Lattice Spinor
Gravity
Quantum gravity


Quantum field theory
Functional integral formulation
Symmetries are crucial

Diffeomorphism symmetry
( invariance under general coordinate transformations )

Gravity with fermions : local Lorentz symmetry
Degrees of freedom less important :
metric, vierbein , spinors , random triangles ,
conformal fields…
Graviton , metric : collective degrees of freedom
in theory with diffeomorphism symmetry
Regularized quantum gravity
①
②
③
④
For finite number of lattice points : functional integral
should be well defined
Lattice action invariant under local Lorentztransformations
Continuum limit exists where gravitational interactions
remain present
Diffeomorphism invariance of continuum limit , and
geometrical lattice origin for this
Spinor gravity
is formulated in terms of fermions
Unified Theory
of fermions and bosons
Fermions fundamental
Bosons collective degrees of freedom




Alternative to supersymmetry
Graviton, photon, gluons, W-,Z-bosons , Higgs scalar :
all are collective degrees of freedom ( composite )
Composite bosons look fundamental at large distances,
e.g. hydrogen atom, helium nucleus, pions
Characteristic scale for compositeness : Planck mass
Massless collective fields
or bound states –
familiar if dictated by symmetries
for chiral QCD :
Pions are massless bound states of
massless quarks !
for strongly interacting electrons :
antiferromagnetic spin waves
Gauge bosons, scalars …
from vielbein components
in higher dimensions
(Kaluza, Klein)
concentrate first on gravity
Geometrical degrees of freedom


Ψ(x) : spinor field ( Grassmann variable)
vielbein : fermion bilinear
Possible Action
contains 2d powers of spinors
d derivatives contracted with ε - tensor
Symmetries




General coordinate transformations
(diffeomorphisms)
Spinor
ψ(x)
: transforms as scalar
Vielbein
: transforms as vector
Action
S
: invariant
K.Akama, Y.Chikashige, T.Matsuki, H.Terazawa (1978)
K.Akama (1978)
D.Amati, G.Veneziano (1981)
G.Denardo, E.Spallucci (1987)
A.Hebecker, C.Wetterich
Lorentz- transformations
Global Lorentz transformations:
 spinor ψ
 vielbein transforms as vector
 action invariant
Local Lorentz transformations:
 vielbein does not transform as vector
 inhomogeneous piece, missing covariant derivative
Two alternatives :
1) Gravity with global and not
local Lorentz symmetry ?
Compatible with observation !
2)
Action with
local Lorentz symmetry ?
Can be constructed !
Spinor gravity with
local Lorentz symmetry
Spinor degrees of freedom

Grassmann variables
Spinor index
Two flavors
Variables at every space-time point

Complex Grassmann variables



Action with local Lorentz
symmetry
A : product of
all eight spinors ,
maximal number ,
totally antisymmetric
D : antisymmetric product
of four derivatives ,
L is totally symmetric
Lorentz invariant tensor
Double index
Symmetric four-index invariant
Symmetric invariant bilinears
Lorentz invariant tensors
Symmetric four-index invariant
Two flavors needed in four dimensions for this construction
Weyl spinors
= diag ( 1 , 1 , -1 , -1 )
Action in terms of Weyl - spinors
Relation to previous formulation
SO(4,C) - symmetry
Action invariant for arbitrary
complex transformation parameters ε !
Real ε : SO (4) - transformations
Signature of time
Difference in signature between
space and time :
only from spontaneous symmetry breaking ,
e.g. by
expectation value of vierbein – bilinear !
Minkowski - action
Action describes simultaneously euclidean and Minkowski theory !
SO (1,3) transformations :
Emergence of geometry
Euclidean vierbein bilinear
Minkowski vierbein bilinear
Global
Lorentz - transformation
vierbein
metric
/Δ
Can action can be reformulated in
terms of vierbein bilinear ?
No suitable W exists
How to get gravitational field
equations ?
How to determine geometry of
space-time, vierbein and metric ?
Functional integral formulation
of gravity
Calculability
( at least in principle)
 Quantum gravity
 Non-perturbative formulation

Vierbein and metric
Generating functional
If regularized functional measure
can be defined
(consistent with diffeomorphisms)
Non- perturbative definition of
quantum gravity
Effective action
W=ln Z
Gravitational field equation for vierbein
similar for metric
Symmetries dictate general form of
effective action and
gravitational field equation
diffeomorphisms !
Effective action for metric :
curvature scalar R + additional terms
Lattice spinor gravity
Lattice regularization




Hypercubic lattice
Even sublattice
Odd sublattice
Spinor degrees of freedom on points of odd
sublattice
Lattice action



Associate cell to each point y of even sublattice
Action: sum over cells
For each cell : twelve spinors located at nearest
neighbors of y ( on odd sublattice )
cells
Local SO (4,C ) symmetry
Basic SO(4,C) invariant building blocks
Lattice action
Lattice symmetries

