1. 'Chance' tacitly invokes certainty

The main Everettian links (Saunders, 2010):

• (C1) Inference: frequencies measure chance
• (C2) Credence: credence should equal chance
• (C3) Uncertainty: chance events are uncertain
  • Do we need this?
  • If all outcomes are actual, maybe we don't want it. (Greaves, 2007)
  • Still, we can make sense of 'not knowing what branch we're on'
  • Even so: does that tell us about a specific measure of chance?
  • One way it might: the principle of indifference.
  • Alas: this principle struggles to give consistent recommendations!

1. 'Chance' tacitly invokes certainty

Full wordings of the chance links:

• (C1) The inferential link. The chance of an event is measured
  (roughly) by (actual) relative frequencies of that event.
• (C2) The credential link. All else being equal, one’s subjective
  degree of belief or credence in an event ought to equal the
  chance of that event.
• (C3) The uncertainty link. Chance events, prior to their
  occurrence, are uncertain.
• (C4) The certainty link. Chance values supervene on the
  totality of physical information that is relevant to the
  (actual) relative frequencies of the event.

2. Certainty of isolation helps (sometimes)

Maybe information about when a system is isolated will help.

2. Certainty of isolation helps with reference class

What goes wrong in the box factory? Maybe it's the reference class problem.

• I.e., we don't know whether the right reference class varies lengths or varies areas.
• Most Everettians follow subjectivists by analyzing chance in terms of credence. Isn't this only a problem frequentists?

2. Certainty of isolation helps with reference class

Doesn't the reference class problem apply only to frequentists?

• For us, physical theory trumps all else.
• We view this as a consistency constraint: if we think QM is authoritative enough to describe the totality of relevant empirical info, it had better also give us our expert credence function!
• First, let's see if we can pin down a 'reference class' (or: unique expert credence function) for the box factory.

2. Certainty of isolation helps with reference class

Doesn't the reference class problem apply only to frequentists?

• Hájek (2007) argues not: subjectivists fill in the credential link, (C2), by asking us to set our credence according to some expert's: $$\textit{cr} (X \mid \textit{cr}_\text{expert}(X) = x) = x,$$ where \(\textit{cr}_\text{expert}(X)\) could be a Lewisian chance function, a function given by a physical theory, a logical probability, etc.
• And these plausibly give different answers: ‘You can’t serve all your masters at once, so you have to play favorites. But who trumps whom, and which trumps which?’ (2007, p. 597).

2. Certainty of isolation helps (sometimes)

Isolation and symmetries will help, sometimes.

• Ontic Separability Principle (OSP) combines steps 1 and 2: $$\textit{ch}(s\mid U) = \textit{ch}(s\mid S) = \textit{ch}(s' \mid S')$$
• Step 3 might uncover some \(S'=S\).

2. Certainty of isolation helps with reference class

Isolation and symmetries will help, sometimes.

• Ontic Separability Principle (OSP). Suppose that a physical theory \(T\) specifies the totality of physical information that is relevant to the (actual) relative frequencies of an event \(X\) in a system \(U\). Moreover, suppose that \(X\) occurs in an isolated subsystem \(S\) of \(U\), and that a map from \(X\) in \(S\) to \(X'\) in \(S'\) preserves its relative frequency. Then \(T\) should assign chance values to \(X\) in \(S\) that are independent of the environment \(U \setminus S\) and preserved by this map: $$\textit{ch}(s\mid U) = \textit{ch}(s\mid S) = \textit{ch}(s' \mid S')$$
• OSP gets steps 1 and 2; step 3 might uncover some \(S'=S\).

2. Certainty of isolation helps with reference class

The isolated system \(S\) in OSP acts as our reference class.

• When OSP gives strong enough constraints to pin down a single measure, it affords a pragmatic resolution of the reference class problem that applies equally well to Lewisians and frequentists (and others besides).
• ...as long as we're clear about that first step of identifying an isolated subsystem...

3. But 'isolation' is ambiguous

Wallace's (2022a; 2022b) theory sectors help us get a handle on recursion.

• Crucially, Wallace doesn't think all sectors are isolated!
• E.g.: Newtonian gravity (NG) defines sectors nearly identical to those of electrostatics.
  • The fields (electric or gravitational) vanish at a sector's boundary.
  • But then no Earth-bound laboratory is a sector of NG!
  • Using different boundary conditions for the gravitational potential field, we get new sectors that include these labs.
  • Doing the same for electrostatics gives a less strict (but still useful) notion of isolation!

