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∥[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}

Why this formula appears here

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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ByB_y

Symbol B_y

ByB_y is a part of this expression. Its role is fixed by the surrounding article and by the operations shown in the formula.

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e−(d(x,y)−cLR t)/ξe^{-\left(d(x,y) - c_{\mathrm{LR}}\, t\right)/\xi}

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

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.

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Published contexts (1)

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∥[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}

Equation 6 · Crossed Fields

Every Crystal Has Its Own Speed of Light

This equation states a bound: one expression must stay on the indicated side of the other under the article’s assumptions.

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…

Meanings in this article

  • AxA_x: the bound says that for two operators.
  • tt: capped by an exponential in the distance between the sites minus the cone’s own reach: [displayed formula].
  • CC: constants fixed by the microscopic couplings.
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