Nilpotent Elements in Physics∗

ЖУРНАЛ ФIЗИЧНИХ ДОСЛIДЖЕНЬ JOURNAL OF PHYSICAL STUDIES т. 11, № 2 (2007) с. 142–147 v. 11, No. 2 (2007) p. 142–147

NILPOTENT ELEMENTS IN PHYSICS∗

A. M. Frydryszak† Institute of Theoretical Physics, University of Wroclaw, pl. M. Borna 9, 50–204 Wroclaw, Poland (Received January 4, 2007) We brieﬂy review some issues of the nilpotent objects in theoretical physics using simple models as an illustration. Nilpotent elements appear at quantum and classical level in several ways. On the one hand there are celebrated BRST operators, external derivative and related with them cohomolo- gy theory, on the other hand there are dual numbers which are a less known structure than complex numbers but important in many approaches, then there are commuting nilpotent variables, some- how generalizing the nilpotent but anticommuting Grassmann variables used in super-mathematics and SUSY models. As an interesting fact we note that nilpotent variables demand the use of the generalized light-cone geometry with the metric of the null signature. Key words: infrared divergence, hydrodynamical approach, renormalization.

PACS number(s): 11.30.Pb, 45.20.Jj

I. INTRODUCTION developed since the 50s of the 20th century when the first appearance of Grassmanian variables in theoreti- Despite the fact that nilpotent objects are more dif- cal physics was noted [1–3]. However, one can look for ficult to handle than regular invertible ones, they play the realization of the nilpotency condition in the simple an important role in theoretical physics. It used to be a commutative structure, not in the graded-commutative customary way in constructing structures, to get rid of or noncommutative one. A possibility is given by the nilpotents. Null divisors complicate life. That is why in construction invented by Clifford [4]. He introduced the the standard approaches in mathematics or physics we “nilpotent unit” avoid nilpotents. However, there are spectacular theories 2 where nilpotent elements are crucial. The BRST formal- ι = 0 (3) ism is the best example were the nontrivial nilpotent and following the construction resembling complex num- operator Q, bers defined the so-called dual numbers D Q2 = 0 (1) z = x + ι y, x, y ∈ R, C, (4) 0 0 0 0 together with the appropriate space allows to quantize a z · z = x · x + ι(x · y + x · y), z ∈ D, (5) gauge field theory system. This highly nontrivial formal- z¯ = x − ιy, z−1 = |z|−2z,¯ |z| = x, (6) ism was being intensively developed in the 90s of the last y y z = x(1 + ι ) = r(1 + ιϕ), Arg(z) = . (7) century. We can say that the meaning of the trivial, at x x the first sight, nilpotency relation strongly depends on the way it is realized. In the case of the BRST theory Such numbers and their generalization to the dual we construct a sophisticated robust structure to repre- quaternions have been known since the 80s of the 19th sent a simple relation of nilpotency. Obviously there are century and they are related to the rotations and trans- other strong demands like hermiticity, etc. Another well lations in the three dimensional Euclidean space. In the known example gives the supersymmetry theory, where case when rotation is composed with translation we get the anticommuting variables are commonly used. Here the so-called screw motion [5, 6]. Dual numbers are a nilpotency is a byproduct of the anticommutativity of common tool in the differential algebra formalism [7]. the Grassmannian variables representing fermionic de- There is a very interesting model constructed by Wald grees of freedom. et al. [8–10] to study quantization of gravity where dual numbers are present. This model avoids the obstruction θθ0 = −θ0θ ⇒ θ2 = 0. (2) coming from the Coleman–Mandula theorem [11], so not only supersymmetry is a solution for this obstruction. A reach branch of mathematics i. e. super-mathematics In the following we will review some applications of the (superalgebra, superanalysis, supergometry) and a vivid dual numbers and discuss new issues of the commuting bundle of physical theories, like super-Yang–Mills, super- nilpotent variables, with the nilpotency of the second gravity, supersymmetric quantum mechanics, have been order. We do not review here the generalization known

∗Paper dedicated to Professor Ivan Vakarchuk on the occasion of his 60th birthday †Supported by Polish KBN grant #1PO3B01828

