Jens Madsen Houlrik

Germain Rousseaux
email: germain.rousseaux@univ-poitiers.fr

Dual kinematics

Keywords: numbers: 03.30.+p, 03.65.-w, 03.50.De

Based on the concept of geometric duality, we argue that uniform synchrony provides non-causal time by symmetric extension. Galilean particle aberration and Doppler effect c → ∞ coexists with conventional time and wave aberration in the Carrollian limit c → 0. Linear canonical dispersion implies internal energy E0 = mc 2 , where antiparticles are space-time inversions and mass/energy conversion is prevented by infinite barrier. The Landé paradox is resolved and de Broglie waves are conceptually demystified as elsewhere time zones. Finally, we derive electric-magnetic duality from pseudoscalar charge symmetry.

I. INTRODUCTION

Concepts of time usually involve some kind of change or motion [START_REF] Barbour | The nature of time[END_REF][START_REF] Smolin | Time reborn[END_REF][START_REF] Rovelli | The order of time[END_REF] such as light pulses bouncing between parallel mirrors with isotropic two-way velocity according to the Michelson-Morley experiment. Given the frequency/time standard f Cs ≡ 9 192 631 770 Hz intrinsic to matter, household devices conveniently provide radar length in light seconds 2l = cτ using a single clock and the assigned value c ≡ 299 792 458 m/s. One-way velocities, on the other hand, are linked to clock synchrony by a vicious circle [START_REF] Winnie | Special relativity without one-way assumptions[END_REF][START_REF] Anderson | Conventionality of synchronization, gauge dependence and test theories of relativity[END_REF][START_REF] Rizzi | Synchronization gauges and the principles of special relativity[END_REF] and the timing of each leg l = c + t + = c -t -remains a priori undefined constrained only by the harmonic mean 2/c = 1/c + + 1/c -. Uniform time is the simplest choice motivated by isotropic space c ± = c as "stipulated" by Einstein in the introduction to special relativity [START_REF] Stachel | Einstein's miraculous year[END_REF]. Nevertheless, wrist-watch time may well disagree with local coordinate time as every traveller exposed to jet-lag will testify.

The subtleties of distributed time keeping were anticipated before 1905 as shown by the writings of Poincaré and Cohn [START_REF] Poincaré | La mesure du temps[END_REF]. Voigt had introduced non-uniform coordinate time in 1887 seeking a covariant wave equation [START_REF] Voigt | Ueber das Doppler'sche Princip[END_REF] and group structure were provided by Larmor [START_REF] Maccrossan | A note on relativity before Einstein[END_REF] and Lorentz [START_REF] Lorentz | Electromagnetic phenomena in a system moving with any velocity smaller than that of light[END_REF] the following decade. While relativity was conceived prior to the idea of gravity as space-time curvature [12,[START_REF] Missner | Gravitation[END_REF], the four-dimensional continuum subsequently introduced by Minkowski [START_REF] Rowe | A look back at Hermann Minkowski's Cologne lecture "Raum und Zeit[END_REF] led to a shift of focus towards metric structure and coordinate based tensor calculus. The metric is, however, a second-order construct. The Lorentz transformation were derived by Furry invoking first-order generators [START_REF] Furry | Lorentz transformation and the Thomas precession[END_REF] including "local time" in the so-called Carrollian limit c → 0 identified by Lévy-Leblond [START_REF] Lévy-Leblond | Une nouvelle limite non-relativiste du groupe de Poincaré[END_REF]. In the following, we sall be concerned with pre-metric and coordinate-free kinematics.

Navigation in smooth manifolds relies on causal signals propagating along geodesics. Assuming isotropic oneway velocity [START_REF] Landau | The classical theory of fields[END_REF][START_REF] Cook | Physical time and physical space in general relativity[END_REF], uniform synchrony is distributed among fixed clocks by converting radar length to light travel-time

s ↔ ct, (1)
introducing a basic space-time symmetry [START_REF] Field | Space-time exchange invariance: Special relativity as a symmetry principle[END_REF] eventually leading to relativistic kinematics. Particle motion in preset time define instant velocities

ds(t) = vdt, (2)
while light-time permutation cdt = (v/c)ds [START_REF] Rovelli | The order of time[END_REF] introduces the inverse function

dt(s) = V -1 ds (4)
with characteristic velocity V = c 2 /v. In this sense, linear elsewhere time set by "through-the-looking-glass" symmetry coexist with inertial motion through the geometric mean c = √ vV and the dimensionless ratio

β = v/c = c/V = v/V . (5)
We emphasize that no tachyon motion [START_REF] Duval | Carroll versus Newton and Galilei: two dual non-Einsteinian concepts of time[END_REF] suggested by relativistic conundrums such as the "scissor" pseudoparadox is involved.

In tangent space, Galilean relativity in absolute time t ′ = t must be accompanied by Carrollian resetting in absolute space x ′ = x according to the symmetry

x ′ = x -v G t, ↔ t ′ = t -x/v C , (6)
where parameters satisfy v G v C = c 2 and modified velocity shifts

v ′ = dx ′ dt ′ = v -v G 1 -v/v C , (7)
allow the speed of light v = c as a true invariant. The transformations differ from Lorentz kinematics only by rescaling

x ′ = γ(x -v G t), t ′ = γ(t -x/v C ) (8)
to second order

γ = 1/ 1 -β 2 ∼ 1 + 1 2 β 2 , (9)
where time dilation dt = γdτ and length contraction is easily demonstrated using light clocks. The kinematics defined by [START_REF] Rizzi | Synchronization gauges and the principles of special relativity[END_REF] does not represent a group at finite c. Symmetric domains x ≪ ct ↔ x ≫ ct extend globally only in singular limits β → 0 and γ → 1, where Galilean relativity is contracted by instant signalling in absolute time

c → ∞, v C → ∞ 2 (10)
while time conventionality is introduced in the Carrollian limit

c → 0, v G → 0 2 (11)
involving stationary frames with free parameter v C . Euclidean geometry is augmented by causal structure in Sec. II and dual vector spaces are treated in Sec. III. Based on geometric optics and linear wave structure, a dynamical principle of maximum aging is introduced in Sec. IV. Classical mechanics is augmented by wave aberration and Doppler effect in Sec. V. Kinetic and canonical momentum coexists identifying the celebrated term E 0 = mc 2 , where mass/energy conversion based on classical antiparticles is prevented by infinite barrier. Matter waves are identified as elsewhere time zones and the Landé paradox is resolved in Sec. VI. Electric-magnetic duality is derived in Sec. VII, while Sec. VIII states our conclusions.

II. EUCLIDEAN AND PREMETRIC KINEMATICS

Locally, geodesics are Euclidean straight lines, where the (0,2) metric tensor converts infinitesimal displacements to distance dl 2 = g ij dx i dx j in arbitrary coordinates. A tangent basis is defined at every point

g ij = e i • e j ( 12
)
keeping all but one variable fixed e i = ∂P/∂x i [12]. The (2,0) inverse given by tensor contraction g ij g jk = δ i k introduces a Laplacian g ij ∂ i ∂ j and a measure of smoothness [START_REF] Neuenschwander | Tensor calculus for physics[END_REF][START_REF] Leonhardt | Geometry and light[END_REF]. The metric is diagonal in Cartesian coordinates and surface normals e i = ǫ i jk e j × e k given by the antisymmetric Levi-Civita symbol coincide with tangents. More generally, directions of steepest ascent e j = ∂x j /∂x i represent transverse surface elements defined by the (1,1) basis e i • e j = δ j i (13) independent of metric. Bearing in mind that neither geometry nor physical law can depend on coordinate bookkeeping [12], dual structure naturally introduces vector reciprocals 1/v = v/v 2 allowing arbitrary rescaling. As an example, independent tangent displacements and distributed time

dx(t) = vdt, dt(x) = v -1 s • dx, (14)
is related by the chain rule

1 = v -1 s • v = (∂t/∂x) • (dx/dt), (15)
where reciprocal phase velocities are referred to as slowness vectors by seismologists.

A. Causal structure

Although space-time navigation require four coordinates, a separate time dimension is not mandatory. Particles are mass points carrying proper de Broglie clocks [START_REF] Lan | A clock directly linking time to a particle's mass[END_REF] with associated light-spheres in Euclidean geometry rather than Minkowski light-cones. The traditional distinction between kinematics and dynamics can hardly be maintained [START_REF] Spekkens | The paradigm of kinematics and dynamics must yield to causal structure[END_REF], because synchrony itself requires a vacuum physics.

Local, coordinate-free foliations relative to a given reference clock provide causal structure of indefinite signature

∓r 2 = x 2 -c 2 t 2 (16)
where x 2 = x • x = xx and metric structure need not be specified. Linear foliations

∓rdr = x • dx -c 2 tdt, (17)
introduce time zones at constant parameter dr = 0,

t(x) = x • v/c 2 , (18)
where dual rays x = v p t intersecting (phase) fronts implies v • v p = c 2 . Provided velocities are parallel, dot products are replaced by algebra, Sec. III, introducing reciprocity

c/v p = v/c. (19)
In this case, space-time permutation along isotropic directions interchanges causal and elsewhere intervals

x ↔ ct, (20)
with no distinction between vector positions and lighttime distance. Signature reversal invariance [START_REF] Duff | Signature reversal invariance[END_REF] appears as a byproduct.

The austerity principle "the boundary of a boundary is zero" coined by Wheeler [START_REF] Kheyfets | The boundary of a boundary principle in field theories and the issue of austerity of the laws of physics[END_REF] also known as Poincaré's lemma implies d(dr) = 0. The resulting quadratic form

∓ds 2 = dx 2 -c 2 dt 2 , (21)
suggests a pseudo-Riemannian continuum, where Euclidean distance is defined instantly meaning dt = 0 and proper time intervals ds 2 /c 2 = dτ 2 are defined at coincidence dx = 0.

B. Vector kinematics

A simple argument demonstrates that Galilean shifts to a new inertial frame

x ′ = x -v G t, t ′ = t, (22)
must be accompanied by distributed elsewhere time.

