The worldline approach to quantum field theorypaginas.fisica.uson.mx/eff.2013/UniSon13WL_OC.pdf ·...

The worldline approach to quantum field theory

Olindo Corradini

CEFyMAP, Universidad Autónoma de Chiapas

VIII Escuela de Física Fundamental – Universidad de Sonora09 Agosto 2013

O. Corradini (UnACh) UniSon 2013 1 / 35

Outline

1 Introduction

2 Worldline formalism in flat spaceQFT from QM models

3 Recent ExtensionsWorldline formalism in curved spaceFields with spinWorldline formalism in manifolds with boundaries

4 Conclusions


Outline

1 Introduction



4 Conclusions


Outline

1 Introduction



4 Conclusions


Outline

1 Introduction



4 Conclusions


Introduction

Worldline methods: Quantum Field Theory results from qznof QM models→ no need to compute momentum integrals orDirac traces explicitly

Main tools in use: particle actions

(schematically) S[x , ψ; G] =

∫ T

0dτ(

x2 + ψψ + V (x , x , ψ; G))

x bosonic ψ fermionic G external

canonical qznpath integral (integral over trajectories)


Introduction

Purely bosonic

Dirichlet boundary conditions (topology of a line)

〈x |e−TH |x ′〉 =

∫ x(T )=x

x(0)=x ′Dx(τ)e−S[x ;G]

x’

x

Periodic boundary conditions (topology of a circle)

Γ[G] =

∫dTT

∫x(0)=x(T )

Dx(τ)e−S[x ;G]


Introduction

Heat Kernel

Heat equation (∂T −D2)K (x , x ′; T ) = 0 , heat kernelΦ(x ,T ) =

∫dx ′K (x , x ′; t)Φ(x ′,0)

Heat kernel ansatz

K (x , x ′; T ) = 〈x |eTD2 |x ′〉 =e−σ(x ,x ′)/2T

(4πT )D/2

∑n

an(x , x ′)T n

path integral representation

K (x , x ′; T ) =

∫ x(T )=x

x(0)=x ′Dx e−S[x ;A] = e−S0[xcl ,A]

∫ q(T )=0

q(0)=0Dq e−Sint [q;A]

an(x , x ′) by loop expansion of S

x’

x

QM path integral with fixed boundary conditions (topology of a line)


Introduction

Heat Kernel


∫dx ′K (x , x ′; t)Φ(x ′,0)

Heat kernel ansatz

K (x , x ′; T ) = 〈x |eTD2 |x ′〉 =e−σ(x ,x ′)/2T

(4πT )D/2

∑n

an(x , x ′)T n


K (x , x ′; T ) =

∫ x(T )=x


∫ q(T )=0


an(x , x ′) by loop expansion of S

x’

x



Introduction

Heat Kernel


∫dx ′K (x , x ′; t)Φ(x ′,0)

Heat kernel ansatz

K (x , x ′; T ) = 〈x |eTD2 |x ′〉 =e−σ(x ,x ′)/2T

(4πT )D/2

∑n

an(x , x ′)T n


K (x , x ′; T ) =

∫ x(T )=x


∫ q(T )=0


an(x , x ′) by loop expansion of S x’

x



Introduction

Heat Kernel

Simple example D2 = α∂2, → S[x ] =∫ T

01

4α x2

K (x , x ′; T ) =e−(x−x ′)2/4αT

(4πT )1/2 =

∫ x(T )=x

x(0)=x ′Dx e−S[x ]

split x(τ) = xcl(τ) + q(τ)

τ

x

x’ x ( )τcl

q( )

⇒ S[x ] = S[xcl ] + S[q] and S[xcl ] = (x − x ′)2/4αT

K (x , x ′; t) = e−S[xcl ]

∫ q(T )=0

q(0)=0Dq e−S[q] = e−(x−x ′)2/4αT 1

(4πT )1/2


Introduction

Heat Kernel


01

4α x2

K (x , x ′; T ) =e−(x−x ′)2/4αT

(4πT )1/2 =

∫ x(T )=x

x(0)=x ′Dx e−S[x ]


