Transition Amplitudes
Quantum states of General Relativity
Spin-foams: Feynman Diagrams for LQG
An open issue
How to calculate transition amplitudes in Loop Quantum Gravity
Jacek Puchta
Instytut Fizyki Teoretycznej Wydziału Fizyki UW
Centre de Physique Théorique, Marseille
March 3, 2014
The project „International PhD Studies at the Faculty of Physics, University of Warsaw” is realized
within the MPD programme of Foundation for Polish Science, cofinanced from European Union,
Regional Development Fund
Transition Amplitudes
Quantum states of General Relativity
Spin-foams: Feynman Diagrams for LQG
Outline
1
Transition Amplitudes
Quantum Mechanics: Probabilities
Quantum Field Theory: Feynman Diagrams
2
Quantum states of General Relativity
Gravity as a curved space-time
Ashtekar quantization
Discrete space in loop quantum gravity
3
Spin-foams: Feynman Diagrams for LQG
Path integrals for the BF -theory
Gravity as a constrained BF -theory
Two examples
4
An open issue
Generalization of Saddle Point Approximation
An open issue
Transition Amplitudes
Quantum states of General Relativity
Spin-foams: Feynman Diagrams for LQG
Quantum Mechanics: Probabilities
Outline
1
Transition Amplitudes
Quantum Mechanics: Probabilities
Quantum Field Theory: Feynman Diagrams
2
Quantum states of General Relativity
Gravity as a curved space-time
Ashtekar quantization
Discrete space in loop quantum gravity
3
Spin-foams: Feynman Diagrams for LQG
Path integrals for the BF -theory
Gravity as a constrained BF -theory
Two examples
4
An open issue
Generalization of Saddle Point Approximation
An open issue
Transition Amplitudes
Quantum states of General Relativity
Spin-foams: Feynman Diagrams for LQG
An open issue
Quantum Mechanics: Probabilities
Probabilities in quantum mechanics
Transition Amplitudes
Given an initial state |ψ0 i and a final state |ψ1 i we cannot determine, whether
they are at the same trajectory. The only think we can say is the probability
that starting the evolution from the state |ψ0 i at the time t0 we can find the
system at the time t1 in the state |ψ1 i:
2 2
ˆ
ı˙ Ht
P ψ0 −→ ψ1 = hψ0 | e ~ |ψ1 i = A ψ0 −→ ψ1 t0 →t1
t0 →t1
Path Integrals
The transition amplitudes can be calculated by a path integral:
!
N ˆ
Y
ˆ
i˙ ˆ t
i˙ ˆ t
ı˙ Ht
hψ0 | e ~ |ψ1 i =
dxi hψ0 | e ~ H N |xN i hxN | · · · |x1 i hx1 | e ~ H N |ψ1 i
i=1
=
=
··· ˆ
´ t2
dt L(x(t))
hψ0 | Dx(t)e t1
|ψ1 i
Transition Amplitudes
Quantum states of General Relativity
Spin-foams: Feynman Diagrams for LQG
Quantum Field Theory: Feynman Diagrams
Outline
1
Transition Amplitudes
Quantum Mechanics: Probabilities
Quantum Field Theory: Feynman Diagrams
2
Quantum states of General Relativity
Gravity as a curved space-time
Ashtekar quantization
Discrete space in loop quantum gravity
3
Spin-foams: Feynman Diagrams for LQG
Path integrals for the BF -theory
Gravity as a constrained BF -theory
Two examples
4
An open issue
Generalization of Saddle Point Approximation
An open issue
Transition Amplitudes
Quantum states of General Relativity
Spin-foams: Feynman Diagrams for LQG
An open issue
Quantum Field Theory: Feynman Diagrams
Quantum Field Theory - an overview
A similar situation we have in the quantum field theory, however here we
calculate
amplitudes between Ethe n-particle states:
E E the transition
~
~
¯(~k3 ), d(~k4 ) etc.
