A Coding of Real Null Four-Momenta into World-Sheet Coordinates

Abstract

The results of minimizing the action for string-like systems on a simply connected world sheet are shown to encode the Cartesian components of real null momentum four-vectors into coordinates on the world sheet. This identification arises consistently from different approaches to the problem.

1. Recapitulation

This paper is based upon an old unpublished article by Fairlie and Roberts [1], which dates back to circa 1972, on a model for amplitudes suggested by string theory, or rather the dual-resonance model as it was then called. (The results in this paper are recorded in the Ph.D. thesis of Roberts [2].) I was so overwhelmed by the evident truth of the famous paper of Goddard et al. quantising the bosonic string in 26 dimensions [3], which I regard as one of the classic papers in string theory, that I never submitted this paper for publication. However since recently, there has appeared an article by Sommerfield and Thorn [4], the 4th section of which is closely related to the model presented in [1], it may be an appropriate time to give these ideas an airing. Also other recent developments have given rise to an interpretation of maximally violating helicity (MVH) amplitudes in Yang Mills theory in terms of topological string amplitudes [5]; the connection between null four-vectors and Koba-Nielsen variables which is at the heart of [1] may not be entirely coincidental. The intention was to construct a viable amplitude for particles living in a strictly four-dimensional space-time, and with zero mass instead of the tachyon ground state which bedevilled the bosonic string dual resonance model. One of the features of tractable models of physical processes which has come to be more appreciated in the intervening years is that there is frequently a mismatch between what is tractable mathematically and what one should like to have; for example, the potential integrability of 𝑁→∞ supersymmetric Yang Mills as against the intractability of QCD, or the Sine Gordon model which displays both solitons and Lorentz invariance, at the cost of working in two dimensions. Here a feature analogous to self-dual Yang-Mills theory, which possesses instantons in a space of even signature, is present; I have realised that the theory presented is more mathematically compelling in a space of signature (2,2), though a Lorentzian interpretation is by no means ruled out. This will be discussed later in relation to the work of Gross and Mende on high energy scattering [6]. The starting point is the famous Koba-Nielsen formula, which gives an elegant expression for the 𝑁 point tree amplitude for 𝑁 particles with incoming momenta 𝑝^𝜇_𝑖 for the ground state of open strings [7],𝐴(𝑠,𝑡)𝑡=^∞𝑁_−∞1𝑑𝑧_𝑘𝑑𝑉_𝑎𝑏𝑐𝜃(𝑧_𝑖−𝑧_𝑖+1)_𝑗>𝑖(𝑧_𝑖−𝑧_𝑗)^{−2𝛼′𝑝𝑖⋅𝑝𝑗},𝑑𝑉_𝑎𝑏𝑐=𝑑𝑧_𝑎𝑑𝑧_𝑏𝑑𝑧_𝑐(𝑧_𝑏−𝑧_𝑎)(𝑧_𝑐−𝑧_𝑎)(𝑧_𝑎−𝑧_𝑐). (1.1)(1.2)(This integration measure is introduced as a consequence of conformal invariance; to account for the property that the real axis along which the integration is performed is invariant under transformations of the Möbius group, provided that 𝛼^′(𝑝^𝜇_𝑖)²=−1.) This means invariance under the mapping:𝑧^′↦𝑎𝑧+𝑏𝑐𝑧+𝑑,𝑎𝑑−𝑏𝑐=1. (1.3)It has been shown that this formula arises as a contribution to string scattering from a simply connected world sheet, thanks to the properties of conformal invariance. Another way of writing (1.1) is as an exponential:^∞_∞exp(_𝑖<𝑗−2𝛼^′𝑝_𝑖⋅𝑝_𝑗log(𝑧_𝑖−𝑧_𝑗))𝑑𝑧_𝑘𝑑𝑉_𝑎𝑏𝑐. (1.4) The exponent in the integrand may be interpreted as the (Euclidean) contribution to the action where the momenta enter the upper half-plane at designated points 𝑧_𝑘 which are then integrated over to give the contribution to the path integral for the amplitude arising from a simply connected world sheet. One of the chief deficiencies in (1.1) is the tachyon condition, namely, that 𝑝^𝜇_𝑖 is light like. This requirement follows from the invariance under mappings which preserve the upper-half complex plane. The radical idea behind [1] was to give the formula for the amplitude a different interpretation; do not integrate, but instead determine the coordinates 𝑧_𝑖 by minimising the integrand; this is tantamount, in the second version to use the method of steepest descents. The equations to be satisfied are, setting 𝛼^′=1,_𝑗𝑝_𝑖⋅𝑝_𝑗(𝑧_𝑖−𝑧_𝑗)=0. (1.5)These equations may be seen to be satisfied, provided that we are in a 4-dimensional space with signature (2,2) with null four-momenta 𝑝^𝜇_𝑗, and the coordinates 𝑧_𝑗 (here on the real line) are given by𝑧_𝑗=𝑝⁰_𝑗+𝑝¹_𝑗𝑝²_𝑗−𝑝³_𝑗=𝑝²_𝑗+𝑝³_𝑗𝑝⁰_𝑗−𝑝¹_𝑗;(𝑝⁰_𝑗)²+(𝑝³_𝑗)²−(𝑝¹_𝑗)²−(𝑝²_𝑗)²=0. (1.6)This works because(𝑧_𝑖−𝑧_𝑗𝑝)=^𝑖⋅𝑝_𝑗+𝑝⁰_𝑖𝑝¹_𝑗−𝑝¹_𝑖𝑝⁰_𝑗+𝑝³_𝑖𝑝²_𝑗−𝑝²_𝑖𝑝³_𝑗(𝑝²_𝑖−𝑝³_𝑖)(𝑝⁰_𝑗−𝑝¹_𝑗),(𝑧_𝑖−𝑧_𝑗)=−𝑝_𝑖⋅𝑝_𝑗+𝑝⁰_𝑖𝑝¹_𝑗−𝑝¹_𝑖𝑝⁰_𝑗−𝑝³_𝑖𝑝²_𝑗−𝑝²_𝑖𝑝³_𝑗(𝑝⁰_𝑖−𝑝¹_𝑖)(𝑝²_𝑗−𝑝³_𝑗). (1.7)The second equation is obtained by using the alternative expression of (𝑧_𝑖,𝑧_𝑗). By subtracting and rationalising, we have(𝑝²_𝑖−𝑝³_𝑖)(𝑝⁰_𝑗−𝑝¹_𝑗)−(𝑝⁰_𝑖−𝑝¹_𝑖)(𝑝²_𝑗−𝑝³_𝑗)=2𝑝_𝑖⋅𝑝_𝑗(𝑧_𝑖−𝑧_𝑗). (1.8)Summing over all particle positions 𝑧_𝑗 except 𝑧_𝑗=𝑧_𝑖 and invoking the conservation of momentum, ∑𝑝^𝜇_𝑗=0, we see that (1.5) is satisfied. If instead of the real line, the integration in (1.1) is performed over the boundary of the unit disc, the points on the boundary where the momenta enter may be parametrised by𝑧_𝑗=𝑝⁰_𝑗+𝑖𝑝³_𝑗𝑝¹_𝑗+𝑖𝑝²_𝑗=𝑝¹_𝑗−𝑖𝑝²_𝑗𝑝⁰_𝑗−𝑝³_𝑗;(𝑝⁰_𝑗)²+(𝑝³_𝑗)²−(𝑝¹_𝑗)²−(𝑝²_𝑗)²=0. (1.9)There is a Möbius transformation (1.3) which connects the two representations, for the plane and the disc,𝑧_disc=𝑖+𝑧_plane𝑖−𝑧_plane. (1.10)Indeed complex Möbius transformations on 𝑧_𝑖 are equivalent to 𝑆𝑈(2,2) transformations on 𝑝^𝜇_𝑖.