τ

x

x’ x ( )τcl

q( )


K (x , x ′; t) = e−S[xcl ]

∫ q(T )=0

q(0)=0Dq e−S[q] = e−(x−x ′)2/4αT 1

(4πT )1/2


Introduction

Heat Kernel


01

4α x2

K (x , x ′; T ) =e−(x−x ′)2/4αT

(4πT )1/2 =

∫ x(T )=x

x(0)=x ′Dx e−S[x ]


τ

x

x’ x ( )τcl

q( )


K (x , x ′; t) = e−S[xcl ]

∫ q(T )=0

q(0)=0Dq e−S[q] = e−(x−x ′)2/4αT 1

(4πT )1/2


Introduction

Heat Kernel

in presence of interaction S[x ] =∫ T

0 ( 14α x2 + V (x , x)), an 6= 0 from

perturbative expansion

V (x , x) = V (xcl , xcl) + V ′x (xcl , xcl)q + V ′x (xcl , xcl)q + . . .

applications:diffusive processes (FP equation)short time: heat kernel expansionLarge time: (improved) Monte Carlo approach OC, Faccioli, Orland 2009

renormalization of QFT’s (worldline formalism: hold on a fewslides!!)


Introduction

Anomalies

classically conserved quantities may be broken at quantum level(e.g. axial symmetry)axial anomaly⟨

∂µJµA⟩

= Tr[γ51

]

≡ limT→0

Tr[γ5eT/D2

], /D2 = D2 + Fµνγµν

= limt→0

tr(γ5

∮Dx e−S[x ,A]

)S[x ; A,F ] =

∫ T

0dτ[

14

x2 + xµAµ + Fµνγµν]

QM path integral with periodic boundary conditions (topology of acircle) Alvarez-Gaumé, Witten, ’84


Introduction

Anomalies


∂µJµA⟩

= Tr[γ51

]≡ lim

T→0Tr[γ5eT/D2


= limt→0

tr(γ5

∮Dx e−S[x ,A]

)S[x ; A,F ] =

∫ T

0dτ[

14




Introduction

Anomalies


∂µJµA⟩

= Tr[γ51

]≡ lim

T→0Tr[γ5eT/D2


= limt→0

tr(γ5

∮Dx e−S[x ,A]

)S[x ; A,F ] =

∫ T

0dτ[

14




Introduction

Anomalies

in 4 dimensions ⟨∂µJµA

⟩=

116π2 ε

µ1···µ4Fµ1µ2Fµ3µ4

Axial anomaly responsible for π0 → γ + γ

Other types of anomalies (gauge, gravitational, conformal,...)computable with the same techniquePath integral on the circle; main application: effective action forquantum field theories and computation of related (1PI) Feynmandiagrams


Worldline formalism in flat space QFT from QM models

Quantum Field Theory

Second quantization: computation of Feynman diagrams fromcorrelation functions of fieldsIn perturbation theory the most generic diagram can be built frompropagators (Green functions) and one particle irreduciblediagrams (effective vertices)

Example: scalar theory with cubic interaction λφ(x)3

*= *

Effective vertices are the key objects for renormalization



Quantum Field Theory

Second quantization: computation of Feynman diagrams fromcorrelation functions of fieldsIn perturbation theory the most generic diagram can be built frompropagators (Green functions) and one particle irreduciblediagrams (effective vertices)Example: scalar theory with cubic interaction λφ(x)3

*= *

Effective vertices are the key objects for renormalization



Worldline Formalism

Tool to computeGreen functions (propagators)effective actions, i.e. functional generators of effective vertices

using particle modelsreview by Schubert ’01



WF: Green function

(− ∂µ∂µ + m2)∆(x , x ′) = δ(x − x ′) ≡ xx’