ν (l) , µ(~k1 ), µ(~k2 ), u
e(k) , ¯
The amplitudes are called the S-matrix elements:
´
Sαβ := hα| T e
3
ˆ
H(x)td
xdt
|βi
We decompose the initial and final states into linear superposition of
one-particle states
X
|Ψi =
Ψα |αi
α
The transition amplitude between complex states is the linear combination of
the transition amplitudes of the basic states:
X
Ψα Ψβ Sαβ
AΨ1 →Ψ2 =
α,β
Transition Amplitudes
Quantum states of General Relativity
Spin-foams: Feynman Diagrams for LQG
Quantum Field Theory: Feynman Diagrams
Feynman diagrams as a tool to compute S-matrix elements
Quantum field theory in two slaids
An open issue
Transition Amplitudes
Quantum states of General Relativity
Spin-foams: Feynman Diagrams for LQG
An open issue
Quantum Field Theory: Feynman Diagrams
Feynman diagrams as a tool to compute S-matrix elements
Quantum field theory in two slaids
Transition amplitude of a Feynman Diagram can be
calculated according to quite simple rules (called Feynman
Rules):
ˆ
ΓD (k1 , . . . kn )
=
!
4
4
d p1 · · · d pm
Y
Γv
·
v
!
!
Y
` internal
(2)
Γ
(p` )
Y
(2)
Γ
` external
Each Feynman diagram represents one of the terms that
contribute to an S-matrix element
X
Sαβ =
ΓD
(k` )
Transition Amplitudes
Quantum states of General Relativity
Spin-foams: Feynman Diagrams for LQG
Gravity as a curved space-time
Outline
1
Transition Amplitudes
Quantum Mechanics: Probabilities
Quantum Field Theory: Feynman Diagrams
2
Quantum states of General Relativity
Gravity as a curved space-time
Ashtekar quantization
Discrete space in loop quantum gravity
3
Spin-foams: Feynman Diagrams for LQG
Path integrals for the BF -theory
Gravity as a constrained BF -theory
Two examples
4
An open issue
Generalization of Saddle Point Approximation
An open issue
Transition Amplitudes
Quantum states of General Relativity
Spin-foams: Feynman Diagrams for LQG
Gravity as a curved space-time
Basic concepts of Einsteins General Relativity
General Relativity:
The principle of relativity
Everything is local
Gravity is a curvature of
spacetime
An open issue
Transition Amplitudes
Quantum states of General Relativity
Spin-foams: Feynman Diagrams for LQG
An open issue
Gravity as a curved space-time
General Relativity
a bit of mathematics
In a nutshell:
A metric field gµν (x) instead of flat constant ηµν
A test particle moves along geodesic curves
The curvature tensor Rµνρσ (x) is the "strength of the gravitational field"
Distribution of matter: Tµν =
δ2 S
δxµ δxν
Relation between gravity and matter: Rµν − 12 gµν R = Tµν
Principle of General Covariance: Every physical law should be defined in
such a way, that it is independent on the coordinate system.
Background-independence: the only thing, that matters, is relative
configurations of fields. The gravitational field is the dynamical
background for the other fields.
Transition Amplitudes
Quantum states of General Relativity
Spin-foams: Feynman Diagrams for LQG
Ashtekar quantization
Outline
1
Transition Amplitudes
Quantum Mechanics: Probabilities
Quantum Field Theory: Feynman Diagrams
2
Quantum states of General Relativity
Gravity as a curved space-time
Ashtekar quantization
Discrete space in loop quantum gravity
3
Spin-foams: Feynman Diagrams for LQG
Path integrals for the BF -theory
Gravity as a constrained BF -theory
Two examples
4
An open issue
Generalization of Saddle Point Approximation
An open issue
Transition Amplitudes
Quantum states of General Relativity
Spin-foams: Feynman Diagrams for LQG
An open issue
Ashtekar quantization
Ashtekar quantization
Ashtekar variables for General Relativity.
Let us introduce the so called Ashtekar Connection and it’s conjugated field
Aia (x) = Γia (x) + γKai (x) Ejb (x) : (det (q)) q ab = Eia Eib kij
They are canonically conjugated: Aia (x), Ejb (y) = −8πGγ δji δab δ(x, y).
Consider a graph embedded in
space:
We can calculate holonomies of Ashtekar
Connection along each link of the graph.
This gives us SU (2) element U` for each
link `. The combination of {U` }`=1,...,L
capture some information about the
gravitational field.
Think of it as a singular configuration of a
gravitational field, (like plane waves of
EM-field).
