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Generation of Curvature Continuous Trajectories for Transport Aircraft using Bezier Curves

With the increase in air traffic, better management and organization of it is crucial for increasing traffic safety and airspace capacity. Hence the need of ad-hoc descriptions of more complex and flexible aircraft trajectories, allowing high traffic densities and limiting environmental impact. The approach consists of a smooth 4D path generation from pre-existing control points by stitching several Bezier curves while assuring G 2 continuity at the joints. Also, as the Euclidian distance between control points and the proposed trajectory is controlled by an optimal reshaping of the trajectory, the tradeoff between curvature-speed-load factor of the trajectory is considered. The generated trajectories are meant to be a complement to regular flight plans, helping to solve air traffic conflicts and enhance air capacity by better timing. Matlab simulations corroborate the feasibility of the approach, showing promising results when a distance range is defined for the reshaping algorithm.

INTRODUCTION

Nowadays, even aircraft with operating ACAS/TCAS are exposed to near mid air collisions, or in worst case scenarios, to events like the one occurred on july 1 over Überlingen, Germany in 2002, where two transportcategory aircraft collided after TCAS instructed one pilot to climb, but the pilot descent in compliance to air traffic controller instructions. Moreover, even if only 1 in 10 Traffic Advisory (TAs) provided by TCAS result in a Resolution Advisory (RAs), this event and others1 , are clear examples of human error and lack of homogeneity in TCAS. As both SESAR and NextGen projects plan to implement new operational concepts which will reduce the spacing between aircraft. TCAS in its current form is not compatible with such concepts and would alert too frequently to be useful. Consequently, in order to allow aircraft to fly closer, quick and efficient 4D trajectory generation when needed is crucial. In this manner, air traffic conflicts can be avoided or addressed if present, taking into account load factor limits to look after passengers comfort. The generation of flyable and efficient trajectories has been considered by several authors [START_REF] Bakolas | Initial guess generation for aircraft landing trajectory optimization[END_REF][START_REF] Anderson | Real-time dynamic trajectory smoothing for unmanned air vehicles[END_REF][START_REF] Judd | Spline based path planning for unmanned air vehicles[END_REF][START_REF] Yang | Generation of dynamically feasible and collision free trajectory by applying six order bezier curve and local optimal reshaping[END_REF][START_REF] Yang | An analytical continuous-curvature path-smoothing algorithm[END_REF][START_REF] Delahaye | Mathematical Models for Aircraft Trajectory Design: A survey[END_REF]. From a general view, reference trajectories are essential for flight plans which meet a large set of overfly or profile constraints, which vary in general from flight to flight. On the other hand, the generation of reference trajectories at short term should allow the implementation of procedures in the case of potential conflicts. Current path generation for transport aircraft is based on a sequence of objective points in a 2D/3D space given to connect two geographical locations [START_REF] Walter | Digital Avionics Handbook[END_REF]. Then, using diagrams like Voronoi, or Delaunay triangulation, or any other method, a piecewise path is constructed using straight lines denominated legs. This path may be offered by a high-level path planner from techniques such as Dijkstra's, A*, probabilistic roadmaps, genetic algorithms, or Rapidly exploring random tree star (RRT*), just to mention a few. However, as path differential requirements have to be satisfied, the initial trajectory needs to be reshaped in order to provide a flyable trajectory for transport aircraft. The problem of designing a flyable path over or close to the objective points while satisfying constraints such as maximum curvature and/or G 2 continuity (curvature continuity), has been addressed using different approaches. As a pioneer, Dubins assured that the shortest path between two points involves circles and straight line path segments [START_REF] Dubins | On curves of minimal length with a constraint on average curvature, and with prescribed initial and terminal positions and tangents[END_REF]). However, curvature continuity at the joints of lines and circular arcs is not satisfied. Techniques to solve this track transition problem are addressed using Clothoids [START_REF] Scheuer | Continuous-curvature path planning for car-like vehicles[END_REF], but as they do not have closed-form expressions, the computation complexity is increased. Circular arcs as transitions tracks have been also proposed [START_REF] Anderson | Real-time dynamic trajectory smoothing for unmanned air vehicles[END_REF], where a 2D real-time trajectory is generated satisfying curvature and velocity constraints. Also, the deviation between the generated circular path and the associated control points is minimized using a parameter κ ∈ [0, 1]. If κ = 0, the distance is zero, so the plane will fly exactly over the waypoint, and if κ = 1, minimum-time transitions between control points are achieved. However, the election of this parameter κ becomes an issue when a range of distance wants to be elected as a permitted deviation from the generated path. Dubins' approaches generate paths limited to straight lines and arcs of circles without parametrization. Thus, the method shows disadvantages being able to generate several arcs without curvature continuity or with high computation complexity. Another tool to generate smooth flyable paths are splines, defined by series of low order polynomials. In [START_REF] Judd | Spline based path planning for unmanned air vehicles[END_REF], after finding the best path from a UAV position location to a target location using a Voronoi diagram and a Dijkstra's algorithm, the 2D path is smoothed using cubic splines. It is worth to say that the optimal locations of the middle knots of a spline are crucial for the shape of the segments. However, the optimal computation of these points is very time-consuming unless a set of cases are defined a priori. As an attractive approach, Bezier curves are able to generate continuous-curvature paths, having the advantage of passing through initial and final points while the whole curve always remains within the convex hull that is built by the control points. Also, the initial and final piecewise straight lines of the control polygon are always tangent to the Bezier curve at the starting and ending control points. An example is given in [START_REF] Lee | Planning and control for collision-free cooperative aerial transportation[END_REF], where the authors present the generation of a 3D path obtained from a combination of Rapidly exploring random tree star (RRT*) using a C 2 class Bezier curve, and Dynamic Movement Primitives (DMP's), that allow cooperative aerial manipulators to avoid known and unknown obstacles. As the final Bezier curve is a barycentric combination of the polygon vertices formed by the control points, it may be thought that a trajectory could not be accurately generated if the control points positions are zig-zags like. However, this aspect can be handled by stitching several Bezier curves to form a bigger path while assuring G 2 continuity at the joints. In [START_REF] Yang | Generation of dynamically feasible and collision free trajectory by applying six order bezier curve and local optimal reshaping[END_REF], after a Guiding Attraction based Random Tree (GART) is used to obtain control points in a 2D plane, a kinematic smoother based on sixth order Bezier curves to achieve second derivative continuity (curvature continuity) is proposed. Finally, a local optimal reshaping of the path, minimizing length and curvature cost is performed. In [START_REF] Yang | An analytical continuous-curvature path-smoothing algorithm[END_REF], it is presented an algorithm based on cubic Bezier curves for 3D path smoothing, satisfying G 2 continuity and maximum curvature constraints, where instead of addressing a direct solution for the 3D path smoothing, a 2D path smoothing for consecutive triplets of control points is applied, considering each triplet as a 2D plane; Nevertheless, deviation from the resulting trajectory w.r.t. the control points is not controlled. In this paper, assuming velocity/time constraints, a time-parametrized smooth trajectory is developed by stitching several Bezier curves while assuring G 2 continuity at the joints, also, the Euclidian distance between the control points and the proposed trajectory is controlled, yielding an optimal 4D trajectory valid for 4D guidance. Finally, since flyable trajectories depend heavily on the load factor, the tradeoff between curvature-speed-load factor is analysed.

TRAJECTORY GENERATION

2.1 Bezier Curves and G 2 Continuity A Bezier curve P (s) of degree n, obtained from n + 1 control points (P 0 , . . . , P n ), is described with its derivatives by

P (s) = n i=0 B n i (s)P i s ∈ [0, 1] (1)
P (s) = n-1 i=0 B n-1 i (s)Q i (2) P (s) = n-2 i=0 B n-2 i (s) (n (n -1) (P i+2 -2P i+1 + P i )) (3)
where Q i = n (P i+1 -P i ) and B n i (s) is the i th Bernstein polynomial of degree n, given by

B n i (s) = n i s i (1 -s) n-i i ∈ {0, 1, . . . , n} (4)
n i = n! i! (n -i)! (5)
Note that

P (0) = P 0 P (1) = P n (6) P (0) = n (P 1 -P 0 ) P (1) = n (P n -P n-1 ) (7)
P (0) = n (n -1) (P 2 -2P 1 + P 0 ) (8)
P (1) = n (n -1) (P n -2P n-1 + P n+2 ) (9)
According to [START_REF] Barsky | Geometric continuity of parametric curves[END_REF], G 2 continuity is the second-order geometric continuity, implying the second derivative continuity of two curves at the joint. Moreover, if P (s) = (x(s), y(s), z(s)) is a regular parametrization of a curve in an interval I = [a, b] → R 3 . An expression for the curvature is obtained by:

κ(s) = P (s)×P (s) P (s) 3 ≥ 0. If κ(s) = 0, P (s) is a straight line.
Otherwise, the curve will have a curvature radius of 1/κ(s). Thus, let a Bezier curve C(s) with m + 1 control points (C 0 , . . . , C m ), and a second Bezier curve D(s) with n + 1 control points (D 0 , . . . , D n ) be joined. C 0 continuity is guaranteed if:

C m = D 0 (10)
A smooth transition is assured if ( 10) is satisfied and C m , D 0 , C m-1 , D 1 are on the same line. Furthermore, C 1 continuity is guaranteed if the tangent vector of the first curve at s = 1 is identical to the tangent vector of the second curve at s = 0, meaning that:

C (1) = m(C m -C m-1 ) = n(D 1 -D 0 ) = D (0) (11)
This states that the ratio Cm-C m-1 D 1 -D 0 must be n m . Since n and m are fixed numbers, the positions of C m-1 and D 1 can be rearranged to be not only at the same line, but also at the proper distance to assure C 1 continuity. In the same tenor, C 2 and G 2 continuity are guaranteed at the joint if it is verified that

C (1) = m(m -1)(C m -2C m-1 + C m-2 ) = n(n -1)(D 2 -2D 1 + D 0 ) = D (0) (12)
Hence, to assure G 2 continuity, the locations of

C m , C m-1 , C m-2 , D 0 , D 1 , D 2
, where (10)-( 12) are satisfied, need to be proposed.

