Paper
0
Contact interactions between particle worldlines
Authors
James P. Edwards,Chris Rehmann
Richard Rediske,Eva Doménech,Lars Östlund,Isabel Escriche,Rebecca Baillie,Günter-Ulrich Tolkiehn,Ivan Murin,Michael Godfrey,Anastasios Petkou,Evgeniya Gorodetskaya,CHUDAMANI PRANESACHAR ANIL KUMAR,Veerabhadra Rao Kotamarthi,Alfonso V. Ramallo,Ramaswamy Jagannathan,Rosa Minoia,Orlando Peres,Richard Gorman,Kirk Jensen,Ewa Wierzbowska,Dian-Yong Chen,Andreas Hamann,Antoine Bourget,Alessandro Pini,Nataliya Dunayevska,Sebastian ```Trojanowski,Martin Goodchild,Cynthia Keeler,Srinivasan Jagannathan,Krzysztof Turzynski,Kristen Smith,Lawrence Kennedy,Aurelien Bigot,Frédéric Dreyer,Rajesh Dachiraju,Aqeel Ahmed,Mikhail Trifonov,Peter Wolfsteiner,Liam McAllister,Mikołaj Baranowski,Benjamin Bellenie,Andy Van Brocklin,Song He,Luis Garcés-Erice,Anatoly Dymarsky,Bernard Gagnon,Digamber Porob,Seok Ki Choi,Matti Jarvinen,Oleg Lunin,Vishnu Ram OV,Felipe Lumbreras,Marie Connett,Aaron Balog,Silviu Pufu,Urfat Nuriyev,Lorenzo Tancredi,Eduardo Peinado,Bo Feng,Nataliya Pryanichnikova,Tetsu Hara,Valery Tishkov,Miquel Riera-Codina,Bernard Binczycki,Pilar G. Estévez,Phillip Szepietowski,Bruno Dufour,Marius de Leeuw,Philip Farrugia,Torsten Gaebel,William Peters,Rong-Xin Miao,Petr Vahalík,Nathaniel Butlin,Elena Rodriguez-Vieitez,Heribertus Bayu Hartanto,Rafael Maestre,Engel Roza,Bindusar Sahoo,Jean-Marc Sparenberg,Yun Jiang,Khairi Zainol,Zamzahaila Mohd Zin,Vladimir Belavin,Francesco Riva,Juan L. Manes,Laura Kenefic,Andrei Zotov,Stephen Ashley,Tomoya Takiwaki,Xiaojun Bi,Monika Blanke,Gautam ,Zhao Li,Savvas Nesseris,Daisuke Yamauchi,Hayato Motohashi,Peter Arnold,Edward Wilson-Ewing,Eduardo Ponton,Aman Mwafy,Tomasz Pawłowski,Ian Shoemaker,Janusz Gluza,Diego Restrepo,Leonardo de Lima,Francis-Yan Cyr-Racine,Alexey Golovnev,HSIN-YI LIN,Fei Teng,Fabio Apruzzi,Michael Black,Jia Liu,Young-Kyun Kwon,Şerafettin Erten,Danning Li,Niko Jokela,Mads Toudal Frandsen,HIND ALKATAN,Richard Jefferson,Taniel Danelian,Luciana Migliore,Marco Alexander Harrendorf,Jaume Garriga,Vincent Tourre,Lukas Witkowski,Roberto Valandro,Aleksandr N. 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Preprint typeset in JHEP style - HYPER VERSION DCPT-15/39
Contact interactions between particle
worldlines
James P. Edwards
Centre for Particle Theory, Department of Mathematical Sciences,
University of Durham, Durham DH1 3LE, UK
Email: j.p.edwards@durham.ac.uk
Abstract: We construct contact interactions for bosonic and fermionic point par-
ticles. We first relate the resulting theories to classical electrostatics by taking func-
tional averages over worldlines whose endpoints are fixed to charged particles. Count-
ing those paths which pass through a space-time point xµ gives the static electric
field at that point, provided we take the limit where the length measured along the
worldlines is large. We also investigate corrections to the classical field that arise
beyond leading order in this limit before constructing a theory of point particles
that interact when their worldlines intersect. We quantise this theory and show that
the partition function contains propagator couplings between the endpoints of the
particles before discussing how this is related to the worldline formalism of quantum
field theory and general action at a distance theories.