Massive scalar field (Feynman) propagator

∆(x , x ′) = 〈φ(x)φ(x ′)〉 =

∫d4p

e−ip·(x−x ′)

p2 + m2 + iε, p2 = −(p0)2 + p2

from anti−Wick rotation∫

d4pe−ip·(x−x ′)

p2 + m2 , p2 = (p4)2 + p2

Schwinger representation

〈φ(x)φ(x ′)〉 =

∫ ∞0

dT∫

d4p e−ip·(x−x ′)−T (p2+m2)

=

∫ ∞0

dT∫

d4p 〈x |e−T (p2+m2)|p〉〈p|x ′〉

Replacing p with p can integrate over p



WF: Green function

(− ∂µ∂µ + m2)∆(x , x ′) = δ(x − x ′) ≡ xx’


∆(x , x ′) = 〈φ(x)φ(x ′)〉 =

∫d4p

e−ip·(x−x ′)

p2 + m2 + iε, p2 = −(p0)2 + p2



p2 + m2 , p2 = (p4)2 + p2


〈φ(x)φ(x ′)〉 =

∫ ∞0

dT∫

d4p e−ip·(x−x ′)−T (p2+m2)

=

∫ ∞0

dT∫

d4p 〈x |e−T (p2+m2)|p〉〈p|x ′〉

Replacing p with p can integrate over p



WF: Green function

(− ∂µ∂µ + m2)∆(x , x ′) = δ(x − x ′) ≡ xx’


∆(x , x ′) = 〈φ(x)φ(x ′)〉 =

∫d4p

e−ip·(x−x ′)

p2 + m2 + iε, p2 = −(p0)2 + p2



p2 + m2 , p2 = (p4)2 + p2


〈φ(x)φ(x ′)〉 =

∫ ∞0

dT∫

d4p e−ip·(x−x ′)−T (p2+m2)

=

∫ ∞0

dT∫

d4p 〈x |e−T (p2+m2)|p〉〈p|x ′〉

Replacing p with p can integrate over pO. Corradini (UnACh) UniSon 2013 12 / 35


WF: Green function

〈φ(x)φ(x ′)〉 =

∫ ∞0

dT e−Tm2〈x |e−Tp2 |x ′〉 , H = p2 = δµνpµpν

Path integral representation of transition element

〈φ(x)φ(x ′)〉 =

∫ ∞0

dT e−Tm2∫ x(T )=x

x(0)=x ′Dx e−S[x ] (1)

S[x(τ)] =14

∫ T

0dτ δµν xµxν (2)

(1) Worldline representation for the scalar field propagator = pathintegral on the line(2) Worldline action

x’

x



Historical details

Feynman (’50): charged scalar particle coupled to vector field

〈φ(x)φ(x ′)〉A =

∫ ∞0

dT e−Tm2∫ x(T )=x

x(0)=x ′Dx e−S[x ,Aµ]

S[x(τ),Aµ] =

∫ T

0dτ

(14δµν xµxν + exµAµ(x(τ))

)+

e2

2

∫ T

0dτ∫ T

0dτ ′ xµx ′νDµν(x(τ)− x(τ ′))

∫ ∞

0ds

∫ x(s)=x′

x(0)=xDx(τ) exp

(−

1

2im2s

)exp

[−

i

2

∫ s

0dτ(

dxµ

dτ)2

− i∫ s

0dτ

dxµ

dτAµ(x(τ))

−i

2e2

∫ s

0dτ

∫ s

0dτ ′ dxµ

dτ

dxν

dτ ′δµν+ (x(τ) − x(τ ′))

](1.7)

That is, for a fixed value of the variable s (which can be identified with Schwinger proper time)one can construct the amplitude as a certain quantum mechanical path integral. This pathintegral has to be performed on the set of all open trajectories running from x to x′ in the fixedproper time s. The action consists of the familiar kinetic term, and two interaction terms. Ofthose the first represents the interaction with the external field, to all orders in the field, whilethe second one describes an arbitrary number of virtual photons emitted and re-absorbed alongthe trajectory of the particle (δ+ denotes the photon propagator). In second quantized fieldtheory, this amplitude would thus correspond to the infinite sequence of Feynman diagramsshown in fig. 4.

