the$real$number$line$is$agood$model$ for$the$physical
TRANSCRIPT
The real number line is a good model for the physical con6nuum
David Tong University of Cambridge
February 2013
Quantum Field Theory
a =R
T 2
F = ma =mR
T 2
F ∼ 1
R2
SE
SB= x
Area(SEF )
Area(SBC)= x2
∆t ∼ R2
∆v = F∆t = constant
�uP · �vP = 0
σ ∼ ω−2/3
φ(x)
ψα(x)
Aµ(x)
References
[1] Gibbons and Rychenkova, 9608085
[2] Kapustin and Strassler, 9902033
5
a =R
T 2
F = ma =mR
T 2
F ∼ 1
R2
SE
SB= x
Area(SEF )
Area(SBC)= x2
∆t ∼ R2
∆v = F∆t = constant
�uP · �vP = 0
σ ∼ ω−2/3
φ(x)
ψα(x)
Aµ(x)
References
[1] Gibbons and Rychenkova, 9608085
[2] Kapustin and Strassler, 9902033
5
a =R
T 2
F = ma =mR
T 2
F ∼ 1
R2
SE
SB= x
Area(SEF )
Area(SBC)= x2
∆t ∼ R2
∆v = F∆t = constant
�uP · �vP = 0
σ ∼ ω−2/3
φ(x)
ψα(x)
Aµ(x)
References
[1] Gibbons and Rychenkova, 9608085
[2] Kapustin and Strassler, 9902033
5
Con6nuum space6me Interes6ng physics
The Trouble: Chiral Fermions
a =R
T 2
F = ma =mR
T 2
F ∼ 1
R2
SE
SB= x
Area(SEF )
Area(SBC)= x2
∆t ∼ R2
∆v = F∆t = constant
�uP · �vP = 0
σ ∼ ω−2/3
φ(x)
ψα(x)
Aµ(x)
{γ5, /D} = 0
P± =1
2(1± γ5)
ψL = P+ψ
ψR = P−ψ
References
[1] Gibbons and Rychenkova, 9608085
[2] Kapustin and Strassler, 9902033
5
a =R
T 2
F = ma =mR
T 2
F ∼ 1
R2
SE
SB= x
Area(SEF )
Area(SBC)= x2
∆t ∼ R2
∆v = F∆t = constant
�uP · �vP = 0
σ ∼ ω−2/3
φ(x)
ψα(x)
Aµ(x)
{γ5, /D} = 0
P± =1
2(1± γ5)
ψL = P+ψ
ψR = P−ψ
References
[1] Gibbons and Rychenkova, 9608085
[2] Kapustin and Strassler, 9902033
5
a =R
T 2
F = ma =mR
T 2
F ∼ 1
R2
SE
SB= x
Area(SEF )
Area(SBC)= x2
∆t ∼ R2
∆v = F∆t = constant
�uP · �vP = 0
σ ∼ ω−2/3
φ(x)
ψα(x)
Aµ(x)
{γ5, /D} = 0
P± =1
2(1± γ5)
ψL = P+ψ
ψR = P−ψ
References
[1] Gibbons and Rychenkova, 9608085
[2] Kapustin and Strassler, 9902033
5
a =R
T 2
F = ma =mR
T 2
F ∼ 1
R2
SE
SB= x
Area(SEF )
Area(SBC)= x2
∆t ∼ R2
∆v = F∆t = constant
�uP · �vP = 0
σ ∼ ω−2/3
φ(x)
ψα(x)
Aµ(x)
{γ5, /D} = 0
P± =1
2(1± γ5)
ψL = P+ψ
ψR = P−ψ
References
[1] Gibbons and Rychenkova, 9608085
[2] Kapustin and Strassler, 9902033
5
!"#$%&'&(!"%)$(*&%+",-#(.&''"-.(/"**"$01'
• A massive fermion needs both chirali6es • A massless fermion needs only one
Anomalies
Symmetries that act differently on fermions of opposite chirality oJen don’t exist in the quantum theory.
An Example in d=1+1
Turn on an electric field E(t) for some 6me
E
p
momenta shiJ
Extra right-‐moving par6cles
Extra leJ-‐moving an6-‐par6cles
Axial charge is violated
Anomalies
Where did the extra charge come from?
From infinity!
The anomaly is an effect arising from the con6nuum
Gauge Anomalies
Anomalies in global symmetries are merely interes6ng.
Anomalies in gauge symmetries are fatal.
A Discrete World
• Replace the con6nuum with a discre6zed la[ce • Now quantum field theory = quantum mechanics • No infini6es. No anomalies.
Anomalies on the La[ce
Anomalous global symmetries are explicitly broken by the discre6za6on procedure.
What about gauge symmetries?
Fermions on the La[ce
a =R
T 2
F = ma =mR
T 2
F ∼ 1
R2
SE
SB= x
Area(SEF )
Area(SBC)= x2
∆t ∼ R2
∆v = F∆t = constant
�uP · �vP = 0
σ ∼ ω−2/3
φ(x)
ψα(x)
Aµ(x)
{γ5, /D} = 0
P± =1
2(1± γ5)
ψL = P+ψ
ψR = P−ψ
S =
�d4x ψ̄(i∂µγ
µ −m)ψ
=
� π/a
−π/ad4p
�i
asin(pµ)γ
µ −m
�ψ̄−pψp
References
[1] Gibbons and Rychenkova, 9608085
[2] Kapustin and Strassler, 9902033
5
BZ la[ce spacing
24 fermions!
• This is fermion doubling. • Try to naively discre6ze a chiral theory and you get a non-‐chiral theory
Overlap Fermions Kaplan, Neuberger, Narayan, Luscher, late 1990s
Project onto leJ/right fermions with
a =R
T 2
F = ma =mR
T 2
F ∼ 1
R2
SE
SB= x
Area(SEF )
Area(SBC)= x2
∆t ∼ R2
∆v = F∆t = constant
�uP · �vP = 0
σ ∼ ω−2/3
φ(x)
ψα(x)
Aµ(x)
{γ5, /D} = 0
P± =1
2(1± γ5)
ψL = P+ψ
ψR = P−ψ
γ̂5 = γ5�1− a /D
�
{γ5, /D} = a /Dγ5 /D
S =
�d4x ψ̄(i∂µγ
µ −m)ψ
=
� π/a
−π/ad4p
�i
asin(pµ)γ
µ −m
�ψ̄−pψp
5
a =R
T 2
F = ma =mR
T 2
F ∼ 1
R2
SE
SB= x
Area(SEF )
Area(SBC)= x2
∆t ∼ R2
∆v = F∆t = constant
�uP · �vP = 0
σ ∼ ω−2/3
φ(x)
ψα(x)
Aµ(x)
{γ5, /D} = 0
P± =1
2(1± γ5)
ψL = P+ψ
ψR = P−ψ
γ̂5 = γ5�1− a /D
�
{γ5, /D} = a /Dγ5 /D
S =
�d4x ψ̄(i∂µγ
µ −m)ψ
=
� π/a
−π/ad4p
�i
asin(pµ)γ
µ −m
�ψ̄−pψp
5
depends on momentum and gauge field.
with