# Decoherence and Phase Transitions in Quantum Dynamics

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## Abstract

A particle with internal degrees of freedom is in contact with a bath of photons (necessitating a relativistic treatment). The occurrence of decoherence is established and the density matrix is found to be diagonal in momentum space. In the case of non-trivial internal degrees of freedom and selection rules there is a first order phase transition separating those degrees of freedom. Finally, because probability amplitudes become probabilities, Einstein’s proposal that more than one detector could respond to a signal is answered.

## Keywords

Decoherence Density matrix Thermal state Phase transition## 1 Introduction

In a recent publication [1] we showed that—subject to technical assumptions—a particle that is bombarded by others tends to a diagonal state in momentum space. Moreover, that diagonal state is just the thermal state \(\propto \exp (-\beta m{\varvec{p}}^2/2)\) with \(\beta \) the inverse temperature (and the Boltzmann constant taken to be unity). In other words it decoheres, but is left with a not-quite-diagonal density matrix in coordinate space, namely there are off-diagonal terms on the scale of the thermal wavelength, \(\lambda _\mathrm {thermal} \sim \hbar \sqrt{\beta /m}\). This result contradicted certain results in the literature [2]. Moreover, it established the spatial limits of decoherence, limits that were not probed when scattering was treated with a no-recoil assumption [3]. In [1], as well as in [4] and [5], the no-recoil assumption was dropped.

In the present paper the bombardment comes from a bath of photons. The conditions suggest a relativistic treatment and it turns out that this is much simpler to implement. Here too we make certain technical assumptions, but again find that the limiting density matrix for a particle in a thermal bath of photons is itself a thermal state, diagonal in momentum space.

We also make the usual physical assumptions. Namely we live with contradictory views of the scattering process. On the one hand, the \(S\)-matrix is used to describe a single scattering; on the other—even though the \(S\)-matrix nominally goes from time \(=-\infty \) to time \(=+\infty \)—we imagine that many scatterings take place. The use of \(S\)-matrix theory is nevertheless justified if the collisions are reasonably separated from one another (obviating the use of this method to draw conclusions about liquids or solids) and if the collisions are exponentially distributed as they are expected to be.

In a concluding section we examine some of the consequences of our calculation, including the possibility of phase transitions, a topic not discussed in [1], since internal degrees of freedom were not considered. We also address an old question of Einstein.

## 2 Defining the States

The photons are considered to form a bath and are characterized by occupation numbers \(\{N_k\}\), with *k* the 4-momentum (so that \(k^2=0\)).

## 3 Evolution of the Particle Density Matrix

*S*is defined by the underlying Hamiltonian dynamics.

*S*satisfies conservation of 4-momentum

*R*on the density matrix of the particle, given by

*S*so as not to clutter our expressions with square roots of the energy. Conservation of momentum gives

*k*. Subtracting Eq. (8) from Eq. (9) gives

*R*induces a stochastic matrix, \(R(p_1,\tau _1|p_0,\tau _0)\) so that

### Proof

## 4 Decoherence and Convergence to the Thermal State

### 4.1 Proof that the Non-diagonal Elements of \(R^N{\rho _{_S}}\) in a Momentum Basis Tend to Zero as \(N\rightarrow \infty \) for a Diagonal Photon Bath

*S*. Therefore

### 4.2 Convergence to Thermal Equilibrium

The diagonal elements of \(R\rho \) are given by Eq. (13). We assume that the stochastic matrix \(R(p_1,\tau _1|p_0,\tau _0)\) in the second term of Eq. (13) is irreducible. In that case the diagonal elements of \(R^N\rho \) converge to the thermal state as \(N\rightarrow \infty \) because the thermal state is a stationary state for the stochastic matrix *R* and is unique if *R* is irreducible. Therefore \(R^N\rho \) converges to the thermal state when \(N\rightarrow \infty \).

### 4.3 Detailed Balance

*R*of Eq. (13) which describes the evolution of the diagonal elements of \(\rho \) satisfies detailed balance

*R*has real eigenvalues.

### 4.4 Phase Transitions

To obtain a phase transition it is necessary and sufficient that the first *m* eigenvalues after 1 be almost degenerate with 1 and that smaller eigenvalues be much smaller [6, 7, 8, 9]. Thus, if the eigenvalues are ordered by magnitude, \(1\equiv \lambda _0\gtrsim \lambda _1 \gtrsim \dots \gtrsim \lambda _m \gg |\lambda _{m+1}|\). For the situation at hand we assume that transitions in the internal parameter are relatively rare so that in this case \(n=m+1\). *R* can then be decomposed in diagonal blocks with small matrix elements connecting different blocks, representing the transitions between internal quantum numbers. Each diagonal block represents a phase, while inside a block the dynamics is given by rapid equilibration, leading to smaller (more rapid) eigenvalues.

*R*is almost degenerate and we have a phase transition, which each phase having a different value of \(\tau \).

### Remark

*k*with \(k^0\ge M_2-M_1\) and thus by Eq. (25)

## 5 Conclusions

We have shown that the eventual state of a bombarded particle leaves it in a thermal state, and in particular one that is not diagonal in coordinate space, but rather in momentum space. The scale of the coordinate space off-diagonal elements is \(\lambda _\mathrm {thermal} \sim \hbar \sqrt{\beta /m}\). The present calculation is performed for bombardment by photons, and although a relativistic treatment is called for and might have been considered troublesome, it actually turns out to be easier than the non-relativistic calculation.

We have also found that one can allow the internal state of the particle to change. Moreover, if there is an approximate conservation law that causes there to be selection rules, the matrix of transition probabilities turns out to exhibit the features of a phase transition—eigenvalues near unity with a subsequent dropoff. (All eigenvalues are real, because of detailed balance, which was also demonstrated.) These different internal states correspond to “channels” in scattering theory.

There is another old question partially answered by this calculation. Back in 1927 Einstein asked [10]^{1} (paraphrasing), “Why don’t we get two detector clicks from the same electron?” “Why when a particle is surrounded by a collection of detectors does it excite only one of them?” In fact, this was eventually tested experimentally [11] and indeed there was but one click per particle. Einstein’s question was asked at the fifth Solvay conference and in our opinion was not satisfactorily answered by those advocating the Copenhagen interpretation. In the present calculation we have shown that a photon bath will convert probability *amplitudes* to probabilities, that is, on the scale of the thermal wavelength the density matrix is diagonal. The thermal wavelength due to cosmic background radiation is already quite small (\(\lambda _\mathrm {thermal} \sim 10^{-34} \sqrt{1/2.7\times 1.4\times 10^{-23} \times 10^{-30}} \sim 15\,\hbox {nm}\) for an electron in MKS units), answering Einstein’s question, at least for electrons and typical detectors.

## Footnotes

- 1.Apropos of seeing “two clicks,” one could also imagine seeing two channels excited in a single scattering event—something never observed. Einstein saw a way out of his rejection of various interpretations through ideas advocated by de Broglie, specifically he said
In my opinion, one can only lift this objection the following way: the process is not only described by the Schrodinger wave, but at the same time one localizes the particle during its propagation. I think M. de Broglie is right to search in this direction. If one uses only Schrödinger waves, then approach II to \(|\psi |^2\) implies, in my opinion, a contradiction to the postulate of relativity.

## Notes

### Acknowledgements

Open access funding provided by the Max Planck Society.

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