LOG#201. Generalized absement.

Absement is generally defined as

(1)   \begin{equation*} \vec{A}=\int \vec{r}(t) dt \end{equation*}

We can even integrate the absement (p-1)-times (position integrated n-times) to get

(2)   \begin{equation*} \vec{A}_n=\int \vec{r}(t) dt^n=\int \vec{A}(t) dt^{p-1}=\int \cdots \int\vec{r}(t) (dt)\underbrace{\cdots}_{n-times} dt \end{equation*}

i.e.

(3)   \begin{equation*} \vec{A}_n=\int \cdots \int\vec{A}(t) (dt)\underbrace{\cdots}_{(p-1)-times} dt \end{equation*}

with dimensions of LT^n and units of ms^n. However, if time is multidimensional, and the time vector is written as

(4)   \begin{equation*} \vec{t}=\left(t_1,\ldots, t_r\right) \end{equation*}

We can indeed define some interesting generalizations of the absement.

Firstly, the scalar absement:

(5)   \begin{equation*} A_s=\int \vec{r}\left(t_1,\ldots,t_r\right)\cdot d\vec{t} \end{equation*}

Second generalization: the pseudovector (bivector) absement (valid in when the cross product is avalaible):

(6)   \begin{equation*} \vec{A}_B=\int \vec{r}\left(t_1,\ldots,t_r\right)\times d\vec{t} \end{equation*}

Thirdly, the exterior (scalar, vector or tensor) absement (valid in arbitrary dimensions!)

(7)   \begin{equation*} A_E=\int X\left(t_1,\ldots,t_r\right)\wedge  dt_1 \wedge \cdots \wedge dt_r \end{equation*}

(8)   \begin{equation*} A_V=\int X^\mu\left(t_1,\ldots,t_r\right)\wedge  dt_1 \wedge \cdots \wedge dt_r \end{equation*}

(9)   \begin{equation*} A_T=\int X^{\mu_1\cdots \mu_s}_{\;\;\;\;\;\;\;\;\;\;\nu_1\cdots \nu_p}\left(t_1,\ldots,t_r\right)\wedge  dt_1 \wedge \cdots \wedge dt_r \end{equation*}

Of course, you can easily include the spatial variables as well in the above definitions, but it would drive us to think absement in terms of field theory, what it is indeed “natural”…Note that separability of variables is a subtle issue in quantum mechanics due to entanglement! Something similar happens in relativity when two tensors are decomposable into pieces with tensor product or when you have an antisymmetric tensor decomposable into exterior products (decomposable p-forms).

Moreover, if X is a 1-vector or 1-form (by duality), but we can even generalize this to an arbitrary antisymmetric q-tensor (q-form, by duality). Thus, in that case, the fourth generalized absement is a (q+r)-form (or (q+r)-vector):

(10)   \begin{equation*} A=\int X_q \wedge  dt_1 \wedge \cdots \wedge dt_r=\int X^{\mu_1\cdots \mu_q} d^r T \end{equation*}

Indeed, we can even go further, by going to the full-relativistic case, defining the speed of light vector (1-form dual)

(11)   \begin{equation*} \vec{C}=(c_1,\ldots,c_r) \end{equation*}

With this definition, in 1T physics, absement has units of L^2\sim LT, as dimension of length is the same of that of time, and hinting that area or absement are MORE fundamental

(12)   \begin{equation*} \vec{A}=\int \vec{r}(t) d(ct) \end{equation*}

Integrating the absement (p-1)-times (the position n-times) to get

(13)   \begin{equation*} \vec{A}_n=\int \vec{r}(t) dt^p=\int \cdots \int\vec{r}(t) (cdt)\underbrace{\cdots}_{n-times} (cdt) \end{equation*}

we get something with units LT^{n}, or using the speed of light (p-1)-times, L^{p} units. If the q-form is separable or not is irrelevant. The generalized absement will have dimensions (after speed of light conversion) of L^{p}. Again, only a fundamental length is important and write:

(14)   \begin{equation*} \vec{A}_B=\int \vec{r}\left(t_1,\ldots,t_r\right)\times d(\vec{ct}) \end{equation*}

(15)   \begin{equation*} A_E=\int X\left(t_1,\ldots,t_r\right)\wedge  d(c_1t_1) \wedge \cdots \wedge d(c_rt_r) \end{equation*}

(16)   \begin{equation*} A=\int X_q \wedge  dt_1 \wedge \cdots \wedge dt_r=\int X^{\mu_1\cdots \mu_q} d^r (CT) \end{equation*}

Finally, from C-space (Clifford spaces) or super-spaces (and even hyper-superspaces!), you can create the most general absement-like object as follows. From the Clifford product AB=A\cdot B+A\wedge B, and

(17)   \begin{equation*} \mathcal{A}=\int X dT=\int X\cdot dT+ X\wedge dT \end{equation*}

or

(18)   \begin{equation*} \mathcal{A}=\int X d(CT)=\int X\cdot (CdT)+ X\wedge d(CT) \end{equation*}

and more generally

(19)   \begin{equation*} \mathcal{A}=\int \langle X d(CT)\rangle_{w} \end{equation*}

where we take the order w-vector from the Clifford product (or just a sum of some grades up to order w). This global generalization for any multivector and polyvector has a very interesting feature. It simplifies the relationship between area and absement. Formally,

(20)   \begin{equation*} \mathcal{A}=\int XdX=\dfrac{X^2}{2}\sim X^2 \end{equation*}

where we have assumed that X is a classical commuting c-number, so absement is like an area or reciprocally, spacetime is the SQUARE root of absement! If X were a grassmannian absement (Oh, God! I love this variation!) field, it would provide absement of order one, since integration acts as derivative for grassmannian variables according to the Berezinian prescriptions. Is out there any generalized absement useful for physmatics? Is there a reason why action is more fundamental than power or energy or are we just biased? If action is fundamental, could absement play a more fundamental role in future physmatics? Surely it could be…After all, action, power or energy are only three possible variables between many of them! Maybe, the privileged role of actergy, a.k.a., action is a consequence of our human perception and we could chose any other kinematical and dynamical variable for describing motion. Absement based dynamics is something it could be even more than possible in the future.

By the way, generalized absement is also related to generalized zilches and helicities of higher dimensional stuff. Ah, yeah…What the hell are zilches and helicities? That is another future blog story!

See you in another blog post!!!!

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