# Extended Born relativity, Nambu 3-form and ternary ($n$-ary) symmetry

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Background: Classical Mechanics is based on the Poincare-Cartan two-form

$$\omega_2=dx\wedge dp$$

where $p=\dot{x}$. Quantum mechanics is secretly a subtle modification of this. By the other hand, the so-called Born-reciprocal relativity is based on the "phase-space"-like metric

$$ds^2=dx^2-c^2dt^2+Adp^2-BdE^2$$

and its full space-time+phase-space extension:

$$ds^2=dX^2+dP^2=dx^\mu dx_\mu+\dfrac{1}{\lambda^2}dp^\nu dp_\nu$$

where $$P=\dot{X}$$

Note: particle-wave duality is something like $x^\mu=\dfrac{h}{p_\mu}$.

In Born's reciprocal relativity you have the invariance group which is the intersection of $SO (4 +4)$ and the ordinary symplectic group $Sp (4)$, related to the invariance under the symplectic transformations leaving the Poincaré-Cartan two-form invariant. The intersection of $SO(8)$ and $Sp(4)$ gives you, essentially, the unitary group $U (4)$, or some "cousin" closely related to the metaplectic group.

We can try to guess an extension of Born's reciprocal relativity based on higher accelerations as an interesting academical exercise (at least it is for me). In order to do it, you have to find a symmetry which leaves spacetime+phasespace invariant, the force-momentum-space-time extended Born space-time+phase-space interval

$ds^2=dx^2+dp^2+df^2$

with $p=\dot{x}$, $f=\dot{p}$ in this set up. Note that is is the most simple extension, but I am also interested in the problem to enlarge it to extra derivatives, like Tug, Yank,...and n-order derivatives of position. Let me continue. This last metric looks invariant under an orthogonal group $SO (4+4+4) = SO (12)$ group (you can forget about signatures at this moment).

One also needs to have an invariant triple wedge product three-form

$$\omega_3=d X\wedge dP \wedge d F$$

something tha seems to be connected with a Nambu structure and where $P=\dot{X}$ and $F=\dot{P}$ and with invariance under the (ternary) 3-ary "symplectic" transformations leaving the above 3-form invariant.

My Question(s): I am trying to discover some (likely nontrivial) Born-reciprocal like generalized transformations for the case of "higher-order" Born-reciprocal like relativities (I am interested in that topic for more than one reason I can not tell you here). I do know what the phase-space Born-reciprocal invariance group transformations ARE (you can see them,e.g., in this nice thesis BornRelthesis) in the case of reciprocal relativity (as I told you above). So, my question, which comes from the original author of the extended Born-phase space relativity, Carlos Castro Perelman in this paper, and references therein, is a natural question in the context of higher-order Finsler-like extensions of Special Relativity, and it eventually would include the important issue of curved (generalized) relativistic phase-space-time. After the above preliminary stuff, the issue is:

What is the intersection of the group $SO (12)$ with the ternary group which leaves invariant the triple-wedge product

$$\omega_3=d X\wedge dP \wedge d F$$

More generally, I am in fact interested in the next problem. So the extra or bonus question is: what is the ($n$-ary?) group structure leaving invariant the ($n+1$)-form

$$\omega_{n+1}=dx\wedge dp\wedge d\dot{p}\wedge\cdots \wedge dp^{(n-1)}$$

where there we include up to ($n-1$) derivatives of momentum in the exterior product or equivalently

$$\omega_{n+1}=dx\wedge d\dot{x}\wedge d\ddot{x}\wedge\cdots \wedge dx^{(n)}$$

contains up to the $n$-th derivative of the position. In this case the higher-order metric would be:

$$ds^2=dX^2+dP^2+dF^2+\ldots+dP^{(n-1)^2}=dX^2+d\dot{X}^2+d\ddot{X}^2+\ldots+dX^{(n)2}$$

This metric is invariant under $SO(4(n+1))$ symmetry (if we work in 4D spacetime), but what is the symmetry group or invariance of the above ($n+1$)-form and whose intersection with the $SO(4(n+1))$ group gives us the higher-order generalization of the $U(4)$/metaplectic invariance group of Born's reciprocal relativity in phase-space?

This knowledge should allow me (us) to find the analogue of the (nontrivial) Lorentz transformations which mix the

$X,\dot{X}=P,\ddot{X}=\dot{P}=F,\ldots$

coordinates in this enlarged Born relativity theory.

Remark: In the case we include no derivatives in the "generalized phase space" of position (or we don't include any momentum coordinate in the metric) we get the usual SR/GR metric. When n=1, we get phase space relativity. When $n=2$, we would obtain the first of a higher-order space-time-momentum-force generalized Born relativity. I am interested in that because one of my main research topics are generalized/enlarged/enhacend/extended theories of relativity. I firmly believe we have not exhausted the power of the relativity principle in every possible direction.

I do know what the transformation are in the case where one only has $X$ and $P$. I need help to find and work out myself the nontrivial transformations mixing $X,P$ and higher order derivatives...The higher-order extension of Lorentz-Born symmetry/transformation group of special/reciprocal relativity.

This post imported from StackExchange Physics at 2022-06-10 17:30 (UTC), posted by SE-user riemannium

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