Completing the square using the algebra defined by the first Eq. ( 31) below [START_REF]The dot product given by Eq[END_REF],

x ′2 = x 2 + v 2 G t 2 -2x • v G t, (23)
implies that isotropic light spheres c 2 t 2 = x 2 are transformed as

t ′ = t 1 + v 2 G /c 2 -2v G • x/c 2 t, (24)
assuming positive time with off-set t 0 = 0 at the common origin x = 0. The expansion

t ′ (x ′ ) ∼ t 1 + v 2 G /2c 2 -v G • x/c 2 . ( 25
)
shows that time remains first-order uniform along the Galilean trajectory x = v G t ≪ ct, where both terms contribute to second-order time dilation t ′ ∼ t/γ. Symmetric events ct ≪ x, on the other hand, allow inertial shifts to be neglected to second order introducing Carrollian resetting

x ′ = x, t ′ = t -v G • x/c 2 , (26)
and the combination

x ′ = x -v G t, t ′ = t -x • v -1 C , (27)
holds to first order, where the algebraic product

v G v C = c 2 , (28)
defines reciprocal parameters. Euclidean tangent space need not be assumed, while global symmetries are defined in singular limits according to Eqs. [START_REF] Maccrossan | A note on relativity before Einstein[END_REF] and [START_REF] Lorentz | Electromagnetic phenomena in a system moving with any velocity smaller than that of light[END_REF]. Dilated clocks passing a stationary ruler or contracted lengths passing a fixed clock introduces comoving velocities γv = dx/dτ . The transformation

v ′ = v -v G 1 -v • v -1 C , (29)
is consistent with Lorentz kinematics at γ G ∼ 1,

γ ′ v ′ = γ(v -v G ), γ ′ = γ(1 -v • v -1 C ), (30)
as can be seen by eliminating Lorentz factors. First-order rescaling of γ-factors for v ≫ c follows from the expansion (9) when exposed to a Galilean shift. Most importantly, Carrollian invariance γ ′ v ′ ∼ γv allows Dopplerlike energy-momentum, Sec. V.

III. DUAL VECTOR SPACES

The concept of geometric duality is most easily developed in generalized vector spaces. Symmetric contraction is augmented by anti-symmetric Grassmann extension as envisioned by Clifford [START_REF] Hestenes | New foundations for classical mechanics[END_REF][START_REF] Doran | Geometric algebra for physicists[END_REF][START_REF] Dorst | Geometric algebra for computer science[END_REF] 2a

• b = ab + ba, 2a ∧ b = ab -ba, (31)
introducing 2-blade parallelograms as geometric primitives. The resulting algebra

ab = a • b + a ∧ b, (32)
compounds scalar projections and simple bivectors into objects of even or axial symmetry under sign reversal a P = -a. In contrast, polar vectors and 3-blade volumes must be odd under parity. As an example, Euclidean trigonometry in the plane

a • b = |a||b| cos θ, a ∧ b = |a||b|ι sin θ, (33)
is spanned by non-parallel vectors providing unit squares ι = e 1 e 2 in Cartesian coordinates, where non-simple bivectors are denoted in boldface italic. It is easily seen that rectangular 2-blades must be pseudovectors with negative square

(a ∧ b) 2 = abab = -a 2 b 2 , (34)
implying ι 2 = -1. The complex-like product

ab/|a||b| = cos θ + i sin θ = e iθ (35)
rotates clockwise as generated by the algebra ie 1 = -e 2 and ie 2 = e 1 , while coordinate positions x = xe 1 + ye 2 define spinors z = e 1 x = x + iy providing a geometric interpretation of imaginary numbers [START_REF] Baylis | Why i?[END_REF].

The concept of parallelism a∧b = 0 and transversality a • b = 0 is independent of second-order metric structure allowing a nonmetric dual basis [START_REF] Missner | Gravitation[END_REF]. Without a meassure of length and angle, transverse 2-blades have indefinite shape insensitive to Galilean shear deformations of the form [START_REF] Leonhardt | Geometry and light[END_REF], but well-defined area, since the product ab remains well-defined. Given a tangent direction, the identity

x = xvv -1 = (x • v)v -1 + (x ∧ v)v -1 , (36)
defines projection and rejection x = x + x ⊥ according to symmetric expressions

2x = x + vxv -1 , 2x ⊥ = x -vxv -1 , (37)
where x •x ⊥ = 0 is readily verified and vxv -1 = x -x ⊥ is the second diagonal. Parallel transport may be defined by premetric constructions such as Schild's ladder [32]. Symmetric and anti-symmetric vector products generalizes to arbitrary grade. Scalar multiplication is even a

• b = ab without extension a ∧ b = 0, while 2a • b = ab -ba = 0, a ∧ b = ab, (38)
shows that there is no scalar-vector contraction. Bivector mutiplication is even a • S = aS and a ∧ S = 0, whereas

2a • S = aS -Sa, 2a ∧ S = aS + Sa, (39)
define contracted vectors and extended volumes proportional to the pseudoscalar unit ι 2 = -1 as shown below.

As a result, the object

aS = a • S + a ∧ S, (40)
is odd under parity supplementing (32).

A. Geometric duality

The distinction between polar and axial vectors is superseded by Hodge duality relating elements of grade n -k and k in n spatial dimensions. Volumes k = 3 are dual to geometric points k = 0 in three dimensions, where the unit cube ι = e 1 e 2 e 3 reverses handedness under parity ι P = -ι. Scalar rescaling is possible a ∧ ι = aι and a • ι = 0, there is no vector extension in three dimensions a ∧ ι = 0, while contraction

s • ι = ιs = S (41)
introduce bivector surface elements superseding normal vectors advocated by Gibbs and Heaviside. Extension provides volume

a ∧ S = ι(a • s), (42)
unless the vector lies within the surface a ∧ S = a • s = 0 and transverse to s. Contracted vectors are dual 2-blades

a • S = ι(a ∧ s), (43)
replacing cross products -a × s. Transverse surfaces are specified by the dual condition a • S = a ∧ s = 0.

B. Multivectors

Following Eqs. ( 32) and ( 40), multigrade primitives in three spatial dimensions can be written [START_REF] Baylis | Geometry of paravector space with applications to relativistic physics[END_REF][START_REF] Chappell | Revisiting special relativity: A natural algebraic alternative to Minkowski spacetime[END_REF][START_REF] Chappell | Time as a geometric property of space[END_REF],

M = a + v + ιw + ιb, (44)
with pre-metric contraction of even grade

M • M = a 2 + v 2 -w 2 -b 2 + 2ι(aw + bv). (45)
The set of quaterions {1, i, j, k} introduced by Hamilton [START_REF] Altmann | Icons and symmetries[END_REF] must be viewed as even bivectors providing surface basis elements. Geometric primitives required by particle motion are point events and tangent vectors

X = ct + x, V = c + v, (46)
where comoving derivation dX/dτ = γV introduces time dilation and individual components are extracted by conjugation (X ± X P )/2. We notice that light-time conversion (20) inevitably links spatial inversion x P = -x to time reversal. Dual fronts of equal time ιX transverse to tangent directions [START_REF] Bossavit | The premetric approach to electromagnetism in the 'waves are not vectors' debate[END_REF] are intersected by rays X p = (c + v p )t defined by v ∧ v p = 0, while contraction V • X P = 0 supersedes Lorentz orthogonality. Isotropic foliations ( 16) can be written ∓XX P and causal motion is restricted by the norm γ 2 V V P = c 2 .

IV. LINEAR WAVES

The geometric optics approximation [12] to complex fields ψ(x, t) ∼ e iϕ is based on scalar phase

ϕ = k • x -ωt, (47)
where the absence of a reference value allow local gauge transformations ϕ → ϕ + ϕ 0 (x, t). Instant wave vector assumed constant over the domain of interest and proper frequency

k = ∂ϕ/∂x, ω = -∂ϕ/∂t, (48)
serve as conjugate variables. Wave length λ = 2π/k and oscillation period T = 2π/ω introduce phase velocities unrelated to transport

v p = λ/T = ω/k, (49)
ranging from the stationary limit ω → 0 to uniform waves k → 0 propagating instantly. The ratio v p /c is reversed under conversion ω ↔ ck and parallelism k ∧ v p = 0 implies algebraic dispersion

kv p = k • v p = ω, (50)
with constant phase along rays. Dual volumes ιt and wave-fronts ιx are suggested by complex phasors invoking extension [START_REF] Arnold | Mathematical methods of classical mechanics[END_REF].

The space-time derivative

∂ X = ∂ ct + ∂ x define conju- gates K = ∂ X ϕ = -ω/c + k, (51)
with scalar contraction ϕ = K • X independent of the speed of light observing gauge freedom K → K + ∂ X ϕ 0 .

The expression K = (1v p /c)k represent linear rays, while second-order wave propagation is ensured by the premetric Lorenz condition

-∂ P X • ∂ X ϕ = (-∂ 2 t /c 2 + ∂ 2 x )ϕ = 0, (52)
rather than Lorentz orthogonality in Minkowski space.

A. Symplectic structure Displacements may represent dual surface elements according to the exact 1-form

dϕ = k • dx -ωdt + x • dk -tdω, (53)
where the projection K • dX vanishes along rays and the second part X • dK vanish at wave packets x = v g t constructed by Fourier superposition according to group dispersion

dk • v g = dω, (54)
that follows from Poincaré's lemma and the closed 2-form

d(dϕ) = 2dk • dx -2dωdt = 0, (55)
introducing symplectic structure dK • dX = 0. Tangent velocities v g = ∂ω/∂k are defined by frequency dispersion ω(k) and reciprocal slowness dk/dω are given by the inverse function k(ω), where dispersion relations ω = Ω(x, k, t) may be viewed as Hamiltonian functions providing a link between optics and mechanics with frequency and wave vector as canonical variables [START_REF] Rousseaux | Horizon effects with surface waves on moving water[END_REF].

B. Causal structure

While symplectic structure is fundamentally a geometric constraint, causal structure is tied to the propagation of physical signals. Second-order vacuum dispersion

∓κ 2 = k 2 -ω 2 /c 2 = KK P , (56)
introduces light shells κ 2 = 0 separating disjunct domains, where linear foliations

∓κdκ = k • dk -ωdω/c 2 , (57)
are transverse K •dK P = 0 at constant parameter dκ = 0. Equivalently,

(c 2 /v p ) • dk/dω = 1, (58)
shows that group dispersion (54) is consistent with dual structure [START_REF] Furry | Lorentz transformation and the Thomas precession[END_REF] identifying v g = v as well as the relation

v g = c 2 /v p .