+ + ...+ +

Figure 4: Sum of Feynman diagrams represented by a single path integral.

As Feynman proceeds to show, this representation extends in an obvious way to the case ofan arbitrary fixed number of scalar particles, moving in an external potential and exchanginginternal photons, and thus to the complete S-matrix for scalar quantum electrodynamics. Everyscalar line or loop is then separately described by a path integral such as the one above. Thepath integrals are coupled by an arbitrary number of photon insertions. The derivation of thistype of path integral will be discussed in detail in chapter 3.

In the present work, we are mainly concerned with path integrals for closed loops. Let ustherefore rewrite Feynman’s formula for the case of a single closed loop in the external field,with no internal photon corrections. What we have at hand then is simply a representation ofthe one-loop effective action for the Maxwell field 2 ,

Γ[A] =∫ ∞

0

dT

Te−m2T

∫Dx exp

[

−∫ T

0dτ

(1

4x2 + ieAµxµ

)]

(1.8)

The path integral runs now over the space of closed trajectories with period T , xµ(T ) = xµ(0).2 The proper time parameter s has been rescaled and Wick rotated, s → −i2T . The spacetime metric

will also be taken as Euclidean, except when stated otherwise (upper and lower indices will be used purely fortypographical convenience). Moreover, we anticipate dimensional regularization and thus usually continue to DEuclidean dimensions.

9

involves virtual photons contributionsinvolves coupling to external field to all orders (arbitrary # ofexternal photons)O. Corradini (UnACh) UniSon 2013 14 / 35


Historical details

Extension to fermions, vectors,... is possible: spin factors orworldline supersymmetry

For fermions, diagram responsible foranomalous contribution to gyromagnetic ratio δg = g − 2



WF: one-loop effective action

Effective action: functional generator of 1PI correlation functionsExample: scalar QED

S[φ; A] =

∫dDx

[| (∂µ + iqAµ)φ|2 + m2|φ|2

]

One-loop effective action (singles out the quadratic part)

e−Γ[A] =

∫DφDφ∗ e−S[φ;A] = Det−1

[−|∂µ + iqAµ|2 + m2

]Γ[A]= Tr ln

[−|∂µ + iqAµ|2 + m2

]=

∫ ∞0

dTT

Tr e−T [−|∂µ+iqAµ|2+m2]

=

∫ ∞0

dTT

∫dx〈x |e−T [−|∂µ+iqAµ|2+m2]|x〉



WF: one-loop effective action

Effective action: functional generator of 1PI correlation functionsExample: scalar QED

S[φ; A] =

∫dDx

[| (∂µ + iqAµ)φ|2 + m2|φ|2

]One-loop effective action (singles out the quadratic part)

e−Γ[A] =

∫DφDφ∗ e−S[φ;A] = Det−1

[−|∂µ + iqAµ|2 + m2

]Γ[A]= Tr ln

[−|∂µ + iqAµ|2 + m2

]=

∫ ∞0

dTT

Tr e−T [−|∂µ+iqAµ|2+m2]

=

∫ ∞0

dTT

∫dx〈x |e−T [−|∂µ+iqAµ|2+m2]|x〉



Scalar QED one-loop effective action

Worldline representation of one-loop effective action

Γ[A] =

∫ ∞0

dTT

e−Tm2∮

Dx e−S[x ,A]

S[x ,A] =

∫ 1

0dτ[

14T

δµν xµxν + iqxµAµ(x(τ))

]

sum over all closed trajectories x(0) = x(1): topology of circle

Γ[A] yields photon amplitudes

Γ[A] =∑�




Worldline representation of one-loop effective action

Γ[A] =

∫ ∞0

dTT

e−Tm2∮

Dx e−S[x ,A]

S[x ,A] =

∫ 1

0dτ[

14T

δµν xµxν + iqxµAµ(x(τ))

