The need to implement time reversal via complex conjugation in quantum theory has always been a bit of a puzzle. Why should i go to – i under temporal reflection when it has no spatiotemporal dimensions? I’ll provide a new insight into this question by showing how the little-appreciated “quantum-looking” classical Schrödinger equation of Schiller and Rosen faces the exact same problem. Since we know how to escape this problem classically, this observation teaches us one way to solve the problem quantum mechanically too. Big picture: if I’m right, the puzzle over quantum time reversal is connected to the interpretation of quantum theory.
The need to implement time reversal via complex conjugation in quantum theory has always been a bit of a puzzle. Why should i go to – i under temporal reflection when it has no spatiotemporal dimensions? I’ll provide a new insight into this question by showing how the little-appreciated “quantum-looking” classical Schrödinger equation of Schiller and Rosen faces the exact same problem. Since we know how to escape this problem classically, this observation teaches us one way to solve the problem quantum mechanically too. Big picture: if I’m right, the puzzle over quantum time reversal is connected to the interpretation of quantum theory.
This article is an introduction to and advertisement of Erwin Schrödinger’s little-known real-valued wave equation, the first published time dependent Schrödinger equation. I argue that this equation is not merely a historical curiosity. Not only does it show that quantum mechanics need not be viewed as essentially complex-valued, but the real formalism also provides a deep insight into the puzzling nature of time reversal in a quantum world. It is hoped that this observation will stimulate the discovery of other areas where ‘keeping it real’ will prove pedagogically or foundationally useful.
The analysis of the reversibility of quantum mechanics depends upon the choice of the time reversal operator for quantum mechanical states. The orthodox choice for the time reversal operator on QM states is known as the Wigner operator, T*, where * performs complex conjugation. The peculiarity is that this is not simply the unitary time reversal operation, but an anti-unitary operator, involving complex conjugation in addition to ordinary time reversal. The alternative choice is the Racah operator, which is simply ordinary time reversal, T. Orthodox treatments hold that it is either logically or empirically necessary to adopt the Wigner operator, and the Racah operator has received little attention. The basis for this choice is analysed in detail, and it is concluded that all the conventional arguments for rejecting the Racah operator and adopting the Wigner operator are mistaken. The additional problem of whether the deterministic part of quantum mechanics should be judged to be reversible or not is also considered. The adoption of the Racah operator for time reversal appears prima facie to entail that quantum mechanics is irreversible. However, it is concluded that the real answer to question depends upon the choice of interpretation of the theory. In any case, the conventional reasons for claiming that quantum mechanics is reversible are incorrect.
It’s time to figure out time! We wish in this paper to answer what is time and how it works, as Einstein changed our whole idea about time, however, not even he answered this one question, which might seem of no importance, in answering it will change all of physics. This paper is bringing a whole new set of ideas, while one might think it is just about time, and that is the start and the original purpose of this paper, one will be shown that this paper will extend to much more beyond that, yielding some very excited and highly unexpected results, the paper will be based on a thought experiment, then going on to arguing and redefining what is time, by asking questions some are previously asked and others are not, then showing for starters how for instance the time dilation effect comes out naturally from this new definition of time both physically and mathematically, the paper then starts using its new ideas and applying it to some of the unsolved problems, however the biggest, unexpected, and most important result that will come out from this paper, The Detection Property, it will be shown how this property will lead not only to the interpretation of quantum mechanics, but a much more powerful effect will be produced, in that the detection property will even require the weird quantum behavior, it will be shown how one can start with simple classical world, and just by considering this new argument of time will be forced to introduce to it – in addition to special relativity - all the weird quantum behavior what will be most exciting about it, is that it will do so by the fact that all the phenomena of quantum mechanics like the measurement problem and Entanglement, and to go even further it will not only explain the phenomena of the Superposition and the wave function collapse, but it will even require its existence, this new definition of time along with the detection property will require that quantum mechanics behave in the way it does.
The first question to ask about the phenomenon of relaxation is why it occurs at all. Both the Newton and the Schrödinger equations are symmetrical under time reversal: The Newton equation, dx/dt = v ; dv/dt = −∂V/∂x, implies that particles obeying this law of motion will retrace their trajectory back in time after changing the sign of both the time t and the particle velocities v. The Schrödinger equation, ∂ψ/∂t = −(i/.h) Ĥ ψ, implies that if (ψ (t) is a solution then ψ *(−t) is also one, so that observables which depend on |ψ|2 are symmetric in time. On the other hand, nature clearly evolves asymmetrically as asserted by the second law of thermodynamics. How does this asymmetry arise in a system that obeys temporal symmetry in its time evolution? Readers with background in thermodynamics and statistical mechanics have encountered the intuitive answer: Irreversibility in a system with many degrees of freedom is essentially a manifestation of the system “getting lost in phase space”:Asystem starts from a given state and evolves in time. If the number of accessible states is huge, the probability that the system will find its way back to the initial state in finite time is vanishingly small, so that an observer who monitors properties associated with the initial state will see an irreversible evolution. The question is how is this irreversible behavior manifested through the reversible equations of motion, and how does it show in the quantitative description of the time evolution. This chapter provides an introduction to this subject using the time-dependent Schrödinger equation as a starting point. Chapter 10 discusses more advanced aspects of this problem within the framework of the quantum Liouville equation and the density operator formalism.
This chapter demonstrates the deduction of quantum physics and, in particular, presents its new applications to eigenvalue problems. Specifically, it derives three of the five axiom presumptions of quantum mechanics (QM), for example, deducing the Schrödinger equation in two general ways. It transforms these three axiom presumptions into three theorems of QM, not only resolving the difficulty in understanding them but also yielding new theories and discoveries. For instance, this paper addresses the puzzle of the origin of wave-particle duality, derives operators, eigenvalues, and eigenstates, deduces commutation relations for coordinate and momentum as well as for time and energy, and reveals that QM is essentially a generalized (mechanics) theory of the complex square root of the (real density function of) classical statistical mechanics. The realization that QM is a generalized theory of the complex square root of classical statistical mechanics represents both a groundbreaking discovery and a revolutionary advancement in physics. This insight profoundly influences philosophical perspectives on the development of modern physics, resolves numerous puzzles in QM and quantum information, and provides QM with a scientifically solid foundation. This foundation eliminates the need for basic axiom presumptions and dispels the so-called “strange incomprehensible properties” of QM, as both classical statistical mechanics and its complex square root that have been scientifically validated. Additionally, this chapter uncovers the reason for not taking the time derivative of spatial coordinates in the Schrödinger equation. Consequently, it offers a solution to the puzzle of the origin of first quantization and primarily deduces quantum physics without the strange and incomprehensible properties currently associated with it.