Transition Amplitudes
Quantum states of General Relativity
Spin-foams: Feynman Diagrams for LQG
An open issue
Ashtekar quantization
Ashtekar quantization
Quantization of Ashtekar variables
In canonical paradigm of quantization of a theory with configuration space C
one uses the Hilbert space L (C)
The configuration space of a discrete SU (2)-gauge field on a graph is
SU (2)L /SU (2)N , thus the Hilbert space of singular configurations of
gravitational field is
HΓ := L2 SU (2)L /SU (2)N
quantization of full configuration space is done by taking the projective limit
limΓ→∞ HΓ .
Using Peter-Weyl theorem we can decompose this Hilbert space in the basis,
given by matrix elements of Wigner matrices contracted with some invariant
tensors. We call it the spin-network basis. The quantum numbers related to
this basis leads us to physical interpretation of these states.
Transition Amplitudes
Quantum states of General Relativity
Spin-foams: Feynman Diagrams for LQG
Discrete space in loop quantum gravity
Outline
1
Transition Amplitudes
Quantum Mechanics: Probabilities
Quantum Field Theory: Feynman Diagrams
2
Quantum states of General Relativity
Gravity as a curved space-time
Ashtekar quantization
Discrete space in loop quantum gravity
3
Spin-foams: Feynman Diagrams for LQG
Path integrals for the BF -theory
Gravity as a constrained BF -theory
Two examples
4
An open issue
Generalization of Saddle Point Approximation
An open issue
Transition Amplitudes
Quantum states of General Relativity
Spin-foams: Feynman Diagrams for LQG
An open issue
Discrete space in loop quantum gravity
Spin-network states
Overview
A spin-network state is determined by one
quantum number j` (half-integer) per each
link of the graph and one quantum number
vn (SU (2) invariant tensor) per each node.
Each node is a quantum polyhedron of
volume vn . Each link is a face (quantum
polygon) between two polyhedra,
and it has
p
the physical area A = a0 `2pl j(j + 1).
Transition Amplitudes
Quantum states of General Relativity
Spin-foams: Feynman Diagrams for LQG
Discrete space in loop quantum gravity
Spin-network states
Details
Area
The spins are interpreted as areas, because:
ˆ
ˆ p
Area (S) :=
dS =
detg (2) dxdy
S
S
An open issue
Transition Amplitudes
Quantum states of General Relativity
Spin-foams: Feynman Diagrams for LQG
Discrete space in loop quantum gravity
Spin-network states
Details
Area
The spins are interpreted as areas, because:
ˆ
ˆ p
ˆ q
Area (S) :=
dS =
detg (2) dxdy =
Eia Ejb kij dxa ∧ dxb
S
S
S
p
Eigenvalues of such operator are exactly a0 · j` (j` + 1) iff the link ` is
transversal to S and 0 iff not. The constant a0 = γ`Pl = γG~.
Volume
Similar situation is for the volume operator
ˆ p
ˆ p
V olume (R) :=
detg (3) d3 x =
|det E|d3 x
R
R
However, here the general form of eigenvalues is not known yet.
An open issue
Transition Amplitudes
Quantum states of General Relativity
Spin-foams: Feynman Diagrams for LQG
An open issue
Discrete space in loop quantum gravity
Spin-network states
Some more properties
Having fixed the area and directions of faces there are many possible
volumes of the polyhedron - in this sense the space is curved.
There is a quanta of area a0 `2pl . We expect the same for the volume (not
proven yet).
Shape of the faces are not well defined - in the sense that length and angle
operators do not commute with volume and area operators. In this sense
spin network defines the quantum geometry.
Transition Amplitudes
Quantum states of General Relativity
Spin-foams: Feynman Diagrams for LQG
Path integrals for the BF -theory
Outline
1
Transition Amplitudes
Quantum Mechanics: Probabilities
Quantum Field Theory: Feynman Diagrams
2
Quantum states of General Relativity
Gravity as a curved space-time
Ashtekar quantization
Discrete space in loop quantum gravity
3
Spin-foams: Feynman Diagrams for LQG
Path integrals for the BF -theory
Gravity as a constrained BF -theory
Two examples
4
An open issue
Generalization of Saddle Point Approximation
An open issue
Transition Amplitudes
Quantum states of General Relativity
Spin-foams: Feynman Diagrams for LQG
An open issue
Path integrals for the BF -theory
Path integral for the BF theory
BF-theory
Consider a theory of a dynamical G-connection A with and a g∗ -valued 2-form
B with action
ˆ
S[A, B] := B ∧ F
where F is the curvature 2-form of A. It is called a BF -theory.
