fractal path

[AntonioLao]

In conventional quantum field theory the path integral for a free point particle quantum mechanics is written in Dirac’s bra and ket notations as áF|exp(iHT)|Iñ where I is the initial state and F is the final state. Exp(iHT) is the unitary operator and H is the Hamiltonian of kinetic energy function plus the potential energy function. T is the total of the time parameter between the quantum transition between I and F, and of course i is the imaginary unity.

However, if the unitary operator is replaced by the real Hadamard operator of square of energy as space-time quantum then the path integral become time independence and I and F can be replaced by the values of the fractal dimension. Since gauge invariance applies to fractal dimensions ½ is equivalent to 1½ or 2½ or 3½, so that the whole number parts can be simply ignored. The new path integral becomes á0|H|½ñ relative to the state of the quantum vacuum 0. Perfect symmetry implies that á0|H|½ñ=á½|H|0ñ. However, spontaneous symmetry breaking start to appear where and when the single Hadamard operator H is replaced by its compositions such that in general á0|HH···HH|½ñ≠á½| HH···HH|0ñ. Symmetry breakings allow 2 things to happen: (1) increase in mass decrease in energy or (2) decrease in mass increase in energy obeying the principle of mass and energy equivalence: E=mc. For the special case of perfect symmetry, c=1 and E=m. An example of a fractal path between fractal dimension ½ and 1/6 is á1/6|HH···HH|½ñ≠á½| HH···HH|1/6ñ. In reference to the algebra of yin-yang compositions, it can be shown that the mass is inverse proportional to the fractal dimension. However, at fractal dimension of ½ the rest mass of the free particle is exactly zero.

[Profpat]

In conventional quantum field theory the path integral for a free point particle quantum mechanics is written in Dirac’s bra .....

I didn't know that Paul wore a bra, much less that he wrote notations in it. What size was he 48 AAA?

[AntonioLao]

The reason you cannot know because he wore them in the imaginary complex plane using spinors and tensor calculus and the fact that they are also invisible to the naked eye can only left everything to the imagination, a nation of virtual images and abstract realities.