Rotations by π/2 in all lattice planes ( invariant )

Reflections of all lattice coordinates ( odd )
12
while we keep t he same t ransformat ion I 1 given in
eq.
(109).
Under t he complex conjugat ion K 5
t he Grassmann variables ψγa t hat change sign are
1
1
1
1
2
ψ5 , ψ6 , ψ7 , ψ8 , ψ3 , ψ42 , ψ52 , ψ62 . For a = 1 we keep t he same
definit ion for complex Grassmann variables ϕ 1α as in eq.
(1), and similar for t he first two component s of ϕ 2 , i.e. ϕ 21
and ϕ 22 . However, inst ead of ϕ 23 and ϕ 24 we use new complex
variables
ξ32 = ψ72 − i ψ32 , ξ42 = ψ82 − i ψ42 .

S = α˜
(118)
T he complex conjugat ion is now realized by a st andard
complex conjugat ion of ξ32 and ξ42 ,
(ξ32 ) ∗ = ψ72 + i ψ32 , (ξ42 ) ∗ = ψ82 + i ψ42 .
a t orus in each “ direct ion” µ, such t hat t he t ot al number of
lat t ice point s is N L = L 4 / 2. Alt ernat ively, we could t ake
some finit e number of lat t ice point s L t in some direct ion,
wit hout imposing a periodicity const raint . T he cont inuum
limit corresponds t o N L → ∞ and is realized by keeping
fixed zµ wit h ∆ → 0.
We writ e t he act ion as a sum over local t erms or Lagrangians L ( y˜ ),
(119)
(Not e t hat under t he complex conjugat ion K 5 t he combinat ions ϕ 23 and ϕ 24 t ransform as K 5 (ϕ 23 ) = K 5 (ψ32 + i ψ72 ) =
− ψ32 + i ψ72 = − (ϕ 23 ) ∗ , K 5 (ϕ 24 ) = K 5 (ψ42 + i ψ82 ) = − ψ42 +
i ψ82 = − (ϕ 24 ) ∗ , such t hat ϕ 23 and ϕ 24 are no longer mapped
t o (ϕ 23 ) ∗ and (ϕ 24 ) ∗ .) T he complex conjugat ion K 5 can
be underst ood as a mult iplicat ion of t he conjugat ion K 1
(corresponding t o c.c. in eq. (2)) wit h t he t ransformat ion
ϕ 2− → − ϕ 2− or ϕ 2 → γ¯ϕ 2 . T he act ion changes sign under
t he involut ion K 5 .
If we writ e t he euclidean act ion S in eq. (33) in t erms
of ϕ 1± , ϕ 2+ and ξ−2 = (ξ32 , ξ42 ), it changes sign under t he new
complex conjugat ion ϕ 1± → (ϕ 1± ) ∗ , ϕ 2+ → (ϕ 2+ ) ∗ , (ξ−2 ) →
(ξ−2 ) ∗ , such t hat S is purely imaginary. Wit h respect t o t he
complex conjugat ion K 5 t he Minkowski act ion SM = i S
is now real and symmet ric, and t herefore hermit ean. We
can use t he involut ion K 1 in order t o est ablish t hat S is
an element of a real Grassmann algebra (wit h Grassmann
variables ψγa (x)). T he complex st ruct ure based on K 5 can
be employed t o define hermit icity of SM , which is relat ed
t o a unit ary t ime evolut ion.
L ( y˜ ) + c.c.
(120)
y˜
Here y˜ µ denot es a posit ion on t he even sublat t ice or “ dual
lat t ice” . It has eight nearest neighbors on t he fundament al
lat t ice, wit h unit dist ance from y˜ . To each point y˜ we
associat e a “ cell” of t hose eight point s x˜ j ( y˜ ), j = 1 . . . 8,
wit h z˜ -coordinat es given by
z˜ µ x˜ j ( y˜ ) = y˜ µ + Vj µ .
(121)
T he eight vect ors Vj obey
Diagonal reflections e.g z1↔z2 ( odd )
V1
V2
V3
V4
D I SC R E T I Z A T I ON
In t his sect ion we formulat e a regularized version of t he
funct ional int egral (3). For t his purpose we will use a lat t ice of space-t ime point s. We recall t hat t he act ion (33) is
(− 1, 0, 0, 0)
(0, − 1, 0, 0)
(0, 0, − 1, 0)
(0, 0, 0, − 1)
,
,
,
,
V5
V6
V7
V8
=
=
=
=
(0, 0, 0, 1)
(0, 0, 1, 0)
(0, 1, 0, 0)
(1, 0, 0, 0),
(122)
and we indicat e t he posit ions x˜ j in t he z0 − z1 plane and
t he z2 − z3 plane, respect ively in Fig. 1. T he dist ance
z3
z1
x~ 7
x~1
D
x~8
D
V I.
=
=
=
=
~
x2
~
x3
D
~
x5
~
x6
D
z0
z2
x~4
FIG. 1: Posit ions of x˜ j . T he cent er of t he squares corresponds
Lattice derivatives
and cell averages
express spinors in terms of derivatives and averages
Bilinears and lattice derivatives
Action in terms of
lattice derivatives
Continuum limit
Lattice distance Δ drops out in continuum limit !
Regularized quantum gravity




