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Equation 6 · Part 5 · Every Crystal Has Its Own Speed of Light

Symbol e^-(d(x,y) - c_LR t)/xi

∥[Ax(t),By]∥≤C e−(d(x,y)−cLR t)/ξ\lVert [A_x(t), B_y] \rVert \le C\, e^{-\left(d(x,y) - c_{\mathrm{LR}}\, t\right)/\xi}
e−(d(x,y)−cLR t)/ξe^{-\left(d(x,y) - c_{\mathrm{LR}}\, t\right)/\xi}

What this part means

e−(d(x,y)e^-(d(x,y) - cLc_LR t)/xi is a part of this expression. Its role is fixed by the surrounding article and by the operations shown in the formula.

Its job in the formula

e−(d(x,y)e^-(d(x,y) - cLc_LR t)/xi is a part of this expression. Its role is fixed by the surrounding article and by the operations shown in the formula.

The passage around this formula

Stated formally, the bound says that for two operators AxA_x and ByB_y acting on well-separated lattice sites x and y , the size of their commutator after evolving one of them forward in time t is capped by an exponential in the distance between the sites minus the cone’s own reach: ∥[Ax(t),By]∥≤C e−(d(x,y)−cLR t)/ξ\lVert [A_x(t), B_y] \rVert \le C\, e^{-\left(d(x,y) - c_{\mathrm{LR}}\, t\right)/\xi}. where C and ξ\xi are constants fixed by the microscopic couplings and d(x,y) is the distance between the two sites on the lattice [ 1 ] . Outside the cone, where d(x,y) exceeds cLRc_{\mathrm{LR}} t , the right-hand side collapses toward zero; inside it, the inequality simply has nothing to say. The velocity cLRc_{\mathrm{LR}} plays exactly the causal role the vacuum speed c plays in relativity, with one…

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Sources cited in the surrounding passage

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