Keywords: Effective field theories, Field theories in lower dimensions,
Confinement.
arXiv:1506.08130v2 [hep-th] 14 Dec 2015
Contact interactions between particle
worldlines
James P. Edwards
Centre for Particle Theory, Department of Mathematical Sciences,
University of Durham, Durham DH1 3LE, UK
Email: j.p.edwards@durham.ac.uk
Abstract: We construct contact interactions for bosonic and fermionic point par-
ticles. We first relate the resulting theories to classical electrostatics by taking func-
tional averages over worldlines whose endpoints are fixed to charged particles. Count-
ing those paths which pass through a space-time point xµ gives the static electric
field at that point, provided we take the limit where the length measured along the
worldlines is large. We also investigate corrections to the classical field that arise
beyond leading order in this limit before constructing a theory of point particles
that interact when their worldlines intersect. We quantise this theory and show that
the partition function contains propagator couplings between the endpoints of the
particles before discussing how this is related to the worldline formalism of quantum
field theory and general action at a distance theories.
Keywords: Effective field theories, Field theories in lower dimensions,
Confinement.
arXiv:1506.08130v2 [hep-th] 14 Dec 2015
Contents
1. Introduction 1
2. Bosonic particles – the classical electric field 4
3. Fermionic particles 13
4. Analysis at finite T 20
4.1 Corrections in the bosonic case 20
4.2 Corrections in the fermionic case 26
5. Contact interactions between particles 28
5.1 Expansion to arbitrary order 39
6. Discussion 41
1. Introduction
Classical electromagnetism is conventionally described by Maxwell’s field theory and
there seems to be little room for debate about its formulation. In [1] and [2], however,
building upon [3] an alternative approach to determining the electromagnetic field
strength tensor for a pair of charged particles led directly to a novel interacting
string theory. This theory contained contact interactions on the string worldsheet
which served to produce expectation values of Wilson lines in Abelian quantum field
theory. In the case of electrostatics, the description given in [3] was in terms of point
particles whose worldlines have their endpoints fixed to the charged particles. The
electric field at a position in space-time was arrived at via a weighted average over all
such worldlines which also pass through the given point. The physical picture which
motivated this approach (and the later interacting string theory) is of Faraday’s lines
of force as fundamental objects which become the physical degrees of freedom of the
electromagnetic field.
To complement this work on contact interactions in string theory it seems ap-
propriate to return to worldline theories to explore the consequences of allowing
point particles to interact when their worldlines intersect. Such theories are of sig-
nificant physical interest, since the so-called worldline formalism of quantum field
– 1 –
1. Introduction 1
2. Bosonic particles – the classical electric field 4
3. Fermionic particles 13
4. Analysis at finite T 20
4.1 Corrections in the bosonic case 20
4.2 Corrections in the fermionic case 26
5. Contact interactions between particles 28
5.1 Expansion to arbitrary order 39
6. Discussion 41
1. Introduction
Classical electromagnetism is conventionally described by Maxwell’s field theory and
there seems to be little room for debate about its formulation. In [1] and [2], however,
building upon [3] an alternative approach to determining the electromagnetic field
strength tensor for a pair of charged particles led directly to a novel interacting
string theory. This theory contained contact interactions on the string worldsheet
which served to produce expectation values of Wilson lines in Abelian quantum field
theory. In the case of electrostatics, the description given in [3] was in terms of point
particles whose worldlines have their endpoints fixed to the charged particles. The
electric field at a position in space-time was arrived at via a weighted average over all
such worldlines which also pass through the given point. The physical picture which
motivated this approach (and the later interacting string theory) is of Faraday’s lines
of force as fundamental objects which become the physical degrees of freedom of the
electromagnetic field.
To complement this work on contact interactions in string theory it seems ap-
propriate to return to worldline theories to explore the consequences of allowing
point particles to interact when their worldlines intersect. Such theories are of sig-
nificant physical interest, since the so-called worldline formalism of quantum field
– 1 –
theory [4, 5, 6] expresses physical quantities in a field theory in terms of the quan-
tum mechanics of point particles which trace out worldlines in space-time [7, 8, 9].
This technique can also be extended to non-Abelian theories and chiral fermions [10]
where it may provide insight to the unification of the fundamental forces [11]. The
coupling between matter fields and the gauge field is described in the point particle
theory as a Wilson-line interaction for the particle worldlines in the presence of a
background field. For example, the partition function for a single scalar field mini-
mally coupled to an Abelian gauge field, A, is expressed in the worldline formalism
(which we derive in more detail later in the paper) by an integral over all closed
curves ω
Z =
∫ ∞
0
dT
T
∮
Dω e−Spoint[ω]W [A] ; W [A] = ei ∫ dω·A, (1.1)
where Spoint [ω] = m ∫ dτ √ ˙ω2 is an action describing the dynamics of a point particle
and W [A] is the Wilson loop describing the interaction of the particle with the
gauge field (we have absorbed the coupling strength into A). The right hand side is
interpreted as quantum mechanics on the worldline of this particle and it is this first
quantised theory which we propose to modify in this paper.
