GAUGE–INVARIANT CHARGED, MONOPOLE AND

DYON FIELDS IN GAUGE THEORIES

J. FRÖHLICH

Theoretical Physics, ETH–Hönggerberg, CH–8093 Zürich, Switzerland

P.A. MARCHETTI

Dipartimento di Fisica, Università di Padova

and

INFN – Sezione di Padova, I–35131 Padova, Italy

Abstract

We propose explicit recipes to construct the euclidean Green functions of gauge–invariant charged, monopole and dyon fields in four–dimensional gauge theories whose phase diagram contains phases with deconfined electric and/or magnetic charges. In theories with only either abelian electric or magnetic charges, our construction is an euclidean version of Dirac’s original proposal, the magnetic dual of his proposal, respectively. Rigorous mathematical control is achieved for a class of abelian lattice theories. In theories where electric and magnetic charges coexist, our construction of Green functions of electrically or magnetically charged fields involves taking an average over Mandelstam strings or the dual magnetic flux tubes, in accordance with Dirac’s flux quantization condition. We apply our construction to ’t Hooft–Polyakov monopoles and Julia–Zee dyons. Connections between our construction and the semiclassical approach are discussed.

1.Introduction

In this paper, we study a variety of gauge theories in four dimensions with the following features: their phase diagrams contain phases with deconfined electric or magnetic charges; their particle spectra thus contain electrically or magnetically charged particles or dyons. They may exhibit phase transitions characterized by condensation of charged particles or magnetic monopoles. Some of them exhibit duality symmetries.

One is interested in studying various aspects of the phase diagram and of the dynamics in particular phases of such theories. Obviously, one would like to study these theories analytically. However, explicit analytical results are often only available for physically rather unrealistic theories with supersymmetries. In order to study more realistic theories, one therefore often resorts to numerical investigations of lattice approximations of such theories. Usually, such investigations are based on a euclidean (imaginary–time) formulation of quantum field theory obtained from a real–time formulation by Wick rotation. In the lattice approximation, one replaces Euclidean space–time by a (finite, but arbitrarily large) lattice.

In order to study the mass spectrum of a quantum field theory, one considers euclidean Green functions of gauge–invariant (physical) fields of the theory that couple a vacuum (ground) state to some one–particle state. Masses can be calculated from exponential decay rates of certain two–point euclidean Green functions.

Signals for a phase transition, e.g., one between a Coulomb– and a confining phase, can be detected in an analysis of asymptotic behaviour of suitable euclidean Green functions. For example, the transition from a Coulomb– to a confining phase in an abelian gauge theory is reflected in the appearance of long–range order in the two–point euclidean Green functions of magnetic monopoles.

On a more foundational level, we are longing for a complete description of quantum field theories in terms of the (euclidean) Green functions of gauge–invariant interpolating fields. For example, to describe the deconfined phase of an abelian gauge theory, we would like to construct Green functions of charged fields.

For QED–like theories without dynamical magnetic monopoles, a proposal for a gauge–invariant charged field has been made, many years ago, by Dirac [1]. For lattice theories of this type, Dirac’s proposal has been studied carefully e.g. in [2,3]. For the convenience of the reader, the main results of this analysis are summarized in sect. 3.

There are, however, plenty of physically interesting theories with deconfined, electrically charged particles and dynamical magnetic monopoles. The problem of constructing gauge–invariant electrically or magnetically charged interpolating fields for such theories has not been solved in adequate generality and is quite non–trivial. The difficulties encountered in studying this problem are a consequence of the Dirac quantization condition. We report partial results towards a solution of this problem in sect. 4.

The main purpose of this paper is to describe constructions of gauge–invariant electrically or magnetically charged interpolating fields and of dyon fields in (lattice) gauge theories with dynamical electric charges and dynamical magnetic monopoles and to exhibit duality transformations converting electrically into magnetically charged fields (and vice versa). Our work is primarily kinematical: we propose fairly explicit recipes for how to construct the euclidean Green functions of such fields. But we do not engage in any mathematically careful analysis of the properties of these Green functions. Instead, we gather evidence supporting various conjectures on their behaviour. Part of this evidence comes from previous, mathematically precise work on lattice gauge theory, another part is based on more heuristic arguments and conventional wisdom.