C. Canonical dynamics

Conjugate or proper frequency defined by [START_REF] Jackson | Classical electromagnetism[END_REF] introduces a Hamilton-Jacobi type equation in phase space ∂ϕ/∂t + Ω(x, k, t) = 0, [START_REF] Jennings | No return to classical reality[END_REF] showing that global phase plays the role of action in geometric optics [12]. Analytical mechanics is anticipated by the exact 1-form

dΩ = -dx • f + dk • v + dt∂Ω/∂t, (60)
equivalent to a Legendre transformation in thermodynamics, where the first term is equivalent of work based on local gradients f = -∂Ω/∂x and the second term defines group motion at fixed event setting v g = v. Group dispersion [START_REF] Recami | Antiparticles from special relativity with ortho-chronous and antichronous Lorentz transformations[END_REF] reveals that only the last term is nonzero in conservative systems allowing geometric points to act as sources or sinks of energy. Second-order group dynamics is introduced by the principle of maximum aging valid also in general relativity [START_REF] Taylor | Exploring black holes[END_REF]. A twin embarking on a non-inertial round trip will return younger than its sibling that stays at home as a consequence time dilation dt = γdτ . However surprising this result may appear, there is nothing paradoxically about the situation, since the two journeys are non-equivalent. Hamiltonian optics introduces a set of first-order equations

dk/dt = -∂Ω/∂x, dx/dt = ∂Ω/∂k, (61)
compatible with Newton's second law dk/dt = f and the comoving derivative

dϕ/dt = k • v p -ω + (x -vt) • f , (62)
plays the role of Lagrangian in configuration space. Wave propagation in least time according to Fermat's Principle [START_REF] Neuenschwander | Emmy Noether's wonderful theorem[END_REF] complements group dynamics, since reciprocal group and phase velocities also define inverse refractive indices 1/n g = n p , where n = c/v.

D. Doppler effect versus aberration

Partial derivatives transforming as

∂ x ′ = ∂ x + v -1 C ∂ t , ∂ t ′ = ∂ t + v G • ∂ x , (63)
under temporal resetting and relative motion, respectively, are responsible for conjugate kinematics

k ′ = k -ω/v C , ω ′ = ω -v G • k, (64)
introducing wave aberration and Doppler effect [START_REF] Houlrik | The relativistic wave vector[END_REF] compatible with first-order special relativity γ G ∼ 1. Factorization

k ′ = k(1 -v p /v C ), ω ′ = ω(1 -v G • v -1 p ), (65)
shows that uniform waves v C = v p are accompanied by stationary groups v G = v g of infinite slowness dω ′ = 0 according to group dispersion [START_REF] Recami | Antiparticles from special relativity with ortho-chronous and antichronous Lorentz transformations[END_REF]. Stationary waves at relative velocity v G = v p are accompanied by elsewhere groups v C = v g of vanishing slowness dk ′ = 0 using the differential dk ′ /dω ′ = dk/dω -1/v C . The number wave crests passing a given event is proportional to phase which must be invariant. it is easily seen that Galilean point aberration is compensated by Doppler effect, while Carrollian wave aberration is compensated by relative time.

V. CLASSICAL MECHANICS

Newton's first law ensures that local inertial frames exists. The second law and the definition of work

dp = Fdt, dE = F • dx (66)
relates differential energy-momentum of a single particle to space-time displacements. Forces may be eliminated in favor of group dispersion

dE = v • dp, (67)
relaxing the assumption of singular mass points, where v = ∂E/∂p is a generalized tangent velocity along canonical momentum. Inverse phase velocities directed along the force are projected using [START_REF] Bossavit | The premetric approach to electromagnetism in the 'waves are not vectors' debate[END_REF],

dp/dE = F/F • v = 1/v p , (68)
allowing local field propagation rather than action at a distance. Rejection v ∧ dp = dx ∧ F introduces axial torsion and exterior derivatives.

A. Canonical dynamics

Newtonian dispersion (67) can just as well be written as a closed 2-form

d(dS) = dp • dx -dEdt = 0 ( 69
)
introducing symplectic structure [START_REF] Arnold | Mathematical methods of classical mechanics[END_REF], where the 1-form dS = Ldt defines a Lagrangian and the action itself

S = p • x -Et (70)
plays the role of wave phase subject to gauge freedom adding a total derivative to the Lagrangian. Premetric dispersion (50) is algebraic,

E = v p • p = v p p, (71)
with rays along canonical momentum

v ∧ v p = 0 ( 72
)
introducing wave-like mechanics with reciprocal parameters [START_REF] Field | Space-time exchange invariance: Special relativity as a symmetry principle[END_REF]. The term "momenergy" coined by Taylor and Wheeler is adequately represented by [START_REF] Dirac | The quantum theory of the electron[END_REF],

P = ∂ X S = -E/c + p, (73)
with scalar contraction S = P • X and symplectic structure dP • dX = 0. Factorization P = (1v p /c)p applies also to differentials using [START_REF] Wignall | Frame dependence of the phase of de Broglie waves[END_REF]. Conjugate energy is responsible for the Hamilton-Jacobi equation

∂S/∂t + H(x, p, t) = 0, (74)
introducing wave mechanics [START_REF] Landauer | Path concepts in Hamilton-Jacobi theory[END_REF][START_REF] Schleich | Schrödinger equation revisited[END_REF][START_REF] Houchmandzadeh | The Hamilton-Jacobi equation: An alternative approach[END_REF], while Lagrangian physics is set in configuration space

L = p • v -E + F • (x -vt), (75)
by Legendre transformation p = ∂L/∂v, where the force F = ∂L/∂x ′ evaluated at the particle position introduces the Euler-Lagrange equation. In terms of multivectors, contraction P • V vanishes at the phase velocity and work dP/dt • X is defined relative to wave packets [START_REF] Mohallem | Galilean invariance in Lagrangian mechanics[END_REF]. Hamiltonian dynamics

dp/dt = -∂H/∂x, dx/dt = ∂H/∂p, (76)
is based on the 1-form

dH(x, p, t) = -dx • F + dp • v + dt∂H/∂t, (77)
consistent with the principle of maximal aging valid also in general relativity.

B. Constitutive relations

As pointed out by van Dantzig [START_REF] Van Dantzig | On the geometrical representations of elementary physical objects and the relations between geometry and physics[END_REF], constitutive relations are required to link energy-momentum to matter. The familiar expressions

p = mv, E = E 0 + 1 2 mv 2 , (78)
are compatible with group dispersion (67) including an arbitrary constant, while canonical dispersion [START_REF] Hernandez-Coronado | Quantum equivalence principle without mass superselection[END_REF],

p = E/v p = Ev/c 2 , (79)
allows mass to be identified as zero-order energy

m = E 0 /c 2 , (80)
observing that wave packets carry internal energy. While avoiding relativistic dynamics, this simple argument does rely on wave/particle duality. Energy dilation E ∼ γE 0 defines kinetic energy, whereas expressions ignoring internal energy H ∼ p 2 /2m and L ∼ 1 2 mv 2 based on the symplectic constant γL = -E 0 are only apparently nonrelativistic, because factors of c 2 cancel.

Energy exchange in geometric points is allowed by the 1-form (77) based on classical antiparticle production and annihilation implying that conversion will take place unless prevented by infinite barrier E 0 → ∞ 2 . In this sense, Galilean physics enforces mass conservation by concealing infinite amounts of energy within matter never to be released.

Photons may be viewed as massless particles with finite product mγ in the limit m → 0 and γ → ∞ subject to light-like dispersion E 0 = cp 0 with Compton momentum p 0 = mc. Massless dispersion also define Galilean photons with finite momentum and infinite, but conserved energy.

C. Aberration versus Doppler effect

Particle aberration and Doppler effect must be complemented by Carrollian time conventionality E 0 → 0 2 in order for kinetic and canonical descriptions to coexist in disjunct domains. A Galilean boost applied to the Newtonian expressions [START_REF] Witten | Duality, spacetime and quantum mechanics[END_REF] implies

p ′ = p -mv G , E ′ = E -v G • p + 1 2 mv 2 G , (81)
and replacing mass by energy

p ′ ∼ p -Ev G /c 2 , E ′ ∼ γ G E -v G • p, (82)
allows canonical transformations of the form [START_REF] Landé | Quantum fact and fiction. IV[END_REF],

p ′ = p -E/v C , E ′ = E -v G • p, (83)
valid to first order γ G ∼ 1. Dual kinematics allows point aberration v ≪ c at constant energy cp ≪ E to coexist with elsewhere wave aberration v p ≫ c, while Doppler shifted waves v p ≪ c at constant momentum cp ≫ E are consistent with elsewhere rays v ≫ c. Factorization

p ′ = p(1 -v p /v C ), E ′ = E(1 -v G • v -1 p ), (84)
shows that momentum vanish in the time frame v C = v p and propagating waves become stationary in the inertial frame v G = v p carrying no energy.

D. Charged particles

Electric charges in units of coulomb interact with external fields. Additional momenergy is provided by magnetic vector and scalar potentials

p = γmv + q e A m , E = γmc 2 + q e φ m , (85)
observing that voltage V=J/C induced by non-stationary magnetic flux in weber V=Wb/s imply products of action CWb=Js in SI-units [START_REF] Jackson | Classical electromagnetism[END_REF]. Canonical transport [START_REF] Itin | An electric charge has no screw sense -a comment on the twist free formulation of electrodynamics by da Rocha and Rodrigues[END_REF] follows by eliminating scaling factors

p -q e A m = (E -q e φ m )v/c 2 (86)
where momentum behaves like velocity given that energy in minimal coupling is scalar. The action must be a global function

S = γE 0 (v • x/c 2 -t) + q e Φ m , (87)
including scalar flux Φ = A • x -φt subject to gauge freedom Φ → Φ+ Φ 0 , where the contribution of a moving particle

S(x = vt) = -mc 2 t/γ -q e φ ′ m t. (88)
can be written in terms of the Doppler shifted scalar potential as given by ( 83),

A ′ = A -φ/v p , φ ′ = φ -v • A, (89)
identifying v G = v and v C = v p . Symmetric contraction Φ s = A s • X may be defined for charges of both species s = e, m introducing conjugate variables

A s = ∂ X Φ s = -φ s /c + A s , (90)
in units of Wb/m and C/m, respectively. As a result, total momenergy can be written

P = mγV ∓ q s A P s , (91)
where symmetric potentials are denoted by tilded indices, A change of sign is induced by pseudosalar magnetic charge ιq m and electric potential ιA e , Sec. VII. Newton's equation of motion based on gradient forces in a comoving frame

d(γmv + q s A s)/dt = -q s ∇(φ s -v • A s), (92)
includes convection in potential fields with corresponding derivative (v

• ∇)A = ∇(v • A) -v × ∇ × A. Doppler shifts cancel d(γmv)/dt = q s (-∇φ s -∂ t A s + v × ∇ × A s), (93)
introducing E + v × B as the electric Lorentz force per unit charge with axial flux density playing the role of vorticity

E = -∇φ m -∂ t A m , B = ∇ × A m . (94)
The magnetic case involves a change of sign

H = ∇φ e + ∂ t A e , D = -∇ × A e , (95)
when reduced to axial vector calculus. The corresponding Lorentz field Hv × D acting on magnetic charges is compatible with reverse electric and magnetic handedness. In both cases, point charges are simply flux singularities [START_REF] Barker | A heuristic potential theory of electric and magnetic monopoles without strings[END_REF].

