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[Xi, Yj ]= δi,j S (4) and all other commutators vanishing. The exponential map exp : hn → Hn respect- ing the multiplication (1) and Heisenberg commutators is n exp : sS + (xkXk + ykYk) 7→ (s, x1,...,xn,y1,...,yn). Xk=1 As any group Hn acts on itself by the conjugation automorphisms A (g)h = g−1hg, which fix the unit e ∈ Hn. The differential Ad : hn → hn of A at e is a linear map which could be differentiated again to the representation ad of the n ∗ Lie algebra h by the commutator: ad (A): B 7→ [B, A]. The dual space hn to the Lie algebra hn is realised by the left invariant first order differential forms Hn n ∗ on . By the duality between h and hn the map Ad generates the co-adjoint ∗ ∗ ∗ representation [21, § 15.1] Ad : hn → hn: ad ∗(s,x,y) : (h,q,p) 7→ (h, q + hy,p − hx), where (s,x,y) ∈ Hn (5) ∗ n and (h,q,p) ∈ hn in bi-orthonormal coordinates to the exponential ones on h . There are two types of orbits in (5) for Ad ∗—Euclidean spaces R2n and single points: 2n Oh = {(h,q,p): for a fixed h 6= 0 and all (q,p) ∈ R }, (6) 2n O(q,p) = {(0,q,p): for a fixed (q,p) ∈ R }. (7) The orbit method of Kirillov [21, § 15] starts from the observation that the above orbits parametrise irreducible unitary representations of Hn. All representations are 2πihs n induced [21, § 13] by a character χh(s, 0, 0) = e of the centre of H generated ∗ by (h, 0, 0) ∈ hn and shifts (5) from the left on orbits. Using [21, § 13.2, Prob. 5] we get a neat formula, which (unlike many other in literature) respects all physical units [24]: −2πi(hs+qx+py) h h ρh(s,x,y): fh(q,p) 7→ e fh(q − 2 y,p + 2 x). (8) n The derived representation dρh of the Lie algebra h defined on the vector fields (3) is: i i dρh(S)= −2πihI, dρh(Xj )= h∂pj + 2 qj I, dρh(Yj )= −h∂qj + 2 pj I (9) j n Operators Dh, 1 ≤ j ≤ n representing vectors from the complexification of h : j h Dh = dρh(−Xj + iYj )= 2 (∂pj + i∂qj )+2π(pj + iqj )I = h∂z¯j +2πzjI (10) where zj = pj + iqj are used to give the following classic result in terms of orbits: Theorem 1 (Stone–von Neumann, cf. [21, § 18.4], [8, Chap. 1, § 5]). All unitary irreducible representations of Hn are parametrised up to equivalence by two classes ∗ of orbits (6) and (7) of co-adjoint representation (5) in hn: (1) The infinite dimensional representations by transformation ρh (8) for h 6=0 j in Fock [8,10] space F2(Oh) ⊂ L2(Oh) of null solutions to the operators Dh (10): j F2(Oh)= {fh(p, q) ∈ L2(Oh) | Dhfh =0, 1 ≤ j ≤ n}. (11) p-MECHANICSANDDEDONDER–WEYLTHEORY 3

(2) The one-dimensional representations as multiplication by a constant on C = L2(O(q,p)) which drops out from (8) for h =0: −2πi(qx+py) ρ(q,p)(s,x,y): c 7→ e c. (12)

2 −|z| /h Note that fh(p, q) is in F2(Oh) if and only if the function fh(z)e , z = p + iq is in the classical Segal–Bargmann space [8, 10], particularly is analytical in z. Furthermore the space F (Oh) is spanned by the Gaussian vacuum vector 2 2 2 e−2π(q +p )/h and all coherent states, which are “shifts” of the vacuum vector by operators (8). Commutative representations (12) correspond to the case h = 0 in the for- mula (8). They are always neglected, however their union naturally (see the ap- pearance of Poisson brackets in (21)) act as the classic phase space:

O0 = O(q,p). (13) R2n (q,p[)∈ ∗ Furthermore the structure of orbits of hn echoes in equation (22) and its symplectic invariance [24].

