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Line discontinuities, local action with both the field and its dual, and spin from no spin in two-dimensional scalar theory

We consider a local action with both the real scalar field and its dual in two Euclidean dimensions. The role of singular line discontinuities is emphasized. Exotic properties of the correlation of the field with its dual, the generation of spin from scalar fields, and quantization of dual charges are pointed out. Wick's theorem and rotation properties of fermions are recovered for half-integer quantization.


Introduction
Duality transformations play important roles in many field theories and statistical physics models. They provide an equivalent description of the system. They often relate two models which appear to be different. They map weak coupling or high temperature regime of one model to the strong coupling or low temperature regime of the other. This provides an understanding of the phases of the model as well as non-perturbative calculational techniques. The degrees of freedom of the dual theory play the role of disorder variables of the original theory. They often provide valuable order parameters to characterize the phases of the model. In addition to all this, order and disorder variables close together have exotic properties. For instance, certain combinations of bosonic variables may behave as a fermion. Such combinations also play a crucial role in providing the description and properties of the model in some phases. Because of these important features, it is useful to have systematic techniques for handling both the field and its dual on an equal footing. In this work we do this for a free massless real scalar field in two (Euclidean) dimensions. This case, though simple, encodes rich physics. It is self dual, so that the dual field is also free scalar field. In Section 2.1, we express the partition function in terms of the dual field. In Section 2.2, we bring out the role of line discontinuities, in close analogy with the order-disorder formulation of the 2-d Ising model [1]. We emphasize that such discontinuities in a field configuration are meaningful without requiring a smoothening, and contribute a finite action.
We want to treat the field and its dual on an equal footing. In particular, we want to handle correlation function of the field with its dual. We thus obtain a local action in Section 2.3, which has both the fields simultaneously present. It is related to Hamiltonians used for describing condensed matter systems in one dimension [2]. The local action is not manifestly invariant under rotations. The underlying reason is the choice of the line of discontinuity. It is also not real in spite of being equivalent to the free massless scalar theory. Even though it has double the number of fields, it represents the original theory.
The correlation of order and disorder variables has been discussed in various contexts in [3].
In our case, the correlation function of the field with its dual has a number of exotic properties, which we explain in Section 2.4. First, it is discontinuous, reflecting that the field introduces a line of discontinuity in the configuration of dual field. The correlation function does not die off at large distances, representing the non-local effect the field has. Indeed the correlation function is proportional to the angle subtended by the vector joining the coordinates of the field and the dual field with the line of discontinuity. Both the field and its dual being scalars, the correlation functdions involving only the field (or only the dual field) depend on the distance and not on the orientation. This is no longer true of the mutual correlations, again reflecting the role of line discontinuities. In addition to all this, the correlation function is pure imaginary even though the fields are real. This is traced to the action which is not real.
The free massless real scalar field in two Euclidean dimensions embodies all minimal models of conformal field theory through the use of boundary charges, the Coulomb gas formalism and the vertex operators [4,5]. In Section 3, we show that the vertex operators of both the field and its dual can be handled together in our formalism. This exhibits the phenomenon of 'spin from no spin'. The correlation functions imply that the vertex operators of the dual field (or equivalently the original field) acquire spin though constructed from a scalar field.
The mixed correlation functions appear to be multivalued. But we emphasize that they are simply discontinuous, a consequence of line discontinuities. For a specific 'quantization' of the 'charges' of the field and its dual, the single valuedness of the correlation function is restored. This is exactly parallel to Dirac quantization of electric and magnetic charges. Just as the Dirac string becomes invisible, the line discontinuities become invisible for the specific vertex operators which acquire spins. The correlation functions are now single valued and rotation covariant with new spins.
The composite operators built out of order and disorder fields often have exotic properties and play important roles in the properties of the theory. In 2-d Ising model, the combination is a fermion as regards to rotational properties and spin-statistics connection [1,6]. Analogous to this is the 2-d bosonization , which builds a fermion from real massless boson [7,8]. Our formalism is well equipped to handle this. In Section 4, we consider half-integer quantization of the dual charges and show how Wick's theorem for fermions is recovered. We point out the role played by the line discontinuities to yield the rotational properties of the fermions.
We choose straight line discontinuities through most of the paper. However, we have the freedom of choosing any path, as discussed in the Appendix.
2 A local action with both the field and its dual, and the role of line discontinuities 2.1 Partition function in terms of dual field Consider a free massless scalar field φ(x) in two Euclidean dimensions. Its correlation functions can be obtained from the generating functional: In (1) ρ(x) is the source coupling locally to φ(x) and the normalization N 1 is defined so that Z[ρ = 0] = 1. We are inserting √ −1 in the source term for two reasons: (i) the duality transformation carried out below is more transparent, (ii) the point sources 1 give the vertex operators 1 Throughout this paper, we use superscripts to label different points in 2-d space.
which play crucial role here as in CFTs. Formally: where (x − y) is the 2-d Coulomb potential satisfying: As the Coulomb potential in 2-d is infrared divergent, we use a finite area of linear dimension R. Also to handle ultraviolet divergences in certain correlation functions, we use the ultraviolet cutoff a to get [9]: Using an auxiliary field b i (x) we linearize the φ(x) terms in (1): A formal integration over φ gives a functional δ function: We solve the 'Gauss law constraint' as follows: where ∂ −1 1 is defined as: For a point source ρ(x) = δ 2 (x − x 0 ), i.e. the flux is carried along a line from x 0 parallel to x 1 axis in the +ve direction. The 'Gauss law' solution (9) implies, dual of the field φ(x). The source ρ, which couples locally to φ, couples in a specific non-local manner to the dual fieldφ. In spite of this, an integration overφ reproduces (4): Now integrating by parts with respect to x 2 and y 2 , and using ∂ , we recover (4).