2. Alternative Approach

Consider a two-dimensional surface embedded in a four-dimensional space and take as parametric representation of the surface the four-vectors 𝑋_𝜇(𝜎,𝜏) where 𝜎 and 𝜏 are intrinsic coordinates on the surface with metric:𝑑𝑠²=𝐸𝑑𝜎²+2𝐹𝑑𝜎𝑑𝜏+𝐺𝑑𝜏², (2.1)where𝐸=(𝜕𝑋_𝜇)𝜕𝜎²,𝐹=(𝜕𝑋_𝜇𝜕𝜎)(𝜕𝑋_𝜇𝜕𝜏),𝐺=(𝜕𝑋_𝜇)𝜕𝜏² (2.2)(see [1]). The Nambu-Goto Lagrangian describing the dynamics of the field 𝑋_𝜇(𝜎,𝜏) is a measure of the area of the world sheet and is the reparameterisation invariant formℒ=𝛼^′𝐸𝐺−𝐹²𝑑𝜎𝑑𝜏. (2.3)On the other hand, it is well known that there exists a transformation to a coordinate system of so-called isometric coordinates in which the Lagrangian takes the simple quadratic formℒ^′=((𝜕𝑋_𝜇)𝜕𝜎²+(𝜕𝑋_𝜇)𝜕𝜏²)𝑑𝜎𝑑𝜏, (2.4)which is invariant only under the subset of reparameterisations of the variables (𝜎,𝜏) which are conformal, that is, those transformations which satisfy the Cauchy-Riemann equations. It is well known that in the coordinate system where 𝜎 and 𝜏 are isometric parameters defined by 𝐸=𝐺;𝐹=0. In this frame, the Euler equation minimising (2.3) becomes linear and is just∇²𝑋_𝜇=0. (2.5) The conditions for an isometric coordinate system may be written in the following form due to Weierstrass(𝜕𝜁_𝜇)𝜕𝑧²=𝐸−𝐺+2𝑖𝐹=0, (2.6)where 𝑋_𝜇 is the real part of 𝜁_𝜇 in view of the fact that (2.5) is satisfied provided that 𝜁 is an analytic function of 𝑧=𝜎+𝑖𝜏. The Weierstrass' condition shows that conformal mappings of coordinate systems preserve the isometric property. We can make a link with the Virasoro conditions for closed strings [8, 9] by noting that this is in fact the gauge condition of the model; writing(𝜕𝜁_𝜇)𝜕𝑧²=^∞_−∞𝐿_𝑛𝑧^𝑛=0. (2.7)This is too stringent to demand as an operator equation. Instead, we require that the matrix elements of (2.7) should vanish for all 𝑧, that is,⟨𝜓^†|𝐿_𝑛|𝜓⟩=0,∀𝑛≥0. (2.8)This is satisfied provided that 𝐿_𝑛=𝐿^†_𝑛=0. These conditions are the familiar Virasoro conditions for closed strings with zero mass ground states. A typical solution of (2.5) with a finite number of singularities is given by𝜁^𝜇=^𝑖=𝑛_𝑖=1𝑝^𝜇_𝑖log(𝑧−𝑧_𝑖). (2.9) By applying the Weierstrass condition, we have_𝑖,𝑗𝑝_𝑖⋅𝑝_𝑗(𝑧−𝑧_𝑖)(𝑧−𝑧_𝑗)=_𝑖,𝑗𝑝_𝑖⋅𝑝_𝑗(1(𝑧−𝑧_𝑖)(𝑧_𝑗−𝑧_𝑖)−1(𝑧−𝑧_𝑗)(𝑧_𝑗−𝑧_𝑖))=0. (2.10)This has to be true for all 𝑧, which evidently requires that ∑_𝑖,𝑗𝑝_𝑖⋅𝑝_𝑗=0 and, with conservation of four-momentum, also requires the same conditions ∑_𝑗(𝑝_𝑖⋅𝑝_𝑗/(𝑧_𝑖−𝑧_𝑗))=0 as before. In the case of the four-point function, the solution of these conditions (1.5) may be readily solved in terms of the cross-ratio 𝜆 to give𝜆=(𝑧_𝑖−𝑧₂)(𝑧₃−𝑧₄)(𝑧₁−𝑧₃)(𝑧₄−𝑧₂)=𝑝₁⋅𝑝₂𝑝₁⋅𝑝₃=𝑠𝑡, (2.11)where𝑠=(𝑝^𝜇₁+𝑝^𝜇₂)²,𝑡=(𝑝^𝜇₁+𝑝^𝜇₃)²,𝑢=(𝑝^𝜇₁+𝑝^𝜇₄)²,𝑠+𝑡+𝑢=0. (2.12)This result, in terms of the cross-ratio, is independent of the metric, so also works in a Lorentz metric with signature (3,1). The resulting amplitude 𝐴(𝑠,𝑡,𝑢) with 𝑠+𝑡+𝑢=0 may be evaluated to give𝐴(𝑠,𝑡,𝑢)=(−𝑠)^{−𝛼′𝑠}(−𝑡)^{−𝛼′𝑡}(−𝑢)^𝛼′𝑢. (2.13)As 𝑠→∞ at fixed 𝑡𝐴(𝑠,𝑡,𝑢)→𝑡^{−𝛼′𝑠}𝑠^{−𝛼′𝑡}, that is, it exhibits Regge asymptotic behaviour. The subject of asymptotic behaviour of high energy string amplitudes was examined to all orders sometime afterwards by Gross and Mende [6] who found the same connection (2.11) between the cross-ratio and the Mandelstam variables.