All in all, we believe we arrive at a fairly consistent picture of how electrically or magnetically charged and dyon fields should be constructed.

We also review duality properties of some gauge theories and analyze how duality transformations act on charged and dyon fields. This part of our analysis makes contact with issues that have been quite topical, during the past few years; (see e.g.[4]).

Finally, we outline a formal construction of Green functions of monopole– and dyon fields in the continuum Georgi–Glashow model and indicate how our construction is related to the (semi–)classical analyses of ’t Hooft and Polyakov [5] and of Julia and Zee [6].

Next, we present brief summaries of the different sections of this paper.

In section 2, we introduce the gauge theories studied in this paper. We consider three classes of models (A, B, and C). The models in class A are non–compact abelian lattice gauge theories with electrically charged matter fields, but without dynamical magnetic monopoles, i.e., models of lattice QED. The models in class B are compact abelian lattice gauge theories with electrically charged matter fields. As originally pointed out by Polyakov [7], they describe dynamical magnetic monopoles coexisting with electrically charged particles. The models in class C are related to (lattice approximations or formal continuum limits of) the Georgi–Glashow model. We define our models in terms of euclidean action functionals. We introduce some key notation and review some facts on the phase diagrams of these models.

In section 3, we construct gauge–invariant charged fields for models of class A, in accordance with Dirac’s proposal and with the result of [2,3]. We also construct monopole fields in compact, pure abelian lattice gauge theories. These theories are dual, in the sense of Kramers–Wannier duality [8], to some models of class A, and this suggests a dual variant of Dirac’s proposal as the right definition of monopole fields.

We start section 3 with a short recapitolation of Osterwalder–Schrader reconstruction and of an analysis of superselection sectors based on euclidean Green functions of charged fields and monopole fields. We then recall (an euclidean version of) Dirac’s proposal [1] for gauge–invariant charged fields,

where is a charged matter field, and is a (non–compact, i.e., real–valued) abelian gauge potential; furthermore, is a c–number one–form related to the electrostatic Coulomb field of a point charge. (Thus, has a source at the point whose charge is the same as the charge of ). We study the “infra–particle nature” of charged particles in such theories.

Our construction of monopole fields and our analysys of their Green functions in compact abelian gauge theories without matter fields follows from the results on electrically charged field by using Kramers–Wannier duality; (this is made explicit in sect. 3.3).

In sect. 4, we study electrically charged particles and dynamical magnetic monopoles in compact abelian lattice gauge theories with matter fields. We start by explaining the origin of Dirac’s quantization condition,

. We then construct gauge–invariant electrically charged fields as averages of charged fields multiplied by Mandelstam string operators (exponential of the gauge field integrated along a path, called Mandelstam string [9]), the average being taken over a suitable space of Mandelstam strings. Among the more subtle points appearing in this paper is the one to come up with a good definition of “averages over Mandelstam strings”. In an analysis of Green functions of these fields based on successively integrating out the large–frequency (short–distance) modes of the fields, one observes that the large–distance effective theory becomes increasingly similar to the one describing a model in class A. In particular, the functional integral expressions for these Green functions resemble the ones for the Green functions of charged fields constructed according to Dirac’s proposal, in models of class A. Suitable averaging over Mandelstam strings yields operators which, under renormalization, approach ones related to (1.1). The magnetic monopoles get suppressed, and Dirac’s flux quantization condition becomes irrelevant.

We then proceed to define monopole fields, or, more precisely, euclidean Green functions of such fields, in terms of certain averages over ’t Hooft disorder operators [10]. We also present a heuristic analysis of properties of euclidean Green functions of electrically charged– and monopole fields and compare our results with these in section 3.

We conclude section 4 with an analysis of dyons and of an duality group in a compact abelian lattice gauge theory with a topological term, related to the instanton number, in the action. Our analysis is based on previous work of Cardy and Rabinovici [11].

In section 5, we outline a construction of physical interpolating fields for ’t Hooft–Polyakov monopoles and Julia–Zee dyons in the Georgi–Glashow model on the lattice and in the formal continuum limit of this model. The role of ’t Hooft’s monopoles and the corresponding disorder operators in our construction in explained in detail.