E. Antiparticles

Leaving symmetry breaking aside [START_REF] Quinn | The asymmetry between matter and antimatter[END_REF] the idea of classical pair production and annihilation is not new. Following Dirac [START_REF] Dirac | The quantum theory of the electron[END_REF], work by Stückelberg [START_REF] Stückelberg | La méchanique du point matériel en théorie de relativité et en théorie des quanta[END_REF] and Feynman [START_REF] Feynman | A relativistic cut-off for classical electrodynamics[END_REF] within special relativity [START_REF] Recami | Antiparticles from special relativity with ortho-chronous and antichronous Lorentz transformations[END_REF][START_REF] Costella | Classical antiparticles[END_REF][START_REF] Tulczyjew | Space-time orientations, electrodynamics, antiparticles[END_REF] were based on time reversal t T = -t introducing anti-symmetry X P T = -X represented by the Klein four-group {1, -1, ι, -ι}. The sign of the positron first observed in cloud chambers must be opposite to that of the electron in order to satisfy charge conservation and the full CP T -theorem of quantum field theory includes charge conjugation q C = -q stating that the distinction between ordinary and antimatter must be conventional [START_REF] Lehnert | CPT symmetry and its violation[END_REF]. Neutral particles are symmetric under space-time inversion, while separate P and T violations are believed to be rare.

While magnetic dipoles are abundant, it is less appreciated that permanent electric dipoles along spin directions have remained unobserved [START_REF] N Fortson | The search for a permanent electric dipole moment[END_REF]. Scalar dipoles d = qr violate P and T symmetry, while the possibility of pseudoscalar charge restores inversion symmetry at the same time classifying magnetic dipoles as classical mesons composed of antiparticle pairs.

VI. MATTER WAVES

Wave-particle duality follows from the identification of mechanical action with wave phase

S = ϕ, (96)
where h = 2π ≡ 6.62607015 × 10 -34 Js is Planck's constant. Term by term comparison P = K implies

E = ω, p = k. ( 97
)
implying that symplectic structure, algebraic dispersion, and geometric duality is all inherited as classical fragments existing within quantum theory [START_REF] Jennings | No return to classical reality[END_REF]. It may be argued that the dichotomy is dissolved rather than resolved by being placed in a proper context [START_REF] Friederich | Interpreting quantum theory: A therapeutic approach[END_REF]. Wave functions ψ(x, t) = √ ρe iS/ introduce conserved probability distributions

∂ t ρ + ∇ • j = 0, ( 98
)
where Fourier transformation

-ωρ + k • j = 0 ( 99
)
introduces the inverse phase velocity v -1 p = k/ω. Dual projection then follows by defining a characteristic velocity j = ρv implying [START_REF] Furry | Lorentz transformation and the Thomas precession[END_REF]. Using J = ρV , continuity may also be written ∂ X • J = 0. The singleparticle Schrödinger equation i ∂ t ψ = Hψ follows from the Hamilton-Jacobi equation [START_REF] Schleich | Schrödinger equation revisited[END_REF] with

H = (p -qA) 2 /2m + qφ, (100)
while the Klein-Gordon equation based on relativistic or quadratic dispersion E 2 = p 2 c 2 +m 2 c 2 including internal energy introduces negative energies and uncontrolled pair production leading to the concept of a quantum field.

A. de Broglie on relativity

The arguments presented in the 1924 thesis relied heavily on special relativity [START_REF] De Broglie | Waves and quanta[END_REF][START_REF] Yilmaz | Lorentz transformations and wave-particle unity[END_REF]. Particles were modelled as localized wave packets moving in uniform time with Fourier components interpreted as phase waves. Guided by a principle of "phase harmony" [START_REF]The wave-particle dualism[END_REF] de Broglie extended the proper oscillations of a point particle defined by the Planck-Einstein relation

mc 2 = ω ′ , (101)
to uniform phase oscillations accompanying uniform time in the rest frame. Particle motion introduces relativistic Doppler effect

ω ′ = γ(ω -v • k), (102)
showing that the frequency scales as ω ′ = ω/γ to second order compatible with energy dilation, when the phase velocity v -1 p = k/ω and light-time reciprocity is invoked. Conjugate momentum is proportional to spatial variations in the dual frame

p = h/λ, (103)
where apparent superluminal motion that worried de Broglie disappears is replaced by elsewhere time zones. Slow groups λ/λ 0 ≫ 1 with λ 0 = h/mc the Compton wave length reproduce the geometric optics approximation in classical physics [12]. Inverse velocities imply inverse refractive indices.

B. Landé paradox

Neglect of Carrollian symmetry leads to the so-called Landé paradox [START_REF] Landé | Quantum fact and fiction. IV[END_REF], when particle kinematics is confronted by wave kinematics

mv = k, (104)
since invariant wave vector is incompatible with Galilean velocity shifts. To avoid a logical paradox both sides must transform the same way. As we have seen, the left-hand-side represents a wave packet with individual Fourier components propagating at the phase velocity. Galilean particle momentum v ≪ c is accompanied by Carrollian phase waves v p ≫ c and the right-hand side must therefore be reset according to Carrollian kinematics allowing wave aberration. The apparent paradox is thus resolved by dual kinematics.

A complementary pseudo-paradox would ask, how both sides of (104) can possibly be invariant for slowly propagating wave components. In this case, Galilean wave kinematics must be accompanied by Carrollian point kinematics leaving the relativistic velocity γv invariant even for γ ∼ 1. Once again, kinetic and canonical momentum coexists peacefully. Non-relativistic wave propagation simply require dual group kinematics.

C. EPR correlations

Relative or frame-dependent simultaneity is a signature of special relativity. Whether non-local quantum correlations could be influenced by time reversal has been investigated in space-like separated, EPR-type experiments with one detector moving. Very accurate timing allowed nearly simultaneous measurements in a frame moving at v G ∼ 100 m/s relative to the laboratory [START_REF] Zbinden | Experimental test of nonlocal quantum correlation in relativistic configurations[END_REF][START_REF] Rembieliński | Einstein-Podolsky-Rosen correlations of spin measurements in two moving inertial frames[END_REF]. Since, β ≪ 1 and γ G ∼ 1, first-order dual kinematics applies, where the dual velocity parameter is of the order v C = c 2 /v G ∼ 10 15 m/s. No difference compared to an identical experiment having both detectors at rest in the laboratory frame could be detected. Although Carrollian kinematics reverses time ordering, a null result is to be anticipated, since physical processes including wave function collapse can not depend on a particular choice of time coordinate.

D. Phase shifts

The simple-minded transformation (81) introduces a Galilean shift S ′ = S -S G of unbalanced terms

S G = mv G • x - 1 2 mv 2 G t, (105)
where the additional phase ψ ′ = e -iSG/ ψ [67-69] generated in the Schrödinger equation should be detectable by interferometer experiments. Bargmann argued that a back and forth trip suddenly reversing velocities combined with spatial translations lead to similar effects [START_REF] Bargmann | On unitary ray representations of continuous groups[END_REF] prompting a spurious selection rule forbidding superposition of different mass states [START_REF] Hernandez-Coronado | Quantum equivalence principle without mass superselection[END_REF][START_REF] Pereira | Galilei covariance and Einstein's equivalence principle in quantum reference frames[END_REF]. Active transformations are restricted by physical laws and non-causal translations must involve elsewhere kinematics. Singleparticle actions are indeed invariant when first-order generators (64) are properly identified.

VII. ELECTROMAGNETISM

Intuitively, the physics of conducting wires, coils, and parallel plates must involve aspects of dual geometry. Point charges define isotropic flux densities

D = q e r/4πr 2 , B = q m r/4πr 2 , ( 106
)
in units of C/m 2 and Wb/m 2 or tesla, respectively, where r is a radial unit vector and 4πr 2 is the area of a sphere.

Although isolated magnetic monopoles have never been observed [START_REF] Rajantie | The search for magnetic monopoles[END_REF], Maxwell's equations allow sources of both specimen, where linear combinations introduce continuous U(1) symmetry responsible for helicity conservation [START_REF] Bliokh | Dual electromagnetism: Helicity, spin, momentum, and angular momentum[END_REF][START_REF] Cameron | Electric-magnetic symmetry and Noether's theorem[END_REF]. Galilean transport define orthogonal field strengths in A/m and V/m, respectively,

H = v × D, E = -v × B, (107)
where the minus sign conforms with Lenz' rule and Maxwell's equations introduced below.

As we have seen, the product of electric and magnetic charge provides action and must be quantized as shown by Dirac considering a semi-infinite string of magnetic dipoles [START_REF] Dirac | The monopole concept[END_REF],

q e q m = nh, (108)
where n is integer. Given the elementary electric unit e ≡ 1.602176634 × 10 -19 C as measured by the Josephson effect with K J = 2e/h the relevant constant, the magnetic unit or flux quantum obtained for n = 1 takes the value Φ 0 = h/e ≃ 2.07 × 10 -15 Wb. The ratio known as von Klitzing's constant

R K = Φ 0 /e = h/e 2 ≃ 25.8 kΩ, (109)
defines a quantum of resistance unrelated to particle transport, but compatible with internal de Broglie oscillations.