]

sum over all closed trajectories x(0) = x(1): topology of circleΓ[A] yields photon amplitudes

Γ[A] =∑�




Easy to do in momentum spaceWrite potential as trivial background plus sum of photons

Aµ(x(τ)) =N∑

i=1

εi,µeipi ·x(τ)

expand e−iq∫

x ·A and pick up terms linear in all polarizations: itinvolves a QM correlation function

Γ[p1, ε1; · · · ; pN , εN ] = qN∫ T

0

dTT

e−Tm2N∏

i=1

∫ 1

0dτi

∮Dx e−

14T

∫x2

× ε1 · x(τ1)eip1·x(τ1) · · · εN · x(τN)eipN ·x(τN )




Easy to do in momentum spaceWrite potential as trivial background plus sum of photons

Aµ(x(τ)) =N∑

i=1

εi,µeipi ·x(τ)

expand e−iq∫

x ·A and pick up terms linear in all polarizations: itinvolves a QM correlation function

Γ[p1, ε1; · · · ; pN , εN ] = qN∫ T

0

dTT

e−Tm2N∏

i=1

∫ 1

0dτi

∮Dx e−

14T

∫x2

× ε1 · x(τ1)eip1·x(τ1) · · · εN · x(τN)eipN ·x(τN )




x(τ) = x0 + y(τ) , y(0) = y(1) = 0 , x0, zero mode∮Dx e−

14T

∫x2

=

∫dx0

∫Dy e−

14T

∫y2

=

∫dx0

x0

Γ[p1, ε1; · · · ; pN , εN ] ∝∫

dx0eix0·∑

pi

(4πT )D/2

⟨V A(ε1,p1) · · ·V A(εN ,pN)

⟩V A(ε,p) =

∫ 10 dτε · yeip·y photon vertex operator

path integral normalization 1(4πT )D/2 =

∫Dy e−

14T

∫y2

momentum conservation∫

dx0eix0·∑

pi = δ(∑

pi)




x(τ) = x0 + y(τ) , y(0) = y(1) = 0 , x0, zero mode∮Dx e−

14T

∫x2

=

∫dx0

∫Dy e−

14T

∫y2

=

∫dx0

x0

Γ[p1, ε1; · · · ; pN , εN ] ∝∫

dx0eix0·∑

pi

(4πT )D/2

⟨V A(ε1,p1) · · ·V A(εN ,pN)

⟩V A(ε,p) =

∫ 10 dτε · yeip·y photon vertex operator

path integral normalization 1(4πT )D/2 =

∫Dy e−

14T

∫y2

momentum conservation∫

dx0eix0·∑

pi = δ(∑

pi)




QM correlation function

⟨· · ·⟩

= expN∑

i,j=1

[12

∆ij pi · pj − i •∆ij εi · pj +12••∆ij εi · εj

]∣∣∣∣lin ε

∆ij ≡ 〈y(τi)y(τj)〉 ,

Yields Bern-Kosower master formulaBern and Kosower (’91) derived it from α′ → 0 of string amplitudesStrassler (’92) rederived BK formula directly from 1-st quantizedQFT (as done above)Extension to curved space subtle: regularization of QM pathintegrals needed




Consider a nontrivial background corresponding to constant e.m.field Aµ(x) = 1

2xρFρµF0m = −Em , Fmn = εmnr Br

Nonperturbative Euler-Heisemberg lagrangian

L(A) =1V

Γ[A] = −∫ ∞

0

dTT 3 e−Tm2 (eaT )(ebT )

tanh(eaT ) tan(ebT )

with invariants a(F , F ) , b(F , F )

B = 0 , E 6= 0 → a = 0 , b = E , integrand has poles

ImL =e2E2

8π3

∑n

1n2 e−

m2πneE

electron-positron production rate = first pole, Γep ∼ e2E2

8π3 e−m2πeE




Consider a nontrivial background corresponding to constant e.m.field Aµ(x) = 1

2xρFρµF0m = −Em , Fmn = εmnr Br

Nonperturbative Euler-Heisemberg lagrangian

L(A) =1V

Γ[A] = −∫ ∞

0

dTT 3 e−Tm2 (eaT )(ebT )

tanh(eaT ) tan(ebT )

with invariants a(F , F ) , b(F , F )

