Time's Arrow and QM

Question

Something puzzles me. It is sometimes said that the fundamental equations of physics are time-reversible, creating the problem of time's arrow. But... isn't Schrodinger's wave equation time-dependent? In particular, doesn't it involve a derivative with respect to time? Now, the wave equation was formulated in order to describe actual reality, and its time derivative, when formulating that description, was used under the assumption of forward-flowing time ("flowing" is only a convenient term to use here, nothing more). An accurate description of reality with backward-flowing time would require a different equation. Therefore, isn't time's arrow implicit in the fundamental equations of physics?

To be clear: one might still worry about what accounts for the directionality of time--but does it make sense to say that the fundamental laws of physics are time-reversible when they were created to be accurately descriptive of the real world and in order to make them descriptive of the real world they use the assumption of forward-flowing time?

Answer 1

The Schrodinger equation is a non-relativistic approximation. Relativistic wave equations, like the Dirac and Klein-Gordon ones, have second-order derivatives in time, so they are time-symmetric. It turns out that if you derive the Schrodinger equation via a non-relativistic approximation to a relativistic wave equation, then you find two Schrodinger equations: one with forward-propagating time and one with backward-propagating time. The conventional treatment then appeals to causality and tosses the reversed-time version, although alternatives exist (see the Transactional Interpretation of quantum mechanics, for example).

Answer 2

Yes it is, in fact the collapse postulate in standard QM determines a direction of time since a collape only occurs after an evolution of the quantum wave.

This, or something similar, was pointed out in Rudolf Haag's book, Local Quantum Physics.

It's also worth noting that t'Hooft considers the time reversability of the fundamental equations of physics to be an apparent illusion.

Answer 3

The Schrödinger equation isn't time-asymmetric in any deep way.

Classically you typically specify the initial conditions of a system as both zeroth and first derivatives. In some cases you can pack them both into a complex number. At least, this is possible with the simple harmonic oscillator, which satisfies $x'' = -ω^2x$ with $x$ real, or $ψ' = iωψ$ with $ψ=x+x'/iω$ complex. If you specify the initial conditions as $x$ and $x'$ and you want to run the oscillator backwards, you negate $x'$ (but not $x$). If you specify the initial conditions as $ψ$, you take the complex conjugate instead.

In quantum mechanics, the zeroth and first derivatives can't be treated as independent because of the uncertainty principle. Nevertheless, it's reasonable to say that the complex wave function incorporates the classical zeroth and first derivatives. To run a system backward, you take the complex conjugate. The time asymmetry in the Schrödinger equation just reflects an arbitrary choice of orientation in the complex plane.

The Schrödinger equation "with backward-propagating time" in Gilbert's answer is just the complex conjugate of the usual one. It doesn't have a reversed thermodynamic arrow of time, which is what people usually have in mind when they talk about time running backward. The same terminological problem shows up in descriptions of antiparticles in relativistic quantum theory. They are "particles moving backward in time" in a certain mathematical sense, but not in the thermodynamic sense.

Answer 4

Reversing the wavefunction in time is like reversing the an expanding gas in a vacuum in time. Like the gas particles will move towards a smaller volume, it will become more compact in space. You could see which way time flew by l seeing it localize (while the momentum representation would spread). The motion of the gas particles is time symmetric though and so is the base of the wavefunction. Like with gas particles, it's very hard to reverse their motions exactly and the same holds for the wavefunction. If it reversed in time and converges, there has to come a moment though it expands discontinuously instead of collapse (inverse collapse), thereby undoing the measurement made back in time. If the measurement was made by a photon, a photon will travel back in time to the time reversed measurement device. For this all to occur though, the hidden variables will have to be reversed with an incredible fine-tuning, like reversing the motion of particles in a gas. And it's exactly that which defines the arrow of time.