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Hola! Introduction Electric field and Coulombs law Generalisations Conclusion Point particle contact interactions and Coulombs law James P. Edwards IFM Morelia Nov 2016 Based on [arXiv:1506.08130 [hep-th]] and [arXiv:1409.4948

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Hola! Introduction Electric field and Coulomb’s law Generalisations Conclusion

Point particle contact interactions and Coulomb’s law James P. Edwards

IFM Morelia Nov 2016 Based on [arXiv:1506.08130 [hep-th]] and [arXiv:1409.4948 [hep-th]]

James P. Edwards Point particle contact interactions and Coulomb’s law

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Hola! Introduction Electric field and Coulomb’s law Generalisations Conclusion

Outline

1

Hola!

2

Introduction Worldline formalism

3

Electric field and Coulomb’s law Faraday’s lines of flux Finite T Results

4

Generalisations Spin 1/2 particles Relativistic contact interactions

5

Conclusion

James P. Edwards Point particle contact interactions and Coulomb’s law

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Hola! Introduction Electric field and Coulomb’s law Generalisations Conclusion

Introducci´

  • n...

James P. Edwards Point particle contact interactions and Coulomb’s law

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Hola! Introduction Electric field and Coulomb’s law Generalisations Conclusion

Introducci´

  • n...

...de mi!! Acabo de empezar una beca posdoctoral aqu´ ı en el IFM – estoy muy contento de haber ingresado en el grupo! Ojal´ a y lleguemos a conocernos muy bien en el futuro... Mi trabajo se trata del formalismo linea de mundo (worldline formalism) de los campos cu´

  • anticos. Particularmente he trabajado en:

James P. Edwards Point particle contact interactions and Coulomb’s law

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Hola! Introduction Electric field and Coulomb’s law Generalisations Conclusion

Introducci´

  • n...

...de mi!! Acabo de empezar una beca posdoctoral aqu´ ı en el IFM – estoy muy contento de haber ingresado en el grupo! Ojal´ a y lleguemos a conocernos muy bien en el futuro... Mi trabajo se trata del formalismo linea de mundo (worldline formalism) de los campos cu´

  • anticos. Particularmente he trabajado en:

La din´ amica de espin cu´ anitico de los electrones para las computadoras cu´ anticas University of Oxford 2009 La ecuaci´

  • n Wheeler-deWitt (Restricci´
  • n de Hamiltonian en relatividad general)

University of Cambridge 2009-2011 Interacciones de contacto dentro de la teor´ ıa de spinning strings sin tensi´

  • n

University of Durham 2011-2015

La anomal´ ıa de la simmetr´ ıa de Weyl y la teor´ ıa de campo de Liouville en dimensiones D < 26 El modelo est´ andar de part´ ıculas y teor´ ıas unificadas SU(5), SO(10) y SO(16).

El formalismo linea de mundo University of Durham 2012-2015 y University of Bath 2015-2016

La ley de Coulomb y la incorporaci´

  • n de espin

Las teor´ ıas non-Abelian en este formalismo (la interacci´

  • n de Wilson)

Los campos cu´ anticos en el espacio no-conmutativo.

James P. Edwards Point particle contact interactions and Coulomb’s law

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Hola! Introduction Electric field and Coulomb’s law Generalisations Conclusion

Trabajo actual

Mi trabajo en Morelia es en colaboraci´

  • n con Christian Schubert y otras personas que

dependen del proyecto: Los propagadores en vac´ ıo y vestidos en campos constantes en el formalismo linea del mundo Fiorenzo Bastianelli (Bologna), Olindo Corradini (Modena), Naser Ahmadiniaz (Institute for Basic Science) La tear´ ıa U(N) en el espacio no-conmutativo con campos de color en la linea de mundo. Olindo Corradini, Pablo Pisani (La Plata), Naser Ahmadiniaz, Idrish Huet (UNACH) C´ alculos num´ ericos de las lineas de mundo – estados ligardos en campos cu´ anticos y mec´ anica cu´ antica Axel Weber (IFM), Anabel Trejo (IFM), Urs Gerber (UNAM)

