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The  real  number  line  is  a  good  model  for  the  physical  con6nuum  

David  Tong  University  of  Cambridge  

February  2013  

The  Punchline  

No  one  knows  how  to  write  down  a  discrete  version  of  the  laws  of  physics  

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  

d=1+1  dimensional  Maxwell  coupled  to  a  massless  Dirac  fermion  

E  

p  

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.  

Chiral  Gauge  Theories  

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  

Nielsen-­‐Ninomiya  Theorem  

You  can’t  do  it*.    

*  up  to  certain  assump6ons  

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  

Just  One  Last  Thing…  

Is  this  defini6on  of  chirality  gauge  invariant?  (i.e.  does  the  theory  exist?)    

Abelian  Theories  

Yes,  if  and  only  if  the  con6nuum  theory  is  non-­‐anomalous.    

Non-­‐Abelian  Theories  

No  one  knows.    

The  Punchline  

The  Standard  Model  is  a  non-­‐Abelian  chiral  gauge  theory.    

No  one  knows  how  to  write  down  a  discrete  version  of  the  Standard  Model.