Interpretation in terms of lines of discontinuity
For the point sources (2) the partition function (12) takes the form: This exhibits the non-local interaction of the dual fieldφ(x) with the local sources of φ(x).

A local action with both field and its dual present
Using an auxiliary fieldπ(x) the partition function (12) can be written as: Notice that ρ couples locally to −∂ −1 1π . Therefore, we identify We can now explore the correlation function of φ(x) with its dual fieldφ(x). For this we introduce a local sourceρ(x) and couple it toφ(x). The partition function (15) takes the form: The partition function (17) represents the original theory (1) with only φ(x) field as an equivalent local theory with both the field and its dual simultaneously present. Moreover it is self-dual as the action itself is invariant under: φ ↔φ and ρ ↔ρ. The action in (17) also has discrete In the absence of sources, the action in (17) is also invariant under the following global transformations: However, it is not manifestly invariant under rotations of the 2-d space. The traditional (∂ 2 φ) 2 and (∂ 2φ ) 2 terms are not present. Also, the action is not real (even in the absence of the imaginary sources we are using). The term bilinear in φ andφ has √ −1 factor. In spite of all this its equivalence to the traditional partition function is seen by integrating over one or the other field (in the absence of sources). The lack of manifest rotation invariance is due to the specific choice of the singular flux line in (10).
We see from (15) that if we regard x 2 as the Euclidean time, the partition function (1) corresponds to the Hamiltonian density (Z = lim T →∞ Tre −T H ): with canonically conjugate variables (φ(x),π(x)). This relates the field φ(x) non-locally to the momentumπ conjugate to the dual fieldφ(x) via (16) and connects our formulation to condensed matter system Hamiltonians [2]. With this interpretation (17) is not so exotic.
Nevertheless, it is very useful as the field and its dual are on the same footing.