2.2. Algebra of Hn and Commutator. Using a left invariant Hn Hn measure dg on the linear space L1( , dg) can be upgraded to an algebra with the convolution multiplication:

−1 −1 (k1 ∗ k2)(g)= k1(g1) k2(g1 g) dg1 = k1(gg1 ) k2(g1) dg1. (14) Hn Hn Z Z Hn Hn Inner derivations Dk, k ∈ L1( ) of L1( ) are given by the commutator for Hn f ∈ L1( ):

−1 −1 Dk : f 7→ [k,f]= k ∗ f − f ∗ k = k(g1) f(g1 g) − f(gg1 ) dg1. (15) Hn Z  Hn Hn A unitary representation ρh of extends to L1( , dg) by the formula:

[ρh(k)f](q,p)= k(g)ρh(g)f(q,p) dg Hn Z = k(s,x,y)e−2πihs ds e−2πi(qx+py)f(q − hy,p + hx) dx dy, (16) R2n R Z Z ! thus ρh(k) for a fixed h 6= 0 depends only from kˆs(h,x,y)—the partial s → h of k(s,x,y). Then the representation of the composition of two convolutions depends only from

′ ′ ′ πih(xy −yx ) ˆ′ ′ ′ ˆ ′ ′ ′ ′ (k ∗ k)ˆs(h,x,y) = e ks(h, x ,y ) ks(h, x − x ,y − y ) dx dy . R2n Z The last expression for the full Fourier transforms of k′ and k turn to be the star product known in deformation quantisation, cf. [28, (9)–(13)]. Consequently the representation of commutator (15) depends only from [24]:

′ ′ ′ ˆ′ ′ ′ ˆ ′ ′ ′ ′ [k , k]ˆs = 2i sin(πh(xy − yx )) ks(h, x ,y )ks(h, x − x ,y − y ) dx dy ,(17) R2n Z 4 VLADIMIR V. KISIL which turn to be exactly the “Moyal brackets” [28] for the full Fourier transforms of k′ and k. Also the expression (17) vanishes for h = 0 as can be expected from the commutativity of representations (12).

2.3. p-Mechanical Brackets on Hn. A multiple A of a right inverse operator to the vector field S (3) on Hn is defined by: 2π e2πihs, if h 6=0, SA =4π2I, where Ae2πihs = ih (18) 4π2s, if h =0.  Hn It can be extended by the linearity to L1( ). We introduce [25] a modified con- Hn volution operation ⋆ on L1( ) and the associated modified commutator:

k1 ⋆ k2 = k1 ∗ (Ak2), {[k1, k2]} = k1 ⋆ k2 − k2 ⋆ k1. (19) −1 Then from (16) one gets ρh(Ak) = (ih) ρh(k) for h 6= 0. Consequently the modification of (17) for h 6= 0 is only slightly different from the original one:

′ 2π ′ ′ ˆ′ ′ ′ ˆ ′ ′ ′ ′ {[k , k]}ˆs = sin(πh(xy − yx )) ks(h, x ,y ) ks(h, x − x ,y − y ) dx dy , (20) R2n h Z However the last expression for h = 0 is significantly distinct from the vanish- 4π ′ ′ 2 ′ ′ ing (17). From the natural assignment h sin(πh(xy − yx )) = 4π (xy − yx ) for h = 0 we get the Poisson brackets for the Fourier transforms of k′ and k defined on O0 (13): ∂kˆ′ ∂kˆ ∂kˆ′ ∂kˆ ρ {[k′, k]} = − . (21) (q,p) ∂q ∂p ∂p ∂q Furthermore the dynamical equation based on the modified commutator (19) with a suitable Hamilton type function H(s,x,y) for an observable f(s,x,y) on Hn by ρ , h 6=0 on O (6) to Moyal’s eq. [28, (8)]; f˙ = {[H,f]} is reduced h h (22) by ρ on O (13) to Poisson’s eq. [1, § 39].  (q,p) 0 The same connections are true for the solutions of these three equations, see [25] for the harmonic oscillator and [4, 5] for forced oscillator examples.