B = 0 , E 6= 0 → a = 0 , b = E , integrand has poles

ImL =e2E2

8π3

∑n

1n2 e−

m2πneE

electron-positron production rate = first pole, Γep ∼ e2E2

8π3 e−m2πeE




Schwinger effect in vacuum

Ecrit =m2

e

∣∣∣∣NU

=m2c3

e~∼ 1016 V cm−1

physical interpretation

eEcrit~

mc∼ mc2

work done on Compton length scale = rest energy of electron

Ecrit (for constant em field) impossibly high..... in vacuum




Inhomogeneous field Ecrit lower; can also be studied withworldline formalism through semiclassical nonperturbativecomputations (worldline instantons) Dunne-Schubert ’05

In some media (e.g. graphene) relativistic-like behaviors withv ∼ c/300: Schwinger effect measurable?In 2+1 dim. J ∝ E3/2 with E ∼ 109 V cm−1

Kao et al ’10-’11

It would be interesting to investigate deviations from standardSchwinger effect. Perhaps due to:

curvature (graphene surfaces aren’t exactly flat)fermion interaction



















SummaryDictionary

QM path integral on the line −→ heat kernel expansion, QFTpropagatorQM path integral on the circle −→ heat kernel trace, partitionfunction, anomalies, QFT effective action and its applicationscanonical qzn of particle models −→ EoM for the field theory(next)

Some extensions:

Worldline formalism in curved spaceamplitudes with gravitons with Bastianelli, Davila, Schubert, Zirotti

Fields with (arbitrary) spin with Bastianelli, Bonezzi, Latini, Waldron

QFT effective actions on manifolds with boundary (Casimir effect)with Bastianelli, Pisani, Schubert



SummaryDictionary


Some extensions:Worldline formalism in curved space

amplitudes with gravitons with Bastianelli, Davila, Schubert, Zirotti





SummaryDictionary








SummaryDictionary







Recent Extensions Worldline formalism in curved space

Scalar field one-loop effective action

Massive scalar field coupled to gravity

S[φ; g] =

∫dDx√

g12

(gµν∂µφ∂νφ+ m2φ2 + λRφ2)

produces one-loop effective action Γ[g] = 12Tr ln(−∇2 + m2 + λR)

It can be represented as

Γ[g] =

∫ ∞0

dTT

e−Tm2∮Dx(τ) e−S[x ;g]

with

S[x ; g] =

∫ 1

0dτ(

14T

gµν xµxν + TλR)

and

Dx ≡∏τ

√g(x(τ))dDx(τ)

Einstein-invariant measureO. Corradini (UnACh) UniSon 2013 25 / 35


One-loop effective action

Γ[g] can be obtained directly in first quantization.Start from 1d Einstein-invariant relativistic point particle action

S[e, xµ; g] =

∫ 1

0dτ

12

[e−1gµν xµxν + e(m2 + λR)

]δe = (ξe)• , δxµ = ξxµ , ⇒ δS =

∫ 1

0dτ(ξL)• = 0

effective action

Γ[g] =

∮DeDx

vol. gaugee−S[e,x ;g]

gauge-fix 1d diff. setting e =∫ 1

0 dτe ≡ 2T and divide out thelength of the circle T = volume of the Killing group