3. Lorentz Signature

As has been remarked, the minimisation condition in the case of the four-point function may be solved in terms of cross-ratios. This suggests that the conditions may be solved directly in terms of the variables 𝑧_𝑗 whatever the metric is. This is indeed the case; for real four-momenta, the solution may be expressed as𝑧_𝑗=𝑝⁰_𝑗+𝑝³_𝑗𝑝¹_𝑗−𝑖𝑝²_𝑗=𝑝^𝑖_𝑗+𝑖𝑝²_𝑗𝑝⁰_𝑗−𝑝³_𝑗,(𝑝⁰_𝑗)²−(𝑝¹_𝑗)²−(𝑝²_𝑗)²−(𝑝³_𝑗)²=0. (3.1)The difference is that in the case of signature (2,2), the four-momenta may be parametrised as 𝑝⁰_𝑗=𝑟cosh(𝜃_𝑗),𝑝¹_𝑗=𝑟sinh(𝜃_𝑗),𝑝²_𝑗=𝑟cosh(𝜙_𝑗),𝑝³_𝑗=𝑟sinh(𝜙_𝑗), which imply that 𝑧_𝑗=exp(𝜃_𝑗+𝜙_𝑗), (3.2)so the variables lie on the real line. Alternatively, a trigonometric parameterisation may be employed, in which case 𝑧_𝑗=exp(𝑖𝜃_𝑗+𝑖𝜙_𝑗). However in the case of signature (1,3), the parameterisation is mixed; 𝑝⁰_𝑗=𝑟cosh(𝜃_𝑗),𝑝³_𝑗=𝑟sinh(𝜃_𝑗),𝑝¹_𝑗=𝑟cos(𝜙_𝑗),𝑝²_𝑗=𝑟sinh(𝜙_𝑗), which imply that 𝑧_𝑗=𝑖exp(𝜃_𝑗+𝑖𝜙_𝑗), so there is no obvious integration contour for (1.1).