Empty space is assigned linear response by generalizing the phenomenology of displaced charge q e = CU with capacitance in farad F=C/V and induced flux Φ m = LI with inductance in henry H=Wb/A. Closed circuits introduce time constants τ = √ CL and the impedance Z = L/C. By analogy, flux densities are related to field strengths by vector constitutive relations

D = ǫ 0 E, B = µ 0 H, (110)
based on electric permittivity ǫ 0 ∼ 8.85 pF/m and magnetic permeability µ 0 ∼ 4π • 10 -7 H/m formerly a defined constant. Virtual charge and current fluctuations allowed by classical pair production have been suggested as a possible mechanism [START_REF] Leuchs | The quantum vacuum at the foundations of classical electrodynamics[END_REF], where the product

ǫ 0 µ 0 = 1/c 2 (111)
defines a characteristic velocity, while the impedance

Z 0 = 1 ǫ 0 c = µ 0 c = µ 0 /ǫ 0 ∼ 377 Ω, (112)
measures the relative strength of electric and magnetic interactions. The fine-structure constant in SI-units

α = e 2 (4πǫ 0 ) c = Z 0 2R K ∼ 1/137 (113)
is the ratio of medium to matter impedance in a vacuum populated by virtual particles,

A. Electric-magnetic duality Dirac quantization is charge symmetric q e ↔ q m reversing the constant (109) and the role of electric and magnetic quantities. Symmetric response ǫ 0 ↔ µ 0 reverses the impedance (112) introducing a dual vacuum α ↔ 1/4α [START_REF] Witten | Duality, spacetime and quantum mechanics[END_REF] with symmetric Coulomb potentials 4πφ e /c = Z 0 q e /r ↔ 4πφ m /c = Z -1 0 q m /r, (114) and vector potentials A s = φ s /v p . Ordinary wave propagation is ensured by the Lorenz constraint

χ = -∂ X • A P = ∂ t φ/c 2 + ∂ x • A = 0 (115)
with Fourier transform K • A P = 0 which may be interpreted as "grounding" the particle at φ ′ = 0 using [START_REF] Field | Space-time exchange invariance: Special relativity as a symmetry principle[END_REF].

The capacitive limit ǫ 0 → ∞ and µ 0 → 0 implies perfect conductivity Z 0 → 0, while the inductive limit ǫ 0 → 0 and µ 0 → ∞ defines a perfect insulator Z 0 → ∞.

At the field level, cross products (107) are replaced by 2-blades and vector contraction using ( 43)

H = v ∧ D, E = ι(v ∧ B) = v • B, (116)
where flux lines intersect transverse surface elements B • B = 0. A choice of spatial orientation need not be specified. Geometric derivation

∂ x A = ∂ x • A + ∂ x ∧ A introduces multivectors reminiscent of complex Riemann- Silberstein fields F = -∂ X A P m = χ m + E/c + B, (117)
superseding Faraday tensors as well as the combination due to Minkowski

G = -∂ X A P e = χ e + H/c + D, (118)
where scalar terms vanish in the Lorenz gauge (115). Scalar electric charge [START_REF] Itin | An electric charge has no screw sense -a comment on the twist free formulation of electrodynamics by da Rocha and Rodrigues[END_REF] is compatible with polar electrics, while dual magnetics implies pseudoscalar magnetic charge [START_REF] Frankel | Scalar and pseudoscalar charges[END_REF]. The terms q m A e and q m φ e then pick up minus signs ... The combination of complementary geometry and charge symmetry is known as electric-magnetic duality

E ↔ H, D ↔ B, q e ↔ ιq m , (119)
implying F ↔ G exchange symmetry. Symmetric Lorentz forces per unit charge acting on monopoles of both species [START_REF] Katz | The magnetic pole in the formulation of electricity and magnetism[END_REF][START_REF] Rindler | Relativity and electromagnetism: The force on a magnetic monopole[END_REF][START_REF] Moulin | Magnetic monopoles and Lorentz force[END_REF] are defined by Galilean relativity [START_REF] Bellac | Galilean Electromagnetism[END_REF][START_REF] Rousseaux | Forty years of Galilean electromagnetism (1973-2013[END_REF] allowing zero forces

E ′ = E -v G • B, H ′ = H -v G ∧ D, (120)
assuming quasi-stationary fields E ≪ cB and H ≪ cD.

Quasi-uniform flux densities E ≫ cB and H ≫ cD introduce Carrollian kinematics

D ′ = D -v -1 C • H, B ′ = B -v -1 C ∧ E, (121)
as follows by invoking vacuum relations (110) and (111) as well as reciprocal parameters [START_REF] Hestenes | New foundations for classical mechanics[END_REF].

B. Maxwell equations

The vector formulation of Gibbs and Heaviside [START_REF] Arthur | The evolution of Maxwell's equations from 1862 to the present day[END_REF] in Euclidean geometry [START_REF] Itin | Backwards on Minkowski's road. From 4D to 3D Maxwellian electromagnetism[END_REF] is easily modified to accommodate magnetic monopoles [START_REF] Jackson | Classical electromagnetism[END_REF]. The laws of Gauss are independent of time

∇ • D = ρ e , ∇ × H = ∂ t D + j e , (122)
∇ • B = ρ m , -∇ × E = ∂ t B + j m , (123)
while non-stationary flux densities and macroscopic currents j s = ρ s w s constituting individual transport velocities induce circulating Ampere and Faraday fields. The equations are linear, uncoupled, and symmetric except for a sign responsible for Lenz' rule. Nowhere does the speed of light appear and properties of the vacuum must be specified by additional constitutive relations such as (110) and (111).

In the absence of source terms, Fourier transformation reveals that flux densities must be orthogonal to phase velocities and field strengths

v -1 p × H = -D, v -1 p × E = B, (124)
introducing linear wave structure. Eliminating cross products is consistent with Carrollian kinematics (121), where flux densities vanish in uniform time k = 0 at v G = v and v C = v p . Eliminating either fluxes or fields using the particle solutions (107) leads to vector relations of the form

E = -v × (v -1 p × E), (125)
where the identity a ×

(b × c) = (a • c)b -(a • b)c and orthogonality v • E = 0 implies E = (v • v -1 p )E (126)
showing that inverse projection (15) define linear dispersion with phase velocity identified as slowness.

Charge and energy conservation

Macroscopic sources must satisfy charge conservation (98) independent of field dynamics and metric, since only counting operations are involved. Kinematical symmetries separate into Galilean relativity w s ≪ c leaving volumes and densities invariant in absolute time

ρ ′ s = ρ s , j ′ s = j s -ρ s v G , (127)
and Carrollian kinematics w s ≫ c in absolute space

ρ ′ s = ρ s -v -1 C • j s , j ′ s = j s , (128)
allowing absolute currents, where the product ρ s w s is first-order invariant, while densities are shifted in relative time and vanish at v C = v p regardless of optional charge cancellations. Charge-current exchange symmetry applies along tangent directions in accordance with the multisource J = cρ s + j s . Field energy S = cu + S symmetric under light-time exchange is conserved ∂ X • S = 0 according to Poynting's theorem

∂ t u + ∇ • S = -E • j e -H • j m , (129)
where the right-hand side is the power density supplied by source currents,

∂ t u = E • ∂ t D + H • ∂ t B, (130)
is the energy density rate of change, and the power flux through a surface

∇ • S = -E • (∇ × H) + H • (∇ × E), (131)
defines the Poynting vector S = E × H = -E • H. In a linear medium such as (110), the energy density is dipolelike and symmetric

u = 1 2 (E • D + H • B), (132)
where H • B = -H • B and electric-magnetic symmetry suggests a corresponding momentum density due to Minkowski that agrees with that of Abraham D × B = S/c 2 in vacuum [START_REF] Jackson | Classical electromagnetism[END_REF].

Dual equations

Algebraic expressions replace coordinate based vector calculus ∇A = ∇ • A + ∇ ∧ A introducing dual relations equivalent to Eqs. ( 42) and ( 43) interchanging contraction and extension

∇ • A = -ι∇ ∧ A, ∇ × A = -ι∇ ∧ A = -∇ • A, (133)
where A = ιA is a dual vector. Applied to Maxwell's equations, Ampere contraction separates from Faraday extension [START_REF] Arthur | Understanding geometric algebra for electromagnetic theory[END_REF][START_REF] Chappell | Geometric algebra for electric and electronic engineers[END_REF][START_REF] Dressel | Spacetime algebra as a powerful tool for electromagnetism[END_REF],

∇ • D = ρ e , -∇ • H = ∂ t D + j e , (134)
∇ ∧ B = ιρ m , -∇ ∧ E = ∂ t B + ιj m , (135)
driven by pseudoscalar magnetic sources. Invoking multivectors (117) and (118) in the Lorenz gauge, only two equations are needed

∇ • G = ∂ ct D + J e /c, ∇ ∧ F = ∂ ct B + ιJ m /c, (136)
where electric-magnetic duality is the equivalence between Ampere and Faraday parts F ↔ G.

C. Macroscopic constitutive relations

Separate electric and magnetic relations are unaffected by dual magnetics. Macroscopic polarization and magnetization is specified by homogeneous and isotropic linear response at rest [START_REF] Van Dantzig | The fundamental equations of electromagnetism, independent of metrical geometry[END_REF], where permittivity ǫ(ω, k) and permeability µ(ω, k) relative to the vacuum are local functions in Fourier space and n 2 = ǫµ is the refractive index [START_REF] Puska | Covariant isotropic constitutive relations in Clifford's geometric algebra[END_REF]. Vector relations in a medium moving at velocity v G were obtained by Minkowski

D + v -1 C × H = ǫǫ 0 (E + v G × B), (137)
B -v -1 C × E = µµ 0 (H -v G × D), (138)
where scaling factors cancel [START_REF] Rousseaux | On the electrodynamics of Minkowski at low velocities[END_REF][START_REF] Ivezić | The constitutive relations and the magnetoelectric effect for moving media[END_REF]. Dual kinematics (120) and (121) thus adequately describe linear electrodynamics. Galilean symmetry eliminates cross products on the left-hand side, where perfect vacuum conductivity is possible at finite permittivity without induction B → 0 2 known as the electric limit, while finite permeability defines a perfect insulator without displacement in the magnetic limit D → 0 2 . Likewise, Carrollian symmetry eliminates cross products on the right-hand side, where finite permeability defines perfect conductivity in the magnetic limit E → 0 2 , while finite permittivity defines a perfect insulator in the electric limit H 0 2 [START_REF] Lindell | Perfect electromagnetic conductor[END_REF]. Singular contraction in reciprocal limits is ultimately a consequence of the speed of light (111) being factorized.