One can construct spin-network-like states for BF -theory for any group G.
Transition amplitudes between such states can be calculated using the path
integral:
ˆ
A (ψin → ψout ) = hψin |
DBDA eı˙S |ψout i
To calculate it one discretises the manifold and takes the limit with refining the
discretisation
ˆ
ˆ
ˆ Y
Y ı˙ ´ B∧F (U )
i
DBDA eı˙S =
lim
DB
dUi
e cell
”#cells→∞”
i
cells
Transition Amplitudes
Quantum states of General Relativity
Spin-foams: Feynman Diagrams for LQG
An open issue
Path integrals for the BF -theory
Spin-foam as the evolution of a spin-network
The amplitude is a product of amplitudes of each cell. To express it one
introduces a picture of spin-foams, that nicely corresponds to a concept of
time-evolution of a spin-network.
Spin-foam is a 2-complex dual to a cellular
decomposition of the manifold. Recall a
spin-network being a graph dual to a granulation of
a 3d-space. A spin-foam does the same but in
4d-spacetime.
Thanks to discretisation of the BF -action, a
spin-foam colored in appropriate way can be used
as a Feynman diagram for the BF theory.
Transition Amplitudes
Quantum states of General Relativity
Spin-foams: Feynman Diagrams for LQG
An open issue
Path integrals for the BF -theory
Spin-foam as a Feynman diagram
Each face is colored by a representation ρf of
the group G - the same, as the representation
on the boundary links incident to it (in case of
G = SU (2) this is represented by spins).
Each vertical edge is a kind of free propagator
- it carries an operator Pe acting on the
Hilbert space of the nodes it links
Transition Amplitudes
Quantum states of General Relativity
Spin-foams: Feynman Diagrams for LQG
An open issue
Path integrals for the BF -theory
Spin-foam as a Feynman diagram
Each face is colored by a representation ρf of
the group G - the same, as the representation
on the boundary links incident to it (in case of
G = SU (2) this is represented by spins).
Each vertical edge is a kind of free propagator
- it carries an operator Pe acting on the
Hilbert space of the nodes it connects.
Each vertex represents an interaction. It
carries an amplitude functional Av (being in
fact a tensor with a complicated structure).
Moreover the faces carry weight factors Af .
The transition amplitude of a spin-foam is

!
! 
O
O
Y
Pe y
Av ·  Af 
W =
e
v
f
Transition Amplitudes
Quantum states of General Relativity
Spin-foams: Feynman Diagrams for LQG
Gravity as a constrained BF -theory
Outline
1
Transition Amplitudes
Quantum Mechanics: Probabilities
Quantum Field Theory: Feynman Diagrams
2
Quantum states of General Relativity
Gravity as a curved space-time
Ashtekar quantization
Discrete space in loop quantum gravity
3
Spin-foams: Feynman Diagrams for LQG
Path integrals for the BF -theory
Gravity as a constrained BF -theory
Two examples
4
An open issue
Generalization of Saddle Point Approximation
An open issue
Transition Amplitudes
Quantum states of General Relativity
Spin-foams: Feynman Diagrams for LQG
An open issue
Gravity as a constrained BF -theory
Gravity as a constrained BF -theory
One of forms of the action of the gravitational field is
ˆ 1
S [e, ω] =
? (e ∧ e) + (e ∧ e) ∧ F [ω]
γ
It is almost the same, as the action of the SL(2, C)-BF theory, except a constraint
on the B-field, called the simplicity constraint:
B = ? (e ∧ e) +
1
(e ∧ e)
γ
This constraint can be expressed in terms of generators of the SL (2, C) group:
~ = −γ L
~
K
It can be solved weakly on the L2 (SL (2, C)). The space of the solutions appears to
be isomorphic with L2 (SU (2)) by so called EPRL map:
Yˆ : H(j) 3 |mij 7→ |j, mi(p,k) ∈ H(p=γj,k=j)
Thus the dynamics of gravity can be encoded in SU (2)-spin-networks and the EPRL
map.