Field theory is the conventional framework in which to introduce interactions and
the local nature of this approach naturally leads to particles interacting upon contact.
However, the worldline formalism can offer substantial calculational advantages over
traditional approaches in field theory, especially since it represents a reorganisation
of the usual perturbative expansion in Feynman diagrams and makes the local gauge
invariance manifest [12]. It is therefore important to develop worldline techniques
further and one of the most basic modifications to the theory must be to introduce
direct interactions between these particle worldlines. As we shall describe below,
a modification to the worldline theory can be interpreted as inducing a change in
the underlying field theory, so the results of this program may provide new tools to
complement the conventional techniques familiar to field theorists. We comment on
this in section 5.
Direct inter-particle interactions can be found in many previous publications.
One of the most well-known approaches is the action at a distance formulation
of electrodynamics by Feynman and Wheeler [13, 14], originally proposed to ad-
dress the problem of radiation reaction. This built upon earlier work in formulating
a consistent theory involving direct inter-particle interactions by Tetrode [15] and
Fokker [16], who described electromagnetic phenomena in terms of interactions be-
tween particles with light-like separation. Ramond generalised this work and found
a set of consistency constraints limiting the form of the interaction that can be
introduced into worldline theories [17]. This was further extended to include di-
rect inter-string interactions as well as interactions between particles and strings
[18, 19, 20, 21, 22]. A number of further theories involving action at a distance have
– 2 –
tum mechanics of point particles which trace out worldlines in space-time [7, 8, 9].
This technique can also be extended to non-Abelian theories and chiral fermions [10]
where it may provide insight to the unification of the fundamental forces [11]. The
coupling between matter fields and the gauge field is described in the point particle
theory as a Wilson-line interaction for the particle worldlines in the presence of a
background field. For example, the partition function for a single scalar field mini-
mally coupled to an Abelian gauge field, A, is expressed in the worldline formalism
(which we derive in more detail later in the paper) by an integral over all closed
curves ω
Z =
∫ ∞
0
dT
T
∮
Dω e−Spoint[ω]W [A] ; W [A] = ei ∫ dω·A, (1.1)
where Spoint [ω] = m ∫ dτ √ ˙ω2 is an action describing the dynamics of a point particle
and W [A] is the Wilson loop describing the interaction of the particle with the
gauge field (we have absorbed the coupling strength into A). The right hand side is
interpreted as quantum mechanics on the worldline of this particle and it is this first
quantised theory which we propose to modify in this paper.
Field theory is the conventional framework in which to introduce interactions and
the local nature of this approach naturally leads to particles interacting upon contact.
However, the worldline formalism can offer substantial calculational advantages over
traditional approaches in field theory, especially since it represents a reorganisation
of the usual perturbative expansion in Feynman diagrams and makes the local gauge
invariance manifest [12]. It is therefore important to develop worldline techniques
further and one of the most basic modifications to the theory must be to introduce
direct interactions between these particle worldlines. As we shall describe below,
a modification to the worldline theory can be interpreted as inducing a change in
the underlying field theory, so the results of this program may provide new tools to
complement the conventional techniques familiar to field theorists. We comment on
this in section 5.
Direct inter-particle interactions can be found in many previous publications.
One of the most well-known approaches is the action at a distance formulation
of electrodynamics by Feynman and Wheeler [13, 14], originally proposed to ad-
dress the problem of radiation reaction. This built upon earlier work in formulating
a consistent theory involving direct inter-particle interactions by Tetrode [15] and
Fokker [16], who described electromagnetic phenomena in terms of interactions be-
tween particles with light-like separation. Ramond generalised this work and found
a set of consistency constraints limiting the form of the interaction that can be
introduced into worldline theories [17]. This was further extended to include di-
rect inter-string interactions as well as interactions between particles and strings
[18, 19, 20, 21, 22]. A number of further theories involving action at a distance have
– 2 –
been proposed to describe various other phenomena within this framework – see for
example [23, 24, 25, 26, 27].