Γ[g] =

∫ ∞0

dT2T

∮Dx e−S[2T ,x ;g] =

∑�





S[e, xµ; g] =

∫ 1

0dτ

12


]δe = (ξe)• , δxµ = ξxµ , ⇒ δS =

∫ 1

0dτ(ξL)• = 0

effective action

Γ[g] =

∮DeDx




Γ[g] =

∫ ∞0

dT2T

∮Dx e−S[2T ,x ;g] =

∑�





S[e, xµ; g] =

∫ 1

0dτ

12


]δe = (ξe)• , δxµ = ξxµ , ⇒ δS =

∫ 1

0dτ(ξL)• = 0

effective action

Γ[g] =

∮DeDx




Γ[g] =

∫ ∞0

dT2T

∮Dx e−S[2T ,x ;g] =

∑�



One-loop effective actionGraviton amplitudes

RecipeRepresent the measure with Lee-Yang ghosts∏τ

√g(x(τ)) ∼

∫DaDbDc e−

14T

∫gµν(aµaν+bµcν)

Γ[g] =

∫ ∞0

dTT

∮DxDaDbDc e−S[2T ,x ,a,b,c,;g]

Write the metric as a sum of gravitons gµν = ηµν +∑

i εiµν eipi ·x

Expand the exponent and pick up all terms multilinear in all εi ’scompute particle correlators



Canonical quantization

Minkowskian action in hamiltonian form

S[e,p, x ; g] =

∫ 1

0dτ[pµxµ − e

2(gµνpµpν + m2 + λR)

]canonical qzn [pµ, xν ] = −i~δνµ and constraint (EoM for e(τ))

(gµνpµpν + m2 + λR)|φ〉 = 0

projecting onto 〈x | : φ(x) = 〈x |φ〉(− 1√

g∂µ√

ggµν∂ν + m2 + λR)φ(x) = 0

Klein-Gordon equation



Comments

For all such computations need to master the calculations of QM pathintegrals in curved space

K (x , x ′; T ) = 〈x |e−TH |x ′〉 =

∫ x(1)=x

x(0)=x ′Dx e−S[x(τ);g]

S[x ; g] =

∫ 1

0dτ[

14T

gµν(x(τ))xµxν + TλR]

Main issue: in general only perturbative (e.g. short time) expansionsare achievable. Expand the path about final point x(τ) = x + y(τ)

gµν(x(τ))xµxν = (gµν(x) + ∂αgµν(x)yα +12∂α∂βgµν(x)yαyβ + · · · )yµyν

=�+�+�+ · · ·



Comments and Issues

single Feynman diagrams can be divergent: need a regularizationscheme and suitable counterterms such that(∂T + H)K (x , x ′; T ) = 0 at all orders in T Bastianelli, OC, van Nieuwenhuizen∼ ’00

generalization to fields with spin is possible: path integrals forO(N) spinning particles (in curved space) describe first qzn ofspin N

2 fields coupled to external gravityN ≤ 2 OK: first qzn of vector fields coupled to gravitysimple O(N) models seem inconsistent in generic curved space;OK in (Anti-)de Sitter

Recents results Bastianelli, Bonezzi, OC, Latini ’11-’12

Studied canonical quantization of spinning particle models thatlead to 1st qzn of fields with (arbitrary) spinWorked out worldline approach to higher spin field theory

Obtained regularization of PI: identify suitable countertermsComputed path integrals on circle in flat space and AdS



Comments and Issues









Comments and Issues








Recent Extensions Fields with spin

Simplest spinning particle

N=1 spinning particle model

S[x ,p] =

∫dτ(

pµ · xµ +i2ηµνψ

µψν − 12ηµνpµpν

)

Hamiltonian and susy charge are conserved

H =12

pµpµ , Q = pµψµ, δf = {f , ξH + iεQ}

{Q,Q}DB = −2iH

Unitarity can be achieved by gauging translations and susy

S[x ,p, ψ; E ] =

∫dτ(


µψν − eH − iχQ)

E = (e(t), χ(t)) gauge fields δe = ξ , δχ = ε





S[x ,p] =

∫dτ(



)Hamiltonian and susy charge are conserved

H =12


{Q,Q}DB = −2iH


S[x ,p, ψ; E ] =

∫dτ(








S[x ,p] =

∫dτ(



)Hamiltonian and susy charge are conserved

H =12


{Q,Q}DB = −2iH


S[x ,p, ψ; E ] =

∫dτ(






N=1 spinning particleCanonical qzn

Canonical qzn yields the constraints (EoM’s for e(τ) and χ(τ))