Path Generation and Time-parametrization

In this paper, the 3D path is decomposed into several 2D planar trajectories laying on the tangent/normal plane of the frenet frame, similar to [START_REF] Yang | An analytical continuous-curvature path-smoothing algorithm[END_REF]. Since three control points are required to form a plane, at least three control points are defined, and then divided into triplets. Several quintic Bezier curves, one for each triplet of control points, are stitched together to form a bigger path while assuring G 2 continuity at the joints. For a triplet of control points (P 1 , P 2 , P 3 ), G 1 continuity can be achieved by interpolating four points

(Q 0 , Q 1 , Q 4 , Q 5 )
, and for G 2 continuity, six points are interpolated (Q i ; i ∈ {0, . . . , 5}). The points are interpolated as follows: Q 0 and Q 5 are defined to be at the middle point of ( P1 P 2 ) and ( P2 P 3 ) respectively, forcing a past and following Bezier curve formed by the past and next triplet of control points, to finish and start at Q 0 and Q 5 . In other words, a next triplet of points conformed by (P 2 , P 3 , P 4 ) will have a Bezier curve starting at Q 5 , so (10) is fulfilled. For Q 1 and Q 4 , they are computed to be also in the same line of ( P1 P 2 ) and

( P2 P 3 ) respectively, but separated by a δ 1 distance from Q 0 in the case of Q 1 , and a distance δ 2 from Q 5 in the case of Q 4 . Note that the number of auxiliary interpolated points to compute the Bezier curves are the same for all the triplet of points, so m and n from (11) are equal. Hence, to fulfill G 1 continuity, only remains to guarantee that the distance δ 2 of a certain triplet of control points is equal to the distance δ 1 of the next triplet of control points. Finally, if the positions of Q 2 and Q 3 are computed also in the same line of ( P1 P 2 ) and ( P2 P 3 ) respectively, separated by the same distances δ 1 from Q 1 , and δ 2 from Q 4 , if it is proposed: δ 1 = P1 P 2 4 , and δ 2 = P2 P 3 4 , equations ( 11), ( 12) are satisfied. Besides, knowing that the first and last Bezier curves conforming the total path will start and end at the middle point of their corresponding control points, as straight lines have zero curvature, the path can be completed with straight lines without affecting the G 2 continuity. After the interpolation of these auxiliary control points, a quintic Bezier curve is adjusted to them in the case of the G 2 continuity path. As several Bezier curves conform the total path, and each Bezier curve is parametrized by s ∈ [0, 1], a timeparametrization needs to be done for the creation of a flyable path for transport aircraft. This re-parametrization consists in perfom the operation: s = t-ti ti+1-ti where i ∈ [0, 1, ..., B n ], considering B n as the number of existent Bezier curves, such that the multiple Bezier curves parametrized by s ∈ [0, 1], can be used at their proper time intervals [t i , t i+1 ]. Note that the times in which each Bezier curve is used need to be known to paramterize the curve, therefore, the velocity for the aircraft to follow the path, or as an alternative way, the timestamps in which the aircraft is supposed to fly over the joints of the Bezier curves, are defined in advance, thus, the computation of the arc length of each Bezier curve is inherent.

OPTIMAL RESHAPING AND LOAD FACTOR OF THE TRAJECTORY

Depending on the application, some aircraft may need to fly directly over the control points or at least near a defined range of distance from the control points. In order to achieve this demand, the Euclidian distance from the G 2 continuity path w.r.t. the nearest control point is controlled. For this special case, an extra auxiliary point is computed such that an optimum path based on sixth order Bezier curves is obtained. The other auxiliary control points are computed as before in order to do not affect curvature continuity, but the position of the heptic auxiliary point is moved gradually until the generated path passes within a distance range defined by the user. Considering that (Q 0 , Q 1 , Q 2 , Q 4 , Q 5 , Q 6 ) are the necessary points to achieve G 2 continuity, the auxiliary point (Q 3 ) is moved away in the direction of:

aux = (Q 2 -Q 1 ) + (Q 4 -Q 5 ) (Q 2 -Q 1 ) + (Q 4 -Q 5 ) (13)
As transport aircraft are designed to flight smooth trajectories, some limitations on the load factor have to be respected for the well being and comfort of passengers. According to (FAA 2017), on a typical flight, the load factor is limited to +2.5g and -1g for regular maneuvers. The expressions relating the load factor and speed of a turning/pitching aircraft, with the horizontal path of radius R, and radius of vertical curvature R are:

R = V 2 a g n 2 z -1 q = (n z -1) g V a = V a R ( 14
)
As the path has a curvature radius of 1/κ, the curvature κ of the path is directly related to ( 14). In this manner, for independent lateral or vertical maneuvers, the path that an aircraft can follow without infringing load factor limits can be computed, taking into account the speed of the aircraft and the curvature of the path generated by the position of the control points. Then, path constraints regarding to maximum curvature can be established for different flyability requirements based on load factor limits. According to ( 14), maximum radius of curvature for circular motions can be computed for different speeds after defining load factor limits. For example, a 2.5g load factor at 200m/s is generated by a circular trajectory of a radius of 1,779.55m, equivalent to a roll angle of 66.5 o , which is also the maximum roll angle permitted for transport aircraft.

SIMULATION RESULTS

In Figure 1, a G 1 /G 2 continuity path for a triplet of points (P 1 , P 2 , P 3 ) is shown. The locations of the auxiliary points is according to Subsection 2.2, as well as the straight lines in the extremes to complete the path without affecting the curvature continuity. Consequently, consider that some control points emulating the avoidance of potential threats are given in Table 1. They generate a G 1 and G 2 path depicted in Figure 2. Regarding to the joints of the G 2 Bezier curves, and assuming that the path is intended to be followed at 200m/s velocity, Figure 2 shows the arc lengths (l i ; i ∈ [1, 6]) and their times (t j ; j ∈ [0, 6]) assigned. The values of l i and t j are given in Table 2. Then, comparing the curvatures of the G 1 and G 2 continuity paths (Fig. 3), it is clearly seen how the G 2 path has continuity in the curvature at the joints of the different Bezier curves, and that these joints occur at the assigned time, represented by asterisks in the correspondent axis. Concerning to the optimal reshaping of the path, in Figure 4, a 100m maximum deviation for a triplet of control points is commanded, meaning that the optimized path will be generated such that it will pass no further than 100m away the control point P 2 . In this case, the optimizing algorithm stopped at a distance of 92.48m. The distance at which the algorithm stops depends on the step size in which the extra auxiliary control point (Q 3 ) is moved away in the direction of aux. For a small step size, the path will be closer to the distance defined by the user, in this case 100m, but the computation time will be increased. Now, for a complete trajectory, using the control points of Table 1, a 100m maximum deviation is commanded, the resultant path is shown in Figure 5. The Euclidian distance at which the trajectory is generated from the closest control points is given by Table 3. Assuring that the maximum deviation distance is achieved. To better exemplify the relation of the load factor w.r.t. the velocity, lets propose a didactic scenario of the trajectory of an aircraft changing airways by changing its heading. Three control points located at the same altitude and separated by 50 nautical miles (nm) from each other are used (Table 4).

The control points are shown in Figure 6, along with the computed Bezier curves of G 2 continuity. Also, the arc lengths of each curve and the points where the Bezier curves are joined, denoted by their timestamp, are shown. Now, lets consider three different speeds for an aircraft to go through these control points, the numerical values of the timestamps where the Bezier curves are joined change according to Table 5. Also, the transition between legs will generate different load factors depending on the speed, shown in Figure 7. On the other hand, independent from the speed, the curvature of the path remains constant. Once that the relation of the loadfactor and velocity is stated, lets explore the curvature constrains due to load factor limits keeping a constant velocity. In the example above, the distance from the trajectory to the point P 2 is 7,161.66m, so, lets propose the trajectory of an aircraft turning closer to P 2 with a constant velocity of 200m/s. If the optimal reshaping of Section 3 is asked to modify the trajectory such that a maximum of 10m deviation from P 2 is achieved, the load factor remains within bounds. This scenario is depicted in Figure 8, where a zoom to P 2 shows how the generated trajectory is bended before and after the control point to satisfy the maximum deviation constraint.

The load factor of the generated trajectory reaches a pike of ≈1.425g and a curvature of ≈2.5x10 -4 (m -1 ). The distance from this trajectory to P 2 is 1.36m. Consequently, in order to have a trajectory close to the control point P 2 while increasing significantly the load factor, two auxiliary control points are added (see Table 6), such that the direction of the path is not bended, like in the case when the optimal reshaping is done. A zoom close to the control point P 2 shows the generated trajectory, which passes at a distance of 1,203.15m from P 2 (Fig. 8). Under this conditions, the load factor reaches the 2.5g limit. The curvature of the path pikes at ≈5.6x10 -4 (m -1 ), close enough to the 1,779.55m radius limit stated at the end of Section 3. Thus, if an aircraft at 200m/s is commanded to pass through "L" shape control points, a single turn maneuver leaves the aircraft at 1,203.15m from the middle control point before infringing load factor limits, on the other hand, the proposed reshaping algorithm generates a flyable trajectory at 1.36m from the same point. Numerical computations can be done for any other velocity and the pitching motion.

CONCLUSIONS

As the curvature of the path is forced to be zero at the joints between Bezier curves, G 2 continuity is assured. As the time interval for each Bezier curve needs to be known for the time-parametrization, velocity/time constraints are assumed, leading to an inherent computation of the arc length for each Bezier curve. The closer the optimal path is to the control points, the bigger the curvature becomes, such that a maximum curvature is defined and controlled by the deviation distance defined by the user, establishing a tradeoff between curvature-speed-load factor of the trajectory. This approach for 4D smooth trajectories generation, opens the way to 4D guidance along the time-parametrized Bezier curves under path and flyability constraints. The adoption of these trajectories should ease air traffic management in congested areas, helping to solve air traffic conflicts and improving on-board guidance systems performance.

