Abstract
Causal fermion systems were introduced in [FGS] as a reformulation and generalization of the setting used in the fermionic projector approach [F7]. In the meantime, the theory of causal fermion systems has evolved to an approach to fundamental physics. It gives quantum mechanics, general relativity and quantum field theory as limiting cases and is therefore a candidate for a unified physical theory. In this chapter, we introduce the mathematical framework and give an overview of the different limiting cases.
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Notes
- 1.
The support of a measure is defined as the complement of the largest open set of measure zero, i.e.
It is by definition a closed set. This definition is illustrated in Exercise 1.5.
- 2.
- 3.
For example, one may choose
as the set of all vectors 
satisfying the conditions
- 4.
It is a general property of the closed chain that if \(\lambda \) is an eigenvalue, then so is \(\overline{\lambda }\); see Exercise 1.15.
- 5.
In order to avoid confusion, we note that the operators \(\hat{\Psi }(x)^\dagger \) which appear in the usual equal-time canonical commutation relations \(\{ \hat{\Psi }^\alpha (t,\vec {x}), \hat{\Psi }^\beta (t,\vec {y})^\dagger \} = \delta ^\alpha _\beta \, \delta ^3(\vec {x}-\vec {y})\) are related to the above operators by \(\hat{\Psi }^\alpha (x)^\dagger = 2 \pi \sum _{\beta =1}^4 \hat{\Psi }^\beta (x)^* \,(\gamma ^0)^\beta _\alpha \).
as the set of all vectors 
satisfying the conditions
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© 2016 Springer International Publishing Switzerland
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Finster, F. (2016). Causal Fermion Systems—An Overview. In: The Continuum Limit of Causal Fermion Systems. Fundamental Theories of Physics, vol 186. Springer, Cham. https://doi.org/10.1007/978-3-319-42067-7_1
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DOI: https://doi.org/10.1007/978-3-319-42067-7_1
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