Transition Amplitudes
Quantum states of General Relativity
Spin-foams: Feynman Diagrams for LQG
An open issue
Gravity as a constrained BF -theory
EPRL spin-foams
The EPRL spin-foams differ from the BF
spin-foams only by the vertex amplitude
functional.
In the BF -model the vertex amplitude is
Av := ψv (1)
where ψv is the spin-network-like state
determined by the coloring of neighbor
elements.
The EPRL amplitude is
AEPRL
:= Yˆ ψv (1)
v
Transition Amplitudes
Quantum states of General Relativity
Spin-foams: Feynman Diagrams for LQG
An open issue
Gravity as a constrained BF -theory
Operator Spin-network diagrams
Together with Marcin Kisielowski, under supervision of prof. Jerzy
Lewandowski, we have introduced a new language to express Loop Quantum
Gravity dynamical processes: the Operator Spin-network Diagrams.
They allow to express the highly complicated structure of the spin foam as a
2-dim drawing.
Transition Amplitudes
Quantum states of General Relativity
Spin-foams: Feynman Diagrams for LQG
An open issue
Gravity as a constrained BF -theory
Feynman-EPRL rules
The Operator Spin-network Diagrams allow to
express processes of spin-foams that would be hard
draw or even hard to imagine.
We can also simply read the formula for transition
amplitude out of them:
A
=
Transition Amplitudes
Quantum states of General Relativity
Spin-foams: Feynman Diagrams for LQG
An open issue
Gravity as a constrained BF -theory
Feynman-EPRL rules
The Operator Spin-network Diagrams allow to
express processes of spin-foams that would be hard
draw or even hard to imagine.
We can also simply read the formula for transition
amplitude out of them:
A
=
hu1 |
Transition Amplitudes
Quantum states of General Relativity
Spin-foams: Feynman Diagrams for LQG
An open issue
Gravity as a constrained BF -theory
Feynman-EPRL rules
The Operator Spin-network Diagrams allow to
express processes of spin-foams that would be hard
draw or even hard to imagine.
We can also simply read the formula for transition
amplitude out of them:
A
=
hu1 | Y † gI−1
Transition Amplitudes
Quantum states of General Relativity
Spin-foams: Feynman Diagrams for LQG
An open issue
Gravity as a constrained BF -theory
Feynman-EPRL rules
The Operator Spin-network Diagrams allow to
express processes of spin-foams that would be hard
draw or even hard to imagine.
We can also simply read the formula for transition
amplitude out of them:
A
=
hu1 | Y † gI−1 gII Y
Transition Amplitudes
Quantum states of General Relativity
Spin-foams: Feynman Diagrams for LQG
An open issue
Gravity as a constrained BF -theory
Feynman-EPRL rules
The Operator Spin-network Diagrams allow to
express processes of spin-foams that would be hard
draw or even hard to imagine.
We can also simply read the formula for transition
amplitude out of them:
A
=
hu1 | Y † gI−1 gII Y · 1
Transition Amplitudes
Quantum states of General Relativity
Spin-foams: Feynman Diagrams for LQG
An open issue
Gravity as a constrained BF -theory
Feynman-EPRL rules
The Operator Spin-network Diagrams allow to
express processes of spin-foams that would be hard
draw or even hard to imagine.
We can also simply read the formula for transition
amplitude out of them:
A
=
hu1 | Y † gI−1 gII Y · 1 · Y † gV−1 gV I Y
Transition Amplitudes
Quantum states of General Relativity
Spin-foams: Feynman Diagrams for LQG
An open issue
Gravity as a constrained BF -theory
Feynman-EPRL rules
The Operator Spin-network Diagrams allow to
express processes of spin-foams that would be hard
draw or even hard to imagine.
We can also simply read the formula for transition
amplitude out of them:
A
=
hu1 | Y † gI−1 gII Y · 1 · Y † gV−1 gV I Y |u10 ij1
Transition Amplitudes
Quantum states of General Relativity
Spin-foams: Feynman Diagrams for LQG
An open issue
Gravity as a constrained BF -theory
Feynman-EPRL rules
The Operator Spin-network Diagrams allow to
express processes of spin-foams that would be hard
draw or even hard to imagine.