Correlations of the field φ with its dualφ
Evaluating (17) and taking functional derivatives with respect to the sources ρ(x) andρ(x) we get correlation functions of field φ(x) with its dualφ(x): Now using we get: which is consistent with (4), and as expected for a massless real scalar field. The correlation function of the field φ(x) with its dualφ(y) is: In (23) the operator ∂ −1 1 ∂ 2 acting on ( x − y) is with respect to the first coordinate x. We define: We now have to directly address the integral to choose the branch for the definition of inverse of tan. The integral is positive definite and well defined as long as x 2 = 0. Even for x 2 = 0, it is well defined if x 1 < 0 because the integrand never blows up. Making a change of variables from x 1 to t ≡ where sgn(x 2 ) equals +1(−1) for x 2 > 0 (x 2 < 0). Therefore, as shown in Figure 1, Θ( x) is the angle subtended by the vector x measured in the anti-clockwise direction with starting value −π from the +ve x-axis. Thus Note that Θ( x) is discontinuous along the +ve x 1 axis because it changes from −π to +π as x 2 changes from a +ve value to −ve value as shown in Figure 1-a. Thus the correlation function of the field with its dual (27) is discontinuous whenever x − y is parallel to the x 1 -axis. As shown in Figure 1-b, it changes by i 2 sgn(x 1 − y 1 ) as (x 2 − y 2 ) changes sign from −ve to +ve. This is a reflection of the boundary condition forced by the line discontinuity generated by the field φ(x) in the configuration space of its dual fieldφ(y). It is convenient to define the standard obeying the standard relation: The correlation function (27) can now be written as: Thus the correlation function of the field φ(x) with its dualφ(y) has the following unusual properties: 1. It does not fall off with the distance, reflecting the non-local effect φ(x) has in the configuration space ofφ(x). We finally have the master formula: We can also consider the action in terms of the linear combinations of the fields φ(x) and its dualφ(x): The action in (17) decouples in ξ + and ξ − : The correlation functions of these combinations are: where z = x 1 + ix 2 . The fields ξ ± are related to holomorphic and anti-holomorphic fields ϕ(z) andφ(z) used in CFT, and are involved in bosonization also (see Section 4).

Vertex operators of the field φ and its dualφ
Point sources (2) correspond to vertex operators e ie i φ(x i ) of well known Coulomb gas formalism for minimal CFTs [4,5]. Their well known correlation functions are read off from the master formula (4) and the regularized propagator (6), When R → ∞, only the net charge zero ( i e i = 0) correlation functions survive. Then, using i<j e i e j = − 1 2 i e 2 i , we can write the a-dependence in Eq. (35) as i a e 2 i /4π . So, to get finite correlations as the ultraviolet regulator a → 0, we 'renormalize' the vertex operators [9]: .
We may define vertex operators for the dual fieldφ in a similar manner: : e igφ(x) :: e −igφ(y) : = 1 The mixed correlation functions can be computed using the master formula (31). We need charge neutrality separately for field φ and dual fieldφ, for finite correlation function in R → ∞ limit. It is as if we have two Coulomb gas partition functions, except that cross-correlations are also present: with i e i = 0, k g k = 0.
Generally these correlation function have jumps in phase whenever a relative vector x i − y k tends to be parallel to the x 1 -axis (see Figure 1-b). We may alternatively interpret that the correlation functions are multivalued. The discontinuity is a reflection of the line discontinuity produced by the field φ(x) in the configuration space of its dualφ. The specific role of x 1 -axis is due to our choice of the solution of the Gauss' law constraint. This parallels the role of Dirac string in Dirac formulation of the magnetic monopole. Let e i = m i e, g k = n k g (where m i and n k are integers) with the Schwinger quantization condition Consider the lowest values m i = 1 and n k = 1. The corresponding phase in (39) is then exp[− 4π φ(x)φ(y) ], and equals e ±iπ across the x 1 -axis (refer to Figure 1-b). Thus the phase is continuous, the 'string' is no longer visible and the correlation functions are single valued.
We now briefly explain when we get the Schwinger quantization condition, and when the Dirac quantization condition eg = 2π. By introducing field φ in (17) and inverting the sym- (27). This is the analogue of the Schwinger potential, since the line discontinuity is distributed to both +ve and −ve x 1 axis ( as in Figure 1-b). Thus this is due to insisting on a symmetric quadratic form in φ andφ.
On the other hand, having only Θ( x − y) in (27) is the analogue of the Dirac potential, since it has the line discontinuity only across the +ve x 1 axis. This is obtained from the partition function in terms of the dual fieldφ alone, after adding a source forφ and integrating overφ.
This has been done in the Appendix. In this case the 'string' becomes invisible for the Dirac quantization condition eg = 2π.
Rotation covariance now manifests in an unusual manner. Though φ andφ were separately scalar fields, the mutual correlation functions (39) i.e., the dual field vertex operators e ±igφ(y) have acquired spin ± 1. Equivalently we may assign spin to e ±ieφ(x) .