Figure 1 :

 1 Figure 1: Bezier curve completed with straight lines.

Figure 2 :

 2 Figure 2: Arc lengths and joints of Bezier curves.

Figure 3 :

 3 Figure 3: Curvature of the G 1 and G 2 continuity paths.

Figure 4 :

 4 Figure 4: 100m deviation optimized path using Q 3 .

Figure 5 :

 5 Figure 5: 100m deviation optimized path.

Figure 6 :

 6 Figure 6: Control points of example with Arc lengths and Times.

Figure 7 :

 7 Figure 7: Load factor at different velocities.

Figure 8 :

 8 Figure 8: Trajectories close to P 2 with a 10m maximum deviation and forcing a maximum load factor.

Table 1 :

 1 Control Points.

	X(m)	Y(m)	Z(m)
	1 0	0	10,000
	2 120,843 16,983	9,300
	3 210,332 -14,779 9,000
	4 272,744 -759	8,200
	5 388,920 -11,130 9,500
	6 478,501 12,964	9,800

Table 2 :

 2 Time and Arc length.

	Arc length(m) Time(s)
			t 0	0
	l 1	61,016	t 1	305.1
	l 2	107,536	t 2	842.8
	l 3	78,523	t 3	1,235.4
	l 4	89,990	t 4	1,685.3
	l 5	104,206	t 5	2,206.4
	l 6	46,383	t 6	2,438.3

Table 3 :

 3 Distance of trajectory from control points.

	From	P 2	P 3	P 4	P 5
	Initial distance (m)	2800.77 2351.04 1449.10 2010.67
	Optimized distance (m)	92.48	99.72	96.09	79.53

Table 4 :

 4 Control points for an aircraft changing airways.

	1nm=1852m	X(m)	Y(m)	Z(m)
	P 1	0	92,600 10,000
	P 2	92,600 92,600 10,000
	P 3	92,600	0	10,000

Table 5 :

 5 Arc lengths and times at different velocities

	Arc length(m)		Time(s)
				170m/s	200m/s 230m/s
			t 0	0	0	0
	l 1	46,300	t 1	272.35	231.5	201.30
	l 2	84,571	t 2	769.83	654.35	569
	l 3	46,300	t 3	1,042.18	885.85	770.3

Table 6 :

 6 Auxiliary control points to force a high load factor

		X(m)	Y(m)	Z(m)
	P aux1	77,043.2	92,600	10,000
	P aux2	92,600	77,043.2 10,000

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We explain, in a slightly modified form, an old construction allowing to reformulate the U (N ) gauge theory defined on a D-dimensional lattice as a theory of lattice strings (a statistical model of random surfaces). The world surface of the lattice string is allowed to have pointlike singularities (branch points) located not only at the sites of the lattice, but also on its links and plaquettes. The strings become noninteracting when N → ∞. In this limit the statistical weight a world surface is given by exp[ -area] times a product of local factors associated with the branch points. In D = 4 dimensions the gauge theory has a nondeconfining first order phase transition dividing the weak and strong coupling phase. From the point of view of the string theory the weak coupling phase is expected to be characterized by spontaneous creation of "windows" on the world sheet of the string.

Talk delivered at the Workshop on string theory, gauge theory and quantum gravity, 28-29 April 1993, Trieste, Italy August 1993

Introduction

The concept of gluons as elementary excitations in QCD is adequate only at distances much less than the correlation length of the theory (the typical hadron radius). Because of the dimensional transmutation, the effective gauge coupling constant seizes to be a small parameter in the infrared domain and the standard perturbation theory collapses at distances comparable with the hadron radius.

An idea which is perhaps as old as QCD is that infrared-stable elementary excitations are described by fluctuating surfaces or strings. The Wilson's strong coupling expansion [1] and the 1/N expansion of 't Hooft [2] are the two basic arguments in favour of the possibility of such a description.

From the point of view of the 1/N expansion, the U (N ) gauge theory resembles a theory of strings with interaction constant 1/N and its multicolor limit N → ∞ is expected to be described by some kind of a noninteracting string. The formation of strings at large distances can be seen as a condensation of dense planar diagrams in the continuum theory (fig. 1). However, the only unambiguous evidence for the relevance of strings in the perturbation theory is the topology of the Feynman diagrams. Because of the divergencies and the need of gauge fixing, it is not easy to make this statement more rigorous. On the other hand, there is the strong coupling expansion in lattice gauge theories [1]. The gauge theory is discretized so that the gauge field is associated with the links =< xx > and the action is a sum over the elementary squares (plaquettes) of the lattice. From the point of view of the strong coupling expansion the lattice gauge theory is statistical mechanics of random surfaces with contact (self)interactions. In the lowest orders of the inverse coupling β = 1/λ, the Wilson loop average is given by the sum over lattice surfaces spanning the loop. In high orders, the dominant contribution comes from overlapping surfaces subjected to strong contact interactions and forming new topologies, and the original string picture is lost.

Some years ago, it has been suggested that the large N limit of the strong coupling expansion might correspond to a theory of noninteracting surfaces. In order to study 1 this possibility, the most commonly used character expansion was not the most efficient mathematical formalism because it produces diagrams for the partition function and not directly for the free energy. In spite of the considerable efforts [3] no evidence for a free string picture was obtained 1 .

An alternative formalism was then suggested by V. Kazakov [6]. The idea of [6], borrowed from the Stanley's solution [7] of the large-N vector model, was to reformulate the U (N ) lattice gauge theory as a kind of Weingarten theory with additional interactions. The Weingarten model [8] is a simple (but rather pathological) gauge theory which is equivalent to a model of noninteracting random surfaces in the large N limit. In the U (N ) lattice gauge theory one can impose the unitarity of the link variable U by means of a Lagrange multiplier and replace the Haar measure on the U (N ) group by a Gaussian measure over complex matrices. The idea of [6] has been accomplished in refs. [9], [10].

In ref. [11] we suggested the possibility that the strong coupling lattice surfaces and planar Feynman-'t Hooft diagrams appear as two extreme cases of the same perturbative expansion. This means that the strong coupling lattice string is perhaps not completely irrelevant to the continuum limit of QCD.

Here we explain the construction of ref. [11] using the standard formalism of the mean field analysis in gauge theories [12]. Our random surface Ansatz will be defined for any choice of the one-plaquette action. In D = 2 dimensions it reproduces the continuum limit of the gauge theory in spite of eventual Gross-Witten transition [13] separating the strong and weak coupling domains. However, the first-order phase transition in D = 4 dimensions is likely to persist and the sum over surfaces has to be modified in the weak coupling phase by allowing windows (free boundaries) to appear on the world sheet. In the continuum limit the world sheet of the string is tore into a network of thin strips which can be interpreted as gluon propagators.

We start the story with a review of the Stanley's solution [7] of the large-N vector model which may help the reader to understand the logic of our construction before dipping into the technical details. The analogy with the vector model can help us to get some more intuition and avoid the rudest conceptual errors in the case of the gauge theory.

The U (N ) vector model

The fluctuating field in this model is an N -component complex vector u x = (u 1

x , ..., u N x ) defined at the points x of the D-dimensional hypercubical lattice and having unit norm

u * x • u x ≡ N i=1 |u i x | 2 = 1 (2.1)
We assume a nearest neighbour interaction βu * x • u y associated with the links =< xy > of the lattice. Then the measure

[du] = N i=1 dū i du i δ( N i=1 |u i x | 2 -1) = du * du δ(u * • u -1) (2.2)
and the interaction are invariant under global U (N ) transformations

u x → Uu x ; U ∈ U (N ) (2.
3)

The partition function

Z = e N F = x [du x ] =<xy> e N βu * x •u y (2.4)
can be calculated order by order in the inverse coupling constant β by expanding the exponential as a series of monomials (strong coupling expansion). This leads to a diagram technique which in the limit N → ∞ involves tree-like clusters of loops and can be summed up explicitly in all orders.

Alternatively one can transform, by introducing auxiliary fields, the integral over the original vector field into a duable Gaussian integral and find an effective field theory in which the number 1/N plays the rôle of a Planck constant.

Introducing a new complex vector field h = (h i , i = 1, 2, ..., N ) we represent the unimodular measure as a flat measure for the unconstrained complex vector field v = (v i , i = 1, ..., N ). We insert the exponential representation of the δ-function

δ(u j -v j ) = 1 2πi i∞ -i∞ dh j e hj (u j -v j ) (2.5)
in the definition (2.4) of the partition function for every component u j x of the vector field. Here h and v are unrestricted complex vectors on which the integration is performed with a flat measure

dv = N j=1 dv j dv j , dh = N j=1 dh j d hj (2.6)
Denoting by

e N F (h * •h) = [du]e N [u * •h+h * •u] (2.7)
the one-site mean-field integral we write the partition function (2.4) in the form

Z = e N F = x dh x dv x e -N (h * x •v x +h x •v * x ) x e N F (h x ) =<xx > e N βv x •v x (2.8)
The potential F (α) is determined by the Ward identity ( i N -2 ∂/∂h i ∂/∂h * i -1)e N F = 0 or, in terms of the α-field,

( 1 N ∂ ∂α + α 1 N 2 ∂ 2 ∂α 2 -1)e N F (α) = 0 (2.9)
which is solved, in the large N limit, by

∂F ∂α = √ 1 + 4α -1 2α
F (α) = √ 1 + 4α -1 -ln 1 + √ 1 + 4α 2 = ∞ n=1 f n α n n (2.10)
Using the integral representation

e N F (h * •h) = dαd αe N [ α(h * •h-α)+F (α)] (2.11)
we finally write the partition function in a form allowing to perform exactly the integral over the vector fields

Z = e N F = x dα x dα x e N [F (α x )-αx α x ] dv x dh x e N [-v * x •h x -h * x •v x ] x e N [ αx h * x •h x ] =<xx > e N βv * x •v x (2.12)
Denoting by Ĉ the connectivity matrix of the D-dimensional hypercubical lattice

Ĉ = x D µ=1 (e ∂/∂x µ + e -∂/∂x µ ) (2.13)
we can write the result of the gaussian integration over the v, h fields as

e N F = x dα x dα x e N [F (α x )-αx α x ] e N F 0 [ α] ≡ e N F 0 [ α] α (2.14)
where

F 0 [α] = - x tr log(1 -β α Ĉ) (2.15)
is the sum over vacuum loops of the vector fields. Formally the sum over vacuum loops can be obtained by expanding the exponential in the last line of (2.12) as a sum of monomials and applying all possible Wick contractions

h * i x v yj = δ x,y δ i j , v * i x h yj = δ x,y δ i j (2.16)
The vertex αx h * x • h x is represented by a node at the point x and contributes a factor αx to the weight of the loop. The vertex βv *

x v y is represented by a segment of the loop covering the link =< xy > and contributes a factor β. The weight of each loop is therefore a product of local factors β and αx associated with its nodes and links, correspondingly. Denoting by Γ any closed loop on the lattice, the r.h.s. of (2.15) can be written as