We can also simply read the formula for transition
amplitude out of them:
A
=
hu1 | Y † gI−1 gII Y · 1 · Y † gV−1 gV I Y |u10 ij1
× hu2 |
Transition Amplitudes
Quantum states of General Relativity
Spin-foams: Feynman Diagrams for LQG
An open issue
Gravity as a constrained BF -theory
Feynman-EPRL rules
The Operator Spin-network Diagrams allow to
express processes of spin-foams that would be hard
draw or even hard to imagine.
We can also simply read the formula for transition
amplitude out of them:
A
=
hu1 | Y † gI−1 gII Y · 1 · Y † gV−1 gV I Y |u10 ij1
× hu2 | |u20 ij2
Transition Amplitudes
Quantum states of General Relativity
Spin-foams: Feynman Diagrams for LQG
An open issue
Gravity as a constrained BF -theory
Feynman-EPRL rules
The Operator Spin-network Diagrams allow to
express processes of spin-foams that would be hard
draw or even hard to imagine.
We can also simply read the formula for transition
amplitude out of them:
A
=
hu1 | Y † gI−1 gII Y · 1 · Y † gV−1 gV I Y |u10 ij1
× hu2 | |u20 ij2
h
i
×
Trjf Y † gV−1III gII Y · 1 · Y † gV−1 gV II Y
Transition Amplitudes
Quantum states of General Relativity
Spin-foams: Feynman Diagrams for LQG
An open issue
Gravity as a constrained BF -theory
Feynman-EPRL rules
The Operator Spin-network Diagrams allow to
express processes of spin-foams that would be hard
draw or even hard to imagine.
We can also simply read the formula for transition
amplitude out of them:
A
=
hu1 | Y † gI−1 gII Y · 1 · Y † gV−1 gV I Y |u10 ij1
× hu2 | |u20 ij2
i
X j h † −1
×
Tr f Y gV III gII Y · 1 · Y † gV−1 gV II Y
{jf }
Transition Amplitudes
Quantum states of General Relativity
Spin-foams: Feynman Diagrams for LQG
An open issue
Gravity as a constrained BF -theory
Feynman-EPRL rules
The Operator Spin-network Diagrams allow to
express processes of spin-foams that would be hard
draw or even hard to imagine.
We can also simply read the formula for transition
amplitude out of them:
ˆ
A =
dgI · · · dgXV I
hu1 | Y † gI−1 gII Y · 1 · Y † gV−1 gV I Y |u10 ij1
× hu2 | |u20 ij2
i
X j h † −1
×
Tr f Y gV III gII Y · 1 · Y † gV−1 gV II Y
{jf }
×···
Transition Amplitudes
Quantum states of General Relativity
Spin-foams: Feynman Diagrams for LQG
Two examples
Outline
1
Transition Amplitudes
Quantum Mechanics: Probabilities
Quantum Field Theory: Feynman Diagrams
2
Quantum states of General Relativity
Gravity as a curved space-time
Ashtekar quantization
Discrete space in loop quantum gravity
3
Spin-foams: Feynman Diagrams for LQG
Path integrals for the BF -theory
Gravity as a constrained BF -theory
Two examples
4
An open issue
Generalization of Saddle Point Approximation
An open issue
Transition Amplitudes
Quantum states of General Relativity
Spin-foams: Feynman Diagrams for LQG
An open issue
Two examples
Dipole Cosmology
the model
A simple model called Dipole Cosmology was
introduced by Bianchi, Rovelli, Vidotto in 2010.
A transition amplitude calculated within this model is
W (z)
=
4
XY
3/2
(2j` + 1) e−2t~j` (j` +1)−˙ıλv0 j`
{j` } `=1
hιN | T |ιS i
z = γ a˙ + ı˙a2
It is peaked precisely at the classical trajectory of
expanding scale factor.
−˙ızj`
×
Transition Amplitudes
Quantum states of General Relativity
Spin-foams: Feynman Diagrams for LQG
An open issue
Two examples
Dipole Cosmology
our input
Using the graph diagrams we have found all contributions to the first order of
the model - and showed, which of them are negligible.
Transition Amplitudes
Quantum states of General Relativity
Spin-foams: Feynman Diagrams for LQG
An open issue
Two examples
Self energy problem
In Spin Foam models there are radiative
corrections to the transition amplitudes similar to those one can find in QFT.
