with that virtual pair since that virtual pair was used by the first real particle. And therefore, the second real particle will have more of tendency to propagate by canceling out with antiparticles in a different direction. On the average, the second real particle will cancel with virtual pairs whose direction is in a direction away from the approaching first particle. From then on the second real particle will tend to cancel with pairs that maintain the new momentum.

If the first real particle has mass, then it has more of a tendency to use up the virtual pairs in its vicinity to maintain that mass. And this removes more virtual pairs from the ability to trade with the second real particle. There is then more of an effect caused by those virtual pairs on the opposite side of the second real particle to move it away from the first. And so on average energy and momentum are conserved.

This suggests, of course, that potential energy is a measure of the availability of virtual pairs needed for propagation. This is rather obvious. The less virtual particle pairs are available in some region, then less likely a particle will travel in that direction. And the closer you are to the center of mass of a particle, the more it will use up virtual particles near the center, and the less likely it will be for other particles to propagate in that direction.

Now suppose that the first particle is real and the second particle is a real antiparticle. Then as the first particle approaches the second, the first particle will have a tendency to cancel with the antiparticle of some virtual pair that is in between the first and second particle. This would leave real a virtual particle. But now this virtual particle left real is more available to annihilate with the second real antiparticle. The first real particle is using up the virtual antiparticles nearby leaving more virtual particles available to the second antiparticle. The tendency will be to draw the first particle and the second antiparticle together. This is an account of the attractive and repulsive charge potential of particles with antiparticles. And it does seem to show that there may be a relationship between a particles charge and its mass, both being a result of the amount of virtual particle trading that it causes. Later I show how gravity that warps space is also a result of this virtual particle trading, though it has much less of an effect than charge.

The usual story is that the charge of a particle is "screened" by the polarization of virtual particles around a bare particle so that the effect of the charge is somewhat shielded more as the distance from the center of charge is increased. Perhaps my description is equivalent since both are a measure of how much the particle is separated from its antiparticle partner.

And since virtual particle pairs pop into and out of existence, they can be said to exist for a brief but finite amount of time. Therefore, it takes time for particle trading to traverse the transition amplitude of one virtual pair before traversing the transition amplitude of the next virtual pair. So there is a natural speed limit to any propagation since it takes time to transit each virtual pair in the succession of pairs in a particle's trajectory. This speed limit is the speed of light. And I assume it can be derived as the average transition distance of a virtual pair divided by its average transition time.

The reason that not everything propagates at the speed of light is because particles with the property of mass do virtual particle trading in all directions. And sometimes those particle trades are done in a direction opposite to its average motion. And of course, the average speed of propagation will be the same through the coordinates of any reference frame. And this alone gives rise to the effects of Special Relativity.

...

More about special and general relativity in terms of virtual particles here.