Spin in the path integral: anti-commuting versus commuting variables
F.G. Scholtz, A.N. Theron, H.B. Geyer (ITP, University of, Stellenbosch, Stellenbosch, South Africa)
TL;DR
This paper explores the equivalence of path integral representations of spin dynamics using anti-commuting and commuting variables, establishing a bosonization dictionary and a free field realization in the path integral framework.
Contribution
It introduces a bosonization dictionary for spin and fermion operators and constructs a Dyson mapping within the path integral formalism.
Findings
Established equivalence between Grassmann and commuting variable path integrals for spin
Developed a bosonization dictionary for spin and fermion operators
Constructed a free field realization (Dyson mapping) in the path integral setting
Abstract
We discuss the equivalence between the path integral representations of spin dynamics for anti-commuting (Grassmann) and commuting variables and establish a bosonization dictionary for both generators of spin and single fermion operators. The content of this construction in terms of the representations of the spin algebra is discussed in the path integral setting. Finally it is shown how a `free field realization' (Dyson mapping) can be constructed in the path integral.
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