James P. Edwards Point particle contact interactions and Coulomb’s law

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Hola! Introduction Electric field and Coulomb’s law Generalisations Conclusion Worldline formalism

Worldline formalism

The so-called worldline formalism of quantum field theory is a first quantised approach

  • riginally motivated by the high tension limit of string theory. It offers significant

computational advantages over the conventional perturbative methods traditionally employed in field theory. It enjoys calculational efficiency and compactness. Feynman diagrams are combined together into gauge invariant combinations. There are indications of a close relationship between string theory, particle worldlines and field theory. Has a very nice physical interpretation. kµ

1

kν

1

p k1 + p kµ

1 kµ 1

p

(a) Two diagrams represent a process in scalar quantum

field theory.

kµ

1

kν

2

(b) Worldline techniques require

  • nly a single diagram.

Figure: Feynman diagrams in field theory and the worldline formalism

James P. Edwards Point particle contact interactions and Coulomb’s law

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Hola! Introduction Electric field and Coulomb’s law Generalisations Conclusion Worldline formalism

Worldline formalism

In the worldline formalism, the basic idea is to integrate out the matter degrees of

  • freedom. The resulting functional determinant is then recast as a transition

amplitude in quantum mechanics[1]. For example, for spinor quantum electrodynamics (D ≡ ∂ + igA) ΓΨ [A] = log

  • D

¯ ΨΨ

  • exp
  • −
  • d4x ¯

Ψ (γ · D − m) Ψ

  • = −1

2Tr log

  • (γ · D)2 + m2

− → ∞ dT T e−m2T

  • x(0)=x(T )

Dx

  • ψ(0)=−ψ(T )

Dψ e−S[x,ψ], where S [x, ψ] = T dτ ˙ x2 4 + 1 2ψ · ˙ ψ + ieA(x) · ˙ x − ieψµFµν(x)ψν

  • (1)

1Strassler, Nucl. Phys. B385 James P. Edwards Point particle contact interactions and Coulomb’s law

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Hola! Introduction Electric field and Coulomb’s law Generalisations Conclusion Worldline formalism

Applications

Worldline approaches have been used successfully to address many problems, including N-point scattering amplitudes at one loop order, extended to incorporate multi-loop diagrams. Determination of effective actions Higher spin fields Graviton / photon production Trace anomalies Non-commutative quantum field theory Grand unification Localisation Landau-Khalatnikov-Fradkin transformations and dressed propagators. Gravitational anomalies Non-Abelian field theory. Worldline approaches are steadily growing in popularity and have an enormous potential to aid our understanding of fundamental interactions between particles.

James P. Edwards Point particle contact interactions and Coulomb’s law

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Hola! Introduction Electric field and Coulomb’s law Generalisations Conclusion Worldline formalism

Higher order interactions

Feynman originally gave the general action for the exchange of an arbitrary number of gauge bosons – for tree level processes in a scalar QED: ∞ dTe−m2T x(T )=x1

x(0)=x0

Dxe

− T

0 dτ

  • ˙

x2 4 +ieA(x)· ˙

x

  • − ie2

2

T

0 dτ

T

0 dτ′ ˙

xµ∆µν(x−x′) ˙ x′ν

(2) This motivates us to study different interactions on the particle worldline. In particular we will consider the infinite mass limit of the propagator and replace e2∆µν

  • x(τ) − x(τ ′)
  • −

→ gδµνδD x(τ) − x(τ ′)

  • (3)

where g absorbs the mass dependence through g = e2µ2

James P. Edwards Point particle contact interactions and Coulomb’s law

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Hola! Introduction Electric field and Coulomb’s law Generalisations Conclusion Faraday’s lines of flux Finite T Results

Particle worldlines

Consider a pair of equal and oppositely charged particles at the spatial points a and b. Note that E′(x) = q

