## David Wallace

Print publication date: 2012

Print ISBN-13: 9780199546961

Published to Oxford Scholarship Online: September 2012

DOI: 10.1093/acprof:oso/9780199546961.001.0001

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# (p.429) Appendix A Proof of the Branching-Decoherence Theorem

Source:
The Emergent Multiverse
Publisher:
Oxford University Press

(p.429) Appendix A

Proof of the Branching-Decoherence Theorem

Recall that the theorem is

Branching-Decoherence Theorem: If $Ƥ = { P ^ j i }$ is a history space and |ψ〉 is a quantum state, then

1. (i) If Ƥ has branching structure (relative to |ψ〉) and α is a history then $C ^ α | ψ 〉 ≠ 0$ iff α is realised (with respect to |ψ〉.

2. (ii) If the set Hist of all histories α such that $C ^ α | ψ 〉 ≠ 0$ has branching structure (that is, if no two histories in Hist agree on their nth index but not on all previous indices), then Ƥ also has branching structure (relative to |ψ〉), and the realised histories in that branching structure are just the histories in Hist.

3. (iii) If Ƥ has branching structure (relative to |ψ〉), Ƥ satisfies the decoherence condition.

4. (iv) If Ƥ satisfies the decoherence condition, it is a coarse-graining of a (decoherent) history space relative which has branching structure relative to |ψ〉.

To develop the proof, it will be helpful to modify Chapter 3’s index notation somewhat. Recall that a history is uniquely specified by a string of indices, one for each time-index. This being the case, I identify each history with that string of indices: a boldface index j denotes a sequence 〈j 0, … j n〉 of indices (one for each time t 0, … t n) such that the projector in the history corresponding to time t k is $P ^ j k k$. I then generalise the notion of history so that j can stand for any sequence of indices for times t 0, …t m, where mn; ℒ(j) will denote the length of the sequence minus one (so if the sequence is defined for times t 0, … t m then ℒ(j) = m). I write ij (p.430) to indicate that one of i and j is an initial segment of the other (so if ℒ(i) = ℒ(j), then ij entails i = j).

I define history operators for our generalised indices in the obvious way: if ℒ(j) = m, then we have

$Display mathematics$
(A.1)

Notice that

$Display mathematics$
(A.2)

It follows from (A.2) that the decoherence criterion may be rewritten for generalized sequences as

$Display mathematics$
(A.3)

Finally, I say that a generalized history is realized (for a branching history space) iff it is an initial segment of a realized history.

Part (i) of the theorem may now be proved by induction on the length of histories. For histories of length 1–2, it is a trivial consequence of the definition of a branching history; for longer histories, let j be an arbitrary history with ℒ(j) = m. We have

$Display mathematics$
(A.4)

where j is the initial segment of j with length ℒ(j ) = m − 1. If the theorem holds for sequences with ℒ(k) 〈 m, then it follows that this vanishes unless j is realized. Assuming j is indeed realized, it also follows that

$Display mathematics$
(A.5)

and therefore, since j is the unique realized history with final element j m−1,

$Display mathematics$
(A.6)

Hence $C ^ j | ψ 〉 = P ^ j m m P ^ j m − 1 m − 1 | ψ 〉$, which, by the definition of branching, vanishes unless j m−1 is the penultimate term on the unique history whose final term is j m.

(p.431) To prove part (ii), notice that if Hist has branching structure, we have

$Display mathematics$
(A.7)

where j is the unique history in Hist with ℒ(j) = m and j m = j. (If there is no such, then $P ^ j m | ψ 〉 = 0$. Then

$Display mathematics$
(A.8)

where j + is the history with final element j′ and initial history j. Since $C ^ j+ | ψ 〉$ is nonzero for at most one such j + $P ^ j m + 1 m + 1 P ^ j m m | ψ 〉$ is nonzero for at most one j.

Part (iii) is a trivial consequence of (i). To prove part (iv), suppose that our history space ${ P ^ j i }$ is decoherent (for given |ψ〉). I then define, for each time index m,

$Display mathematics$
(A.9)

where j is any sequence with ℒ(j) = m. If $P ^ j m$ is any projector in the mth PVM in the history space, then

$Display mathematics$
(A.10)

Hence the set of operators

$Display mathematics$
(A.11)

is a PVM and ${ P ^ j m }$ is a coarse-graining of that PVM.

To see that the history space consisting of the family of these PVMs (for each m) has branching structure, it suffices to note that:

1. 1. Since |ψ〉 can always be written as a superposition of the $| j , m 〉 , O ^ j m | ψ 〉 = 0.$

2. 2.

$Display mathematics$
(A.12)

so $O ^ j m + 1 Q ^ k m | ψ 〉 = 0$. (p.432)

3. 3.

$Display mathematics$
(A.13)

and

$Display mathematics$
(A.14)

so $Q ^ k m + 1 Q ^ j m$ vanishes unless j is an initial segment of k.