Realistic definition of Grassmann numbers and fermionic states

Details Realistic definition of Grassmann numbers and fermionic states Realistic definition of Grassmann numbers and fermionic states

2012-02-20

arxiv.org/abs/1202.4449v1

Physics

General Physics

Scientific paper

In this paper we attempt to define Grassmann numbers as elements living in multidimensional space, that exist independently of path integral. Furthermore, we define path integral as literally a "limit of the sum" in such a way that it approximates the expected mathematical properties of Grassmann integration. We then proceed to show that "Harmonic oscillator" in Fermionic space produces a linear function. The latter has two degrees of freedom per variable (the slope and the height at the origin), just as expected from Fermi exclusion principle. At the same time, the linear function is living in a continuum space; in other words, only two out of uncountably many degrees of freedom are "utilized". However, we argue that we would be able to utilize the rest of the degrees of freedom if we were to modify the structure of our Lagrangian. Such modification involves the use of functions that can not be generated from the usual wedge products of Grassmann numbers. Thus, it requires an appleal to the fact that Grassmann numbers are well defined mathematical objects, independently of standardly-accepted formalism.

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