The general principle is to couple the worldlines of the particles together by
adding extra terms to the free particle action, Spoint. An illustrative example would
be to consider a theory of two particles whose worldlines are described by ωµ
a (τa)
and ωµ
b (τb) and to introduce [28, 25]
Sint = gagb
∫
ωa
dτa
∫
ωb
dτb ˙ωa (τa) · ˙ωb (τb) D (ωa − ωb) (1.2)
as an interaction term in the action1. Here the function D must be symmetric and
accounts for the relative strength of interaction as a function of particle separation.
In the literature discussed above D (ωa − ωb) has been taken to be supported for
light-like, time-like and space-like separations, the last of which is the relativistic
generalisation of an instantaneous interaction. In Feynman-Wheeler theory, for ex-
ample, D (ωa − ωb) is taken to be the sum of advanced and retarded Green functions
of the (space-time) Laplacian. As is now well known, these action at a distance the-
ories are often cast into a form reminiscent of a field theory, although the “fields” are
not independent variables but are rather defined in terms of the particle trajectories
and the choice of D. For instance, if we take [29]
D (ωa − ωb) = 1
4π δ ((ωa − ωb)2) − m
8π
θ ((ωa − ωb)2)
√
(ωa − ωb)2
J1
(
m
√
(ωa − ωb)2
)
(1.3)
which is the time symmetric Green satisfying (∂µ∂µ − m2) D (ωa − ωb) = −δ4 (ωa − ωb)
then we may define
Aµ (x) = ∑
a
ga
∫
dτa ˙ωµ
a (τa) D (ωa (τa) − x) . (1.4)
This satisfies the Maxwell equations and gauge condition
(∂ν ∂ν − m2) Aµ (x) = jµ (x) ; jµ (x) = − ∑
a
ga
∫
dτa ˙ωµ
a (τa) δ4 (x − ωa) ,
∂µAµ (x) = 0 (1.5)
and in terms of A the interaction between the particles takes the form
∑
a
ga
∫
ωa
dτa ˙ωa (τa) · A (ωa (τa)) (1.6)
which shows that the action at a distance formalism contains the same equations of
motion and interactions as more traditional approaches using field theory.
1To be precise this provides vector-like interactions between point particles. Scalar interactions
can be produced by replacing each ˙ωiµ by ( ˙ω2
i
) 1
2 .
– 3 –
example [23, 24, 25, 26, 27].
The general principle is to couple the worldlines of the particles together by
adding extra terms to the free particle action, Spoint. An illustrative example would
be to consider a theory of two particles whose worldlines are described by ωµ
a (τa)
and ωµ
b (τb) and to introduce [28, 25]
Sint = gagb
∫
ωa
dτa
∫
ωb
dτb ˙ωa (τa) · ˙ωb (τb) D (ωa − ωb) (1.2)
as an interaction term in the action1. Here the function D must be symmetric and
accounts for the relative strength of interaction as a function of particle separation.
In the literature discussed above D (ωa − ωb) has been taken to be supported for
light-like, time-like and space-like separations, the last of which is the relativistic
generalisation of an instantaneous interaction. In Feynman-Wheeler theory, for ex-
ample, D (ωa − ωb) is taken to be the sum of advanced and retarded Green functions
of the (space-time) Laplacian. As is now well known, these action at a distance the-
ories are often cast into a form reminiscent of a field theory, although the “fields” are
not independent variables but are rather defined in terms of the particle trajectories
and the choice of D. For instance, if we take [29]
D (ωa − ωb) = 1
4π δ ((ωa − ωb)2) − m
8π
θ ((ωa − ωb)2)
√
(ωa − ωb)2
J1
(
m
√
(ωa − ωb)2
)
(1.3)
which is the time symmetric Green satisfying (∂µ∂µ − m2) D (ωa − ωb) = −δ4 (ωa − ωb)
then we may define
Aµ (x) = ∑
a
ga
∫
dτa ˙ωµ
a (τa) D (ωa (τa) − x) . (1.4)
This satisfies the Maxwell equations and gauge condition
(∂ν ∂ν − m2) Aµ (x) = jµ (x) ; jµ (x) = − ∑
a
ga
∫
dτa ˙ωµ
a (τa) δ4 (x − ωa) ,
∂µAµ (x) = 0 (1.5)
and in terms of A the interaction between the particles takes the form
∑
a
ga
∫
ωa
dτa ˙ωa (τa) · A (ωa (τa)) (1.6)
which shows that the action at a distance formalism contains the same equations of
motion and interactions as more traditional approaches using field theory.
1To be precise this provides vector-like interactions between point particles. Scalar interactions
can be produced by replacing each ˙ωiµ by ( ˙ω2
i
) 1
2 .