VIII. CONCLUSIONS

Perceived by our senses everyday phenomena seem to be connected by instantly propagating light signals as argued by Descartes. Rømer's observation that the speed of light is finite eventually led to the idea of causal structure, where only two-way velocities measured by a single clock are physically meaningful. One-way propagation is linked to synchrony by a vicious circle introducing kinematical redundancy comparable to gauge freedom. We have argued that uniform synchrony set by light-time conversion must coexist with equal-time zones transverse to tangent directions. The question raised by Dyson [START_REF] Dyson | Feynman's proof of the Maxwell equations[END_REF] as to how special relativity seemingly arises out of thin air starting from Galilean symmetry c → ∞ has thus been answered: Relative time defined by non-causal extension at finite c may ultimately be traced to kinematical gauge freedom introducing Carrollian time conventionality in the global limit c → 0.

Fundamental insights are highlighted by the following simple facts: Geometry and physical laws transcend coordinate bookkeeping, kinematics and dynamics can not be sharply distinguished, and metric structure is second order. Massive particles carry internal de Broglie clocks and dynamics is provided by the principle of maximum aging circumventing action at a distance. As demonstrated, dual geometric structure is implied by macro-scopic continuity involving only counting operations. Relying on dual vector spaces, we have shown that lighttime permutation introduces first-order relativity and wave-particle duality in classical mechanics, where the celebrated contribution E 0 = mc 2 provides internal energy to localized wave packets. As a result canonical and kinetic particle momenergy coexists resolving the Landé paradox by wave aberration. Remarkably, the concept of de Broglie matter waves reduces to elsewhere time zones in ordinary space whether Euclidean or premetric with no extra time dimension required.

The awkward distinction between polar electrics and axial magnetics required by vector calculus is superseded by electric-magnetic duality based on charge symmetry and geometric duality. Galilean kinematics provides symmetric Lorentz forces acting on point charges, while Carrollian kinematics applies to linear flux propagation. Macroscopic Maxwell equations augmented by the simplest possible constitutive relations define dual vacuum limits that are either perfectly conducting with no electric field or magnetic induction or perfectly insulating lacking either magnetic field or electric displacement.

Pair production and is predicted at finite c with antiparticles simply defined as classical CP Tinversions linked by light-time conversion. Save rare exceptions, separate inversion symmetries are expected to hold separately [START_REF] Pavsic | External inversion, internal inversion, and reflection invariance[END_REF] as suggested by the absence of permanent electric dipoles. Most interestingly, pseudoscalar magnetic dipoles then qualify as classical antiparticle pairs or mesons explaining the observed elusiveness of free monopoles.

ACKNOWLEDGMENTS

Helpful discussions and correspondence with Ulf Leonhardt, Richard Cleyton, and Gary Gibbons is gratefully acknowledged.


Dr Manuel Barday

Dr Pierre Bouillac

Dr Yoann Coquerel

Prof Muriel T Amatore
email: muriel.amatore@univ-amu.fr

Thierry Constantieux
email: thierry.constantieux@univ-amu.fr

Jean Rodriguez

Prof K C Peter

Vollhardt

Prof J Constantieux

Rodriguez

Enantioselective Organocatalytic Syntheses and Ring-Expansions of Cyclobutane Derivatives

Keywords: Medium-ring compounds, Organocatalysis, Ring expansion, Small ring systems, Strained molecules

published or not. The documents may come   L'archive ouverte pluridisciplinaire

Introduction

Cyclobutane and cyclobutene are four-membered cyclic hydrocarbons that embed strain due to geometrical constraints. [1] Nevertheless, these motifs are present in naturally occurring products, [2] which has led organic chemists to develop numerous synthetic methodologies for their synthesis and use as molecular platforms to access more complex, or biologically relevant products. [3] Taking advantage of the ring strain, [4] progress in homogeneous catalysis with metal complexes have enabled the development of original synthetic approaches based on the selective cleavage of one of the C C bonds, unlocking new opportunities for synthesis. [5] Comparatively, enantioselective organocatalysis has only been used scarcely, and it is only recently that its potential was revealed. The objectives of this mini-review are to highlight selected representative recent organocatalytic methods, involving at least the formation/cleavage of one C C bond, that allow i) for the enantioselective synthesis of cyclobutane derivatives, and ii) for the enantioselective, sometimes enantiospecific, ring-expansion reactions of cyclobutane derivatives towards five-to eightmembered rings.

Enantioselective organocatalytic synthesis of cyclobutane derivatives

Although recent advances have allowed efficient stereocontrol, the organocatalytic enantioselective synthesis of structurally diversified cyclobutanes is still challenging. By far, (2 + 2) annulation reactions represent the most straightforward strategy. [6] For categorisation purpose, one can distinguish between organocatalytic methodologies involving covalent activation through iminium cation or enamine intermediates, and approaches relying on non-covalent activation such as hydrogen-bonds. Complementarily, dual catalysis based on the combination of organocatalysis and photoactivation also emerged as a highly enantioselective approach.

Covalent activation

via iminium cation intermediate

In this section, most of the organocatalytic enantioselective (2 + 2) annulations are initiated by FriedelCrafts-type processes. As pioneers, Ishihara's group described in 2007 the reaction between unactivated alkenes and α-acyloxyacroleins catalysed by an organoammonium salt derived from a chiral triamine, affording the desired cycloadducts in moderate yield, but with excellent diastereo-and enantioselectivity (Scheme 1). [7] A stepwise mechanism involves in situ iminium activation of the acrolein derivative, triggering a Michael addition with the alkene followed by intramolecular trapping of the generated carbocation by the resulting enamine. To account for the observed absolute and relative stereochemistry, the authors postulated the crucial role of a hydrogen bond between the carbonyl group of the acyloxy moiety and the protonated form of the catalyst at the pyrrolidine ring, stabilizing the iminium cation in its (Z)-configuration.

A related olefin (2 + 2) annulation approach, using diarylprolinol silyl ethers as catalysts, was developed in 2013 by Xu and co-workers for a highly enantioselective access to functionalised cyclobutanes from 2-vinylpyrroles and α,β-unsaturated aldehydes (Scheme 2). [8] This domino reaction proceeds via the addition of the electron-rich 2-vinylpyrrole onto the conjugated iminium cation intermediate, leading to a transient enamine having a distal aza-fulvenyl cation. Rearomatisation by conjugate addition followed by reduction delivered the desired products in good yields and with excellent diastereo-and enantioselectivity. The overall process advantageously exploits the concept of organocatalysed domino reactions based on the iminium-enamine double activation of enals. [9] Another related domino approach was developed in 2016 by the group of McNulty, using an alkenylphenol as electronrich alkene, i. e. a cinnamyl alcohol derivative in which the alkene is activated by mesomeric effect from the hydroxyl group in 4-position of the aryl (Scheme 3). [10] As many naturally occurring cyclobutanes appear to be derived from the dimerisation of cinnamyl or coumaryl diene amides, this elegant methodology offers a stereoselective entry to the total synthesis of biologically active compounds with this strained carbocyclic core. From a mechanistic point of view, after conjugate addition of the activated alkene to the iminium cation intermediate, subsequent cyclisation occurs via enamine conjugate addition to the distal p-quinomethide cation. More recently, Xu and co-workers reported a highly enantioselective access to very strained and relatively unexplored spiro[2.3]hexane derivatives, by introducing methylenecyclopropanes as electron-rich alkenes in a formal (2 + 2) annulation operating by a domino Michael addition/ring expansion/cyclisation reaction (Scheme 4). [11] Compared to the above-mentioned reports, one of the key points for the success of this transformation is the modification of the organocatalyst that activates the enal, by using a difluoro-substituted diarylsilylprolinol ether (TDS = thexyldimethylsilyl). After Michael addition of the alkene to the β position of the conjugated iminium cation, rearrangement of the resulting carbocation intermediate by ring expansion of the cyclopropane occurs to install the new four-membered core. Finally, a new cyclopropane is formed by cyclisation of the resulting enamine with the rearranged carbocation. The reaction can be scaled up easily and the products can be selectively post-functionalised, offering an interesting route to more complex skeletons present in natural products and compounds of pharmaceutical interest.

via dienamine intermediate

The reaction between a secondary amine with an α,β-unsaturated aldehyde bearing an aliphatic group on the γ-position results in the formation of a dienamine product. [12] In this way, amino-catalysis offers effective solutions to the functionalisation of unsaturated aldehydes, either in α-position or in γ-position. The latter usually results in a more difficult stereocontrol due to the greater distance between the catalyst and the reacting carbon atom. In this context, Jørgensen and co-workers addressed in 2012 the highly challenging development of an enantioselective (2 + 2) annulation between the remote double bond of a dienamine and a nitroolefin (Scheme 5). [13] This methodology offers a direct access to polysubstituted cyclo- Shortly after, Vicario's group reported a conceptually similar (2 + 2) annulation based on a dual catalysis approach combining a diarylprolinol silyl ether and an achiral thiourea catalysts (Scheme 6). [14] The former covalently activates the enal and the latter activates the α-hydroxymethylnitrostyrene derivative through hydrogen bonding. A Michael/Michael/hemiacetalisation domino sequence, followed by an oxidation step, results in the formation of cyclobutane-containing bicyclic lactones with excellent diastereo-and enantioselectivity. These pioneering results, together with subsequent contributions from Wang [15] and Jørgensen, [16] illustrate the high potential of cooperative dienamine/hydrogen-bonding activation.