142 NILPOTENT ELEMENTS IN PHYSICS as paragrassmann relations where the nilpotency is of a B. Numerical differential algebra higher n-order i. e. ϑn = 0. We will not touch upon usual supersymmetry and BRST formalism. The main goal of this text is to call attention to the less known aspects of Despite the simplicity of the form of dual differentiable the presence of nilpotent objects in physical theories. function these objects are of great use in practical explicit calculations of derivatives for functions of real variables. The area of applications of this kind is named Numerical II. DUAL NUMBERS IN THEORETICAL Differential Algebra and is used in accelerator physics PHYSICS and optics for solving highly nonlinear systems [14]. In the most general terms one considers an equation of the type In the introduction we have recalled basic definitions related to the dual numbers. As is easily seen, due to the nilpotency of ι the geometry of the dual plane is different zf = M(zi, a), (12) from that of the complex plane. It is interesting to ob- serve that a circle on the dual plane with the centre in z0 where a denotes a set of parameters and z relevant fields consists of the points of two vertical parallel lines locat- (zi initial, zf final). Parameters are from chosen ed symmetrically on both sides of z0. Dual numbers are intervals and are known with some precision. The tran- + obviously nonarchimedean. Namely, let z = (x, y) ∈ D , sition matrix ∂zM and sensitivity ∂aM are usually ob- when the first nonzero coordinate is positive. Then: tained using computer analysis and the so-called TPSA Truncated Power Series Algebra. To explain this idea • z = 0, or z ∈ D+, or −z ∈ D+, let us consider a very simple example, namely Veronese Algebra which is denoted by 1D1. Let for the elements • x, y ∈ D+ ⇒ x + y ∈ D+. of 1D1 the multiplication be of the dual numbers’ type This allows to define the order in D in the following sense: x < y ⇔ y − x ∈ D+. Now with respect to the order de- (a0, a1)(b0, b1) = (a0b0, a0b1 + a1b0) (13) fined by the sets D+ we see that the ι is infinitesimal (infinitely small) in the sense that and the derivation be given by

ι < 1 ⇒ n · ι < 1, n ∈ N (8) (D, ∂): ∂(a0, a1) = (0, a1). (14) (n is a natural number). This observation gives justifi- We introduce a notion of standard functions F in the cation to the calculus of infintesimals used frequently in form physics when making linear approximations (product of two infinitesimals is negligible). Dual numbers are a for- 0 mal structure behind this popular practice. This fact is F (a0, a1) = (f(a0), a1f (a0)) (15) frequently neglected. hence choosing a specific point (x, 1) for evaluation we get A. Dual functions F (x, 1) = (f(x), f 0(x)). (16) One can define dual (number) functions analogously to the complex functions [12, 13] This gives a practical algoritm for the “algebraic” differentiation. 1 f(z) = u(x, y) + ιv(x, y), Example: let f(x) = 1 , then 1+ x f(z + h) − f(z) = f 0(z) · h + O (9) id(x, 1) ⇒ ((x, 1)−1 + (x, 1))−1 and in this case analogs of the Cauchy–Riemann equa- 1 1 tions have the following form = (( , − ) + (x, 1))−1 x x2 1 ∂u ∂v ∂u 1 1 − 2 = , − x . (17) = , = 0, (10) 1 1 2 ∂x ∂y ∂y 1 + x (1 + x2 ) and the dual differentiable function has the following This idea can be generalized to the case of n-variables simple form th and m order of the derivative i. e. to the algebra mDn: f(z) = u(x) + ι(v(x) + y · u0(x)). (11) (f, ∂f, ∂2f, . . . , ∂mf), (18) For example, the dual exponent is ez = ex(1 + ιy) and the trigonometric function is sin(z) = sin(x) + ιy cos(x) which is a very practical tool in computer analysis [14].

