$$X^\mu(u,\Sigma)$$

Equation 22 · Evolutionary Physics

The Atlas That Refuses to Close

This mathematical expression combines the displayed quantities; its precise role follows from the surrounding article text.

$X^\mu(u,\Sigma)$ is this article’s first chart. It takes two pieces of auxiliary data, an observer and a slice, and returns a point. Different choices of (u,$\Sigma$) for the identical physical body — the identical $T^{\alpha\beta}$ field, the identical total four-momentum $P^\mu$ = $\int_\Sigma$ $T^{\mu\nu}$\, d$\Sigma_\nu$ and total angular momentum $J^{\mu\nu}$ — can and do return different points, and the question this article is built to answer precisely is how different, and under what conditions the difference means something rather than nothing.

Meanings in this article

• $X^\mu$: this article’s first chart.
• $\Sigma$: a spacelike hypersurface and $u^\mu$ the four-velocity of the observer whose notion of simultaneity $\Sigma$ represents.

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Equation 34 · Evolutionary Physics

The Atlas That Refuses to Close

This mathematical expression combines the displayed quantities; its precise role follows from the surrounding article text.

Fix the slicing convention that anchors the centroid to the body’s own rest frame — the choice, standard in the spinning-body literature, in which the spin tensor about the centroid satisfies $S^{\alpha\beta}$$P_\beta$ = 0 — and ask how $X^\mu(u,\Sigma)$ moves as u is varied away from the body’s own rest-frame observer to some other inertial observer boosted at velocity $\mathbf v$ relative to it. Costa and Natário’s synthesis of the classical spinning-body literature gives the answer in closed form: the centroid measured by the boosted observer is displaced from the rest-frame centroid by

Meanings in this article

• $u$: varied away from the body’s own rest-frame observer to some other inertial observer boosted at velocity $\mathbf v$ relative to it.

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Equation 44 · Evolutionary Physics

The Atlas That Refuses to Close

This mathematical expression combines the displayed quantities; its precise role follows from the surrounding article text.

a shift transverse to both the spin and the relative velocity [ 13 ] . This is an exact identity of special relativity, not an approximation valid only for small v ; it holds all the way to v $\to$ c , at which point |$\Delta$$\mathbf x$| $\to$ |$\mathbf S$|/(Mc) = $\rho_M$ exactly, since |$\mathbf S$ $\times$ $\mathbf v$| $\le$ |$\mathbf S$|\,v saturates when the spin is perpendicular to the boost. No boosted observer, however extreme, can push a body’s own centroid further from its rest-frame value than $\rho_M$ . As u ranges over every inertial observer, $X^\mu(u,\Sigma)$ traces out a disk of exactly that radius, orthogonal to $\mathbf S$ — the world-tube first identified by Møller and reproduced in modern language…

Meanings in this article

• $u$: varied away from the body’s own rest-frame observer to some other inertial observer boosted at velocity $\mathbf v$ relative to it.

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Equation 59 · Evolutionary Physics

The Atlas That Refuses to Close

This mathematical expression combines the displayed quantities; its precise role follows from the surrounding article text.

The first, the covariant centre of energy, transforms simply as the spatial part of a four-vector under a change of Lorentz frame — exactly the property $X^\mu(u,\Sigma)$ was built with above — but its three spatial components fail to commute with each other once spin is present, by an amount proportional to the spin itself. The second, constructed by Newton and Wigner for exactly this purpose and reproduced in Pryce’s classification as a distinct case, has strictly commuting components, [$X_{\mathrm{NW}}^i$, $X_{\mathrm{NW}}^j$] = 0 , the property any operator called “position” ought to have if it is to support ordinary probability densities and wavefunction localization at all, at the price of…

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Equation 65 · Evolutionary Physics

The Atlas That Refuses to Close

This mathematical expression combines the displayed quantities; its precise role follows from the surrounding article text.

It is worth being precise about what kind of object each chart in this pairing actually is, because the two are not interchangeable answers to the same experiment. $X^\mu(u,\Sigma)$ , built from a classical or semiclassical stress-energy distribution, is a number: a spacetime point associated with one physical configuration. $\mathbf X_{\mathrm{NW}}$ is an operator on a Hilbert space, and what a quantum experiment actually reports is not $\mathbf X_{\mathrm{NW}}$ itself but expectation values and, eventually, click statistics built from it. The two charts answer “where is it” for two different kinds of description of the same underlying physics, classical field and quantum state, and the atlas has…

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Equation 111 · Evolutionary Physics

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This mathematical expression combines the displayed quantities; its precise role follows from the surrounding article text.

Applied to position, the same object, $x_w$ = $\langle$ f|$\hat x$|$\psi$$\rangle$/$\langle$ f|$\psi$$\rangle$ , is this atlas’s fifth chart, and it is admitted with a label the other four do not carry. $x_w$ answers a genuinely different question from every chart above: not “what does this state’s position operator return,” but “what does a weakly coupled pointer’s mean reading become, once conditioned on an event that filters the ensemble after the fact.” Because the conditioning can make $\langle$ f|$\psi$$\rangle$ arbitrarily small, $x_w$ can be pushed arbitrarily far from anything the Møller disk or the Pryce trade-off constrains; it is not bounded by $\rho_M$ , and it does not need to be, because it was never…

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Equation 114 · Evolutionary Physics

The Atlas That Refuses to Close

This mathematical expression combines the displayed quantities; its precise role follows from the surrounding article text.

Collect the pieces. Three charts, the Newtonian centroid $\mathbf R_N$ , the observer-dependent stress-energy centroid $X^\mu(u,\Sigma)$ , and the canonical Newton–Wigner operator $\mathbf X_{\mathrm{NW}}$ , form a genuine, well-behaved sub-atlas. Define, for any two of them evaluated on the same hypersurface $\Sigma$ ,

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