Some technical points are relegated to two short appendices.

2. The models

The models we consider in this paper are lattice gauge theories.( Some comments on a continuum gauge theory are added in the last section, for the Georgi–Glashow model.)

Our (euclidean space–time) lattice is , where the subscript indicates that the coordinates of the sites are half–integers. We will also have to consider sublattices (more precisely cell subcomplexes) of .

Lattice fields can be defined as follows:

A scalar field is a map from the sites, , of the lattice to a normed vector space .

A fermion field is an anticommuting map from the sites to the orthonormal frames of a vector space , the fermion space, where , the spin space, carries a representation of the Dirac–Clifford algebra.

A gauge field is a map from the links of the lattice to a group , the gauge group.

If is an additive abelian group and is a positive integer, one can define forms with values in as maps, , from oriented -dimensional cells, of the lattice to satisfying , where denotes the cell obtained from by reversing the orientation.

We denote by the lattice exterior differential

and by the Hodge–star:

Let denote the cell in the dual lattice, , dual to . Then

We also introduce the codifferential and the Laplacian . If is a sublattice we denote the restriction of lattice operators to forms over by a subindex , e.g. instead of .

If is a Hilbert space with scalar product one can define a scalar product among forms and by

The restriction of the scalar product (2.1) to forms defined on a sublattice is denoted by . We define the –norm of by

The vacuum functional of a gauge theory with gauge field and with matter fields is given in terms of formal integration “measures”

where is the Haar measure on , is the Lebesgue measure on denotes Berezin integration, is the total action, and is a normalization factor, the partition function. [Mathematically, the measure (2.2) is first defined for a finite lattice , with suitable boundary conditions at ; subsequently, one takes the limit ].

Expectation values w.r.t. the measure (2.2) are denoted by .

2.1 Actions

We consider the following classes of models.

Class A: Non–compact abelian gauge theories with matter fields.

As an example we discuss the abelian Higgs model .

The fields of this model are a real gauge field, , and a complex scalar field, .

The action is given by

is a gauge fixing term, e.g.

From now on, this gauge fixing term will usually not be written explicitly in our formulas, anymore. In the limit this model reduces to the Stückelberg model. For simplicity, we only discuss the model at , in this paper. It will be referred to as model A.

Remark 2.1 A second interesting model in class A is spinor QED on the lattice; (see [2] for a rather detailed discussion). The fields of QED are the real gauge field and a four–component fermion field . The action is given by

where are matrices on given by

if is directed in the direction, , where are Euclidean Dirac matrices.

Remark 2.2 In these models, we can omit gauge fixing by working directly with a measure defined on gauge equivalence classes

where is a real scalar field [12].

Class B: Compact abelian gauge theories with matter fields.

As an example we analyze the compact abelian Higgs model (model B).

In the “Villain formulation”, the basic fields are a –valued –form , a -valued 1–form , a –valued 1–form and a -valued 2–form . The action is given by

where is the charge of the matter field; it is an integer.

Alternatively, we can use the “Wilson formulation”:

Class C: Non abelian gauge theories coupled to matter fields breaking the gauge

group to a Cartan subgroup containing .

As an example we analyze the Georgi–Glashow model (model C).

Its basic fields are an –valued gauge field and a Higgs field , of unit length, in the adjoint representation of . The action is given by

where denotes the character of the fundamental representation, and denotes the adjoint representation.

2.2 Phase diagrams

For small and , model A has a Coulomb phase with a massless photon [13,12,14]. Furthermore, it is known that, for small and suficiently large , it has a superconducting Higgs phase [15].

Model B, with , has a confining / Higgs phase, for small or large . Furthermore one expects a massless phase for and small [16].

For , a phase transition line is expected to separate the confining and the Higgs phases, furthermore, for sufficiently large, a Coulomb phase is known to appear [13,12] for intermediate values of , for arbirary large . If we add a topological -term,

where the wedge product on the lattice can be defined as in [17], to the action of model B, in the limit , we recover the model discussed by Cardy and Rabinovici, which is expected [11] to exhibit Higgs, Coulomb, confinement and “oblique confinement” phases , related to an (approximate) symmetry under an duality group.

