The Theory of Causal Fermion Systems

Surface Layer Integrals

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In daily life we experience space and objects therein. These objects are typically described by densities, and integrating these densities over space gives particle numbers, charges, the total energy, etc. In mathematical terms, the densities are usually described as the normal components of vector fields on a Cauchy surface, and conservation laws express that the values of these integrals do not depend on the choice of the Cauchy surface, i.e.

\[ \int_{\scrN_1} J^k \nu_k\: d\mu_{\scrN_1}(x) = \int_{\scrN_2} J^k \nu_k\: d\mu_{\scrN_2}(x) \:, \]

where $\scrN_1$ and $\scrN_2$ are two Cauchy surfaces, $\nu$ is the future-directed normal, and $d\mu_{\scrN_{1/2}}$ is the induced volume measure (see the left of the figure below).

In the setting of causal fermion systems, surface integrals are undefined. Instead, one considers so-called surface layer integrals, as we now explain. In general terms, a surface layer integral is a double integral of the form

\[ \int_\Omega \bigg( \int_{M \setminus \Omega} \cdots\: \L(x,y)\: d\rho(y) \bigg)\, d\rho(x) \:, \]

where one variable is integrated over a subset $\Omega \subset M$, and the other variable is integrated over the complement of $\Omega$. In order to explain the basic idea, we make the assumption that the Lagrangian is of short range in the following sense. We let $d \in C^0(M \times M, \R^+_0)$ be a distance function on $M$. The assumption of short range means that $\L$ vanishes on distances larger than $\delta$, i.e.

\[ d(x,y) > \delta \quad \Longrightarrow \quad \L(x,y) = 0 \:. \]

Then the above surface layer integral only involves pairs $(x,y)$ of distance at most $\delta$, where $x$ is in $\Omega$ and $y$ is in the complement $M \setminus \Omega$. Thus the integral only involves points in a layer around the boundary of $\Omega$ of width $\delta$. Therefore, a surface layer integral can be regarded as an approximation of a surface integral on the length scale $\delta$. This is illustrated on the right of the next figure.

A surface integral and a corresponding surface layer integral

For surface layer integrals to be a sensible concept, it must be possible to express the usual conservation laws for charge, energy, … in terms of surface layer integrals. The corresponding conservation laws should be a consequence of the EL equations of the causal action. We now outline the different conservation laws.

Noether's Theorem

The classical Noether theorem gives a connection between symmetries and conservation laws. In the setting of causal fermion systems, a symmetry of the Lagrangian is described by a one-parameter variation $(\Phi_\tau)_{\tau \in (-\delta, \delta)}$ with

$\Phi_\tau : M \rightarrow \F$    and    $\Phi_0 = \1$

which leaves the Lagrangian invariant in the sense that

\[ \L \big( x, \Phi_\tau(y) \big) = \L \big( \Phi_{-\tau}(x), y \big) \]

for all $\tau \in (-\delta, \delta)$ and all $x, y \in M$. Under these assumptions, it is proven in [noether15] that the surface layer integral

\[ \frac{d}{d\tau} \int_\Omega d\rho(x) \int_{M \setminus \Omega} d\rho(y)\: \Big( \L \big( \Phi_\tau(x),y \big) – \L \big( \Phi_{-\tau}(x), y \big) \Big) \Big|_{\tau=0} \]

is conserved in the sense that its value is preserved if $\Omega$ is changed by a compact set.

In the example of symmetries described by unitary transformations of $\H$,

$\Phi(\tau, x) = U_\tau \,x\, U_\tau^{-1}$    and    $U_\tau = \exp(i \tau A)$ ,

this conservation law generalizes the conservation of the Dirac current. Symmetries under translations in spacetime give rise to the conservation of energy and momentum.

The Symplectic Form

In [jet16] it is shown that the linearized field equations also give rise to a conservation law. Indeed, for two solutions $\mathfrak{u}$ and $\mathfrak{v}$, the symplectic form $\sigma_\Omega$ defined by\[ \sigma^\Omega_\rho(\mathfrak{u}, \mathfrak{v}) = \int_\Omega d\rho(x) \int_{M \setminus \Omega} \Big( \nabla_{1,\mathfrak{u}} \nabla_{2,\mathfrak{v}} \L(x,y) – \nabla_{1,\mathfrak{v}} \nabla_{2,\mathfrak{u}} \L(x,y) \Big) \: d\rho(y) \]is conserved (again in the sense that its value is preserved if $\Omega$ is changed by a compact set).In [action17] it is shown that in the example of causal fermion systems in Minkowsi space and the Maxwell field, the above surface layer integral reduces to the symplectic form of classical field theory.

The Surface Layer Inner Product

In [osi18] a general class of conserved surface layer integrals is derived. Most interesting for the applications is the surface layer inner product defined by

\[ (\mathfrak{u}, \mathfrak{v})^\Omega_\rho = \int_\Omega d\rho(x) \int_{M \setminus \Omega} \Big( \nabla_{1,\mathfrak{u}} \nabla_{1,\mathfrak{v}} \L(x,y) – \nabla_{2,\mathfrak{v}} \nabla_{2,\mathfrak{u}} \L(x,y) \Big) \: d\rho(y) \]

As a consequence of the second derivatives acting on the same variable, this surface layer integral depends on the choice of coordinates on $\F$. A related issue is that it is conserved only approximately. In [action17] it is shown that in the example of causal fermion systems in Minkowsi space and the Maxwell field, it reduces to the scalar product used in quantum field theory obtained by inserting a frequency splitting into the symplectic form. The freedom in choosing the frequency splitting corresponds to the freedom in choosing the coordinates of $\F$.

The surface layer inner product is used in the existence theory for the linearized field equations for getting energy estimates (see [linhyp18]). Moreover, it is used in [fockbosonic18] for giving the linearized solutions an almost complex structure.

A Nonlinear Surface Layer Integral

In [fockbosonic18] another surface layer integral was introduced which is nonlinear in the sense that it compares two measures $\rho$ and $\tilde{\rho}$ on $\F$ which do not need to be related to each other by jet transformations. It takes the form

\[ \gamma^\Omega(\tilde{\rho}, \rho) = \int_{\tilde{\Omega}} d\tilde{\rho}(x) \int_{M \setminus \Omega} d\rho(y)\: \L(x,y) – \int_{\Omega} d\rho(x) \int_{\tilde{M} \setminus \tilde{\Omega}} d\tilde{\rho}(y)\: \L(x,y) \]

The corresponding conservation law can be arranged by constructing a suitable mapping $F : M \rightarrow \tilde{M}$ and setting $\tilde{\Omega}=F(\Omega)$. This conservation law is crucial for obtaining a norm-preserving dynamics on Fock spaces. It is also used in [pmt19] for defining the total mass of a static causal fermion system.

Positive functionals