F 0 [h * ] = Γ β #links x∈Γ αx
(2.17)

The action for the fields α, α is proportional to N and in the limit N → ∞ these fields freeze at their expectation values determined by the classical equations of motion

α = ∂F (α) ∂α = √ 1 + 4α -1 2α (2.18) α = ∂F 0 (α) ∂ α = β Ĉ 1 -αβ Ĉ (2.19)
or, equivalently,

α + α α2 = 1, α 1 -β α Ĉ = 1 (2.20)
The last equation has a simple geometrical interpretation in terms of a random walk on the lattice. It can be obtained also directly from the unimodularity condition u * x • u x = 1. Consider the two-point Greens function G xy = u * x • u y . Repeating the same steps as for the partition function we find

G xy = v * x • v y = x| α 1 -β Ĉ |y = Γ xy α#nodes β #links (2.21)
Therefore the second of the equations (2.20) is just the condition

G xx = 1. (2.22)
We see that the original vector field u x acquires a longitudinal degree of freedom and instead of 2N -1 we have 2N degrees of freedom at each point x. These describe N identical noninteracting particles with dynamically generated mass. The constraint u * x • u x = 1 is imposed on dynamical level by the condition (2.22) which determines the mass gap of the model. After diagonalysing the connectivity matrix we write it as an integral in the momentum space

π -π 1 m 2 + D µ=1 [2 sin(p µ /2)] 2 = β (2.23) m 2 = 1 αβ -2D (2.24)
The continuum limit (infinite correlation length) is achieved when m 2 → 0, i.e., 2Dβ α = 1.

Let us examine more closely this equation depending on the dimension D.

D ≤ 2

When D ≤ 2, the integral is infrared divergent when m → 0 and gives

β = d D p (2π) D 1 p 2 + m 2 = m D-2 , D < 2 (2.25)
In the marginal case D = 2 we find the asymptotic freedom law for the dependence of the renormalized coupling β = β(m) as a function of the correlation length 1/m

β = 1 4π ln 1 m 2 (2.26) 2.2. D > 2
Above two dimensions the integral is convergent at m → 0 and the limit of zero mass is achieved at finite value of β = β c .

In the weak coupling phase β > β c the equations of motion (2.23) have no solution which means that something in our construction is wrong. This contradiction is resolved if we admit that the vector fields acquire nonzero vacuum expectation values v x = v cl , h x = h cl determined by the saddle-point equations

h cl = β Ĉv cl = 2Dβv cl , v cl = αh cl (2.27)
which are solved by h cl = v cl = 0 ( strong coupling U (N )-symmetric phase) or by m 2 = 1 -2Dβ α = 0 with h cl , v cl = 0 ( weak coupling phase with spontaneously broken U (N ) symmetry). Geometrically the weak coupling phase is characterized by possibility of the the random walk to break into two pieces. Then the two-point Greens function is a sum of connected and disconnected parts

G xy = v 2 cl + x| α 1 -αβ Ĉ |y (2.28)
The unimodularity condition now reads

v 2 cl + 1 β d D p (2π) D 1 D µ=1 (2 sin(p µ /2)) 2 = v 2 cl + β c β = 1 (2.29)
Thus in the weak coupling phase there is a spontaneous magnetization v cl = 1 -β c /β < 1 and 2N -1 Goldstone excitations with zero mass due to the spontaneous breaking of the U (N ) symmetry. The critical value β c is related to the probability Π 0 that a Brownian walk returns to the origin [START_REF] Drouffe | Statistical Field Theory[END_REF]

2Dβ c = 1 1 -Π 0 (2.30)
and its large behaviour is given by

2Dβ c = 1 + 1 2D + 3 1 2d 2 + ... (2.31)
The series diverges at D = 2 and as

D → 2, β c → ∞ 2Dβ c ∼ 2 π(D -2) (2.32)
and below the lower critical dimension D = 2 no transition occurs. In D < 2 there exists only a high temperature symmetric phase.

D = 1

The D = 1 vector model has no dynamical degrees of freedom and is analogous to the D = 2 gauge theory.

In terms of the weak coupling parameter λ related to β by 2/β = sinh(λ/2) we find

α = tanh λ 2 , β = 1 2sinhλ/2 (2.33)
and the two-point correlator is given by

G xy = e -(λ/2)|x-y| (2.34)
The two-point correlator can be interpreted either as the sum over all random walks (with backtrackings) connecting the points x and y, or as the only minimal path (without backtrackings) connecting the two points, with a weight factor G 1 = e -λ/2 associated with each of its links. In the first interpretation the mean value of α is tuned so that it compensates completely the entropy due to backtrackings. In the case of the gauge theory the analog of the backtrackings are the folds of the world sheet of the string. We shall see that the D = 2 gauge theory can be formulated in terms of minimal nonfolding surfaces.

The U (N ) gauge theory

In the Wilson-Polyakov formulation of the U (N ) lattice theory [1] the independent fluctuating variable is a unitary matrix

U = {(U ) j i , i, j = 1, ..., N } U † U = I (3.1)
associated with the oriented links =< xy > of the hypercubical lattice2 . The measure and the interaction are invariant with respect to local gauge transformations

U <xy> → U x U <xy> U † y (3.2)
The amplitude U(Γ) associated with the parallel transport along the loop Γ = { 1 2 ... n } is given by the product of the link variables

U(Γ) = U 1 U 2 ...U n (3.3)
The action in the gauge theory is a functional on the loop fields (3.3). The simplest nontrivial loop Γ is the boundary ∂p of a plaquette p, an elementary square on the hypercubical lattice. In what follows we will denote by U p the corresponding loop variable, the ordered product of the 4 link variables along the boundary of the plaquette p. Let us denote by S λ (U) the one-plaquette action where λ is the coupling constant of the lattice gauge theory. Then the partition function is defined by the integral over all link variables

Z = e N 2 F = ∈S [dU ] p e S λ (U p ) (3.4)
where

[dU] = dUdU † δ(U † U -I) ≡ N i,j=1 dU j i d(U † ) j i δ k (U † ) k i U j k -δ j i (3.5)
is the invariant measure on the group U (N ). By F we denoted the free energy per degree of freedom. The one-plaquette action is subjected to the following two requirements. First, it should be a real function defined on the conjugacy classes of the group U (N ). This means that it can be expanded in the characters of the irreducible representations of the group

U (N ) e N 2 S λ (U) = -∞≤n 1 ≤...≤n N ≤∞ χ n (I)χ n (U)e N 2 S n λ (3.6)
The character of the representation with signature n = {n 1 , ..., n N } depends on the eigenvalues u 1 , ..., u N of the unitary matrix U as

χ n (U) = det ik (u n k +k-1 i ) det ik (u k-1 i ) . (3.7)
Second, we assume that the continuum limit is achieved when λ → 0 and in this limit

S λ (U) → - 1 2 tr N A 2 , U = e i √ λA . (3.8)
The Fourier image of the action in this limit is proportional to the second Cazimir operator

S n λ = λ 2N C 2 (R) = λ 2N N i=1 n 2 i + i<j (n i -n j ) (3.9)
If we assume that eq. (3.9) remains true also for finite values of λ, the corresponding one-plaquette action is known as heat kernel action [START_REF] Polyakov | [END_REF], S HK λ (U). The exponential of this action is a solution of the heat kernel equation on the group manifold

2 ∂ ∂λ + (U∂/∂U) 2 e N 2 S HK λ (U) = 0 (3.10)
The heat kernel action has the nice property of reproducing itself after group integration [16] [dU]e N 2 S HK λ 1

(U 1 U) e N 2 S HK λ 2 (U 2 U -1 ) = e N 2 S HK λ 1 +λ 2 (U 1 U 2 ) (3.11) Another simple choice is the Wilson action [1] S Wils λ (U) = 1 2λ tr N (U + U † ).
Its Fourier image is given by two different analytic expressions in the limit N → ∞, depending on the value of the coupling constant. For example, for the fundamental representation

n f = [0, 0, ..., 0, 1] S n f λ = 1 2λ , if λ > 1; 1 -1 2 λ, if λ < 1 (3.12)
The nonanalyticity at the point λ = 1 known as Gross-Witten phase transition [13] is due to the fact that the integral for the inverse Fourier transform

e N 2 S n λ = 1 χ n (I) [dU]e N 2 S λ (U) χ n (U) (3.13)
is saturated by the vicinity of a saddle point and the saddle-point solution can have two analytic branches. In the strong coupling domain λ > 1 the eigenvalues of the unitary matrix U are distributed all along the unit circle, and in the weak coupling domain λ < 1 the density of the phases of the eigenvalues is supported by an interval [-φ, φ] with φ < π.

The gauge theory in terms of nonrestricted complex matrices

Now let us convert this model into a theory with flat measure and regular interaction using the Laplace transform of the measure in the fields as in the U (N ) vector model. This has been actually done in the mean field analysis of lattice gauge theories [12]. We insert the exponential representation of the δ-function

δ(U j i -V j i ) = 1 2πi i∞ -i∞ dH i j e Hi j (U j i -V j i ) (3.14)
in the definition of the partition function for every matrix element (U ) j i of the gauge fields. Here V and H are unrestricted complex matrices on which the integration is performed with a flat measure

dV = N i,j=1 dV i j d V i j , dH = N i,j=1 dH i j d Hi j (3.15)
For each oriented link =< xy > only one pair of these fields is introduced and the convention

V <yx> = V -1 = V † = V † <xy> , H <yx> = H -1 = H † = H † <xy> (3.16)
is assumed. Denoting by

e N 2 F (H) = [dU]e N tr(H † U+HU † ) (3.17)
the one-link mean-field integral we write the partition function (3.4) in the form

Z = e N 2 F = dH dV e -N tr(H † V +H V † ) e N 2 F (H ) p e N 2 S(V p ) (3.18)
where V p denotes the ordered product of link variables along the boundary ∂p of the elementary square p

V p = ∈∂p V . (3.19)
The measure is now flat and the integrand is regular. The constraint (3.1) is satisfied in the sense of mean values

(V † V -I)... = 0 (3.20)
In order to give topological meaning of the 1/N expansion we need to express the two potentials F and S λ in terms of the moments tr( • ) n , n = 1, 2, ..., of their arguments.

