Limiting cases Causal fermion system

1 limiting cases

1.1 lorentzian spin geometry of globally hyperbolic space-times
1.2 quantum mechanics , classical field equations
1.3 einstein field equations
1.4 quantum field theory in minkowski space

limiting cases

causal fermion systems have mathematically sound limiting cases give connection conventional physical structures.

lorentzian spin geometry of globally hyperbolic space-times

starting on globally hyperbolic lorentzian spin manifold



(



m
^



,
g
)


{\displaystyle ({\hat {m}},g)}

spinor bundle



s



m
^





{\displaystyle s{\hat {m}}}

, 1 gets framework of causal fermion systems choosing



(


h


,

⟨

.

|

.


⟩



h



)


{\displaystyle ({\mathcal {h}},{\langle }.|.{\rangle }_{\mathcal {h}})}

subspace of solution space of dirac equation. defining so-called local correlation operator



f
(
p
)


{\displaystyle f(p)}





p
∈



m
^





{\displaystyle p\in {\hat {m}}}

by

⟨

ψ

|

f
(
p
)
ϕ


⟩



h



=
−

≺

ψ

|

ϕ


≻


p




{\displaystyle {\langle }\psi |f(p)\phi {\rangle }_{\mathcal {h}}=-{\prec }\psi |\phi {\succ }_{p}}

(where




≺

ψ

|

ϕ


≻


p




{\displaystyle {\prec }\psi |\phi {\succ }_{p}}

inner product on fibre




s

p





m
^





{\displaystyle s_{p}{\hat {m}}}

) , introducing universal measure push-forward of volume measure on






m
^





{\displaystyle {\hat {m}}}

,

ρ
=

f

∗


d
μ



,


{\displaystyle \rho =f_{*}d\mu {\,},}

one obtains causal fermion system. local correlation operators well-defined,





h




{\displaystyle {\mathcal {h}}}

must consist of continuous sections, typically making necessary introduce regularization on microscopic scale



ε


{\displaystyle \varepsilon }

. in limit



ε
↘
0


{\displaystyle \varepsilon \searrow 0}

, intrinsic structures on causal fermion system (like causal structure, connection , curvature) go on corresponding structures on lorentzian spin manifold. geometry of space-time encoded in corresponding causal fermion systems.

quantum mechanics , classical field equations

the euler-lagrange equations corresponding causal action principle have well-defined limit if space-times



m
:=

supp


ρ


{\displaystyle m:={\text{supp}}\,\rho }

of causal fermion systems go on minkowski space. more specifically, 1 considers sequence of causal fermion systems (for example





h




{\displaystyle {\mathcal {h}}}

finite-dimensional in order ensure existence of fermionick fock state of minimizers of causal action), such corresponding wave functions go on configuration of interacting dirac seas involving additional particle states or holes in seas. procedure, referred continuum limit, gives effective equations having structure of dirac equation coupled classical field equations. example, simplified model involving 3 elementary fermionic particles in spin dimension two, 1 obtains interaction via classical axial gauge field



a


{\displaystyle a}

described coupled dirac- , yang-mills equations

(
i
∂




/

 
+

γ

5


a




/

 
−
m
)
ψ



=
0





c

0


(

∂

j


k



a

j


−
◻

a

k


)
−

c

2



a

k





=
12

π

2





ψ
¯




γ

5



γ

k


ψ

.






{\displaystyle {\begin{aligned}(i\partial \!\!\!/\ +\gamma ^{5}a\!\!\!/\ -m)\psi &=0\\c_{0}(\partial _{j}^{k}a^{j}-\box a^{k})-c_{2}a^{k}&=12\pi ^{2}{\bar {\psi }}\gamma ^{5}\gamma ^{k}\psi \,.\end{aligned}}}

taking non-relativistic limit of dirac equation, 1 obtains pauli equation or schrödinger equation, giving correspondence quantum mechanics. here




c

0




{\displaystyle c_{0}}

,




c

2




{\displaystyle c_{2}}

depend on regularization , determine coupling constant rest mass.

likewise, system involving neutrinos in spin dimension 4, 1 gets massive



s
u
(
2
)


{\displaystyle su(2)}

gauge field coupled left-handed component of dirac spinors. fermion configuration of standard model can described in spin dimension 16.

the einstein field equations

for just-mentioned system involving neutrinos, continuum limit yields einstein field equations coupled dirac spinors,

r

j
k


−


1
2



r


g

j
k


+
Λ


g

j
k


=
κ


t

j
k


[
Ψ
,
a
]

,


{\displaystyle r_{jk}-{\frac {1}{2}}\,r\,g_{jk}+\lambda \,g_{jk}=\kappa \,t_{jk}[\psi ,a]\,,}

up corrections of higher order in curvature tensor. here cosmological constant



Λ


{\displaystyle \lambda }

undetermined, ,




t

j
k




{\displaystyle t_{jk}}

denotes energy-momentum tensor of spinors ,



s
u
(
2
)


{\displaystyle su(2)}

gauge field. gravitation constant



κ


{\displaystyle \kappa }

depends on regularization length.

quantum field theory in minkowski space

starting coupled system of equations obtained in continuum limit , expanding in powers of coupling constant, 1 obtains integrals correspond feynman diagrams on tree level. fermionic loop diagrams arise due interaction sea states, whereas bosonic loop diagrams appear when taking averages on microscopic (in non-smooth) space-time structure of causal fermion system (method of microscopic mixing). detailed analysis , comparison standard quantum field theory work in progress.