Model C has been rigorously shown to have a Coulomb phase for and large. The Coulomb phase is expected [18] to extend into a region of large values of and . A confining phase is also known to exist [15], for small values of .

Charged particles and monopoles are expected to exist in the Coulomb phase of these models as (infra–)particle excitations. This is the main issue discussed in this paper.

For models in class C, magnetic monopoles are expected to exist as massive particles also in the continuum limit (provided it exists). These are the celebrated at’t Hooft–Polyakov monopoles.In the models of class B, monopoles are a lattice version of the (singular) Dirac monopoles.

3. Charges or monopoles

To motivate our constructions of charged and monopole fields for the models introduced in the previous section, we outline a general strategy exploited in [19,20,21,2](see also [22]) to construct charged (and soliton) fields and superselection sectors directly from correlation functions defined through euclidean functional integrals. We discuss this construction on the lattice; but similar ideas have been applied to continuum models, although only formally.

3.1 Reconstruction of charged and soliton quantum fields from euclidean Green functions

Let denote a euclidean “observable”, i.e., a neutral, gauge–invariant, local function of the basic fields with support on a compact connected set of cells , such as a Wilson loop, a ”string field”, etc.

We consider expectation values of such euclidean observables

Let denote reflection in the time–zero plane, a complex–valued function and its complex conjugate. We define the Osterwalder-Schrader (O.S.) involution as an antilinear map satisfying

Let denote the set of linear combinations of euclidean observables with supported in the positive–time lattice. If, for every ,

then the correlation functions are said to be O.S. positive.

Theorem 3.1 (O.S. reconstruction). If the correlation functions satisfy

i) lattice–translation invariance, and

ii) O.S. positivity

then one can reconstruct from

a) a separable Hilbert space, of physical states,

b) a vector of unit norm, , the vacuum,

c) a self–adjoint transfer matrix , with unit norm, and unitary space translation operators , acting on and leaving invariant.We define .

If, moreover, the correlation functions satisfy

iii) cluster properties

then

d) is the unique vector in invariant under and .

From the explicit proof of the theorem it follows that there is a set of vectors in , with contained in the positive time lattice, ,such that the set of linear combinations, denoted , is dense in . On these vectors, the scalar product is defined by

Field operators , with contained in the strip , can be defined on by setting

where denotes translation by in the time direction. The operators are the quantum mechanical operators reconstructed from the euclidean observables . The algebra, , generated by the operators defined above is called ”lattice observable algebra”.

If the space of the physical states of the model contains charged or soliton states, such states are not contained in the Hilbert space constructed above.

The construction underlying Theorem 3.1 reproduces only the vacuum sector of such models. Suppose, however, that the Hilbert space,, of physical states of the model can be decomposed into orthogonal sectors invariant under and the lattice observable algebra , i.e.,

Here is defined to be the subspace given by . It is called vacuum sector. A sector is called a “charged sector” if there are no lattice translation– invariant vectors in .

To construct the charged sectors, suppose that we can find an enlarged set of correlation functions obtained by taking expectation values involving, besides euclidean observables, “charged” order– and disorder–fields, denoted by and by , respectively, with support in a connected (but, in general, non–compact) set of cells . Correlation functions with non vanishing total “charge” are obtained from those of vanishing total charge,by removing the support of the charge of a “compensating” field of charge to infinity.

Assume that the correlation functions still satisfy the hypotheses of the O.S. reconstruction theorem.

Denote by the Hilbert space, the transfer matrix, the translation operators, the vacuum and the quantum fields obtained via O.S. reconstruction from the correlation functions . If all the correlation functions of non-vanishing total “charge” vanish, and clustering holds, then the Hilbert space contains charged sectors, because the scalar product between two charged states of unequal charge, defined in analogy with (3.2), vanishes; (for more precise definitions and proofs see [ ]).

The quantum fields map the vacuum sector to a charged sector.

Remark 3.1 Some of the quantum fields might couple the vacuum to one–particle states, or infra–particle states. This can be seen as follows: Provided that , the mass operator can be defined by

where denotes the fibre of of zero total momentum, (i.e. a generalized vector belongs to iff ). Then we have the following result.

