Interpretations of Quantum Theory in the Light of Modern Cosmology

Abstract

The difficult issues related to the interpretation of quantum mechanics and, in particular, the “measurement problem” are revisited using as motivation the process of generation of structure from quantum fluctuations in inflationary cosmology. The unessential mathematical complexity of the particular problem is bypassed, facilitating the discussion of the conceptual issues, by considering, within the paradigm set up by the cosmological problem, another problem where symmetry serves as a focal point: a simplified version of Mott’s problem.

Appendix

In this appendix, we discuss some specific issues that arise in the attempt to use decoherence related arguments in the context of the problem at hand.

The first issue is that connected to the implication of symmetry regarding the choice of a preferential basis or so called pointer states.

The simplest example exhibiting this problem is provided by a standard EPR-R setup: Consider the decay of a spin 0 particle into two spin 1/2 particles. Take the direction of the decay as being the x axis (the particles momenta are $\vec P= \pm P \hat{x}$ with $\hat{x}$ the unit vector in the $\vec x$ direction)¹⁵. Now, we characterize the two particle states, that emerges after the decay in terms of the $\vec {z}$ polarization states of the two Hilbert spaces of individual particles. As it is known, the conservation of the angular momentum of the system indicates that the state must be:

$$\begin{aligned} |\chi \rangle = \frac{1}{\sqrt{2}}( |+ \rangle ^{(1)}_{z} |- \rangle ^{(2)}_{z} + |+ \rangle ^{(2)}_{z} |- \rangle ^{(1)}_{z} ) \end{aligned}$$

(19)

The state is clearly invariant under rotations around the x axis (simply because it is an eigen-state with zero angular momentum along that axis). The density matrix for the system is thus $\rho = |\chi \rangle \langle \chi |$. Now assume we decide we are not interested in one of the particles (call it 1), and thus we regard it as an environment for the system of interest (particle 2). The reduced density matrix is then:

$$\begin{aligned} \rho ^{(2)} = \rho _{Reduced}= Tr_{(1)} \rho = \frac{1}{ 2}( |+ \rangle ^{(2)}_{z} \langle +|^{(2)}_{z}+|- \rangle ^{(2)}_{z} \langle -|^{(2)}_{z} ) \end{aligned}$$

(20)

Now suppose we want to say that as the reduced density matrix is diagonal, we have found the pointer basis and that somehow the particle must be considered as having its spin along the z axis defined to be either $+1/2$ or $-1/2$.

The problem is that the symmetry of the state $|\chi \rangle$ regarding rotations around the x axis is reflected in the fact that we could have written this density matrix also as

$$\begin{aligned} \rho ^{(2)} = \frac{1}{ 2}( |+ \rangle ^{(2)}_{y} \langle +|^{(2)}_{y}+|- \rangle ^{(2)}_{y} \langle -|^{(2)}_{y} ) \end{aligned}$$

(21)

leading, this time, to the conclusion that the particle must be considered as having its spin along the y axis defined to be either $+1/2$ or $-1/2$.

In fact, as the density matrix is proportional to the identity ( i.e. $\rho ^{(2)} = \frac{1}{ 2} I$) it would have the same form in any orthogonal basis.

One might be inclined to consider that this problem occurs only in very simple situations, such as the one of the above example, and that, in general, we will not encounter such difficulty. However that consideration is mistaken as can be seen from the general result encapsulated in the following:

1.1 Theorem

Consider a quantum system made of a subsystem S and an environment E, with corresponding Hilbert spaces $H_{S}$ and $H_{E}$ so that the complete system is described by states in the product Hilbert space $H_{S}\otimes H_{E}$. Let G be a symmetry group acting on the Hilbert space of the full system in a way that does not mix the system and environment. That is, the unitary representation O of G on $H_{S}\otimes H_{E}$ is such that $\forall g \in G$, $\hat{O}(g) = \hat{O}^{S}(g)\otimes \hat{O}^{E}(g)$, where $\hat{O}^{S}(g)$ and $\hat{O}^{E}(g)$ act on $H_{S}$ and $H_{E}$ respectively.

Let the system be characterized by a density matrix $\hat{\rho }$ which is invariant under G. Then the reduced density matrix of the subsystem is a multiple of the identity in each invariant subspace of $H_{S}$.

1.2 Proof

The reduced density matrix $\hat{\rho }_{S}= Tr_{E} ( \hat{\rho })$.The trace over the environment of any operator $\hat{A}$ in $H_{S}\otimes H_{E}$ is obtained by taking any orthonormal basis $\lbrace |e_{j}\rangle \rbrace$ of $H_{E}$ and evaluating $\Sigma _{j} \langle e_{j}| \hat{A} |e_{j}\rangle$.

Now, by assumption, we have $\hat{\rho }= {\hat{O}(g)}^{\dagger }\hat{\rho }\hat{O}(g)$, $\forall g \in G$. Then, for all $g \in G$, we have $\hat{\rho }_{S} = \Sigma _{j} \langle e_{j}| \hat{\rho }|e_{j}\rangle = \Sigma _{j} \langle e_{j}| {\hat{O}^{S}(g)}^{\dagger }\otimes { \hat{O}^{E}(g)}^{\dagger }\hat{\rho }\hat{O}^{S}(g)\otimes \hat{O}^{E}(g) |e_{j}\rangle = \Sigma _{j} {{\hat{O}}^{S} (g)}^{\dagger }\langle e^{\prime }_{j}| \hat{\rho }|e^{\prime }_{j}\rangle {\hat{O}}^{S}(g)$, where $|e^{\prime }_{j}\rangle \equiv O^{E}(g) |e_{j}\rangle$. However, the fact that the operator ${\hat{O}}^{E}(g)$ is unitary implies that the transformed states $\lbrace |e^{\prime }_{j}\rangle \rbrace$ form also an orthonormal basis of $H_{E}$.

Thus we have $\hat{\rho }_{S} = {{\hat{O}}^{S}(g)}^{\dagger }( \Sigma _{j} \langle e^{\prime }_{j}| \hat{\rho }|e^{\prime }_{j}\rangle ) {\hat{O}}^{S}(g ) = {{\hat{O}}^{S}(g)}^{\dagger }\hat{\rho }_{S} {\hat{O}}^{S}(g )$ or equivalently $\hat{\rho }_{S} {{\hat{O}}^{S}(g)} = {\hat{O}}^{S}(g ) \hat{\rho }_{S}$. So we have found that $[ \hat{\rho }_{S} , {\hat{O}}^{S}(g )]=0$, $\forall g \in G$, and thus by Schur’s lemma it follows that $\hat{\rho }_{S}$ must be a multiple of the identity in each invariant subspace of $H_{S}$, QED.

In particular, this result indicates that, if we start with a pure state invariant under the symmetry group, the reduced density matrix must be a multiple of the identity in each invariant subspace of $H_{S}$. This is exemplified by the well known case of a standard EPR setting, where a spinless particle decays into two photons, and where one considers the photons’ spin degrees of freedom. The reduced density matrix describing one of the photons is a multiple of the identity, and thus the decoherence that results from tracing over the first photon’s spin does not determine a preferential basis for the characterization of the spin of the second photon. Decoherence then fails under these conditions to provide a well defined preferential context for the interpretation of the reduced density matrix, as representing the various alternatives for the state of the subsystem after decoherence.