The potential for the H-field

The function F can be determined using the Ward identity

[tr(∂/∂H † ∂/∂H) -N ]e F (H † H) = 0 (3.21)
which is the matrix analogue of (2.9). In the large-N limit this function was calculated explicitly by Brezin and Gross [17] in terms of the eigenvalues of the Hermitean matrix H † H. For sufficiently small field H it can be expanded as a series in the momenta

α n = tr N (H † H) n , (3.22
)

F [α] = ∞ n=1 k 1 ,...,k n ≥1 f [k 1 ,...,k n ] α k 1 ...α k n n! (3.23)
The coefficients of the series were found in the large N limit by O'Brien and Zuber [18] as a solution of a system of algebraic relations suggested originally by Kazakov in [6]. These relations are equivalent to eq. (3.21) written in terms of loop variables

n≥1 nα n-1 ∂ n N 2 + k,n≥1 [(n + k + 1)α n α k ∂ k+n+1 N 2 + nkα n+k-1 ∂ n N 2 ∂ k N 2 ] -1 e N 2 F [α] = 0
(3.24) where we denoted α 0 = 1, ∂ n = ∂/∂α n . Eq. (3.24) is sufficient to determine the 1/N expansion of the link-vertices f [k 1 ,...,k n ] . In the large N limit the identity (3.24) is equivalent to a system of recurrence relations [9] f

[1] = 1 f [k 1 ,...,k n ] + n j=2 k j f [k 1 +k j ,k 2 ,..., kj ,...,k n ] + k 1 -1 k=1 α∈P (L) f [k 1 -k,L\α] f [k,α] = 0 (3.25)
where L = k 2 , ..., k n and P (L) is the set of all subsets of L, including the emply set, and the bar means omitting the argument below it. The equations for the weights f [k] of the disks [6] f

[1] = 1; f [n] + n-1 k=1 f [k] f [n-k] = 0, n = 1, 2, ... (3.26)
are solved by the Catalan numbers

f [n] = (-) (n-1) (2n -2)! n!(n -1)! (3.27)

The potential for the V field

The potential S λ (V ) is defined by the analytic continuation of the r.h.s. of (3.6) from the hyperplane V = U, U † U = I to the space of all complex matrices V.

Using the explicit formula for the characters (3.7) we can express the r.h.s. of (3.6) as a series in the moments

β n = tr N V n , β -n = tr N V †n ; n = 1, 2, ... (3.28)
The series is convergent for small V and can be represented as exponential of another series

N 2 S λ [β] = N 2 ∞ n=1 k 1 ,...,k n =0 s [k 1 ...k n ] β k 1 ...β k n n! (3.29)
In the continuum limit λ → 0 the expansion (3.29) takes the form3

S 0 (V) = ∞ n=1 1 n (β n + β -n -β n β -n ) = N i,j-1 1 -v i vj (1 -v i ) N (1 -vj ) N (3.30)
which tends to the δ-function δ(V, I) when V approaches the hypersurface V † V = I.

Note that the potential (3.29) for the Wilson action has the form tr(V + V † ) only in the strong coupling domain. In the weak coupling domain λ < 1 it is given by the most general expansion (3.29) with complicated coefficients. Therefore the heat kernel action is simpler to work with in the weak coupling regime.

The lowest-order coefficients for the heat kernel action are, in the large N limit

s [1] = s [-1] = e -λ/2 , s [2] = s [-2] (1 -λ)e -λ , s [1,1] = s [-1,-1] = (λ -λ 2 /2)e -λ , s [1,-1] = -e -λ
(3.31)

The U (N ) gauge theory à la Weingarten

The interpretation of the functional integral as a sum over surfaces is possible if the action is linear in the traces of the matrix fields. This is achieved by the integral transformation

e N 2 F (H † H) = ∞ n=1 (dα n dα n e N 2 [ αn n tr N (H † H) n -αn α n n ] ) e N 2 F [α] (3.32)
By means of another system of parameters we represent the exponential of the plaquette action in the form

e N 2 S(V) = ∞ n=1 (dβ n d βn dβ -n d β-n e N 2 [ βn n ( tr N V n -β n )+ β-n n (trV †n -β -n )] )e N 2 S[β] (3.33)
We therefore introduce at each link and at each elementary square p a set of auxiliary loop variables coupled to the moments of the matrix fields The integration over the matrix fields in (3.36) will produce an effective action which will be interpreted in the next section as a sum over closed connected surfaces on the lattice. The leading term in this action is proportional to N 2 . Therefore, in the limit N → ∞ the integral over the scalar fields (3.34) and (3.35) is saturated by a saddle point and these fields freeze at their vacuum expectation values αn ( ) = αn , βn (p) = β-n (p) = βn (3.38) This is natural to expect since the auxiliary scalar fields are coupled to gauge invariant collective excitations of the original matrix fields (the moments (•) n ) which, due to selfaveraging, become classical in the large N limit. The classical values of the fields αn can be determined from the stationarity condition for the effective action, or, alternatively, from the Ward identity (3.20). Thus in the limit N → ∞ we obtain a theory of noninteracting planar (G=0) random surfaces with Boltzmann weights determined dynamically. This picture is conceptually the same as in the U (N ) vector model. Since we are interested only in the large N limit, the fields (3.34) and (3.35) can be replaced with their vacuum expectation values even before the integration over the matrix fields. We therefore find the following expression for the free energy of the U (∞) gauge theory The values α n , βn should be kept as free parameters. The parameters αn are determined by the condition of unitarity (3.20) and βn can be considered as coupling constants defining the plaquette action.

F = lim N →∞ 1 N 2 log dH dV e -N tr(H † V +H V † ) ;n>0

Branched random surfaces

Feynman rules in the large N limit

Assuming that the saddle point for the integral (3.18) is the trivial field H = V = 0, we can express the free energy as a sum of vacuum Feynman diagrams.

The Feynman rules are obtained as follows. Inverting the bilinear part of the action we find the propagators

(H ) i j (V † ) k l = (H † ) i j (V ) k l = 1 N δ i l δ k j ; (V † ) i j (V ) k l = α1 N δ i l δ k j (4.1)
The vertex βn trV n p is associated with the loop (∂p) n going n times around the boundary ∂p of the elementary plaquette p. We attache to the loop (∂p) n a little planar surface p n called n-plaquette. The n-plaquette p n has area n (we assume that the elementary plaquette has unit area) and a boundary going n times along the boundary of the plaquette p. The first three multi-plaquettes (n = 1, 2, 3) are shown in fig. 2 where for eye's convenience we have displaced the edges of the boundary.

The n-plaquette with n > 1 has a singularity (branch point of order n) containing Gaussian curvature 2π(n -1) at some point, say, at its centre. The n-plaquettes will serve as building blocks for constructing the world sheet of the lattice string. The Boltzmann weight of an n-plaquette in βn . The propagator V -V has simple geometrical meaning: it glues two edges of multiplaquettes having opposite orientations.

The absence of a H -H propagator means that the H field just play the role of glue for identifying the edges of the multiplaquettes. Besides the simple contraction V -V there are cyclic contractions of n pairs of oppositely oriented edges, n = 2, 3, ..., which are represented by the vertices αn tr(H † H) n . The edges are glued half-by-half in a cyclic way. The surface obtained in this way has a singularity at the middle of the link where a curvature 2π(n -1) is concentrated. We will call as before this singularity a branch point of order n. The Boltzmann factor associated with a cyclic contraction of order n is αn ( ). The simple contraction and the first nontrivial ones are shown in fig. 3. The vacuum Feynman diagrams are surfaces composed of multiplaquettes glued together along their edges by means of cyclic contractions. An important feature of these surfaces is that they can have branch points (that is, local singularities of the curvature) not only at the sites, as it is the case in the original Weingarten model, but also at the links and plaquettes of the surface. These singularities can be interpreted as processes of splitting and joining of strings. The weight of a closed surface is a product of the weights of its elements times a power of N which, with our normalizations, is equal to the Euler characteristics of the surface.

n = 1 n = 2 n = 3
The free energy of the U (∞) gauge theory is given by the sum over all closed connected surfaces with the topology of a sphere.

The connection with the traditional Feynman-like diagrams is by duality. Sometimes it is mire convenient to use the traditional diagrammatical notations. Then a surface bounded by a loop Γ = [ 1 2 ... n ] will be represented by a planar Feynman graph with n external legs dual to the links 1 , ..., n . The cyclic contraction of 2n is represented by a vertex with 2n lines as is shown in fig. 4. N V(Γ) we find a representation of W (Γ) as the sum over all planar surfaces bounded by the contour Γ. The Boltzmann weight of a surface is the product of the mean values (3.38) associated with the multiplaquettes and cyclic contractions. The local Boltzman weights can be decomposed into factors contributing to the total area of the surface and the length of its boundary, and factors related to the branch points. We denote

βn = αn = αn 1 ω (1) n , βn 1 ω (2) n (4.2)
Then the string path integral for the Wilson loop reads

W (Γ) = e -m 0 L[Γ] S:∂S=Γ e -M 0 A[S] Ω[S] (4.3)
where

Ω[S] = ∞ n=1 σ=1,2 (ω (σ) n ) N (σ) n (4.5)
ant the area of the surface is

A = ∞ n=1 nN (2) n (4.6)

Irreducible surfaces (a miracle)

Unlike the random walk, the two-dimensional surface can exist in configurations with very uneven intrinsic geometry. The typical syngularity is a "neck" representing an intermediate closed string state of small connecting a "baby universe" with the main body of the surface. The most singular configurations with necks can be readily removed from the path integral of the string. Their contribution can be taken into account by modifying the local Boltzmann weights associated with cyclic contractions of edges. Miraculously, the new Boltzmann weights become easily calculable and do not depend neither on the dimension D nor on the choice of the one plaquette action! This is why we will restrict the sum over surfaces to irreducible ones which will render the random surface Ansatz much simpler than it its original version.

The configurations to be excluded are surfaces with necks occupying a single link (fig. 5). We call such a surface reducible with respect to this link.. In what follows by sum over surfaces we will understand a sum over irreducible surfaces defined as follows.

Definition:

A surface S is reducible with respect to given link ∈ L if it splits into two or more disconnected pieces after being cut along this link. A surface which is not reducible with respect to any ∈ L is is called irreducible.

