# Why are observables hermitian operators in the Everett interpretation?

Observables correspond to hermitian operators on the quantum state.

But in the Everett interpretation, the wave function doesn’t collapse since we consider the entire universe as a single quantum state at time $$t$$, so observation only happens “within that quantum state” (or something like that).

This confuses me. Given that observations are not treated differently than other events in the Everett interpretation, how does this relate to observations being hermitian operators?

Even in traditional interpretations of quantum mechanics it is easier to define what is an observable than what is an observation: it is a property of a quantum system or subsystem that we in principle have access to through observation (whatever the latter means).

If we say that this is a decomposition in orthogonal subspaces of the state space, which are distinguished by a real numerical value assigned to it, called the outcome of the measurement, as does the measurement postulate, we don't need to know anymore what we mean by an observation.

In Everett's interpretation an observation is described by unitary evolution just like everything else. A decomposition of the state space in orthogonal subspaces however remains meaningful, just like in traditional interpretations. Let's still call the associate number an outcome. Disregarding subtleties when working in infinite dimensional state spaces, orthogonal decompositions in subspaces indexed by some real numbers are exactly equivalent to hermitian operators (just pick your orthonormal basis $|\lambda\rangle$ with corresponding outcome $\lambda$. The corresponding hermitian operator is $\sum\lambda|\lambda\rangle\langle\lambda|$. This happens to be one of the occasions where I really like the Dirac notation). It could be argued that the former, the decomposition with outcomes, actually is the more fundamental definition of an observable.

• This post is a bit unclear. Just because you can decompose the Hilbert space into orthogonal subspaces doesn't mean you have explained which outcome or outcomes we (and our many worlds counterparts) actually experience. Furthermore, the decomposition of the Hilbert space is not unique so the question can be asked why it appears to us, after measurement, that we have measured in a particular basis when the underlying structure prefers no particular basis. – jgerber Sep 25 '18 at 2:32
• @jgerber Decoherence does have the ability to pick out a preferred basis. Typical decoherence processes such as particle scattering have been extensively studied starting in 1970s. They show that due to the local nature of QED interactions, the position basis is typically preferred - a process call localization. This is what justifies describing a particle using a wave packet. – Bruce Greetham Sep 25 '18 at 3:25
• @jgerber thanks. What I describe is what observables are and why they are encoded in hermitian operators, independent of whether we can actually make an observation in that basis, and independent of what an observer would experience. An observable encodes a basis, but doesn't prefer one. An experimental setup tries to mould the world to (temporarily and locally) "prefer" one decomposition over another, e.g. in a Stern-Gerlach experiment two orthogonal spins are preferred, in an idealized CCD array you have one subspace for each pixel and a large component for whatever misses the array etc. – doetoe Sep 25 '18 at 6:13

While many worlds interpretation provides a complete and thorough description of the state of quantum systems (including observers as quantum system), it doesn't provide a prescription for how the physical state of the system is related to the experience of observers.

In many worlds there is no significance to any operators on the Hilbert space unless those operators happen to be part of the Hamiltonian (which is Hermitian) which is used to determine the time evolution of the universal wavefunction.

Otherwise there is no special significance to Hermitian operators. One might say that many worlds doesn't give a prescription for how observations happen so there are no such thing as observations in the many worlds interpretation and thus there is no need for observables.