Friday, June 04, 2010

Axioms for the many-worlds interpretation

Quantum theory is conventionally thought to be basis-independent. The state of a quantum system can be represented by a vector Ψ in a special type of vector space, called a Hilbert space H, and a basis is simply a collection of vectors {ψ} which enable any element to be decomposed as a linear combination of those basis vectors:

Ψ = c1 ψ1 + ⋅ ⋅ ⋅ + cn ψn.

The elements of the general linear group GL(H) transform from one basis to another, and by virtue of being basis-independent, the same quantum state Ψ can be expressed as a different linear combination in a different basis {ψ'}:

Ψ = c1' ψ1' + ⋅ ⋅ ⋅ + cn' ψn'.

In this sense, quantum theory can be said to be a GL(H)-invariant theory.

The many-worlds interpretation of quantum theory, however, suggests that: (a) there is a process called decoherence which selects a preferred basis; and (b) the universe splits into the branches selected by decoherence. Thus, whilst quantum theory per se does not identify a branching structure for the universe, the many-worlds interpretation does. A measurement-like interaction, as a special case of decoherence, can be said to extrude a collection of branches from a GL(H)-invariant structure, much like a rose-bush emerging from a thicket of brambles.

To elaborate, let us attempt to define some axioms for the many-worlds interpretation of quantum theory:

(i) Quantum theory is fundamental and universal.

(ii) A pure quantum state provides a maximal specification of the state (or history, in the Heisenberg picture) of a physical system.

(iii) Each type of physical system is represented by a unitary representation of the local space-time symmetry group on a Hilbert Space H.

(iv) The time evolution of a physical system in a local reference frame is represented by a continuous one-parameter group of unitary linear transformations U(t):H → H of the Hilbert space. This corresponds to the representation of the time-translation subgroup of the local space-time symmetry group.

(v) The interaction Hamiltonian between a macroscopic system and its environment is such that any macroscopic observable commutes with the interaction Hamiltonian.

(vi) Approximate GL(H) symmetry-breaking selects a preferred basis in the Hilbert space. Given a superposition of macroscopically distinguishable eigenstates of a macroscopic observable, the interaction between the macroscopic system and its environment is such that the reduced state of the macroscopic system evolves very rapidly towards a state which is empirically indistinguishable from a mixture of the states which were initially superposed. In other words, the eigenbasis of the macroscopic observable almost diagonalizes the reduced density operator. This process is referred to as decoherence. In effect, a preferred basis is selected by the interaction Hamiltonian. Given that there is a one-one mapping between the bases of a Hilbert space and the general linear group GL(H), decoherence approximately breaks the GL(H) symmetry of quantum theory.

(vii) Each decohering macroscopic state can be treated as a Gaussian wave-packet Ψ(x,p), of mean position 〈x〉 and mean momentum 〈p〉. A free Gaussian wave-packet initially minimises the position and momentum uncertainty. i.e., Δx Δ p = 1/2 ℏ. By virtue of being a wave-packet, the mean 〈x〉 of the position probability distribution will move with a velocity equal to the mean 〈p〉/m of the velocity probability distribution. A free Gaussian wave-packet is such that Δp remains constant, but Δx increases with time, a process referred to as the spreading of the wave-packet. Macroscopic Gaussian wave-packets, however, are constrained from spreading due to the continual interaction of the macroscopic system with its environment.

(viii) For each measurement-like interaction, the universe branches, and it is the branching which transforms potentiality into actuality. The branches are those selected by decoherence, and each branch realises one and only one of the states in the mixed state produced by decoherence.

(ix) The squared modulus of the complex amplitudes in the initial superposition correspond to the relative frequencies with which the different outcomes occur in most branches of the universe.

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