For finite number of lattice points : functional integral
should be well defined
Lattice action invariant under local Lorentztransformations
Continuum limit exists where gravitational interactions
remain present
Diffeomorphism invariance of continuum limit , and
geometrical lattice origin for this
Lattice diffeomorphism
invariance




Lattice equivalent of diffeomorphism symmetry in
continuum
Action does not depend on positioning of lattice points
in manifold , once formulated in terms of lattice
derivatives and average fields in cells
Arbitrary instead of regular lattices
Continuum limit of lattice diffeomorphism invariant
action is invariant under general coordinate
transformations
Lattice action and functional measure
of spinor gravity are
lattice diffeomorphism invariant !
act lat t ice 0 1
µ
0
1
ce
y
˜
=
(
y
˜
,
y
˜
),
wit
h
y
˜
int
eger
and
y
˜
+
y
˜
even,
point s on
DH =
.
(3)
nit h cart e- bosonic discret e lat
before
t ice= fields
H
(
z
˜
)
=
±
1
,
k
=
1,
2, 3. (3)
DH
.
k
z˜
k H ( z˜ ) = ± 1
t he=funct
we define t he
e- do not For t he formulat ion of S
we
LH (k y(˜z˜int
).) = egral
(5)
z˜
k ional
±1
T his measure
does obviously
not depend
on t he posit ionce
for- A measure
as a product
of independent
sums
st ence.
ot
ing. For a finit e number ofy˜ lat t ice point s t he part it ion
eatpoint
sTinhis measure
t
ice
obviously
not depend on t he posit ionAposit ion- funct ion isdoes
a
finit
e
sum
nt s on
DH = of lat t ice point
. s t he part
(3)
ing. cell
For consist
a finit es number
itnearest
ion
nirregular
Each
of
four
lat
t
ice
point
s
t
hat
are
ecart
ez˜
k H k ( z˜ ) = ± 1
Z
=
DH
S),j = 1 . . . 4. (4)
funct
ion
is
a
finit
e
sum
e
abst
ract
n˜ , denot ed by xexp(−
˜ j ( y˜ ),
T heir lato notneighbors of y
two posi- his measure does obviously not depend
0
1 posit ionar
on
ty
he
ce. At ice T
coordinat
es
are
z
˜
x
˜
(
y
˜
)
=
(
y
˜
−
1,
˜
) , z˜ x˜ 2 ( y˜ ) =
1 for finit e S.
finit esimal
and t herefore well defined
For a finit eZnumber
of lat
t ice1 S),
point s t he part it ion (4)
=ion asDH
exp(−
nt s in 0 ing.
ct
1 We
0over
finitesimal
define
t
he
act
a
sum
local
cellsand
locat edz
atx
(
y
˜
,
y
˜
−
1)
,
z
˜
x
˜
(
y
˜
)
=
(
y
˜
,
y
˜
+
1),
˜
˜ 4 ( y˜ ) =
3
funct
ion
is
a
finit
e
sum
0
1
µ
0
1
sit
ionind cannot
y˜ = 1
( y˜ , y˜ ), wit h y˜ int eger and y˜ + y˜ even,
0
1, y˜ ). well
T hedefined
local tfor
ermfinit
L e( y˜S.
) involves lat t ice fields
egular
oint
al s (z˜ y.˜and+ t herefore
= Scell
exp(−
S),denot e by H(5)(4)
stisract
=DHt hat
L ( y˜ ).we
also
re-t he four sit es of Z
on
t
he
x˜ j ).
We
k (ed
al
We
define
t
he
act
ion
as
a
sum
over
local
cells
locat
at
can
define
y˜
posi0
1
µ
0
1
choose
ot
y
˜
=
(
y
˜
,
y
˜
),
wit
h
y
˜
int
eger
and
y
˜
+
y
˜
even,
and
z
˜
in
esimal 2 and t herefore well defined for finit e S.
Lattice action for bosons in d=2
k
Each cell consist s of four lat t ice point s t hat are nearest
ill
t
hen
be
α kof
esimal
We
define
the
ion ed
as by
a sum
over
locatlat
ed- at
l my˜ ,act
neighbors
denot
x˜ j ( y˜ ),
j =local
1 . . . cells
4. T heir
at
neigh˜y˜ )=t ice
=( y˜ 0coordinat
(S
x˜x˜eger
H
(˜00x˜).−+2 )1,
+
˜x˜ 3()y˜ )+ =H k ((5)
x˜ 4 )
1 1 H k (x
1 )( y˜+
k((y˜y˜y
=
L
eannot L ( y
, 48
y˜ 1 ), wites
hH
yare
˜ µk int
and
y
˜
even,
z
˜
)
=
y
˜
)
,
z
˜
1
2
of a lat t ice
0
1
z˜ .
ne
y˜( y˜ 0 , y˜ 1 + 1), and z˜ x˜ 4 ( y˜ ) =
(
y
˜
,
y
˜
−
1)
,
z
˜
x
˜
(
y
˜
)
=
3
