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Fermionic spinfoam models and TQFTs Steven Kerr University of - - PowerPoint PPT Presentation
Fermionic spinfoam models and TQFTs Steven Kerr University of - - PowerPoint PPT Presentation
Fermionic spinfoam models and TQFTs Steven Kerr University of Nottingham Quantum Gravity Sum over Histories Quantum Gravity Sum over Histories D g e iS Z = Quantum Gravity Sum over Histories D g e iS Z =
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Quantum Gravity
◮ Sum over Histories
Z = “
- Dg eiS”
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Quantum Gravity
◮ Sum over Histories
Z = “
- Dg eiS”
◮ Fundamental length scale
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Quantum Gravity
◮ Sum over Histories
Z = “
- Dg eiS”
◮ Fundamental length scale
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Problems
◮ Triangulation independence
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Problems
◮ Triangulation independence ◮ Absence of matter
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Problems
◮ Triangulation independence ◮ Absence of matter ◮ We take the point of view that matter and triangulation
independence are crucial!
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Induced actions
Z =
- DψDψ ei
- ψ /
Dψ
/ D = γµ(dµ − iAµ)
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Induced actions
Z =
- DψDψ ei
- ψ /
Dψ
/ D = γµ(dµ − iAµ) = det(i / D) = etr ln i /
D
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Induced actions
Z =
- DψDψ ei
- ψ /
Dψ
/ D = γµ(dµ − iAµ) = det(i / D) = etr ln i /
D
= eiSeff
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Induced actions
Z =
- DψDψ ei
- ψ /
Dψ
/ D = γµ(dµ − iAµ) = det(i / D) = etr ln i /
D
= eiSeff
◮ John Barret has suggested that the Standard Model can be
induced in this way: arXiv:1101.6078v2 [hep-th]
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A one dimensional fermionic TQFT
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A one dimensional fermionic TQFT
ψ(t), ¯ ψ(t), t ∈ [0, 2π] ψi, ¯ ψi, i = 1..N
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A one dimensional fermionic TQFT
ψ(t), ¯ ψ(t), t ∈ [0, 2π] ψi, ¯ ψi, i = 1..N
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A one dimensional fermionic TQFT
ψ(t), ¯ ψ(t), t ∈ [0, 2π] ψi, ¯ ψi, i = 1..N A(t) Qi = Pei
- Adt
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A one dimensional fermionic TQFT
ˆ Z =
- N
- i=1
dψid ¯ ψi e
N
i=1 ¯
ψi(ψi−Qi+1ψi+1)
ψN+1 = ψ1 QN+1 = Q1
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A one dimensional fermionic TQFT
ˆ Z =
- N
- i=1
dψid ¯ ψi e
N
i=1 ¯
ψi(ψi−Qi+1ψi+1)
ψN+1 = ψ1 QN+1 = Q1 = det(1 − Q) Q =
N
- i=1
Qi
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A one dimensional fermionic TQFT
ˆ Z =
- N
- i=1
dψid ¯ ψi e
N
i=1 ¯
ψi(ψi−Qi+1ψi+1)
ψN+1 = ψ1 QN+1 = Q1 = det(1 − Q) Q =
N
- i=1
Qi
◮ ˆ
Z is triangulation independent - a topological invariant!
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Action
What is the significance of this theory? It is a discretisation of a
- ne dimensional Dirac theory,
ˆ S = −i
N
- i=1
¯ ψi (ψi − Qi+1ψi+1) = i∆t
N
- i=1
¯ ψi Qi+1ψi+1 − ψi ∆t
- ∆t = 2π
N
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Action
What is the significance of this theory? It is a discretisation of a
- ne dimensional Dirac theory,
ˆ S = −i
N
- i=1
¯ ψi (ψi − Qi+1ψi+1) = i∆t
N
- i=1
¯ ψi Qi+1ψi+1 − ψi ∆t
- ∆t = 2π
N lim
∆t→0 i
Qi+1ψi+1 − ψi ∆t
- = /
Dtψ(t) lim
∆t→0 ∆t N
- i=1
= 2π dt lim
∆t→0
ˆ S =
- dtψ /
Dψ
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Continuum theory
One can calculate the partition function of the continuum theory exactly
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Continuum theory
One can calculate the partition function of the continuum theory exactly Z =
- DψDψ ei
- dt ψ /
Dψ
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Continuum theory
One can calculate the partition function of the continuum theory exactly Z =
- DψDψ ei
- dt ψ /
Dψ
We find that ˆ Z = Z!
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Continuum theory
One can calculate the partition function of the continuum theory exactly Z =
- DψDψ ei
- dt ψ /
Dψ
We find that ˆ Z = Z! Naturally, one would like to try do something similar in higher
- dimensions. This is the subject of current investigation.
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