4. Minimal Surface Interpretation

Further insight may be gained by a parameterisation of minimal surfaces embedded in four-dimensional Euclidean space, originally due to Eisenhart [10], but rediscovered by Shaw [11] and quoted in [1, 12]. It is given by𝑋⁰=Re(𝑓(𝑧)−𝑧𝑓^′(𝑧)+𝑔(𝑧)^′𝑋),₃=Im(𝑓(𝑧)−𝑧𝑓^′(𝑧)−𝑔(𝑧)^′𝑋),₁=Re(𝑔(𝑧)−𝑧𝑔^′(𝑧)+𝑓(𝑧)^′𝑋),₂=Im(𝑧𝑔^′(𝑧)−𝑔(𝑧)+𝑓(𝑧)^′), (4.1)where a prime denotes the derivative with respect to the argument. Suppose we seek a parameterisation where 𝑋^𝜇+𝑎^𝜇 is the real part of 𝜁^𝜇=∑_𝑖𝑝^𝜇_𝑖𝐺_𝑖(𝑧), and 𝑎^𝜇 is an arbitrary origin. Then, thanks to the linearity of the above equations, we can split 𝑓(𝑧) and 𝑔(𝑧) into sums of independent components, that is, 𝑓(𝑧)=∑𝑓_𝑖(𝑧),𝑔(𝑧)=∑𝑔_𝑖(𝑧), and deduce, up to shifts of origin,(𝑝⁰_𝑖+𝑖𝑝³_𝑖)𝐺_𝑖(𝑧)=2(𝑓_𝑖(𝑧)−𝑧𝑓^′_𝑖(𝑧)+𝑔^′_𝑖(𝑧)),(𝑝¹_𝑖−𝑖𝑝²_𝑖)𝐺_𝑖(𝑧)=2(𝑔_𝑖(𝑧)−𝑧𝑔^′_𝑖(𝑧)+𝑓^′_𝑖(𝑧)),(𝑝¹_𝑖+𝑖𝑝²_𝑖)𝐺_𝑖(𝑧)=2(𝑔_𝑖(𝑧)−𝑔^′_𝑖(𝑧)+𝑓^′_𝑖(𝑧))^∗,(𝑝⁰_𝑖−𝑖𝑝³_𝑖)𝐺_𝑖(𝑧)=2(𝑓_𝑖(𝑧)−𝑧𝑓^′_𝑖(𝑧)+𝑧𝑔^′(𝑧))^∗. (4.2)If one postulates, using the same relations as obtained before, then 𝑧_𝑖=𝑝⁰_𝑖+𝑖𝑝³_𝑖𝑝¹_𝑖−𝑖𝑝²_𝑖=𝑝¹_𝑖+𝑖𝑝²_𝑖𝑝⁰_𝑖−𝑖𝑝³_𝑖. (4.3)These equations possess basic solutions of the form2𝑓_𝑖(𝑧)=(𝑎−𝑏)ln(𝑧+1)(𝑧+1)2(𝑧₁++1)(𝑎+𝑏)ln(𝑧−1)(𝑧−1)2(𝑧₁+−1)(𝑧₁𝑎+𝑏)ln(𝑧−𝑧₁)(𝑧−𝑧₁)(1−𝑧²₁)−𝑎,2𝑔_𝑖(𝑧)=(𝑏−𝑎)ln(𝑧+1)(𝑧+1)2(𝑧₁++1)(𝑎+𝑏)ln(𝑧−1)(𝑧−1)2(𝑧₁+−1)(𝑧₁𝑏+𝑎)ln(𝑧−𝑧₁)(𝑧−𝑧₁)(1−𝑧²₁)𝐺−𝑏,_𝑖=𝑐(−1+ln(𝑧−𝑧₁)), (4.4)where 𝑎=(𝑝⁰_𝑖+𝑖𝑝³_𝑖)𝑐 and 𝑏=𝑧₁(𝑝¹_𝑖−𝑖𝑝²_𝑖)𝑐, and 𝑐 is a real parameter. If the real parameterisation (1.8) is employed, then the 𝑧_𝑖 lie on the real axis𝑋^𝜇=𝑖𝑝^𝜇_𝑖log(𝑧−𝑧_𝑖)=𝜋𝑝^𝜇_𝑖Θ(𝑧−𝑧_𝑖)for𝑧ontherealaxis. (4.5)As 𝑧 moves from +∞ to −∞,𝑋^𝜇 jumps by 𝜋𝑝^𝜇_𝑖 at the point 𝑧_𝑖, so the skew polygon formed by the partial sums of momenta (closed on account of momentum conservation) is mapped into intervals on the real line.

5. Conclusion

The principal message of this paper is to draw attention to the link between the Cartesian components of real null four-momenta in four-dimensional flat space and complex variables on a simply connected world sheet, associated with a minimal surface, or a form of string evolution. The set of four momenta are also required to sum to zero, that is, momentum is conserved in the system. Various aspects leading to this identification are explored. The minimisation of the Koba-Nielsen integrand, the consequence of the Weierstrass condition upon a linear combination of elementary solutions to the free equations of motion, to guarantee a minimal surface solution, and the direct determination of this class of minimal surface solution from the Eisenhart parameterisation are all shown to entail the same identification of a complex variable in terms of the components of a null four-momentum. In a space of even signature (2,2); in one representation, the complex variables lie on the real line; in another on a circle; in the case of odd signature (Lorentz metric), there is no specific curve on which the variables lie. 𝑆𝐿(2,𝐶) transformations of the complex variable implement homogeneous Lorentz transformations upon the momentum.

In our original paper, as is standard practice, the optimistic anticipation of further development of these ideas was raised, but it must be admitted that neither author has been able to add anything substantially new in the intervening 35 years! However, as T.S. Eliot has said, “A poem may have meanings which are hidden from its author.” It may be that the further examination of solutions to the four-dimensional minimal surface equations originally proposed by Eisenhart will be fruitful. The ideas of this paper seem rooted in four dimensions; the parameterisation of classical string solutions proposed in [12, 13] based upon the division algebras may contain the clue to extend the connection between momenta and world sheet coordinates to 10 dimensions. The recent paper of Sommerfield and Thorn [4] extends their ideas to AdS space-time, and the picture of world sheets bounded by a closed polygon of null lines which is presented therein and is also contained in [14] is essentially the same as that in Section 4 of the present paper. In addition, the treatment of high energy string amplitudes by Gross and Mende [6] extends some aspects of this analysis to multiply connected world sheets.

In this spirit, this revised and rewritten version of [1] is offered in the hope that some deeper connection between momentum space and the world sheet will be discovered.