nt ity wit h
×
Hy˜l 1().x˜ 4T) he
− local
HSl (=x˜t erm
(involves
x˜ 3 ) − H
( x˜ fields
+ c.c.,(6)
0
1 ) [H
m
2 ) (5)
LL((y˜ym
).
so
re(
y
˜
+
1,
˜
)
lat
t
ice
n
osit ioning
define
Each
cellt heconsists
four
t ice
nearest
on
four sit es of
of the
celllat
ty˜hat
we point
denot esbyt hat
H k ( x˜ are
j ). We
we use
a
be
choose
such
t of
hey˜act
ion specifies
model.
z˜ 2 in
neighbors
, denot
ed by x˜ (a
y˜ ),general
j = 1 . .Ising
. 4. Ttype
heir lat
quiring
for t hat
of the conjugation K 1
etric
wit h respect to global
SO(3)-rotations between
t h t he t ransformation
V1 = (− 1, 0, 0, 0) , V5 = (0, 0, 0, 1)
on changes
sign under
ree
components
H k , Vprovided
the func, V6 =replace
(0, 0, 1, 0)
2 = (0, − 1, 0, 0) we
V3 = (0, 0, − 1, 0)SO(3)-invariant
, V7 = (0, 1, 0, 0)
meaSmeasure
in eq. (33) in (3)
terms by the standard
V
=
(0,
0,
0,
−
1)
,
V
=
(1,
0,
0,
0),
(122)
4
8
ges sign under the new
µ
rϕ 2+ every
˜
.
to
unbounded
fields,
e.g.
2 ∗ z
2 A generalization
wit
h
=
−
=
1
and
x
01
10
→ (ϕ + ) , (ξ− ) →
j short hands for t h
0
1
we indicat e the posit ions x˜ j in t he z − z plane and
y.HWit(hz
respect
to
the, isand
˜
)
<
∞
more
of
the
t he z2 − z3problemat
plane,
inbecause
Fig.
distcell
ance y
wskikact ion SM = i S
of respect
t he ively
sitices
x˜ j 1.of Tthehe
˜un, i.e. x µj = x µp z˜ x˜ j (
erefore
hermitof
ean.the
We act ion (6) even for a finit e number of
edness
z volume corresponds
z
t o establish t hat S is
t o t he
surface inclosed by st r
x
sit(wit
es.h Grassmann
A further generation
uses complex
fields H k
ebra
~
~
joining
t
he
four
lat
t
ice
point s x˜ j x( y˜ ) of t he cell in
x
x
ct ure based on K 5 can+
onstraint
H k ( z˜ )H k (Dz˜ ) =x˜ ,1.x˜ ,For
a .suit
able
SU(3)-we rest rict t he dis
SM , which is related
D
x
˜
,
x
˜
For
simplicity
~
2
4 ~3
~
~
x
x
x1 partit
x function
ant measure the action
and
ion
are
in- lat t ice, x µ = z˜
“
deformat
ions”
of
t
he
regular
p
z
D
D
z
x SU(3).
under unit ary t ransformations
of
the
group
V ( y˜ ) remains always positxive and t he pat h of
T I ON
~
ction (5), (6) is realx and during
we concentrate
on real
α. t ouches anot her
~
t he deformat
ion never
x
ularized proceed
version of the to an (almost) arbit rary posit ioning of
now
st raight line between two ot her point s at t he b
pose we will use a latFIG.piece
1: Posit ions
ofhe
x˜ . 2Tsurface.
he
cent er
of t he
squares
corresponds
ttice
hat t hepoints
action (33) on
is
a
of
by
specifying
posit
ions V ( y˜ ) for t he d
t
We
use
t he volume
t
o
y
˜
.
) transformat ions and
Thisionassociates
t o each an
cellintaegral
“ volume”
V (relevant
y˜ ),
over t he
region of t he mani
gularizat
will t herex µ ( z˜ ) = x µ ( z˜ ) + ξ µ ( z˜ ),
(256)
H 1 ( x˜ 8 ) − H 1 ( x˜ 1 ) H 2 ( x˜ 7 ) − H 2 ( x˜ 2 )
and ask how t he act ion depends on t he assignment of po− H 2 ( x˜ 8 ) − H 2 ( x˜ 1 ) H 1 ( x˜ 7 ) − H 1 ( x˜ 2 ) . (257)
sit ions. More precisely, since we use a fixed manifold described by t he coordinat es x µ , we want t o know how L¯( y˜ ),
(Act ually, two fields H 1,2 would be sufficient and H 3 could
expressed in t erms of average fields and lat t ice derivat ives,
be a linear combinat ion of t hem. We keep here an indepenµ
depends on ξ ( z˜ ). If L¯( y˜ ) is found t o be independent of
dent H 3 for t he sake of analogy wit h our set t ing of spinor
ξ µ ( z˜ ) t he act ion is lat t ice diffeomorphism invariant . In