Evaluation of the weights of the cyclic contractions

Consider the Wilson loop average for the contour Γ = ( 1 2 ...) with coinciding endpoints.

We will exploit the unitarity condition VV † = I, applied to the loop amplitude. It means that the Wilson average W (Γ) will not change if a backtracking piece -1 is added to the contour Γ W (Γ -1 ) = W (Γ) (4.7)

The sum over surfaces spanning the loop Γ -1 can be divided into two pieces

W (Γ -1 ) = W (Γ)W ( -1 ) + W conn (Γ -1 ) (4.8)
The first term is the sum over all surfaces made of two disconnected parts spanning the loops Γ and -1 . The second term contains the rest. The constraint (4.7) is satisfied for all loops if W ( -1 ) = 1 and W conn (Γ -1 ) = 0. But there is only one irreducible surface spanning the loop -1 ; it contains a single contraction α1 ( ) between and -1 . Therefore α1 = 1. Now consider a surface contributing to the second term W I . The links and -1 may be connected to the rest of the surface by means of the same contraction or by two cyclic contractions The condition that their total contribution is zero is αn +

W [( -1 ) n ] = tr N (V V † ) n = 1, n = 1, 2, ... (4.11)
For each of these Wilson loops the sum over irreducible surfaces contains only finite number of terms, namely, the link-vertices contracting directly the edges of the loop -1 . For example,

W ( -1 ) = α1 , W [( -1 ) 2 ] = 2 α2 1 + α2 , ... (4.12)
Introducing the generating functions

w(t) = ∞ n=0 t n W [( -1 ) n ] = 1 1 -t , f (t) = 1 + ∞ n=1 t n (α n ) n (4.13)
we easily find the relation [11]

w(t) = f [tw 2 (t)] (4.14)
which is solved by the function generating the Catalan numbers

f (t) = 1 + √ 1 + 4t 2 (4.15)
In this way, unlike the U (N ) vector model, the classical values of the auxiliary fields do not depend on the dimension of the space-time. The expectation value α in the vector model is determined by the long wave excitations of the random walk. Here in the gauge theory, the expectation values of the fields αn are determined in purely local way, as it is clear from their derivation. One way to explain this difference is the local character of the U (N ) invariance in the case of the gauge theory.

On the contrary, the weights of the multiplaquettes will depend on the dimension as well as on the choice of the one-plaquette action. If wee are interested only in the large N limit, it is more convenient not to try to calculate them by solving the equations of motion but just to take them as independent coupling constants. The continuum limit (if it exists) then will be achieved along a trajectory βn = βn (λ, D), λ → 0 (4.16)

where λ is the coupling constant.

String representation of the Wilson loop average. Resumé

Let us summarize what we have achieved by now. The Wilson loop average W (Γ) in the U (∞) gauge theory defined on a D-dimensional lattice is equal to the sum of all planar irreducible surfaces having as a boundary the loop Γ. These surfaces are allowed to have branch points of all orders located at the sites, links and plaquettes of the space-time lattice. Introducing unified notations ω (k) n for the weights of the branch points associated with the k-cells of the space-time lattice (sites are 0-cells, links ate 1-cells, and plaquettes are 2-cells) where ω (0) n = 1

ω (1) n = f [n] = (-) (n-1) (2n -2)! n!(n -1)! ω (2) n = βn ( β1 ) n (4.17)
we can write the sum over surfaces as

W (Γ) = ∂S=Γ Ω(S)e -M 0 Area(S) (4.18)
where the factor Ω(S) is a product of the weights of all branched points of the world sheet and M 0 = -ln β1 .

The contribution of the surfaces with folds is zero

The weights of the branch points provide a mechanism of suppressing the backtracking motion of the strings or, which is the same, world surfaces having folds. The surfaces with folds have both positive and negative weights and their total contribution to the string path integral is zero. We have checked that in many particular cases but the general proof is missing.

It is perhaps instructive to give one example. Consider the surface in fig. 7.

Figure 7. A piece of surface having a fold It covers three times the interiour of the nonselfintersecting loop C and once -the rest of the lattice. There are two specific points on the loop C at which the curvature has a conical singularity; they can be thought of as the points where the fold is created and annihilated. Each of these points can occur either at a site or at a link (in the last case it is associated with a weight factor ω

(1) 2 = f [2] = -1).
Let us evaluate the total contribution of all irreducible surfaces distinguished by the positions of the two singular points. Denoting by n 0 and n 1 (= n 0 ) the number of sites and links along the loop C, and remembering that the a singular point located at a link has to be taken with a weight f [2] = -1, we find that the contribution of these surfaces is proportional to

[ n 0 (n 0 -1) 2 + f [2] n 0 n 1 + f 2 [2] n 1 (n 1 -1) 2 ] -[n 1 + 2f [2] n 1 ] = 0 (4.19)
The second term on the left hand side contains the contribution of the reducible surfaces which had to be subtracted. A reducible surface arises when the two points are located at the extremities of the same link (n 1 configurations) or when one of the points is a branch point and the other is located at one of the extremities of the same link (2n 1 configurations).

Loop equations

In ref. [11] we proved that the loop equations in the Wilson lattice gauge theory [20] are satisfied by the the sum over surfaces (4.18). Let us only write here the general formula which is derived in the same fashion. For any link ∈ Γ

∞ n=1 p:∂p βn [(W (Γ(∂p) n ) -W (Γ(∂p) -n )] = ∈Γ W (Γ )W (Γ )[δ( , ) -δ( -1 , )] (4.20)
The sum on the l.h.s. goes over the 2(D -1) plaquettes adjacent with the link and the closed loops Γ , Γ in the contact term are obtained by cutting the links and and reconnecting them in the other possible way. The case D = 2 was considered recently in details in [21]. For the heat kernel action the string tension M 0 and the weights ω Let us illustrate how the string Ansatz (4.18) works for the simplest nontrivial example of a Wilson loop (fig. 8) which have been calculated previously using the Migdal-Makeenko loop equations [22]. The nonfolding branched surfaces spanning the loop will cover the three petals of the flower (denoted by 1 in fig. 8 ) once, and the head (denoted by 2) -twice. Therefore the total area will be always A = A 1 + 2A 2 , where A i is the area enclosed by the domain i (i = 1, 2). Each of these surfaces will have a branch point located at some site, link or cell of the overlapping area 2.

Let us denote by n 0 , n 1 , n 2 the numbers of sites, links, cells belonging to the domain 2. Then the sum over the Ω-factors due to the branch points reads (4.23)

Strong versus weak coupling

Let us first discuss the case of the Wilson action where the mean field problem is solved exactly in the large N limit [17]. It is well known that when N ≥ 4 and D = 4, the strong coupling phase of a theory with Wilson action is separated from the continuum limit by a nondeconfining first-order phase transition. The assumption that H cl = V cl = 0 is a local minimum of the free energy is justified in the strong coupling phase of the gauge theory. In the weak coupling phase the matrix fields will develop vacuum expectation values. The classical fields form an orbit of the gauge group

V <xy> = U x V cl U -1 y , H <xy> = U x H cl U -1 y (5.1)
The diagram technique of sect. 2 then has to be modified according to the general rules explained in [12]. The surface elements will contain free edges, and, as a consequence, windows will appear spontaneously on the world sheet of the string. Whether these windows destroy or not the world sheet in the continuum limit is a dynamical question. Believing in the confinement, we expect that in D ≤ 4 dimensions the windows are still not sufficiently large for that, and the world sheet will have in the continuum limit the structure of a dense network of thin strips separating the windows (fig. 1). These strips will correspond to the gluon propagators in the standard Feynman rules. At the critical dimension D = 4 the effective string tension of the string (with windows on the world sheet) should scale with the coupling λ according to the asymptotic freedom law, just as the mass in the vector model does in D = 2 dimensions. In D > 4 dimensions the world sheet of the string is eaten by one or several large windows and the Wilson loop behaves as exponential of the length of the contour.

Whether this picture is true in geleral, can be decided by studying the mean field problem for a general one-plaquette action.

U

  (N ) Gauge Theory and Lattice Strings Ivan K. Kostov * # Service de Physique Théorique † de Saclay CE-Saclay, F-91191 Gif-Sur-Yvette, France

Figure 1 .

 1 Figure 1. A dence planar diagram for the Wilson loop average in the multicolour QCD

  α n ( ), αn ( ); n = 1, 2, ... (3.34) β n (p), βn (p); n = ±1, ±2, ... (3.35) and represent the integral (3.18) as a theory of Weingarten type described by the partition function Z = e N 2 F = dH dV e -N tr(H † V +H V † ) ;n>0 e N n αn ( )tr(H † H ) n p;n>0 e N n [ βn (p)trV n p + β-n trV †n p ] α, β (3.36) where the Boltzmann weights of the surfaces are themselves quantum fields and the average with respect to them is defined by * α, β = ∈L;n>0 dα n ( )dα n ( )e -N 2 αn ( )α n ( ) n +F [α( )] p;n =0 d βn (p)dβ n (p)e -N 2 βn (p)β n (p)

Figure 2 . 3 n = 1 n = 2 n = 3 Figure 3 .

 2333 Figure 2. Multiplaquettes of orders 1, 2, 3

Figure 4 .

 4 Figure 4. Vertices dual to cyclic contractions

1 β 1 e

 11 L[Γ] is the length of the contour Γ, A[S] is the area of the surface S, the bare string tension M 0 and the boundary mass m 0 are related to the original variables α1 , β1 by e -M 0 = α2 -m o = α1 (4.4) and the Ω-factor is the product of the local factors ω (σ) n associated with the branch points. The Ω-factor of the surface S containing N (σ) n n-plaquettes, n = 1, 2, ..., and N (1) n cyclic contractions of order n , n = 1, 2, ..., is given by

Figure 5 .

 5 Figure 5. A reducible surface

n- 1 k=1Figure. 6 .

 16 Figure. 6. Graphical representation of the unitarity conditionThe second derivation is based only on the the sum over surfaces for the trivial Wilson loops

4. 7 .

 7 The trivial D = 2 gauge theory as a nontrivial model of random surfaces

  ) m = (1 -n(n -1) 2 λ + ...) (4.21)

Figure 8 .