3. De Donder–Weyl Theory We extend p-mechanics to the De Donder–Weyl field theory, see [11–19] for de- tailed exposition and further references. We will be limited here to the preliminary discussion which extends the comment 5.2.(1) from the earlier paper [24]. Our nota- tions will slightly different from the used in the papers [11–19] to make it consistent with the used above and avoid clashes.

3.1. Hamiltonian Form of Field Equation. Let the underlying space-time have dimension and n + 1 parametrised by coordinates uµ, µ =0, 1,...,n (with u0 pa- rameter traditionally associated with a time-like direction). Let a field be described by m component qa, a = 1,...,m. For a system defined by a Lagrangian a a µ density L(q , ∂µq ,u ) De Donder–Weyl theory suggests new set of polymomenta µ a µ µ pa and DW Hamiltonian function H(q ,pa ,u ) defined as follows: a a µ µ ∂L(q , ∂µq ,u ) a µ µ µ a a a µ pa := a and H(q ,pa ,u )= pa ∂µq − L(q , ∂µq ,u ). (23) ∂(∂µy ) p-MECHANICSANDDEDONDER–WEYLTHEORY 5

Consequently the Euler–Lagrange field equations could be transformed to the Hamil- ton form: a µ ∂q ∂H ∂pa ∂H µ = µ , µ = − a , (24) ∂u ∂pa ∂u ∂q with the standard summation (over repeating indexes) agreement. The main dis- tinction from a particle mechanics is the existence of n + 1 different polymomenta µ a pa associated to each field variable q . Correspondingly particle mechanics could be considered as a particular case when n + 1 dimensional space-time degenerates for n = 0 to “time only”. The next two natural steps [11–19] inspired by particle mechanics are: (1) Introduce an appropriate Poisson structure, such that the Hamilton equa- tions (24) will represent the Poisson brackets. (2) Quantise the above Poisson structure by some means, e.g. Dirac-Heisenberg- Shr¨odinger-Weyl technique or geometric quantisation. We use here another path: first to construct a p-mechanical model for equa- tions (24) and then deduce its quantum and classical derivatives as was done for the particle mechanics above. To simplify presentation we will start from the scalar a µ µ field, i.e. m = 1. Thus we drop index a in q and pa and simply write q and p instead. We also assume that underlying space-time is flat with a constant metric ten- sor ηµν . This metric define a related Clifford algebra [?,3,6] with generators eµ satisfying the relations eµeν + eν eµ = ηµν . (25) Remark 2. For the Minkowski space-time (i.e. in the context of special relativity) a preferable choice is [27] with generators i, j, k instead the general Clifford algebra. Since q and pµ look like conjugated variables p-mechanics suggests that they should generate a Lie algebra with relations similar to (4). The first natural as- sumption is the n +3(= 1+(n + 1) + 1)-dimensional Lie algebra spanned by X, Yµ, and S with the only non-trivial commutators [X, Yµ]= S. However as follows from the Kirillov theory [20] any its unitary irreducible representation is limited to a representation of H1 listed by the Stone–von Neumann Theorem 1. Consequently there is little chances that we could obtain the field equations (24) in this way. 3.2. p-Mechanicanical Approach to the Field Theory. The next natural can- didate is the Galilean group Gn+1, i.e. a nilpotent step 2 Lie group with the 2n +3(=1+(n +1)+(n + 1))-dimensional Lie algebra. It has a basis X, Yµ, and Sµ with n+1-dimensional centre spanned by Sµ. The only non-trivial commutators are [X, Yµ]= Sµ, where µ =0, 1, . . . , n. (26) Again the Kirillov theory [20] assures that any its complex valued irreducible repre- sentation is a representation of H1, but multidimensionality of the centre offers an option [7,23] to consider Clifford valued representations of Gn+1. Thus we proceed with this group. Remark 3. The appearance of Clifford algebra in connection with field theory and space-time geometry is natural. For example, the conformal invariance of space- time has profound consequences in astrophysics [26] and, in their turn, conformal 6 VLADIMIR V. KISIL