gravity.)
t his case t he part icular posit ioning of lat t ice point s plays
To each posit ion x˜ j ( y˜ ) wit hin a cell a lat t ice point z˜ j
no role for t he form of t he act ion in t erms of average fields
is associat ed by eq. (121), and t o every z˜ j we associat e
and lat t ice derivat ives. Lat t ice diffeomorphism invariance
a posit ion x( z˜ j ) on t he manifold. In an int uit ive not at ion
is a property of a part icular class of lat t ice act ions and will
we will denot e t hese four posit ions by x 1 , x 2 , x 7 , x 8 . As
not be realized for some arbit rary lat t ice act ion. For examdepict ed in fig. 2 t hese posit ions are at (almost ) arbit rary
ple, t he st andard Wilson act ion for lat t ice gauge t heories
point s3in t he plane.
1
is not lat t ice diffeomorphism invariant .
T he cont inuum limit of a lat t ice diffeomorphism invari7
ant act ion for spinor gravity is rat her st raight forward. Lat t ice derivat ives are replaced7 by ordinary derivat ives. T he
5
8
independence on t he posit ioning of t he variables ϕ(x) result s in t he independence of t he cont inuum act ion S =
6
L¯(x) on t he1variable change ϕ(x)8 → ϕ(x − ξ) = ϕ(x)3−
x
µ
ξ ∂ µ ϕ(x). T his amount s t o diffeomorphism symmet ry of
2
0 inuum diffeot he cont inuum act ion. Indeed, in t he cont
morphisms can be formulat ed as a genuine symmet ry cor1
responding t o a map in t he space of variables. T his is due
2
t o t he fact t hat for any
given point x t here will be some
2
variable, originally locat ed at x − ξ, which will be moved t o
4
FIG. 2: Posit ions of variables in a cell
t his posit ion by t he change of t he posit ioning (256). T his
property is only realized in t he cont inuum. For a lat t ice
typically no variable will be posit ioned precisely at x afT he volume V ( y˜ ) associat ed t o t he cell corresponds t o
t er t he change of posit ions (256), if j before a variable was
t he surface enclosed by t he solid lines in fig. 2. It is given
locat ed t here.
by
We emphasize t hat we employ here a fixed “ coordinat e
1 0
V ( y˜ ) =
|(x − x 01 )(x 17 − x 12 ) − (x 18 − x 11 )(x 07 − x 02 )|
manifold” for t he coordinat es x µ . No met ric is int roduced
2 8
at t his st age. Physical dist ances do not correspond t o
1
µ
µ
Minkowski and a euν
ν
√ will
=
(258)
cart esian dist ances on t he coordinat e manifold. T hey
µ ν (x 8 − x 1 )(x 7 − x 2 ).
2
neighboring
˜ jfunct
is ions,2,asand each point in t heµ µ
rat her bebetween
induced bytwo
appropriat
e correlat ionx
For t he part icular case x = z˜ ∆ t his yields
describedcell
in general
t erm
by “ geomet
ry from general
st at is-is further
has six
nearest
neighbors.
T here
an “ opposit e
t ics” [15]. T he appropriat e met ric for t he comput at ion of
x 08 − x 01 = 2∆ , x 18 − x 11 = 0 , x 17 − x 12 = 2∆ , x 07 − x 02 = 0,
point
” at distance
2, wit
h pairs
e point s given by
infinit esimal
physical
dist ances swill
be associat
ed of
t o opposit
t he
(259)
nal hypercubic latt ice
expect at ion
field.
and t herefore V ( y˜ ) = 2∆ 2 . For simplicity we rest rict t he
( x˜ 1value
, x˜ 8 ),of(ax˜ suit
˜able
( x˜ 3 , x˜ive
( x˜ 4 , x˜ 5 ).
2, x
7 ), collect
6 ),
ish between the “ even
discussion t o deformat ions from t his simple case where
Lagrangian
L (iy˜ant
) isact
given
a sum
ofµ “ hyperloops”
.
µ
2. L at t i ceTdihe
ffeom
or phi sm i nvar
i on i nby
t wo
ν
ν
µ
µ
