The Theory of Causal Fermion Systems

Causal fermion systems evolved from an attempt to combine several physical principles in a coherent mathematical framework. As a result, these principles appear in a specific way:

• The principle of causality. A causal fermion system gives rise to a causal structure and a time direction. The causal action principle is compatible with this notion of causality in the sense that the pairs of points with spacelike separation do not enter the Euler-Lagrange equations. In simple terms, points with spacelike separation do not interact.
• The local gauge principle. Local gauge freedom becomes apparent when representing the physical wave functions in bases of the spin spaces. More precisely, choosing a pseudo-orthonormal basis $(\mathfrak{e}_\alpha(x))_{\alpha=1,\ldots, \text{dim}(S_x)}$ of each spin space $(S_x, \prec .|. \succ_x)$, a physical wave function $\psi^u$ can be represented as

                    $\displaystyle \psi^u(x) = \sum_{\alpha=1}^{\text{dim} S_x} \psi^\alpha(x)\: \mathfrak{e}_\alpha(x)$

  with component functions $\psi^1, \ldots, \psi^{\text{dim} S_x}$. The freedom in choosing the basis $(\mathfrak{e}_\alpha)$ is described by the group of unitary transformations with respect to the indefinite spin inner product. This gives rise to the transformations

                    $\displaystyle \mathfrak{e}_\alpha(x) \rightarrow \sum_\beta U^{-1}(x)^\beta_\alpha\; \mathfrak{e}_\beta(x)$    and    $\displaystyle \psi^\alpha(x) \rightarrow \sum_\beta U(x)^\alpha_\beta\: \psi^\beta(x)$ .

  As the basis $(\mathfrak{e}_\alpha)$ can be chosen independently at each spacetime point, one obtains local gauge transformations of the wave functions, where the gauge group is determined to be the isometry group of the spin scalar product. The causal action is gauge invariant in the sense that it does not depend on the choice of spinor bases.

• The Pauli exclusion principle. This can be seen in various ways. One formulation of the Pauli exclusion principle states that every fermionic one-particle state can be occupied by at most one particle. In this formulation, the Pauli exclusion principle is respected because every wave function can either be represented in the form $\psi^u$ (the state is occupied) with $u \in \H$ or it cannot be represented as a physical wave function (the state is not occupied). Via these two conditions, the fermionic projector encodes for every state the occupation numbers $1$ and $0$, respectively, but it is impossible to describe higher occupation numbers.

  More technically, one may obtain the connection to the fermionic Fock space formalism by choosing an orthonormal basis $u_1, \ldots, u_f$ of ${\mathcal{H}}$ and forming the $f$-particle Hartree-Fock state

                    $\Psi := \psi^{u_1} \wedge \cdots \wedge \psi^{u_f}$ .

  Clearly, the choice of the orthonormal basis is unique only up to the unitary transformations

                    $\displaystyle u_i \rightarrow \tilde{u}_i = \sum_{j=1}^f U_{ij} \,u_j$    with    $U \in \text{U}(f)$ .

  Due to the anti-symmetrization, this transformation changes the corresponding Hartree-Fock state only by an irrelevant phase factor,

                    $\psi^{\tilde{u}_1} \wedge \cdots \wedge \psi^{\tilde{u}_f} = \det U \: \psi^{u_1} \wedge \cdots \wedge \psi^{u_f}$ .

  Thus the configuration of the physical wave functions can be described by a fermionic multi-particle wave function. The Pauli exclusion principle becomes apparent in the total anti-symmetrization of this wave function.
  Clearly, the above Hartree-Fock state $\Psi$ does not account for quantum entanglement. Indeed, the description of entanglement requires more general Fock space constructions (→ connection to quantum field theory in the mathematics section).

• The equivalence principle. Starting from a causal fermion system $({\mathcal{H}}, {\mathcal{F}}, \rho)$, spacetime $M:= \text{supp} \rho$ is given as the support of the universal measure. Thus spacetime is a topological space (with the topology on $M$ induced by sup-norm on $\text{L}({\mathcal{H}}$). In situations when spacetime has a smooth manifold structure, one can describe spacetime by choosing coordinates. However, there is no distinguished coordinate systems, giving rise to the freedom of performing general coordinate transformations. The causal action as well as all the constraints are invariant under such transformations. In this sense, the equivalence principle is implemented in the Theory of Causal Fermion Systems.

However, other physical principles are missing. For example, the principle of locality is not included. Indeed, the causal action principle is non-local, and locality is recovered only in the continuum limit. Moreover, our concept of causality is quite different from causation (in the sense that the past determines the future) or microlocality (stating that the observables of spacelike separated regions must commute).