Theorem 3.2 A field operator acting on couples the vacuum to a stable massive one–particle state iff

with , for any . This decay law is called Ornstein–Zernike decay.(For infra–particles the exponent in the denominator on the r.h.s. of (3.5) has a positive small correction.)

For many lattice models, the large behaviour involved in (3.5) can be analysed in terms of expansions methods [23,19,21].

3.2 Charged fields in non–compact abelian models

In this section we recapitulate the basic steps of the construction of charged fields in the models of class A [2,3], following the scheme outlined above. We do this for the sake of completeness and in order to elucidate the difficulties arising in attempts to extend our construction to models in class B.

Let be a real–valued 1–form with support on an infinite, connected sublattice of the time–zero hyperplane,, of the lattice .

Furthermore we assume that

where

and

Here is the origin in and denotes the euclidean distance between and . As an example, one may consider

We denote by the 1-form translated by ; describes the electrostatic Coulomb field surrounding a source of charge 1 located at . We define the charged fields of the Higgs model by

(See fig.1) Typical observables, , are Wilson loops

where is a loop, , and “string fields”

where is a path form to .

Correlation functions are defined by

For later purposes it is convenient to define a generalization of (3.12): For two different 1-form satisying (3.6), we set

The correlation functions can be expressed as sums over configurations of “electric currents”, by first integrating over the matter fields and then integrating over . For example

where is an integer valued 1-current, is a certain statistical weight determined by the action of the model, and

Since, in the denominator of (3.18), , the support of the currents is given by a set of loops. Hence the denominator can be interpreted as the partition function of a gas of current loops interacting via the four–dimensional (lattice) Coulomb potential . In the numerator of (3.18), open currents appear, with sources at , besides current loops. At the sources, the electric currents spread out in fixed time planes, as described by . The currents can be interpreted as the Euclidean worldlines of charged particles; currents supported on loops correspond to worldlines of virtual particle–antiparticle pairs, open currents with connected support correspond to the worldlines of particles created at one end of the line and annihilated at the other one.

Theorem 3.3 The correlation functions defined in (3.12) are lattice translation invariant and O.S. positive. Furthermore, for large enough or for strictly positive and small enough, clustering holds and all correlation functions with non–zero total charge vanish.

Idea of proof Invariance under lattice translations follows from the existence of the thermodynamic limit of the measure corresponding to (2.3) derived by correlation inequalities [15,16]. O.S. positivity of those measures can be proved as in [15], and O.S. positivity of charged correlations follows from the fact that is localized in a fixed–time plane, so that e.g.

for .(There is a slight subtlety cocerning the choice of boundary conditions for correlation functions of charged fields in a bounded space–time volume.It can be dealt with in a way similar to that explained in [21]; see also section 4.2)

Cluster properties are a consequence of (generalizations of) the bounds stated in the next theorem,for details see [2,3].

Theorem 3.4 In model A, for large enough or sufficiently small, and large enough,

and for

where tends to 0 exponentially, as , and, for fixed , , as , , and , , as ; is a current of minimal length and flux 1 connecting to , and

The upper bound in Theorem 3.2 shows that charged (infra–)particles in the Higgs model have strictly positive mass in the range of coupling constants indicated in the theorem. The lower bound proves that, in the same range of coupling constants, their mass is finite.

The method of proof is based on a combination of a Peierls– and renormalization group argument, following [12,24]. For the lower bound in (3.16) can be obtained more easily from Jensen’s inequality, using representation (3.14).

Remark 3.2 It can be shown, following [25], that, for sufficiently small and sufficiently large, clustering holds, and

uniformly in . Hence correlation functions of non zero total charge do not vanish. This is a manifestation of charged–particle condensation typical for the Higgs phase.

From the O.S. reconstruction theorem we obtain a Hilbert space, denoted by , a transfer matrix and a dense set of vectors:

corresponding to charged fields of charges inserted at points and local observables located at , with and in the positive time lattice.

From (generalizations of) the lower bounds (3.16) it follows that , i.e. the states (3.19) have finite energy.

One can define non–local charged fields by

with .

We now consider the region of coupling constant space where all correlation functions of non–zero total charge vanish. Then decomposes into orthogonal sectors labelled by the total electric charge :

Cluster properties show that the sectors , , are charged sectors, in the sense described in sect.3.1.