˜ ) > 0.
µ ν (x 8 − x 1 )(x 7 − x 2 ) remains posit ive, i.e. V ( y
int eger, Σ µ y˜ even,
di m ensiA
onshyperloop is a product of an even number
of
Grassmann
We
also
assume
t
hat
in
t
he
course
of
such
deformat
ions
∆ , z˜ µ int eger, Σ µ z˜ µT he latvariables
t ice act ion (253), (131) is lat t ice diffeomorphism
no
posit
ion
ever
crosses
any
boundary
line
of
t
he
surface
located
posit ions
˜under˜ ) wit hin t he cell at y˜ .
j (y
. T he basic const
ruct ionatprinciple
can bex
between two ot her posit ions.
ed as t he fundamentinvariant
al
st ood in In
two accordance
dimensions. T hewit
concept
of lat(2)
t ice diffeomorh eq.
we will consider hyperloops
Positioning of lattice points
1
V ( y˜ ) =
2
µ
(x
µν
4
−
x µ1 )(x ν3
−
x ν2 ),
d2 x (7)
=
V ( y˜ ),
y˜
where we define the region by t he surface cove
wit h
∂ˆ 0 H k ( y˜ ) =
01
= −
10
= 1 and x j short hands for t he posit
ions
t he
for
11
1
µ
1cell
µ
y˜ ) es
=(x
H
x
˜
(
y
˜
)
of H
t he
x˜ j 3of−t he
y
˜
,
i.e.
x
=
x
˜˜ 1x(9)
˜)j ( y˜ ) .comm
T he
k (sit
k
j
x
)
H
(
x
˜
)
−
H
x
pk (z
j
k
4
2
2V ( y˜ corresponds
) 4 j t o t he surface inclosed by st raightpart
icu
volume
lines
Lattice
derivatives
ioni
joining1 t he four
order
1 lat t ice point s x˜ ( y˜ ) of t he cell in t hesit
j
and lat t ice derivat
ives
(2)
associat
ed
t
o
t
he
cell
−
(x
−
x
)
H
(
x
˜
)
−
H
( x˜ 2rict
) t ,he discussion
k
3
lat ttice
x˜ 1 , x˜ 2 ,4x˜ 4 , x˜ 3 .1 For simplicity
wek rest
o
“ deformat
ions” of t he regular lat t ice, x µp = z˜ µ ∆ , where
L¯( y˜ ; x
1
(x013always
− x 120) posit
H k (ive
x˜ 4 )and
( x˜ 1pat
) h of one known
)1remains
point
ˆ∂1 H∂kˆ 0(Hy˜ )k ( y˜=) =V ( y˜2V
˜ 3−) H−tkhe
H
˜ 2)
( y˜ ) (x 4 − x 1 ) H k ( x
k (x
during
or a
2V ( y˜t)he deformat ion never t ouches anot her pointt heorie
1
st raight
her
− (x 14line
− xbetween
˜two
˜point
3 ) −otH
k (x
2 ) s, at t he boundary of
1 ) H k (x
0
0We use t he volume V ( y˜ ) for t he definit ion
I nterp
t−
he(x
surface.
of
−
x
)
H
(
x
˜
)
−
H
(
x
˜
)
.
(10)
k
4
k
1
31
2
0 t he 0relevant region of t he manifold t inuum
an int egral over
ˆ∂ 1 H k ( y˜ ) =
(x 4 − x 1 ) H k ( x˜ 3 ) − H k ( x˜ 2 )
exhibi
2V ( y˜ )
2
For the pairs ( x˜ j 1 , x˜ j 2 )0 = (0x˜ 4 , x˜ 1 ) dand
( x˜ 3V,(x˜y˜ ),2 ) the lattice
will(8)be
x
=
− (x 3 − x 2 ) H k ( x˜ 4 ) − H k ( x˜ 1 ) .
(10)
sider a
derivatives obey
y˜
t ermin
For t he pairs (where
x˜ j 1 , x˜ we
= ( x˜ 4 t, he
x˜ 1µ)region
and (µby
x˜ 3 ,t x
˜ 2 )surface
t he latcovered
t ice by 2t,he
he
j 2 ) define
wh
ˆ
H
(
x
˜
)
−
H
(
x
˜
)
=
(x
−
x
)
∂
H
(
y
˜
).
(11)
k
j2
derivatkivesj obey
1 cells appearing
j 1 sum. j 2 µ k
in t he
t hose
We next express tµhe actµion (5), (6) in terms of average
t ice po
ˆ
H
(
x
˜
)
−
H
(
x
˜
)
=
(x
−
x
)
∂
H
(
y
˜
).
(11)
k
j fields
µ
k
ink t hejderivatives
In terms of average
and
quantit
ies in L ( y˜ t)hat al
1
2cell
j1
jall
2
depend
on the
cell variable
y˜ orives
theall1associated
posit
ion
ofpoint s
In t erms
of average
and derivat
quant
it
ies
in
L
(
y
˜
)
H k ( y˜ ) =
H k x˜ j ( y˜ )
(9)
µ
t
he
4
the cell
x p ( yon
˜ ) that
wevariable
take somewhere
inside
the surface
depend
t he cell
y˜ or t he associat
ed posit
ion of of po
j
wit h t
t he cell x µ ( y˜ ) t hat we t ake somewhere inside t he surface of
Cell average
the cell, :the pprecise assignment
being unimportant at this
discretLˆe(x)
point
s independs
the manifold
corresponding
t o t he on
cell
