Paper VI: Packet blocks of a semigroup

2 Packet blocks of a semigroup

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The central idea

A packet block remembers both the origin and the destination. Composition must sum over every intermediate packet, because an unobserved intermediate coordinate can carry a real contribution. The original formulas below express this bookkeeping exactly.

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Two rational transfer steps resolved through all three intermediate states. The displayed products are computed exactly before plotting. This finite illustration explains the intermediate-packet sum.

Let

\[ P_\alpha P_\beta=\delta_{\alpha\beta}P_\alpha, \qquad \sum_{\alpha\in I}P_\alpha=I. \]

For any operator \(B\), define

\[ B_{\beta\alpha}=P_\beta BP_\alpha. \]

Then

\[ B=\sum_{\alpha,\beta}B_{\beta\alpha}. \]

For the semigroup, write

\[ T_{\beta\alpha}(t)=P_\beta e^{tA}P_\alpha. \]

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Theorem 2.1: Packet semigroup convolution

For every \(s,t\) and every pair of packets \(\alpha,\beta\),

\[ \boxed{ T_{\beta\alpha}(s+t) =\sum_{\gamma\in I}T_{\beta\gamma}(s)T_{\gamma\alpha}(t). } \]

Thus the semigroup law becomes ordinary matrix multiplication in the packet indices.

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Proof of Theorem 2.1

Insert the resolution of the identity between the two semigroup factors:

\[ \begin{aligned} P_\beta e^{(s+t)A}P_\alpha &=P_\beta e^{sA}\left(\sum_\gamma P_\gamma\right)e^{tA}P_\alpha\\ &=\sum_\gamma (P_\beta e^{sA}P_\gamma)(P_\gamma e^{tA}P_\alpha). \end{aligned} \]

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Corollary 2.2: Reducing packets

If \(P_\beta AP_\alpha=0\) for every \(\beta\ne\alpha\), then

\[ P_\beta e^{tA}P_\alpha=0 \quad(\beta\ne\alpha) \]

for every \(t\). Each packet evolves independently under \(P_\alpha AP_\alpha\).

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Proof of Corollary 2.2

The hypothesis makes \(A\) block diagonal. Every power \(A^m\) is block diagonal with respect to the same projectors, and therefore so is the exponential series.

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Norms of the packet blocks P β e tA P α . Off-diagonal entries quantify transfer between packets. — Norms of the packet blocks \(P_\beta e^{tA}P_\alpha\). Off-diagonal entries quantify transfer between packets.

Remark 2.3: Path expansion

Expanding the exponential and inserting \(I=\sum_\alpha P_\alpha\) between every pair of generators gives

\[ P_{\alpha_m}A^mP_{\alpha_0} = \sum_{\alpha_1,\ldots,\alpha_{m-1}} P_{\alpha_m}AP_{\alpha_{m-1}}\cdots P_{\alpha_1}AP_{\alpha_0}. \]

Every coefficient is therefore a sum over packet-transfer paths. The numerical library verifies that truncating this path series converges to the ordinary matrix exponential.

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