143 A. M. FRYDRYSZAK

C. Rotations and robotics represents translation. Again a composition of pure rotation given by the unit quaternion q0 and pure translation t gives a screw motion Kinematics is one of the oldest topics studied. For many centuries it was regarded as one of the basic sci- 1 tq = q + ι(t i + t j + t k) q , (23) ences that explained observed physical phenomena. It 0 0 2 0 1 2 0 was used to engineer machines. Its recent developments that can be compactly written in the following form yield the application of the modern robot kinematics to study biological systems and their functions at the mi- θˆ θˆ q = cos + s sin , (24) croscopic level and to the engineering of new diagnos- 2 2 tic tools. Dual numbers are essential in describing and ˆ solving the movements of rigid bodies especially in con- where θ = θ0+ιθ1 is, as before, dual angle and s = s0+ιs1 1 structing the parallel or serial robots, were special con- is a dual unit vector. Geometrically 2 θ0 is the angle of 1 ditions on pivots and spatial constraints are assumed. rotation around the axis s0 and 2 θ1 is the translation The finite screw displacement combination of transla- along this axis. The s1 is a unique moment of the axis, tion and rotation is a basic tool in the theoretical kine- giving position of the axis in affine space, for the giv- matics and its representation involves dual numbers. A en position vector p we have s1 = p × s0. A formalism considerable theoretical development of the description based on dual quaternions is widely used in robotics and of rigid body displacements was taking place from the in computer 3D-simulations of body motion, it is easier early 19th century up till the late 20th century. One of to implement, eliminates artifacts and is much faster. the classical results, the Chasles theorem states that the There is a rich area of applications of a dual valued ob- most general rigid body displacement can be produced ject in practical computer analysis calculations and simu- by a translation along the line followed (or preceded) by lations of movements of complicated rigid bodies [15–17]. rotation about that line. Such a displacement is called a But the formalism with the use of dual quaternions is ap- screw motion. Other essential constructions involve Cay- plied as well in classical electrodynamics [18]. ley’s formula, Hamilton’s quaternions, the Clifford dual quaternions and Study’s dual angle [5, 6]. The dual angle θˆ, proposed in the late 19th century, is used in the D. Dual numbers in field theory realization of the screw motion in R3 As is well known, in local quantum field theory it is ~p 0 = ~p + s~u, impossible within the conventional structure, to combine ~s = s~u, (19) nontrivially space-time symmetries with internal ones. This fact was recognized in 1967 in the famous Cole- ˆ θ = θ + ιs, man and Mandula theorem [11]. A new kind of symmetries outside Lie group type were to be introduced. They where ~s denotes the translation vector from ~p to p~0 and were uper symmetries. The work of Haag–L opuszański– s is its length, ~u is the unit vector and θ rotation an- Sohnius [19] showed that the graded algebra of genera- gle. The dual number θˆ is called a dual angle. The screw tors of symmetries of the S matrix should be admitted transformation is given in unique way by θˆ and ~u. The in QFT. It is little known that despite supersymmetry composition of screw transformations is realized by du- there was considered another possibility to go around al number multiplication. The explicit realization of the the obstruction of the Coleman–Mandula theorem. In a idea of a screw motion is with the use of dual quater- series of papers of Wald et al. [8–10] in the context of 4 nions [4]. Let Q denote a set of quaternions. By the dual the gravity theory a simplified model of the scalar λφ quaternion we understand a dual number constructed with the fields taking values in the dual numbers was over quaternion algebra suggested. φ(x) = ψ(x) + ιχ(x). (25) q = q0 + ιq1, q0, q1 ∈ Q. (20) The model given by the action The norm of dual quaternion can be written as Z a 2 3 S[φ, χ] = (−∂aψ∂ χ − m ψχ − λψ χ) (26) hq0, q1i k q k=k q0 k +ι . (21) k q0 k exhibits interesting symmetry properties. The field equations for component fields are of the form As is well known, rotation in the R3 can be represented 2 3 by a unit quaternion, translations are naturally embed- ψ − m ψ − λψ = 0 2 2 (27) ded into the dual quaternions χ − (m + 3λψ )χ = 0, where a nontrivial extension of the Poincarégroup is in- 3 3 v −→ v = 1 + ι(v i + v j + v k), (22) R 0 1 2 troduced. where v = (v0, v1, v2) and i, j, k are quaternion units. ψ → Pgψ 1 . (28) Now, the dual unit quaternion t = 1 + 2 ι(t0i + t1j + t2k) χ → Pgχ + £h Pg ψ

