# On Quantum Collapse as a Basis for the Second Law of Thermodynamics

## Abstract

**:**

## 1. Introduction

## 2. Reversible vs. Non-Reversible Processes

_{i}is the probability that the system is in the pure state $|{\mathsf{\Psi}}_{i}\rangle $, and the P

_{i}sum to unity. The states $|{\mathsf{\Psi}}_{i}\rangle $ need not be orthogonal, so in general {$|{\mathsf{\Psi}}_{i}\rangle $} is not a basis.

_{i}, elements of a particular basis, with no causal mechanism describing the occurrence of the observed output state O

_{k}. (Of course, hidden variables theories attempt to provide a causal mechanism by “completing” quantum theory, but here we consider quantum mechanics as already complete and simply in need of a direct-action (transactional) interpretation.) The different possible outcomes are statistically weighted by probabilities P

_{i}according to the Born Rule. As a result of the measurement transition, the system is represented by a mixed state $\tilde{\rho}$. This one-to-many transition is inherently irreversible; once a final state occurs, the original state is not accessible to it through simple time reversal.

## 3. Standard Approaches to the Second Law; “Smuggling In” Non-Unitarity

_{i}that a system is in state |i> to the transition rates R

_{ij}between that state and other states |j>. Specifically:

_{ij}between states 1 and 2 are both ½. The solutions for P

_{i}(i = 1, 2) will be:

_{VN}is defined in terms of the density operator in a basis-independent way as:

_{B}) to the Gibbs entropy, which is still conserved in any unitary, deterministic process. Therefore, entropy cannot increase unless there is an element of randomness along with the underlying Liouville (deterministic) evolution. The latter corresponds to the “coarse graining” or “blurring” of the fine-level trajectories resulting from Liouville evolution. The question now obviously arises: how is this “blurring” related to the non-unitarity inherent from master equations such as (3)? In the context of classical systems, the traditional answer is that in order to obtain the relevant rates used in master equations, one has to deal in practice with an approximate description, owing to the enormous complexity of the macroscopic system under study. This is thought of as “throwing out information”—an epistemic interpretation of the “coarse-graining”. At the classical level, that is the only possible source of the “blurring”.

_{i}/dt used in master equations, any phase coherence in quantum states is lost. This is a loss of information that can be understood in ontological (rather than epistemic) terms, in contrast to the classical level, if there is ongoing real non-unitary projection into the basis {i}; i.e., repeated transformations of any initial pure state to an epistemic (proper) mixed state. Thus, if such non-unitary projection actually occurs during the evolution of a given system (such as a gas), then its entropy does increase, despite the governing deterministic Hamiltonian dynamics; the phase-space conserving evolution of the Liouville equation is physically broken at the micro-level.

## 4. The Transactional Interpretation

In the kinetic theory of molecules, for every process in which only a few elementary particles participate (e.g., molecular collisions), the inverse process also exists. But that is not the case for the elementary processes of radiation. According to our prevailing theory, an oscillating ion generates a spherical wave that propagates outwards. The inverse process does not exist as an elementary process. A converging spherical wave is mathematically possible, to be sure; but to approach its realization requires a vast number of emitting entities. The elementary process of emission is not invertible. In this, I believe, our oscillation theory does not hit the mark. Newton’s emission theory of light seems to contain more truth with respect to this point than the oscillation theory since, first of all, the energy given to a light particle is not scattered over infinite space, but remains available for an elementary process of absorption.[17] (emphasis added)

**k**of a given energy) ends up being delivered to only a single absorbing system; thus the process acquires a final anisotropy (i.e., a specific wave vector

**k**) not present initially. The latter is a feature of the particle-like aspect of light, and that is what makes the process non-invertible (this microscopic origin of irreversibility was also pointed out by Doyle [18]). As we will see, TI acknowledges both a wavelike and particlelike aspect to light, and it is the latter that brings about the irreversibility, just as Einstein noted.

#### 4.1. Background

[The Wheeler-Feynman direct-action model] swept the electromagnetic field from between the charged particles and replaced it with “half-retarded, half advanced direct interaction” between particle and particle. It was the high point of this work to show that the standard and well-tested force of reaction of radiation on an accelerated charge is accounted for as the sum of the direct actions on that charge by all the charges of any distant complete absorber. Such a formulation enforces global physical laws, and results in a quantitatively correct description of radiative phenomena, without assigning stress-energy to the electromagnetic field.([30], p. 427)

#### 4.2. Measurement in the Transactional Interpretation

_{i}to the component of the offer received by them. The advanced responses of absorbers are termed “confirmation waves” (CW). (As their names indicate, both of these objects are wavelike entities—specifically, they are deBroglie waves.) Specifically, an absorber X

_{k}will receive an offer wave component $\langle {X}_{k}|\Psi \rangle |{X}_{k}\rangle $ and will respond with a matching adjoint confirmation $\langle \Psi |\text{}{X}_{k}\rangle \langle {X}_{k}|$. The product of the offer/confirmation exchange is a weighted projection operator, $\langle {X}_{k}|\Psi \rangle \text{}\langle \Psi |\text{}{X}_{k}\rangle \text{}\left|{X}_{k}\rangle \langle {X}_{k}\right|={|\langle {X}_{k}|\Psi \rangle |}^{2}\left|{X}_{k}\rangle \langle {X}_{k}\right|$. Clearly, the weight is the Born Rule, and this is how TI provides a physical origin for this formerly ad hoc rule. When one takes into account the responses of all the other absorbers {X

_{i}}, what we have is the von Neumann measurement transition from a pure state to a mixed state $\tilde{\rho}$:

_{i}}, the linearity of this deterministic propagation is broken, and we get the non-unitary transformation (8).

**k**corresponds to the particle-like aspect or photon. (Of course, in this respect, “particle-like” does not mean having a localized corpuscular quality. Rather, the “particle” is a discrete quantum of energy/momentum. The directionality of the final received photon momentum is what localizes the expanding spherical wave to a particular final individual absorbing gas molecule, resulting in approximate localization of the transferred photon.) Since the latter process exchanges a determinate quantity of energy/momentum—a photon Fock state—the energy/momentum basis can be understood as distinguished. We return to the latter issue when we consider the relativistic level, in Section 5 below.

**k**corresponding to that absorber (i.e., as noted above, the spherically emitted offer wave collapses to only one momentum component). The emitter loses a quantum of energy/momentum and the absorber gains the same, leaving an imprint of the interaction (at least in the short term), which thus establishes the time-asymmetric conditions of the Stosszahlansatz. Thus, the time-asymmetric statistical description that Boltzmann assumed in order to derive the Second Law is justified, based on a real physical process.

## 5. The Relativistic Level: Further Roots of the Arrow of Time

## 6. Conclusions

## Acknowledgments

## Conflicts of Interest

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Kastner, R.E. On Quantum Collapse as a Basis for the Second Law of Thermodynamics. *Entropy* **2017**, *19*, 106.
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Kastner RE. On Quantum Collapse as a Basis for the Second Law of Thermodynamics. *Entropy*. 2017; 19(3):106.
https://doi.org/10.3390/e19030106

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Kastner, Ruth E. 2017. "On Quantum Collapse as a Basis for the Second Law of Thermodynamics" *Entropy* 19, no. 3: 106.
https://doi.org/10.3390/e19030106