**Authors:** Shuang-ren Zhao

The path integral in quantum mechanics is a very important mathematical tools. It is widely applied in quantum electrodynamics and quantum field theory. But its basic concepts confuse all of us. The first thing is the propagation of the probability. The second is the path can be any paths you can draw. How this can work? In this article, a new definition of energy pipe streamline integral is introduced in which the mutual energy theorem and the mutual energy flow theorem, mutual energy principle, self-energy principle, Huygens principle, and surface integral inner product of the electromagnetic fields are applied to offer a meaningful and upgraded path integral. The mutual energy flow is the energy flow from the emitter to the absorber. This energy flow is built by the retarded wave radiates from the emitter and the advanced wave radiates from the absorber. The mutual energy flow theorem guarantees that the energy go through any surface between the emitter and the absorber are all equal. This allow us to build many slender flow pipes to describe the energy flow. The path integral can be defined on these pipes. This is a updated path integral is referred as the energy pip streamline integral. The Huygens principle allow us to insert virtual current sources on any place of the pipes. Self-energy principle tell us that any particles are consist of 4 waves: the retarded wave, the advanced wave and another two time-reversal waves. All these waves are canceled and hence the waves do not carry or transfer any energy. Energy is only carried and transferred by the mutual energy flow. Hence the mutual energy flow theorem is actually the energy flow theorem. Wave looks like probability wave, but mutual energy flow are real energy flow is not a probability flow. In this article the streamline integral is applied to electromagnetic field or photon or other particle for example electrons.

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[v1] 2018-08-19 18:13:25

[v2] 2018-09-01 10:49:25

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