 8 Figure 8. A contour with the form of a flower and a branched surface bounded by it

1 2 (A 1

 11 n 0 -n 1 + (1 -λ)n 2 ) = 1 -n 2 λ (4.22)(Here we used the Euler relation n 0 -n 1 + n 2 = 1.) ThereforeW (C) = (1 -A 2 )e -+2A 2 )

Nevertheless, very interesting results concerning QCD

have been obtained in a series of recent papers[4][5] using the method of characters.

We will use the notation -1 =< yx > for the link obtained from =< xy > by reversing the orientation. More generally, if Γ 1 and Γ 2 are two oriented lines, the product Γ 1 Γ 2 is defined, in case it exists, as the line obtained by identifying the end of Γ 1 with the beginning of Γ 2 .

This formula resembles eq. (22) in the paper by M. Douglas[19]. We tried to establish an exact correspondence between his formalism and ours, but haven't succeed in this


Aimée Lahaussois

Alexis Michaud

Problèmes de phonologie diachronique: 21-28

As it developed, linguistics clarified its object of research; its perspective came to be modified as it extended its scope to all of the world's languages. Linguistics moved from a naturalistic to a historical perspective, then to a sociological perspective. An earlier conception of language as organism shifted to a conception of language as structure.

[1. Historical linguistics and the Neogrammarians]

Among the human sciences, linguistics is one of the earliest fields of study: it has been developing for almost 150 years. Moreover, it is a discipline whose modern orientations are of general epistemological interest.

Like any science, it has progressed by applying its methods to an ever broadening domain of observable facts; by developing its conceptual foundation; by demanding greater rigour; and by constantly re-examining the validity of its postulates. It has also built on progress that was being made in neighbouring fields. During the first years of the 19th century, the study of European languages did not yield a satisfactory account of the origin of vocabulary and grammatical forms. The discovery of Sanskrit, and the comparison of Sanskrit with classical languages, gave a decisive impetus to the development of a type of explanation that would dominate throughout the 19th century: historical explanation. According to F. [START_REF] Bopp | Vergleichende Grammatik des Sanskrit, Zend, Griechischen, Lateinischen, Litthauischen, Gothischen und Deutschen [Comparative grammar of Sanskrit, Zend, Greek, Latin, Lithuanian, Gothic and German[END_REF] program for the missions of linguistics, linguistics aims "to provide a description of the organism of the languages listed in the book's title; to André-Georges Haudricourt 2 compare facts of the same nature, to study the physical and mechanical laws that govern these languages, and to look for the origin of the forms that express grammatical relationships" [cited from Bréal 1867]. This program would still be considered valid by modern linguists, except that Bopp based his ideas on a set of postulates [some of which would not be accepted in the current state of the discipline]. Bopp postulated that comparison concerned comparable elements in different languages, and that an explanation for grammatical forms in daughter languages was to be found in the mother language [proto-language]. In Bopp's days, there was an implicit assumption that the mother language must be simple and regular. This led Bopp to believe that there were only three original vowels (proto-vowels), *i, *u, and *a [in proto-Indo-European], and that the short vowels ĕ and ŏ in Greek must be derived from a short ă in Sanskrit: "Among simple vowels, the Ancient Indian alphabet lacks vowels corresponding to Greek ɛ and . In case these sounds developed at a stage when Sanskrit was still a living language, they must have developed from the short a after the time when the alphabet was fixed, because this alphabet represents the finest differences in sound, and would certainly have reflected the difference between a, ĕ and ŏ if it had existed." [START_REF] Bopp | Vergleichende Grammatik des Sanskrit, Zend, Griechischen, Lateinischen, Litthauischen, Gothischen und Deutschen [Comparative grammar of Sanskrit, Zend, Greek, Latin, Lithuanian, Gothic and German[END_REF]:3, cited from Bréal 1867:31] At the time, it was not conceivable that *ĕ and *ŏ could have existed and later merged into *a before Sanskrit writing became fixed. Work from this period aimed to explain the origin of the rich morphology of Indo-European languages; it was implicitly assumed that a language with invariable elements is more basic than a language with inflectional morphology (whose root vowels vary according to declension and conjugation). It was believed that the natural development of languages consisted of evolution from a stage in which all elements are clearly distinct from one another, to a stage in which they merge into a single word.

Between 1871 and 1880, a whole series of discoveries called the earlier postulates into question. One such example was the fact that short *ĕ and *ŏ had left traces in Sanskrit: in this language, earlier *kĕ and *kŏ yielded ca and ka, respectively. These findings led to the development of a school of linguistics known as the Neogrammarian school (Junggrammatik), which questioned any sort of explanatory and evolutionary postulates, and only recognized facts and laws, in a perspective that may be called

The transformations of linguistic science (1957) 3 positivist. What the Neogrammarians referred to as "laws" were essentially phonetic laws, which Brugmann characterized as follows:

Every sound change, inasmuch as it occurs mechanically, takes place according to laws that admit no exception. (...) all words in which the sound subjected to the change appears in the same relationship are affected by the change, without exception. [Osthoff & Brugmann 1878:13] For example, any Latin c [k] followed by an a yields ch [ʃ] Instead of searching for a rational origin for grammatical forms, the Neogrammarians are content to explain through which phonetic laws or analogical processes these forms may be derived from earlier forms. The principle of consistency of phonetic laws led to increased methodological rigour, leading researchers to clarify which cases constitute exceptions to phonetic laws and require an explanation in terms of analogy.

[2. Saussure and the development of general linguistics]

The Neogrammarians' method had a major shortcoming, however: it projected automatically into the past any anomaly that was found in the data. Ferdinand de Saussure was the only linguist at the time who reintroduced the notion of language as an organism. In his celebrated Memoir on the Primitive System of Vowels in Indo-European Languages (1879), he attempted to treat the reconstructed Indo-European language, not as a repository for aberrant forms, but as a full-fledged language possessing a structure of its own. This led him to reconstruct consonants for which no attestation was known at the time, and whose existence was proved forty years later when Hittite was deciphered.

Only in the 20th century, particularly after the posthumous publication of Saussure's Course in General Linguistics (1916), did linguistics, which until then had been practically synonymous with comparative grammar, become a much broader field, General Linguistics. Saussure showed that linguistics should first and foremost adopt a synchronic point of view: studying language in use, as a system of signs used by a given human group. Diachronic or historical studies only come next, and are subordinate to synchronic studies. Since Saussure, a significant step forward has consisted of studying the history of languages in terms of the evolution of their structure: taking into account the overall state of the language's structure when studying individual sound changes [START_REF] Martinet | Economie des changements phonétiques. Traité de phonologie diachronique [The economy of phonetic changes[END_REF]. But to take stock of the progress realized in linguistics during this period, we need to consider the relationship of linguistics with related sciences: phonetics and ethnology, which played the role of auxiliaries 1 in the development of linguistics.

[3. The role of phonetics/phonology]

In the initial phase of comparative work on classical languages, the material under study consisted of letters; the first chapters of Bopp's book on Indo-European linguistics are accordingly entitled "Sanskrit alphabet", "Slavic alphabet", and so on. It was only in the second half of the 19th century that an increasing familiarity with foreign languages in their spoken form, along with the need to transcribe languages that did not have a writing system, led linguists to distinguish clearly between sounds and letters. This distinction led to the creation of phonetic alphabets, providing a direct representation of sounds, independent from spelling habits associated with a specific language. With this system, a single, unchanging letter is consistently used to represent a single sound, for instance using [ʃ] for a sound which is variously written as ch in French, sh in English, sch in German, ski in Swedish, sci in Italian, sz in Polish, and so on. From that point on, there was far clearer awareness than before of the physical, material aspect of languages. Physicists set out to study the nature of these sounds. The sounds of language, considered in themselves and defined by their physical and physiological properties, seemed to offer a field of study akin to that of 1. [The term 'auxiliary', auxiliaire, is by no means derogatory: it is applied to disciplines that are distinct from linguistics proper, but which make a contribution to its development. In Haudricourt's view, this includes experimental phonetics, which is concerned with acoustics and physiology, and thus belongs to the natural sciences.] the Natural Sciences. The rigour of the phonetic laws discovered by the Neogrammarians seemed to confirm the biological nature of human language. One of the best-known phoneticians of the end of the 19th century, Pierre-Jean Rousselot ["l'abbé Rousselot"], believed that changes in pronunciation were caused by a disease in the nerve endings of muscles in the tongue and the phonatory apparatus. Rev. Van Ginneken tried to demonstrate that "brachycephalic races" [human groups supposedly characterized by shorter skulls] mostly used sounds articulated at the back of the mouth.

As against these extreme proponents of a biological view of language, the majority of linguists recognized the fundamentally social nature of language. Among phoneticians too, some authors (such as Henry Sweet, Paul Passy, and Daniel Jones) recognized that the existence of a certain set of sounds in a language was a social phenomenon. The term 'phoneme' was borrowed from the Slavicist Baudoin de Courtenay by the Russian phonetician Ščerba, to refer to a sound in a language that has a social and differentiating value, that is, a sound which distinguishes different words within a given social group. Thus, in French as spoken in Marseilles, From the point of view of the Natural Sciences, there is an infinite range of sounds that humans are capable of producing. From the point of view of linguistics, on the other hand, there is only a limited and fixed number of sounds that are used to differentiate words in a given language: these are the language's phonemes. When early linguists spoke about "letters", they actually meant phonemes; it is the existence of a limited number of phonemes in each language that makes it possible to write it down with an alphabet that contains a fixed number of letters.

The discontinuous nature of oppositions between phonemes explains the regularity of phonetic laws. Phonemes have a differentiating [i.e. contrastive] function, which forces them to set themselves up in opposition to one another. This leads the linguist to pay attention to the entire set that they constitute: the language's phoneme system. The major contribution of phonology to modern linguistics consisted of bringing out the importance of the notion of system, or rather, in making the notion of system fully explicit.

[4. The role of ethnology and sociology]

From the very beginnings of linguistics, W. von Humboldt suggested that there exists a tight relationship between language, mentality and psychology. Since then, a number of scholars have been tempted to adduce linguistic data to conduct studies in comparative psychology. While phonetic changes were attributed to biology, analogical changes were attributed to psychology. At the end of the 19th century, the clear distinction that was being established between psychology and sociology freed linguistics from Wundt's overly ambitious explanations, which rested on general proposals about the "psychology of peoples". Psychology can nonetheless provide some assistance to linguistics in its broadest definition: as a science of language.