[xµ,pν ] = i~δµν , [ψµ, ψν ]+ = ~ηµν

H|φ〉 = 0 , Q|φ〉 = 0 , |φ〉 ∈ physical Hilbert space

ψµ are realized by the gamma matrices γµ ∼√

2~ ψ

µ

Using wave functions ψα(x) = 〈x , α|φ〉

Q|φ〉 = ψµpµ|φ〉 = 0 ⇒ (γµ)αβ ∂µ φβ(x) = 0

i.e. massless Dirac equation γµ∂µφ = 0can be coupled to gravity. Circle path integral gives effective actionfor Dirac fermion in curved space: yields graviton amplitudesModels with N local supersymmetries −→ first quantization of spinN2 field Bastianelli, Bonezzi, OC, Latini and Waldron JHEP 2007, 2008, 2009, 2010, 2011, 2012








2~ ψ

µ











2~ ψ

µ





Recent Extensions Worldline formalism in manifolds with boundaries

Casimir effect

Free massive scalar field

e−VE =

∫Dφe−

∫dx [(∂φ)2+m2φ2+V (x)φ2] , E =

∫ ∞0

dTTV

∮Dx e−S[x ,m]

V (x) sets the interaction with the boundary: e.g. DBCV (x) = λ

∫Σ dσδ(x − xσ)

Casimir energy Ec = E − E1 − E2: paths that touch bothboundaries have weight one, other paths have weights zeroWorldline Monte Carlo technique Gies, Langfeld, Moyaerts ’03

â geometry independentâ different boundary conditions?

Σ1

Σ2


Recent Extensions Worldline formalism in manifolds with boundaries

Analytic approach

Flat manifold with boundary; simplest case half lineHeat kernel on the half line (∂t −D2)KR+

(x , x ′; t) = 0 withD2 = ∂2 + V (x)and KR+

(x ,0′; t) = 0 DBC, ∂x ′KR+(x ,0; t) = 0 NBC

Worldline approach, via image charge methodKR+

(x , x ′; t) = K (x , x ′; t)∓ K (x ,−x ′; t) andV (x) = θ(x)V (x) + θ(−x)V (−x)

implemented different b.c.’s (e.g. Neumann or Robin)might be able to implement fields with spin and curved space


Conclusions

Summary and outlook

Worldline formalism efficient alternative to standard QFT

Obtained several new applications of the method, in flat spaceand curved space:

Calculation of graviton(-photon) amplitudesFields with arbitrary spin: the method provides a way to quantizehigher spin fields (outstanding problem)Manifolds with boundaryWorldline formalism for non-commutative QFT’s with Bonezzi, Pisani,

Franchino-Viñas

Analytic approach to Casimir effect for a more generic set ofboundariesExtension to massive higher spins (via KK reduction)More on Schwinger effect (curvature corrections,...)Systematics of graviton amplitude: seek for a Bern-Kosower-likeformulaGauge-gravity dualities in the WF


Conclusions

Summary and outlook

Worldline formalism efficient alternative to standard QFTObtained several new applications of the method, in flat spaceand curved space:


Franchino-Viñas

Analytic approach to Casimir effect for a more generic set ofboundariesExtension to massive higher spins (via KK reduction)More on Schwinger effect (curvature corrections,...)Systematics of graviton amplitude: seek for a Bern-Kosower-likeformulaGauge-gravity dualities in the WF


Conclusions

Summary and outlook

Worldline formalism efficient alternative to standard QFTObtained several new applications of the method, in flat spaceand curved space:


Franchino-Viñas

Analytic approach to Casimir effect for a more generic set ofboundariesExtension to massive higher spins (via KK reduction)More on Schwinger effect (curvature corrections,...)Systematics of graviton amplitude: seek for a Bern-Kosower-likeformulaGauge-gravity dualities in the WFO. Corradini (UnACh) UniSon 2013 35 / 35

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