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This theory is consistent classically. However, as the au- The nilpotent component can be integrated in particular thors state: “Thus to summarize the result [. . . ], we find p that the conventional S-matrix theory, constructed using xN (x0) = E0 − U0 (x0) the Feynman rules, for our model quantum field theory does not make sense, and there do not appear to be ! 1 Z E − U (x ) any promising alternative approaches. This strongly sug- × dx N N 0 + C . (36) 0 3/2 gests that a similar conclusion will apply to the new class 2 (E0 − U0 (x0)) of gauge theories for massless spin-2 fields, even though at the classical level many of these theories have well- posed initial value formulation and have asymptotically well-behaved solutions to the equations of motion”. This result indirectly supports the distinguished role of su- 2. Hamiltonian formalism persymmetry whose presence especially at the quantum level is particularly fruitfull. For a dual mechanical system it is possible to perform a passage to phase space. The dual number valued Hamil- tonian H = H +ιH has two components and there are E. Dual mechanics 0 N three dual component momenta p0, p1, pN therefore the analogue of the Legendre transformation is more compli- Dual number valued mechanical systems were consid- cated and we obtain ered in [20] as an on-shell limit of a supersymmetric system. A more general formalism of dual classical mechan- ∂L0 ∂LN p0 = =x ˙ 0 = = p1, ics was considered there as well. Let us briefly discuss its ∂x˙ 0 ∂x˙ N elements. ∂LN pN = =x ˙ N , (37) ∂x˙ 0 1. Lagrangian formalism 2 p0 A general Lagrangian of a dual classical mechani- H0 = p0x˙ 0 − L0 = + U0 (x0) , (38) 2 cal system can be taken in the following form L = 0 HN = pN x˙ 0 + p0x˙ N − LN = p0pN + xN U0 L0 + ιLN . Assuming that time is double number valued × (x ) + U (x ) . (39) T = t+ι tN , one obtains the following general expansion 0 N 0 for coordinate functions Hence finally we obtain six Hamiltonian equations of mo- x (T) = x0 (T) + ι xN (T) . (29) tion

∂H0 ∂HN ∂HN and expansion of dual valued velocity x˙ 0 = , x˙ 0 = , x˙ N = , ∂p0 ∂pN ∂p0 (40) dx (T) ∂H0 ∂HN ∂HN p˙0 = − , p˙N = − , p˙0 = − . =x ˙ (t) + ι [x ˙ (t) +x ¨ (t) t ] . (30) ∂x0 ∂x0 ∂xN dT 0 N 0 N Finally the expanded Lagrangian has the form One can introduce generalized Poisson brackets for both sectors of the phase space [20]: 1 dx (T)2 L = − U (x (T)) , ∂A ∂B ∂A ∂B 2 dT {A, B}0 = − (41) ∂x0 ∂p0 ∂p0 ∂x0 U (x) = U0 (x) + ι UN (x) . (31) ∂A ∂B ∂A ∂B When we restrict to the case with tN = 0 i. e., to the the- {A, B}N = − , ories with the usual time parameter, the above formulas ∂x0 ∂pN ∂pN ∂x0 simplify to the following form ∂A ∂B ∂A ∂B + − . (42) ∂x ∂p ∂p ∂x 1 N 0 0 N L = x˙ 2 (t) − U (x (t)) , (32) 0 2 0 0 0 In the “nilpotent” sector of the phase space the Poisson bracket is related to an unusual conjugation of canonical 0 LN =x ˙ 0 (t)x ˙ N (t) − xN (t) U0 (x0 (t)) variables − UN (x0 (t)) . (33) x0 ↔ pN and xN ↔ p0. (43) The relevant equations of motion take the form

The second type of the Poisson bracket { , }N needs an x¨ + U 0 (x ) = 0, (34) 0 0 0 additional quantization rule with dual partner ~N of the 00 0 x¨N + xN U0 (x0) + UN (x0) = 0. (35) Planck constant ~.

