Abstract
The theory of causal fermion systems is an approach to describe fundamental physics. We here introduce the mathematical framework and give an overview of the objectives and current results.
Mathematics Subject Classification (2010). 81-02, 81-06, 81V22
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Notes
- 1.
- 2.
For completeness, we derive the inequality (\(\star\)): Since the operator \(\sqrt{\vert y\vert }-\sqrt{\vert x\vert }\) is symmetric and has finite rank, there is a normalized vector \(u \in \mathcal{H}\) such that
$$\displaystyle{ \Big(\sqrt{\vert y\vert }-\sqrt{\vert x\vert }\Big)u = \pm \Big\|\sqrt{\vert y\vert }-\sqrt{\vert x\vert }\Big\|\,u\:. }$$(17)Possibly by exchanging the roles of x and y we can arrange the plus sign. Then
$$\displaystyle{\Big\|\sqrt{\vert y\vert }-\sqrt{\vert x\vert }\Big\| =\big\langle u\,\big\vert \,\Big(\sqrt{\vert y\vert }-\sqrt{\vert x\vert }\Big)u\big\rangle \leq \big\langle u\,\big\vert \,\Big(\sqrt{\vert y\vert } + \sqrt{\vert x\vert }\Big)u\big\rangle \:,}$$where in the last step we used that the operator \(\sqrt{\vert x\vert }\) is positive. Multiplying by \(\big\|\sqrt{\vert y\vert }-\sqrt{\vert x\vert }\big\|\) and using (17) with the plus sign, we obtain
$$\displaystyle\begin{array}{rcl} & & \Big\|\sqrt{\vert y\vert }-\sqrt{\vert x\vert }\Big\|^{2} {}\\ & & \quad \leq \frac{1} {2}\bigg(\big\langle u\,\big\vert \,\Big(\sqrt{\vert y\vert } + \sqrt{\vert x\vert }\Big)\Big(\sqrt{\vert y\vert }-\sqrt{\vert x\vert }\Big)u\big\rangle +\big\langle \Big (\sqrt{\vert y\vert }-\sqrt{\vert x\vert }\Big)u\,\big\vert \,\Big(\sqrt{\vert y\vert } + \sqrt{\vert x\vert }\Big)u\big\rangle \bigg) {}\\ & & \quad = \frac{1} {2}\:\big\langle u\,\big\vert \,\left \{\Big(\sqrt{\vert y\vert } + \sqrt{\vert x\vert }\Big),\Big (\sqrt{\vert y\vert }-\sqrt{\vert x\vert }\Big)\right \}u\big\rangle =\big\langle u\,\big\vert \,\big(\vert y\vert -\vert x\vert \big)\,u\big\rangle \leq \big\|\vert y\vert -\vert x\vert \big\|\:. {}\\ \end{array}$$We thus obtain the inequality \(\big\|\sqrt{\vert y\vert }-\sqrt{\vert x\vert }\big\|^{2} \leq \big\|\vert y\vert -\vert x\vert \big\|\). Applying this inequality with x replaced by x 2 and y replaced by y 2, it also follows that \(\big\|\vert y\vert -\vert x\vert \big\|^{2} \leq \big\| y^{2} - x^{2}\big\| \leq \big\| y - x\big\|\,\big\|y + x\big\|\). Combining these inequalities gives (\(\star\)).
- 3.
For example, one may choose \(\mathcal{D}(P)\) as the set of all vectors \(\psi \in \mathcal{K}\) satisfying the conditions
$$\displaystyle{\phi:=\int _{M}x\,\psi (x)\,d\rho (x)\: \in \:\mathcal{H}\qquad \mbox{ and}\qquad \vert \hspace{-0.85005pt}\vert \hspace{-0.85005pt}\vert \phi \vert \hspace{-0.85005pt}\vert \hspace{-0.85005pt}\vert <\infty \:.}$$
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Finster, F. (2016). Causal Fermion Systems: An Overview. In: Finster, F., Kleiner, J., Röken, C., Tolksdorf, J. (eds) Quantum Mathematical Physics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-26902-3_15
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