Aldo Riello found recently an expression for
the melonic radiative correction to the free
propagator
W = ln Λ · T2
but he could not describe the operator T.
Transition Amplitudes
Quantum states of General Relativity
Spin-foams: Feynman Diagrams for LQG
An open issue
Two examples
Self energy problem
I investigated this operator and found it equals
ˆ
T
dη
=
R+
sinh η
4πη
2 Y
N
e−(ji +mi +1−˙ıγji )η ×
i=1
−2η
2 F1 ji + 1 − ı˙γji , ji + 1 + mi ; 2ji + 2; 1 − e
I also found the leading order of this integral
T=
1
4π
2 6π
P
J · (1 + γ 2 ) xi
3
2
5
I + O J− 2
This operator in the Spin-foam theory has meaning similar to the renormalized
propagator in QFT. It is significant, that it is proportional to identity.
[ JP, arXiv:1307.4747 ]
Transition Amplitudes
Quantum states of General Relativity
Spin-foams: Feynman Diagrams for LQG
Generalization of Saddle Point Approximation
Outline
1
Transition Amplitudes
Quantum Mechanics: Probabilities
Quantum Field Theory: Feynman Diagrams
2
Quantum states of General Relativity
Gravity as a curved space-time
Ashtekar quantization
Discrete space in loop quantum gravity
3
Spin-foams: Feynman Diagrams for LQG
Path integrals for the BF -theory
Gravity as a constrained BF -theory
Two examples
4
An open issue
Generalization of Saddle Point Approximation
An open issue
Transition Amplitudes
Quantum states of General Relativity
Spin-foams: Feynman Diagrams for LQG
An open issue
Generalization of Saddle Point Approximation
An open issue
Let us recall the T operator. It is given by the integral
ˆ
T=
d3 ηµ (|η|)
R3
N
Y
(j)
fm
(|η|)
i=1
for µ (η) :=
sinh η
4πη
2
and
(j)
fm
(η) = e−(ji +mi +1−˙ıγji )η 2 F1 ji + 1 − ı˙γji , ji + 1 + mi ; 2ji + 2; 1 − e−2η
where 2 F1 (a, b, c; z) is the Gauss Hypergeometric function, defined by the
series:
2 F1
(a, b, c; z) :=
∞
X
an bn n
z
cn n!
n=0
where xn := x · (x + 1) · · · (x + n − 1)
Such integral is horrible to calculate directly, but it can be estimated.
IF the integrand satisfies appropriate conditions, one can apply to it so called
Saddle Point Approximation, thanks to which the integration would be given by
a derivative of appropriate function. Derivatives (even of Hypergeometric
functions) are relatively easy to calculate, especially when compared to the
above integral.
Transition Amplitudes
Quantum states of General Relativity
Spin-foams: Feynman Diagrams for LQG
An open issue
Generalization of Saddle Point Approximation
Saddle Point Approximation.
Given two functions f, g : Ω →´C defined on a compact region Ω ⊂ Rn one can
estimate the integral I (Λ) := Ω dn x g (x) e−Λf (x) for Λ 1 using the
following analogy:
´ n
1 Obviously
d x g (x) δ n (x) = g (0)
Ω
2
3
Consider now g (x) = 1 and f (x) = |x|2 . Then
n/2
´
2
I (Λ) = Ω dn x e−Λ|x| = 2π
Λ
2
Note now, that e−Λx mimics n-dimensional δ-distribution (scaled by
2π n/2
) thus one can guess, that
Λ
ˆ
2
dn x g (x) e−Λ|x| =
I (Λ) =
Ω
2π
Λ
n/2
1
g (0) 1 + O
1
Λ2
In fact one can prove, that for a wide class of functions f, g : Ω → C the
following estimation is true
2 n/2
ˆ
∂ f 2π
1
I (Λ) =
dn x g (x) e−Λf (x) = n g (x0 ) e−Λf (x0 ) 1 + O
1
∂x x=x0 Λ
Λ2
Ω
where x0 is the point such that ∇f (x0 ) = 0 and the real part of Hessian
2
f matrix < ∂x∂i ∂x
is positive defined.
j
x=x
Transition Amplitudes
Quantum states of General Relativity
Spin-foams: Feynman Diagrams for LQG
An open issue
Generalization of Saddle Point Approximation
Application of SPA to our case
Q
(j)
Unfortunately our function Φ (η, j) := µ (|η|) N
i=1 fm (|η|) is neither of form
−Λf (x)
g (x) e
, nor defined on a compact support. The second issue is not a
problem - it is a simple exercise to show, that if Φ (x, Λ) is bounded by e−|x|Λ
outside some compact region Ω (which is the case for our integrand), then the
part outside Ω has no influence on the the estimation of the integral. However
the form of the integrand IS a problem.