3. But 'isolation' is ambiguous

The gravitational potential field

$$ \frac{\mathrm{d}^2}{\mathrm{d}t^2} x^i_J = \nabla_i \Phi(x^i_J,t)$$

satisfies the Poisson equation

$$\nabla^2 \Phi(x,t) = -4\pi G \sum_J \delta (x-x_J) m_J.$$

and is linear at the boundary:

$$\lim_{|x|\to \infty} \Phi(x,t) = \Phi_0 (t) + a_i(t) x^i(t)$$

If it is constant at the boundary, this theory recovers Newton's law.

3. But 'isolation' is ambiguous

• So, we flesh out the recursion link with Wallace's (2022a; 2022b) theory sectors, and the case of Newtonian gravity motivates the control link.
  • (I1) The recursion link. Roughly, 'isolated subsystems' should instantiate a recursive structure (like Wallace's theory sectors).
  • (I2) The control link. We can study them in a laboratory.
• There are plausibly many valid ways of fleshing out these links—just like Lewisians and frequentists differ on how to fill in (C1)–(C4).
• Using theory sectors for (I1) meshes nicely with OSP.

3. But 'isolation' is ambiguous

• Using theory sectors for (I1) meshes nicely with OSP. Recall from OSP:$$\textit{ch}(s\mid U) = \textit{ch}(s\mid S) = \textit{ch}(s' \mid S')$$
• The second equality gives a function of states \(S\) and outcomes \(s\) that is preserved by applying a symmetry map to both.
• In other words, it says the chances are the same up to a re-labeling of outcomes. 
• This closely matches what Wallace would call an intrinsic property of a sector.
• Thus, we'll refer to chance's invariance under re-labeling as OSP's intrinsicality condition.

4. Probing Everettian 'isolation' with theory sectors

• Microstates 1 and 2 are isolated iff
  • their kinematical states are separable: $$\psi_{12} = \psi_1\otimes \psi_2$$
  • and their dynamics are separable: $$U_{12} = U_1\otimes U_2$$
• If macrostates 1 and 2 are isolated,
  • their kinematical states are orthogonal: $$\psi_1\bot\psi_2$$
  • their dynamics are completely described by their own unitaries
    • Orthodoxy adds: either 1 or 2 occurs after a projection
    • Everettians add: both 1 and 2 occur but are dynamically isolated

4. Probing Everettian 'isolation' with theory sectors

• According to orthodoxy about quantum mechanics, there are two dynamical rules:
  • unitary dynamics describes self-evolution
  • projective dynamics describes measurement
• In contrast, a traditional selling point of Everettian
  approaches is that they treat all dynamics as unitary.
• Thus, one might think that orthodoxy will frustrate attempts
  to define a theory sector, while Everettian approaches will
  accommodate sectors more straightforwardly.
  • Both of these impressions are false!

4. Probing Everettian 'isolation' with theory sectors

• Thus, one might think that orthodoxy will frustrate attempts
  to define a theory sector, while Everettian approaches will
  accommodate sectors more straightforwardly.
  • Both of these impressions are false!
    • On the orthodox approach, one can helpfully use von Neumann’s measurement scheme to relegate projective dynamics to interactions with macroscopic objects.
    • Similarly, Everettians rely on projections to specify the sense in which their macrostates—branches—are dynamically isolated.
• Everettian or not, use two sectors: one for microstates and one for macrostates.

4. Probing Everettian 'isolation' with theory sectors

How defensible is the orthodox approach?

• Nice notions of extension and restriction in both 'space' (smaller and larger systems) and 'time' (before and after measurement)
• Two problems:
  • Using projections means measurements must be repeatable; no room for noise.
    • Too strict: like how Earth-bound labs couldn't test our first pass at Newtonian gravity. (Measurement theory addresses this!)
  • It's not clear why some projections are allowed and not others.
    • Too liberal: like saying we don't need the electric field to vanish in electrostatics.

4. Probing Everettian 'isolation' with theory sectors

How defensible is the orthodox approach?