From the above construction it is clear that, a priori, the charged sectors depend on the choice of the distribution . It is natural to ask if is orthogonal to , for . In order to answer this question , one considers the generalized correlation functions (3.16).

One can define a scalar product between states in and by

From generalizations of the upper bounds (3.19), (3.20 ) it follows that

diverges and one easily realizes that this divergence occurs if and do not have the same “behaviour at infinity”.

There is an interesting choice of and which naturally leads to a divergence in (3.21). For example, if one chooses to be supported in a spatial cone with apex in 0 and opening solid angle less than , the states obtained via O.S. reconstruction are the lattice approximation of the states discussed by Buchholz [26] in the algebraic approach to Q.E.D.. If the field is chosen to be localized in a disjoint spatial cone , then (3.21) diverges. In particular, if we choose to be the cone obtained by a rotation of then this divergence shows that in the continuum limit (if it exists) the rotations cannot be unitarily implemented on “Buchholz states”. For more details see [2,3,19].

We conclude with some remarks about particle structure analysis on charge sectors. By ispection of the proof of Theorem 3.2 (see [2,3]), one can argue that for large :

where is a current of flux 1 and connected support is a statistical weight and is a positive measure of the form

In (3.23), is a gaussian measure with mean 0 and covariance (+ gauge fixing), where is a renormalized coupling constant and is a sum of gauge–invariant “irrelevant” terms, in the jargon of the renormalization group. Hence the large distance behaviour of , which is independent of the irrelevant terms, , should essentially be given by a product of two factors: one is due to the fluctualing current line , and, from the analysis in terms of excitation expansions of [23], it is expected to produce an Ornstein–Zernike decay

corresponding to a particle of mass . The second factor can be argued to contribute another power correction to the exponential law

It is due to the soft photons accompanying an infra–particle. Therefore, as , one expects that

Equation (3.24) exhibits the infraparticle nature of the charged particles in the Higgs model. In fact, the vacuum expectation value of a charged field vanishes in the region of coupling constant space considered, and a comparison with the general formula (3.5) shows that the mass operator , as defined in (3.4), does not have a discrete eigenvalue corresponding to a sharp one–particle state.

Remark 3.3 (Q.E.D.) In lattice Q.E.D., the electron field is defined by

and the conjugate euclidean field is

Typical local observables are Wilson loops and string fields. Correlation functions of charged fields and local observables satisfy the hypotheses of the Reconstruction Theorem 3.1 and, for sufficiently small, bounds analogous to those in Theorem œ[5 3.3 (as discussed in [2]).

This permits one to reconstruct non–local electron–positron field operators

, which are defined by

for .

The field can be viewed as the lattice approximation of the formal operator (1.1) introduced by Dirac. An analogue of equation (3.24) is expected to hold for small enough, on the basis of the proof of bounds on electron-positron correlations. It would exhibit the infraparticle nature of electrons and positrons.

3.3 Monopoles and duality

Model B, in Villain form, at is dual to model A in the limit and it describes an interacting theory of “photons” and Dirac monopoles. Since monopoles can be viewed as solitons in such a model, we may appeal to the strategy outlined in sect. 3.1 to construct a monopole field operator by introducing disorder fields, whose expectation values are the Euclidean Green functions of the Dirac monopoles [2,3,19]. For this purpose we introduce a real 3–form on the lattice , given by and define the disorder field by

where

and is a –valued 3–form satisfying

We now explain why expectation values of are correlation functions of monopoles.

First, we use the Hodge decomposition to rewrite

where

Let be an integer–valued solution of the cohomological equation (3.28). Then every other solution is of the form , where is a to -valued 1-form. We define a real-valued gauge field by setting

and one can easily verify that

is the euclidean “photon field”, and the Hodge–dual of is supported on a set of lines in the dual lattice , which can be interpreted as Euclidean worldlines of Dirac monopoles; itself can be viewed as a Dirac string if we take

where has support in an open line at constant time with one end at and the other end joining a compensating current at infinity (monopole b.c.).

Then is exactly the lattice approximations of the magnetic field of a monopole located at , together with its Dirac string .

In the presence of the disorder field the Euclidean observables of the model must be modified so that expectation values do not depend on the choice of the “Dirac string” . In particular the Wilson loop is now replaced by

where is a surface; see [19,2].

