Quantum Entropy

arXiv (Cornell University), 2021

All the laws of physics are time-reversible. Time arrow emerges only when ensembles of classical particles are treated probabilistically, outside of physics laws, and the entropy and the second law of thermodynamics are introduced. In quantum physics, no mechanism for a time arrow has been proposed despite its intrinsic probabilistic nature. In consequence, one cannot explain why an electron in an excited state will "spontaneously" transition into a ground state as a photon is created and emitted, instead of continuing in its reversible unitary evolution. To address such phenomena, we introduce an entropy for quantum physics, which will conduce to the emergence of a time arrow. The entropy is a measure of randomness over the degrees of freedom of a quantum state. It is dimensionless; it is a relativistic scalar, it is invariant under coordinate transformation of position and momentum that maintain conjugate properties and under CPT transformations; and its minimum is positive due to the uncertainty principle. To excogitate why some quantum physical processes cannot take place even though they obey conservation laws, we partition the set of all evolutions of an initial state into four blocks, based on whether the entropy is (i) increasing but not a constant, (ii) decreasing but not a constant, (iii) a constant, (iv) oscillating. We propose a law that in quantum physics entropy (weakly) increases over time. Thus, evolutions in the set (ii) are disallowed, and evolutions in set (iv) are barred from completing an oscillation period by instantaneously transitioning to a new state. This law for quantum physics limits physical scenarios beyond conservation laws, providing causality reasoning by defining a time arrow.

Maximum Entropy and Bayesian Methods Garching, Germany 1998, 1999

Entropic arguments are shown to play a central role in the foundations of quantum theory. We prove that probabilities are given by the modulus squared of wave functions, and that the time evolution of states is linear and also unitary.

Journal of Physics A: Mathematical and General, 1990

We use a simple generating function to calculate exactly the entropy of random quantum states for finite-dimensional Hilbert spaces over real, complex and quaternionic scalars. This allows us to extend our previous formula for the Quantum Correlation Information of a state determination apparatus to include real and quaternionic von Neumann analysers.

The British Journal for the Philosophy of Science, 2003

Shenker has claimed that Von Neumann's argument for identifying the quantum mechanical entropy with the Von Neumann entropy, SðrÞ ¼ Àktrðr log rÞ, is invalid. Her claim rests on a misunderstanding of the idea of a quantum mechanical pure state. I demonstrate this, and provide a further explanation of Von Neumann's argument.

arXiv (Cornell University), 2021

A quantum coordinate-entropy formulated in quantum phase space has been recently proposed together with an entropy law that asserts that such entropy can not decrease over time. The coordinate-entropy is dimensionless, a relativistic scalar, and it is invariant under coordinate and CPT transformations. We study here the time evolution of this entropy. We show that the entropy associated with coherent states evolving under a Dirac Hamiltonian is increasing. However, for the collisions of two particles, where each is evolving as a coherent state, as they come closer to each other their spatial entanglement causes the total system's entropy to oscillate. We augment time reversal with time translation and show that CPT with time translation can transform particles with decreasing entropy for a finite time interval into anti-particles with increasing entropy for the same finite time interval. We then analyze the impact of the entropy law for the evolution scenarios described above and explore the possibility that entropy oscillations trigger the annihilation and the creation of particles.

Statistical Mechanics and Scientific Explanation: Determinism, Indeterminism and Laws of Nature (World Scientific), 2020

In a quantum universe with a strong arrow of time, it is standard to postulate that the initial wave function started in a particular macrostate-the special low-entropy macrostate selected by the Past Hypothesis. Moreover, there is an additional postulate about statistical mechanical probabilities according to which the initial wave function is a "typical" choice in the macrostate (the Statistical Postulate). Together, they support a probabilistic version of the Second Law of Thermodynamics: typical initial wave functions will increase in en-tropy. Hence, there are two sources of randomness in such a universe: the quantum-mechanical probabilities of the Born rule and the statistical mechanical probabilities of the Statistical Postulate. I propose a new way to understand time's arrow in a quantum universe. It is based on what I call the Thermodynamic Theories of Quantum Mechanics. According to this perspective, there is a natural choice for the initial quantum state of the universe, which is given by not a wave function but by a density matrix. The density matrix plays a microscopic role: it appears in the fundamental dynamical equations of those theories. The density matrix also plays a macro-scopic / thermodynamic role: it is exactly the (normalized) projection operator onto the Past Hypothesis subspace (of the Hilbert space of the universe). Thus, given an initial subspace, we obtain a unique choice of the initial density matrix. I call this property the conditional uniqueness of the initial quantum state. The conditional uniqueness provides a new and general strategy to eliminate statistical mechanical probabilities in the fundamental physical theories, by which we can reduce the two sources of randomness to only the quantum mechanical one. I also explore the idea of an absolutely unique initial quantum state, in a way that might realize Penrose's idea (1989) of a strongly deterministic universe.

Annalen der Physik

Entropic Dynamics (ED) is a framework in which Quantum Mechanics (QM) is derived as an application of entropic methods of inference. The magnitude of the wave function is manifestly epistemic: its square is a probability distribution. The epistemic nature of the phase of the wave function is also clear: it controls the flow of probability. The dynamics is driven by entropy subject to constraints that capture the relevant physical information. The central concern is to identify those constraints and how they are updated. After reviewing previous work I describe how considerations from information geometry allow us to derive a phase space geometry that combines Riemannian, symplectic, and complex structures. The ED that preserves these structures is QM. The full equivalence between ED and QM is achieved by taking account of how gauge symmetry and charge quantization are intimately related to quantum phases and the single-valuedness of wave functions.

Physics Letters A, 2001

European Review, 2010

Physical Review Letters, 2002

We consider a single free spin-1 2 particle. The reduced density matrix for its spin is not covariant under Lorentz transformations. The spin entropy is not a relativistic scalar and has no invariant meaning.