where
only
on quantit
ies wit h support
int erpolat ion
t
erpolating
fun
posit
ions.
We
indicat
e
explicit
ly
t
he
dependence
on
t
he
discrete points in the manifold corresponding to the cell
T his reflect sa
interpolation
choice
of t he
ioningexplicit
by t helyargument
xp.
posit
ions.
Weposit
indicate
t he dependence
on the
int erpolat
ing
T
his
reflect
s
th
T
he
act
ion
appears
now
in
a
form
referring
t
o
t
he
posichoice of the posit ioning by the argument x p .
t he point s xfu
n
interpolating
t ions
the manifold
Theon
action
appears now in a form referring to t he posi3
t ion
descript
t
he
point
s xn ,
t ions on the manifold
lat ion t he fun
S(x p ) =
d2 x L¯( y˜ ; x p ) =
d2 x L¯(x; x p ),
(12) t ion description
t he posit ions
posit ioning, or 2independent of 2ξp for infinit esimal changes
t akes
he v
latition
t hetfunct
¯
¯
S(x
)
=
d
x
L
(
y
˜
;
x
)
=
d
x
L
(x;
x
),
(12)
p ions x = x + pξ ,
p
of posit
p
p
p
it different
takes theposi
valu
x˜ j ( y˜ ) . T he
wit h
t o two differe
different
posit i
3
st raight lines
¯
¯
wit h
L ( y˜ ; x p + ξp ) = L ( y˜ ; x p ) , LˆS(x
(14)
In different
cont rast
px p+) ξp ) = S(x p ).
(
y
˜
;
t
o two
¯
¯
l in t he order
L ( y˜ ;ioning,
x p ) = L (x;
. ξ for infinit
(13) esimal
p) = ˆ
format
ion abt
s for the posit ions
posit
or xindependent
of
chang
In contrast
p
V
(
y
˜
;
x
)
L
(
y
˜
;
x
)
p
p
discussion t o
¯( y˜ ; xion
¯(x; x p )t he
T his definit
provides
necessary
tcontained
he
L
)
=
L
=
. specification
(13) forformation
p
in
abou
of
posit
ions
x
=
x
+
ξ
,
µ
p
p
p ice diffeomorphism
V
(
y
˜
;
x
)
z
˜
∆
,
where
p
z
˜
x
˜
(
y
˜
)
.
T
he
precise
meaning
of
latt
invariance.
j
ing, or
independent
of
ξ
for
infinitesimal
Lat t ice diffeomorphism
invariance
stat es t hatchanges
for fixed contained
pends onint he
p
th
however,
t
he
notion
of
smoot
h
lat
t
ice
fields
is
no
longer
¯
of one pointH ( y˜ ) and
The
ξ
-independence
of
L
(
y
˜
;
x
)
means
t
hat
t
he
depenˆ
¯
∂,µ Hp ( y˜ )¯t he rat
io L ( y˜ ; x p¯)states
is pindependent
of t he pends
int erpolat
ions
ed
by
st
raight
lines
Lattice
diffeomorphism
invariance
t hat
for fixed
on the
ons
x
=
x
+
ξ
available.
L
(
y
˜
;
x
+
ξ
)
=
L
(
y
˜
;
x
)
,
S(x
+
ξ
)
=
S(x
).
(1
p
er pointp or a p dence
p
p
p
p
p
p
ˆ
of
V
(
y
˜
;
x
)
and
L
(
y
˜
;
x
)
on
t
he
posit
ioning
x
must
ˆ
¯
p
p
p
H (order
y˜ ) and ∂µ H ( y˜ ) the
ratio L ( y˜ ; x p )symmetry
is independent
of the limit.
interpolation
cell in the
Diffeomorphism
of continuum
Coming m
eeboundary
of
cancel. Insert ing eqs. (9), (11) in eq. (6) yields
the
discussion
to
back
to the
latprovides
t ice modelthe
(5),necessary
(6) we associate
int erpolatThis
definit
ion
specification
for th
e definit ion of
¯
µx + µξ ) = L
(˜
y; xprecise
, kS(x
)lattice
=to S(x
(14)
functions
t hediffeomorphism
latt
H k ( z˜invariance.
). Wit h t he
kp(x)
˜ ∆p , where
p )αing
p + Hξof
pˆ).ice fields
anifold
l mmeaning
µν ˆ
p p= z
ˆ
L ( y˜ ) = choice V
(
y
˜
)H
(
y
˜
)
∂
H
(
y
˜
)
∂
H
(
y
˜
)
+
c.c.,
(15)
k ives (16),
µ l (17) νand
m eq. (15) we
of
derivat
can define
¯
pat h of one point
12 ξp -independence of L ( y˜ ; x p ) means that
The
t he depe
the cont inuum act ion as a functional of t he int erpolating
nother point
(8) or a the
ˆ ( y˜ ;Vx(py˜)) on
finition
provides
necessary
for
ofions
V ( y˜t hat
; x p )specification
and
thethe
posit
funct
and wedence
find
indeed
t he
factLor
cancels
in L¯(ioning