Correlation functions in the gauge theory are then defined by

By duality e.g.

where denotes the expectation value in the Higgs model A and . O.S. reconstruction theorem applied to provide us with non–local Dirac monopole fields and, for large enough, Dirac monopole sectors . Furthermore a particle–structure analysis along the line of sect. 3.2 exhibits the infraparticle nature of the Dirac monopole of charge .

Remark 3.3 The monopole construction outlined above can be applied to every –gauge theory without matter fields. This constructions and variants thereof have been applied in numerical simulations of lattice theories in [27], with the aim of detecting phase transitions between phases corresponding to different behaviour at large distances of the monopole Green functions, which has been described in the dual picture in Theorem 3.3 and Remark 3.2.

4. Charges and monopoles We start this section with an outline of the new problems appearing in an attempt to extend the construction sketched in the previous sections to models where charges and monopoles coexist (class B,C).

4.1 Dirac quantization condition

We have seen that, for , model B (in Villain form) can be written explicitly in terms of a “photon field” and a “monopole field” . The Dirac strings of the virtual monopole loops in the partition function sweep out surfaces described by the support of . The term in the action introduced in eq.(2.6) couples the gauge field to a charged matter field. In our expression for the partition function, the charged matter gives rise to charged loops describing the worldlines of virtual particle–antiparticle pairs. We must ask whether the statistical weights of these charged loops depend on the location of the Dirac strings of virtual monopoles.

To answer this question, let us rewrite the action (2.6) in terms of . We obtain, for ,

Independence of the Dirac strings corresponds, as recalled above, to independence of the choice of the two–form satisfying , as in (3.28).

Let be a -valued 1-form. Then by Poisson summation formula,

Integrating over we obtain

We conclude, using the Poincaré lemma, that there exists a -valued 2-form, , such that

so that

where the last equality follows from Hodge decomposition.

The second term depends only on and

This is nothing but the Dirac quantization condition, since the electric charges appearing in the partition functions are and the magnetic charges are , so that

From the above proof it is clear that if we consider the expectation value of , where the “electric current” 1-form is not integer valued, as introduced in sect 3.2, we would encounter an inconsistent dependence on the choice of Dirac strings. This problem is often neglected in the physics literature, wher virtual monopole loops are sometimes ignored, assuming that monopoles are “very heavy”, see e.g. [28]. But, in order to arrive at a fully consistent definition of gauge–invariant charged field Green functions, one must cope with it.

The natural suggestion is to replace the electric field of sect. 3.2 by an integer–valued electric current, a “Mandelstam string”, starting at and ending at the location of some compensating charge which will eventually be sent to infinity.

A naive idea would be to simply replace the variable , used in the construction of charged states in model A, by

in model B, and then take the limit . In (4.3), is a unit 1-form with support on a straight line in a fixed–time plane from to some point at a distance and then joning that point to a fixed point in the time–zero plane, and is some normalization factor.

Consider the 2-point function: Integrating out we obtain

where are -valued currents whose statistical wheight is denoted by , and is the partition function. The phase factors appearing in (4.4) define Wilson loops, and, in the region of coupling constant space where monopoles are particle excitations, the expectation value of the Wilson loop is known to exhibit perimeter decay. Hence we expect (4.4) to vanish exponentially fast as .

This decay is dominantly due to the self–energy of the strings . This effect could eliminated by adjusting . But we would then be left with an interaction term between the strings which, being attractive and extending over the full string, tends to infinity in the limit , and this appears to render the renormalized Green function divergent. Furthermore, since the interaction term depends on the time distance between strings, it cannot be renormalized away without violating O.S. positivity. This problem could be solved if we replace a “straight Mandelstam string” by a sum over “fluctuating Mandelstam strings”, weighted by a measure which is concentrated on strings fluctuating so strongly that, with probability one, their “interaction energy” remains finite in the limit .

A proposal for a “natural” measure can be inferred from a representation of correlation functions of a scalar field, , with coupling constant , coupled to an external gauge field , in terms of random walks. E.g., in dimensions, with denoting the corresponding expectation value,

where is a path (“string”) from to , its length, is the partition function of the system, and