  • C

dτ dω dτ δ3 (ω(τ) − x) (4) satisfies Gauss’ law: ∇ · E′ = qδ3(x − a) − qδ3(x − b). It does not, however, satisfy ∇ × E′ = 0, so it is not (yet!) a physical electric field. So we examine its functional average over curves, C,

  • E′(x)
  • T = q

ω(T )=b

ω(0)=a

Dω

  • C

dτ dω dτ δ3 (ω(τ) − x) e−

T

˙ ω2 4 dτ

(5) In Fourier space the insertion is familiar to worldline theorists and analogous to the vertex operators of string theory:

  • E′i(k)
  • T = q

T dτ

  • V i

k(τ)

  • ;

V i

k(τ) = ˙

wi(τ)eik·ω(τ) (6)

James P. Edwards Point particle contact interactions and Coulomb’s law

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Hola! Introduction Electric field and Coulomb’s law Generalisations Conclusion Faraday’s lines of flux Finite T Results

Interpretation

There are complementary ways to interpret this average: One may interpret this action as similar to a thermal average (at temperature T) against a Hamiltonian H =

  • ˙

ω2 4 dτ.

The average is over all trajectories that pass through the spatial point x and whose endpoints are at a and b.

Alternatively, one can think of T as setting the intrinsic length measured along the particle worldline

This worldline is allowed to fluctuate (with fixed endpoints) and we count only the contribution of those fluctuations that pass through the point x.

In the high temperature limit, T → ∞, the contributions to the average are localised to the endpoints and we find lim

T →∞

  • E′(k)
  • T = qik

k2

  • eik·a − eik·b

+ O 1 k2T

  • (7)

which in three dimensions gives the classical dipole field at the point x: lim

T →∞

  • E′(x)
  • = q

4π ∇

  • −1

|x − a| + 1 |x − b|

  • .

(8)

James P. Edwards Point particle contact interactions and Coulomb’s law

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Hola! Introduction Electric field and Coulomb’s law Generalisations Conclusion Faraday’s lines of flux Finite T Results

What about finite temperature?

At large finite T we find a deviation from the Coulomb field: in momentum space these are suppressed in inverse powers of k2T. There are three techniques:

1

Expand E′(k)T in inverse powers of k2T and compute the inverse Fourier transform term-by-term. Finite order and algebraically tedious.

2

Numerically compute the inverse Fourier transform for E′(k)T at finite T. Arbitrary order but numerical errors caused by poles.

3

Discretise the path integral: ”Worldline Monte-Carlo” simulation. Large number

  • f worldlines and discretisation of action.
  • Dωe−
  • ˙

ω2 4 → (2πT)−1

Nloops

  • C

At small T we carry out a similar expansion in powers of k2T or we could simulate worldlines that satisfy the Gaussian weight and our boundary conditions. Numerically integration breaks down without a regularisation of the contact interaction.

James P. Edwards Point particle contact interactions and Coulomb’s law

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Hola! Introduction Electric field and Coulomb’s law Generalisations Conclusion Faraday’s lines of flux Finite T Results

Results I

We start at large

T |b−a|2 ≫ 1 and show the evolution of the field lines as we gradually

reduce T. Recall that the field at each point in space is found by averaging over all trajectories that pass through that point (with endpoints fixed to the charges).

(a) The stream lines of the electric

field for large (but finite) T .

(b) The stream lines of the electric

field change as T is reduced.

Figure: Numerically simulated field lines for the dipole field.

James P. Edwards Point particle contact interactions and Coulomb’s law

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Hola! Introduction Electric field and Coulomb’s law Generalisations Conclusion Faraday’s lines of flux Finite T Results

Results II

At the other extreme,

T |b−a|2 ≪ 1, the field is supported only on a small region close

to the straight line joining the charges. Analytically, it is most convenient to first carry

  • ut the inverse Fourier transform to exploit the T dependence:

I (x)T = −q (2π)

3 2

1 dτ1

  • T ˜

G (τ1, τ1) 3

2

˙ ˜ G (τ1, τ1) 2 ˜ G (τ1, τ1) (x − ωc) − ˙ ωc

  • exp
  • − (x − ωc)2

2T ˜ G (τ1, τ1)

  • ,

(9)

(a) The stream lines of the

electric field for small T .