Non-covalent activation

Another major activation strategy within enantioselective organocatalysis is based on the use of chiral hydrogen-bond donors, [17] including essentially Cinchona alkaloid derivatives, diols, ureas, thioureas, squaramides and phosphoric acids. Among them, bifunctional catalysts possessing both H-bonding function and basic/nucleophilic moiety have emerged as platforms capable of simultaneously activating electrophilic and nucleophilic reaction partners, respectively. In many cases, the non-covalent interactions allowed reaching nearly perfect spatial organisation in the approaches to the transition states, leading to excellent regio-and stereoselectivity. [18] In this context, Hong's group showed in 2018 that a bifunctional catalyst of the squaramide/cinchona alkaloid family could promote the (2 + 2) annulation between nitroolefins and β,γ-unsaturated ketones in a highly regio-, diastereo-and enantioselective manner (Scheme 7). [19] This Michael/Michael domino sequence provides diversely functionalised cyclobutanes with four contiguous stereogenic carbon atoms, including a quaternary one. In this formal annulation, the nitroolefin is activated by H-bonding interactions with the squaramide moiety and the vinylogous ketone enolate is generated by the basic part of the organocatalyst. A conjugate addition/nitro-Michael cyclisation sequence afforded the four-membered core in excellent diastereo-and enantioselectivity.

More recently, a stepwise related (2 + 2) annulation between nitroolefins and β,γ-unsaturated pyrazolyl amides, leading to the same family of cyclobutanes, has been reported by Huang and co-workers (Scheme 8). [20] The strategy involves first a chiral squaramide-catalysed vinylogous Michael addition, delivering the corresponding adducts with high diastereo-and enantiose-lectivity, which are then enantiospecifically cyclised by action of a catalytic amount of trimethylguanidine.

Chiral phosphoric acids are alternative popular organocatalysts operating by non-covalent interactions. [21] However, their application to the formation of cyclobutane derivatives through (2 + 2) annulations has only been reported once, in combination with a Lewis acid, by Ishihara and co-workers. They developed the reaction between phenyl vinyl sulfide and various α-and/or β-substituted acroleins to lead to the synthesis of cyclobutane adducts with high enantiomeric excesses (Scheme 9). [22] To explain the regio-and the enantioselectivity, the authors proposed a transition state in which the acrolein derivative is activated, on the one hand, by hydrogen bonding with the proton of the Brønsted acid catalyst, and on the other hand by additive hydrogen bonds developed by one of the bromine atoms of the boron tribromide Lewis acid catalyst, itself coordinated to the oxo group of the phosphoric acid catalyst (catalytic Lewis pair).

Dual photo-and organocatalytic activation

via non-covalent activation

[2 + 2] Cycloadditions are suitable for photochemical activation, [23] but their stereochemical control remained challenging until Bach and co-workers demonstrated in 2009 the first highly enantioselective organocatalytic photocycloaddition. [24] They designed a chiral sensitizer combining a chromophore from the xanthone family and an amide moiety able to develop hydrogen-bonding type interactions. Using 10 mol% of this new dual activator, the intramolecular [2 + 2] photocycloaddition of a simple prochiral quinolone proceeded with excellent regio-and enantiocontrol (Scheme 10). This agrees with the catalyst-substrate complex absorbing UV light at longer wavelength than the free substrate. These pioneering results, together with subsequent contributions from the same group, [25] perfectly fulfil the requirements of a chiral triplet sensitizer for enantioselective photoreactions.

Contemporarily, Sibi, Sivaguru and co-workers demonstrated that atropisomeric binaphthyl-based thiourea organocatalysts can efficiently absorb photons and promote similar intramolecular transformations in the coumarin series, with good enantiocontrol (Scheme 11). [26] Under UV irradiation and with relatively low catalyst loading, the reaction gives access to fused four-membered ring frameworks with good to excellent enantioselectivities. In this transformation, the photon-excited thiourea catalyst activates the coumarin carbonyl moiety though hydrogen-bonding, and the generated exciplex results in the formation of the desired photo-cycloadduct.

Although UV irradiation is frequently used in photochemistry, the energy associated to the corresponding photons may be, in some cases, significantly higher than the triplet energy of many photochemical substrates. This limitation prompted researchers to move towards more suitable photocatalysts active under visible-light irradiation. Along these lines, Bach and his team modified their previously designed chiral sensitizer, switching the chromophore part from the xanthone to the thioxanthone series. [27] By this way, they were able, in particular, to perform a highly enantioselective intermolecular [2 + 2] photocycloaddition between 2(1H)-quinolones and electron-deficient olefins, with excellent regio-, diastereoselectivity, and enantiomeric excesses (Scheme 12). [28] Alternatively, the same group also developed a dual catalytic system, combining the use of a chiral bis-thiourea as a template for H-bonding activation of the substrate and an achiral thioxanthone as triplet sensitizer for enantioselective intramolecular [2 + 2] photocycloaddition of dihydropyridones, however with lower enantioselectivity. [29]

via covalent activation

Recently, Alemán and co-workers reported an original visiblelight photo-and aminocatalysed strategy to obtain highly enantioenriched cyclobutane derivatives through [2 + 2] cycloaddition between enones and olefins (Scheme 13). [30] They postulated that the in situ generation of an iminium ion-based charge transfer complex allows its preferential light excitation over the carbonyl species, thus avoiding any racemic background reaction. In the presence of the alkene partner, the [2 + 2] photocycloaddition gives a biradical intermediate, that evolves to the final product after hydrolysis of the resulting transient cyclobutyl iminium ion. Supported by DFT calculations, this approach was validated leading to the desired fourmembered cyclic compounds in high yields, with good diastereomeric ratios and enantiomeric excesses.

Enantioselective ring-expansion to five-membered ring systems

Using cyclobutenones

In 2015, the highly regio-and stereoselective 1,3-dipolar annulation of monosubstituted γ-chlorocyclobutenones with azomethine imines affording original heterocyclic compounds has been reported by Chi and co-workers (Scheme 14). [31] Using a chiral isothiourea organocatalyst, the formation of a reactive dienolate intermediate is favoured by rupture of a weakened C C bond in the zwitterionic adduct obtained by addition of the Lewis base catalyst to the ketone group in the substrate. The catalytic dienolate species then undergoes α-selective addition to the 1,3-dipole followed by cyclisation. Optimisation studies have revealed that the presence of a chlorine atom at the γ-position of the cyclobutenone partner was crucial for the reaction to occur. [32] The corresponding 1,3-dipolar annulation products were obtained in moderate to good yields and high enantioselectivities.

Using cyclobutanol derivatives

The semipinacol rearrangement of strained allylic cyclobutanols bearing a trichloroacetimidate moiety as potential leaving group can furnish the corresponding α-vinyl cyclopentanones in an efficient manner with high enantioselectivity through organocatalytic activation (Scheme 15). [33] In the presence of a chiral Brønsted acid, the substrate undergoes a semipinacol rearrangement leading to optically active cyclopentanones with a quaternary carbon atom. This methodology has been previously developed with cyclic vinylogous γ-ketols to furnish spirocyclic diketones featuring a quaternary stereogenic carbon atom with good to excellent yields under dual catalysis in the presence of a primary amine chiral organocatalyst and a chiral acid co-catalyst (N-Boc-L-phenylglycine, NBLP, Scheme 16). [34] Several cyclic enones having a cyclobutanol substituent at C3 were subjected to the reaction conditions and afforded the corresponding spiro-1,4-diketones via a semipinacol-type 1,2-migration in the chiral intermediate. Both diastereomers of the cyclobutanol substrates were amenable to the transformation with comparable rates (no kinetic resolution observed), and in most of the reported examples, the R 1 and R 2 substituents were identical, avoiding match/mismatch issues. However, with either R 1 or R 2 being a hydrogen atom, it was demonstrated that the match diastereomer is with R 2 < C = > H (cis isomer). Recent computational studies provided some mechanistic insights and confirmed that intrinsic steric repulsions in the catalytic chiral iminium cation were at the origin of enantioselectivity. [35] The chiral carboxylic acid additive was found necessary to control the conformation of the cyclohexene core in the approach to the transition state and to serve as a proton shuttle, resulting in an increased enantiocontrol.

The access to valuable spiro [4.4]nonane-1,6-dione cores has been investigated using a Nazarov cyclisation/semipinacol rearrangement sequence. [36] Contrary to tedious multi-step approaches generally described for the synthesis of such compounds, the proposed methodology advantageously offers a one-step procedure and allows the construction of four controlled stereogenic carbon atoms, with high yields and enantioselectivities (Scheme 17). Mechanistically, a chiral Ntriflylphosphoramide Brønsted acid is able to trigger the enantioselective Nazarov cyclisation of unactivated racemic substrates (no kinetic resolution was observed). At this point, the cationic oxyallyl intermediate induces a stereocontrolled semipinacol rearrangement to furnish the spiro [4.4]nonane-1,6diones.

Another ring-expansive semipinacol rearrangement was developed using the SOMO organocatalytic activation mode. [37] In this case, strained allylic cyclobutanols were appropriate SOMO-nucleophiles affording a new carbocation intermediate amenable to the rearrangement (Scheme 18). [38] According to this strategy, a large panel of 1,5-dicarbonyl compounds bearing a quaternary stereogenic carbon atom have been obtained with satisfactory results in terms of yield and stereoselectivity. Some intramolecular versions of this reaction afforded the corresponding bicyclic products, which are valuable synthetic platforms as illustrated with an elegant enantioselective total synthesis of (+)-cerapicol.

The semipinacol rearrangement of cyclobutanols induced by the formation of a halonium cation intermediate has been described by Alexakis and co-workers as an elegant approach to tricyclic β-fluorospiroketones. [39] Excellent yields and stereoselectivities have been reported using a chiral BINOL-based phosphoric acid sodium salt and Selectfluor as the cationic fluorinating agent (Scheme 19).

In the course of their optimisation studies, the authors confirmed that the chiral organocatalyst was responsible for both the stereocontrolled transfer of fluorine atom and the alkyl migration step through an ion-pairing mechanism. Moreover, reactions were conducted in a basic mixture of non-polar solvents in which the chiral phosphoric acid anion behaved as a homogeneous anionic phase-transfer catalyst for insoluble cationic Selectfluor, producing a catalytic soluble chiral fluori-nating reagent. [40] Later, alongside mechanistic studies, the substrate scope of this reaction was extended and successfully applied to the deracemisation of chiral substrates. [41] The synthesis of valuable chiral organosulfur compounds has also been investigated based on a similar strategy with excellent results (Scheme 20). [42] A catalytic combination of chiral Lewis base selenide and chiral phosphoric acid was used for the stereocontrolled transfer of an electrophilic sulfur atom released from N-phenylthiosaccharin. Under these conditions, a catalytic chiral sulfenylating agent was generated in situ and the chiral phosphate directed the allylic alcohol substrate for the formation of the favored thiiranium ion intermediate and subsequent semipinacol rearrangement.