y˜ ) = x p mu
t the boundary of
Inserting
(9), (6)
(11)is in
yields
Lˆ ( y˜ )/ Vcancel.
(diffeomorphism
y˜ ). T hus
t he acteqs.
ion invariance.
(5),
lateq.
t ice (6)
diffeomormeaning
of
lattice
α
2 kl m µν
r the definit ion phism
of
S
=
d
xproperty
H kis
(x)∂
c.c. (22)
µ H l (x)∂
ν Ha
m (x)
overed by t he
invariant.
T
his
specific
for
cert+ain
¯
12
α
(˜y; x ) means
that
the
manifold
kl m ˆ
pe-independence
ˆ µ Hdepenclassof
of LˆL
actions
( y˜ ) = -p fork l mexample
V ( y˜ )H kadding
( y˜ ) µ ν ∂to
˜ )a∂νquant
H m ( yity
˜ ) + c.c., (1
l (y
Lattice diffeomorphism invariance
Continuum
Limit :
kl m
p
µ˜
attice
derivattives.
For
the funct
a
Wit h respect
o infinit
esimal
repositions
ionsaone
has
µ (x)
Lattice
diffeomorphism
ioning
result
s
in
µ
µ
µ
µ
µ
µ
δp d0 = ξp ( x˜ 4 ) − transformation
ξp ( x˜ 1 ) , δp d1 = ξp ( x˜ 3 ) − ξp ( x˜ 2 ),
δp aµµ˜ (x) = − ∂ˆ µ ξpν ( y˜ )aµν˜ (x).
cha
(3
nd infers for t he change of t he cell volume
µ
e pµ˜ does not δdepend
on
the
posit
ioning
on
t
he
ˆ
V
(
y
˜
)
=
∂
ξ
(
y
˜
)V
(
y
˜
),
(3
p
µ p
ng one obtains from eq. (26)
t h ∂ˆ µ ξpν ( y˜ ) formed wit h t he st andard prescript ion (2
ν
ˆ
ˆ
ˆ
µ
˜y
δ
∂
f
(
y
˜
)
=
−
∂
ξ
(
y
˜
)
∂
f
(
˜
).
p µ
ν
r lat t ice derivat
ives. For t he µfunct
ions
a
(x)
a change
p
µ
osit ioning result s in
δp aµµ˜ (x) = − ∂ˆ µ ξpν ( y˜ )aµν˜ (x).
(3
nce pµ˜ does not depend on t he posit ioning on t he po
variant funct ional of k, in t he same sense as S[H ]. If yes,
we can employ for smoot h enough lat t ice fields h( y˜ ) t he
cont inuum limit wit h int erpolat ing funct ions h(x). T he
effect ive act ion Γ h(x) will t hen be a diffeomorphism invariant funct ional of h(x). If we choose int erpolat ions for
h(x) and j (x) such t hat eqs. (23) and (41) hold, we can
writ e
Effective action is
diffeomorphism symmetric
exp
− Γ h(x), j (x) =
H k (x) − hk (x)
+
p
l
G
DH ( z˜ ) exp{ − SH (x)
j k∗ (x)
+ c.c ,
(43)
w
x
wit h S H (x) t he diffeomorphism symmet ric cont inuum
act ion (22). T he int egrand on t he r.h.s. is diffeomorphism symmet ric if H (x) t ransforms as a scalar and j (x)
as a scalar density. T he funct ional measure DH ( z˜ ) is
invariant
under
diffeomorphism,
suchfor
t hatgraviton
Γ h(x), j (x)
is
Same
for
effective
action
!
formally diffeomorphism symmet ric, and t his ext ends t o
Γ]bi g[h(x) .
a
O
t
Lattice action and functional measure
of spinor gravity are
lattice diffeomorphism invariant !
Gauge symmetries
Proposed action for lattice spinor gravity has also
chiral SU(2) x SU(2) local gauge symmetry
in continuum limit ,
acting on flavor indices.
Lattice action :
only global gauge symmetry realized
Next tasks


Compute effective action for composite metric
Verify presence of Einstein-Hilbert term
( curvature scalar )
Conclusions



Unified theory based only on fermions seems
possible
Quantum gravity –
functional measure can be regulated
Does realistic higher dimensional unified
model exist ?
end
Gravitational field equation
and energy momentum tensor
Special case : effective action depends only on metric
Unified theory in higher dimensions
and energy momentum tensor



Only spinors , no additional fields – no genuine source
Jμm : expectation values different from vielbein
and incoherent fluctuations
Can account for matter or radiation in effective four
dimensional theory ( including gauge fields as higher
dimensional vielbein-components)