(b) The field lines are

compressed as T is reduced.

Figure: Numerically simulated field lines for the dipole field in the low T limit.

James P. Edwards Point particle contact interactions and Coulomb’s law

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Hola! Introduction Electric field and Coulomb’s law Generalisations Conclusion Faraday’s lines of flux Finite T Results

Results II

At the other extreme,

T |b−a|2 ≪ 1, the field is supported only on a small region close

to the straight line joining the charges. Analytically, it is most convenient to first carry

  • ut the inverse Fourier transform to exploit the T dependence:

I (x)T = −q (2π)

3 2

1 dτ1

  • T ˜

G (τ1, τ1) 3

2

˙ ˜ G (τ1, τ1) 2 ˜ G (τ1, τ1) (x − ωc) − ˙ ωc

  • exp
  • − (x − ωc)2

2T ˜ G (τ1, τ1)

  • ,

(9)

(c) The stream lines of the

electric field for small T .

(d) The field lines are

compressed as T is reduced.

Figure: Numerically simulated field lines for the dipole field in the low T limit.

James P. Edwards Point particle contact interactions and Coulomb’s law

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Hola! Introduction Electric field and Coulomb’s law Generalisations Conclusion Spin 1/2 particles Relativistic contact interactions

Fermionic particle worldlines

These ideas can be generalised to the case that the point particles also have spin degrees of freedom. The (N = 1 SUSY) action becomes S [ω, ψ] = T dτ ˙ ω2 4 + 1 2ψ · ˙ ψ (10) and the vertex operators become V µ

k (τ) = ( ˙

ωµ(τ) − iψµ(τ)k · ψ(τ)) eik·ω (11) which arise from gauge fixing a 1D super-gravity with contact interactions and two super-symmetric invariant constraints on the einbein and gravitino.

James P. Edwards Point particle contact interactions and Coulomb’s law

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Hola! Introduction Electric field and Coulomb’s law Generalisations Conclusion Spin 1/2 particles Relativistic contact interactions

Fermionic particle worldlines

These ideas can be generalised to the case that the point particles also have spin degrees of freedom. The (N = 1 SUSY) action becomes S [ω, ψ] = T dτ ˙ ω2 4 + 1 2ψ · ˙ ψ (10) and the vertex operators become V µ

k (τ) = ( ˙

ωµ(τ) − iψµ(τ)k · ψ(τ)) eik·ω (11) which arise from gauge fixing a 1D super-gravity with contact interactions and two super-symmetric invariant constraints on the einbein and gravitino. To illustrate the new ingredients, the dipole field picks up an extra “spin-factor” lim

T →∞ Iµ(x)T =

q 4π2 γµγ · ∇

  • 1

|x − a|2 − 1 |x − b|2

  • (12)

James P. Edwards Point particle contact interactions and Coulomb’s law

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Hola! Introduction Electric field and Coulomb’s law Generalisations Conclusion Spin 1/2 particles Relativistic contact interactions

Interacting worldlines

We finally return to the worldline formalism of quantum field theory and construct a new worldline action. For a fixed number, N of scalar particle worldines with endpoints aµ

i and bµ i the total gauge fixed worldline action is:

S =

N

  • i=1

S0 [ωi] + g 2

N

  • i,j=1

T T dτidτj ˙ ωi(τi) · ˙ ωj(τj)δ4 (ωi(τi) − ωj(τj)) (13) The classical equations of motion can be re-written to resemble a particle coupled to an external current: let Jµ(x) ≡

  • i
  • ωi

dωµ

i δ4 (x − ωi) =

⇒ ∂µJµ(x) =

  • i

δ4(x − ai) − δ4(x − bi), (14) then the equations of motion become m d dτi ˙ ωaµ √ ˙ ω2

a

  • = g

2 ˙ ων

a∂[νJµ]