There is so called Euclidean version of spin-foam theory, where the map Y
maps SU (2) representations into SO (4) representations (instead of SL (2, C)
representations). In this theory there is an analogous integral, however the
(j)
analogs of functions fm are slightly different, i.e. they are:
(j)
−j ln(a(g)b(g))
˜
fm (g) = e
where a and b are two functions on SU (2)-group.
Then j becomes the largeness parameter and the integral can be calculated
using the SPA formula.
Unfortunately in the full, Lorentzian theory, the integrals are as they are. There
is no obvious decomposition of Φ (η, j) into functions exponentially decaying
in j.
Transition Amplitudes
Quantum states of General Relativity
Spin-foams: Feynman Diagrams for LQG
An open issue
Generalization of Saddle Point Approximation
The issue
Thus I am interested in generalizing the Saddle Point approximation theorem.
There must be a way of extending the class of functions that can be estimated
by the above estimation.
Generalized saddle point hypothesis
Let Φ (x, Λ) : Ω × R+ → C satisfying appropriate conditions. Then
2 n/2
ˆ
∂ φ
2π
1
I (Λ) :=
Φ (x0 , Λ) 1 + O
dn xΦ (x, Λ) = n 1
∂x x=x0 Λ
Λ2
Ω
for φ (x) := limΛ→∞ Λ1 Φ (x, Λ) and x0 ∈ Ω such that ∇φ (x0 ) = 0 and
2
φ < ∂x∂i ∂x
negatively defined.
j
x=x0
The main issue is to define, what does appropriate mean. Possibly in terms of
some analytic conditions (i.e. estimations) for behavior with respect to x − x0
(obviously the traditional SPA theorem puts some conditions on Φ, but they
are of algebraic nature and they are very hard to check for our function).
Transition Amplitudes
Quantum states of General Relativity
Spin-foams: Feynman Diagrams for LQG
An open issue
Generalization of Saddle Point Approximation
Properties of the integrand
Well, I do not need the most general form of the theorem (of course it would
be nice to find it, but well, it sometimes is just to difficult). It is enough to find
such class of functions, for which it holds, that contains my integrand. Here are
some properties of it. Q
(ji )
Let Φ (η, j) = µ (|η|) N
i=1 fmi (|η|). Then:
1
2
The largeness parameter is j := maxi (ji )
1
2 J PN
i=1 xi + j
η
1−2η−e−2η 12
|Φ(η, j)| ≤ sinh
e
where the equality is only
4πη
for η = 0
3
The point η = 0 is the only relevant candidate for the η0 point.
4
The functions fm (η) are analytic in η (so in fact all the function Φ (η, j)
is analytic).
(j)
Some further properties can be derived from the properties of the
Hypergeometric Function 2 F1 .
Transition Amplitudes
Quantum states of General Relativity
Spin-foams: Feynman Diagrams for LQG
An open issue
Generalization of Saddle Point Approximation
Proposition of strategy
Of course there are many strategies to address the above problem. I can tell
you, how would I try to solve it:
1
Find the proof of the original SPA theorem
2
At each step of the prove check, which properties of the functions f (x)
and g (x) one uses.
3
At each step consider, can one translate such assumption into an
assumption on the integrand function Φ (j, Λ) as a whole.
4
Then synthetise the observations made at 2 and 3.
And obviously if you have a working which is different from mine - that’s great!
Transition Amplitudes
Quantum states of General Relativity
Spin-foams: Feynman Diagrams for LQG
An open issue
Generalization of Saddle Point Approximation
Thank you for your involvement!
Contact to me:
Jacek Puchta
Instytut Fizyki Teoretycznej, Hoża 69, pok. 053
e-mail: jpa_na_serwerze_fuw.edu.pl or jpa.puchta_na_serwerze_gmail.com
tel. 22 55 32 303