– 3 –
The proposal we will make will follow the same form as (1.2) except that we shall
choose D (ωa − ωb) = δ4 (ωa − ωb) so as to provide contact interactions between the
worldlines. This also ensures that, although in principle (1.2) implies the interaction
is non-local on the worldlines, the particles only communicate when they meet so
that the theory is local in space-time. In other words we are no longer considering
action at a distance but we allow particles to interact when they find themselves at
the same space-time position. As stated above, we have previously considered such
contact interactions between strings, where the theory found application to classical
electromagnetism and quantum electrodynamics. We now intend to explore the same
ideas for the case of point particles.
This article revisits and extends the results of [3] and also generalises that work
to the case of fermionic particles. It then goes beyond leading order in the coupling
strength to demonstrate that in fact the full quantum theory of a set of interacting
point particles is consistent and free of unwanted divergences. We develop the func-
tional approach to one dimensional field theory for consistency with [1, 2] and for the
generalisation to fermionic particles we will find it most natural to form the theory
in superspace. We will first consider a single particle worldline with fixed endpoints
that is constrained to pass through a given point in space and will relate it to clas-
sical electrostatics and the well-known phenomenon of confinement. We then repeat
a similar calculation for spin 1/2 particles to explore the fermionic version of the
theory before considering an arbitrary set of interacting worldlines. We will see that
the partition function of this theory is related to propagators of the Klein-Gordon
operator.
The structure of this article is as follows. In section 2 the bosonic theory pre-
sented at lowest order in [3] is reviewed before we generalise it to include spin degrees
of freedom in section 3. In section 4 we also carry out the first analysis of the theory
beyond the classical limit to explore higher order corrections to the result in [3]. Fol-
lowing this a full quantum theory of interacting worldlines is described and quantised
in section 5. Some supporting calculations on our regularisation scheme are given in
the Appendix.
2. Bosonic particles – the classical electric field
We begin by working in D spatial dimensions and consider a static charged particle
at position a and an oppositely charged particle at b. The classical electric dipole
field for this configuration is the well known solution to Maxwell’s equations in the
presence of these point particles. In [3] an alternative proposal was made which
generates the electric field by carrying out an average over a set of curves joining
the two sources. This concept goes some way to reviving Faraday’s notion of electric
flux lines and it is this calculation that we now review and extend.
– 4 –
choose D (ωa − ωb) = δ4 (ωa − ωb) so as to provide contact interactions between the
worldlines. This also ensures that, although in principle (1.2) implies the interaction
is non-local on the worldlines, the particles only communicate when they meet so
that the theory is local in space-time. In other words we are no longer considering
action at a distance but we allow particles to interact when they find themselves at
the same space-time position. As stated above, we have previously considered such
contact interactions between strings, where the theory found application to classical
electromagnetism and quantum electrodynamics. We now intend to explore the same
ideas for the case of point particles.
This article revisits and extends the results of [3] and also generalises that work
to the case of fermionic particles. It then goes beyond leading order in the coupling
strength to demonstrate that in fact the full quantum theory of a set of interacting
point particles is consistent and free of unwanted divergences. We develop the func-
tional approach to one dimensional field theory for consistency with [1, 2] and for the
generalisation to fermionic particles we will find it most natural to form the theory
in superspace. We will first consider a single particle worldline with fixed endpoints
that is constrained to pass through a given point in space and will relate it to clas-
sical electrostatics and the well-known phenomenon of confinement. We then repeat
a similar calculation for spin 1/2 particles to explore the fermionic version of the
theory before considering an arbitrary set of interacting worldlines. We will see that
the partition function of this theory is related to propagators of the Klein-Gordon
operator.
The structure of this article is as follows. In section 2 the bosonic theory pre-
sented at lowest order in [3] is reviewed before we generalise it to include spin degrees
of freedom in section 3. In section 4 we also carry out the first analysis of the theory
beyond the classical limit to explore higher order corrections to the result in [3]. Fol-
lowing this a full quantum theory of interacting worldlines is described and quantised
in section 5. Some supporting calculations on our regularisation scheme are given in
the Appendix.
2. Bosonic particles – the classical electric field
We begin by working in D spatial dimensions and consider a static charged particle
at position a and an oppositely charged particle at b. The classical electric dipole
field for this configuration is the well known solution to Maxwell’s equations in the
presence of these point particles. In [3] an alternative proposal was made which
generates the electric field by carrying out an average over a set of curves joining
the two sources. This concept goes some way to reviving Faraday’s notion of electric
flux lines and it is this calculation that we now review and extend.
– 4 –
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