Enantioselective ring-expansion to six-membered ring systems 4.1. Using cyclobutenones

Chi and his group reported in 2015 the first example of Nheterocyclic carbenes (NHC)-catalysed stereoselective ring-expansion reaction of cyclobutenones (Scheme 21). [43] Some computational mechanistic investigations showed that a concerted asynchronous process takes place with isatin imines (true aza-Diels-Alder cycloaddition), while a stepwise 1,2addition/lactamisation is preferred in the case of cyclic sulfonylimines (formal aza-Diels-Alder cycloaddition). [44] The same year, Zhang and co-workers developed another Lewis base-catalysed ring opening of cyclobutenones, now using an amino-acid derived phosphine-based catalyst bearing a thiourea moiety in a (4 + 2) annulation (Scheme 22). [45] According to the authors, activation of the dienophile via hydrogen-bonding allows the reaction to proceed through the topicity depicted in scheme 22. The use of sodium iodide as an additive is believed to stabilize the zwitterionic dienolate intermediate.

Using cyclobutanol silyl ethers

As a unique example, the Lupton group developed a highly stereoselective NHC-catalysed, indirect fluoride-induced ringopening/aldolisation/lactonisation cascade from some cyclobutanol trimethylsilyl ethers to form bicyclic six-membered products in excellent yields (Scheme 23). [46] The cascade process is initiated upon addition of the NHC catalyst onto the α,βunsaturated acyl fluoride which liberates the required catalytic fluoride anion atom needed for the deprotection of the cyclobutanol substrate. A charge-accelerated ring-opening step occurs to generate the corresponding acyclic stabilised ester enolate having a distal aldehyde moiety amenable to further cyclisation by aldolisation. Mechanistic studies have demonstrated that O-addition takes place leading to the ketal intermediate which undergoes a [3,3]-sigmatropic rearrangement. Subsequent aldolisation/lactonisation regenerates the NHC-catalyst and releases the desired product.

Using cyclobutanones

Our group recently published the organocatalysed enantiospecific ring opening of α-disubstituted cyclobutanones (Scheme 24). [47] Relying on a previously reported methodology in which we were able to access α-disubstituted cyclobutanones enantioselectively, [48] those substrates were reacted with the Okamoto isothiourea catalyst to trigger the ring opening. A series of highly enantioenriched glutarimides was thus isolated in moderate to excellent yields. Excellent enantiospecificity was observed though with low diastereoselectivities in some cases. A cross-experiment showed that with alternative basic catalysts, some retro-Michael/Michael processes occur at significant rate, leading to partial racemisation in the products. A computational mechanistic study was performed in order to fully elucidate the reaction pathway.

Enantioselective ring-expansion to medium size ring systems

In 2019, Coquerel, Rodriguez, and co-workers reported a Michael/cyclisation/fragmentation domino reaction from cyclobutanones, which allowed preparing benzo-fused eight membered lactams through an original four-atom ring expansion approach (Scheme 25). [49] The benzazocinone derivatives were obtained in moderate to good yields with high enantioselectivities and good diastereoselectivities. Interestingly, it was demonstrated that diastereoselectivity, i. e. the relative configuration of the carbon atom bearing the secondary amide substituent, is governed by thermodynamic factors.

Inspired by this work, Yang, Deng, and co-workers have recently reported a related original three-atom ring expansion of cyclobutanones based on similar Michael/cyclisation/fragmentation domino reaction, now using 2-nitrovinyl indoles instead of the previously employed ortho-aminonitrostyrenes (Scheme 26). [50] It resulted in an elegant and efficient synthetic Scheme 23. NHC-catalysed ring-opening/aldolisation/lactonisation sequence.

Scheme 24. Enantiospecific two-atom ring-expansion of cyclobutanones.

approach to highly enantioenriched azepino [1,2-a]indole derivatives. Following a different approach, Jiang et al. have reported recently an original four-atom ring expansion of cyclobutenones with 1-azadienes (Scheme 27). [51] Using a nucleophilic organocatalyst the reaction is initiated by fragmentation of the cyclobutenone substrate into a chiral zwitterionic dienolate that can then undergo (4 + 4) annulation through a Michael addition/cyclisation sequence. The products are obtained in high yields with excellent enantioselectivities. Remarkably, this is a unique example for a bulky tertiary amine catalyst to promote the catalytic fragmentation of cyclobutenones (see above for examples with other nucleophilic catalysts).

Summary and Outlook

Recent progress in the enantioselective organocatalytic syntheses and reactions of cyclobutane derivatives are highlighted in this mini-review. The synthesis of cyclobutane derivatives usually involves stepwise processes, typically Friedel-Crafts initiated (2 + 2) annulations, or photochemically-promoted concerted [2 + 2] cycloadditions. Cyclobutane derivatives have been successfully used in efficient ring expansion reactions. For one-atom ring-expansion reactions, the most productive approach has been the semipinacol rearrangement of cyclobutanol derivatives using either aminocatalysts operating by iminium ion activation or Brønsted acid catalysts providing hydrogen-bonding interactions. For two-or-more-atom ringexpansion reactions, two promising approaches emerged to realize (4 + n) annulations: one relies on the fragmentation of cyclobutenones using nucleophilic catalysts to produce zwitterionic dienolate intermediates amenable to add to electrophilic moieties followed by cyclisation, and the other involves Michael addition-initiated processes of cyclobutanones followed by cyclisation and fragmentation. Taking full advantage of the ring strain, it can reasonably be anticipated that more of these enantioselective organocatalytic ring expansive (4 + n) annulation reactions will be discovered soon.

Manuel

  Barday was born in Chalon-sur-Saône (France). After studying chemistry at the UPMC (Paris, France), he graduated in 2018 with a Ph.D. from the University of Liverpool (United Kingdom) under the supervision of Christophe Aïssa. He worked as a postdoctoral fellow at the IRCOF (Rouen, France) with Tatiana Besset and joined the group in 2020. His interests focus on organo-and organometallic catalysis. Pierre Bouillac was born in Strasbourg, France, in 1993. He obtained his master's degree in 2018 in University of Bordeaux where he conducted research for Pr. Stéphane Quideau and Pr. Graham K. Murphy in Waterloo, Canada. He is now pursuing his Ph.D. under the guidance of Pr. Thierry Constantieux and Dr. Muriel Amatore in Aix-Marseille University. His main research interests focus on stereoselective NHC-and transition-metal-catalysed processes. Yoann Coquerel received a Ph.D. degree from Grenoble-Alpes University with Prof. Jean-Pierre Deprés in 2001. After a post-doctoral appointment at Florida State University, he began his academic career as a CNRS researcher in 2003 at Aix-Marseille University with Prof. Jean Rodriguez, where he received his habilitation in 2009. In 2015, he was promoted Research Director at CNRS and became a "Junior Distinguished Member" of the French Chemical Society. Dr. Coquerel's research focuses on molecular chirality and the development of stereoselective methods for chemical synthesis, including applications with short-lived synthetic intermediates, in enantioselective organocatalysis, and more recently in the field of chiral polycyclic aromatic molecules. Muriel Amatore was born in Paris in 1979 and obtained her Ph.D. in 2006 with Dr. C. Gosmini (UPEC, France). She carried out three postdoctoral research stays in enantioselective organocatalysis (Princeton University, Prof. D. W. C. MacMillan) and organometallic catalysis (Ecole Polytechnique, Dr C. Gosmini/Université Pierre et Marie Curie, Dr. Corinne Aubert) before achieving a permanent position as assistant professor at UPMC (2010). In 2018, she moved to Aix-Marseille Université to join the group of Prof. Thierry Constantieux (iSm2, STeRéO). Her current research interests center on the development of transition metal-catalysed and organocatalysed processes for their application in stereoselective synthesis. Thierry Constantieux was born in Pau, France, in 1968. After studying chemistry at the University Bordeaux I (France), he received a Ph.D. degree in 1994. He completed his Habilitation in 2004, at Aix-Marseille Université (France), where he is currently Professor, and Director of the Doctoral School of Chemical Sciences. His main research interest is focused on the development of new eco-compatible synthetic methodologies, especially organocatalytic enantioselective domino and multicomponent reactions from 1,3-dicarbonyl compounds, and their applications in heterocyclic chemistry. Jean Rodriguez was born in Cieza (Spain) in 1958, and in 1959 his family emigrated to France. After studying chemistry at the University of Aix-Marseille (France), he completed his Ph.D. as a CNRS researcher in 1987, and his Habilitation in 1992, also at Marseille, where he is currently Professor and Director of the UMR-CNRS-7313-iSm2. His research interests include the development of multiple bondforming transformations, including domino and multicomponent reactions, and their application in stereoselective organocatalysed synthesis. In 1998, he was awarded the ACROS prize in Organic Chemistry, and in 2009 the prize of the Division of Organic Chemistry from the French Chemical Society (SCF). In 2013, he became "Distinguished member" of the French Chemical Society.

Scheme 1 .

 1 Scheme 1. Pioneering results from Ishihara's group.

  Scheme 5. (2 + 2) Annulation via dienamine activation with bifunctional catalyst (DEA: N,N-diethylacetamide)

Scheme 8 .

 8 Scheme 8. Base-mediated enantiospecific cyclisation toward cyclobutanes.

Scheme 11 .

 11 Scheme 11. Organo-photocatalytic enantioselective [2 + 2] cycloaddition.

Scheme 14 .

 14 Scheme 14. Lewis base-catalysed 1,3-dipolar annulation.

Scheme 20 .

 20 Scheme 20. Sulfenylation/semipinacol rearrangement of allylic cyclobutanols.

Acknowledgements

M. B. thanks the Fondation de la Maison de la Chimie for a postdoctoral fellowship. P. B. thanks the French government for a doctoral studentship (2018)(2019)(2020)(2021). Financial support from Aix-Marseille University, Centrale Marseille and the CNRS is also gratefully acknowledged.

Conflict of Interest

The authors declare no conflict of interest.