(15)

James P. Edwards Point particle contact interactions and Coulomb’s law

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Hola! Introduction Electric field and Coulomb’s law Generalisations Conclusion Spin 1/2 particles Relativistic contact interactions

Physical content

Expanding the interaction we must compute expectation values of products of vertex

  • perators
  • V µ

k1(τ1) · V−k1µ(τ2) . . . V α KN (τ2N−1)V−KN α(τ2N)

  • T →∞

(16) There are three physical effects in the limit that T → ∞:

1

A finite renormalisation of the free action (mass renormalisation) due to vanishing

  • f the δ function for ωi = ωj at τi = τj.

2

An infinite renormalisation of the (worldline) cosmological constant for the same reason (previously implicit in the worldline action).

3

A physical pair-wise interaction localised to the end-points of the worldlines: g 4π2

′

  • jk
  • 1

|aj − ak|2 − 1 |aj − bk|2

  • −
  • 1

|bj − ak|2 − 1 |bj − bk|2

  • (17)

James P. Edwards Point particle contact interactions and Coulomb’s law

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Hola! Introduction Electric field and Coulomb’s law Generalisations Conclusion Spin 1/2 particles Relativistic contact interactions

Quantisation to all orders

Interestingly, this pair-wise interaction holds to arbitrary order in the coupling. For example, for the self interaction at order gN the momentum space expectation value in the high T limit becomes gN

2N

  • i=1

T dτi

  • V µ

k1(τ1) · V−k1µ(τ2) . . . V α KN (τ2N−1)V−KN α(τ2N)

  • T →∞

(18) =(−2)NgN

N

  • i=1

eiki·(b−a) k2

i

+ renormalisation absorbed by local counter terms. (19) Factorisation techniques allow straightforward exponentiation of interactions between distinct worldlines so we find at last (in the limit T → ∞): e−g ′

j,k[(G(aj−ak)−G(aj−bk))−(G(bj−ak)−G(bj−bk))]

(20)

James P. Edwards Point particle contact interactions and Coulomb’s law

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Hola! Introduction Electric field and Coulomb’s law Generalisations Conclusion

Conclusion

Inspired by the worldline formalism of quantum field theory we have considered a theory in which point particles interact when their worldlines intersect.

1

In the non-relativistic limit, Coulomb’s law and electric Dipole fields arose from a functional average over macroscopic worldlines with fixed endpoints constrained to pass through a given spatial point.

2

The finite T limit provided corrections to these fields – the small temperature limit showed an effect similar to confining flux tubes.

3

The relativistic theory provided scalar propagator interactions between the end points of the dynamical particle worldlines which can be exponentiated.

4

Allows for the creation and study of novel interactions in quantum field theory and classical gravity.

5

Future generalisation to non-Abelian interactions, higher spin fields, non-commutative geometry and constant background fields possible using existing worldline techniques.

James P. Edwards Point particle contact interactions and Coulomb’s law

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Hola! Introduction Electric field and Coulomb’s law Generalisations Conclusion

Conclusion

Inspired by the worldline formalism of quantum field theory we have considered a theory in which point particles interact when their worldlines intersect.

1

In the non-relativistic limit, Coulomb’s law and electric Dipole fields arose from a functional average over macroscopic worldlines with fixed endpoints constrained to pass through a given spatial point.

2

The finite T limit provided corrections to these fields – the small temperature limit showed an effect similar to confining flux tubes.

3

The relativistic theory provided scalar propagator interactions between the end points of the dynamical particle worldlines which can be exponentiated.

4

Allows for the creation and study of novel interactions in quantum field theory and classical gravity.

5

Future generalisation to non-Abelian interactions, higher spin fields, non-commutative geometry and constant background fields possible using existing worldline techniques. Muchas gracias por su atenci´

  • n!

James P. Edwards Point particle contact interactions and Coulomb’s law