145 A. M. FRYDRYSZAK

III. NILPOTENT MECHANICS The generalization of the Hamiltonian function to the nilpotent phase-space PN is conventional A recently introduced formalism of the nilpotent clas- H = pkη˙ − L (50) sical mechanics is based on the nilpotent commuting k variables η [22, 23]. Contrary to the Grassmannian vari- and we get for this formalism the equations of motion ables nilpotency is not automatical here and a relevant analogous to the standard Hamilton’s equations of mo- differential calculus for the functions of η-variables dif- tion fers from the usual one. In particular, the Leibniz rule p˙k = − ∂H , (51) is with an additional term and then the properties of ∂ηk the generalized Poisson brackets are not a direct conse- ∂H η˙k = k , (52) quence of derivations. To obtain a nontrivial theory with ∂p η-variables it is necessary to consider a particular form that for the η-harmonic oscillator gives of hyperbolic geometry [22], which is related directly to i 1 −1 ij the light cone formalism in spaces with the metric of a x˙ = m (s ) pj 2 j . (53) zero signature i. e. diag (+,+,. . . +,−, −,. . . ,−) with the p˙i = −mω sijx same number of pluses and minuses. However, the only The extension of the time derivative to the phase-space admissible form of a metric in the so-called s-spaces is functions f(η, p) is given in the following form (we omit off-diagonal e. g. here explicit time dependence) 0 In 2 T d ∂ s = ; s = I2n; s = s. (44) k k ¯ k n 0 =η ˙k∂ +p ˙ ∂k, where ∂ = I dt ∂ηk ¯ ∂ The above matrix is strictly traceless and symmetric and and ∂k = . (54) analogously to the antisymmetric matrix in superspace ∂pk it is a good “metric” in linear space (module) for vectors Defining the η-Poisson bracket as with nilpotent commuting coordinates. ¯ k {f(η, p), g(η, p)}0 = ∂kf(η, p) · ∂ g(η, p) k ¯ − ∂ f(η, p) · ∂kg(η, p) (55) A. Configuration space we can realize the time derivative in the following form To define a mechanical system one introduces a La- d f(η, p) = {H, f(η, p)} . (56) grangian with the terms being analogs of kinetic and dt 0 potential energy. Analogs only, because the s-form is not positively definite and η coordinates are nilpotent and It is worth noting that for η-functions time derivative commuting does not fulfill the usual Leibniz rule. The η-Poisson bracket has an analogous property, m L = s(η, ˙ η˙) − V (η), (45) 2 (i) {f, g}0 = −{g, f}0, (57)

mω mω (ii) {f + f , g} = {f , g} + {f , g} , (58) V (x) = s(η, η) = s ηiηj. (46) 1 2 0 1 0 2 0 2 2 ij

One can obtain the Euler–Lagrange equations of motion (iii) {f, g · h}0 = {f, g}0 · h + g ·{f, h}0 for the nilpotent mechanical system in the form [23] − 2♦(f|g, h), (59) ∂L d ∂L X − = 0. (47) (iv) {f, {g, h} } = 0, (60) ∂ηk dt ∂η˙i 0 0 cycl For the η-harmonic oscillator it gives where the para-Leibniz term is of the form 2 ¯k k ¯k η¨i = −ω ηi. (48) ♦(f|g, h) = ∂ f · ∇kg · ∂kh − ∂kf · ∇¯ g · ∂ h. (61) It is worth noting that one of the consequences of the B. Phase space “non-derivative” character of this bracket is violation of the Jacobi identity. The Jacobiator J(f, g, h) is, in general, not vanishing. It has the following form It appears that it is possible to make a passage to the phase space via the Legendre transformation with the X X i ¯i J(f, g, h) = 2 (η {∂if, ∂ig}∂ h deﬁnition of momenta cycl i ∂L − p {∂¯if, ∂¯ig}∂ h). (62) pk = . (49) i i ∂η˙k

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CONCLUSIONS very complicated like in the case of the BRST theory or fairy simple as for dual numbers. Nilpotent mechan- We have brieﬂy reviewed selected areas of theoretical ics provides an interesting example of the theory having physics were the nilpotent object plays an essential role some properties of the supersymmetry, where only the and is not something that one usually wants to get rid exclusion principle for fermions is realized by nilpotency of. For the physical content it is very important how the of relevant η-ﬁelds, but the anticommutativity condition nilpotency is realized, the underlying structure can be is relaxed.

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НIЛЬПОТЕНТНI ЕЛЕМЕНТИ У ФIЗИЦI

А. М. Фридришак Iнститут теоретичної фiзики Вроцлавського унiверситету, пл. М. Борна, 9, 50–204, м. Вроцлав, Польща Подано короткий огляд деяких питань, що стосуються нiльпотентних об’єктiв у теоретичнiй фiзицi, для iлюстрацiй використано простi моделi. Нiльпотентнi об’єкти виникають на квантовому i класичному рiвнях у деяких випадках. З одного боку, iснують вiдомi BRST-оператори, зовнiшня похiдна та пов’язанi з ними когомологiчнi теорiї. З iншого дуальнi числа, якi є менш вiдомими структурами порiвняно з комплекс- ними числами. Також є комутуючi нiльпотентнi змiннi, що певним чином узагальнюють нiльпотентнi, або некомутуючi, ґрассмановi змiннi, якi використовують у суперматематицi та суперсиметричних моделях. Як цiкавий факт вiдзначено, що нiльпотентнi змiннi вимагають використання узагальненої геометрiї свiтлового конуса з метрикою з нульовою сиґнатурою.

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