Fermions from Bosons
In the previous section, we obtained vertex operators with spin from scalar fields. With eg 2π = 1, the spins were integer valued (in particular ±1 in (41)). In this section, we consider the case of half-integer quantization, viz eg 2π = 1 2 . In this case, the discontinuity in phase is π for the correlation functions (39), and so the line of discontinuity is now visible. Despite this, the case is of significance. We generate operators that have spin 1 2 . They change sign under rotation by 2π. Indeed in the case of e 2 = g 2 = π we recover the correlation functions of the free fermions [9,11].
The action for the free massless (Dirac) fermions is two Euclidean dimensions is: where ∂ ± = ∂ 1 ± i∂ 2 . Under rotation by an angle ω, The corresponding transformations on ψ ± (x) that leave the action (42) invariant are: Thus ψ + (x) andψ + (x) transform the same way by half the angle. (ψ andψ are independent Grassmann fields in the Euclidean functional integral). Thus ψ ± ,ψ ± change by a sign under full rotation by 2π and carry spin ± 1 2 . The propagators: Here x ± = x 1 ± ix 2 and θ( x) is the angle subtended by x with the x 1 -axis. We also have: All other correlation functions vanish: The 2n-point functions are obtained by Wick's theorem for fermions: · · · 1 (x n − y n ) + + · · · all possible permutations of y 1 , y 2 , · · · y n .
The right hand side of (48) contains n! terms and each term appears with ± sign depending upon the corresponding sign of permutation of (y 1 , y 2 , · · · y n ). Therefore, the above correlation functions are simply: This can be further written as the Cauchy determinant formula (see equation (12.195) of [5]): We have similar correlation functions for ψ − ,ψ − with x + , y + replaced by x − , y − .
All these correlation functions are reproduced by combinations of vertex operators of φ and the dual fieldφ 'close together' in the special case of To show this we consider the composite operators: This fact leads to the rotation property by half angle under θ( ) → θ( ) + ω just like ψ + (ψ − ) in (44). Under rotation by 2π, Ψ ± andΨ ± all change by a sign as θ( ) changes by 2π and sgn ( 2 ) does not change.
From (39) and (52) we get: Note that the phase factors in (52) have been chosen to cancel out the contribution of the selfcorrelation φ( x+ )φ( x) within each composite operator (compare (39)). Thus the correlations of Ψ defined as in (52) are insensitive to , and we drop this superscript henceforth. For the special case (51) we get: as e i π 2 sgn(x 2 −y 2 ) = isgn(x 2 − y 2 ). We can now reproduce the fermion propagator (45) with the identification: The correlator ψ + (y)ψ + (x) is identified as the negative of the right-hand side of (55), as ψ's are Grassmann variables.
The factor sgn(x 2 − y 2 ) provides the crucial anti-symmetric term to relate the bosonic correlation functions to fermionic ones. The Klein factor required in bosonization [12] comes from this factor. Consider the multi-fermion correlation function We can now reproduce (50), by identifying Let us first choose {x i } and {y i } such that x 1 2 > x 2 2 > x 3 2 > · · · > x n 2 and y 1 2 > y 2 2 > y 3 2 > · · · > y n 2 .
In this case, all sgn x i 2 − x j 2 = +1 for i < j and all sgn y k 2 − y l 2 = +1 for k < l. Then any other ordering of {x i } and {y i } on the left-hand side of (58) will change it by sgnP x sgnP y , where sgnP is the sign of the permutation P and sgnP x (sgnP y ) is the permutation which takes (x 1 , x 2 , · · · x n ) (respectively (y 1 , y 2 , · · · y n )) to the choice (59).
Finally, we note that the two fermion number conservations, i.e. conservation of ψ + and of ψ − , can be traced to the two global symmetries (18) of the bosonic fields φ(x) andφ(x) [11].
As a consequence of these symmetries, the correlations i e ie i φ(x i ) k e ig kφ (x k ) are zero unless i e i = 0 and k g k = 0. Thus these charge neutrality conditions are valid even for a finite area, though in Section 3 we obtained them by requiring infrared finiteness.