Explanations based on analogy can be made more precise by using mathematics, with calculations of frequency of use. This strand of research, which was initiated by Zipf, is now being pursued in modern cybernetics and information theory. This type of explanation shows, for example, that irregular verbs are immune to the power of analogy due to their frequency of use, and that frequent use tends to lead to a decrease in a word's length. This field is evolving rapidly.

In the second half of the 19th century, the study of social phenomena from the perspectives of sociology and ethnology began in earnest. The social nature of languages is strongly asserted by Saussure and Meillet, who build on Durkheim's approach to sociology.

A widely different approach was applied by Rev. Wilhelm Schmidt: he attempted to pair grammatical elements and clusters of ethnographic facts. Schmidt was thus following the formalist methods of German ethnography, which focused on the examination of clusters of cultural facts found within the same geographical area (the "Kulturkreis" approach). Another attempt consisted in applying Morgan and Engels's evolutionary ethnography to language: Nikolai Marr, a Caucasologist, classified grammatical types along different stages in social evolution. Looking back on these various attempts, they all appear to be completely arbitrary: there does not seem to exist a causal link between social structure and grammatical structure.

On the other hand, a fruitful collaboration has developed between linguistics and ethnology in the course of dialectal-lexical surveys. The elaboration of linguistic atlases -by Gilliéron in France, then by Swiss scholars in Italy -provided an impetus for this collaboration. In this type of work, linguistics was always a step ahead of ethnography, because comparative technology -the study of objects and tools -had only attracted very little attention until then. For linguistics, an important outcome of dialectal-lexical surveys was that the notion of structure came to be applied to the study of the lexicon, as the role played by homonymy became clear: the existence of a pressure to resist sound changes that threaten lexical distinctions. There would be no point in studying the history and the geographic distribution of a word without considering the other words in the language that are similar to it in either sound or meaning. The study of lexical systems was carried out by W. van Wartburg and his students as they put together historical and etymological dictionaries.

Collaboration between linguistics and ethnography can arguably be credited with fostering the development of studies of indigenous languages in North America, which in turn played a decisive role in shaping the general orientation of linguistics in America, ushering in structural theories in general linguistics. The study of American languages was undertaken by American scholars (many of whom were born in Europe) among whom the most famous are Boas, Sapir and Bloomfield. The positivist method of collecting supposedly objective raw data was contradicted by contact with these languages. It became fully clear that objectivity of the observer was an illusion. The phonetic alphabets devised in Europe, which were meant to represent physical sounds, only represented the phonemes of European languages. The most amusing case of creativity in devising new tools to approach indigenous languages of the Americas is that of the phonetic alphabet devised for the Southern Fuegians by Rev. Bridges. 2 Bridges's alphabet not only offered means to represent the sounds of the Indian language: it could also transcribe the sounds found in the author's dialect of English, including unitary symbols for the diphthongs in 'house', 'mice', and 'boy'.

The description of Indian languages of North America showed how different the grammatical categories and lexical concepts in these languages were from those of European languages. It revealed the extent to which the influence of models developed for European languages could introduce errors into the description and interpretation of the linguistic facts. This is why Bloomfield, who was the group's main theoretician, was led to define a "non-mentalist" attitude in linguistics, namely to base description on 2. [Rev. Thomas Bridges (1842-1898) in fact used the English phonotypic alphabet devised by Alexander John Ellis (1814-1890) and Isaac Pitman (1813-1897) in his grammar and dictionary of Yámana (Yaghan).] form, without concerning oneself with meaning, as our understanding of meaning is inevitably influenced by our linguistic categories. Indeed, a grammatical category in a European language is often defined by its meaning. For instance, in German, 'verb' is referred to as Zeitwort, literally 'time-word'; the verb is thus seen as a word which has tense (<Fr. temps 'time'). In French verbal morphology, 'indicative mode' (indicatif) is the one which 'indicates'. Other labels for grammatical categories refer to properties of form, such as adverb, referring to a word occurring 'next to the verb' (from Latin ad-verbum), and subjunctive, referring to a mode only used in subordinate clauses. There exists a tight, reciprocal relationship between form and meaning, so that it is understandable that they should sometimes become blended in linguistic terminology; but one must bear in mind that the reason why we make a strong difference between nouns and verbs as word classes is because verbs and nouns are very different in form in the languages with which we are most familiar: the languages of Europe. In many exotic languages, this difference is much weaker, so much so that the word 'father', referring to a notion which to us appears squarely nominal, can carry inflection for tense, e.g. to refer to 'my late father'; for person, to refer to 'my father, your father'; and for voice, to convey 'I am a father' vs. 'This is my father'.

[5. Bloomfield, Harris, and Pike]

In his first book, 3 Bloomfield based his work on Wundt's psychological linguistics. He then turned to the behaviourist concepts that held sway in the American scholarship of the time. 4 Behaviourism was precisely a psychology of behaviour, studying behaviour in response to an environment, without concern for its significance for the subject. This point of view was taken to excess in the Chicago school of linguistics, currently headed by Zellig Harris. In contrast, Kenneth Pike and his disciples -most of whom are Protestant missionaries -developed and taught a good descriptive method; they are working the world over to describe the last exotic languages that have so far remained undocumented.

3. [Bloomfield, Leonard. 1914 Among modern currents in linguistics, Hjelmslev and the Copenhagen school also need to be mentioned. Hjelmslev's attempt continues a strand of thought that can be traced back to Saussure: that language must be studied for itself and in itself, without reference to anything external to it.

In order to achieve this, Hjelmslev came up with a set of new disciplines which are exclusively concerned with combinatorial possibilities for phonemes (cenematics) and morphemes (plerematics).

[7. Concluding remarks: central and non-central issues in the scientific study of language]

A lingering problem is that of language classification, and the relationship between language genealogy (i.e., natural classification) and typology (artificial classification). We believe that a classification based on genealogy cannot encompass all of the world's languages, despite attempts by a few overenthusiastic researchers. 5 Calculations of statistical probability will be necessary to delineate the limits within which genealogical classification is possible: to determine the respective probability of chance and inheritance when similarities are found between languages. In order to perform calculations, one would need to be able to quantify the degree of interdependency of elements in a language's structure. Swadesh attempted to translate genealogical relationships into absolute chronology, using lexical similarities across languages. But this method, called glottochronology, is based on the controversial assumption that the rate of renewal of basic vocabulary is constant in time. Furthermore, the application of glottochronological tools is often extended to cases where family relationships are not established.

All linguists now acknowledge the social nature of language and accept that its essential functions are communication and expression. The possibility of studying language in a truly scientific way is more and more widely recognized. Linguistics can uncover laws: language-specific laws -"phonetic laws" in the classical sense: regular sound changes that have taken place at one point in history -and also general laws, referring to language structures independent of time and place. 6 Attempts at explaining the evolution of languages, and the diversity of attested languages, are becoming more and more precise; they make reference either to external phenomena (such as bilingualism, and the action of social groups on one another) or to internal phenomena. The latter result from changes in the nature of communications between individuals within society; they consist of changes in the frequency of use of given elements and of the various combinations in which these elements appear. Auxiliary sciences 7 to linguistics have multiplied: in addition to acoustics, and to the physiological study of the organs of the speaker and listener, new fields are developing, such as cybernetics and machine translation.

The program from the Oslo linguistics conference of August 1957 is a good indication of the current topics of interest in modern linguistics: of the fifteen conference topics, only one concerns genealogical relationships between languages.

Comment

Haudricourt provides the following foreword to this article in the French edition of his collected papers on historical phonology:

This article was written under the spur of indignation, after reading in Scientia a paper by Oddone [START_REF] Assirelli | La linguistica nel cinquantenario della rivista[END_REF]. The author heaps praise on the most debatable linguists: W. Schmidt, Van Ginneken, and most of all his former teacher, Trombetti. The editorial board of Scientia deleted the name of Trombetti, which was cited in brackets at the end of the sentence, 'We believe that a classification based on genealogy cannot encompass all of the world's languages, despite attempts by a few overenthusiastic researchers.' (Haudricourt 1972:21) 6. [Haudricourt is alluding here to the panchronic programme in linguistics, which aims to formulate generalizations about language change that are independent of any particular language or language group. See Haudricourt 1940 "Towards empirical laws in general linguistics" (this volume).] 7. [As noted above, the term 'auxiliary', auxiliaire, is by no means derogatory: it is applied to disciplines that Haudricourt considers distinct from linguistics proper, but which make a contribution to its development.] Condominas (1997:13) reports that Haudricourt liked to cite colonel Henri Frey's Annamese as the mother of languages: on the common origin of the Celtic, Semitic, Sudanese and Indochinese races (1892) as a typical example of wild speculations about the origin of languages and ethnic groups. The propensity to reveries of this type endures to this day: cherrypicking similarities between languages (or cultures) and proposing a common ancestry on the basis of what is in fact no more than chance resemblances. From time to time, a historical linguist stands up to warn against the errors of the authors who press all of the world's languages into a single family tree on the basis of "Pre-Copernican" methods [START_REF] Salmons | Global etymology" as pre-Copernican linguistics[END_REF]. But Haudricourt's article is not restricted to this issue, which only constitutes a very minor part of the discussion. In keeping with the stated orientation of Scientia, a non-technical, multilingual, multidisciplinary journal emphasizing links between the sciences, Haudricourt insists on the relation of linguistics to other disciplines and currents of thought.

  [e] and [ɛ] constitute one single phoneme, since [e] only appears in open syllables, and [ɛ] in closed syllables. On the other hand, in French as spoken in Paris, [e] and [ɛ] constitute distinct phonemes: Parisians distinguish [ʁe] ré (the musical note D) and [ʁɛ] raie ('ray, skate': a species of fish).

  5. [In a header added to the reprint of this article in Problèmes de phonologie diachronique, Haudricourt indicates that the name of the linguist Trombetti was cited in brackets (after the word 'enthusiasts') in the original manuscript; it was removed by the editorial board of the journal Scientia. See the Introduction to the article.]

.

  Introduction to the Study of Language.

	[6.
	New 4. [Bloomfield's second, behaviourist book was published in 1933: Language. York: Holt.] New York: Holt.]

Hjelmslev and the Copenhagen school]