(M¨obius) transformations are most naturally represented by linear-fractional trans- formations in Clifford algebras [6]. Some other links between nilpotent Lie groups and Clifford algebras are listed in [23]. The Lie group Gn+1 as homeomorphic to R2n+3 with coordinates (s,x,y), where x ∈ R and s,y ∈ Rn+1. The group multiplication in this coor- dinates defined by, cf. (1): (s,x,y) ∗ (s′, x′,y′) (27) 1 1 = (s + s′ + ω(x, y ; x′,y′ ),...,s + s′ + ω(x, y ; x′,y′ ), x + x′,y + y′). 0 0 2 0 0 n n 2 n n Gn+1 Observables are again defined as convolution operators on L2( ). To define an appropriate brackets of two observables k1 and k2 we will again modify their commutator [k1, k2] by antiderivative operators A0, A1,..., An which are multiples of right inverse to the vector fields S0, S1,..., Sn, cf. (18):

2π e2πihsµ , if h 6=0, S A =4π2I, where A e2πihsµ = ih and µ =0, 1, . . . , n. µ µ µ 4π2s , if h =0,  µ (28) The definition of the brackets follows the ideas of [7, § 3.3]: to each vector field Sµ should be associated a generator eµ of Clifford algebra (25). Thus our brackets are as follows, cf. (19):

µ {[B1,B2]} = B1 ∗ AB2 − B2 ∗ AB1, where A = e Aµ. (29) These brackets will be used in the right-hand side of the p-dynamic equation. Its left-hand side should contain a replacement for the time derivative. As was already mentioned in [11–19] the space-time play a rˆole of multidimensional time in the De Donder–Weyl construction. Thus we replace time derivative by the symmetric µ pairing D◦ with the [?, 3, 6] D = e ∂µ as follows: 1 ∂f ∂f D ◦ f = − eµ + eµ , where D = eµ∂ . (30) 2 ∂uµ ∂uµ µ   Finally the p-mechanical dynamic equation, cf. (22): D ◦ f = {[H,f]} , (31) is defined through the brackets (29) and the Dirac operator (30). To “verify” the equation (31) we will find its classical representation and compare it with (24). Similarly to calculations in section 2.2 we find, cf. (21):

ˆ′ ˆ ˆ′ ˆ ′ ∂k µ ∂k ∂k µ ∂k ρ µ {[k , k]} = e − e . (32) (q,p ) ∂q ∂pµ ∂pµ ∂q Consequently the dynamic of field observable q from the equation (31) with a scalar- valued Hamiltonian H is given by: ∂H ∂ ∂H ∂ ∂q ∂H D ◦ q = eµ − eµ q ⇐⇒ eµ = eµ, (33) ∂q ∂pµ ∂pµ ∂q ∂uµ ∂pµ   i.e. after separation of components with different generators eµ we get first n +1 equations from (24). p-MECHANICSANDDEDONDER–WEYLTHEORY 7

To get the last equation for polymomenta (24) we again use the Clifford algebra ν generators to construct the combined polymomenta p = eν p . For them: 1 ∂e pν ∂e pν ∂pµ D ◦ p = − eµ ν + ν eµ = − eµe , 2 ∂uµ ∂uµ ∂uµ µ   ∂H ∂e pν ∂H ∂e pν ∂H {[H,f]} = eµ ν − eµ ν = eµe . ∂q ∂pµ ∂pµ ∂q ∂q µ ν Thus the equation (31) for the combined polymomenta p = eν p becomes: ∂pµ ∂H eµe = − eµe , (34) ∂uµ µ ∂q µ µ i.e. coincides with the last equation in (24) up to a constant factor e eµ. Consequently images of the equation (31) under the infinite dimensional repre- sentation of the group Gn+1 could stand for quantisations of its classical images in (24), (32). A further study of quantum images of the equation (31) as well as extension to vector or spinor fields should follow in subsequent papers.

Remark 4. To consider vector or spinor fields with components qa, a = 1,...,m it worths to introduce another Clifford algebra with generators ca and consider a a composite field q = c qa. There are different ways to link Clifford and Grassmann algebras, see e.g. [2,9]. Through such a link the Clifford algebra with generators eµ corresponds to horizontal differential forms in the sense of [11–19] and the Clifford algebra generated by ca—to the vertical. Acknowledgement. I am very grateful to Prof. I.V. Kanatchikov for introducing me to the De Donder–Weyl theory and many useful discussions and comments.

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