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Currently things are becoming more and more technical. Although I want to make this as accessible for the general reader as possible I fear that I may have already delved so deep into the technical aspects of the topic of quantum mechanics that any casual reader would simply loose interest. Please bare with me. I’ll try my best to make this simple.

At the heart of any scientific endeavour lies the desire to understand something about our universe. The sad fact is that the scientific method sometimes provides us with a mindless mathematical process of making successful predictions without a clear understanding of what this mathematical process tells us about nature. Quantum mechanics is such an example.

Some people believe that quantum mechanics cannot be a description of nature at the fundamental level. Even if this is true quantum mechanics must be true at some level. As a result the mathematical formulation of quantum mechanics must mean something. One should distinguish between the mathematical formulation of quantum mechanics and the interpretation of quantum mechanics. The latter includes such things as vacuum fluctuations and the existence of particles, which I previously argued may be wrong interpretations. What one then needs to do is to discard the current interpretation of quantum mechanics and consider only the mathematical formulation of quantum mechanics and then ask oneself what this mathematical formulation tells us about nature.

This brings me to the notion of virtual particles, which is based on both vacuum fluctuations and the existence of particles. There are many measurable physical effects that seem to point to the existence of virtual particles. Yet can that be true if there are no vacuum fluctuations and no particles? The point is that virtual particles are not necessarily the only possible explanation of these measurable physical effects. To see this we need to consider the mathematical formulation of quantum mechanics more carefully. (Here I include quantum field theory.)

Let’s take the Coulomb field of a charged particle as our example for this discussion. In classical electromagnetic theory this Coulomb field is taken as a static field (for a stationary particle), described by a smooth function of space. If another charge particle comes close to this particle it will experience a force due to this Coulomb field. The latter is something that we know from experiments.

In quantum mechanics (or rather quantum field theory) the Coulomb field is replaced by a cloud of virtual particles (virtual photons). The force that another particle would feel is now interpreted as an exchange of these virtual photons. At least that is according to the interpretation of the quantum physics. It does not follow directly from the mathematical formulation of (in this case) quantum field theory.

To explain why I’ll use a little diagram, called a Feynman diagram, after Richard Feynman, its inventor.
This type of diagram is used as an aid in quantum field theory calculations. This particular Feynman diagram represents the scattering between two charged particles (electrons) via the exchange of a virtual photon. Each line represents a particle. There are two incoming electrons, two outgoing electrons and one virtual photon that is exchanged between the two electrons. As such the diagram is interpreted as the interaction among particles. However, if one looks at the mathematical expression that this diagram represents, one finds that the lines actually represent plane waves. Each of the particles is actually a field that is expanded in terms of plane waves using Fourier theory. The diagram therefore represents the interaction among the plane waves of the different fields. It also involves the summations (or integrations) over all such plane waves. There would in general be one such summation for each line in the diagram, five in total. However, one can restrict the direction of propagation of the ingoing and outgoing fields, which would effectively remove the summations associated with these field. The summation over the plane waves of the (virtual) photon field (Coulomb field) would however still remain.

Richard Feynman was a staunched believer in particles, but I am sure he know full well that these diagrams are actually representations of the interaction among the plane waves that make up the different fields.

The mathematical calculation expands the Coulomb field in terms of plane waves and determines their interaction with the plane waves of the electrons fields. This Coulomb field is the same one that we find in the classical theory. Nowhere in the actual mathematical quantum field theory calculation does the existence of any particle, virtual or not, appear. By looking at these calculations we see that the existence of virtual particles does not as such play any role in any prediction made by quantum field theory. The fields that mediate the interaction in quantum physics do not need to consist of particles.

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