Discussion
Our aim in this paper is to develop techniques for handling a field and its dual on an equal footing. We have both fields present in a local, self-dual action. The dual field is interpreted in terms of line discontinuities in the configurations of the original field (or vice versa). We have emphasized that the dual field explores new configurations of the original field. In many contexts the dual field and also composite operators made of both fields play an important role.
Here we illustrate this in the simple case of a free massless real scalar field in two Euclidean dimensions. In this case it is related to Hamiltonians useful in one-dimensional condensed matter physics.
Our local action is not manifestly invariant under rotations. The underlying reason is the specific choice of the line discontinuity to be made. This also gives unusual features of the correlations of the field with its dual. Even though the correlation of either field with itself is as expected for scalar fields, the mutual correlations have a dependence on the orientation. This leads to the generation of spin from scalar fields. This is in analogy with the Saha-Wilson contribution of the electromagnetic field to the angular momentum of a charge-monopole system. For correlations of vertex operators rotation covariance is recovered at the cost of assigning a spin to these operators. Demanding single valuedness of the correlaton functions, or equivalently, invisibilty of the line discontinuity, gives quantization condition for the charges of the field and the dual field. This parallels Dirac's argument for a charge-monopole system. The dual field is non-locally related to the conjugate momentum. This is the reason that even with double the number of fields, we are not introducing new degrees of freedom but only reinterpreting the original model. Even though we considered a free theory here our framework can accomodate interacting theories as well. The crucial point is that the dual field is realted to the conjugate momentum and therefore involves only the kinetic energy part of the action and not the potential energy part. Thus for the theory L = 1 2 (∂ i φ) 2 + V (φ) our local theory with both the field and its dual is simply: The presence of V (φ) makes important differences. In order that the action be finite in the presence of the line discontinuity, φ can only take the values corresponding to the minima of V (φ). For V (φ) = λ(φ 2 − v 2 ) 2 , these are only ±v and not arbitrary as in the absence of V (φ).
Also there is only one global invarianceφ →φ +σ and only one conserved fermion number. We also want to emphasise that our considerations are not restricted to two dimensions or scalar theories. We can handle even non-abelian gauge theories in higher dimensions. This will be explored elsewhere.

Appendix
On the choice of the line of discontinuity In Section 2, when solving the Gauss law (9), we chose the line of discontinuity to be along the x 1 axis. We could have chosen an arbitray path to carry the conserved flux. The general solution of the Gauss law constraint is: where brings the flux from X(−∞) to y along an arbitrary path X(τ ) parametrized by a parameter τ . As y is the end point, X(0) = y. Indeed, In the last step above we have taken the other end point X(−∞) of the flux line to be located outside the region of interest. For different point sources located at y r (r = 1, 2, · · · N ) we may choose different discontinuity lines X r (τ ) ending at y r . For a distributed source ρ(x), with the boundary conditions X r (τ = 0) = y r . Using (60) in (8) and introducing source term forφ we get: (x) (x−y)ρ(y)+iρ(x) ( x− y)(∂ 1 β 2 (y)−∂ 2 β 1 (y))) .
We also have  (68) where dθ is the angle subtended by d X at x, as in Figure 2, with the following sign convention: dθ is positive (negative) according as the angle between ( X − x) and d X (measured in the anticlockwise sense from the former to latter) is > π (< π). This convention is shown in Figure 3.
Thus (67) becomes: where is the net change in the angle θ along the infinite path P. This effectively measures the angle from the asymptote of the path. If we choose the negative x 1 -axis along this asympote, then (70) is the same as Θ( x− y) of Section 2.4. We want to point out that here we got only Θ( x− y), and not the symmetric combination 1 2 (Θ( x − y) + Θ( y − x)) as in (27), because we have not explicitly introduced the field φ and integrated over a symmetric quadratic form involving φ andφ.