• It's not clear why some projections are allowed and not others.
  • Too liberal: like saying we don't need the electric field to vanish in electrostatics.
• Everettians address this with decoherence (at least in part).
• This gives a dynamical explanation both for which projections define macrostates and how macrostates stay isolated.
  • The 'how' is the branching criterion: only one prior branch \(i\) contributes to any future branch \(j\),
    $$|\psi_{ij}^{t_1 t_2}|\neq 0, \,|\psi_{i'j}^{t_1 t_2}|\neq 0 \Longleftrightarrow i= i' $$  

4. Probing Everettian 'isolation' with theory sectors

• Branching doesn't ensure quasi-classicality, but it's still a useful notion of isolation (like that for generalized electrostatics)
• Realistic systems only branch approximately; appeal to the \(\hbar\to 0\) limit to idealize without chance (Dawid and Thébault, 2025)

4. Probing Everettian 'isolation' with theory sectors

4. Probing Everettian 'isolation' with theory sectors

Branching criterion: only one prior branch \(i\) contributes to any future branch \(j\),

$$|\psi_{ij}^{t_1 t_2}|\neq 0, \,|\psi_{i'j}^{t_1 t_2}|\neq 0 \Longleftrightarrow i= i' $$  

Where

$$ \Psi_{ij}^{{t_1}{t_2}} = {\hat{X}_j^{t_2} \hat{U}^{t_2\leftarrow t_1} {\hat{X}_i^{t_1}}  \hat{U}^{t_1\leftarrow t_0} \Psi^{t_0}.} $$

\(\hat{X}\) is a macroscopic projection (a coarse-graining of a projection onto a microstate).

5. Sectors unite and strengthen Born rule derivations

• Comparisons with existing approaches:
  • Wallace (2012) uses erasures to send symmetric branches to the same state
    • In tension with subsystem recursivity: erasures can't be performed at the same time (Mandolesi, 2019)
  • Sebens and Carroll (2018) use global branching to identify microstates belonging to the same (agent) macrostate
    • This only applies a short time before decoherence reaches them (McQueen and Vaidman, 2019)
  • McQueen and Vaidman (2019) use symmetry and locality
    • We only need the symmetries of NRQM

5. Sectors unite and strengthen Born rule derivations

The core issue in S&C's setup:

$$| \Psi_0 \rangle = | R_0 \rangle_A  |R \rangle_{D1} |R \rangle_{D2} |\!\uparrow_x \rangle |E_R \rangle$$

must evolve to

$$| \Psi_1  \rangle = \frac{1}{\sqrt{2}} {\color{orange}| R \rangle_A } |\!\uparrow \rangle_{D1} |\heartsuit \rangle_{D2} |\!\uparrow_z \rangle \left |E_{\uparrow\heartsuit} \right\rangle + \frac{1}{\sqrt{2}}  {\color{orange}| R \rangle_A } |\!\downarrow \rangle_{D1} |\diamondsuit \rangle_{D2} |\!\downarrow_z \rangle \left |E_{\downarrow\diamondsuit} \right\rangle$$

but not to

$$ | \Psi_1 ' \rangle = \frac{1}{\sqrt{2}} {\color{Magenta}| R_\uparrow \rangle_A } |\!\uparrow \rangle_{D1} |\heartsuit \rangle_{D2} |\!\uparrow_z \rangle \left |E_{\uparrow\heartsuit} \right\rangle + \frac{1}{\sqrt{2}} {\color{Magenta} | R_\downarrow \rangle_A } |\!\downarrow \rangle_{D1} |\diamondsuit \rangle_{D2} |\!\downarrow_z \rangle \left |E_{\downarrow\diamondsuit} \right\rangle$$

to get identical reduced \(\rho\) for agent \(A\) and detector \(D2\) in the symmetric setup that flips \(\heartsuit\) and \(\diamondsuit\) labels.

5. Sectors unite and strengthen Born rule derivations

S&C's (2018) ESP: Suppose that the universe \(U\) contains within it a set of subsystems \(S\) such that every agent in an internally qualitatively identical state to agent \(A\) is located in some subsystem that is an element of \(S\). The probability that \(A\) ought to assign to being located in a particular subsystem \(X\in S\) given that they are in \(U\) is identical in any possible universe which also contains subsystems \(S\) in the same exact states (and does not contain any copies of the agent in an internally qualitatively identical state that are not located in \(S\)):
$$ c\left( X \mid U\right) = c\left(X\mid S\right). $$ 

5. Sectors unite and strengthen Born rule derivations

S&C's (2018) ESP-QM: Suppose that an experiment has just measured observable \(\hat{O}\) of system \(S\) and registered some eigenvalue \(O_i\) on each branch of the wavefunction. The probability that agent \(A\) ought to assign to the detector \(D\) having registered \(O_i\) in their branch when the universal wavefunction is \(\Psi\), \(c\left( O_i | \Psi \right)\), only depends on the reduced density matrix of \(A\) and \(D\), \(\hat{\rho}_{AD}\):

$$ c\left( O_i \, | \, \Psi \right) = c \left( O_i \, |\, \hat{\rho}_{AD} \right) .$$